'203.252' ************** MAGMA ***************** Host 203.252.48.247 (203.252.48.247) Time: Sat Dec 31 14:53:23 2005 Input: nat:=5; pairs:=[{i,j}: i in [1..j-1], j in [1..nat]]; G:=PermutationGroup<#pairs | [[Index(pairs,p^g): p in pairs]: g in [(1,2),(1,2,3),(4,5)]]>; Rat:=Rationals(); R0:=InvariantRing(G,Rat); PI:=PrimaryInvariants(R0); print "*** Primary invariants"; PI; #PS:=SecondaryInvariants(R0); #print "*** Secondary invariants"; #PS; Output: ** WARNING: Computation time exceeded 20 seconds, so computation was terminated after 20 seconds. ** Magma V2.11-10 Sat Dec 31 2005 14:53:03 on modular [Seed = 350336662] ------------------------------------- Errors: /bin/sh: line 1: 24073 Alarm clock nice -n 19 /usr/local/bin/magma '203.252' ************** MAGMA ***************** Host 203.252.48.247 (203.252.48.247) Time: Sat Dec 31 14:31:06 2005 Input: nat:=5; pairs:=[{i,j}: i in [1..j-1], j in [1..nat]]; G:=PermutationGroup<#pairs | [[Index(pairs,p^g): p in pairs]: g in [(1,2),(1,2,3),(4,5)]]>; Rat:=Rationals(); R0:=InvariantRing(G,Rat); PI:=PrimaryInvariants(R0); print "*** Primary invariants"; PI; PS:=SecondaryInvariants(R0); print "*** Secondary invariants"; PS; Output: ** WARNING: Computation time exceeded 20 seconds, so computation was terminated after 20 seconds. ** Magma V2.11-10 Sat Dec 31 2005 14:30:45 on modular [Seed = 4161628035] ------------------------------------- Errors: /bin/sh: line 1: 24038 Alarm clock nice -n 19 /usr/local/bin/magma '203.252' ************** MAGMA ***************** Host 203.252.48.247 (203.252.48.247) Time: Sat Dec 31 14:13:07 2005 Input: nat:=4; pairs:=[{i,j}: i in [1..j-1], j in [1..nat]]; G:=PermutationGroup<#pairs | [[Index(pairs,p^g): p in pairs]: g in [(1,2),(3,4)]]>; Rat:=Rationals(); R0:=InvariantRing(G,Rat); PI:=PrimaryInvariants(R0); print "*** Primary invariants"; PI; PS:=SecondaryInvariants(R0); print "*** Secondary invariants"; PS; Output: Magma V2.11-10 Sat Dec 31 2005 14:13:07 on modular [Seed = 3409858710] ------------------------------------- *** Primary invariants [ x1, x2 + x3 + x4 + x5, x6, x2^2 + x3^2 + x4^2 + x5^2, x2*x3 + x4*x5, x2*x4 + x3*x5 ] *** Secondary invariants [ 1, x2^3 + x3^3 + x4^3 + x5^3 ] Total time: 0.210 seconds, Total memory usage: 3.43MB '203.252' ************** MAGMA ***************** Host 203.252.48.247 (203.252.48.247) Time: Sat Dec 31 14:03:34 2005 Input: nat:=3; pairs:=[{i,j}: i in [1..j-1], j in [1..nat]]; G:=PermutationGroup<#pairs | [[Index(pairs,p^g): p in pairs]: g in [(1,2),(1,2,3)]]>; Rat:=Rationals(); R0:=InvariantRing(G,Rat); PI:=PrimaryInvariants(R0); print "*** Primary invariants"; PI; PS:=SecondaryInvariants(R0); print "*** Secondary invariants"; PS; Output: Magma V2.11-10 Sat Dec 31 2005 14:03:33 on modular [Seed = 3744090820] ------------------------------------- *** Primary invariants [ x1 + x2 + x3, x1^2 + x2^2 + x3^2, x1^3 + x2^3 + x3^3 ] *** Secondary invariants [ 1 ] Total time: 0.200 seconds, Total memory usage: 3.43MB '212.138' ************** MAGMA ***************** Host 212.138.47.21 (212.138.47.21) Time: Tue Dec 20 09:25:33 2005 Input: P:=PolynomialRing(RationalField(),5); I:=ideal

; Radical(I); Output: Magma V2.11-10 Tue Dec 20 2005 09:25:32 on modular [Seed = 2940532906] ------------------------------------- Ideal of Polynomial ring of rank 5 over Rational Field Lexicographical Order Variables: x, y, z, u, v Dimension 0, Radical Groebner basis: [ x - v^16 - 8*v^14 - 32*v^12 - 80*v^10 - 138*v^8 - 168*v^6 - 144*v^4 - 80*v^2 - 26, y - v^8 - 4*v^6 - 8*v^4 - 8*v^2 - 5, z - v^4 - 2*v^2 - 2, u - v^2 - 1, v^32 + 16*v^30 + 128*v^28 + 672*v^26 + 2580*v^24 + 7664*v^22 + 18208*v^20 + 35296*v^18 + 56472*v^16 + 74944*v^14 + 82432*v^12 + 74624*v^10 + 54792*v^8 + 31776*v^6 + 13888*v^4 + 4160*v^2 - v + 677 ] Total time: 0.190 seconds, Total memory usage: 3.43MB '212.138' ************** MAGMA ***************** Host 212.138.113.12 (212.138.113.12) Time: Tue Dec 20 09:24:52 2005 Input: P:=PolynomialRing(RationalField(),5); I:=ideal

; Radical(I); Output: Magma V2.11-10 Tue Dec 20 2005 09:24:51 on modular [Seed = 2975524571] ------------------------------------- Ideal of Polynomial ring of rank 5 over Rational Field Lexicographical Order Variables: x, y, z, u, v Dimension 0, Radical Groebner basis: [ x - v^16 - 8*v^15 - 40*v^14 - 140*v^13 - 390*v^12 - 884*v^11 - 1702*v^10 - 2790*v^9 - 3980*v^8 - 4900*v^7 - 5282*v^6 - 4876*v^5 - 3910*v^4 - 2580*v^3 - 1440*v^2 - 567*v - 183, y - v^8 - 4*v^7 - 12*v^6 - 22*v^5 - 35*v^4 - 38*v^3 - 37*v^2 - 21*v - 13, z - v^4 - 2*v^3 - 4*v^2 - 3*v - 3, u - v^2 - v - 1, v^32 + 16*v^31 + 144*v^30 + 920*v^29 + 4620*v^28 + 19208*v^27 + 68348*v^26 + 212732*v^25 + 588380*v^24 + 1462760*v^23 + 3297580*v^22 + 6786000*v^21 + 12814320*v^20 + 22292560*v^19 + 35837420*v^18 + 53355230*v^17 + 73679935*v^16 + 94452240*v^15 + 112430520*v^14 + 124216240*v^13 + 127251670*v^12 + 120654560*v^11 + 105615510*v^10 + 85034690*v^9 + 62677680*v^8 + 42006568*v^7 + 25385078*v^6 + 13653832*v^5 + 6434290*v^4 + 2579820*v^3 + 849969*v^2 + 208088*v + 33673 ] Total time: 0.210 seconds, Total memory usage: 3.43MB '212.138' ************** MAGMA ***************** Host 212.138.47.24 (212.138.47.24) Time: Tue Dec 20 09:24:05 2005 Input: P:=PolynomialRing(RationalField(),3); I:=ideal

; Radical(I); Output: Magma V2.11-10 Tue Dec 20 2005 09:24:05 on modular [Seed = 3176590049] ------------------------------------- >> P:=PolynomialRing(RationalField(),3); I:=ideal

> -x + (y^2+y+1), ^ User error: Identifier 'x' has not been declared or assigned >> Radical(I);; ^ User error: Identifier 'I' has not been declared or assigned Total time: 0.190 seconds, Total memory usage: 3.24MB '212.138' ************** MAGMA ***************** Host 212.138.47.17 (212.138.47.17) Time: Tue Dec 20 09:19:52 2005 Input: P:=PolynomialRing(RationalField(),3); I:=ideal

; Radical(I); Output: Magma V2.11-10 Tue Dec 20 2005 09:19:51 on modular [Seed = 2321668820] ------------------------------------- Ideal of Polynomial ring of rank 3 over Rational Field Lexicographical Order Variables: x, y, z Dimension 0, Radical Groebner basis: [ x - z^4 - 2*z^3 - 4*z^2 - 3*z - 3, y - z^2 - z - 1, z^8 + 4*z^7 + 12*z^6 + 22*z^5 + 35*z^4 + 38*z^3 + 37*z^2 + 20*z + 13 ] Total time: 0.190 seconds, Total memory usage: 3.34MB '212.138' ************** MAGMA ***************** Host 212.138.113.12 (212.138.113.12) Time: Tue Dec 20 09:18:42 2005 Input: P:=PolynomialRing(RationalField(),3); I:=ideal

; Radical(I); Output: Magma V2.11-10 Tue Dec 20 2005 09:18:42 on modular [Seed = 2625096967] ------------------------------------- >> -x + y^2+y+1), ^ User error: bad syntax >> -y + z^2+z+1), ^ User error: bad syntax >> -z + (x^2+x+1) >; ^ User error: bad syntax >> Radical(I);; ^ User error: Identifier 'I' has not been declared or assigned Total time: 0.190 seconds, Total memory usage: 3.24MB '66.80.2' ************** MAGMA ***************** Host 66.80.213.2 (66.80.213.2) Time: Mon Dec 19 18:33:57 2005 Input: 123 Output: Magma V2.11-10 Mon Dec 19 2005 18:33:57 on modular [Seed = 3375221216] ------------------------------------- 123 Total time: 0.200 seconds, Total memory usage: 3.24MB '66.80.2' ************** MAGMA ***************** Host 66.80.213.2 (66.80.213.2) Time: Mon Dec 19 18:33:54 2005 Input: "Replace this by some code, then click [PARI] or [MAGMA]!" Output: Magma V2.11-10 Mon Dec 19 2005 18:33:54 on modular [Seed = 3425225954] ------------------------------------- Replace this by some code, then click [PARI] or [MAGMA]! Total time: 0.200 seconds, Total memory usage: 3.24MB '130.83.' ************** MAGMA ***************** Host 130.83.2.27 (130.83.2.27) Time: Mon Dec 19 09:54:32 2005 Input: F2 := GF(2^163); a := 00000000 28194bf9 c17b1b81 9a71b20f 5ef0b51d 67cb6830; b := 00000009 8e59a5bf 2e0edbab 1487fcce ba428afa 855dfd12; E := EllipticCurve([1, 0, 0, a, b]); time #E; FactoredOrder(E); Output: Magma V2.11-10 Mon Dec 19 2005 09:54:32 on modular [Seed = 1653074050] ------------------------------------- >> a := 00000000 28194bf9 c17b1b81 9a71b20f 5ef0b51d 67cb6830; ^ User error: bad syntax >> b := 00000009 8e59a5bf 2e0edbab 1487fcce ba428afa 855dfd12; ^ User error: bad syntax >> E := EllipticCurve([1, 0, 0, a, b]); ^ User error: Identifier 'a' has not been declared or assigned >> time #E; ^ User error: Identifier 'E' has not been declared or assigned >> FactoredOrder(E);; ^ User error: Identifier 'E' has not been declared or assigned Total time: 0.190 seconds, Total memory usage: 3.24MB '130.83.' ************** MAGMA ***************** Host 130.83.2.27 (130.83.2.27) Time: Mon Dec 19 09:53:09 2005 Input: F2 := GF(2^163); a := 0000000028194bf9c17b1b819a71b20f5ef0b51d67cb6830; b := 000000098e59a5bf2e0edbab1487fcceba428afa855dfd12; E := EllipticCurve([1, 0, 0, a, b]); time #E; FactoredOrder(E); Output: Magma V2.11-10 Mon Dec 19 2005 09:53:09 on modular [Seed = 1820976753] ------------------------------------- >> a := 0000000028194bf9c17b1b819a71b20f5ef0b51d67cb6830; ^ User error: bad syntax >> b := 000000098e59a5bf2e0edbab1487fcceba428afa855dfd12; ^ User error: bad syntax >> E := EllipticCurve([1, 0, 0, a, b]); ^ User error: Identifier 'a' has not been declared or assigned >> time #E; ^ User error: Identifier 'E' has not been declared or assigned >> FactoredOrder(E);; ^ User error: Identifier 'E' has not been declared or assigned Total time: 0.200 seconds, Total memory usage: 3.24MB '136.206' ************** MAGMA ***************** Host 136.206.1.20 (136.206.1.20) Time: Mon Dec 19 08:46:38 2005 Input: 2^123497 Output: Magma V2.11-10 Mon Dec 19 2005 08:46:38 on modular [Seed = 98631666] ------------------------------------- 2001587198193684625170316366565213998770374765249539510968102874320528369280493\ 8986637440863966655910880436057015983629544805127690562348522105614355084980855\ 8100817350081063197408828461733641385699501604461211045438847140673688794520554\ 2154078678320893488245862834270199856499880144810356878526556017678836917907254\ 1046165766739171527647053135746574000765241140167521092967514930946037638159715\ 7324103041307853303707793081576249426290030160858776795634913065666764828870166\ 0817659761875008825402341766867988536473966628066571208495055021474203862053389\ 5619540415131014622090395555693412034725579822183946548196555711278388954151119\ 0335085512829608797342262217110627469824475552779402624494276981138185048459139\ 8524065478614504954527189393774015733718549520302500730232792551080567537098078\ 8504520403789845365307705116991604735772774428558035953536010649938224410067292\ 8726555743983504829486577211989987455967309277828078519623595338742342437569964\ 5480459542625040887895090424933868304713939583880575630273834790051610134516840\ 6227136764211875488062082160793844857874605530551098666226714806818667434144772\ 1063136347457836471378902998209687326426084007814921488032709718444818718797563\ 1141813287986845869664517980768746940850528726695163905943110109392108359161327\ 1435209214981086265090134155848586057194572973901021662468515412261193920494090\ 9274509060742943695708606635951092816028314742513588178522068898450751376914728\ 2105774082292189829524094815261817282380214547134492188670089396881892310558668\ 1075138822851095257150058382118481180777512842216174253916023460239766019544100\ 3440843242728325628214429228738744714732818782903251958599530525220184027857144\ 8494766691817545549179063010274278690882471994516169263645095088224017259735556\ 5460231370789994586701620182457752827395050864683700385365475052736126239627619\ 4565459534267572124445589815767503559146817323763563208395710939760067521823093\ 1039478071755774924299386829359477219344654557781020906230508052676142868199375\ 8820931570513731276070453033810404956787631704576192896431835499331291546070722\ 7190201851621353358192174806118358472970591935359590026199558874379595660861858\ 9696012980229179034987320784459133545806797826719157383137889513193005979305814\ 9517009315897690318613758217204056551173667110384474025519330720383291584977125\ 1501477939505655648279305065857100625251200428822652269455045440687939400145210\ 5944109328090321327466102115295188267167908443558655805705652964037232040454723\ 3214525907425420448703413811844845355431305416273212490421465116252485072159739\ 5832728124959630867540680772887561691838912831388869988133638291883062606868532\ 7010553445866529825347265106311408970375411578581162354756181776766855863863910\ 3666405469774824822749893808976093004702012502632596092309146114879185980357730\ 3861342700363386765252454917887675646729006062822382731919402870677014607812709\ 6742863782944785309137485591391473520882675261639169157028134563998259122606414\ 1214061808380859475240005407144224706478896212736978049847274888875101819616029\ 9966056275586323932325057051732694804116544104519152361253824143726585366108886\ 7824403554589285284935611921649878883484648078963332754980560806878465118995176\ 8636996763145767446119826236229282709824730386049555511195740119353423233856730\ 3003546375862251574339462096898282492902092773335958671071365209990805102574092\ 0636142630104451906526838644223997241041911264846371619293496929717047691412369\ 1587346265740663090731687558728136079480707562808197369049864379159635030104833\ 5462063207330037679739569439800054740440906773030574364446455020404814972154764\ 8465659046448645981163111625351420897515029106902997014825073863782367924689987\ 3076456293857228619376979994449524671977219099392394476306453997067225647326307\ 9651244784572003798516669285898768736293284876231569011061165312951792196750664\ 8004754964130050168046555775625521612295094815319045752147740812584490059471702\ 9110508498720322371497180824853974807955076774415446379857426101140477331235232\ 5282219526154347465870231715669282321911900759173373302053902742843137071740045\ 3621167367929877315269281203851500653681929449277541612917321658168275189795276\ 2412503712674455797837236292478679948347530840728542976552595796030279570515203\ 7940542968833411862734488251943840515303069923090930754867030519999763927008753\ 4109278412879725343113493554581197681691396093950913079134280924275077184989217\ 1982956692149258192788201462843811643821788250046018873375736899319270437924928\ 5540099696779485485716894358744992122125312773996048150216323732658153945679472\ 2320977755208176552178482693983039701785708172180465823200429017613355668581316\ 9197409885823806913479548221920578462725001628320343226209425531737323271123249\ 9727126380340937411653379777029732265276325290672904798107737192524653101768400\ 9963726349210748613853596548808286287341587314433611182352932800970657575890482\ 9602907447189590499440728343982860489151634746504481128663366540410107769278962\ 2181978575636973826070190685730400287078130504620330974191383418411304598280217\ 6289175130935471020191822476327340127497769105439927288109432302920619396686207\ 5563266955573972135825134090714013167104886818196119505018915027954165916798660\ 5640685233817787209455553922776298303907978125459096002683296118532244383049939\ 2102581707216392827636621552640316689749473277663821087276462030534324285817090\ 1574039834128452828945043423138163615088396652676945031102616772537236027362572\ 9093550129291730074222552412449785060754950844397709165842732972161473559312128\ 1648177665545877001929057577821788966777334571056138060542028896643985137026462\ 8061525933974099968310206635290264404637756991509321664429344972626116684300533\ 7861106196797563625824671581750709088611111005393211508348946836662320709164744\ 4854402831452730245851694998484779312170342406956127253906007800990005060969124\ 2395898623324386522359041849344412397966880334086445922750564087082180538353448\ 9634389254021811690315680739893729453852192126826862284872702040996640300451513\ 8200522354863683731681011043122873336107797215878814823980958952036686642194726\ 3706921381635816998209832545626576977827117750452679746326170192207125326473365\ 8497931921864125246498435153832638634428500668609805760957351666370173186152489\ 8549196644441257764954206799410775690220693547389493395063894074243026894236255\ 6121951407299610469418576123455522867323117795789526865250121923123020730121861\ 6624481869097437255443211332437606329843916191606975531157236542756283294761097\ 4454018830191740393255424759617907227542814955807285179342469298454284434333124\ 2188046685850140650249332877897138070156412249953068010125972434512590631995781\ 6632292281816295180543030856988598755698719014935611610497899707334483953087418\ 5979872236654381900555278741541091036775020178274565847573095539888643891558305\ 0175424376912130256217972415808265422685233550897840075159979797826139100476420\ 4214518939826524611784332641689623503135283070093056061865458136633828822285069\ 0004425896765346972392541315554962782742115932728784942482810982493570350990537\ 5108530943317744846866596119617581606715824753315301552311532186271382016305676\ 9063360495161630225801269986315318362731038779141167969552641858086429656721362\ 9599772990492455970027177634852018878282322897538817547405848057532086774798145\ 5364151801960828568783263237919368509672246292670138632820604703249416046299906\ 6309128599020726672129686873000548278929749049292091715882893019593031702086657\ 2174198190541394814812140410431243892023851682561822293927876173223740753378627\ 4994641265344197339284929991571856502679410265220369215765028835049768624721877\ 0121988303345502026704739350854447208331960688620845925110719853167768327430021\ 4142932824672056999929554415787065542872881312583864621789852055601802352448366\ 9108631443716501593765756505629326189956506734462415837555130395960452986739287\ 7660151949506573367420849284827494218187419458582547237607859031878846049644312\ 7920103300044594753464102678439477091221172525537648927003459027346572134757383\ 1030600313929525796904581198077505596731390132687144156670231248193846340305888\ 8787917882570263282367890190436422838181163776957771953426461883287295449828723\ 3488289675215560589564372515907540699914660691853939721373454211606692217598553\ 6584081788667858602130179979038955672511395920461593442738819291451583509219039\ 0618919475111553422287107453358753977381991894543719718939258589010970555112756\ 4846282563174724721875722821448159738351297436685228054903187961451796566642003\ 8877582017402536367794132418674778384181958606150038938978716935404757178232108\ 9544886249853827024538758904911767797137081414281149102511949914183355860551613\ 3966655353632315135931193933915921426295256616267943129810039331850010374647879\ 5019398749091731961928686112450404317715640599039825046649445559552364442288050\ 4878208618632646277400281860369252007926862480622634852930180193126283373580905\ 2188932240739236342043990025893410560025923506161687442534685002077189502865770\ 8628684577211773410041575281787067835156771184738929787375294329344608955759281\ 6377654751461037940825426469015541257395655125850006534845483689697461507641197\ 6033519972257906660048537471491539843946831599600103967246479411724790599911674\ 1836370606906770341983903573287845621138941743228357128906245733541012474702751\ 8453701557805174085939366323849083212393050852348898855243496187309225102337985\ 3152423482372965525069167089113730940204405410575250750673485929713480182810589\ 8682924721845452017908478942531854379243327576779228936408223207821032121141629\ 8026545748202577851026072354880919535700504372738944562621789406059507848845670\ 9314010730492365706988794320639036441537104845093729304487948798494150514394443\ 3321460335522184092101043554017852228116475405922692692379589150422884131353808\ 329 ** WARNING: Output too long, hence truncated. '136.206' ************** MAGMA ***************** Host 136.206.1.20 (136.206.1.20) Time: Mon Dec 19 08:46:24 2005 Input: 2^12349 Output: Magma V2.11-10 Mon Dec 19 2005 08:46:24 on modular [Seed = 250344695] ------------------------------------- 2626736171012131461696331866622504022562035768491774100353857649406801506742242\ 3084825445525830313237579267060342464266751281652973705149653235374277824190887\ 1189531386079630150950885566776783139898209075387224197253278610227046686860285\ 2146772936587820191698239338478093635758535173421076624226668386717908021430077\ 2575959758867004070058358245154844488213421278411858599405003098389395982774741\ 5207523985664026620091887697430049224315869743164135546469569593156053308579763\ 4263347484501060259378983778512308193286625242724234283776294831969816375131722\ 8936983092780315563728112542813599636673151675373431312055279054784023549919569\ 1219311126509583342457083167117071756897339079913318716330389289773213619097648\ 5462291545382776372231239190460511893638750161514624836332728442123550337942797\ 0107763595947903341784814752544645876919755965073714519271381674895702373091853\ 3215436670680437605890497876295398853415101607642404800691586567280726998586113\ 3155635070581153975745361022821609701475154342525458848449138355080543080276328\ 8562238102868156174523151272995385551848413365281813064332695534895707693916077\ 3662839142437371930375144772683967311099712589747556416445072489722773502081757\ 3932762811380394736879531383074542888986949490172385899865512107245330358348141\ 2002536006877334990455229759352997623012123459286967364906330023656745029008423\ 1318591798112344953149789799176371875210520880011310213974786327498190170775194\ 0246174540950463559914962203857965870347514848862996573094556026725122248029488\ 6275846517739215502335403747057022256585935251030078500156821873918117987513970\ 4426698969743209570301505361020908142913826815198427238143354973528877107200224\ 1362276706671793015984396682901357490556654981141630717756138713107208664121793\ 5964945552831824348164195179552274467535374454989803677496230655131121084486570\ 0261226878948470170102082465535705353261735614453198125197698538320433876443585\ 5548730497312029569853044174888346610679682782177024305897164452504267762524472\ 8942106845930309209375265735037991439661066606147717307209666084147940792076368\ 5347488239038281717789732344573987600622333170050711612768838066021919121172723\ 7211642627359164577023720988405242216072270697121621110267174598217721659879750\ 6359722942347060658884300635157725165894578694825025691943617203907024115252487\ 2604789418047694030384602063079611761683791030363420595196990670237588860702010\ 6169944921970214471348265581404547669212797420512416967703949848988112509424036\ 1134688460311879742839068107057958734150381319733809695520201181756924313844696\ 1918204395509649687931010893331484714802246461165949708658831409554486264656020\ 2841600039327715319442826732033633872044383688769435144495559623035113598100509\ 0075714762536621556465045676759336653654216176823133677119064993039589879343991\ 1804372050260066046422880519351824998305832815771105572582031735088076992900746\ 8766513825643038069839131037767895599272614757883512126120210656816117647905093\ 1080232106668263738912090292701279801420121666281544640796361809598122771169016\ 7769682855286385498577015573401574654445955805783975120408618104937011191205360\ 0034286567891044077209771318409439621839771975487282097247546270122766333673139\ 7818925249813795323067965555218133003053559443872659882156008658054660949373786\ 5912057785114351497720960768672901692108186937013204195723476497603434228407112\ 0818914716884554084155256682143282018872321808954078627677339016574811674828448\ 4545103570576976166155059054557530744414713978466406343423134640612927620214279\ 5177691324413778261863862669992600426858077025874394518306583020390601349897083\ 3017916526700157169696336864604125724185399555853360649287157768704465038886601\ 1190907156678096345969954990068832490888864044127228955581257071464844705313975\ 17312 Total time: 0.200 seconds, Total memory usage: 3.24MB '136.206' ************** MAGMA ***************** Host 136.206.1.20 (136.206.1.20) Time: Mon Dec 19 08:46:15 2005 Input: 2^1234 Output: Magma V2.11-10 Mon Dec 19 2005 08:46:14 on modular [Seed = 200208868] ------------------------------------- 2958112246080986290600446957161035907863396871353729922395562070506573507962389\ 2426105383724837805018644364775907095599312082089933038176093702721248284094494\ 1362110665443775183495726811929203861182015218323892077355983393191208928867652\ 6559936024879031137085494026686245211006117942703402327660993170980488874938090\ 23127398253860618772619035009883272941129544640111837184 Total time: 0.190 seconds, Total memory usage: 3.24MB '136.206' ************** MAGMA ***************** Host 136.206.1.20 (136.206.1.20) Time: Mon Dec 19 08:46:04 2005 Input: 2^1234324 Output: ** WARNING: Computation time exceeded 20 seconds, so computation was terminated after 20 seconds. ** Magma V2.11-10 Mon Dec 19 2005 08:45:44 on modular [Seed = 1955696309] ------------------------------------- Errors: /bin/sh: line 1: 2621 Alarm clock nice -n 19 /usr/local/bin/magma '136.206' ************** MAGMA ***************** Host 136.206.1.20 (136.206.1.20) Time: Mon Dec 19 08:45:33 2005 Input: 2^123 Output: Magma V2.11-10 Mon Dec 19 2005 08:45:33 on modular [Seed = 1905560389] ------------------------------------- 10633823966279326983230456482242756608 Total time: 0.190 seconds, Total memory usage: 3.24MB '136.206' ************** MAGMA ***************** Host 136.206.1.20 (136.206.1.20) Time: Mon Dec 19 08:45:26 2005 Input: eval("2^123") Output: Magma V2.11-10 Mon Dec 19 2005 08:45:26 on modular [Seed = 2057273413] ------------------------------------- >> eval("2^123"); ^ User error: Identifier 'eval' has not been declared or assigned Total time: 0.190 seconds, Total memory usage: 3.24MB '136.206' ************** MAGMA ***************** Host 136.206.1.20 (136.206.1.20) Time: Mon Dec 19 08:45:09 2005 Input: gap.eval("2^123") Output: Magma V2.11-10 Mon Dec 19 2005 08:45:08 on modular [Seed = 1736613194] ------------------------------------- >> gap.eval("2^123"); ^ User error: Identifier 'gap' has not been declared or assigned Total time: 0.200 seconds, Total memory usage: 3.24MB '82.58.6' ************** MAGMA ***************** Host 82.58.68.160 (82.58.68.160) Time: Mon Dec 19 08:11:01 2005 Input: 268165538371456392121266295429649186242633 Output: Magma V2.11-10 Mon Dec 19 2005 08:10:58 on modular [Seed = 2675600823] ------------------------------------- 268165538371456392121266295429649186242633 Total time: 0.210 seconds, Total memory usage: 3.24MB '220.214' ************** MAGMA ***************** Host 220.214.76.207 (220.214.76.207) Time: Mon Dec 19 07:10:23 2005 Input: n:=20;d:=8; ////////////////////////// circulant:=function(r) n:=#r; m:=r; row:=r; for j in [1..n-1] do Rotate(~row,1); m cat:=row; end for; return Matrix(Parent(r[1]),n,n,m); end function; ////////////////////////// ////////////////////////// x1:=Vector(GF(2),1,[0]); x3:=Vector(GF(2),1,[1]); all1:=Matrix(GF(2),1,n-1,[1 : i in [1..n-1]]); t:=Transpose(Matrix(GF(2),1,n,Eltseq(x1) cat Eltseq(all1))); V:=EvenWeightCode(n-2); x2:=Random(V); x2; x:=Eltseq(x1) cat Eltseq(x2); M1:=VerticalJoin(all1,circulant(x)); M2:=HorizontalJoin(t,M1); M2; Output: Magma V2.11-10 Mon Dec 19 2005 07:10:23 on modular [Seed = 3425250472] ------------------------------------- (0 1 1 1 1 0 0 0 1 1 0 1 1 0 0 1 1 0) [0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1] [1 0 0 1 1 1 1 0 0 0 1 1 0 1 1 0 0 1 1 0] [1 0 0 0 1 1 1 1 0 0 0 1 1 0 1 1 0 0 1 1] [1 1 0 0 0 1 1 1 1 0 0 0 1 1 0 1 1 0 0 1] [1 1 1 0 0 0 1 1 1 1 0 0 0 1 1 0 1 1 0 0] [1 0 1 1 0 0 0 1 1 1 1 0 0 0 1 1 0 1 1 0] [1 0 0 1 1 0 0 0 1 1 1 1 0 0 0 1 1 0 1 1] [1 1 0 0 1 1 0 0 0 1 1 1 1 0 0 0 1 1 0 1] [1 1 1 0 0 1 1 0 0 0 1 1 1 1 0 0 0 1 1 0] [1 0 1 1 0 0 1 1 0 0 0 1 1 1 1 0 0 0 1 1] [1 1 0 1 1 0 0 1 1 0 0 0 1 1 1 1 0 0 0 1] [1 1 1 0 1 1 0 0 1 1 0 0 0 1 1 1 1 0 0 0] [1 0 1 1 0 1 1 0 0 1 1 0 0 0 1 1 1 1 0 0] [1 0 0 1 1 0 1 1 0 0 1 1 0 0 0 1 1 1 1 0] [1 0 0 0 1 1 0 1 1 0 0 1 1 0 0 0 1 1 1 1] [1 1 0 0 0 1 1 0 1 1 0 0 1 1 0 0 0 1 1 1] [1 1 1 0 0 0 1 1 0 1 1 0 0 1 1 0 0 0 1 1] [1 1 1 1 0 0 0 1 1 0 1 1 0 0 1 1 0 0 0 1] [1 1 1 1 1 0 0 0 1 1 0 1 1 0 0 1 1 0 0 0] [1 0 1 1 1 1 0 0 0 1 1 0 1 1 0 0 1 1 0 0] Total time: 0.220 seconds, Total memory usage: 3.24MB '220.214' ************** MAGMA ***************** Host 220.214.76.207 (220.214.76.207) Time: Mon Dec 19 07:10:00 2005 Input: n:=20;d:=8; ////////////////////////// circulant:=function(r) n:=#r; m:=r; row:=r; for j in [1..n-1] do Rotate(~row,1); m cat:=row; end for; return Matrix(Parent(r[1]),n,n,m); end function; ////////////////////////// ////////////////////////// x1:=Vector(GF(2),1,[0]); x3:=Vector(GF(2),1,[1]); all1:=Matrix(GF(2),1,n-1,[1 : i in [1..n-1]]); t:=Transpose(Matrix(GF(2),1,n,Eltseq(x1) cat Eltseq(all1))); V:=EvenWeightCode(n-2); x2:=Random(V); x2; x:=Eltseq(x1) cat Eltseq(x2); M1:=VerticalJoin(all1,circulant(x)); M2:=HorizontalJoin(t,M1); M2; gen:=HorizontalJoin(ScalarMatrix(n,GF(2)!1),M2); Output: WARNING: MAGMA command contains unsafe command 'alarm', so it will not be executed. '220.214' ************** MAGMA ***************** Host 220.214.76.207 (220.214.76.207) Time: Mon Dec 19 07:09:51 2005 Input: n:=20;d:=8; ////////////////////////// circulant:=function(r) n:=#r; m:=r; row:=r; for j in [1..n-1] do Rotate(~row,1); m cat:=row; end for; return Matrix(Parent(r[1]),n,n,m); end function; ////////////////////////// ////////////////////////// x1:=Vector(GF(2),1,[0]); x3:=Vector(GF(2),1,[1]); all1:=Matrix(GF(2),1,n-1,[1 : i in [1..n-1]]); t:=Transpose(Matrix(GF(2),1,n,Eltseq(x1) cat Eltseq(all1))); V:=EvenWeightCode(n-2); x2:=Random(V); x2; x:=Eltseq(x1) cat Eltseq(x2); M1:=VerticalJoin(all1,circulant(x)); M2:=HorizontalJoin(t,M1); M2; gen:=HorizontalJoin(ScalarMatrix(n,GF(2)!1),M2); C:=LinearCode(gen); gen; Output: WARNING: MAGMA command contains unsafe command 'alarm', so it will not be executed. '220.214' ************** MAGMA ***************** Host 220.214.76.207 (220.214.76.207) Time: Mon Dec 19 07:09:32 2005 Input: n:=20;d:=8; ////////////////////////// circulant:=function(r) n:=#r; m:=r; row:=r; for j in [1..n-1] do Rotate(~row,1); m cat:=row; end for; return Matrix(Parent(r[1]),n,n,m); end function; ////////////////////////// ////////////////////////// x1:=Vector(GF(2),1,[0]); x3:=Vector(GF(2),1,[1]); all1:=Matrix(GF(2),1,n-1,[1 : i in [1..n-1]]); t:=Transpose(Matrix(GF(2),1,n,Eltseq(x1) cat Eltseq(all1))); V:=EvenWeightCode(n-2); x2:=Random(V); x2; x:=Eltseq(x1) cat Eltseq(x2); M1:=VerticalJoin(all1,circulant(x)); M2:=HorizontalJoin(t,M1); M2; gen:=HorizontalJoin(ScalarMatrix(n,GF(2)!1),M2); C:=LinearCode(gen); IsSelfDual(C); MinimumWeight(C); Output: WARNING: MAGMA command contains unsafe command 'alarm', so it will not be executed. '220.214' ************** MAGMA ***************** Host 220.214.76.207 (220.214.76.207) Time: Mon Dec 19 07:08:52 2005 Input: n:=20;d:=8; ////////////////////////// circulant:=function(r) n:=#r; m:=r; row:=r; for j in [1..n-1] do Rotate(~row,1); m cat:=row; end for; return Matrix(Parent(r[1]),n,n,m); end function; ////////////////////////// ////////////////////////// x1:=Vector(GF(2),1,[0]); x3:=Vector(GF(2),1,[1]); all1:=Matrix(GF(2),1,n-1,[1 : i in [1..n-1]]); t:=Transpose(Matrix(GF(2),1,n,Eltseq(x1) cat Eltseq(all1))); V:=EvenWeightCode(n-2); x2:=Random(V); x2; x:=Eltseq(x1) cat Eltseq(x2); M1:=VerticalJoin(all1,circulant(x)); M2:=HorizontalJoin(t,M1); M2; Output: Magma V2.11-10 Mon Dec 19 2005 07:08:52 on modular [Seed = 667076280] ------------------------------------- (0 0 1 0 0 0 1 0 1 1 0 0 0 1 1 1 1 0) [0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1] [1 0 0 0 1 0 0 0 1 0 1 1 0 0 0 1 1 1 1 0] [1 0 0 0 0 1 0 0 0 1 0 1 1 0 0 0 1 1 1 1] [1 1 0 0 0 0 1 0 0 0 1 0 1 1 0 0 0 1 1 1] [1 1 1 0 0 0 0 1 0 0 0 1 0 1 1 0 0 0 1 1] [1 1 1 1 0 0 0 0 1 0 0 0 1 0 1 1 0 0 0 1] [1 1 1 1 1 0 0 0 0 1 0 0 0 1 0 1 1 0 0 0] [1 0 1 1 1 1 0 0 0 0 1 0 0 0 1 0 1 1 0 0] [1 0 0 1 1 1 1 0 0 0 0 1 0 0 0 1 0 1 1 0] [1 0 0 0 1 1 1 1 0 0 0 0 1 0 0 0 1 0 1 1] [1 1 0 0 0 1 1 1 1 0 0 0 0 1 0 0 0 1 0 1] [1 1 1 0 0 0 1 1 1 1 0 0 0 0 1 0 0 0 1 0] [1 0 1 1 0 0 0 1 1 1 1 0 0 0 0 1 0 0 0 1] [1 1 0 1 1 0 0 0 1 1 1 1 0 0 0 0 1 0 0 0] [1 0 1 0 1 1 0 0 0 1 1 1 1 0 0 0 0 1 0 0] [1 0 0 1 0 1 1 0 0 0 1 1 1 1 0 0 0 0 1 0] [1 0 0 0 1 0 1 1 0 0 0 1 1 1 1 0 0 0 0 1] [1 1 0 0 0 1 0 1 1 0 0 0 1 1 1 1 0 0 0 0] [1 0 1 0 0 0 1 0 1 1 0 0 0 1 1 1 1 0 0 0] [1 0 0 1 0 0 0 1 0 1 1 0 0 0 1 1 1 1 0 0] Total time: 0.200 seconds, Total memory usage: 3.24MB '220.214' ************** MAGMA ***************** Host 220.214.76.207 (220.214.76.207) Time: Mon Dec 19 07:08:36 2005 Input: n:=20;d:=8; ////////////////////////// circulant:=function(r) n:=#r; m:=r; row:=r; for j in [1..n-1] do Rotate(~row,1); m cat:=row; end for; return Matrix(Parent(r[1]),n,n,m); end function; ////////////////////////// ////////////////////////// x1:=Vector(GF(2),1,[0]); x3:=Vector(GF(2),1,[1]); all1:=Matrix(GF(2),1,n-1,[1 : i in [1..n-1]]); t:=Transpose(Matrix(GF(2),1,n,Eltseq(x1) cat Eltseq(all1))); V:=EvenWeightCode(n-2); x2:=Random(V); x2; x:=Eltseq(x1) cat Eltseq(x2); M1:=VerticalJoin(all1,circulant(x)); M2:=HorizontalJoin(t,M1); M1; Output: Magma V2.11-10 Mon Dec 19 2005 07:08:36 on modular [Seed = 751422883] ------------------------------------- (0 1 0 0 0 0 0 0 1 1 1 1 0 0 0 0 1 0) [1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1] [0 0 1 0 0 0 0 0 0 1 1 1 1 0 0 0 0 1 0] [0 0 0 1 0 0 0 0 0 0 1 1 1 1 0 0 0 0 1] [1 0 0 0 1 0 0 0 0 0 0 1 1 1 1 0 0 0 0] [0 1 0 0 0 1 0 0 0 0 0 0 1 1 1 1 0 0 0] [0 0 1 0 0 0 1 0 0 0 0 0 0 1 1 1 1 0 0] [0 0 0 1 0 0 0 1 0 0 0 0 0 0 1 1 1 1 0] [0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 1 1 1 1] [1 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 1 1 1] [1 1 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 1 1] [1 1 1 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 1] [1 1 1 1 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0] [0 1 1 1 1 0 0 0 0 1 0 0 0 1 0 0 0 0 0] [0 0 1 1 1 1 0 0 0 0 1 0 0 0 1 0 0 0 0] [0 0 0 1 1 1 1 0 0 0 0 1 0 0 0 1 0 0 0] [0 0 0 0 1 1 1 1 0 0 0 0 1 0 0 0 1 0 0] [0 0 0 0 0 1 1 1 1 0 0 0 0 1 0 0 0 1 0] [0 0 0 0 0 0 1 1 1 1 0 0 0 0 1 0 0 0 1] [1 0 0 0 0 0 0 1 1 1 1 0 0 0 0 1 0 0 0] [0 1 0 0 0 0 0 0 1 1 1 1 0 0 0 0 1 0 0] Total time: 0.200 seconds, Total memory usage: 3.24MB '220.214' ************** MAGMA ***************** Host 220.214.76.207 (220.214.76.207) Time: Mon Dec 19 07:08:23 2005 Input: n:=20;d:=8; ////////////////////////// circulant:=function(r) n:=#r; m:=r; row:=r; for j in [1..n-1] do Rotate(~row,1); m cat:=row; end for; return Matrix(Parent(r[1]),n,n,m); end function; ////////////////////////// ////////////////////////// x1:=Vector(GF(2),1,[0]); x3:=Vector(GF(2),1,[1]); all1:=Matrix(GF(2),1,n-1,[1 : i in [1..n-1]]); t:=Transpose(Matrix(GF(2),1,n,Eltseq(x1) cat Eltseq(all1))); V:=EvenWeightCode(n-2); x2:=Random(V); x2; x:=Eltseq(x1) cat Eltseq(x2); M1:=VerticalJoin(all1,circulant(x)); M2:=HorizontalJoin(t,M1); Output: Magma V2.11-10 Mon Dec 19 2005 07:08:23 on modular [Seed = 701286613] ------------------------------------- (1 0 1 1 1 1 0 0 0 1 1 1 1 1 0 1 1 0) Total time: 0.210 seconds, Total memory usage: 3.24MB '220.214' ************** MAGMA ***************** Host 220.214.76.207 (220.214.76.207) Time: Mon Dec 19 07:08:14 2005 Input: n:=20;d:=8; ////////////////////////// circulant:=function(r) n:=#r; m:=r; row:=r; for j in [1..n-1] do Rotate(~row,1); m cat:=row; end for; return Matrix(Parent(r[1]),n,n,m); end function; ////////////////////////// ////////////////////////// x1:=Vector(GF(2),1,[0]); x3:=Vector(GF(2),1,[1]); all1:=Matrix(GF(2),1,n-1,[1 : i in [1..n-1]]); t:=Transpose(Matrix(GF(2),1,n,Eltseq(x1) cat Eltseq(all1))); V:=EvenWeightCode(n-2); x2:=Random(V); x2; x:=Eltseq(x1) cat Eltseq(x2); M1:=VerticalJoin(all1,circulant(x)); M2:=HorizontalJoin(t,M1); gen:=HorizontalJoin(ScalarMatrix(n,GF(2)!1),circulant(x)); Output: WARNING: MAGMA command contains unsafe command 'alarm', so it will not be executed. '220.214' ************** MAGMA ***************** Host 220.214.76.207 (220.214.76.207) Time: Mon Dec 19 07:07:57 2005 Input: n:=20;d:=8; ////////////////////////// circulant:=function(r) n:=#r; m:=r; row:=r; for j in [1..n-1] do Rotate(~row,1); m cat:=row; end for; return Matrix(Parent(r[1]),n,n,m); end function; ////////////////////////// ////////////////////////// x1:=Vector(GF(2),1,[0]); x3:=Vector(GF(2),1,[1]); all1:=Matrix(GF(2),1,n-1,[1 : i in [1..n-1]]); t:=Transpose(Matrix(GF(2),1,n,Eltseq(x1) cat Eltseq(all1))); V:=EvenWeightCode(n-2); x2:=Random(V); x2; x:=Eltseq(x1) cat Eltseq(x2); M1:=VerticalJoin(all1,circulant(x)); M2:=HorizontalJoin(t,M1); gen:=HorizontalJoin(ScalarMatrix(n,GF(2)!1),circulant(x)); C:=LinearCode(gen); IsSelfDual(C); MinimumWeight(C); Output: WARNING: MAGMA command contains unsafe command 'alarm', so it will not be executed. '220.214' ************** MAGMA ***************** Host 220.214.76.207 (220.214.76.207) Time: Mon Dec 19 07:07:37 2005 Input: n:=20;d:=8; ////////////////////////// circulant:=function(r) n:=#r; m:=r; row:=r; for j in [1..n-1] do Rotate(~row,1); m cat:=row; end for; return Matrix(Parent(r[1]),n,n,m); end function; ////////////////////////// ////////////////////////// x1:=Vector(GF(2),1,[0]); x3:=Vector(GF(2),1,[1]); all1:=Matrix(GF(2),1,n-1,[1 : i in [1..n-1]]); t:=Transpose(Matrix(GF(2),1,n,Eltseq(x1) cat Eltseq(all1))); V:=EvenWeightCode(n-2); x2:=Random(V); x2; Output: Magma V2.11-10 Mon Dec 19 2005 07:07:37 on modular [Seed = 519831516] ------------------------------------- (1 0 1 1 0 1 1 0 0 1 0 1 1 0 1 1 1 1) Total time: 0.210 seconds, Total memory usage: 3.24MB '220.214' ************** MAGMA ***************** Host 220.214.76.207 (220.214.76.207) Time: Mon Dec 19 07:07:27 2005 Input: n:=20;d:=8; ////////////////////////// circulant:=function(r) n:=#r; m:=r; row:=r; for j in [1..n-1] do Rotate(~row,1); m cat:=row; end for; return Matrix(Parent(r[1]),n,n,m); end function; ////////////////////////// ////////////////////////// x1:=Vector(GF(2),1,[0]); x3:=Vector(GF(2),1,[1]); all1:=Matrix(GF(2),1,n-1,[1 : i in [1..n-1]]); t:=Transpose(Matrix(GF(2),1,n,Eltseq(x1) cat Eltseq(all1))); V:=EvenWeightCode(n-2); x2:=Random(V); Output: Magma V2.11-10 Mon Dec 19 2005 07:07:26 on modular [Seed = 469695022] ------------------------------------- Total time: 0.230 seconds, Total memory usage: 3.24MB '220.214' ************** MAGMA ***************** Host 220.214.76.207 (220.214.76.207) Time: Mon Dec 19 07:07:12 2005 Input: n:=20;d:=8; ////////////////////////// circulant:=function(r) n:=#r; m:=r; row:=r; for j in [1..n-1] do Rotate(~row,1); m cat:=row; end for; return Matrix(Parent(r[1]),n,n,m); end function; ////////////////////////// Output: Magma V2.11-10 Mon Dec 19 2005 07:07:09 on modular [Seed = 15065382] ------------------------------------- Total time: 0.260 seconds, Total memory usage: 3.24MB '220.214' ************** MAGMA ***************** Host 220.214.76.207 (220.214.76.207) Time: Mon Dec 19 07:06:09 2005 Input: n:=20;d:=8; ////////////////////////// circulant:=function(r) n:=#r; m:=r; row:=r; for j in [1..n-1] do Rotate(~row,1); m cat:=row; end for; return Matrix(Parent(r[1]),n,n,m); end function; ////////////////////////// x1:=Vector(GF(2),1,[0]); x3:=Vector(GF(2),1,[1]); all1:=Matrix(GF(2),1,n-1,[1 : i in [1..n-1]]); t:=Transpose(Matrix(GF(2),1,n,Eltseq(x1) cat Eltseq(all1))); V:=EvenWeightCode(n-2); x2:=Random(V); x:=Eltseq(x1) cat Eltseq(x2); M1:=VerticalJoin(all1,circulant(x)); M2:=HorizontalJoin(t,M1); gen:=HorizontalJoin(ScalarMatrix(n,GF(2)!1),circulant(x)); C:=LinearCode(gen); IsSelfDual(C); MinimumWeight(C); Output: WARNING: MAGMA command contains unsafe command 'alarm', so it will not be executed. '60.225.' ************** MAGMA ***************** Host 60.225.128.201 (60.225.128.201) Time: Mon Dec 19 00:36:02 2005 Input: Factorization(00000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000090000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000); Output: Magma V2.11-10 Mon Dec 19 2005 00:36:02 on modular [Seed = 3926912878] ------------------------------------- [ <2, 94>, <3, 2>, <5, 94> ] Total time: 0.190 seconds, Total memory usage: 3.24MB '60.225.' ************** MAGMA ***************** Host 60.225.128.201 (60.225.128.201) Time: Mon Dec 19 00:34:19 2005 Input: Factorization(000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000900000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000007000000000000000000000000000000000000000000000000000000000000000000000000000000000); Output: ** WARNING: Computation time exceeded 20 seconds, so computation was terminated after 20 seconds. ** Magma V2.11-10 Mon Dec 19 2005 00:33:59 on modular [Seed = 3758482913] ------------------------------------- Errors: /bin/sh: line 1: 1446 Alarm clock nice -n 19 /usr/local/bin/magma '60.225.' ************** MAGMA ***************** Host 60.225.128.201 (60.225.128.201) Time: Mon Dec 19 00:30:20 2005 Input: Factorization(0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000009000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000070000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000); Output: ** WARNING: Computation time exceeded 20 seconds, so computation was terminated after 20 seconds. ** Magma V2.11-10 Mon Dec 19 2005 00:29:59 on modular [Seed = 3661618379] ------------------------------------- Errors: /bin/sh: line 1: 1436 Alarm clock nice -n 19 /usr/local/bin/magma '60.225.' ************** MAGMA ***************** Host 60.225.128.201 (60.225.128.201) Time: Mon Dec 19 00:28:12 2005 Input: Factorization(33333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333); Output: ** WARNING: Computation time exceeded 20 seconds, so computation was terminated after 20 seconds. ** Magma V2.11-10 Mon Dec 19 2005 00:27:51 on modular [Seed = 3543585858] ------------------------------------- Errors: /bin/sh: line 1: 1429 Alarm clock nice -n 19 /usr/local/bin/magma '82.55.7' ************** MAGMA ***************** Host 82.55.71.167 (82.55.71.167) Time: Sun Dec 18 17:30:04 2005 Input: 10! Output: Magma V2.11-10 Sun Dec 18 2005 17:30:04 on modular [Seed = 3160022383] ------------------------------------- >> 10!; ^ User error: bad syntax Total time: 0.190 seconds, Total memory usage: 3.24MB '155.207' ************** MAGMA ***************** Host 155.207.209.176 (155.207.209.176) Time: Sun Dec 18 16:51:33 2005 Input: R := PolynomialRing(Integers()); > > f :=x^4-26*x^2+1; > > K := NumberField(f); > > O := MaximalOrder(K); > > I := IntegralBasis(K); > > print I; p:=3; q:=7; n:=p*q; > >NormEquation(O, 4*n^4); K := NumberField(f); u1:=-9842-1526*a-16289/2*(a^2-1)-8036/4*(a^3 +a^2 -a -1); u2:=651+189*a+1113/2*(a^2-1)+532/4*(a^3 +a^2 -a -1); u3:=-42+0*a-63/2*(a^2-1)-42/4*(a^3 +a^2 -a -1); u4:=-63+21*a-161/2*(a^2-1)+56/4*(a^3 +a^2 -a -1); u5:=994-154*a+2457/2*(a^2-1)-812/4*(a^3 +a^2 -a -1); UnitEquation(a*u1,-a*u1,2*a*n); UnitEquation(a*u2,-a*u2,2*a*n); UnitEquation(a*u3,-a*u3,2*a*n); UnitEquation(a*u4,-a*u4,2*a*n); UnitEquation(a*u5,-a*u5,2*a*n); Output: Magma V2.11-10 Sun Dec 18 2005 16:51:32 on modular [Seed = 266966483] ------------------------------------- [ 1, y, 1/2*(y^2 - 1), 1/4*(y^3 + y^2 - y - 1) ] true [ [-9842, -1526, -16289, -8036], [651, 189, 1113, 532], [-42, 0, -63, -42], [-63, 21, -161, 56], [994, -154, 2457, -812] ] [] [] [] [] [] Total time: 1.199 seconds, Total memory usage: 3.72MB '155.207' ************** MAGMA ***************** Host 155.207.209.176 (155.207.209.176) Time: Sun Dec 18 16:46:21 2005 Input: R := PolynomialRing(Integers()); > > f :=x^4-26*x^2+1; > > K := NumberField(f); > > O := MaximalOrder(K); > > I := IntegralBasis(K); > > print I; p:=3; q:=7; n:=p*q; > >NormEquation(O, 4*n^4); K := NumberField(f); u1:=-21-42*a-21/2*(a^2+1)+42/44*(a^3 + 11*a^2 + 43*a + 33); UnitEquation(a*u1,-a*u1,2*a*n); Output: Magma V2.11-10 Sun Dec 18 2005 16:46:21 on modular [Seed = 2958437156] ------------------------------------- [ 1, y, 1/2*(y^2 - 1), 1/4*(y^3 + y^2 - y - 1) ] true [ [-9842, -1526, -16289, -8036], [651, 189, 1113, 532], [-42, 0, -63, -42], [-63, 21, -161, 56], [994, -154, 2457, -812] ] >> UnitEquation(a*u1,-a*u1,2*a*n); ^ Runtime error in 'UnitEquation': Elements must all be integral Total time: 0.440 seconds, Total memory usage: 3.60MB '155.207' ************** MAGMA ***************** Host 155.207.209.124 (155.207.209.124) Time: Sun Dec 18 16:32:15 2005 Input: R := PolynomialRing(Integers()); > > f :=x^4-34*x^2+121; > > K := NumberField(f); > > O := MaximalOrder(K); > > I := IntegralBasis(K); > > print I; p:=3; q:=7; n:=p*q; > >NormEquation(O, 4*n^4); K := NumberField(f); u1:=-21-42*a-21/2*(a^2+1)+42/44*(a^3 + 11*a^2 + 43*a + 33); UnitEquation(a*u1,-a*u1,2*a*n); Output: Magma V2.11-10 Sun Dec 18 2005 16:32:13 on modular [Seed = 1269685551] ------------------------------------- [ 1, y, 1/2*(y^2 + 1), 1/44*(y^3 + 11*y^2 + 43*y + 33) ] true [ [-21, -42, -21, 42] ] [ [[84, 73, 18, -44] [81, 63, 15, -38]], [[-40, 73, 26, -44] [-43, 63, 23, -38]], [[0, 7, 2, -4] [-3, -3, -1, 2]], [[4, 7, 2, -4] [1, -3, -1, 2]], [[9, 5, 1, -3] [6, -5, -2, 3]], [[-6, 5, 2, -3] [-9, -5, -1, 3]], [[-1, 3, 1, -2] [-4, -7, -2, 4]], [[3, 3, 1, -2] [0, -7, -2, 4]], [[43, -63, -23, 38] [40, -73, -26, 44]], [[-81, -63, -15, 38] [-84, -73, -18, 44]] ] Total time: 1.229 seconds, Total memory usage: 3.72MB '155.207' ************** MAGMA ***************** Host 155.207.209.124 (155.207.209.124) Time: Sun Dec 18 16:31:23 2005 Input: R := PolynomialRing(Integers()); > > f :=x^4-34*x^2+121; > > K := NumberField(f); > > O := MaximalOrder(K); > > I := IntegralBasis(K); > > print I; p:=3; q:=7; n:=p*q; > >NormEquation(O, 4*n^4); K := NumberField(f); u1:=-21-42*a-21/2*(a^2+1)+42/44*(y^3 + 11*y^2 + 43*y + 33); UnitEquation(a*u1,-a*u1,2*a*n); Output: Magma V2.11-10 Sun Dec 18 2005 16:31:23 on modular [Seed = 1421001286] ------------------------------------- [ 1, y, 1/2*(y^2 + 1), 1/44*(y^3 + 11*y^2 + 43*y + 33) ] true [ [-21, -42, -21, 42] ] >> u1:=-21-42*a-21/2*(a^2+1)+42/44*(y^3 + 11*y^2 + 43*y + 33); ^ Runtime error in '+': Arguments are not compatible Argument types given: FldNumElt, FldNumElt >> UnitEquation(a*u1,-a*u1,2*a*n); ^ User error: Identifier 'u1' has not been declared or assigned Total time: 0.350 seconds, Total memory usage: 3.63MB '155.207' ************** MAGMA ***************** Host 155.207.209.124 (155.207.209.124) Time: Sun Dec 18 16:28:33 2005 Input: R := PolynomialRing(Integers()); > > f :=x^4-34*x^2+121; > > K := NumberField(f); > > O := MaximalOrder(K); > > I := IntegralBasis(K); > > print I; p:=3; q:=7; n:=p*q; > >NormEquation(O, 4*n^4); K := NumberField(f); z:=a/2; u1:=56-105*z-63*z^2+49*z^3; u2:=14-35*z+7*z^2-7*z^3; u3:=-21*z+21*z^2+21*z^3; u4:=14-7*z-77*z^2-35*z^3; u5:=-56-49*z+217*z^2+105*z^3; UnitEquation(a*u1,-a*u1,2*a*n); UnitEquation(a*u2,-a*u2,2*a*n); UnitEquation(a*u3,-a*u3,2*a*n); UnitEquation(a*u4,-a*u4,2*a*n); UnitEquation(a*u5,-a*u5,2*a*n); Output: Magma V2.11-10 Sun Dec 18 2005 16:28:33 on modular [Seed = 1589168614] ------------------------------------- [ 1, y, 1/2*(y^2 + 1), 1/44*(y^3 + 11*y^2 + 43*y + 33) ] true [ [-21, -42, -21, 42] ] >> UnitEquation(a*u1,-a*u1,2*a*n); ^ Runtime error in 'UnitEquation': Elements must all be integral >> UnitEquation(a*u2,-a*u2,2*a*n); ^ Runtime error in 'UnitEquation': Elements must all be integral >> UnitEquation(a*u3,-a*u3,2*a*n); ^ Runtime error in 'UnitEquation': Elements must all be integral >> UnitEquation(a*u4,-a*u4,2*a*n); ^ Runtime error in 'UnitEquation': Elements must all be integral >> UnitEquation(a*u5,-a*u5,2*a*n); ^ Runtime error in 'UnitEquation': Elements must all be integral Total time: 0.350 seconds, Total memory usage: 3.63MB '155.207' ************** MAGMA ***************** Host 155.207.209.221 (155.207.209.221) Time: Sun Dec 18 14:30:21 2005 Input: R := PolynomialRing(Integers()); > > f :=x^4-20*x^2+16; > > K := NumberField(f); > > O := MaximalOrder(K); > > I := IntegralBasis(K); > > print I; p:=3; q:=7; n:=p*q; > >NormEquation(O, 4*n^4); K := NumberField(f); z:=a/2; u1:=56-105*z-63*z^2+49*z^3; u2:=14-35*z+7*z^2-7*z^3; u3:=-21*z+21*z^2+21*z^3; u4:=14-7*z-77*z^2-35*z^3; u5:=-56-49*z+217*z^2+105*z^3; UnitEquation(a*u1,-a*u1,2*a*n); UnitEquation(a*u2,-a*u2,2*a*n); UnitEquation(a*u3,-a*u3,2*a*n); UnitEquation(a*u4,-a*u4,2*a*n); UnitEquation(a*u5,-a*u5,2*a*n); Output: Magma V2.11-10 Sun Dec 18 2005 14:30:20 on modular [Seed = 266990835] ------------------------------------- [ 1, 1/2*y, 1/4*y^2, 1/8*y^3 ] true [ [56, -105, -63, 49], [14, -35, 7, -7], [0, -21, 21, 21], [14, -7, -77, -35], [-56, -49, 217, 105] ] [] [] [ [[62, 72, -13, -15] [67, 91, -14, -19]], [[5, 0, -1, 0] [10, 19, -2, -4]], [[2, -4, -1, 1] [7, 15, -2, -3]], [[-7, -15, 2, 3] [-2, 4, 1, -1]], [[-10, -19, 2, 4] [-5, 0, 1, 0]], [[-67, -91, 14, 19] [-62, -72, 13, 15]] ] [] [] Total time: 1.250 seconds, Total memory usage: 3.70MB '155.207' ************** MAGMA ***************** Host 155.207.209.221 (155.207.209.221) Time: Sun Dec 18 14:29:01 2005 Input: R := PolynomialRing(Integers()); > > f :=x^4-20*x^2+16; > > K := NumberField(f); > > O := MaximalOrder(K); > > I := IntegralBasis(K); > > print I; p:=3; q:=7; n:=p*q; > >NormEquation(O, 4*n^4); K := NumberField(f); z:=a/2; u1:=56-105*z-63*z^2+49*z^3; u2:=14-35*z+7*z^2-7*z^3; u3:=-21*z+21*z^2+21*z^3; UnitEquation(a*u1,-a*u1,2*a*n); UnitEquation(a*u2,-a*u2,2*a*n); UnitEquation(a*u3,-a*u3,2*a*n); Output: Magma V2.11-10 Sun Dec 18 2005 14:29:00 on modular [Seed = 166720069] ------------------------------------- [ 1, 1/2*y, 1/4*y^2, 1/8*y^3 ] true [ [56, -105, -63, 49], [14, -35, 7, -7], [0, -21, 21, 21], [14, -7, -77, -35], [-56, -49, 217, 105] ] [] [] [ [[62, 72, -13, -15] [67, 91, -14, -19]], [[5, 0, -1, 0] [10, 19, -2, -4]], [[2, -4, -1, 1] [7, 15, -2, -3]], [[-7, -15, 2, 3] [-2, 4, 1, -1]], [[-10, -19, 2, 4] [-5, 0, 1, 0]], [[-67, -91, 14, 19] [-62, -72, 13, 15]] ] Total time: 1.260 seconds, Total memory usage: 3.70MB '155.207' ************** MAGMA ***************** Host 155.207.209.221 (155.207.209.221) Time: Sun Dec 18 14:27:53 2005 Input: R := PolynomialRing(Integers()); > > f :=x^4-20*x^2+16; > > K := NumberField(f); > > O := MaximalOrder(K); > > I := IntegralBasis(K); > > print I; p:=3; q:=7; n:=p*q; > >NormEquation(O, 4*n^4); K := NumberField(f); z:=a/2; u1:=56-105*z-63*z^2+49*z^3; u2:=14-35*z+7*z^2-7*z^3; UnitEquation(a*u1,-a*u1,2*a*n); UnitEquation(a*u2,-a*u2,2*a*n); Output: Magma V2.11-10 Sun Dec 18 2005 14:27:53 on modular [Seed = 81985293] ------------------------------------- [ 1, 1/2*y, 1/4*y^2, 1/8*y^3 ] true [ [56, -105, -63, 49], [14, -35, 7, -7], [0, -21, 21, 21], [14, -7, -77, -35], [-56, -49, 217, 105] ] [] [] Total time: 0.420 seconds, Total memory usage: 3.60MB '155.207' ************** MAGMA ***************** Host 155.207.209.221 (155.207.209.221) Time: Sun Dec 18 14:27:32 2005 Input: R := PolynomialRing(Integers()); > > f :=x^4-20*x^2+16; > > K := NumberField(f); > > O := MaximalOrder(K); > > I := IntegralBasis(K); > > print I; p:=3; q:=7; n:=p*q; > >NormEquation(O, 4*n^4); K := NumberField(f); z:=a/2; u1:=56-105*z-63*z^2+49*z^3; u2:=14-35*z+7*z^2-7*z^3; UnitEquation(a*u1,a*u1,2*a*n); UnitEquation(a*u2,a*u2,2*a*n); Output: Magma V2.11-10 Sun Dec 18 2005 14:27:31 on modular [Seed = 536470540] ------------------------------------- [ 1, 1/2*y, 1/4*y^2, 1/8*y^3 ] true [ [56, -105, -63, 49], [14, -35, 7, -7], [0, -21, 21, 21], [14, -7, -77, -35], [-56, -49, 217, 105] ] [] [] Total time: 0.430 seconds, Total memory usage: 3.60MB '155.207' ************** MAGMA ***************** Host 155.207.209.221 (155.207.209.221) Time: Sun Dec 18 14:26:43 2005 Input: R := PolynomialRing(Integers()); > > f :=x^4-20*x^2+16; > > K := NumberField(f); > > O := MaximalOrder(K); > > I := IntegralBasis(K); > > print I; p:=3; q:=7; n:=p*q; > >NormEquation(O, 4*n^4); K := NumberField(f); z:=a/2; u1:=56-105*z-63*z^2+49*z^3; u1:=56-105*z-63*z^2+49*z^3; UnitEquation(a*u1,a*u1,2*a*n); Output: Magma V2.11-10 Sun Dec 18 2005 14:26:42 on modular [Seed = 452783832] ------------------------------------- [ 1, 1/2*y, 1/4*y^2, 1/8*y^3 ] true [ [56, -105, -63, 49], [14, -35, 7, -7], [0, -21, 21, 21], [14, -7, -77, -35], [-56, -49, 217, 105] ] [] Total time: 0.430 seconds, Total memory usage: 3.60MB '155.207' ************** MAGMA ***************** Host 155.207.209.221 (155.207.209.221) Time: Sun Dec 18 14:26:27 2005 Input: R := PolynomialRing(Integers()); > > f :=x^4-20*x^2+16; > > K := NumberField(f); > > O := MaximalOrder(K); > > I := IntegralBasis(K); > > print I; p:=3; q:=7; n:=p*q; > >NormEquation(O, 4*n^4); K := NumberField(f); z:=a/2; u1:=56-105*z-63*z^2+49*z^3; u1:=56-105*z-63*z^2+49*z^3; UnitEquation(a*u1,a*u1,2*a*n)); Output: Magma V2.11-10 Sun Dec 18 2005 14:26:27 on modular [Seed = 368304582] ------------------------------------- [ 1, 1/2*y, 1/4*y^2, 1/8*y^3 ] true [ [56, -105, -63, 49], [14, -35, 7, -7], [0, -21, 21, 21], [14, -7, -77, -35], [-56, -49, 217, 105] ] >> UnitEquation(a*u1,a*u1,2*a*n));; ^ User error: bad syntax Total time: 0.370 seconds, Total memory usage: 3.60MB '155.207' ************** MAGMA ***************** Host 155.207.209.221 (155.207.209.221) Time: Sun Dec 18 14:25:59 2005 Input: R := PolynomialRing(Integers()); > > f :=x^4-20*x^2+16; > > K := NumberField(f); > > O := MaximalOrder(K); > > I := IntegralBasis(K); > > print I; p:=3; q:=7; n:=p*q; > >NormEquation(O, 4*n^4); K := NumberField(f); z:=a/2; u1:=56-105*z-63z^2+49*z^3; u1:=56-105*z-63z^2+49*z^3; UnitEquation(a*u1,a*u1,2*a*n)); Output: Magma V2.11-10 Sun Dec 18 2005 14:25:58 on modular [Seed = 284615910] ------------------------------------- [ 1, 1/2*y, 1/4*y^2, 1/8*y^3 ] true [ [56, -105, -63, 49], [14, -35, 7, -7], [0, -21, 21, 21], [14, -7, -77, -35], [-56, -49, 217, 105] ] >> u1:=56-105*z-63z^2+49*z^3; ^ User error: bad syntax >> u1:=56-105*z-63z^2+49*z^3; ^ User error: bad syntax >> UnitEquation(a*u1,a*u1,2*a*n));; ^ User error: bad syntax Total time: 0.420 seconds, Total memory usage: 3.60MB '155.207' ************** MAGMA ***************** Host 155.207.209.183 (155.207.209.183) Time: Sun Dec 18 13:14:25 2005 Input: R := PolynomialRing(Integers()); > > f :=x^4-20*x^2+16; > > K := NumberField(f); > > O := MaximalOrder(K); > > I := IntegralBasis(K); > > print I; p:=3; q:=7; > >NormEquation(O, 4*(p*q)^4); Output: Magma V2.11-10 Sun Dec 18 2005 13:14:24 on modular [Seed = 1118086763] ------------------------------------- [ 1, 1/2*y, 1/4*y^2, 1/8*y^3 ] true [ [56, -105, -63, 49], [14, -35, 7, -7], [0, -21, 21, 21], [14, -7, -77, -35], [-56, -49, 217, 105] ] Total time: 0.380 seconds, Total memory usage: 3.60MB '155.207' ************** MAGMA ***************** Host 155.207.209.81 (155.207.209.81) Time: Sun Dec 18 13:09:52 2005 Input: R := PolynomialRing(Integers()); > > f :=x^4-4*x^2+2; > > K := NumberField(f); > > O := MaximalOrder(K); > > I := IntegralBasis(K); > > print I; p:=1; q:=5; > >NormEquation(O, 4*(p*q)^4); Output: Magma V2.11-10 Sun Dec 18 2005 13:09:51 on modular [Seed = 1286253194] ------------------------------------- [ 1, y, y^2, y^3 ] true [ [0, 0, -5, 0] ] Total time: 0.330 seconds, Total memory usage: 3.63MB '155.207' ************** MAGMA ***************** Host 155.207.209.81 (155.207.209.81) Time: Sun Dec 18 13:09:40 2005 Input: R := PolynomialRing(Integers()); > > f :=x^4-4*x^2+2; > > K := NumberField(f); > > O := MaximalOrder(K); > > I := IntegralBasis(K); > > print I; p=1; q=5; > >NormEquation(O, 4*(p*q)^4); Output: Magma V2.11-10 Sun Dec 18 2005 13:09:39 on modular [Seed = 1236250499] ------------------------------------- [ 1, y, y^2, y^3 ] >> p=1; ^ User error: Identifier 'p' has not been declared or assigned >> q=5; ^ User error: Identifier 'q' has not been declared or assigned >> NormEquation(O, 4*(p*q)^4);; ^ User error: Identifier 'p' has not been declared or assigned Total time: 0.200 seconds, Total memory usage: 3.24MB '212.138' ************** MAGMA ***************** Host 212.138.47.15 (212.138.47.15) Time: Sun Dec 18 10:08:42 2005 Input: P:=PolynomialRing(RationalField(),3); I:=ideal

; Radical(I); Output: Magma V2.11-10 Sun Dec 18 2005 10:08:41 on modular [Seed = 1387555329] ------------------------------------- Ideal of Polynomial ring of rank 3 over Rational Field Lexicographical Order Variables: x, y, z Dimension 0, Radical Groebner basis: [ x + 216*z^4 - 504*z^3 - 24*z^2 + 371*z + 110, y + 6*z^2 - 7*z - 5, z^9 + 279936*z^8 - 1306368*z^7 + 1461888*z^6 + 1106784*z^5 - 1953720*z^4 - 775656*z^3 + 793998*z^2 + 492317*z + 73365 ] Total time: 0.200 seconds, Total memory usage: 3.34MB '212.138' ************** MAGMA ***************** Host 212.138.47.23 (212.138.47.23) Time: Sun Dec 18 10:08:04 2005 Input: P:=PolynomialRing(RationalField(),3); I:=ideal

; Radical(I); Output: Magma V2.11-10 Sun Dec 18 2005 10:08:04 on modular [Seed = 1589143072] ------------------------------------- Ideal of Polynomial ring of rank 3 over Rational Field Lexicographical Order Variables: x, y, z Dimension 0, Radical Groebner basis: [ x + 216*z^4 - 504*z^3 - 24*z^2 + 371*z + 110, y + 6*z^2 - 7*z - 5, z^8 - 14/3*z^7 + 47/9*z^6 + 427/108*z^5 - 335/48*z^4 - 775655/279936*z^3 + 44111/15552*z^2 + 492317/279936*z + 24455/93312 ] Total time: 0.190 seconds, Total memory usage: 3.34MB '212.138' ************** MAGMA ***************** Host 212.138.47.29 (212.138.47.29) Time: Sun Dec 18 10:07:22 2005 Input: P:=PolynomialRing(RationalField(),3); I:=ideal

; Radical(I); Output: Magma V2.11-10 Sun Dec 18 2005 10:07:21 on modular [Seed = 1151499335] ------------------------------------- Ideal of Polynomial ring of rank 3 over Rational Field Lexicographical Order Variables: x, y, z Dimension 0, Radical Groebner basis: [ x - 20670318404676804121590341679907342944438721410236055716516252786259218\ 420711433810584139369555228744398094045155409199104/1393743412806712329\ 99759010297844161860871496969373562963049485474223528276610984064461508\ 093880757021543538634046896781115514291717975*z^26 + 52817691742356886370701872211589394258641546772944677761355469155105730\ 384820419602679329873674311460101684589053757063168/1393743412806712329\ 99759010297844161860871496969373562963049485474223528276610984064461508\ 093880757021543538634046896781115514291717975*z^25 + 44084189102656559761036896914171515230703918700362814196736477697748482\ 349329165388194181519205694144483399773379046731904/1393743412806712329\ 99759010297844161860871496969373562963049485474223528276610984064461508\ 093880757021543538634046896781115514291717975*z^24 + 10282605380629114589617264989256557708549756853994112065698656467656094\ 04406261142299579793912180745541265316913236203077632/13937434128067123\ 29997590102978441618608714969693735629630494854742235282766109840644615\ 08093880757021543538634046896781115514291717975*z^23 - 22228881978492193293559859093936405405939009088196569488635554540757211\ 37017756817169737946803181622067694970688957503668224/13937434128067123\ 29997590102978441618608714969693735629630494854742235282766109840644615\ 08093880757021543538634046896781115514291717975*z^22 - 17181692219196114429115811806207940681105897576614572215614886254607317\ 41173496080747558758974447442649000940198623520859392/13937434128067123\ 29997590102978441618608714969693735629630494854742235282766109840644615\ 08093880757021543538634046896781115514291717975*z^21 - 61640289475596731450679340165759399618431062957953433768199378491567206\ 565511264862842657794696865313209618370001141849358336/1393743412806712\ 32999759010297844161860871496969373562963049485474223528276610984064461\ 508093880757021543538634046896781115514291717975*z^20 - 66095752571214459624518529004011368779461729417153411440699393722538310\ 17246088969923789514916084342288456155181781001857089536/13937434128067\ 12329997590102978441618608714969693735629630494854742235282766109840644\ 61508093880757021543538634046896781115514291717975*z^19 + 25238028072639823696679503339288656807992425850973474585354246877644706\ 187685801574587471601563331251578311852754405108094443328/1393743412806\ 71232999759010297844161860871496969373562963049485474223528276610984064\ 461508093880757021543538634046896781115514291717975*z^18 + 15100774264212872360384730726830227678249159853860787662574702196479871\ 49631337678553773700882486776776263487029256690143371264/13937434128067\ 12329997590102978441618608714969693735629630494854742235282766109840644\ 61508093880757021543538634046896781115514291717975*z^17 + 44841353739392875941013024676041792726615548527879166766758623056926826\ 3013268190022444874591316028252564801406550386388655271936/139374341280\ 67123299975901029784416186087149696937356296304948547422352827661098406\ 4461508093880757021543538634046896781115514291717975*z^16 - 14801931190140762886015675654974862129502652192610372634978790059234754\ 947512399230178102320439394185071889427571268295241900385632/1393743412\ 80671232999759010297844161860871496969373562963049485474223528276610984\ 064461508093880757021543538634046896781115514291717975*z^15 + 64118150688406168899806736311366044438266976175150900955009941229687123\ 359613473660565416868586665235987375007859422298128871370752/1393743412\ 80671232999759010297844161860871496969373562963049485474223528276610984\ 064461508093880757021543538634046896781115514291717975*z^14 - 59766235433460852251769712174872729369152399253394915317285117833345339\ 512065855473683081139767348030618064160139820193734351419392/1393743412\ 80671232999759010297844161860871496969373562963049485474223528276610984\ 064461508093880757021543538634046896781115514291717975*z^13 - 28951070792768735987119578603040775907790667817768845226622240582260527\ 922476126045937075626141100552572245765618249291256309909688/2787486825\ 61342465999518020595688323721742993938747125926098970948447056553221968\ 12892301618776151404308707726809379356223102858343595*z^12 + 76499372079855456323013037441293431543773048046253148902645677881534577\ 4660856190940754360644032963410442949681497376069584715073536/139374341\ 28067123299975901029784416186087149696937356296304948547422352827661098\ 4064461508093880757021543538634046896781115514291717975*z^11 - 18265271834715021921507366802108499920741026425191081094680295339668321\ 8637033273500632516798684292743702866140155367781483358671872/139374341\ 28067123299975901029784416186087149696937356296304948547422352827661098\ 4064461508093880757021543538634046896781115514291717975*z^10 + 81234711366392755150707194044826385840652588795161583303875060921131616\ 16613798195316541777619380893565306500576380034572591855769816/13937434\ 12806712329997590102978441618608714969693735629630494854742235282766109\ 84064461508093880757021543538634046896781115514291717975*z^9 - 76301446847565195312748120513029598866156889065065860133024518164894305\ 3105777355729600924867262813718143714436913502675351522258163712/139374\ 34128067123299975901029784416186087149696937356296304948547422352827661\ 0984064461508093880757021543538634046896781115514291717975*z^8 - 82348805467619018020877218869999199242072232530941549101334278979740312\ 08331342268339335248205917407685848265330600950653722177270555648/13937\ 43412806712329997590102978441618608714969693735629630494854742235282766\ 10984064461508093880757021543538634046896781115514291717975*z^7 + 84412521960472857160911216066242401851090200344030094679739981190800010\ 375965825632433557714526225507464746855468694951312740791718236002/1393\ 74341280671232999759010297844161860871496969373562963049485474223528276\ 610984064461508093880757021543538634046896781115514291717975*z^6 - 19683600898511070798075127987078432887095168620485541000219071579591808\ 6564629037654229865133236791460268215512602009300843346282091915264/139\ 37434128067123299975901029784416186087149696937356296304948547422352827\ 6610984064461508093880757021543538634046896781115514291717975*z^5 + 10666103245020463025460572567380480364748019759231156004389783614618856\ 4278509574124073487675029961578117423850555852445478175878833243136/139\ 37434128067123299975901029784416186087149696937356296304948547422352827\ 6610984064461508093880757021543538634046896781115514291717975*z^4 + 96184597403301187882098000111345017149388757160701053527995880628846239\ 321327676865418804844922873757179730046636417630135402601301240819/1393\ 74341280671232999759010297844161860871496969373562963049485474223528276\ 610984064461508093880757021543538634046896781115514291717975*z^3 - 23484072682953374360602270413978122643907931573358680446186791183604617\ 35064254014182229707643805925968450130018541132374369207378926592/55749\ 73651226849319990360411913766474434859878774942518521979418968941131064\ 439362578460323755230280861741545361875871244620571668719*z^2 - 26306871523575483715460693403165967104582376597576450916239167773516248\ 016865114672591341164780430740062095698813097584417659321598050816/1393\ 74341280671232999759010297844161860871496969373562963049485474223528276\ 610984064461508093880757021543538634046896781115514291717975*z - 17685951105934495101334685727679688500362705972794161671957590971299254\ 4928960696571368311776218030826201622816796879764966588533652253/278748\ 68256134246599951802059568832372174299393874712592609897094844705655322\ 196812892301618776151404308707726809379356223102858343595, y - 52327418222015905902570988536026443305400447176512318167424696241950974\ 64867103815670348284947982736369877307627698015104/13937434128067123299\ 97590102978441618608714969693735629630494854742235282766109840644615080\ 93880757021543538634046896781115514291717975*z^26 + 53322713168254937049374753704315866133748322761477018891704908145697222\ 36175698315958165401476066818724050651331474596288/13937434128067123299\ 97590102978441618608714969693735629630494854742235282766109840644615080\ 93880757021543538634046896781115514291717975*z^25 + 16200344963969739629738687977305337274986451247440630132666003014205689\ 06154724466997188357122175984508399767599792176664/13937434128067123299\ 97590102978441618608714969693735629630494854742235282766109840644615080\ 93880757021543538634046896781115514291717975*z^24 + 24267663491107314919267151607004192719477817031022986301215056432744760\ 4502999413958546800557457229730073541614908067430272/139374341280671232\ 99975901029784416186087149696937356296304948547422352827661098406446150\ 8093880757021543538634046896781115514291717975*z^23 - 22941289203469812166392436718685805905743949242949535266547366634353399\ ** WARNING: Output too long, hence truncated. '212.138' ************** MAGMA ***************** Host 212.138.47.18 (212.138.47.18) Time: Sun Dec 18 10:06:52 2005 Input: P:=PolynomialRing(RationalField(),3); I:=ideal

; Radical(I); Output: ** WARNING: Computation time exceeded 20 seconds, so computation was terminated after 20 seconds. ** Magma V2.11-10 Sun Dec 18 2005 10:06:32 on modular [Seed = 1084651922] ------------------------------------- Errors: /bin/sh: line 1: 26986 Alarm clock nice -n 19 /usr/local/bin/magma '212.138' ************** MAGMA ***************** Host 212.138.47.17 (212.138.47.17) Time: Sun Dec 18 10:05:35 2005 Input: P:=PolynomialRing(RationalField(),3); I:=ideal

; Radical(I); Output: ** WARNING: Computation time exceeded 20 seconds, so computation was terminated after 20 seconds. ** Magma V2.11-10 Sun Dec 18 2005 10:05:15 on modular [Seed = 1219391945] ------------------------------------- Errors: /bin/sh: line 1: 26978 Alarm clock nice -n 19 /usr/local/bin/magma '212.138' ************** MAGMA ***************** Host 212.138.47.24 (212.138.47.24) Time: Sun Dec 18 10:04:18 2005 Input: P:=PolynomialRing(RationalField(),3); I:=ideal

; Radical(I); Output: Magma V2.11-10 Sun Dec 18 2005 10:04:17 on modular [Seed = 1905739818] ------------------------------------- Ideal of Polynomial ring of rank 3 over Rational Field Lexicographical Order Variables: x, y, z Dimension 0, Radical Groebner basis: [ x - 607747819788532155708373696835866320181177034137286756794368/1534183140\ 25015422015563440134202740864381613046837395925*z^26 - 1644194756182732188695998750143313136448830566796183794089984/767091570\ 125077110077817200671013704321908065234186979625*z^25 + 1519370344618022681485678701207219405338713755668400813637632/697355972\ 84097919097983381879183064029264369566744270875*z^24 + 5657683761495134333516311523334543610877895279826294418178048/153418314\ 025015422015563440134202740864381613046837395925*z^23 - 54919850897379097829474394140593057612213076270900171882102784/76709157\ 0125077110077817200671013704321908065234186979625*z^22 - 23749346132955892738971647263493830752398879458430423918444544/15341831\ 4025015422015563440134202740864381613046837395925*z^21 - 20700681958857365834122812251195046038933464222145607741997056/76709157\ 0125077110077817200671013704321908065234186979625*z^20 + 329797041412983216677707986578040306567545389152753038938669056/7670915\ 70125077110077817200671013704321908065234186979625*z^19 + 5743093752282330081008122287919690804163733840955113232662528/153418314\ 025015422015563440134202740864381613046837395925*z^18 - 223440592050718037577946396909580575534889719118640118530048/1450078582\ 467064480298331192194732900419485945622281625*z^17 + 105410529530167997137721778655638855754188729832104137565487104/3068366\ 2805003084403112688026840548172876322609367479185*z^16 + 885277150983208290850600622134945917710089103724880750058029056/7670915\ 70125077110077817200671013704321908065234186979625*z^15 - 12228856033159216097966918255958697335709910281377823340045697024/76709\ 1570125077110077817200671013704321908065234186979625*z^14 - 2428919581438507999580094225861501397229411784479549789109817344/153418\ 314025015422015563440134202740864381613046837395925*z^13 + 19686636770492932197315778634758016279089740344967393996509011968/76709\ 1570125077110077817200671013704321908065234186979625*z^12 + 37439000525180257727455377920560461618208300249854315692199601152/76709\ 1570125077110077817200671013704321908065234186979625*z^11 - 242567485796071338802342670543220142016253736503063297377912832/1534183\ 14025015422015563440134202740864381613046837395925*z^10 - 1851283411800758364534308335130407783249267591238748229467858432/306836\ 62805003084403112688026840548172876322609367479185*z^9 - 32281291693399996932626040560041370875138715107061454029379634176/76709\ 1570125077110077817200671013704321908065234186979625*z^8 + 15187976308103132481649219442943042891239263040607663202857682752/76709\ 1570125077110077817200671013704321908065234186979625*z^7 + 32541311772827410438962148277497379673569947455678018687923473506/76709\ 1570125077110077817200671013704321908065234186979625*z^6 + 13913681728440239381658207089017885127837169844456124071384306152/76709\ 1570125077110077817200671013704321908065234186979625*z^5 - 5201437434931523590484162612412046914431897286598813426355149518/767091\ 570125077110077817200671013704321908065234186979625*z^4 - 8369631173418546797554673043809526207557975516145664666830070596/767091\ 570125077110077817200671013704321908065234186979625*z^3 - 367049861123048438708477131899166019080891143075089642030257163/6973559\ 7284097919097983381879183064029264369566744270875*z^2 - 952768348549232830699641549698525626444725223809958208866600116/7670915\ 70125077110077817200671013704321908065234186979625*z - 18715587866479416200274107374916451473634359600279292058148243/15341831\ 4025015422015563440134202740864381613046837395925, y + 8588783796645609270041309550413627734427158955259021244760064/767091570\ 125077110077817200671013704321908065234186979625*z^26 - 637476922202747849091768753421039549247886904737543430864896/7670915701\ 25077110077817200671013704321908065234186979625*z^25 - 2991662828937587719651268641902300229276057309871396005292802048/767091\ 570125077110077817200671013704321908065234186979625*z^24 - 3905324324791053203620496116512887656336880862025760883867648/697355972\ 84097919097983381879183064029264369566744270875*z^23 + 27504966074299938066699391828241118157638650281304603241981935616/76709\ 1570125077110077817200671013704321908065234186979625*z^22 + 16954503077996297980763933818251459139545426272372724218478723072/76709\ 1570125077110077817200671013704321908065234186979625*z^21 - 10148400141775193758023270288734527185830514208009811283475562496/69735\ 597284097919097983381879183064029264369566744270875*z^20 - 136748565042632957498398260907316114042218819459638218594974040064/7670\ 91570125077110077817200671013704321908065234186979625*z^19 + 218073368412696644786151217621427176878167387891939188253975248896/7670\ 91570125077110077817200671013704321908065234186979625*z^18 + 476420567037107074864416424418893515283464984044841541861899829248/7670\ 91570125077110077817200671013704321908065234186979625*z^17 - 87911783741774450433038258838115842490336235736309248584654553088/76709\ 1570125077110077817200671013704321908065234186979625*z^16 - 865393107630329396744064599190108622066096619174761478073516490752/7670\ 91570125077110077817200671013704321908065234186979625*z^15 - 491732366580711670721311407822861406520038434484615375661704249344/7670\ 91570125077110077817200671013704321908065234186979625*z^14 + 734818272058152712148400269178951592247749820810396870290521808896/7670\ 91570125077110077817200671013704321908065234186979625*z^13 + 1053964300460031535169533803184852097290543328887561757855461595136/767\ 091570125077110077817200671013704321908065234186979625*z^12 + 8634030541199246468005771144699878375129632040218392070043614208/153418\ 314025015422015563440134202740864381613046837395925*z^11 - 76063092256498361515666815473156851226821587577053074339657521152/69735\ 597284097919097983381879183064029264369566744270875*z^10 - 605969915846052348237289545733521947789758114815909673885915143168/7670\ 91570125077110077817200671013704321908065234186979625*z^9 + 103462238301384026891084346638351778208358212884489589520021951616/7670\ 91570125077110077817200671013704321908065234186979625*z^8 + 206006578234046558080680416999501373559639471746243171719113212/4047976\ 62335133039618900897451722271409977870835982575*z^7 + 213087953931560709344296477834302537326554381040028500222583661956/7670\ 91570125077110077817200671013704321908065234186979625*z^6 - 1551500011152425928327059620628005368730724758729742956668117102/153418\ 314025015422015563440134202740864381613046837395925*z^5 - 14331700909217848590119018556215107062277900112783826180653965517/15341\ 8314025015422015563440134202740864381613046837395925*z^4 - 43953099780249835711295595369687919026992869322936517188129514756/76709\ 1570125077110077817200671013704321908065234186979625*z^3 - 13801578868727311991345220593097358202836764355928486394382755206/76709\ 1570125077110077817200671013704321908065234186979625*z^2 - 2357219767421710505084804774312577449297980257986286253398662112/767091\ 570125077110077817200671013704321908065234186979625*z - 35062495528309440485031693067359094546187759778808876771992076/15341831\ 4025015422015563440134202740864381613046837395925, z^27 - 21/2*z^25 - 19/3*z^24 + 49*z^23 + 532/9*z^22 - 49939/432*z^21 - 6517/27*z^20 + 228487/2592*z^19 + 74759341/139968*z^18 + 1224167/5184*z^17 - 86467843/139968*z^16 - 3761045885/5038848*z^15 + 290329361/1679616*z^14 + 1419975504253/1632586752*z^13 + 13760646589/30233088*z^12 - 131206787713/362797056*z^11 - 4113563481113/7346640384*z^10 - 12868110131287/78364164096*z^9 + 110838315489477121/609359740010496*z^8 + 827650776663089/4231664861184*z^7 + 8940368413909147/152339935002624*z^\ 6 - 652513311785843/25389989167104*z^5 - 4957091804726761/152339935002624*z^4 - 511975131257783/33853318889472*z^3 - 151452934564537/38084983750656*z^2 - 358514838196123/609359740010496*z - 23491990397845/609359740010496 ] Total time: 0.210 seconds, Total memory usage: 3.43MB '65.191.' ************** MAGMA ***************** Host 65.191.73.113 (65.191.73.113) Time: Sun Dec 18 09:57:59 2005 Input: K := FiniteField(2); > C := LinearCode; S, f := StandardForm(C); D := Dual(S); (D meet S) eq S; M:=VerticalJoin(GeneratorMatrix(S), GeneratorMatrix(D)); M1:=EchelonForm(M); M2:=Submatrix(M1,22,22,14,35); C2:=LinearCode(M2); WeightDistribution(C2); L:=MinimumWords(C2); C3:=LinearCode(M); aut3 := AutomorphismGroup(C3); Order(aut3); Generators(aut3); aut2 := AutomorphismGroup(C2); Order(aut2); Generators(aut2); Output: Magma V2.11-10 Sun Dec 18 2005 09:57:58 on modular [Seed = 954074084] ------------------------------------- true [ <0, 1>, <8, 23>, <10, 175>, <12, 780>, <14, 2227>, <16, 3898>, <18, 4285>, <20, 3135>, <22, 1377>, <24, 410>, <26, 64>, <28, 9> ] 7 { (1, 5, 2, 6, 3, 7, 4)(8, 12, 9, 13, 10, 22, 11)(14, 23, 15, 24, 16, 25, 17)(18, 26, 19, 27, 20, 28, 21)(29, 33, 30, 34, 31, 35, 32)(36, 40, 37, 41, 38, 42, 39)(43, 47, 44, 48, 45, 49, 46)(50, 54, 51, 55, 52, 56, 53) } 1 {} Total time: 1.330 seconds, Total memory usage: 3.93MB '65.191.' ************** MAGMA ***************** Host 65.191.73.113 (65.191.73.113) Time: Sun Dec 18 09:56:58 2005 Input: K := FiniteField(2); > C := LinearCode; S, f := StandardForm(C); D := Dual(S); (D meet S) eq S; M:=VerticalJoin(GeneratorMatrix(S), GeneratorMatrix(D)); M1:=EchelonForm(M); M2:=Submatrix(M1,22,22,14,35); C2:=LinearCode(M2); WeightDistribution(C2); L:=MinimumWords(C2); C3:=LinearCode(M); aut3 := AutomorphismGroup(C3); Order(aut3); Generators(aut3); aut2 := AutomorphismGroup(C2); Order(aut2); Generators(aut3); Output: Magma V2.11-10 Sun Dec 18 2005 09:56:57 on modular [Seed = 666967220] ------------------------------------- true [ <0, 1>, <8, 23>, <10, 175>, <12, 780>, <14, 2227>, <16, 3898>, <18, 4285>, <20, 3135>, <22, 1377>, <24, 410>, <26, 64>, <28, 9> ] 7 { (1, 5, 2, 6, 3, 7, 4)(8, 12, 9, 13, 10, 22, 11)(14, 23, 15, 24, 16, 25, 17)(18, 26, 19, 27, 20, 28, 21)(29, 33, 30, 34, 31, 35, 32)(36, 40, 37, 41, 38, 42, 39)(43, 47, 44, 48, 45, 49, 46)(50, 54, 51, 55, 52, 56, 53) } 1 { (1, 5, 2, 6, 3, 7, 4)(8, 12, 9, 13, 10, 22, 11)(14, 23, 15, 24, 16, 25, 17)(18, 26, 19, 27, 20, 28, 21)(29, 33, 30, 34, 31, 35, 32)(36, 40, 37, 41, 38, 42, 39)(43, 47, 44, 48, 45, 49, 46)(50, 54, 51, 55, 52, 56, 53) } Total time: 1.320 seconds, Total memory usage: 3.93MB '65.191.' ************** MAGMA ***************** Host 65.191.73.113 (65.191.73.113) Time: Sun Dec 18 09:51:22 2005 Input: K := FiniteField(2); > C := LinearCode; S, f := StandardForm(C); D := Dual(S); (D meet S) eq S; M:=VerticalJoin(GeneratorMatrix(S), GeneratorMatrix(D)); M1:=EchelonForm(M); M2:=Submatrix(M1,22,22,14,35); C2:=LinearCode(M2); WeightDistribution(C2); L:=MinimumWords(C2); C3:=LinearCode(M); aut3 := AutomorphismGroup(C3); Order(aut3); Generators(aut3); Output: Magma V2.11-10 Sun Dec 18 2005 09:51:21 on modular [Seed = 801708908] ------------------------------------- true [ <0, 1>, <8, 23>, <10, 175>, <12, 780>, <14, 2227>, <16, 3898>, <18, 4285>, <20, 3135>, <22, 1377>, <24, 410>, <26, 64>, <28, 9> ] 7 { (1, 5, 2, 6, 3, 7, 4)(8, 12, 9, 13, 10, 22, 11)(14, 23, 15, 24, 16, 25, 17)(18, 26, 19, 27, 20, 28, 21)(29, 33, 30, 34, 31, 35, 32)(36, 40, 37, 41, 38, 42, 39)(43, 47, 44, 48, 45, 49, 46)(50, 54, 51, 55, 52, 56, 53) } Total time: 1.320 seconds, Total memory usage: 3.93MB '65.191.' ************** MAGMA ***************** Host 65.191.73.113 (65.191.73.113) Time: Sun Dec 18 09:50:08 2005 Input: K := FiniteField(2); > C := LinearCode; D := Dual(S); (D meet S) eq S; M:=VerticalJoin(GeneratorMatrix(S), GeneratorMatrix(D)); M1:=EchelonForm(M); M2:=Submatrix(M1,22,22,14,35); C2:=LinearCode(M2); WeightDistribution(C2); L:=MinimumWords(C2); C3:=LinearCode(M); aut3 := AutomorphismGroup(C3); Order(aut3); Generators(aut3); Output: Magma V2.11-10 Sun Dec 18 2005 09:50:08 on modular [Seed = 3560104159] ------------------------------------- >> D := Dual(S); ^ User error: Identifier 'S' has not been declared or assigned >> (D meet S) eq S; ^ User error: Identifier 'D' has not been declared or assigned >> M:=VerticalJoin(GeneratorMatrix(S), GeneratorMatrix(D)); ^ User error: Identifier 'S' has not been declared or assigned >> M1:=EchelonForm(M); ^ User error: Identifier 'M' has not been declared or assigned >> M2:=Submatrix(M1,22,22,14,35); ^ User error: Identifier 'M1' has not been declared or assigned >> C2:=LinearCode(M2); ^ User error: Identifier 'M2' has not been declared or assigned >> WeightDistribution(C2); ^ User error: Identifier 'C2' has not been declared or assigned >> L:=MinimumWords(C2); ^ User error: Identifier 'C2' has not been declared or assigned >> C3:=LinearCode(M); ^ User error: Identifier 'M' has not been declared or assigned >> aut3 := AutomorphismGroup(C3); ^ User error: Identifier 'C3' has not been declared or assigned >> Order(aut3); ^ User error: Identifier 'aut3' has not been declared or assigned >> Generators(aut3); ; ^ User error: Identifier 'aut3' has not been declared or assigned Total time: 0.190 seconds, Total memory usage: 3.24MB '65.191.' ************** MAGMA ***************** Host 65.191.73.113 (65.191.73.113) Time: Sun Dec 18 09:37:24 2005 Input: K := FiniteField(2); > C := LinearCode; Output: Magma V2.11-10 Sun Dec 18 2005 09:37:24 on modular [Seed = 3274044620] ------------------------------------- Total time: 0.190 seconds, Total memory usage: 3.24MB '212.138' ************** MAGMA ***************** Host 212.138.47.14 (212.138.47.14) Time: Sun Dec 18 09:16:17 2005 Input: P:=PolynomialRing(RationalField(),3); I:=ideal

; Radical(I); Output: Magma V2.11-10 Sun Dec 18 2005 09:16:17 on modular [Seed = 4061454056] ------------------------------------- Ideal of Polynomial ring of rank 3 over Rational Field Lexicographical Order Variables: x, y, z Dimension 0, Radical Groebner basis: [ x - 1296*z^9 + 4536*z^7 + 3240*z^6 - 5292*z^5 - 7560*z^4 - 600*z^3 + 4410*z^2 + 3101*z + 710, y + 6*z^3 - 7*z - 5, z^27 - 21/2*z^25 - 15/2*z^24 + 49*z^23 + 70*z^22 - 7811/72*z^21 - 1715/6*z^20 + 12985/432*z^19 + 133685/216*z^18 + 14063/32*z^17 - 274925/432*z^16 - 8654347/7776*z^15 - 62965/864*z^14 + 54259387/46656*z^13 + 88401125/93312*z^12 - 81855529/279936*z^11 - 260804915/279936*z^10 - 402754685/839808*z^9 + 224599585/1119744*z^8 + 1933044449/5038848*z^7 + 1197004745/6718464*z^6 - 823553731/40310784*z^5 - 8293758725/120932352*z^4 - 29056568867/725594112*z^3 - 3016860665/241864704*z^2 - 781606447/362797056*z - 715820345/4353564672 ] Total time: 0.190 seconds, Total memory usage: 3.34MB '212.138' ************** MAGMA ***************** Host 212.138.47.14 (212.138.47.14) Time: Sun Dec 18 09:13:54 2005 Input: P:=PolynomialRing(RationalField(),3); I:=ideal

; Radical(I); Output: Magma V2.11-10 Sun Dec 18 2005 09:13:52 on modular [Seed = 4279882356] ------------------------------------- Ideal of Polynomial ring of rank 3 over Rational Field Lexicographical Order Variables: x, y, z Dimension 0, Radical Groebner basis: [ x + 216*z^4 + 288*z^3 - 288*z^2 - 256*z + 165, y + 6*z^2 + 4*z - 5, z^8 + 8/3*z^7 - 8/9*z^6 - 160/27*z^5 + 23/162*z^4 + 1262/243*z^3 - 458/729*z^2 - 505855/279936*z + 162685/279936 ] Total time: 0.220 seconds, Total memory usage: 3.34MB '144.137' ************** MAGMA ***************** Host 144.137.185.243 (144.137.185.243) Time: Sun Dec 18 04:05:26 2005 Input: 2+1 divided by x = 4x Output: Magma V2.11-10 Sun Dec 18 2005 04:05:26 on modular [Seed = 3340870312] ------------------------------------- >> 2+1 divided by x = 4x; ^ User error: bad syntax Total time: 0.200 seconds, Total memory usage: 3.24MB '216.239' ************** MAGMA ***************** Host 216.239.167.217 (216.239.167.217) Time: Sat Dec 17 21:24:05 2005 Input: "Replace this by some code, then click [PARI] or [MAGMA]!" Output: Magma V2.11-10 Sat Dec 17 2005 21:24:05 on modular [Seed = 3910034073] ------------------------------------- Replace this by some code, then click [PARI] or [MAGMA]! Total time: 0.190 seconds, Total memory usage: 3.24MB '212.138' ************** MAGMA ***************** Host 212.138.47.15 (212.138.47.15) Time: Sat Dec 17 13:59:08 2005 Input: Table[Timing[FullSimplify[{i, Cos[Pi/i], Sin[Pi/i]}]], {i, 1, 22}] Output: Magma V2.11-10 Sat Dec 17 2005 13:59:07 on modular [Seed = 1686337705] ------------------------------------- >> Table[Timing[FullSimplify[{i, Cos[Pi/i], Sin[Pi/i]}]], {i, 1, 22}] ; ^ User error: Identifier 'Table' has not been declared or assigned Total time: 0.190 seconds, Total memory usage: 3.24MB '85.89.1' ************** MAGMA ***************** Host 85.89.168.182 (85.89.168.182) Time: Sat Dec 17 13:57:50 2005 Input: "Replace this by some code, then click [PARI] or [MAGMA]!" Output: Magma V2.11-10 Sat Dec 17 2005 13:57:50 on modular [Seed = 2005953512] ------------------------------------- Replace this by some code, then click [PARI] or [MAGMA]! Total time: 0.190 seconds, Total memory usage: 3.24MB '212.138' ************** MAGMA ***************** Host 212.138.47.15 (212.138.47.15) Time: Sat Dec 17 13:52:57 2005 Input: P:=PolynomialRing(RationalField(),4); I:=ideal

; Radical(I); Output: Magma V2.11-10 Sat Dec 17 2005 13:52:56 on modular [Seed = 250155959] ------------------------------------- >> P:=PolynomialRing(RationalField(),4); I:=ideal

: Rhs argument 4 is invalid for this constructor >> Radical(I);; ^ User error: Identifier 'I' has not been declared or assigned Total time: 0.200 seconds, Total memory usage: 3.34MB '212.138' ************** MAGMA ***************** Host 212.138.47.24 (212.138.47.24) Time: Sat Dec 17 13:50:38 2005 Input: P:=PolynomialRing(RationalField(),4); I:=ideal

; Radical(I); Output: Magma V2.11-10 Sat Dec 17 2005 13:50:38 on modular [Seed = 183309509] ------------------------------------- >> -1 + m1 + m2*m3/(1+m4^2)^(1/2) + (m3 + m3/(1+m4^2)^(1/2))^(1/2), ^ Runtime error in '^': Bad argument types Argument types given: RngMPolElt, RngMPolElt >> Radical(I);; ^ User error: Identifier 'I' has not been declared or assigned Total time: 0.190 seconds, Total memory usage: 3.24MB '212.138' ************** MAGMA ***************** Host 212.138.47.23 (212.138.47.23) Time: Sat Dec 17 13:50:07 2005 Input: P:=PolynomialRing(RationalField(),4); I:=ideal

; Radical(I); Output: Magma V2.11-10 Sat Dec 17 2005 13:50:06 on modular [Seed = 132127208] ------------------------------------- >> -t1 + m1 + m2*m3/(1+m4^2)^(1/2) + (m3 + m3/(1+m4^2)^(1/2))^(1/2), ^ User error: Identifier 't1' has not been declared or assigned >> Radical(I);; ^ User error: Identifier 'I' has not been declared or assigned Total time: 0.190 seconds, Total memory usage: 3.24MB '212.138' ************** MAGMA ***************** Host 212.138.47.15 (212.138.47.15) Time: Sat Dec 17 13:48:52 2005 Input: P:=PolynomialRing(RationalField(),4); I:=ideal

; Radical(I); Output: Magma V2.11-10 Sat Dec 17 2005 13:48:52 on modular [Seed = 15146661] ------------------------------------- >> 1 = m1 + m2*m3/(1+m4^2)^(1/2) + (m3 + m3/(1+m4^2)^(1/2))^(1/2), ^ Runtime error in '^': Bad argument types Argument types given: RngMPolElt, RngMPolElt >> Radical(I);; ^ User error: Identifier 'I' has not been declared or assigned Total time: 0.200 seconds, Total memory usage: 3.24MB '212.138' ************** MAGMA ***************** Host 212.138.47.17 (212.138.47.17) Time: Sat Dec 17 13:48:05 2005 Input: P:=PolynomialRing(RationalField(),4); I:=ideal

; Radical(I); Output: Magma V2.11-10 Sat Dec 17 2005 13:48:05 on modular [Seed = 486212168] ------------------------------------- >> t1 = m1 + m2*m3/(1+m4^2)^(1/2) + (m3 + m3/(1+m4^2)^(1/2))^(1/2), ^ User error: Identifier 't1' has not been declared or assigned >> Radical(I);; ^ User error: Identifier 'I' has not been declared or assigned Total time: 0.200 seconds, Total memory usage: 3.24MB '212.138' ************** MAGMA ***************** Host 212.138.47.24 (212.138.47.24) Time: Sat Dec 17 13:45:43 2005 Input: P:=PolynomialRing(RationalField(),1); I:=ideal

; Radical(I); Output: Magma V2.11-10 Sat Dec 17 2005 13:45:42 on modular [Seed = 351472627] ------------------------------------- Ideal of Polynomial ring of rank 1 over Rational Field Lexicographical Order Variables: y Dimension 0, Radical Groebner basis: [ y^21 - 12*y^20 + 55*y^19 - 107*y^18 + 17*y^17 + 245*y^16 - 175*y^15 - 535*y^14 + 715*y^13 + 844*y^12 - 2407*y^11 + 1075*y^10 + 1459*y^9 - 503*y^8 - 2999*y^7 + 4018*y^6 - 2002*y^5 + 37*y^4 + 489*y^3 - 311*y^2 + 86*y - 13 ] Total time: 0.200 seconds, Total memory usage: 3.24MB '212.138' ************** MAGMA ***************** Host 212.138.113.12 (212.138.113.12) Time: Sat Dec 17 13:44:41 2005 Input: P:=PolynomialRing(RationalField(),1); I:=ideal

; Radical(I); Output: Magma V2.11-10 Sat Dec 17 2005 13:44:41 on modular [Seed = 301338297] ------------------------------------- Ideal of Polynomial ring of rank 1 over Rational Field Lexicographical Order Variables: x Dimension 0, Radical Groebner basis: [ x^7 - 2*x^6 - x^5 + x^4 + x^3 + x^2 - x - 1 ] Total time: 0.200 seconds, Total memory usage: 3.24MB '83.26.1' ************** MAGMA ***************** Host 83.26.157.172 (83.26.157.172) Time: Sat Dec 17 10:35:39 2005 Input: pi(51) Output: Magma V2.11-10 Sat Dec 17 2005 10:35:39 on modular [Seed = 301359720] ------------------------------------- >> pi(51) ^ User error: Identifier 'pi' has not been declared or assigned Total time: 0.190 seconds, Total memory usage: 3.24MB '83.26.1' ************** MAGMA ***************** Host 83.26.157.172 (83.26.157.172) Time: Sat Dec 17 10:35:28 2005 Input: piprime(51) Output: Magma V2.11-10 Sat Dec 17 2005 10:35:28 on modular [Seed = 751525726] ------------------------------------- >> piprime(51) ^ User error: Identifier 'piprime' has not been declared or assigned Total time: 0.190 seconds, Total memory usage: 3.24MB '83.26.1' ************** MAGMA ***************** Host 83.26.157.172 (83.26.157.172) Time: Sat Dec 17 10:29:43 2005 Input: prime(51) Output: Magma V2.11-10 Sat Dec 17 2005 10:29:43 on modular [Seed = 3442122777] ------------------------------------- >> prime(51) ^ User error: Identifier 'prime' has not been declared or assigned Total time: 0.190 seconds, Total memory usage: 3.24MB '83.26.1' ************** MAGMA ***************** Host 83.26.157.172 (83.26.157.172) Time: Sat Dec 17 10:28:07 2005 Input: pi(100) Output: Magma V2.11-10 Sat Dec 17 2005 10:28:07 on modular [Seed = 3257249654] ------------------------------------- >> pi(100) ^ User error: Identifier 'pi' has not been declared or assigned Total time: 0.200 seconds, Total memory usage: 3.24MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Sat Dec 17 03:18:15 2005 Input: G:=DirichletGroup(432);G; X :=Elements(G);X; X; Y :=X[1]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 2, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; qEigenform(D[1],99); Output: Magma V2.11-10 Sat Dec 17 2005 03:18:13 on modular [Seed = 684237507] ------------------------------------- Group of Dirichlet characters of modulus 432 over Rational Field [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3 ] [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3 ] 1 1 [ Modular symbols space for Gamma_0(432) of weight 2 and dimension 1 over Rational Field, Modular symbols space for Gamma_0(432) of weight 2 and dimension 1 over Rational Field, Modular symbols space for Gamma_0(432) of weight 2 and dimension 1 over Rational Field, Modular symbols space for Gamma_0(432) of weight 2 and dimension 1 over Rational Field, Modular symbols space for Gamma_0(432) of weight 2 and dimension 1 over Rational Field, Modular symbols space for Gamma_0(432) of weight 2 and dimension 1 over Rational Field, Modular symbols space for Gamma_0(432) of weight 2 and dimension 1 over Rational Field, Modular symbols space for Gamma_0(432) of weight 2 and dimension 1 over Rational Field ] q - 5*q^7 - 7*q^13 + q^19 - 5*q^25 + 4*q^31 - q^37 - 8*q^43 + 18*q^49 - 13*q^61 - 11*q^67 + 17*q^73 + 13*q^79 + 35*q^91 + 5*q^97 + O(q^99) Total time: 1.399 seconds, Total memory usage: 6.08MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Sat Dec 17 03:15:53 2005 Input: G: =DirichletGroup(432);G; X :=Elements(G);X; X; Y :=X[1]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 2, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; qEigenform(D[1],99); Output: Magma V2.11-10 Sat Dec 17 2005 03:15:52 on modular [Seed = 3509672902] ------------------------------------- >> G: =DirichletGroup(432);G; ^ User error: bad syntax >> X :=Elements(G);X; ^ User error: Identifier 'G' has not been declared or assigned >> X :=Elements(G);X; ^ User error: Identifier 'X' has not been declared or assigned >> X; ^ User error: Identifier 'X' has not been declared or assigned >> Y :=X[1]; Conductor(Y); Order(Y); ^ User error: Identifier 'X' has not been declared or assigned >> Y :=X[1]; Conductor(Y); Order(Y); ^ User error: Identifier 'Y' has not been declared or assigned >> Y :=X[1]; Conductor(Y); Order(Y); ^ User error: Identifier 'Y' has not been declared or assigned >> M := ModularSymbols(Y, 2, 1); ^ User error: Identifier 'Y' has not been declared or assigned >> D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); ^ User error: Identifier 'M' has not been declared or assigned >> D; ^ User error: Identifier 'D' has not been declared or assigned >> qEigenform(D[1],99); ^ User error: Identifier 'D' has not been declared or assigned Total time: 0.180 seconds, Total memory usage: 3.24MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Sat Dec 17 03:15:32 2005 Input: G: = DirichletGroup(432);G; X :=Elements(G);X; X; Y :=X[1]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 2, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; qEigenform(D[1],99); Output: Magma V2.11-10 Sat Dec 17 2005 03:15:32 on modular [Seed = 3593886941] ------------------------------------- >> G: = DirichletGroup(432);G; ^ User error: bad syntax >> X :=Elements(G);X; ^ User error: Identifier 'G' has not been declared or assigned >> X :=Elements(G);X; ^ User error: Identifier 'X' has not been declared or assigned >> X; ^ User error: Identifier 'X' has not been declared or assigned >> Y :=X[1]; Conductor(Y); Order(Y); ^ User error: Identifier 'X' has not been declared or assigned >> Y :=X[1]; Conductor(Y); Order(Y); ^ User error: Identifier 'Y' has not been declared or assigned >> Y :=X[1]; Conductor(Y); Order(Y); ^ User error: Identifier 'Y' has not been declared or assigned >> M := ModularSymbols(Y, 2, 1); ^ User error: Identifier 'Y' has not been declared or assigned >> D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); ^ User error: Identifier 'M' has not been declared or assigned >> D; ^ User error: Identifier 'D' has not been declared or assigned >> qEigenform(D[1],99); ^ User error: Identifier 'D' has not been declared or assigned Total time: 0.190 seconds, Total memory usage: 3.24MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Sat Dec 17 03:15:27 2005 Input: G: =DirichletGroup(432);G; X :=Elements(G);X; X; Y :=X[1]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 2, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; qEigenform(D[1],99); Output: Magma V2.11-10 Sat Dec 17 2005 03:15:27 on modular [Seed = 3678097899] ------------------------------------- >> G: =DirichletGroup(432);G; ^ User error: bad syntax >> X :=Elements(G);X; ^ User error: Identifier 'G' has not been declared or assigned >> X :=Elements(G);X; ^ User error: Identifier 'X' has not been declared or assigned >> X; ^ User error: Identifier 'X' has not been declared or assigned >> Y :=X[1]; Conductor(Y); Order(Y); ^ User error: Identifier 'X' has not been declared or assigned >> Y :=X[1]; Conductor(Y); Order(Y); ^ User error: Identifier 'Y' has not been declared or assigned >> Y :=X[1]; Conductor(Y); Order(Y); ^ User error: Identifier 'Y' has not been declared or assigned >> M := ModularSymbols(Y, 2, 1); ^ User error: Identifier 'Y' has not been declared or assigned >> D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); ^ User error: Identifier 'M' has not been declared or assigned >> D; ^ User error: Identifier 'D' has not been declared or assigned >> qEigenform(D[1],99); ^ User error: Identifier 'D' has not been declared or assigned Total time: 0.180 seconds, Total memory usage: 3.24MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Sat Dec 17 03:15:16 2005 Input: G:=DirichletGroup(432);G; X :=Elements(G);X; X; Y :=X[1]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 2, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; qEigenform(D[1],99); Output: Magma V2.11-10 Sat Dec 17 2005 03:15:14 on modular [Seed = 3223341821] ------------------------------------- Group of Dirichlet characters of modulus 432 over Rational Field [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3 ] [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3 ] 1 1 [ Modular symbols space for Gamma_0(432) of weight 2 and dimension 1 over Rational Field, Modular symbols space for Gamma_0(432) of weight 2 and dimension 1 over Rational Field, Modular symbols space for Gamma_0(432) of weight 2 and dimension 1 over Rational Field, Modular symbols space for Gamma_0(432) of weight 2 and dimension 1 over Rational Field, Modular symbols space for Gamma_0(432) of weight 2 and dimension 1 over Rational Field, Modular symbols space for Gamma_0(432) of weight 2 and dimension 1 over Rational Field, Modular symbols space for Gamma_0(432) of weight 2 and dimension 1 over Rational Field, Modular symbols space for Gamma_0(432) of weight 2 and dimension 1 over Rational Field ] q + 4*q^5 + 3*q^7 - 4*q^11 + q^13 - 4*q^17 + q^19 - 4*q^23 + 11*q^25 + 4*q^31 + 12*q^35 - 9*q^37 + 8*q^43 + 12*q^47 + 2*q^49 - 8*q^53 - 16*q^55 - 4*q^59 - 5*q^61 + 4*q^65 - 11*q^67 - 8*q^71 + q^73 - 12*q^77 + 5*q^79 - 8*q^83 - 16*q^85 + 12*q^89 + 3*q^91 + 4*q^95 + 5*q^97 + O(q^99) Total time: 1.409 seconds, Total memory usage: 6.10MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Sat Dec 17 03:15:02 2005 Input: G: =DirichletGroup(432);G; X :=Elements(G);X; X; Y :=X[1]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 2, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; qEigenform(D[1],99); Output: Magma V2.11-10 Sat Dec 17 2005 03:15:02 on modular [Seed = 3307556844] ------------------------------------- >> G: =DirichletGroup(432);G; ^ User error: bad syntax >> X :=Elements(G);X; ^ User error: Identifier 'G' has not been declared or assigned >> X :=Elements(G);X; ^ User error: Identifier 'X' has not been declared or assigned >> X; ^ User error: Identifier 'X' has not been declared or assigned >> Y :=X[1]; Conductor(Y); Order(Y); ^ User error: Identifier 'X' has not been declared or assigned >> Y :=X[1]; Conductor(Y); Order(Y); ^ User error: Identifier 'Y' has not been declared or assigned >> Y :=X[1]; Conductor(Y); Order(Y); ^ User error: Identifier 'Y' has not been declared or assigned >> M := ModularSymbols(Y, 2, 1); ^ User error: Identifier 'Y' has not been declared or assigned >> D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); ^ User error: Identifier 'M' has not been declared or assigned >> D; ^ User error: Identifier 'D' has not been declared or assigned >> qEigenform(D[1],99); ^ User error: Identifier 'D' has not been declared or assigned Total time: 0.180 seconds, Total memory usage: 3.24MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Sat Dec 17 03:14:49 2005 Input: G:=DirichletGroup(432);G; X :=Elements(G);X; X; Y :=X[1]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 2, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; qEigenform(D[1],99); Output: Magma V2.11-10 Sat Dec 17 2005 03:14:48 on modular [Seed = 3391766813] ------------------------------------- Group of Dirichlet characters of modulus 432 over Rational Field [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3 ] [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3 ] 1 1 [ Modular symbols space for Gamma_0(432) of weight 2 and dimension 1 over Rational Field, Modular symbols space for Gamma_0(432) of weight 2 and dimension 1 over Rational Field, Modular symbols space for Gamma_0(432) of weight 2 and dimension 1 over Rational Field, Modular symbols space for Gamma_0(432) of weight 2 and dimension 1 over Rational Field, Modular symbols space for Gamma_0(432) of weight 2 and dimension 1 over Rational Field, Modular symbols space for Gamma_0(432) of weight 2 and dimension 1 over Rational Field, Modular symbols space for Gamma_0(432) of weight 2 and dimension 1 over Rational Field, Modular symbols space for Gamma_0(432) of weight 2 and dimension 1 over Rational Field ] q + 3*q^5 + q^7 + 3*q^11 - 4*q^13 - 2*q^19 + 6*q^23 + 4*q^25 + 6*q^29 - 5*q^31 + 3*q^35 + 2*q^37 - 6*q^41 + 10*q^43 - 6*q^47 - 6*q^49 + 9*q^53 + 9*q^55 - 12*q^59 + 8*q^61 - 12*q^65 - 14*q^67 - 7*q^73 + 3*q^77 - 8*q^79 + 3*q^83 - 18*q^89 - 4*q^91 - 6*q^95 - q^97 + O(q^99) Total time: 1.429 seconds, Total memory usage: 6.00MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Sat Dec 17 03:12:34 2005 Input: G :=DirichletGroup(432); G; X :=Elements(G); X; Y :=X[1]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 2, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; qEigenform(D[1],99); Output: Magma V2.11-10 Sat Dec 17 2005 03:12:32 on modular [Seed = 4027612079] ------------------------------------- Group of Dirichlet characters of modulus 432 over Rational Field [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3 ] 1 1 [ Modular symbols space for Gamma_0(432) of weight 2 and dimension 1 over Rational Field, Modular symbols space for Gamma_0(432) of weight 2 and dimension 1 over Rational Field, Modular symbols space for Gamma_0(432) of weight 2 and dimension 1 over Rational Field, Modular symbols space for Gamma_0(432) of weight 2 and dimension 1 over Rational Field, Modular symbols space for Gamma_0(432) of weight 2 and dimension 1 over Rational Field, Modular symbols space for Gamma_0(432) of weight 2 and dimension 1 over Rational Field, Modular symbols space for Gamma_0(432) of weight 2 and dimension 1 over Rational Field, Modular symbols space for Gamma_0(432) of weight 2 and dimension 1 over Rational Field ] q - 5*q^7 - 7*q^13 + q^19 - 5*q^25 + 4*q^31 - q^37 - 8*q^43 + 18*q^49 - 13*q^61 - 11*q^67 + 17*q^73 + 13*q^79 + 35*q^91 + 5*q^97 + O(q^99) Total time: 1.399 seconds, Total memory usage: 6.00MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Sat Dec 17 03:11:15 2005 Input: G :=DirichletGroup(432); G; X :=Elements(G); X; Y :=X[1]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 2, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; Output: Magma V2.11-10 Sat Dec 17 2005 03:11:14 on modular [Seed = 4111826680] ------------------------------------- Group of Dirichlet characters of modulus 432 over Rational Field [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3 ] 1 1 [ Modular symbols space for Gamma_0(432) of weight 2 and dimension 1 over Rational Field, Modular symbols space for Gamma_0(432) of weight 2 and dimension 1 over Rational Field, Modular symbols space for Gamma_0(432) of weight 2 and dimension 1 over Rational Field, Modular symbols space for Gamma_0(432) of weight 2 and dimension 1 over Rational Field, Modular symbols space for Gamma_0(432) of weight 2 and dimension 1 over Rational Field, Modular symbols space for Gamma_0(432) of weight 2 and dimension 1 over Rational Field, Modular symbols space for Gamma_0(432) of weight 2 and dimension 1 over Rational Field, Modular symbols space for Gamma_0(432) of weight 2 and dimension 1 over Rational Field ] Total time: 1.379 seconds, Total memory usage: 6.00MB '200.177' ************** MAGMA ***************** Host 200.177.28.205 (200.177.28.205) Time: Fri Dec 16 21:51:05 2005 Input: 0^0; Output: Magma V2.11-10 Fri Dec 16 2005 21:51:04 on modular [Seed = 2153823931] ------------------------------------- 1 Total time: 0.190 seconds, Total memory usage: 3.24MB '212.138' ************** MAGMA ***************** Host 212.138.47.14 (212.138.47.14) Time: Fri Dec 16 21:08:24 2005 Input: P:=PolynomialRing(RationalField(),3); I:=ideal

; Radical(I); Output: Magma V2.11-10 Fri Dec 16 2005 21:08:23 on modular [Seed = 1437951329] ------------------------------------- Ideal of Polynomial ring of rank 3 over Rational Field Lexicographical Order Variables: x, y, z Dimension 0, Radical Groebner basis: [ x + 24021947384028469294199051557281326175896046567/88390501855296726402302\ 4091659442585600000000*z^7 - 499887372604618086734751651977842907689699\ 89/6313607275378337600164457797567447040000000*z^5 - 3578323456874350741013274769517430940050603/450971948241309828583175556\ 96910336000000*z^3 - 7781918205774041731639575981199074243/257698256137\ 8913306189574611252019200*z, y - 463586742170926250485324993944801190173/1287206326036293892078321948672\ 0000000*z^7 + 1085903501599971128651516196294585591/9194330900259242086\ 2737282048000000*z^5 + 9873044499317982446687031775259151/9381970306386\ 9817206874777600000*z^3 + 128601565742559180668707869379/11258364367664\ 3780648249733120*z, z^8 - 36195915798000255920/96199778220194782809*z^6 - 22658347845807961565600/7792182035835777407529*z^4 + 432194167813940000000/4125272842501293921633*z^2 + 16295122460522500000000/631166744902697970009849 ] Total time: 0.200 seconds, Total memory usage: 3.34MB '155.207' ************** MAGMA ***************** Host 155.207.209.210 (155.207.209.210) Time: Fri Dec 16 19:36:19 2005 Input: R := PolynomialRing(Integers()); > > f :=x^4-4*x^2+2; > > K := NumberField(f); > > O := MaximalOrder(K); > > I := IntegralBasis(K); > > print I; > >NormEquation(O,5^4); Output: Magma V2.11-10 Fri Dec 16 2005 19:36:18 on modular [Seed = 3476017489] ------------------------------------- [ 1, y, y^2, y^3 ] true [ [5, 0, 0, 0] ] Total time: 0.320 seconds, Total memory usage: 3.63MB '155.207' ************** MAGMA ***************** Host 155.207.209.210 (155.207.209.210) Time: Fri Dec 16 19:35:13 2005 Input: R := PolynomialRing(Integers()); > > f :=x^4-4*x^2+2; > > K := NumberField(f); > > O := MaximalOrder(K); > > I := IntegralBasis(K); > > print I; > >NormEquation(O,2*5^4); Output: Magma V2.11-10 Fri Dec 16 2005 19:35:13 on modular [Seed = 3526544527] ------------------------------------- [ 1, y, y^2, y^3 ] true [ [0, -5, 0, 0] ] Total time: 0.330 seconds, Total memory usage: 3.63MB '155.207' ************** MAGMA ***************** Host 155.207.209.210 (155.207.209.210) Time: Fri Dec 16 19:35:00 2005 Input: R := PolynomialRing(Integers()); > > f :=x^4-4*x^2+2; > > K := NumberField(f); > > O := MaximalOrder(K); > > I := IntegralBasis(K); > > print I; > >NormEquation(O,5); Output: Magma V2.11-10 Fri Dec 16 2005 19:35:00 on modular [Seed = 3577072637] ------------------------------------- [ 1, y, y^2, y^3 ] false Total time: 0.310 seconds, Total memory usage: 3.63MB '155.207' ************** MAGMA ***************** Host 155.207.209.210 (155.207.209.210) Time: Fri Dec 16 19:32:52 2005 Input: R := PolynomialRing(Integers()); > > f :=x^4-4*x^2+2; > > K := NumberField(f); > > O := MaximalOrder(K); > > I := IntegralBasis(K); > > print I; > >NormEquation(O, 5^4); Output: Magma V2.11-10 Fri Dec 16 2005 19:32:51 on modular [Seed = 3745497200] ------------------------------------- [ 1, y, y^2, y^3 ] true [ [5, 0, 0, 0] ] Total time: 0.320 seconds, Total memory usage: 3.63MB '66.14.9' ************** MAGMA ***************** Host 66.14.95.197 (66.14.95.197) Time: Fri Dec 16 18:40:23 2005 Input: sin(3.141549); Output: Magma V2.11-10 Fri Dec 16 2005 18:40:23 on modular [Seed = 2659093989] ------------------------------------- >> sin(3.141549);; ^ User error: Identifier 'sin' has not been declared or assigned Total time: 0.190 seconds, Total memory usage: 3.24MB '66.14.9' ************** MAGMA ***************** Host 66.14.95.197 (66.14.95.197) Time: Fri Dec 16 18:39:33 2005 Input: sin(pi); Output: Magma V2.11-10 Fri Dec 16 2005 18:39:33 on modular [Seed = 2040097360] ------------------------------------- >> sin(pi);; ^ User error: Identifier 'pi' has not been declared or assigned Total time: 0.190 seconds, Total memory usage: 3.24MB '66.14.9' ************** MAGMA ***************** Host 66.14.95.197 (66.14.95.197) Time: Fri Dec 16 18:38:29 2005 Input: 5! Output: Magma V2.11-10 Fri Dec 16 2005 18:38:29 on modular [Seed = 1071592406] ------------------------------------- >> 5!; ^ User error: bad syntax Total time: 0.190 seconds, Total memory usage: 3.24MB '155.207' ************** MAGMA ***************** Host 155.207.209.110 (155.207.209.110) Time: Fri Dec 16 18:36:14 2005 Input: R := PolynomialRing(Integers()); > > f :=x^4-4*x^2+2; > > K := NumberField(f); > > O := MaximalOrder(K); > > I := IntegralBasis(K); > > print I; > >NormEquation(O, 4*5^4); Output: Magma V2.11-10 Fri Dec 16 2005 18:36:12 on modular [Seed = 14683248] ------------------------------------- [ 1, y, y^2, y^3 ] true [ [0, 0, -5, 0] ] Total time: 0.330 seconds, Total memory usage: 3.63MB '128.138' ************** MAGMA ***************** Host 128.138.240.175 (128.138.240.175) Time: Fri Dec 16 16:41:06 2005 Input: 1234123412341234341111/2714351253451253424 Output: Magma V2.11-10 Fri Dec 16 2005 16:41:06 on modular [Seed = 3459168674] ------------------------------------- 411374470780411447037/904783751150417808 Total time: 0.180 seconds, Total memory usage: 3.24MB '131.156' ************** MAGMA ***************** Host 131.156.3.93 (131.156.3.93) Time: Fri Dec 16 16:09:01 2005 Input: P:=PolynomialRing(RationalField(),3); I:=ideal

; Radical(I); Output: Magma V2.11-10 Fri Dec 16 2005 16:09:01 on modular [Seed = 2040075914] ------------------------------------- Ideal of Polynomial ring of rank 3 over Rational Field Lexicographical Order Variables: x, y, z Dimension 0, Radical Groebner basis: [ x + 24021947384028469294199051557281326175896046567/88390501855296726402302\ 4091659442585600000000*z^7 - 499887372604618086734751651977842907689699\ 89/6313607275378337600164457797567447040000000*z^5 - 3578323456874350741013274769517430940050603/450971948241309828583175556\ 96910336000000*z^3 - 7781918205774041731639575981199074243/257698256137\ 8913306189574611252019200*z, y - 463586742170926250485324993944801190173/1287206326036293892078321948672\ 0000000*z^7 + 1085903501599971128651516196294585591/9194330900259242086\ 2737282048000000*z^5 + 9873044499317982446687031775259151/9381970306386\ 9817206874777600000*z^3 + 128601565742559180668707869379/11258364367664\ 3780648249733120*z, z^8 - 36195915798000255920/96199778220194782809*z^6 - 22658347845807961565600/7792182035835777407529*z^4 + 432194167813940000000/4125272842501293921633*z^2 + 16295122460522500000000/631166744902697970009849 ] Total time: 0.190 seconds, Total memory usage: 3.34MB '131.156' ************** MAGMA ***************** Host 131.156.3.93 (131.156.3.93) Time: Fri Dec 16 16:06:07 2005 Input: P:=PolynomialRing(RationalField(),3); I:=ideal

; GroebnerBasis(I); Output: Magma V2.11-10 Fri Dec 16 2005 16:06:07 on modular [Seed = 2124288351] ------------------------------------- [ x + 24021947384028469294199051557281326175896046567/88390501855296726402302\ 4091659442585600000000*z^7 - 499887372604618086734751651977842907689699\ 89/6313607275378337600164457797567447040000000*z^5 - 3578323456874350741013274769517430940050603/450971948241309828583175556\ 96910336000000*z^3 - 7781918205774041731639575981199074243/257698256137\ 8913306189574611252019200*z, y - 463586742170926250485324993944801190173/1287206326036293892078321948672\ 0000000*z^7 + 1085903501599971128651516196294585591/9194330900259242086\ 2737282048000000*z^5 + 9873044499317982446687031775259151/9381970306386\ 9817206874777600000*z^3 + 128601565742559180668707869379/11258364367664\ 3780648249733120*z, z^8 - 36195915798000255920/96199778220194782809*z^6 - 22658347845807961565600/7792182035835777407529*z^4 + 432194167813940000000/4125272842501293921633*z^2 + 16295122460522500000000/631166744902697970009849 ] Total time: 0.190 seconds, Total memory usage: 3.34MB '131.156' ************** MAGMA ***************** Host 131.156.3.93 (131.156.3.93) Time: Fri Dec 16 16:02:52 2005 Input: P:=PolynomialRing(RationalField(),3); I:=ideal

; GroebnerBasis(I); Output: Magma V2.11-10 Fri Dec 16 2005 16:02:48 on modular [Seed = 1168452898] ------------------------------------- [ x + 24021947384028469294199051557281326175896046567/88390501855296726402302\ 4091659442585600000000*z^7 - 499887372604618086734751651977842907689699\ 89/6313607275378337600164457797567447040000000*z^5 - 3578323456874350741013274769517430940050603/450971948241309828583175556\ 96910336000000*z^3 - 7781918205774041731639575981199074243/257698256137\ 8913306189574611252019200*z, y - 463586742170926250485324993944801190173/1287206326036293892078321948672\ 0000000*z^7 + 1085903501599971128651516196294585591/9194330900259242086\ 2737282048000000*z^5 + 9873044499317982446687031775259151/9381970306386\ 9817206874777600000*z^3 + 128601565742559180668707869379/11258364367664\ 3780648249733120*z, z^8 - 36195915798000255920/96199778220194782809*z^6 - 22658347845807961565600/7792182035835777407529*z^4 + 432194167813940000000/4125272842501293921633*z^2 + 16295122460522500000000/631166744902697970009849 ] Total time: 0.230 seconds, Total memory usage: 3.34MB '222.124' ************** MAGMA ***************** Host 222.124.19.133 (222.124.19.133) Time: Fri Dec 16 14:01:39 2005 Input: if ver eq 1 then M := Matrix(8,8,[ [2, 0, 0, 0, 0, 0, 0, 0], [0, 396,-214,-386, 36, 25,-144, 426], [1,-205, -34,-196, 230, 83,-662, 19], [1,-305, 528,-358,-250, 73, 38, 277], [1, 38, -45,-282, 584, 122, -24,-476], [0, 127, 131, 119, 369,-633, 152,-275], [0, 436, -54,-138,-442, 330,-312,-350], [1, 82, 757, 102, 372, 111,-248, 258]]); else M := Matrix(8,8,[ [ 2, 0, 0, 0, 0, 0, 0, 0], [ 0, 36, 221, 5, -64, 23, -32, 352], [ 0,129,-108,-193,-285, 97, 146,-178], [ 0, 46, -89,-166, 66, -46,-374,-241], [ 0,274, -85, 254, 212, 175, 166, 36], [ 1, 99,-185, 145, 145,-400, -19, 98], [ 1, 82,-367, 197, -42, 197,-191, 80], [ 0, 23, 40, 218,-182,-214,-224,-330]]); end if; Output: Magma V2.11-10 Fri Dec 16 2005 14:01:35 on modular [Seed = 2170670481] ------------------------------------- >> if ver eq 1 then ^ User error: Identifier 'ver' has not been declared or assigned Total time: 0.230 seconds, Total memory usage: 3.24MB '144.122' ************** MAGMA ***************** Host 144.122.137.67 (144.122.137.67) Time: Fri Dec 16 07:55:33 2005 Input: F:=GF(2^4); DefiningPolynomial(F); P := PolynomialRing(F); f := x^3+3*x+1; Order($.1); Output: Magma V2.11-10 Fri Dec 16 2005 07:55:32 on modular [Seed = 2124383264] ------------------------------------- $.1^4 + $.1 + 1 >> Order($.1);; ^ Runtime error: Bad dollar structure Total time: 0.200 seconds, Total memory usage: 3.34MB '144.122' ************** MAGMA ***************** Host 144.122.137.67 (144.122.137.67) Time: Fri Dec 16 07:55:15 2005 Input: F:=GF(2^4); DefiningPolynomial(F); P := PolynomialRing(F); f := x^3+3*x+1; Order(f); Output: Magma V2.11-10 Fri Dec 16 2005 07:55:15 on modular [Seed = 2073723219] ------------------------------------- $.1^4 + $.1 + 1 >> Order(f);; ^ Runtime error in 'Order': Bad argument types Argument types given: RngUPolElt[FldFin] Total time: 0.200 seconds, Total memory usage: 3.34MB '144.122' ************** MAGMA ***************** Host 144.122.137.67 (144.122.137.67) Time: Fri Dec 16 07:52:31 2005 Input: F:=GF(2^4); P := PolynomialRing(F); f := x^3+3*x+1; Order(f); Output: Magma V2.11-10 Fri Dec 16 2005 07:52:31 on modular [Seed = 1252766174] ------------------------------------- >> Order(f);; ^ Runtime error in 'Order': Bad argument types Argument types given: RngUPolElt[FldFin] Total time: 0.200 seconds, Total memory usage: 3.34MB '144.122' ************** MAGMA ***************** Host 144.122.137.67 (144.122.137.67) Time: Fri Dec 16 07:52:16 2005 Input: F:=GF(2^4); P := PolynomialRing(F); f := x^3+3*x+1; Exponent(f); Output: Magma V2.11-10 Fri Dec 16 2005 07:52:15 on modular [Seed = 1471589427] ------------------------------------- >> Exponent(f);; ^ Runtime error in 'Exponent': Bad argument types Argument types given: RngUPolElt[FldFin] Total time: 0.200 seconds, Total memory usage: 3.34MB '144.122' ************** MAGMA ***************** Host 144.122.137.67 (144.122.137.67) Time: Fri Dec 16 07:52:08 2005 Input: F:=GF(2^4); P := PolynomialRing(F); f := x^3+3*x+1; Exponenet(f); Output: Magma V2.11-10 Fri Dec 16 2005 07:52:08 on modular [Seed = 1555932991] ------------------------------------- >> Exponenet(f);; ^ User error: Identifier 'Exponenet' has not been declared or assigned Total time: 0.190 seconds, Total memory usage: 3.34MB '144.122' ************** MAGMA ***************** Host 144.122.137.67 (144.122.137.67) Time: Fri Dec 16 07:51:51 2005 Input: F:=GF(2^4); P := PolynomialRing(F); f := x^3+3*x+1; 2*f; Output: Magma V2.11-10 Fri Dec 16 2005 07:51:50 on modular [Seed = 1505272356] ------------------------------------- 0 Total time: 0.190 seconds, Total memory usage: 3.34MB '144.122' ************** MAGMA ***************** Host 144.122.137.67 (144.122.137.67) Time: Fri Dec 16 07:51:29 2005 Input: F:=GF(2^4); P := PolynomialRing(F); f := x^3+3*x+1; 3*f; Output: Magma V2.11-10 Fri Dec 16 2005 07:51:29 on modular [Seed = 650632556] ------------------------------------- x^3 + x + 1 Total time: 0.190 seconds, Total memory usage: 3.34MB '144.122' ************** MAGMA ***************** Host 144.122.137.67 (144.122.137.67) Time: Fri Dec 16 07:51:18 2005 Input: F:=GF(2^4); P := PolynomialRing(F); f := x^3+3*x+1; 3*f; Output: Magma V2.11-10 Fri Dec 16 2005 07:51:17 on modular [Seed = 599971965] ------------------------------------- x^3 + x + 1 Total time: 0.190 seconds, Total memory usage: 3.34MB '144.122' ************** MAGMA ***************** Host 144.122.137.67 (144.122.137.67) Time: Fri Dec 16 07:51:09 2005 Input: F:=GF(2^4); P := PolynomialRing(F); f := x^3+3*x+1; f; Output: Magma V2.11-10 Fri Dec 16 2005 07:51:04 on modular [Seed = 684315403] ------------------------------------- x^3 + x + 1 Total time: 0.260 seconds, Total memory usage: 3.34MB '213.84.' ************** MAGMA ***************** Host 213.84.213.14 (213.84.213.14) Time: Fri Dec 16 02:41:40 2005 Input: FindGroupOrder2 := function (p, s) K := GF(p); v := K ! (4*s); u := K ! (s^2-5); x := u^3; b := 4*x*v; a := (v-u)^3*(3*u+v); A := a/b-2; x := x/v^3; b := x^3 + A*x^2 + x; E := EllipticCurve([0,b*A,0,b^2,0]); return FactoredOrder(E); end function; p := 140853945410621700611366248656986006762214430713643; s := 941728572; FindGroupOrder2(p, s); Output: Magma V2.11-10 Fri Dec 16 2005 02:41:38 on modular [Seed = 920066706] ------------------------------------- [ <2, 2>, <3, 1>, <5, 1>, <137, 1>, <251, 1>, <317, 1>, <331, 1>, <7307, 1>, <13339, 1>, <15667, 1>, <29209, 1>, <137209, 1>, <332933, 1>, <393571, 1>, <811351, 1> ] Total time: 2.390 seconds, Total memory usage: 5.31MB '60.225.' ************** MAGMA ***************** Host 60.225.131.213 (60.225.131.213) Time: Fri Dec 16 02:29:36 2005 Input: Factorization(00000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001); Output: Magma V2.11-10 Fri Dec 16 2005 02:29:36 on modular [Seed = 3994009881] ------------------------------------- [] Total time: 0.180 seconds, Total memory usage: 3.24MB '60.225.' ************** MAGMA ***************** Host 60.225.131.213 (60.225.131.213) Time: Fri Dec 16 02:29:14 2005 Input: Factorization(0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000); Output: Magma V2.11-10 Fri Dec 16 2005 02:29:14 on modular [Seed = 3808611390] ------------------------------------- >> Factorization(0000000000000000000000000000000000000000000000000000000000000 ^ Runtime error in 'Factorization': Argument 1 is not non-zero Total time: 0.190 seconds, Total memory usage: 3.24MB '60.225.' ************** MAGMA ***************** Host 60.225.131.213 (60.225.131.213) Time: Fri Dec 16 00:56:30 2005 Input: Factorization(50000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000); Output: Magma V2.11-10 Fri Dec 16 2005 00:56:30 on modular [Seed = 616902717] ------------------------------------- [ <2, 832>, <5, 833> ] Total time: 0.190 seconds, Total memory usage: 3.24MB '60.225.' ************** MAGMA ***************** Host 60.225.131.213 (60.225.131.213) Time: Fri Dec 16 00:55:25 2005 Input: Factorization(500000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000012355); Output: ** WARNING: Computation time exceeded 20 seconds, so computation was terminated after 20 seconds. ** Magma V2.11-10 Fri Dec 16 2005 00:55:05 on modular [Seed = 970468503] ------------------------------------- Errors: /bin/sh: line 1: 21550 Alarm clock nice -n 19 /usr/local/bin/magma '60.225.' ************** MAGMA ***************** Host 60.225.131.213 (60.225.131.213) Time: Fri Dec 16 00:53:55 2005 Input: Factorization(50000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000005678901234567890); Output: ** WARNING: Computation time exceeded 20 seconds, so computation was terminated after 20 seconds. ** Magma V2.11-10 Fri Dec 16 2005 00:53:34 on modular [Seed = 852699003] ------------------------------------- Errors: /bin/sh: line 1: 21541 Alarm clock nice -n 19 /usr/local/bin/magma '60.225.' ************** MAGMA ***************** Host 60.225.131.213 (60.225.131.213) Time: Fri Dec 16 00:06:55 2005 Input: Factorization(4000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000); Output: Magma V2.11-10 Fri Dec 16 2005 00:06:55 on modular [Seed = 2090693635] ------------------------------------- [ <2, 2351>, <5, 2349> ] Total time: 0.190 seconds, Total memory usage: 3.24MB '60.225.' ************** MAGMA ***************** Host 60.225.131.213 (60.225.131.213) Time: Fri Dec 16 00:06:08 2005 Input: Factorization(0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000004); Output: Magma V2.11-10 Fri Dec 16 2005 00:06:07 on modular [Seed = 1437883770] ------------------------------------- [ <2, 2> ] Total time: 0.190 seconds, Total memory usage: 3.24MB '60.225.' ************** MAGMA ***************** Host 60.225.131.213 (60.225.131.213) Time: Fri Dec 16 00:05:06 2005 Input: Factorization(0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000); Output: Magma V2.11-10 Fri Dec 16 2005 00:05:05 on modular [Seed = 48433683] ------------------------------------- >> Factorization(0000000000000000000000000000000000000000000000000000000000000 ^ Runtime error in 'Factorization': Argument 1 is not non-zero Total time: 0.190 seconds, Total memory usage: 3.24MB '65.191.' ************** MAGMA ***************** Host 65.191.73.113 (65.191.73.113) Time: Thu Dec 15 19:21:11 2005 Input: K := FiniteField(2); > C := LinearCode; S, f := StandardForm(C); D := Dual(S); (D meet S) eq S; M:=VerticalJoin(GeneratorMatrix(S), GeneratorMatrix(D)); M1:=EchelonForm(M); M2:=Submatrix(M1,22,22,14,35); M2; C2:=LinearCode(M2); WeightDistribution(C2); L:=MinimumWords(C2); C3:=LinearCode; C3; Output: Magma V2.11-10 Thu Dec 15 2005 19:21:11 on modular [Seed = 3576995498] ------------------------------------- true [1 0 0 0 0 0 0 0 0 0 1 1 0 1 0 0 0 1 0 1 0 0 0 0 0 0 0 1 1 0 1 1 1 1 0] [0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 1 1 0 0 0 0 0 1 1 0 0 1 0 0 1 0] [0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 1 1 1 0 0 0 0 0 0 1 1 1 1 0 1 0 0 1 0 1] [0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 1 1 1 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 1] [0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 1 1 1 1 0 0 0 0 0 0 1 0 0 1 0 1 1 1 0 0] [0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 1 1 1 1 0 0 0 0 0 0 1 0 0 1 0 1 1 1 0] [0 0 0 0 0 0 1 0 0 0 1 1 1 1 0 1 1 1 1 1 0 0 0 0 0 0 0 1 0 1 1 1 0 1 1] [0 0 0 0 0 0 0 1 0 0 1 1 1 0 0 1 1 0 0 0 1 0 0 0 0 0 0 0 0 1 0 1 1 0 0] [0 0 0 0 0 0 0 0 1 0 0 1 1 1 0 1 0 0 1 1 1 0 0 0 0 0 0 0 1 1 0 1 0 0 1] [0 0 0 0 0 0 0 0 0 1 1 1 0 1 0 0 1 1 1 0 1 0 0 0 0 0 0 0 0 1 0 0 1 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 0 0 0 0 0 0 0 1 1 1 1 1 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 1 0 1 0 1 0 0 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 1 1 1 0 1 0 0 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 1 0 1 0 0 1 1 1] [ <0, 1>, <8, 23>, <10, 175>, <12, 780>, <14, 2227>, <16, 3898>, <18, 4285>, <20, 3135>, <22, 1377>, <24, 410>, <26, 64>, <28, 9> ] [35, 14, 8] Linear Code over GF(2) Generator matrix: [1 0 0 0 0 0 0 0 0 0 1 1 0 1 0 0 0 1 0 1 0 0 0 0 0 0 0 1 1 0 1 1 1 1 0] [0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 1 1 0 0 0 0 0 1 1 0 0 1 0 0 1 0] [0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 1 1 1 0 0 0 0 0 0 1 1 1 1 0 1 0 0 1 0 1] [0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 1 1 1 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 1] [0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 1 1 1 1 0 0 0 0 0 0 1 0 0 1 0 1 1 1 0 0] [0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 1 1 1 1 0 0 0 0 0 0 1 0 0 1 0 1 1 1 0] [0 0 0 0 0 0 1 0 0 0 1 1 1 1 0 1 1 1 1 1 0 0 0 0 0 0 0 1 0 1 1 1 0 1 1] [0 0 0 0 0 0 0 1 0 0 1 1 1 0 0 1 1 0 0 0 1 0 0 0 0 0 0 0 0 1 0 1 1 0 0] [0 0 0 0 0 0 0 0 1 0 0 1 1 1 0 1 0 0 1 1 1 0 0 0 0 0 0 0 1 1 0 1 0 0 1] [0 0 0 0 0 0 0 0 0 1 1 1 0 1 0 0 1 1 1 0 1 0 0 0 0 0 0 0 0 1 0 0 1 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 0 0 0 0 0 0 0 1 1 1 1 1 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 1 0 1 0 1 0 0 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 1 1 1 0 1 0 0 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 1 0 1 0 0 1 1 1] Total time: 0.200 seconds, Total memory usage: 3.34MB '65.191.' ************** MAGMA ***************** Host 65.191.73.113 (65.191.73.113) Time: Thu Dec 15 19:20:21 2005 Input: K := FiniteField(2); > C := LinearCode; S, f := StandardForm(C); D := Dual(S); (D meet S) eq S; M:=VerticalJoin(GeneratorMatrix(S), GeneratorMatrix(D)); M1:=EchelonForm(M); M2:=Submatrix(M1,22,22,14,35); M2; C2:=LinearCode(M2); WeightDistribution(C2); L:=MinimumWords(C2); C3:=LinearCode; L; Output: Magma V2.11-10 Thu Dec 15 2005 19:20:21 on modular [Seed = 3745420434] ------------------------------------- true [1 0 0 0 0 0 0 0 0 0 1 1 0 1 0 0 0 1 0 1 0 0 0 0 0 0 0 1 1 0 1 1 1 1 0] [0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 1 1 0 0 0 0 0 1 1 0 0 1 0 0 1 0] [0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 1 1 1 0 0 0 0 0 0 1 1 1 1 0 1 0 0 1 0 1] [0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 1 1 1 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 1] [0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 1 1 1 1 0 0 0 0 0 0 1 0 0 1 0 1 1 1 0 0] [0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 1 1 1 1 0 0 0 0 0 0 1 0 0 1 0 1 1 1 0] [0 0 0 0 0 0 1 0 0 0 1 1 1 1 0 1 1 1 1 1 0 0 0 0 0 0 0 1 0 1 1 1 0 1 1] [0 0 0 0 0 0 0 1 0 0 1 1 1 0 0 1 1 0 0 0 1 0 0 0 0 0 0 0 0 1 0 1 1 0 0] [0 0 0 0 0 0 0 0 1 0 0 1 1 1 0 1 0 0 1 1 1 0 0 0 0 0 0 0 1 1 0 1 0 0 1] [0 0 0 0 0 0 0 0 0 1 1 1 0 1 0 0 1 1 1 0 1 0 0 0 0 0 0 0 0 1 0 0 1 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 0 0 0 0 0 0 0 1 1 1 1 1 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 1 0 1 0 1 0 0 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 1 1 1 0 1 0 0 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 1 0 1 0 0 1 1 1] [ <0, 1>, <8, 23>, <10, 175>, <12, 780>, <14, 2227>, <16, 3898>, <18, 4285>, <20, 3135>, <22, 1377>, <24, 410>, <26, 64>, <28, 9> ] { (1 1 0 0 0 1 0 1 1 1 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0), (0 0 1 0 0 0 1 1 1 0 0 0 1 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0), (1 1 1 0 0 1 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0), (0 0 1 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 1 0 0 0 1 0 1), (0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 1 1 1 1 1 0 1 0 0), (0 0 0 1 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 1 0 1 1 0 0 0 0 1 0 0 0), (0 1 0 0 0 0 1 1 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 1), (0 0 0 0 1 0 1 0 0 0 1 0 0 1 0 0 0 0 0 1 0 0 0 1 1 0 0 0 1 0 0 0 0 0 0), (0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 1 1 1 0 1 0 0 1), (0 1 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 1 0 1 1), (0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 1 1 1 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 1 0), (0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 1 0 1 0 1 0 0 1 1), (0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 0 0 1 0 1 1 1 0 1 0), (0 0 1 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 1 0 1 1 0 0 0 0 1 0 0 0 0), (0 0 0 1 0 0 1 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 1 1 0 0 0 1 0), (0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 1 0 1 0 0 1 1 1), (0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 1 0 0 0 0 1 1 1 0 1), (0 1 0 0 0 0 0 0 0 0 1 0 0 0 1 1 1 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 1 0 0), (0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 1 1 1 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 1), (0 1 0 0 0 0 1 0 1 0 0 0 0 0 0 0 1 0 1 1 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0), (0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 1 0 1 0 0 1 1 1 0), (1 0 0 0 0 0 0 0 0 1 0 0 0 0 1 1 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 1 0), (0 1 0 0 0 1 0 1 0 0 0 1 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0) } Total time: 0.190 seconds, Total memory usage: 3.34MB '65.191.' ************** MAGMA ***************** Host 65.191.73.113 (65.191.73.113) Time: Thu Dec 15 19:17:48 2005 Input: K := FiniteField(2); > C := LinearCode; S, f := StandardForm(C); D := Dual(S); (D meet S) eq S; M:=VerticalJoin(GeneratorMatrix(S), GeneratorMatrix(D)); M1:=EchelonForm(M); M2:=Submatrix(M1,22,22,14,35); M2; C2:=LinearCode(M2); WeightDistribution(C2); L:=MinimumWords(C2); C3:=LinearCode(L); L; Output: Magma V2.11-10 Thu Dec 15 2005 19:17:47 on modular [Seed = 3644366202] ------------------------------------- true [1 0 0 0 0 0 0 0 0 0 1 1 0 1 0 0 0 1 0 1 0 0 0 0 0 0 0 1 1 0 1 1 1 1 0] [0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 1 1 0 0 0 0 0 1 1 0 0 1 0 0 1 0] [0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 1 1 1 0 0 0 0 0 0 1 1 1 1 0 1 0 0 1 0 1] [0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 1 1 1 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 1] [0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 1 1 1 1 0 0 0 0 0 0 1 0 0 1 0 1 1 1 0 0] [0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 1 1 1 1 0 0 0 0 0 0 1 0 0 1 0 1 1 1 0] [0 0 0 0 0 0 1 0 0 0 1 1 1 1 0 1 1 1 1 1 0 0 0 0 0 0 0 1 0 1 1 1 0 1 1] [0 0 0 0 0 0 0 1 0 0 1 1 1 0 0 1 1 0 0 0 1 0 0 0 0 0 0 0 0 1 0 1 1 0 0] [0 0 0 0 0 0 0 0 1 0 0 1 1 1 0 1 0 0 1 1 1 0 0 0 0 0 0 0 1 1 0 1 0 0 1] [0 0 0 0 0 0 0 0 0 1 1 1 0 1 0 0 1 1 1 0 1 0 0 0 0 0 0 0 0 1 0 0 1 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 0 0 0 0 0 0 0 1 1 1 1 1 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 1 0 1 0 1 0 0 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 1 1 1 0 1 0 0 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 1 0 1 0 0 1 1 1] [ <0, 1>, <8, 23>, <10, 175>, <12, 780>, <14, 2227>, <16, 3898>, <18, 4285>, <20, 3135>, <22, 1377>, <24, 410>, <26, 64>, <28, 9> ] >> C3:=LinearCode(L); ^ Runtime error in 'LinearCode': Bad argument types Argument types given: SetEnum[ModTupFldElt] { (1 1 0 0 0 1 0 1 1 1 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0), (0 0 1 0 0 0 1 1 1 0 0 0 1 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0), (1 1 1 0 0 1 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0), (0 0 1 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 1 0 0 0 1 0 1), (0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 1 1 1 1 1 0 1 0 0), (0 0 0 1 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 1 0 1 1 0 0 0 0 1 0 0 0), (0 1 0 0 0 0 1 1 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 1), (0 0 0 0 1 0 1 0 0 0 1 0 0 1 0 0 0 0 0 1 0 0 0 1 1 0 0 0 1 0 0 0 0 0 0), (0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 1 1 1 0 1 0 0 1), (0 1 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 1 0 1 1), (0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 1 1 1 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 1 0), (0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 1 0 1 0 1 0 0 1 1), (0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 0 0 1 0 1 1 1 0 1 0), (0 0 1 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 1 0 1 1 0 0 0 0 1 0 0 0 0), (0 0 0 1 0 0 1 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 1 1 0 0 0 1 0), (0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 1 0 1 0 0 1 1 1), (0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 1 0 0 0 0 1 1 1 0 1), (0 1 0 0 0 0 0 0 0 0 1 0 0 0 1 1 1 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 1 0 0), (0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 1 1 1 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 1), (0 1 0 0 0 0 1 0 1 0 0 0 0 0 0 0 1 0 1 1 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0), (0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 1 0 1 0 0 1 1 1 0), (1 0 0 0 0 0 0 0 0 1 0 0 0 0 1 1 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 1 0), (0 1 0 0 0 1 0 1 0 0 0 1 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0) } Total time: 0.190 seconds, Total memory usage: 3.34MB '65.191.' ************** MAGMA ***************** Host 65.191.73.113 (65.191.73.113) Time: Thu Dec 15 18:58:17 2005 Input: K := FiniteField(2); > C := LinearCode; S, f := StandardForm(C); D := Dual(S); (D meet S) eq S; M:=VerticalJoin(GeneratorMatrix(S), GeneratorMatrix(D)); M1:=EchelonForm(M); M2:=Submatrix(M1,22,22,14,35); M2; C2:=LinearCode(M2); WeightDistribution(C2); MinimumWords(C2); Output: Magma V2.11-10 Thu Dec 15 2005 18:58:17 on modular [Seed = 3193962159] ------------------------------------- true [1 0 0 0 0 0 0 0 0 0 1 1 0 1 0 0 0 1 0 1 0 0 0 0 0 0 0 1 1 0 1 1 1 1 0] [0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 1 1 0 0 0 0 0 1 1 0 0 1 0 0 1 0] [0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 1 1 1 0 0 0 0 0 0 1 1 1 1 0 1 0 0 1 0 1] [0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 1 1 1 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 1] [0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 1 1 1 1 0 0 0 0 0 0 1 0 0 1 0 1 1 1 0 0] [0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 1 1 1 1 0 0 0 0 0 0 1 0 0 1 0 1 1 1 0] [0 0 0 0 0 0 1 0 0 0 1 1 1 1 0 1 1 1 1 1 0 0 0 0 0 0 0 1 0 1 1 1 0 1 1] [0 0 0 0 0 0 0 1 0 0 1 1 1 0 0 1 1 0 0 0 1 0 0 0 0 0 0 0 0 1 0 1 1 0 0] [0 0 0 0 0 0 0 0 1 0 0 1 1 1 0 1 0 0 1 1 1 0 0 0 0 0 0 0 1 1 0 1 0 0 1] [0 0 0 0 0 0 0 0 0 1 1 1 0 1 0 0 1 1 1 0 1 0 0 0 0 0 0 0 0 1 0 0 1 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 0 0 0 0 0 0 0 1 1 1 1 1 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 1 0 1 0 1 0 0 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 1 1 1 0 1 0 0 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 1 0 1 0 0 1 1 1] [ <0, 1>, <8, 23>, <10, 175>, <12, 780>, <14, 2227>, <16, 3898>, <18, 4285>, <20, 3135>, <22, 1377>, <24, 410>, <26, 64>, <28, 9> ] { (1 1 0 0 0 1 0 1 1 1 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0), (0 0 1 0 0 0 1 1 1 0 0 0 1 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0), (1 1 1 0 0 1 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0), (0 0 1 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 1 0 0 0 1 0 1), (0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 1 1 1 1 1 0 1 0 0), (0 0 0 1 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 1 0 1 1 0 0 0 0 1 0 0 0), (0 1 0 0 0 0 1 1 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 1), (0 0 0 0 1 0 1 0 0 0 1 0 0 1 0 0 0 0 0 1 0 0 0 1 1 0 0 0 1 0 0 0 0 0 0), (0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 1 1 1 0 1 0 0 1), (0 1 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 1 0 1 1), (0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 1 1 1 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 1 0), (0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 1 0 1 0 1 0 0 1 1), (0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 0 0 1 0 1 1 1 0 1 0), (0 0 1 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 1 0 1 1 0 0 0 0 1 0 0 0 0), (0 0 0 1 0 0 1 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 1 1 0 0 0 1 0), (0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 1 0 1 0 0 1 1 1), (0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 1 0 0 0 0 1 1 1 0 1), (0 1 0 0 0 0 0 0 0 0 1 0 0 0 1 1 1 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 1 0 0), (0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 1 1 1 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 1), (0 1 0 0 0 0 1 0 1 0 0 0 0 0 0 0 1 0 1 1 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0), (0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 1 0 1 0 0 1 1 1 0), (1 0 0 0 0 0 0 0 0 1 0 0 0 0 1 1 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 1 0), (0 1 0 0 0 1 0 1 0 0 0 1 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0) } Total time: 0.190 seconds, Total memory usage: 3.34MB '65.191.' ************** MAGMA ***************** Host 65.191.73.113 (65.191.73.113) Time: Thu Dec 15 18:56:30 2005 Input: K := FiniteField(2); > C := LinearCode; S, f := StandardForm(C); D := Dual(S); (D meet S) eq S; M:=VerticalJoin(GeneratorMatrix(S), GeneratorMatrix(D)); M1:=EchelonForm(M); M2:=Submatrix(M1,22,22,14,35); M2; C2:=LinearCode(M2); WeightDistribution(C2); Output: Magma V2.11-10 Thu Dec 15 2005 18:56:30 on modular [Seed = 3092904222] ------------------------------------- true [1 0 0 0 0 0 0 0 0 0 1 1 0 1 0 0 0 1 0 1 0 0 0 0 0 0 0 1 1 0 1 1 1 1 0] [0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 1 1 0 0 0 0 0 1 1 0 0 1 0 0 1 0] [0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 1 1 1 0 0 0 0 0 0 1 1 1 1 0 1 0 0 1 0 1] [0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 1 1 1 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 1] [0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 1 1 1 1 0 0 0 0 0 0 1 0 0 1 0 1 1 1 0 0] [0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 1 1 1 1 0 0 0 0 0 0 1 0 0 1 0 1 1 1 0] [0 0 0 0 0 0 1 0 0 0 1 1 1 1 0 1 1 1 1 1 0 0 0 0 0 0 0 1 0 1 1 1 0 1 1] [0 0 0 0 0 0 0 1 0 0 1 1 1 0 0 1 1 0 0 0 1 0 0 0 0 0 0 0 0 1 0 1 1 0 0] [0 0 0 0 0 0 0 0 1 0 0 1 1 1 0 1 0 0 1 1 1 0 0 0 0 0 0 0 1 1 0 1 0 0 1] [0 0 0 0 0 0 0 0 0 1 1 1 0 1 0 0 1 1 1 0 1 0 0 0 0 0 0 0 0 1 0 0 1 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 0 0 0 0 0 0 0 1 1 1 1 1 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 1 0 1 0 1 0 0 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 1 1 1 0 1 0 0 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 1 0 1 0 0 1 1 1] [ <0, 1>, <8, 23>, <10, 175>, <12, 780>, <14, 2227>, <16, 3898>, <18, 4285>, <20, 3135>, <22, 1377>, <24, 410>, <26, 64>, <28, 9> ] Total time: 0.190 seconds, Total memory usage: 3.34MB '65.191.' ************** MAGMA ***************** Host 65.191.73.113 (65.191.73.113) Time: Thu Dec 15 18:55:19 2005 Input: K := FiniteField(2); > C := LinearCode; S, f := StandardForm(C); D := Dual(S); (D meet S) eq S; M:=VerticalJoin(GeneratorMatrix(S), GeneratorMatrix(D)); M1:=EchelonForm(M); M2:=Submatrix(M1,22,22,14,35); M2; Output: Magma V2.11-10 Thu Dec 15 2005 18:55:19 on modular [Seed = 2739069644] ------------------------------------- true [1 0 0 0 0 0 0 0 0 0 1 1 0 1 0 0 0 1 0 1 0 0 0 0 0 0 0 1 1 0 1 1 1 1 0] [0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 1 1 0 0 0 0 0 1 1 0 0 1 0 0 1 0] [0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 1 1 1 0 0 0 0 0 0 1 1 1 1 0 1 0 0 1 0 1] [0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 1 1 1 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 1] [0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 1 1 1 1 0 0 0 0 0 0 1 0 0 1 0 1 1 1 0 0] [0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 1 1 1 1 0 0 0 0 0 0 1 0 0 1 0 1 1 1 0] [0 0 0 0 0 0 1 0 0 0 1 1 1 1 0 1 1 1 1 1 0 0 0 0 0 0 0 1 0 1 1 1 0 1 1] [0 0 0 0 0 0 0 1 0 0 1 1 1 0 0 1 1 0 0 0 1 0 0 0 0 0 0 0 0 1 0 1 1 0 0] [0 0 0 0 0 0 0 0 1 0 0 1 1 1 0 1 0 0 1 1 1 0 0 0 0 0 0 0 1 1 0 1 0 0 1] [0 0 0 0 0 0 0 0 0 1 1 1 0 1 0 0 1 1 1 0 1 0 0 0 0 0 0 0 0 1 0 0 1 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 0 0 0 0 0 0 0 1 1 1 1 1 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 1 0 1 0 1 0 0 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 1 1 1 0 1 0 0 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 1 0 1 0 0 1 1 1] Total time: 0.200 seconds, Total memory usage: 3.34MB '65.191.' ************** MAGMA ***************** Host 65.191.73.113 (65.191.73.113) Time: Thu Dec 15 18:54:58 2005 Input: K := FiniteField(2); > C := LinearCode: S, f := StandardForm(C); D := Dual(S); (D meet S) eq S; M:=VerticalJoin(GeneratorMatrix(S), GeneratorMatrix(D)); M1:=EchelonForm(M); M2:=Submatrix(M1,22,22,14,35); M2; Output: Magma V2.11-10 Thu Dec 15 2005 18:54:58 on modular [Seed = 2924474356] ------------------------------------- >> 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]>: ^ User error: bad syntax >> S, f := StandardForm(C); ^ User error: Identifier 'C' has not been declared or assigned >> D := Dual(S); ^ User error: Identifier 'S' has not been declared or assigned >> (D meet S) eq S; ^ User error: Identifier 'D' has not been declared or assigned >> M:=VerticalJoin(GeneratorMatrix(S), GeneratorMatrix(D)); ^ User error: Identifier 'S' has not been declared or assigned >> M1:=EchelonForm(M); ^ User error: Identifier 'M' has not been declared or assigned >> M2:=Submatrix(M1,22,22,14,35); ^ User error: Identifier 'M1' has not been declared or assigned >> M2; ^ User error: Identifier 'M2' has not been declared or assigned Total time: 0.190 seconds, Total memory usage: 3.24MB '65.191.' ************** MAGMA ***************** Host 65.191.73.113 (65.191.73.113) Time: Thu Dec 15 18:53:27 2005 Input: K := FiniteField(2); > C := LinearCode; S, f := StandardForm(C); S; D := Dual(S); (D meet S) eq S; M:=VerticalJoin(GeneratorMatrix(S), GeneratorMatrix(D)); M1:=EchelonForm(M); M1; M2:=Submatrix(M1,22,22,14,35); M2; Output: Magma V2.11-10 Thu Dec 15 2005 18:53:27 on modular [Seed = 2823416407] ------------------------------------- [56, 21] Linear Code over GF(2) Generator matrix: [1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 0 0 0 0 1 1 1 0 1 0 1 0 0 1 1 0 0 1 1 0 0 0 1 1 0 1 0 0] [0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 1 0 1 0 0 1 0 0 0 1 0 0 1 0 1 1 1 0 0 0 1 1 0 0 1 1 0 1 0] [0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 1 0 0 0 1 1 0 0 0 0 0 0 0 1 1 1 1 1 0 0 0 0 0 0 0 1 1 1 1 0 0] [0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 1 1 1 1 0 1 1 0 0 1 1 0 1] [0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 1 0 1 1 0 0 1 1 1 1 1 1 0 1 1 1 0 0 1 1 0 1 1 1 1 0 0 1 1 0] [0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 0 0 0 0 1 0 1 1 1 0 0 0 1 1 1 0 0 0 1 0 0 1 0 0 1 0 1 0 0 0 1] [0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 1 1 1 0 0 1 0 1 0 1 1 0 1 0 0 1 1 1 1 1 0 1 1 0 0 0 1 0 1 0] [0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 1 1 0 0 1 1 0 0 0 1 0 0 1 0 1 1 0 0 1 1 1 0 1 1 0 1 0 0 1 1] [0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 1 1 1 0 1 0 1 1 1 0 0 1 1 1 0 0 0 1 1 1 1 1 1 1 1 0 1 1 0 0 0] [0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 0 1 1 1 1 0 0 0 0 1 1 0 1 1 0 1 0 1 1 1 0 1 0 0 1 0 1 1 0 0] [0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 1 1 1 0 0 1 0 1 1 1 1 1 0 0 0 0 1 1 1 1 0 1 1 0 0 0 0 0 1 0 1 1 0] [0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 1 0 1 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 0 1 0 1 1 1 0 1 0 0 1] [0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 1 1 1 1 1 0 0 1 1 1 0 1 1 1 0 0 1 1 1 1 1 0 1 0 0 1 1 1 0 0 0 1 0 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 1 0 0 0 1 0 0 0 1 0 1 0 1 0 1 1 0 0 1 1 1 0 1 0 0 0 1 1 0 0 0 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 1 1 0 0 0 0 1 0 0 1 1 1 0 1 1 1 0 1 0 1 0 0 1 1 1 0 1 1 0 1 0 0 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0 1 1 0 0 1 1 0 0 0 1 1 0 1 0 0 1 1 1 0 0 0 1 0 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0 1 1 1 0 1 1 0 0 0 1 1 0 1 0 0 1 1 1 0 0 0 1 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 1 0 1 1 1 0 1 0 0 0 1 0 1 1 0 0 1 0 1 1 0 0 1 1 0 0 0 1 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 1 0 1 0 0 1 1 1 0 0 1 1 1 0 1 0 1 1 1 0 1 0 1 0 1 0 0 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 1 0 1 1 0 1 0 0 1 1 0 1 1 0 0 0 0 1 1 0 0 0 1 0 0 0 1 0 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 1 1 1 1 0 1 0 0 0 1 0 1 1 0 0 1 0 1 1 0 0 0 1 0 0 0 1 0 1] true [1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 1 0 1 0 0 0 0 0 0 0 0 1 1 0 0 0 1 1 0] [0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 1 0 1 0 0 0 0 1 1 1 0 1 1 1 0 0 0 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'65.191.' ************** MAGMA ***************** Host 65.191.73.113 (65.191.73.113) Time: Thu Dec 15 18:47:10 2005 Input: K := FiniteField(2); > C := LinearCode; S, f := StandardForm(C); S; D := Dual(S); (D meet S) eq S; M:=VerticalJoin(GeneratorMatrix(S), GeneratorMatrix(D)); M1:=EchelonForm(M); M1; Output: Magma V2.11-10 Thu Dec 15 2005 18:47:10 on modular [Seed = 2675891433] ------------------------------------- [56, 21] Linear Code over GF(2) Generator matrix: [1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 0 0 0 0 1 1 1 0 1 0 1 0 0 1 1 0 0 1 1 0 0 0 1 1 0 1 0 0] [0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 1 0 1 0 0 1 0 0 0 1 0 0 1 0 1 1 1 0 0 0 1 1 0 0 1 1 0 1 0] [0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 1 0 0 0 1 1 0 0 0 0 0 0 0 1 1 1 1 1 0 0 0 0 0 0 0 1 1 1 1 0 0] [0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 1 1 1 1 0 1 1 0 0 1 1 0 1] [0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 1 0 1 1 0 0 1 1 1 1 1 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0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0] Total time: 0.190 seconds, Total memory usage: 3.34MB '65.191.' ************** MAGMA ***************** Host 65.191.73.113 (65.191.73.113) Time: Thu Dec 15 18:45:08 2005 Input: K := FiniteField(2); > C := LinearCode; S, f := StandardForm(C); S; D := Dual(S); (D meet S) eq S; M:=VerticalJoin(GeneratorMatrix(S), GeneratorMatrix(S)); M1:=EchelonForm(M); M1; Output: Magma V2.11-10 Thu Dec 15 2005 18:45:08 on modular [Seed = 2591807894] ------------------------------------- [56, 21] Linear Code over GF(2) Generator matrix: [1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 0 0 0 0 1 1 1 0 1 0 1 0 0 1 1 0 0 1 1 0 0 0 1 1 0 1 0 0] [0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 1 0 1 0 0 1 0 0 0 1 0 0 1 0 1 1 1 0 0 0 1 1 0 0 1 1 0 1 0] [0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 1 0 0 0 1 1 0 0 0 0 0 0 0 1 1 1 1 1 0 0 0 0 0 0 0 1 1 1 1 0 0] [0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 1 1 1 1 0 1 1 0 0 1 1 0 1] [0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 1 0 1 1 0 0 1 1 1 1 1 1 0 1 1 1 0 0 1 1 0 1 1 1 1 0 0 1 1 0] [0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 0 0 0 0 1 0 1 1 1 0 0 0 1 1 1 0 0 0 1 0 0 1 0 0 1 0 1 0 0 0 1] [0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 1 1 1 0 0 1 0 1 0 1 1 0 1 0 0 1 1 1 1 1 0 1 1 0 0 0 1 0 1 0] [0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 1 1 0 0 1 1 0 0 0 1 0 0 1 0 1 1 0 0 1 1 1 0 1 1 0 1 0 0 1 1] [0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 1 1 1 0 1 0 1 1 1 0 0 1 1 1 0 0 0 1 1 1 1 1 1 1 1 0 1 1 0 0 0] [0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 0 1 1 1 1 0 0 0 0 1 1 0 1 1 0 1 0 1 1 1 0 1 0 0 1 0 1 1 0 0] [0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 1 1 1 0 0 1 0 1 1 1 1 1 0 0 0 0 1 1 1 1 0 1 1 0 0 0 0 0 1 0 1 1 0] [0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 1 0 1 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 0 1 0 1 1 1 0 1 0 0 1] [0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 1 1 1 1 1 0 0 1 1 1 0 1 1 1 0 0 1 1 1 1 1 0 1 0 0 1 1 1 0 0 0 1 0 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 1 0 0 0 1 0 0 0 1 0 1 0 1 0 1 1 0 0 1 1 1 0 1 0 0 0 1 1 0 0 0 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 1 1 0 0 0 0 1 0 0 1 1 1 0 1 1 1 0 1 0 1 0 0 1 1 1 0 1 1 0 1 0 0 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0 1 1 0 0 1 1 0 0 0 1 1 0 1 0 0 1 1 1 0 0 0 1 0 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0 1 1 1 0 1 1 0 0 0 1 1 0 1 0 0 1 1 1 0 0 0 1 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 1 0 1 1 1 0 1 0 0 0 1 0 1 1 0 0 1 0 1 1 0 0 1 1 0 0 0 1 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 1 0 1 0 0 1 1 1 0 0 1 1 1 0 1 0 1 1 1 0 1 0 1 0 1 0 0 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 1 0 1 1 0 1 0 0 1 1 0 1 1 0 0 0 0 1 1 0 0 0 1 0 0 0 1 0 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 1 1 1 1 0 1 0 0 0 1 0 1 1 0 0 1 0 1 1 0 0 0 1 0 0 0 1 0 1] true [1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 0 0 0 0 1 1 1 0 1 0 1 0 0 1 1 0 0 1 1 0 0 0 1 1 0 1 0 0] [0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 1 0 1 0 0 1 0 0 0 1 0 0 1 0 1 1 1 0 0 0 1 1 0 0 1 1 0 1 0] [0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 1 0 0 0 1 1 0 0 0 0 0 0 0 1 1 1 1 1 0 0 0 0 0 0 0 1 1 1 1 0 0] [0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 1 1 1 1 0 1 1 0 0 1 1 0 1] [0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 1 0 1 1 0 0 1 1 1 1 1 1 0 1 1 1 0 0 1 1 0 1 1 1 1 0 0 1 1 0] [0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 0 0 0 0 1 0 1 1 1 0 0 0 1 1 1 0 0 0 1 0 0 1 0 0 1 0 1 0 0 0 1] [0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 1 1 1 0 0 1 0 1 0 1 1 0 1 0 0 1 1 1 1 1 0 1 1 0 0 0 1 0 1 0] [0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 1 1 0 0 1 1 0 0 0 1 0 0 1 0 1 1 0 0 1 1 1 0 1 1 0 1 0 0 1 1] [0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 1 1 1 0 1 0 1 1 1 0 0 1 1 1 0 0 0 1 1 1 1 1 1 1 1 0 1 1 0 0 0] [0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 0 1 1 1 1 0 0 0 0 1 1 0 1 1 0 1 0 1 1 1 0 1 0 0 1 0 1 1 0 0] [0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 1 1 1 0 0 1 0 1 1 1 1 1 0 0 0 0 1 1 1 1 0 1 1 0 0 0 0 0 1 0 1 1 0] [0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 1 0 1 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 0 1 0 1 1 1 0 1 0 0 1] [0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 1 1 1 1 1 0 0 1 1 1 0 1 1 1 0 0 1 1 1 1 1 0 1 0 0 1 1 1 0 0 0 1 0 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 1 0 0 0 1 0 0 0 1 0 1 0 1 0 1 1 0 0 1 1 1 0 1 0 0 0 1 1 0 0 0 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 1 1 0 0 0 0 1 0 0 1 1 1 0 1 1 1 0 1 0 1 0 0 1 1 1 0 1 1 0 1 0 0 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0 1 1 0 0 1 1 0 0 0 1 1 0 1 0 0 1 1 1 0 0 0 1 0 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0 1 1 1 0 1 1 0 0 0 1 1 0 1 0 0 1 1 1 0 0 0 1 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 1 0 1 1 1 0 1 0 0 0 1 0 1 1 0 0 1 0 1 1 0 0 1 1 0 0 0 1 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 1 0 1 0 0 1 1 1 0 0 1 1 1 0 1 0 1 1 1 0 1 0 1 0 1 0 0 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 1 0 1 1 0 1 0 0 1 1 0 1 1 0 0 0 0 1 1 0 0 0 1 0 0 0 1 0 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 1 1 1 1 0 1 0 0 0 1 0 1 1 0 0 1 0 1 1 0 0 0 1 0 0 0 1 0 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0] Total time: 0.180 seconds, Total memory usage: 3.34MB '65.191.' ************** MAGMA ***************** Host 65.191.73.113 (65.191.73.113) Time: Thu Dec 15 18:05:15 2005 Input: K := FiniteField(2); > C := LinearCode; S, f := StandardForm(C); S; D := Dual(S); (D meet S) eq S; Output: Magma V2.11-10 Thu Dec 15 2005 18:05:14 on modular [Seed = 1538911993] ------------------------------------- [56, 21] Linear Code over GF(2) Generator matrix: [1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 0 0 0 0 1 1 1 0 1 0 1 0 0 1 1 0 0 1 1 0 0 0 1 1 0 1 0 0] [0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 1 0 1 0 0 1 0 0 0 1 0 0 1 0 1 1 1 0 0 0 1 1 0 0 1 1 0 1 0] [0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 1 0 0 0 1 1 0 0 0 0 0 0 0 1 1 1 1 1 0 0 0 0 0 0 0 1 1 1 1 0 0] [0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 1 1 1 1 0 1 1 0 0 1 1 0 1] [0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 1 0 1 1 0 0 1 1 1 1 1 1 0 1 1 1 0 0 1 1 0 1 1 1 1 0 0 1 1 0] [0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 0 0 0 0 1 0 1 1 1 0 0 0 1 1 1 0 0 0 1 0 0 1 0 0 1 0 1 0 0 0 1] [0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 1 1 1 0 0 1 0 1 0 1 1 0 1 0 0 1 1 1 1 1 0 1 1 0 0 0 1 0 1 0] [0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 1 1 0 0 1 1 0 0 0 1 0 0 1 0 1 1 0 0 1 1 1 0 1 1 0 1 0 0 1 1] [0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 1 1 1 0 1 0 1 1 1 0 0 1 1 1 0 0 0 1 1 1 1 1 1 1 1 0 1 1 0 0 0] [0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 0 1 1 1 1 0 0 0 0 1 1 0 1 1 0 1 0 1 1 1 0 1 0 0 1 0 1 1 0 0] [0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 1 1 1 0 0 1 0 1 1 1 1 1 0 0 0 0 1 1 1 1 0 1 1 0 0 0 0 0 1 0 1 1 0] [0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 1 0 1 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 0 1 0 1 1 1 0 1 0 0 1] [0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 1 1 1 1 1 0 0 1 1 1 0 1 1 1 0 0 1 1 1 1 1 0 1 0 0 1 1 1 0 0 0 1 0 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 1 0 0 0 1 0 0 0 1 0 1 0 1 0 1 1 0 0 1 1 1 0 1 0 0 0 1 1 0 0 0 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 1 1 0 0 0 0 1 0 0 1 1 1 0 1 1 1 0 1 0 1 0 0 1 1 1 0 1 1 0 1 0 0 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0 1 1 0 0 1 1 0 0 0 1 1 0 1 0 0 1 1 1 0 0 0 1 0 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0 1 1 1 0 1 1 0 0 0 1 1 0 1 0 0 1 1 1 0 0 0 1 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 1 0 1 1 1 0 1 0 0 0 1 0 1 1 0 0 1 0 1 1 0 0 1 1 0 0 0 1 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 1 0 1 0 0 1 1 1 0 0 1 1 1 0 1 0 1 1 1 0 1 0 1 0 1 0 0 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 1 0 1 1 0 1 0 0 1 1 0 1 1 0 0 0 0 1 1 0 0 0 1 0 0 0 1 0 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 1 1 1 1 0 1 0 0 0 1 0 1 1 0 0 1 0 1 1 0 0 0 1 0 0 0 1 0 1] true Total time: 0.190 seconds, Total memory usage: 3.34MB '65.191.' ************** MAGMA ***************** Host 65.191.73.113 (65.191.73.113) Time: Thu Dec 15 18:03:40 2005 Input: K := FiniteField(2); > C := LinearCode; S, f := StandardForm(C); S; D := Dual(S); Output: Magma V2.11-10 Thu Dec 15 2005 18:03:40 on modular [Seed = 1168633857] ------------------------------------- [56, 21] Linear Code over GF(2) Generator matrix: [1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 0 0 0 0 1 1 1 0 1 0 1 0 0 1 1 0 0 1 1 0 0 0 1 1 0 1 0 0] [0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 1 0 1 0 0 1 0 0 0 1 0 0 1 0 1 1 1 0 0 0 1 1 0 0 1 1 0 1 0] [0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 1 0 0 0 1 1 0 0 0 0 0 0 0 1 1 1 1 1 0 0 0 0 0 0 0 1 1 1 1 0 0] [0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 1 1 1 1 0 1 1 0 0 1 1 0 1] [0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 1 0 1 1 0 0 1 1 1 1 1 1 0 1 1 1 0 0 1 1 0 1 1 1 1 0 0 1 1 0] [0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 0 0 0 0 1 0 1 1 1 0 0 0 1 1 1 0 0 0 1 0 0 1 0 0 1 0 1 0 0 0 1] [0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 1 1 1 0 0 1 0 1 0 1 1 0 1 0 0 1 1 1 1 1 0 1 1 0 0 0 1 0 1 0] [0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 1 1 0 0 1 1 0 0 0 1 0 0 1 0 1 1 0 0 1 1 1 0 1 1 0 1 0 0 1 1] [0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 1 1 1 0 1 0 1 1 1 0 0 1 1 1 0 0 0 1 1 1 1 1 1 1 1 0 1 1 0 0 0] [0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 0 1 1 1 1 0 0 0 0 1 1 0 1 1 0 1 0 1 1 1 0 1 0 0 1 0 1 1 0 0] [0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 1 1 1 0 0 1 0 1 1 1 1 1 0 0 0 0 1 1 1 1 0 1 1 0 0 0 0 0 1 0 1 1 0] [0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 1 0 1 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 0 1 0 1 1 1 0 1 0 0 1] [0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 1 1 1 1 1 0 0 1 1 1 0 1 1 1 0 0 1 1 1 1 1 0 1 0 0 1 1 1 0 0 0 1 0 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 1 0 0 0 1 0 0 0 1 0 1 0 1 0 1 1 0 0 1 1 1 0 1 0 0 0 1 1 0 0 0 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 1 1 0 0 0 0 1 0 0 1 1 1 0 1 1 1 0 1 0 1 0 0 1 1 1 0 1 1 0 1 0 0 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0 1 1 0 0 1 1 0 0 0 1 1 0 1 0 0 1 1 1 0 0 0 1 0 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0 1 1 1 0 1 1 0 0 0 1 1 0 1 0 0 1 1 1 0 0 0 1 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 1 0 1 1 1 0 1 0 0 0 1 0 1 1 0 0 1 0 1 1 0 0 1 1 0 0 0 1 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 1 0 1 0 0 1 1 1 0 0 1 1 1 0 1 0 1 1 1 0 1 0 1 0 1 0 0 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 1 0 1 1 0 1 0 0 1 1 0 1 1 0 0 0 0 1 1 0 0 0 1 0 0 0 1 0 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 1 1 1 1 0 1 0 0 0 1 0 1 1 0 0 1 0 1 1 0 0 0 1 0 0 0 1 0 1] Total time: 0.190 seconds, Total memory usage: 3.34MB '152.6.1' ************** MAGMA ***************** Host 152.6.19.192 (152.6.19.192) Time: Thu Dec 15 12:35:48 2005 Input: K := FiniteField(2); > C := LinearCode; IsSelfOrthogonal(C); > aut := AutomorphismGroup(C); Order(aut); > FactoredOrder(aut); > CompositionFactors(aut); Generators(aut); WeightDistribution(C); Output: Magma V2.11-10 Thu Dec 15 2005 12:35:48 on modular [Seed = 3177142192] ------------------------------------- false 7 [ <7, 1> ] G | Cyclic(7) 1 { (1, 2, 3, 4, 5, 6, 7)(8, 9, 10, 11, 12, 13, 14)(15, 16, 17, 18, 19, 20, 21)(22, 23, 24, 25, 26, 27, 28)(29, 30, 31, 32, 33, 34, 35)(36, 37, 38, 39, 40, 41, 42)(43, 44, 45, 46, 47, 48, 49)(50, 51, 52, 53, 54, 55, 56) } [ <0, 1>, <16, 5082>, <20, 91168>, <24, 507045>, <28, 890560>, <32, 507045>, <36, 91168>, <40, 5082>, <56, 1> ] Total time: 0.310 seconds, Total memory usage: 5.51MB '199.89.' ************** MAGMA ***************** Host 199.89.64.177 (199.89.64.177) Time: Thu Dec 15 11:15:37 2005 Input: FindGroupOrder2 := function (p, s) K := GF(p); v := K ! (4*s); u := K ! (s^2-5); x := u^3; b := 4*x*v; a := (v-u)^3*(3*u+v); A := a/b-2; x := x/v^3; b := x^3 + A*x^2 + x; E := EllipticCurve([0,b*A,0,b^2,0]); return FactoredOrder(E); end function; p := 140853945410621700611366248656986006762214430713643; s := 941728572; FindGroupOrder2(p, s); Output: Magma V2.11-10 Thu Dec 15 2005 11:15:30 on modular [Seed = 1636021435] ------------------------------------- [ <2, 2>, <3, 1>, <5, 1>, <137, 1>, <251, 1>, <317, 1>, <331, 1>, <7307, 1>, <13339, 1>, <15667, 1>, <29209, 1>, <137209, 1>, <332933, 1>, <393571, 1>, <811351, 1> ] Total time: 2.439 seconds, Total memory usage: 5.31MB '129.13.' ************** MAGMA ***************** Host 129.13.186.1 (129.13.186.1) Time: Thu Dec 15 05:37:39 2005 Input: 9+100000000000000000000000000000 Output: Magma V2.11-10 Thu Dec 15 2005 05:37:39 on modular [Seed = 2338951053] ------------------------------------- 100000000000000000000000000009 Total time: 0.180 seconds, Total memory usage: 3.24MB '148.87.' ************** MAGMA ***************** Host 148.87.1.172 (148.87.1.172) Time: Thu Dec 15 00:38:27 2005 Input: for(x=1,50,print1("d(",x,")=",numdiv(x),", ")) Output: Magma V2.11-10 Thu Dec 15 2005 00:38:27 on modular [Seed = 1669407046] ------------------------------------- >> for(x=1,50,print1("d(",x,")=",numdiv(x),", ")); ^ User error: bad syntax Total time: 0.200 seconds, Total memory usage: 3.24MB '159.18.' ************** MAGMA ***************** Host 159.18.221.196 (159.18.221.196) Time: Wed Dec 14 17:22:35 2005 Input: 3^-1%0xF5DD146F31C94C0F7BF8CE66A63E86D47FB68E7C3F00BFB2232B848ECCBB0AF5 Output: WARNING: MAGMA command contains unsafe command '%', so it will not be executed. '159.18.' ************** MAGMA ***************** Host 159.18.221.196 (159.18.221.196) Time: Wed Dec 14 17:22:14 2005 Input: 3^-1mod0xF5DD146F31C94C0F7BF8CE66A63E86D47FB68E7C3F00BFB2232B848ECCBB0AF5 Output: Magma V2.11-10 Wed Dec 14 2005 17:22:14 on modular [Seed = 3143132098] ------------------------------------- >> 3^-1mod0xF5DD146F31C94C0F7BF8CE66A63E86D47FB68E7C3F00BFB2232B848ECCBB0AF5; ^ User error: bad syntax Total time: 0.190 seconds, Total memory usage: 3.24MB '159.18.' ************** MAGMA ***************** Host 159.18.221.196 (159.18.221.196) Time: Wed Dec 14 17:22:06 2005 Input: 3^-1%0xF5DD146F31C94C0F7BF8CE66A63E86D47FB68E7C3F00BFB2232B848ECCBB0AF5 Output: WARNING: MAGMA command contains unsafe command '%', so it will not be executed. '159.18.' ************** MAGMA ***************** Host 159.18.221.196 (159.18.221.196) Time: Wed Dec 14 17:21:51 2005 Input: 3^-1 Output: Magma V2.11-10 Wed Dec 14 2005 17:21:51 on modular [Seed = 2738911795] ------------------------------------- 1/3 Total time: 0.190 seconds, Total memory usage: 3.24MB '159.18.' ************** MAGMA ***************** Host 159.18.221.196 (159.18.221.196) Time: Wed Dec 14 17:21:26 2005 Input: "Replace this by some code, then click [PARI] or [MAGMA]!" Output: Magma V2.11-10 Wed Dec 14 2005 17:21:25 on modular [Seed = 2823516450] ------------------------------------- Replace this by some code, then click [PARI] or [MAGMA]! Total time: 0.200 seconds, Total memory usage: 3.24MB '147.96.' ************** MAGMA ***************** Host 147.96.20.233 (147.96.20.233) Time: Wed Dec 14 14:00:13 2005 Input: P<[x]>:=PolynomialRing(GF(2),3,"grevlex"); g1:=x[1]*x[3]+x[2]*x[3]+x[3]; g2:=x[1]*x[2]+x[1]*x[3]+x[2]*x[3]+x[1]; g3:=x[2]^2+x[2]*x[3]+x[3]^2+1; g4:=x[2]+x[3]+1; g5:=x[3]^2+x[3]; pol:=x[2]*x[3]*g4; g:=[g1,g2,g3,g4,g5]; I:=ideal; T1:=ideal; T2:=ideal; T3:=ideal; T4:=ideal; T5:=ideal; T:=[T1,T2,T3,T4,T5]; deg:=1; equis:=x cat [1]; Mons:={a:a in x} join {1}; for j in [2..deg] do Mons:={mon*vble:vble in equis, mon in Mons}; end for; Mons2:=Mons; for i in [1,2] do Mons2:={mon*vble:vble in equis, mon in Mons2}; end for; Mons2:=[mon:mon in Mons2]; polinomios:=[mon*g[i]: i in [1..5], mon in Mons | not(mon in T[i])]; dim1:=#polinomios+1; dim2:=#Mons2+1; //G:=GF(2,dim1+1); //AssertAttribute(G,"PowerPrinting",false); M:=RMatrixSpace(P,dim1,dim2); Mat:=M!0; for i in [1..dim1-1] do lista:=Monomials(polinomios[i]); for j in [1..dim2-1] do if Mons2[j] in lista then Mat[i,j]:=1; end if; end for; end for; lista:=Monomials(pol); for j in [1..dim2-1] do if Mons2[j] in lista then Mat[dim1,j]:=1; end if; end for; Mat[dim1,dim2]:=1; //Mat[dim1,dim2]:=1; B:=EchelonForm(Mat); //newpol:=&+[Mons2[j]*B[dim1,j]:j in [1..dim2-1]]; Output: Magma V2.11-10 Wed Dec 14 2005 14:00:12 on modular [Seed = 2356126809] ------------------------------------- >> B:=EchelonForm(Mat); ^ Runtime error in 'EchelonForm': Argument 1 has no echelon algorithm Total time: 0.190 seconds, Total memory usage: 3.24MB '66.50.2' ************** MAGMA ***************** Host 66.50.2.109 (66.50.2.109) Time: Wed Dec 14 11:20:13 2005 Input: 7102277187569652249869562285153253710645556161842592536300347670222640516361186363486317189963554176580264215037288238394086254424175854090684343545452440 Output: Magma V2.11-10 Wed Dec 14 2005 11:20:12 on modular [Seed = 3728595848] ------------------------------------- 7102277187569652249869562285153253710645556161842592536300347670222640516361186\ 363486317189963554176580264215037288238394086254424175854090684343545452440 Total time: 0.190 seconds, Total memory usage: 3.24MB '129.20.' ************** MAGMA ***************** Host 129.20.36.132 (129.20.36.132) Time: Wed Dec 14 10:32:34 2005 Input: K:=CyclotomicField(13); G,phi:=UnitGroup(K); G; Output: Magma V2.11-10 Wed Dec 14 2005 10:32:33 on modular [Seed = 301099370] ------------------------------------- Abelian Group isomorphic to Z/26 + Z (5 copies) Defined on 6 generators Relations: 26*G.1 = 0 Total time: 1.260 seconds, Total memory usage: 3.82MB '67.62.1' ************** MAGMA ***************** Host 67.62.112.123 (67.62.112.123) Time: Wed Dec 14 07:16:04 2005 Input: a:=1; a: Output: Magma V2.11-10 Wed Dec 14 2005 07:16:03 on modular [Seed = 3442377647] ------------------------------------- >> a:; ^ User error: bad syntax Total time: 0.190 seconds, Total memory usage: 3.24MB '83.250.' ************** MAGMA ***************** Host 83.250.129.152 (83.250.129.152) Time: Wed Dec 14 06:40:51 2005 Input: 5*5 Output: Magma V2.11-10 Wed Dec 14 2005 06:40:49 on modular [Seed = 717843850] ------------------------------------- 25 Total time: 0.260 seconds, Total memory usage: 3.24MB '217.24.' ************** MAGMA ***************** Host 217.24.144.35 (217.24.144.35) Time: Wed Dec 14 06:30:58 2005 Input: jnbkjnlml Output: Magma V2.11-10 Wed Dec 14 2005 06:30:57 on modular [Seed = 1972778086] ------------------------------------- >> jnbkjnlml; ^ User error: Identifier 'jnbkjnlml' has not been declared or assigned Total time: 0.200 seconds, Total memory usage: 3.24MB '65.191.' ************** MAGMA ***************** Host 65.191.73.113 (65.191.73.113) Time: Tue Dec 13 22:39:18 2005 Input: K := FiniteField(2); > C := LinearCode; M:=VerticalJoin(GeneratorMatrix(C), ParityCheckMatrix(C)); M1:=EchelonForm(M); M2:=Submatrix(M1,22,22,14,35); M2; C2:=LinearCode(M2); WeightDistribution(C2); C2; L:=MinimumWords(C2); C3:= LinearCode; C3; C4:=EvenWeightCode(15); WeightDistribution(C4); Output: Magma V2.11-10 Tue Dec 13 2005 22:39:17 on modular [Seed = 670824447] ------------------------------------- [1 0 0 0 0 0 0 0 0 0 1 0 0 1 0 1 0 1 1 1 0 0 0 0 1 1 1 0 0 1 0 0 0 1 0] [0 1 0 0 0 0 0 0 0 0 1 0 1 0 0 1 0 0 1 1 0 0 0 0 0 1 1 1 0 1 1 1 1 0 1] [0 0 1 0 0 0 0 0 0 0 0 1 0 1 0 0 1 0 0 1 1 0 0 0 1 1 0 1 0 0 0 1 1 0 1] [0 0 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 1 1 0 0 0 1 0 0 0 1 0 0 0 1 1 0] [0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 1 1 1 1 0 0 0 0 0 0 1 0 0 1 0 1 1 1 0 0] [0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 1 1 1 1 0 0 0 0 0 0 1 0 0 1 0 1 1 1 0] [0 0 0 0 0 0 1 0 0 0 1 1 1 1 0 1 1 1 1 1 0 0 0 0 0 0 0 1 0 1 1 1 0 1 1] [0 0 0 0 0 0 0 1 0 0 1 1 1 0 0 1 1 0 0 0 1 0 0 0 0 0 0 0 0 1 0 1 1 0 0] [0 0 0 0 0 0 0 0 1 0 0 1 1 1 0 1 0 0 1 1 1 0 0 0 0 0 0 0 1 1 0 1 0 0 1] [0 0 0 0 0 0 0 0 0 1 1 1 0 1 0 0 1 1 1 0 1 0 0 0 0 0 0 0 0 1 0 0 1 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 0 0 0 0 0 0 0 1 1 1 1 1 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 1 0 1 0 1 0 0 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 1 1 1 0 1 0 0 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 1 0 1 0 0 1 1 1] [ <0, 1>, <8, 14>, <10, 182>, <12, 805>, <14, 2200>, <16, 3885>, <18, 4340>, <20, 3066>, <22, 1400>, <24, 420>, <26, 70>, <28, 1> ] [35, 14, 8] Linear Code over GF(2) Generator matrix: [1 0 0 0 0 0 0 0 0 0 1 0 0 1 0 1 0 1 1 1 0 0 0 0 1 1 1 0 0 1 0 0 0 1 0] [0 1 0 0 0 0 0 0 0 0 1 0 1 0 0 1 0 0 1 1 0 0 0 0 0 1 1 1 0 1 1 1 1 0 1] [0 0 1 0 0 0 0 0 0 0 0 1 0 1 0 0 1 0 0 1 1 0 0 0 1 1 0 1 0 0 0 1 1 0 1] [0 0 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 1 1 0 0 0 1 0 0 0 1 0 0 0 1 1 0] [0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 1 1 1 1 0 0 0 0 0 0 1 0 0 1 0 1 1 1 0 0] [0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 1 1 1 1 0 0 0 0 0 0 1 0 0 1 0 1 1 1 0] [0 0 0 0 0 0 1 0 0 0 1 1 1 1 0 1 1 1 1 1 0 0 0 0 0 0 0 1 0 1 1 1 0 1 1] [0 0 0 0 0 0 0 1 0 0 1 1 1 0 0 1 1 0 0 0 1 0 0 0 0 0 0 0 0 1 0 1 1 0 0] [0 0 0 0 0 0 0 0 1 0 0 1 1 1 0 1 0 0 1 1 1 0 0 0 0 0 0 0 1 1 0 1 0 0 1] [0 0 0 0 0 0 0 0 0 1 1 1 0 1 0 0 1 1 1 0 1 0 0 0 0 0 0 0 0 1 0 0 1 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 0 0 0 0 0 0 0 1 1 1 1 1 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 1 0 1 0 1 0 0 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 1 1 1 0 1 0 0 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 1 0 1 0 0 1 1 1] [35, 10, 8] Linear Code over GF(2) Generator matrix: [1 0 0 0 0 0 1 0 1 1 1 1 0 0 0 1 0 1 0 1 0 0 0 0 1 1 1 1 1 0 1 0 1 1 1] [0 1 0 0 0 0 1 0 1 0 0 0 1 0 0 1 1 1 1 1 1 0 0 0 0 1 1 0 1 1 0 1 1 1 1] [0 0 1 0 0 0 1 0 1 0 1 1 0 1 0 0 0 1 0 1 0 0 0 0 1 1 0 0 1 0 1 1 1 1 1] [0 0 0 1 0 0 1 1 1 0 1 0 1 0 0 1 0 1 1 1 1 0 0 0 1 0 0 1 0 1 1 1 0 0 0] [0 0 0 0 1 0 1 0 0 0 1 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 1 1 1 0 0 1 1 1] [0 0 0 0 0 1 1 1 1 1 1 0 0 0 0 1 0 1 0 1 1 0 0 0 0 0 1 1 1 1 1 0 1 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 0 0 0 0 0 0 0 1 1 1 1 1 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 1 0 1 0 1 0 0 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 1 1 1 0 1 0 0 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 1 0 1 0 0 1 1 1] [ <0, 1>, <2, 105>, <4, 1365>, <6, 5005>, <8, 6435>, <10, 3003>, <12, 455>, <14, 15> ] Total time: 0.190 seconds, Total memory usage: 3.34MB '65.191.' ************** MAGMA ***************** Host 65.191.73.113 (65.191.73.113) Time: Tue Dec 13 22:23:21 2005 Input: K := FiniteField(2); > C := LinearCode; M:=VerticalJoin(GeneratorMatrix(C), ParityCheckMatrix(C)); M1:=EchelonForm(M); M2:=Submatrix(M1,22,22,14,35); M2; C2:=LinearCode(M2); WeightDistribution(C2); C2; L:=MinimumWords(C2); C3:= LinearCode; C3; Output: Magma V2.11-10 Tue Dec 13 2005 22:23:21 on modular [Seed = 1038216745] ------------------------------------- [1 0 0 0 0 0 0 0 0 0 1 0 0 1 0 1 0 1 1 1 0 0 0 0 1 1 1 0 0 1 0 0 0 1 0] [0 1 0 0 0 0 0 0 0 0 1 0 1 0 0 1 0 0 1 1 0 0 0 0 0 1 1 1 0 1 1 1 1 0 1] [0 0 1 0 0 0 0 0 0 0 0 1 0 1 0 0 1 0 0 1 1 0 0 0 1 1 0 1 0 0 0 1 1 0 1] [0 0 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 1 1 0 0 0 1 0 0 0 1 0 0 0 1 1 0] [0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 1 1 1 1 0 0 0 0 0 0 1 0 0 1 0 1 1 1 0 0] [0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 1 1 1 1 0 0 0 0 0 0 1 0 0 1 0 1 1 1 0] [0 0 0 0 0 0 1 0 0 0 1 1 1 1 0 1 1 1 1 1 0 0 0 0 0 0 0 1 0 1 1 1 0 1 1] [0 0 0 0 0 0 0 1 0 0 1 1 1 0 0 1 1 0 0 0 1 0 0 0 0 0 0 0 0 1 0 1 1 0 0] [0 0 0 0 0 0 0 0 1 0 0 1 1 1 0 1 0 0 1 1 1 0 0 0 0 0 0 0 1 1 0 1 0 0 1] [0 0 0 0 0 0 0 0 0 1 1 1 0 1 0 0 1 1 1 0 1 0 0 0 0 0 0 0 0 1 0 0 1 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 0 0 0 0 0 0 0 1 1 1 1 1 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 1 0 1 0 1 0 0 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 1 1 1 0 1 0 0 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 1 0 1 0 0 1 1 1] [ <0, 1>, <8, 14>, <10, 182>, <12, 805>, <14, 2200>, <16, 3885>, <18, 4340>, <20, 3066>, <22, 1400>, <24, 420>, <26, 70>, <28, 1> ] [35, 14, 8] Linear Code over GF(2) Generator matrix: [1 0 0 0 0 0 0 0 0 0 1 0 0 1 0 1 0 1 1 1 0 0 0 0 1 1 1 0 0 1 0 0 0 1 0] [0 1 0 0 0 0 0 0 0 0 1 0 1 0 0 1 0 0 1 1 0 0 0 0 0 1 1 1 0 1 1 1 1 0 1] [0 0 1 0 0 0 0 0 0 0 0 1 0 1 0 0 1 0 0 1 1 0 0 0 1 1 0 1 0 0 0 1 1 0 1] [0 0 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 1 1 0 0 0 1 0 0 0 1 0 0 0 1 1 0] [0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 1 1 1 1 0 0 0 0 0 0 1 0 0 1 0 1 1 1 0 0] [0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 1 1 1 1 0 0 0 0 0 0 1 0 0 1 0 1 1 1 0] [0 0 0 0 0 0 1 0 0 0 1 1 1 1 0 1 1 1 1 1 0 0 0 0 0 0 0 1 0 1 1 1 0 1 1] [0 0 0 0 0 0 0 1 0 0 1 1 1 0 0 1 1 0 0 0 1 0 0 0 0 0 0 0 0 1 0 1 1 0 0] [0 0 0 0 0 0 0 0 1 0 0 1 1 1 0 1 0 0 1 1 1 0 0 0 0 0 0 0 1 1 0 1 0 0 1] [0 0 0 0 0 0 0 0 0 1 1 1 0 1 0 0 1 1 1 0 1 0 0 0 0 0 0 0 0 1 0 0 1 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 0 0 0 0 0 0 0 1 1 1 1 1 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 1 0 1 0 1 0 0 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 1 1 1 0 1 0 0 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 1 0 1 0 0 1 1 1] [35, 10, 8] Linear Code over GF(2) Generator matrix: [1 0 0 0 0 0 1 0 1 1 1 1 0 0 0 1 0 1 0 1 0 0 0 0 1 1 1 1 1 0 1 0 1 1 1] [0 1 0 0 0 0 1 0 1 0 0 0 1 0 0 1 1 1 1 1 1 0 0 0 0 1 1 0 1 1 0 1 1 1 1] [0 0 1 0 0 0 1 0 1 0 1 1 0 1 0 0 0 1 0 1 0 0 0 0 1 1 0 0 1 0 1 1 1 1 1] [0 0 0 1 0 0 1 1 1 0 1 0 1 0 0 1 0 1 1 1 1 0 0 0 1 0 0 1 0 1 1 1 0 0 0] [0 0 0 0 1 0 1 0 0 0 1 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 1 1 1 0 0 1 1 1] [0 0 0 0 0 1 1 1 1 1 1 0 0 0 0 1 0 1 0 1 1 0 0 0 0 0 1 1 1 1 1 0 1 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 0 0 0 0 0 0 0 1 1 1 1 1 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 1 0 1 0 1 0 0 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 1 1 1 0 1 0 0 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 1 0 1 0 0 1 1 1] Total time: 0.190 seconds, Total memory usage: 3.34MB '65.191.' ************** MAGMA ***************** Host 65.191.73.113 (65.191.73.113) Time: Tue Dec 13 22:20:16 2005 Input: K := FiniteField(2); > C := LinearCode; M:=VerticalJoin(GeneratorMatrix(C), ParityCheckMatrix(C)); M1:=EchelonForm(M); M2:=Submatrix(M1,22,22,14,35); M2; C2:=LinearCode(M2); WeightDistribution(C2); C2; L:=MinimumWords(C2); C3:= LinearCode; C3; Output: Magma V2.11-10 Tue Dec 13 2005 22:20:16 on modular [Seed = 3323595400] ------------------------------------- [1 0 0 0 0 0 0 0 0 0 1 0 0 1 0 1 0 1 1 1 0 0 0 0 1 1 1 0 0 1 0 0 0 1 0] [0 1 0 0 0 0 0 0 0 0 1 0 1 0 0 1 0 0 1 1 0 0 0 0 0 1 1 1 0 1 1 1 1 0 1] [0 0 1 0 0 0 0 0 0 0 0 1 0 1 0 0 1 0 0 1 1 0 0 0 1 1 0 1 0 0 0 1 1 0 1] [0 0 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 1 1 0 0 0 1 0 0 0 1 0 0 0 1 1 0] [0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 1 1 1 1 0 0 0 0 0 0 1 0 0 1 0 1 1 1 0 0] [0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 1 1 1 1 0 0 0 0 0 0 1 0 0 1 0 1 1 1 0] [0 0 0 0 0 0 1 0 0 0 1 1 1 1 0 1 1 1 1 1 0 0 0 0 0 0 0 1 0 1 1 1 0 1 1] [0 0 0 0 0 0 0 1 0 0 1 1 1 0 0 1 1 0 0 0 1 0 0 0 0 0 0 0 0 1 0 1 1 0 0] [0 0 0 0 0 0 0 0 1 0 0 1 1 1 0 1 0 0 1 1 1 0 0 0 0 0 0 0 1 1 0 1 0 0 1] [0 0 0 0 0 0 0 0 0 1 1 1 0 1 0 0 1 1 1 0 1 0 0 0 0 0 0 0 0 1 0 0 1 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 0 0 0 0 0 0 0 1 1 1 1 1 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 1 0 1 0 1 0 0 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 1 1 1 0 1 0 0 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 1 0 1 0 0 1 1 1] [ <0, 1>, <8, 14>, <10, 182>, <12, 805>, <14, 2200>, <16, 3885>, <18, 4340>, <20, 3066>, <22, 1400>, <24, 420>, <26, 70>, <28, 1> ] [35, 14, 8] Linear Code over GF(2) Generator matrix: [1 0 0 0 0 0 0 0 0 0 1 0 0 1 0 1 0 1 1 1 0 0 0 0 1 1 1 0 0 1 0 0 0 1 0] [0 1 0 0 0 0 0 0 0 0 1 0 1 0 0 1 0 0 1 1 0 0 0 0 0 1 1 1 0 1 1 1 1 0 1] [0 0 1 0 0 0 0 0 0 0 0 1 0 1 0 0 1 0 0 1 1 0 0 0 1 1 0 1 0 0 0 1 1 0 1] [0 0 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 1 1 0 0 0 1 0 0 0 1 0 0 0 1 1 0] [0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 1 1 1 1 0 0 0 0 0 0 1 0 0 1 0 1 1 1 0 0] [0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 1 1 1 1 0 0 0 0 0 0 1 0 0 1 0 1 1 1 0] [0 0 0 0 0 0 1 0 0 0 1 1 1 1 0 1 1 1 1 1 0 0 0 0 0 0 0 1 0 1 1 1 0 1 1] [0 0 0 0 0 0 0 1 0 0 1 1 1 0 0 1 1 0 0 0 1 0 0 0 0 0 0 0 0 1 0 1 1 0 0] [0 0 0 0 0 0 0 0 1 0 0 1 1 1 0 1 0 0 1 1 1 0 0 0 0 0 0 0 1 1 0 1 0 0 1] [0 0 0 0 0 0 0 0 0 1 1 1 0 1 0 0 1 1 1 0 1 0 0 0 0 0 0 0 0 1 0 0 1 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 0 0 0 0 0 0 0 1 1 1 1 1 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 1 0 1 0 1 0 0 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 1 1 1 0 1 0 0 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 1 0 1 0 0 1 1 1] >> C3:= LinearCode: Rhs argument 1 is invalid for this constructor >> C3;; ^ User error: Identifier 'C3' has not been declared or assigned Total time: 0.190 seconds, Total memory usage: 3.34MB '200.177' ************** MAGMA ***************** Host 200.177.46.74 (200.177.46.74) Time: Tue Dec 13 22:05:47 2005 Input: p := 1461501624496790265145448589920785493717258890819; n := 1461501624496790265145447380994971188499300027613; Fp := GF(p); b := Fp!0; repeat repeat b := b + 1; until IsSquare(b + 1); y := Root(b + 1, 2); E := EllipticCurve([Fp | 0, b]); G := E![1, y]; until IsZero(n*G); "b =", b; "G =", G; rho := (p - n)^2; "rho =", rho; "rho^6 mod n =", rho^6 mod n; H := rho*G; "H =", H; zeta := H[1]; "zeta =", zeta; "zeta^3 =", zeta^3; lambda := Fp!2; while IsPower(lambda, 3) do lambda +:= 1; end while; "lambda =", lambda; Fp2 := ExtensionField; mu := i + 1; while IsSquare(mu) do mu +:= 1; end while; "mu =", mu; xi := 1/(lambda^2*mu^3); Fp12 := ExtensionField; Et := EllipticCurve([Fp2 | 0, b/xi]); xt := 1; while not IsSquare(xt^3 + b/xi) do xt +:= 1; end while; yt := Sqrt(xt^3 + b/xi); Gt := Et![xt, yt]; "Gt =", Gt; k := 12; assert (p^k - 1) mod n eq 0; chi := (p^k - 1) div n; Ek := EllipticCurve([Fp12 | 0, Fp12!b]); g := function(U, V, Q) if IsZero(U) or IsZero(V) or (U eq -V) or IsZero(Q) then return Fp12!1; end if; /* m := (U eq V) select 3*U[1]^2/(2*U[2]) else (V[2] - U[2])/(V[1] - U[1]); return m*(Q[1] - U[1]) + U[2] - Q[2]; */ if U eq V then m := 3*U[1]^2; s := 2*U[2]; else m := V[2] - U[2]; s := V[1] - U[1]; end if; return m*(Q[1] - U[1]) + s*(U[2] - Q[2]); end function; length := function(r) n := 0; v := 1; while v le r do n +:= 1; v +:= v; end while; return n; end function; bit := function(r, i) return (r div 2^i) mod 2; end function; tate := function(P, Qt) Q := Ek![Qt[1]*z^2, Qt[2]*z^3]; f := 1; A := P; for i := length(n) - 2 to 0 by -1 do f := f^2*g(A, A, Q); A := 2*A; if bit(n, i) eq 1 then f *:= g(A, P, Q); A +:= P; end if; end for; return f^chi; end function; g := tate(G, Gt); "g =", g; /* for j in [1..11] do print "g^(p^" * Sprint(j) * "-1): " * Sprint(g^(p^j-1) ne 1); end for; print "g^(p^12-1): " * Sprint(g^(p^12-1) eq 1); "g^n =", g^n; */ /* for j in [1..100] do "--", j; u := Random(n-1); v := Random(n-1); w := tate(u*G, v*Gt); h := g^((u*v) mod n); if w ne h then "Failure!"; print "u = " * Sprint(u); print "v = " * Sprint(v); print "e(u*P, v*Q) = " * Sprint(w); print "e(P, Q)^(u*v) = " * Sprint(h); quit; end if; end for; "Success!"; */ trace4 := function(g) t := g^p; t ^:= p; t ^:= p; t ^:= p; // g^(p^4) s := t^p; s ^:= p; s ^:= p; s ^:= p; // g^(p^8) return g + t + s; end function; trace6 := function(g) t := g^p; t ^:= p; t ^:= p; t ^:= p; t ^:= p; t ^:= p; // g^(p^6) return g + t; end function; conj4 := function(g) s := Eltseq(g); s[4] := -s[4]; s[5] := -s[5]; s[6] := -s[6]; return Seqelt(s, Fp12); end function; t40 := trace4(g); t41 := conj4(t40); "trace4(g) =", Eltseq(t40); "trace4(g)' =", Eltseq(t41); "sum4 =", t40 + t41; "prod4 =", t40*t41; Output: Magma V2.11-10 Tue Dec 13 2005 22:05:46 on modular [Seed = 4246811337] ------------------------------------- b = 3 G = (1 : 1461501624496790265145448589920785493717258890817 : 1) rho = 1461501624493534334825397658811989710051820598436 rho^6 mod n = 1 H = (1627965160026674480212199743920457792 : 2 : 1) zeta = 1627965160026674480212199743920457792 zeta^3 = 1 lambda = 2 mu = i + 1 Gt = (8 : 816263181872116351510202985179226587277470764815*i + 295865244505705705023665406736615173923424579851 : 1) g = (1113252570408097904801784205204725186435037650621*i + 828892838102560531997994710291403383901103286823)*z^5 + (78288732951012207123651737359843385303103429616*i + 1421790144970157374993435834657786699116722047068)*z^4 + (1334693519950620886403708174450030523556565932446*i + 65995713479010101585078190425689016349129829691)*z^3 + (1151508433446340267838152948068030563616071711737*i + 1331505104782800437119636772308446788364866291823)*z^2 + (1167306608241793402349182741773283684304074553579*i + 813404015744428633410120092529896632595508931142)*z + 800403695404043311146437378782593638877310314601*i + 901249395544166733264846949825665528694219338474 trace4(g) = [ 939709461715339668293863546426995422914672052984*i + 1242246562135709934649092259556211092365399124603, 0, 0, 1081077310858282128920227343508520583235180015700*i + 197987140437030304755234571277067049047389489073, 0, 0 ] trace4(g)' = [ 939709461715339668293863546426995422914672052984*i + 1242246562135709934649092259556211092365399124603, 0, 0, 380424313638508136225221246412264910482078875119*i + 1263514484059759960390214018643718444669869401746, 0, 0 ] sum4 = 417917298933889071442278502933205352112085215149*i + 1022991499774629604152735929191636691013539358387 prod4 = 1290910566114678872655078479583819895277125832347*i + 1222447203379043640445950747036893708432248781076 Total time: 0.870 seconds, Total memory usage: 3.34MB '200.177' ************** MAGMA ***************** Host 200.177.46.74 (200.177.46.74) Time: Tue Dec 13 22:02:17 2005 Input: p := 1461501624496790265145448589920785493717258890819; n := 1461501624496790265145447380994971188499300027613; Fp := GF(p); b := Fp!0; repeat repeat b := b + 1; until IsSquare(b + 1); y := Root(b + 1, 2); E := EllipticCurve([Fp | 0, b]); G := E![1, y]; until IsZero(n*G); "b =", b; "G =", G; rho := (p - n)^2; "rho =", rho; "rho^6 mod n =", rho^6 mod n; H := rho*G; "H =", H; zeta := H[1]; "zeta =", zeta; "zeta^3 =", zeta^3; lambda := Fp!2; while IsPower(lambda, 3) do lambda +:= 1; end while; "lambda =", lambda; Fp2 := ExtensionField; mu := i + 1; while IsSquare(mu) do mu +:= 1; end while; "mu =", mu; xi := 1/(lambda^2*mu^3); Fp12 := ExtensionField; Et := EllipticCurve([Fp2 | 0, b/xi]); xt := 1; while not IsSquare(xt^3 + b/xi) do xt +:= 1; end while; yt := Sqrt(xt^3 + b/xi); Gt := Et![xt, yt]; "Gt =", Gt; k := 12; assert (p^k - 1) mod n eq 0; chi := (p^k - 1) div n; Ek := EllipticCurve([Fp12 | 0, Fp12!b]); g := function(U, V, Q) if IsZero(U) or IsZero(V) or (U eq -V) or IsZero(Q) then return Fp12!1; end if; /* m := (U eq V) select 3*U[1]^2/(2*U[2]) else (V[2] - U[2])/(V[1] - U[1]); return m*(Q[1] - U[1]) + U[2] - Q[2]; */ if U eq V then m := 3*U[1]^2; s := 2*U[2]; else m := V[2] - U[2]; s := V[1] - U[1]; end if; return m*(Q[1] - U[1]) + s*(U[2] - Q[2]); end function; length := function(r) n := 0; v := 1; while v le r do n +:= 1; v +:= v; end while; return n; end function; bit := function(r, i) return (r div 2^i) mod 2; end function; tate := function(P, Qt) Q := Ek![Qt[1]*z^2, Qt[2]*z^3]; f := 1; A := P; for i := length(n) - 2 to 0 by -1 do f := f^2*g(A, A, Q); A := 2*A; if bit(n, i) eq 1 then f *:= g(A, P, Q); A +:= P; end if; end for; return f^chi; end function; g := tate(G, Gt); "g =", g; /* for j in [1..11] do print "g^(p^" * Sprint(j) * "-1): " * Sprint(g^(p^j-1) ne 1); end for; print "g^(p^12-1): " * Sprint(g^(p^12-1) eq 1); "g^n =", g^n; */ /* for j in [1..100] do "--", j; u := Random(n-1); v := Random(n-1); w := tate(u*G, v*Gt); h := g^((u*v) mod n); if w ne h then "Failure!"; print "u = " * Sprint(u); print "v = " * Sprint(v); print "e(u*P, v*Q) = " * Sprint(w); print "e(P, Q)^(u*v) = " * Sprint(h); quit; end if; end for; "Success!"; */ trace4 := function(g) t := g^p; t ^:= p; t ^:= p; t ^:= p; // g^(p^4) s := t^p; s ^:= p; s ^:= p; s ^:= p; // g^(p^8) return g + t + s; end function; trace6 := function(g) t := g^p; t ^:= p; t ^:= p; t ^:= p; t ^:= p; t ^:= p; // g^(p^6) return g + t; end function; conj6 := function(g) s := Eltseq(g); s[4] := -s[4]; s[5] := -s[5]; s[6] := -s[6]; return Seqelt(s, Fp12); end function; t40 := trace4(g); t41 := t40^p; t42 := t41^p; "trace4(g) =", Eltseq(t40); "trace4(g)' =", Eltseq(t41); "trace4(g)'' =", Eltseq(t42); "sum4 =", t40 + t41 + t42; "prod4 =", t40*t41*t42; t60 := trace6(g); t61 := conj6(t60); "trace6(g) =", t60; "trace6(g)' =", t61; "sum6 =", t60 + t61; "prod6 =", t60*t61; Output: Magma V2.11-10 Tue Dec 13 2005 22:02:16 on modular [Seed = 2204060549] ------------------------------------- b = 3 G = (1 : 1461501624496790265145448589920785493717258890817 : 1) rho = 1461501624493534334825397658811989710051820598436 rho^6 mod n = 1 H = (1627965160026674480212199743920457792 : 2 : 1) zeta = 1627965160026674480212199743920457792 zeta^3 = 1 lambda = 2 mu = i + 1 Gt = (8 : 816263181872116351510202985179226587277470764815*i + 295865244505705705023665406736615173923424579851 : 1) g = (1113252570408097904801784205204725186435037650621*i + 828892838102560531997994710291403383901103286823)*z^5 + (78288732951012207123651737359843385303103429616*i + 1421790144970157374993435834657786699116722047068)*z^4 + (1334693519950620886403708174450030523556565932446*i + 65995713479010101585078190425689016349129829691)*z^3 + (1151508433446340267838152948068030563616071711737*i + 1331505104782800437119636772308446788364866291823)*z^2 + (1167306608241793402349182741773283684304074553579*i + 813404015744428633410120092529896632595508931142)*z + 800403695404043311146437378782593638877310314601*i + 901249395544166733264846949825665528694219338474 trace4(g) = [ 939709461715339668293863546426995422914672052984*i + 1242246562135709934649092259556211092365399124603, 0, 0, 1081077310858282128920227343508520583235180015700*i + 197987140437030304755234571277067049047389489073, 0, 0 ] trace4(g)' = [ 521792162781450596851585043493790070802586837835*i + 1242246562135709934649092259556211092365399124603, 0, 0, 600370227937686771113197478282678333920097663482*i + 1221575564612301174446849985825970354859774772881, 0, 0 ] trace4(g)'' = [ 939709461715339668293863546426995422914672052984*i + 1242246562135709934649092259556211092365399124603, 0, 0, 380424313638508136225221246412264910482078875119*i + 1263514484059759960390214018643718444669869401746, 0, 0 ] sum4 = (600370227937686771113197478282678333920097663482*i + 1221575564612301174446849985825970354859774772881)*z^3 + 939709461715339668293863546426995422914672052984*i + 803736437413549273656379598827062289661679592171 prod4 = (351960067217720226471026218912124823215388842998*i + 636471885525581132299463326642541502123540418918)*z^3 + 1073254876702081306986584990089982603797367449521*i + 1088939299807643406655341764511290451420861288604 trace6(g) = (156577465902024414247303474719686770606206859232*i + 1382078665443524484841423079394787904516185203317)*z^4 + (841515242395890270530857306215275633514884532655*i + 1201508585068810609093824954696108083012473692827)*z^2 + 139305766311296357147426167644401784037361738383*i + 340997166591543201384245309730545563671179786129 trace6(g)' = (1304924158594765850898145115201098723111052031587*i + 79422959053265780304025510525997589201073687502)*z^4 + (841515242395890270530857306215275633514884532655*i + 1201508585068810609093824954696108083012473692827)*z^2 + 139305766311296357147426167644401784037361738383*i + 340997166591543201384245309730545563671179786129 sum6 = (221528860294990275916266022509765773312510174491*i + 941515545640830953042201319471430672307688494835)*z^2 + 278611532622592714294852335288803568074723476766*i + 681994333183086402768490619461091127342359572258 prod6 = (1199217906511468853397684229168281220558780854227*i + 1030765899693116805073366946226732872257120333608)*z^4 + (378575975826376771203186488360822049552104323606*i + 472503248255084275151742862775344953200669638852)*z^2 + 230957764124266000451333381176639619321109453643*i + 684491156455614923026389854551594178711792858992 Total time: 1.100 seconds, Total memory usage: 3.34MB '72.19.1' ************** MAGMA ***************** Host 72.19.126.33 (72.19.126.33) Time: Tue Dec 13 21:59:13 2005 Input: F := FunctionField(GF(5,2)); A := AffineAlgebra; P := PolynomialRing(F); y^-1; > MinimalPolynomial(w); Output: ** WARNING: Computation time exceeded 20 seconds, so computation was terminated after 20 seconds. ** Magma V2.11-10 Tue Dec 13 2005 21:58:53 on modular [Seed = 1134529040] ------------------------------------- Errors: /bin/sh: line 1: 32125 Alarm clock nice -n 19 /usr/local/bin/magma '72.19.1' ************** MAGMA ***************** Host 72.19.126.33 (72.19.126.33) Time: Tue Dec 13 21:58:26 2005 Input: F := FunctionField(GF(5,2)); A := AffineAlgebra; P := PolynomialRing(F); y^-1; > MinimalPolynomial(w); Output: Magma V2.11-10 Tue Dec 13 2005 21:58:25 on modular [Seed = 1184534313] ------------------------------------- 1/t^17*w^100 + (4*t^4 + 4)/t^17*w^80 + 4/t^13*w^76 + (t^8 + 2*t^4 + 1)/t^17*w^60 + (2*t^4 + 2)/t^13*w^56 + 1/t^9*w^52 + (4*t^12 + 2*t^8 + 2*t^4 + 4)/t^17*w^40 + (2*t^8 + 4*t^4 + 2)/t^13*w^36 + (2*t^4 + 2)/t^9*w^32 + 4/t^5*w^28 + (t^16 + 4*t^12 + t^8 + 4*t^4 + 1)/t^17*w^20 + (4*t^12 + 2*t^8 + 2*t^4 + 4)/t^13*w^16 + (t^8 + 2*t^4 + 1)/t^9*w^12 + (4*t^4 + 4)/t^5*w^8 + 1/t*w^4 + 1/t z^125 + (t^20 + t^16 + 1)*z^25 + (2*t^20 + t^16)*z^5 + t^20*z + 4*t^21 Total time: 0.200 seconds, Total memory usage: 3.72MB '72.19.1' ************** MAGMA ***************** Host 72.19.126.33 (72.19.126.33) Time: Tue Dec 13 21:58:12 2005 Input: F := FunctionField(GF(5,2)); A := AffineAlgebra; P := PolynomialRing(F); y^-1; > MinimalPolynomial(w); Output: ** WARNING: Computation time exceeded 20 seconds, so computation was terminated after 20 seconds. ** Magma V2.11-10 Tue Dec 13 2005 21:57:52 on modular [Seed = 1284539714] ------------------------------------- Errors: /bin/sh: line 1: 32114 Alarm clock nice -n 19 /usr/local/bin/magma '72.19.1' ************** MAGMA ***************** Host 72.19.126.33 (72.19.126.33) Time: Tue Dec 13 21:56:51 2005 Input: F := FunctionField(GF(5,2)); A := AffineAlgebra; P := PolynomialRing(F); y^-1; > MinimalPolynomial(w); Output: Magma V2.11-10 Tue Dec 13 2005 21:56:51 on modular [Seed = 1401919902] ------------------------------------- 1/t^17*w^100 + (4*t^4 + 4)/t^17*w^80 + 4/t^13*w^76 + (t^8 + 2*t^4 + 1)/t^17*w^60 + (2*t^4 + 2)/t^13*w^56 + 1/t^9*w^52 + (4*t^12 + 2*t^8 + 2*t^4 + 4)/t^17*w^40 + (2*t^8 + 4*t^4 + 2)/t^13*w^36 + (2*t^4 + 2)/t^9*w^32 + 4/t^5*w^28 + (t^16 + 4*t^12 + t^8 + 4*t^4 + 1)/t^17*w^20 + (4*t^12 + 2*t^8 + 2*t^4 + 4)/t^13*w^16 + (t^8 + 2*t^4 + 1)/t^9*w^12 + (4*t^4 + 4)/t^5*w^8 + 1/t*w^4 + 1/t z^125 + (t^20 + t^16 + 1)*z^25 + (2*t^20 + t^16)*z^5 + t^20*z + 4*t^21 Total time: 0.210 seconds, Total memory usage: 3.72MB '72.19.1' ************** MAGMA ***************** Host 72.19.126.33 (72.19.126.33) Time: Tue Dec 13 21:56:42 2005 Input: F := FunctionField(GF(5,2)); A := AffineAlgebra; P := PolynomialRing(F); y^-1; > MinimalPolynomial(w); Output: Magma V2.11-10 Tue Dec 13 2005 21:56:42 on modular [Seed = 1451924610] ------------------------------------- t^3*w^100 + 3*t^3*w^80 + 4*t^3*w^76 + 4*t^3*w^60 + 4*t^3*w^56 + t^3*w^52 + 2*t^3*w^40 + 3*t^3*w^36 + 4*t^3*w^32 + 4*t^3*w^28 + t^3*w^20 + 2*t^3*w^16 + 4*t^3*w^12 + 3*t^3*w^8 + t^3*w^4 + 1/t z^125 + (2*t^4 + 1)/t^4*z^25 + (t^4 + 2)/t^4*z^5 + 1/t^4*z + 4/t^4 Total time: 0.200 seconds, Total memory usage: 3.82MB '72.19.1' ************** MAGMA ***************** Host 72.19.126.33 (72.19.126.33) Time: Tue Dec 13 21:56:18 2005 Input: F := FunctionField(GF(5,2)); A := AffineAlgebra; P := PolynomialRing(F); y^-1; > MinimalPolynomial(x); Output: Magma V2.11-10 Tue Dec 13 2005 21:56:18 on modular [Seed = 1568774065] ------------------------------------- t^3*w^100 + 3*t^3*w^80 + 4*t^3*w^76 + 4*t^3*w^60 + 4*t^3*w^56 + t^3*w^52 + 2*t^3*w^40 + 3*t^3*w^36 + 4*t^3*w^32 + 4*t^3*w^28 + t^3*w^20 + 2*t^3*w^16 + 4*t^3*w^12 + 3*t^3*w^8 + t^3*w^4 + 1/t z^25 + (t^4 + 1)/t^4*z^5 + 1/t^4*z + 4/t^4 Total time: 0.200 seconds, Total memory usage: 3.53MB '72.19.1' ************** MAGMA ***************** Host 72.19.126.33 (72.19.126.33) Time: Tue Dec 13 21:56:15 2005 Input: F := FunctionField(GF(5,2)); A := AffineAlgebra; P := PolynomialRing(F); y^-1; > MinimalPolynomial(x); Output: Magma V2.11-10 Tue Dec 13 2005 21:56:15 on modular [Seed = 1622977169] ------------------------------------- t^3*w^100 + 3*t^3*w^80 + 4*t^3*w^76 + 4*t^3*w^60 + 4*t^3*w^56 + t^3*w^52 + 2*t^3*w^40 + 3*t^3*w^36 + 4*t^3*w^32 + 4*t^3*w^28 + t^3*w^20 + 2*t^3*w^16 + 4*t^3*w^12 + 3*t^3*w^8 + t^3*w^4 + 1/t z^25 + (t^4 + 1)/t^4*z^5 + 1/t^4*z + 4/t^4 Total time: 0.200 seconds, Total memory usage: 3.53MB '72.19.1' ************** MAGMA ***************** Host 72.19.126.33 (72.19.126.33) Time: Tue Dec 13 21:56:01 2005 Input: F := FunctionField(GF(5,2)); A := AffineAlgebra; P := PolynomialRing(F); y^-1; > MinimalPolynomial(x); Output: Magma V2.11-10 Tue Dec 13 2005 21:56:00 on modular [Seed = 1740352921] ------------------------------------- 1/t*y^4 + 1/t z^25 + (t^4 + 1)/t^4*z^5 + 1/t^4*z + 4/t^4 Total time: 0.190 seconds, Total memory usage: 3.34MB '72.19.1' ************** MAGMA ***************** Host 72.19.126.33 (72.19.126.33) Time: Tue Dec 13 21:55:53 2005 Input: F := FunctionField(GF(5,2)); A := AffineAlgebra; P := PolynomialRing(F); y^-1; > MinimalPolynomial(x); Output: Magma V2.11-10 Tue Dec 13 2005 21:55:52 on modular [Seed = 1790353644] ------------------------------------- >> A := AffineAlgebra; ^ User error: Identifier 'w' has not been declared or assigned >> y^-1; ^ User error: Identifier 'y' has not been declared or assigned >> MinimalPolynomial(x); ^ User error: Identifier 'x' has not been declared or assigned Total time: 0.190 seconds, Total memory usage: 3.34MB '200.177' ************** MAGMA ***************** Host 200.177.46.74 (200.177.46.74) Time: Tue Dec 13 21:53:18 2005 Input: p := 1461501624496790265145448589920785493717258890819; n := 1461501624496790265145447380994971188499300027613; Fp := GF(p); b := Fp!0; repeat repeat b := b + 1; until IsSquare(b + 1); y := Root(b + 1, 2); E := EllipticCurve([Fp | 0, b]); G := E![1, y]; until IsZero(n*G); "b =", b; "G =", G; rho := (p - n)^2; "rho =", rho; "rho^6 mod n =", rho^6 mod n; H := rho*G; "H =", H; zeta := H[1]; "zeta =", zeta; "zeta^3 =", zeta^3; lambda := Fp!2; while IsPower(lambda, 3) do lambda +:= 1; end while; "lambda =", lambda; Fp2 := ExtensionField; mu := i + 1; while IsSquare(mu) do mu +:= 1; end while; "mu =", mu; xi := 1/(lambda^2*mu^3); Fp12 := ExtensionField; Et := EllipticCurve([Fp2 | 0, b/xi]); xt := 1; while not IsSquare(xt^3 + b/xi) do xt +:= 1; end while; yt := Sqrt(xt^3 + b/xi); Gt := Et![xt, yt]; "Gt =", Gt; k := 12; assert (p^k - 1) mod n eq 0; chi := (p^k - 1) div n; Ek := EllipticCurve([Fp12 | 0, Fp12!b]); g := function(U, V, Q) if IsZero(U) or IsZero(V) or (U eq -V) or IsZero(Q) then return Fp12!1; end if; /* m := (U eq V) select 3*U[1]^2/(2*U[2]) else (V[2] - U[2])/(V[1] - U[1]); return m*(Q[1] - U[1]) + U[2] - Q[2]; */ if U eq V then m := 3*U[1]^2; s := 2*U[2]; else m := V[2] - U[2]; s := V[1] - U[1]; end if; return m*(Q[1] - U[1]) + s*(U[2] - Q[2]); end function; length := function(r) n := 0; v := 1; while v le r do n +:= 1; v +:= v; end while; return n; end function; bit := function(r, i) return (r div 2^i) mod 2; end function; tate := function(P, Qt) Q := Ek![Qt[1]*z^2, Qt[2]*z^3]; f := 1; A := P; for i := length(n) - 2 to 0 by -1 do f := f^2*g(A, A, Q); A := 2*A; if bit(n, i) eq 1 then f *:= g(A, P, Q); A +:= P; end if; end for; return f^chi; end function; g := tate(G, Gt); "g =", g; /* for j in [1..11] do print "g^(p^" * Sprint(j) * "-1): " * Sprint(g^(p^j-1) ne 1); end for; print "g^(p^12-1): " * Sprint(g^(p^12-1) eq 1); "g^n =", g^n; */ /* for j in [1..100] do "--", j; u := Random(n-1); v := Random(n-1); w := tate(u*G, v*Gt); h := g^((u*v) mod n); if w ne h then "Failure!"; print "u = " * Sprint(u); print "v = " * Sprint(v); print "e(u*P, v*Q) = " * Sprint(w); print "e(P, Q)^(u*v) = " * Sprint(h); quit; end if; end for; "Success!"; */ trace4 := function(g) t := g^p; t ^:= p; t ^:= p; t ^:= p; // g^(p^4) s := t^p; s ^:= p; s ^:= p; s ^:= p; // g^(p^8) return g + t + s; end function; trace6 := function(g) t := g^p; t ^:= p; t ^:= p; t ^:= p; t ^:= p; t ^:= p; // g^(p^6) return g + t; end function; t40 := trace4(g); t41 := t40^p; t42 := t41^p; "trace4(g) =", t40; "trace4(g) =", Eltseq(t40); "trace4(g)' =", Eltseq(t41); "trace4(g)'' =", Eltseq(t42); "sum4 =", t40 + t41 + t42; "prod4 =", t40*t41*t42; t60 := trace6(g); t61 := t60^p; "trace6(g) =", Eltseq(t60); "trace6(g)' =", Eltseq(t61); "sum6 =", t60 + t61; "prod6 =", t60*t61; Output: Magma V2.11-10 Tue Dec 13 2005 21:53:17 on modular [Seed = 1907211090] ------------------------------------- b = 3 G = (1 : 1461501624496790265145448589920785493717258890817 : 1) rho = 1461501624493534334825397658811989710051820598436 rho^6 mod n = 1 H = (1627965160026674480212199743920457792 : 2 : 1) zeta = 1627965160026674480212199743920457792 zeta^3 = 1 lambda = 2 mu = i + 1 Gt = (8 : 816263181872116351510202985179226587277470764815*i + 295865244505705705023665406736615173923424579851 : 1) g = (1113252570408097904801784205204725186435037650621*i + 828892838102560531997994710291403383901103286823)*z^5 + (78288732951012207123651737359843385303103429616*i + 1421790144970157374993435834657786699116722047068)*z^4 + (1334693519950620886403708174450030523556565932446*i + 65995713479010101585078190425689016349129829691)*z^3 + (1151508433446340267838152948068030563616071711737*i + 1331505104782800437119636772308446788364866291823)*z^2 + (1167306608241793402349182741773283684304074553579*i + 813404015744428633410120092529896632595508931142)*z + 800403695404043311146437378782593638877310314601*i + 901249395544166733264846949825665528694219338474 trace4(g) = (1081077310858282128920227343508520583235180015700*i + 197987140437030304755234571277067049047389489073)*z^3 + 939709461715339668293863546426995422914672052984*i + 1242246562135709934649092259556211092365399124603 trace4(g) = [ 939709461715339668293863546426995422914672052984*i + 1242246562135709934649092259556211092365399124603, 0, 0, 1081077310858282128920227343508520583235180015700*i + 197987140437030304755234571277067049047389489073, 0, 0 ] trace4(g)' = [ 521792162781450596851585043493790070802586837835*i + 1242246562135709934649092259556211092365399124603, 0, 0, 600370227937686771113197478282678333920097663482*i + 1221575564612301174446849985825970354859774772881, 0, 0 ] trace4(g)'' = [ 939709461715339668293863546426995422914672052984*i + 1242246562135709934649092259556211092365399124603, 0, 0, 380424313638508136225221246412264910482078875119*i + 1263514484059759960390214018643718444669869401746, 0, 0 ] sum4 = (600370227937686771113197478282678333920097663482*i + 1221575564612301174446849985825970354859774772881)*z^3 + 939709461715339668293863546426995422914672052984*i + 803736437413549273656379598827062289661679592171 prod4 = (351960067217720226471026218912124823215388842998*i + 636471885525581132299463326642541502123540418918)*z^3 + 1073254876702081306986584990089982603797367449521*i + 1088939299807643406655341764511290451420861288604 trace6(g) = [ 139305766311296357147426167644401784037361738383*i + 340997166591543201384245309730545563671179786129, 0, 841515242395890270530857306215275633514884532655*i + 1201508585068810609093824954696108083012473692827, 0, 156577465902024414247303474719686770606206859232*i + 1382078665443524484841423079394787904516185203317, 0 ] trace6(g)' = [ 1322195858185493907998022422276383709679897152436*i + 340997166591543201384245309730545563671179786129, 0, 96267089637067702603160574155571500808560497740*i + 198193976097141083414977909683245959979351734515, 0, 39750298548322008993974458110694835814573932369*i + 170269780948936534327757016306807241564764251208, 0 ] sum6 = (196327764450346423241277932830381606420780791601*i + 90846821895670754023731505780809652363690563706)*z^4 + (937782332032957973134017880370847134323445030395*i + 1399702561165951692508802864379354042991825427342)*z^2 + 681994333183086402768490619461091127342359572258 prod6 = (403224252430258213018808164592556921112551430842*i + 1072501587439756383839928276680242651041055479001)*z^4 + (515192785473442154214411326985372045692209216551*i + 1264811833155190234052567165579767753345088245117)*z^2 + 933753153541436149967745700541145775019625660786*i + 579210675113629004884943196866573303442263067154 Total time: 1.080 seconds, Total memory usage: 3.34MB '72.19.1' ************** MAGMA ***************** Host 72.19.126.33 (72.19.126.33) Time: Tue Dec 13 21:46:47 2005 Input: F := FunctionField(GF(5,2)); A := AffineAlgebra; Q := PolynomialRing(A); B := ext; P := PolynomialRing(F); y^-1; > MinimalPolynomial(x+y+w); Output: Magma V2.11-10 Tue Dec 13 2005 21:46:47 on modular [Seed = 2074591186] ------------------------------------- >> B := ext; ^ Runtime error: No constructor provided for this type of object 1/t*y^4 + 1/t >> MinimalPolynomial(x+y+w); ^ Runtime error in 'MinimalPolynomial': Bad argument types Argument types given: RngUPolElt[RngMPolRes] Total time: 0.290 seconds, Total memory usage: 7.84MB '72.19.1' ************** MAGMA ***************** Host 72.19.126.33 (72.19.126.33) Time: Tue Dec 13 21:46:34 2005 Input: F := FunctionField(GF(5,2)); A := AffineAlgebra; Q := PolynomialRing(A); B := ext; P := PolynomialRing(F); y^-1; > MinimalPolynomial(x+y+w); Output: Magma V2.11-10 Tue Dec 13 2005 21:46:33 on modular [Seed = 2124593865] ------------------------------------- >> B := ext; ^ Runtime error: No constructor provided for this type of object 1/t*y^4 + 1/t >> MinimalPolynomial(x+y+w); ^ Runtime error in 'MinimalPolynomial': Bad argument types Argument types given: RngUPolElt[RngMPolRes] Total time: 0.300 seconds, Total memory usage: 7.84MB '72.19.1' ************** MAGMA ***************** Host 72.19.126.33 (72.19.126.33) Time: Tue Dec 13 21:45:47 2005 Input: F := FunctionField(GF(5,2)); A := AffineAlgebra; B := ext; P := PolynomialRing(F); y^-1; > MinimalPolynomial(x+y+w); Output: Magma V2.11-10 Tue Dec 13 2005 21:45:47 on modular [Seed = 98156896] ------------------------------------- >> B := ext; ^ User error: Identifier 'w' has not been declared or assigned 1/t*y^4 + 1/t >> MinimalPolynomial(x+y+w); ^ User error: Identifier 'w' has not been declared or assigned Total time: 0.190 seconds, Total memory usage: 3.34MB '72.19.1' ************** MAGMA ***************** Host 72.19.126.33 (72.19.126.33) Time: Tue Dec 13 21:45:13 2005 Input: F := FunctionField(GF(5,2)); A := AffineAlgebra; B := ext; P := PolynomialRing(F); y^-1; > MinimalPolynomial(x+y+w); Output: Magma V2.11-10 Tue Dec 13 2005 21:45:13 on modular [Seed = 148163599] ------------------------------------- >> B := ext; ^ User error: bad syntax 1/t*y^4 + 1/t >> MinimalPolynomial(x+y+w); ^ User error: Identifier 'w' has not been declared or assigned Total time: 0.180 seconds, Total memory usage: 3.34MB '72.19.1' ************** MAGMA ***************** Host 72.19.126.33 (72.19.126.33) Time: Tue Dec 13 21:44:54 2005 Input: F := FunctionField(GF(5,2)); A := AffineAlgebra; B := Ext; P := PolynomialRing(F); y^-1; > MinimalPolynomial(x+y+w); Output: Magma V2.11-10 Tue Dec 13 2005 21:44:54 on modular [Seed = 248695318] ------------------------------------- >> B := Ext; ^ User error: bad syntax 1/t*y^4 + 1/t >> MinimalPolynomial(x+y+w); ^ User error: Identifier 'w' has not been declared or assigned Total time: 0.190 seconds, Total memory usage: 3.34MB '72.19.1' ************** MAGMA ***************** Host 72.19.126.33 (72.19.126.33) Time: Tue Dec 13 21:44:44 2005 Input: F := FunctionField(GF(5,2)); A := AffineAlgebra; B := Ext; P := PolynomialRing(F); y^-1; > MinimalPolynomial(x+y+w); Output: Magma V2.11-10 Tue Dec 13 2005 21:44:44 on modular [Seed = 365547831] ------------------------------------- >> B := Ext; ^ User error: bad syntax 1/t*y^4 + 1/t >> MinimalPolynomial(x+y+w); ^ User error: Identifier 'w' has not been declared or assigned Total time: 0.190 seconds, Total memory usage: 3.34MB '72.19.1' ************** MAGMA ***************** Host 72.19.126.33 (72.19.126.33) Time: Tue Dec 13 21:43:39 2005 Input: F := FunctionField(GF(5,2)); A := AffineAlgebra; B := AffineAlgebra; P := PolynomialRing(F); y^-1; > MinimalPolynomial(x+y+w); Output: ** WARNING: Computation time exceeded 20 seconds, so computation was terminated after 20 seconds. ** Magma V2.11-10 Tue Dec 13 2005 21:43:19 on modular [Seed = 415555041] ------------------------------------- Errors: /bin/sh: line 1: 32038 Alarm clock nice -n 19 /usr/local/bin/magma '72.19.1' ************** MAGMA ***************** Host 72.19.126.33 (72.19.126.33) Time: Tue Dec 13 21:42:55 2005 Input: F := FunctionField(GF(5,2)); A := AffineAlgebra; B := AffineAlgebra; P := PolynomialRing(F); y^-1; > MinimalPolynomial(x+y+w); Output: ** WARNING: Computation time exceeded 20 seconds, so computation was terminated after 20 seconds. ** Magma V2.11-10 Tue Dec 13 2005 21:42:35 on modular [Seed = 516086715] ------------------------------------- Errors: /bin/sh: line 1: 32032 Alarm clock nice -n 19 /usr/local/bin/magma '72.19.1' ************** MAGMA ***************** Host 72.19.126.33 (72.19.126.33) Time: Tue Dec 13 21:41:57 2005 Input: F := FunctionField(GF(5,2)); A := AffineAlgebra; B := AffineAlgebra; P := PolynomialRing(F); y^-1; > MinimalPolynomial(x+y+w); Output: ** WARNING: Computation time exceeded 20 seconds, so computation was terminated after 20 seconds. ** Magma V2.11-10 Tue Dec 13 2005 21:41:36 on modular [Seed = 637132923] ------------------------------------- Errors: /bin/sh: line 1: 32027 Alarm clock nice -n 19 /usr/local/bin/magma '72.19.1' ************** MAGMA ***************** Host 72.19.126.33 (72.19.126.33) Time: Tue Dec 13 21:40:43 2005 Input: F := FunctionField(GF(5,2)); A := AffineAlgebra; B := AffineAlgebra; P := PolynomialRing(F); y^-1; > MinimalPolynomial(x+y+w); Output: Magma V2.11-10 Tue Dec 13 2005 21:40:43 on modular [Seed = 687140157] ------------------------------------- >> B := AffineAlgebra= 0 1/t*y^4 + 1/t >> MinimalPolynomial(x+y+w); ^ User error: Identifier 'w' has not been declared or assigned Total time: 0.190 seconds, Total memory usage: 3.34MB '72.19.1' ************** MAGMA ***************** Host 72.19.126.33 (72.19.126.33) Time: Tue Dec 13 21:40:31 2005 Input: F := FunctionField(GF(5,2)); A := AffineAlgebra; B := AffineAlgebra; P := PolynomialRing(F); y^-1; > MinimalPolynomial(x+y+w); Output: Magma V2.11-10 Tue Dec 13 2005 21:40:31 on modular [Seed = 804514318] ------------------------------------- >> B := AffineAlgebra> MinimalPolynomial(x+y+w); ^ User error: Identifier 'w' has not been declared or assigned Total time: 0.190 seconds, Total memory usage: 3.34MB '72.19.1' ************** MAGMA ***************** Host 72.19.126.33 (72.19.126.33) Time: Tue Dec 13 21:39:31 2005 Input: F := FunctionField(GF(5,2)); A := AffineAlgebra; B := AffineAlgebra, w | w^5+w-x*t/y>; P := PolynomialRing(F); y^-1; > MinimalPolynomial(x+y+w); Output: Magma V2.11-10 Tue Dec 13 2005 21:39:31 on modular [Seed = 854521071] ------------------------------------- >> B := AffineAlgebra, w | w^5+w-x*t/y>; ^ User error: bad syntax 1/t*y^4 + 1/t >> MinimalPolynomial(x+y+w); ^ User error: Identifier 'w' has not been declared or assigned Total time: 0.190 seconds, Total memory usage: 3.34MB '72.19.1' ************** MAGMA ***************** Host 72.19.126.33 (72.19.126.33) Time: Tue Dec 13 21:39:13 2005 Input: F := FunctionField(GF(5,2)); A := AffineAlgebra; B := AffineAlgebra; P := PolynomialRing(F); y^-1; > MinimalPolynomial(x+y+w); Output: Magma V2.11-10 Tue Dec 13 2005 21:39:13 on modular [Seed = 971374550] ------------------------------------- >> B := AffineAlgebra; ^ Runtime error in 'AssignNames': Argument 2 should have length at most 1 1/t*y^4 + 1/t >> MinimalPolynomial(x+y+w); ^ User error: Identifier 'w' has not been declared or assigned Total time: 0.300 seconds, Total memory usage: 7.84MB '72.19.1' ************** MAGMA ***************** Host 72.19.126.33 (72.19.126.33) Time: Tue Dec 13 21:38:49 2005 Input: F := FunctionField(GF(5,2)); A := AffineAlgebra; B := AffineAlgebra; P := PolynomialRing(F); y^-1; > MinimalPolynomial(x+y+w); Output: Magma V2.11-10 Tue Dec 13 2005 21:38:48 on modular [Seed = 3373600590] ------------------------------------- 1/t*y^4 + 1/t $.1^5 + $.1 + 4*x*y^4 + 4*x + 4/t*y + 4*t Total time: 0.300 seconds, Total memory usage: 7.84MB '72.19.1' ************** MAGMA ***************** Host 72.19.126.33 (72.19.126.33) Time: Tue Dec 13 21:37:38 2005 Input: F := FunctionField(GF(5,2)); A := AffineAlgebra; B := AffineAlgebra; P := PolynomialRing(F); y^-1; > MinimalPolynomial(x-y); Output: Magma V2.11-10 Tue Dec 13 2005 21:37:37 on modular [Seed = 3574141851] ------------------------------------- 1/t*y^4 + 1/t z^25 + (t^4 + 1)/t^4*z^5 + 1/t^4*z + (t^9 + t + 4)/t^4 Total time: 0.300 seconds, Total memory usage: 7.84MB '72.19.1' ************** MAGMA ***************** Host 72.19.126.33 (72.19.126.33) Time: Tue Dec 13 21:37:22 2005 Input: F := FunctionField(GF(5,2)); A := AffineAlgebra; B := AffineAlgebra; P := PolynomialRing(F); y^-1; > MinimalPolynomial(x-y); Output: Magma V2.11-10 Tue Dec 13 2005 21:37:21 on modular [Seed = 3657833604] ------------------------------------- 1/t*y^4 + 1/t z^25 + (t^4 + 1)/t^4*z^5 + 1/t^4*z + (t^9 + t + 4)/t^4 Total time: 0.300 seconds, Total memory usage: 7.84MB '72.19.1' ************** MAGMA ***************** Host 72.19.126.33 (72.19.126.33) Time: Tue Dec 13 21:36:38 2005 Input: F := FunctionField(GF(5,2)); A := AffineAlgebra; B := AffineAlgebra; P := PolynomialRing(F); y^-1; > MinimalPolynomial(x-y); Output: Magma V2.11-10 Tue Dec 13 2005 21:36:38 on modular [Seed = 3741522390] ------------------------------------- 1/t^17*w^100 + (4*t^4 + 4)/t^17*w^80 + 4/t^13*w^76 + (t^8 + 2*t^4 + 1)/t^17*w^60 + (2*t^4 + 2)/t^13*w^56 + 1/t^9*w^52 + (4*t^12 + 2*t^8 + 2*t^4 + 4)/t^17*w^40 + (2*t^8 + 4*t^4 + 2)/t^13*w^36 + (2*t^4 + 2)/t^9*w^32 + 4/t^5*w^28 + (t^16 + 4*t^12 + t^8 + 4*t^4 + 1)/t^17*w^20 + (4*t^12 + 2*t^8 + 2*t^4 + 4)/t^13*w^16 + (t^8 + 2*t^4 + 1)/t^9*w^12 + (4*t^4 + 4)/t^5*w^8 + 1/t*w^4 + 1/t z^25 + (t^4 + 1)/t^4*z^5 + 1/t^4*z + (t^9 + t + 4)/t^4 Total time: 0.250 seconds, Total memory usage: 3.63MB '200.177' ************** MAGMA ***************** Host 200.177.7.38 (200.177.7.38) Time: Tue Dec 13 21:36:30 2005 Input: p := 1461501624496790265145448589920785493717258890819; n := 1461501624496790265145447380994971188499300027613; Fp := GF(p); b := Fp!0; repeat repeat b := b + 1; until IsSquare(b + 1); y := Root(b + 1, 2); E := EllipticCurve([Fp | 0, b]); G := E![1, y]; until IsZero(n*G); "b =", b; "G =", G; rho := (p - n)^2; "rho =", rho; "rho^6 mod n =", rho^6 mod n; H := rho*G; "H =", H; zeta := H[1]; "zeta =", zeta; "zeta^3 =", zeta^3; lambda := Fp!2; while IsPower(lambda, 3) do lambda +:= 1; end while; "lambda =", lambda; Fp2 := ExtensionField; mu := i + 1; while IsSquare(mu) do mu +:= 1; end while; "mu =", mu; xi := 1/(lambda^2*mu^3); Fp12 := ExtensionField; Et := EllipticCurve([Fp2 | 0, b/xi]); xt := 1; while not IsSquare(xt^3 + b/xi) do xt +:= 1; end while; yt := Sqrt(xt^3 + b/xi); Gt := Et![xt, yt]; "Gt =", Gt; k := 12; assert (p^k - 1) mod n eq 0; chi := (p^k - 1) div n; Ek := EllipticCurve([Fp12 | 0, Fp12!b]); g := function(U, V, Q) if IsZero(U) or IsZero(V) or (U eq -V) or IsZero(Q) then return Fp12!1; end if; /* m := (U eq V) select 3*U[1]^2/(2*U[2]) else (V[2] - U[2])/(V[1] - U[1]); return m*(Q[1] - U[1]) + U[2] - Q[2]; */ if U eq V then m := 3*U[1]^2; s := 2*U[2]; else m := V[2] - U[2]; s := V[1] - U[1]; end if; return m*(Q[1] - U[1]) + s*(U[2] - Q[2]); end function; length := function(r) n := 0; v := 1; while v le r do n +:= 1; v +:= v; end while; return n; end function; bit := function(r, i) return (r div 2^i) mod 2; end function; tate := function(P, Qt) Q := Ek![Qt[1]*z^2, Qt[2]*z^3]; f := 1; A := P; for i := length(n) - 2 to 0 by -1 do f := f^2*g(A, A, Q); A := 2*A; if bit(n, i) eq 1 then f *:= g(A, P, Q); A +:= P; end if; end for; return f^chi; end function; g := tate(G, Gt); "g =", g; /* for j in [1..11] do print "g^(p^" * Sprint(j) * "-1): " * Sprint(g^(p^j-1) ne 1); end for; print "g^(p^12-1): " * Sprint(g^(p^12-1) eq 1); "g^n =", g^n; */ /* for j in [1..100] do "--", j; u := Random(n-1); v := Random(n-1); w := tate(u*G, v*Gt); h := g^((u*v) mod n); if w ne h then "Failure!"; print "u = " * Sprint(u); print "v = " * Sprint(v); print "e(u*P, v*Q) = " * Sprint(w); print "e(P, Q)^(u*v) = " * Sprint(h); quit; end if; end for; "Success!"; */ trace4 := function(g) t := g^p; t ^:= p; t ^:= p; t ^:= p; // g^(p^4) s := t^p; s ^:= p; s ^:= p; s ^:= p; // g^(p^8) return g + t + s; end function; trace6 := function(g) t := g^p; t ^:= p; t ^:= p; t ^:= p; t ^:= p; t ^:= p; // g^(p^6) return g + t; end function; t40 := trace4(g); t41 := t40^p; t42 := t41^p; "trace4(g) =", Eltseq(t40); "trace4(g)' =", Eltseq(t41); "trace4(g)'' =", Eltseq(t42); "sum4 =", t40 + t41 + t42; "prod4 =", t40*t41*t42; t60 := trace6(g); t61 := t60^p; "trace6(g) =", Eltseq(t60); "trace6(g)' =", Eltseq(t61); "sum6 =", t60 + t61; "prod6 =", t60*t61; Output: Magma V2.11-10 Tue Dec 13 2005 21:36:29 on modular [Seed = 3845727223] ------------------------------------- b = 3 G = (1 : 1461501624496790265145448589920785493717258890817 : 1) rho = 1461501624493534334825397658811989710051820598436 rho^6 mod n = 1 H = (1627965160026674480212199743920457792 : 2 : 1) zeta = 1627965160026674480212199743920457792 zeta^3 = 1 lambda = 2 mu = i + 1 Gt = (8 : 816263181872116351510202985179226587277470764815*i + 295865244505705705023665406736615173923424579851 : 1) g = (1113252570408097904801784205204725186435037650621*i + 828892838102560531997994710291403383901103286823)*z^5 + (78288732951012207123651737359843385303103429616*i + 1421790144970157374993435834657786699116722047068)*z^4 + (1334693519950620886403708174450030523556565932446*i + 65995713479010101585078190425689016349129829691)*z^3 + (1151508433446340267838152948068030563616071711737*i + 1331505104782800437119636772308446788364866291823)*z^2 + (1167306608241793402349182741773283684304074553579*i + 813404015744428633410120092529896632595508931142)*z + 800403695404043311146437378782593638877310314601*i + 901249395544166733264846949825665528694219338474 trace4(g) = [ 939709461715339668293863546426995422914672052984*i + 1242246562135709934649092259556211092365399124603, 0, 0, 1081077310858282128920227343508520583235180015700*i + 197987140437030304755234571277067049047389489073, 0, 0 ] trace4(g)' = [ 521792162781450596851585043493790070802586837835*i + 1242246562135709934649092259556211092365399124603, 0, 0, 600370227937686771113197478282678333920097663482*i + 1221575564612301174446849985825970354859774772881, 0, 0 ] trace4(g)'' = [ 939709461715339668293863546426995422914672052984*i + 1242246562135709934649092259556211092365399124603, 0, 0, 380424313638508136225221246412264910482078875119*i + 1263514484059759960390214018643718444669869401746, 0, 0 ] sum4 = (600370227937686771113197478282678333920097663482*i + 1221575564612301174446849985825970354859774772881)*z^3 + 939709461715339668293863546426995422914672052984*i + 803736437413549273656379598827062289661679592171 prod4 = (351960067217720226471026218912124823215388842998*i + 636471885525581132299463326642541502123540418918)*z^3 + 1073254876702081306986584990089982603797367449521*i + 1088939299807643406655341764511290451420861288604 trace6(g) = [ 139305766311296357147426167644401784037361738383*i + 340997166591543201384245309730545563671179786129, 0, 841515242395890270530857306215275633514884532655*i + 1201508585068810609093824954696108083012473692827, 0, 156577465902024414247303474719686770606206859232*i + 1382078665443524484841423079394787904516185203317, 0 ] trace6(g)' = [ 1322195858185493907998022422276383709679897152436*i + 340997166591543201384245309730545563671179786129, 0, 96267089637067702603160574155571500808560497740*i + 198193976097141083414977909683245959979351734515, 0, 39750298548322008993974458110694835814573932369*i + 170269780948936534327757016306807241564764251208, 0 ] sum6 = (196327764450346423241277932830381606420780791601*i + 90846821895670754023731505780809652363690563706)*z^4 + (937782332032957973134017880370847134323445030395*i + 1399702561165951692508802864379354042991825427342)*z^2 + 681994333183086402768490619461091127342359572258 prod6 = (403224252430258213018808164592556921112551430842*i + 1072501587439756383839928276680242651041055479001)*z^4 + (515192785473442154214411326985372045692209216551*i + 1264811833155190234052567165579767753345088245117)*z^2 + 933753153541436149967745700541145775019625660786*i + 579210675113629004884943196866573303442263067154 Total time: 1.070 seconds, Total memory usage: 3.34MB '72.19.1' ************** MAGMA ***************** Host 72.19.126.33 (72.19.126.33) Time: Tue Dec 13 21:36:22 2005 Input: F := FunctionField(GF(5,2)); A := AffineAlgebra; B := AffineAlgebra; P := PolynomialRing(F); x^-1; > MinimalPolynomial(x-y); Output: Magma V2.11-10 Tue Dec 13 2005 21:36:22 on modular [Seed = 3946262002] ------------------------------------- 1/t^20*w^120 + 4/t^20*w^116 + 1/t^20*w^112 + 4/t^20*w^108 + 1/t^20*w^104 + 4/t^20*w^100 + 1/t^20*w^96 + 4/t^20*w^92 + 1/t^20*w^88 + 4/t^20*w^84 + 1/t^20*w^80 + 4/t^20*w^76 + 1/t^20*w^72 + 4/t^20*w^68 + 1/t^20*w^64 + 4/t^20*w^60 + 1/t^20*w^56 + 4/t^20*w^52 + 1/t^20*w^48 + 4/t^20*w^44 + 1/t^20*w^40 + 4/t^20*w^36 + 1/t^20*w^32 + 4/t^20*w^28 + 1/t^20*w^24 + (t^4 + 1)/t^4*w^20 + (4*t^4 + 4)/t^4*w^16 + (t^4 + 1)/t^4*w^12 + (4*t^4 + 4)/t^4*w^8 + (t^4 + 1)/t^4*w^4 + 1 z^25 + (t^4 + 1)/t^4*z^5 + 1/t^4*z + (t^9 + t + 4)/t^4 Total time: 0.240 seconds, Total memory usage: 3.53MB '72.19.1' ************** MAGMA ***************** Host 72.19.126.33 (72.19.126.33) Time: Tue Dec 13 21:35:19 2005 Input: F := FunctionField(GF(5,2)); A := AffineAlgebra; B := AffineAlgebra; P := PolynomialRing(F); x^-1; > MinimalPolynomial(x-y); Output: Magma V2.11-10 Tue Dec 13 2005 21:35:19 on modular [Seed = 4029429544] ------------------------------------- >> B := AffineAlgebra: Rhs argument 1 is invalid for this constructor x^4*y^4 + x^4 + y^4 + 1 z^25 + (t^4 + 1)/t^4*z^5 + 1/t^4*z + (t^9 + t + 4)/t^4 Total time: 0.190 seconds, Total memory usage: 3.34MB '72.19.1' ************** MAGMA ***************** Host 72.19.126.33 (72.19.126.33) Time: Tue Dec 13 21:34:52 2005 Input: F := FunctionField(GF(5,2)); A := AffineAlgebra; B := AffineAlgebra; P := PolynomialRing(F); x^-1; > MinimalPolynomial(x-y); Output: Magma V2.11-10 Tue Dec 13 2005 21:34:52 on modular [Seed = 4113117702] ------------------------------------- >> B := AffineAlgebra; ^ Runtime error in AffineAlgebra< ... >: Rhs argument 1 is invalid for this constructor x^4*y^4 + x^4 + y^4 + 1 z^25 + (t^4 + 1)/t^4*z^5 + 1/t^4*z + (t^9 + t + 4)/t^4 Total time: 0.200 seconds, Total memory usage: 3.34MB '200.177' ************** MAGMA ***************** Host 200.177.7.38 (200.177.7.38) Time: Tue Dec 13 21:20:51 2005 Input: p := 1461501624496790265145448589920785493717258890819; n := 1461501624496790265145447380994971188499300027613; Fp := GF(p); b := Fp!0; repeat repeat b := b + 1; until IsSquare(b + 1); y := Root(b + 1, 2); E := EllipticCurve([Fp | 0, b]); G := E![1, y]; until IsZero(n*G); "b =", b; "G =", G; rho := (p - n)^2; "rho =", rho; "rho^6 mod n =", rho^6 mod n; H := rho*G; "H =", H; zeta := H[1]; "zeta =", zeta; "zeta^3 =", zeta^3; lambda := Fp!2; while IsPower(lambda, 3) do lambda +:= 1; end while; "lambda =", lambda; Fp2 := ExtensionField; mu := i + 1; while IsSquare(mu) do mu +:= 1; end while; "mu =", mu; xi := 1/(lambda^2*mu^3); Fp12 := ExtensionField; Et := EllipticCurve([Fp2 | 0, b/xi]); xt := 1; while not IsSquare(xt^3 + b/xi) do xt +:= 1; end while; yt := Sqrt(xt^3 + b/xi); Gt := Et![xt, yt]; "Gt =", Gt; k := 12; assert (p^k - 1) mod n eq 0; chi := (p^k - 1) div n; Ek := EllipticCurve([Fp12 | 0, Fp12!b]); g := function(U, V, Q) if IsZero(U) or IsZero(V) or (U eq -V) or IsZero(Q) then return Fp12!1; end if; /* m := (U eq V) select 3*U[1]^2/(2*U[2]) else (V[2] - U[2])/(V[1] - U[1]); return m*(Q[1] - U[1]) + U[2] - Q[2]; */ if U eq V then m := 3*U[1]^2; s := 2*U[2]; else m := V[2] - U[2]; s := V[1] - U[1]; end if; return m*(Q[1] - U[1]) + s*(U[2] - Q[2]); end function; length := function(r) n := 0; v := 1; while v le r do n +:= 1; v +:= v; end while; return n; end function; bit := function(r, i) return (r div 2^i) mod 2; end function; tate := function(P, Qt) Q := Ek![Qt[1]*z^2, Qt[2]*z^3]; f := 1; A := P; for i := length(n) - 2 to 0 by -1 do f := f^2*g(A, A, Q); A := 2*A; if bit(n, i) eq 1 then f *:= g(A, P, Q); A +:= P; end if; end for; return f^chi; end function; g := tate(G, Gt); "g =", g; /* for j in [1..11] do print "g^(p^" * Sprint(j) * "-1): " * Sprint(g^(p^j-1) ne 1); end for; print "g^(p^12-1): " * Sprint(g^(p^12-1) eq 1); "g^n =", g^n; */ /* for j in [1..100] do "--", j; u := Random(n-1); v := Random(n-1); w := tate(u*G, v*Gt); h := g^((u*v) mod n); if w ne h then "Failure!"; print "u = " * Sprint(u); print "v = " * Sprint(v); print "e(u*P, v*Q) = " * Sprint(w); print "e(P, Q)^(u*v) = " * Sprint(h); quit; end if; end for; "Success!"; */ trace4 := function(g) t := g^p; t ^:= p; t ^:= p; t ^:= p; // g^(p^4) s := t^p; s ^:= p; s ^:= p; s ^:= p; // g^(p^8) return g + t + s; end function; trace6 := function(g) t := g^p; t ^:= p; t ^:= p; t ^:= p; t ^:= p; t ^:= p; // g^(p^6) return g + t; end function; t40 := trace4(g); t41 := t40^p; t42 := t41^p; "trace4(g) =", t40; "trace4(g)' =", t41; "trace4(g)'' =", t42; "sum4 =", t40 + t41 + t42; "prod4 =", t40*t41*t42; t60 := trace6(g); t61 := t60^p; "trace6(g) =", t60; "trace6(g)' =", t61; "sum6 =", t60 + t61; "prod6 =", t60*t61; Output: Magma V2.11-10 Tue Dec 13 2005 21:20:50 on modular [Seed = 2170372924] ------------------------------------- b = 3 G = (1 : 1461501624496790265145448589920785493717258890817 : 1) rho = 1461501624493534334825397658811989710051820598436 rho^6 mod n = 1 H = (1627965160026674480212199743920457792 : 2 : 1) zeta = 1627965160026674480212199743920457792 zeta^3 = 1 lambda = 2 mu = i + 1 Gt = (8 : 645238442624673913635245604741558906439788126004*i + 1165636379991084560121783183184170319793834310968 : 1) g = (348249054088692360343664384716060307282221240198*i + 632608786394229733147453879629382109816155603996)*z^5 + (78288732951012207123651737359843385303103429616*i + 1421790144970157374993435834657786699116722047068)*z^4 + (126808104546169378741740415470754970160692958373*i + 1395505911017780163560370399495096477368129061128)*z^3 + (1151508433446340267838152948068030563616071711737*i + 1331505104782800437119636772308446788364866291823)*z^2 + (294195016254996862796265848147501809413184337240*i + 648097608752361631735328497390888861121749959677)*z + 800403695404043311146437378782593638877310314601*i + 901249395544166733264846949825665528694219338474 trace4(g) = (380424313638508136225221246412264910482078875119*i + 1263514484059759960390214018643718444669869401746)*z^3 + 939709461715339668293863546426995422914672052984*i + 1242246562135709934649092259556211092365399124603 trace4(g)' = (861131396559103494032251111638107159797161227337*i + 239926059884489090698598604094815138857484117938)*z^3 + 521792162781450596851585043493790070802586837835*i + 1242246562135709934649092259556211092365399124603 trace4(g)'' = (1081077310858282128920227343508520583235180015700*i + 197987140437030304755234571277067049047389489073)*z^3 + 939709461715339668293863546426995422914672052984*i + 1242246562135709934649092259556211092365399124603 sum4 = (861131396559103494032251111638107159797161227337*i + 239926059884489090698598604094815138857484117938)*z^3 + 939709461715339668293863546426995422914672052984*i + 803736437413549273656379598827062289661679592171 prod4 = (1109541557279070038674422371008660670501870047821*i + 825029738971209132845985263278243991593718471901)*z^3 + 1073254876702081306986584990089982603797367449521*i + 1088939299807643406655341764511290451420861288604 trace6(g) = (156577465902024414247303474719686770606206859232*i + 1382078665443524484841423079394787904516185203317)*z^4 + (841515242395890270530857306215275633514884532655*i + 1201508585068810609093824954696108083012473692827)*z^2 + 139305766311296357147426167644401784037361738383*i + 340997166591543201384245309730545563671179786129 trace6(g)' = (39750298548322008993974458110694835814573932369*i + 170269780948936534327757016306807241564764251208)*z^4 + (96267089637067702603160574155571500808560497740*i + 198193976097141083414977909683245959979351734515)*z^2 + 1322195858185493907998022422276383709679897152436*i + 340997166591543201384245309730545563671179786129 sum6 = (196327764450346423241277932830381606420780791601*i + 90846821895670754023731505780809652363690563706)*z^4 + (937782332032957973134017880370847134323445030395*i + 1399702561165951692508802864379354042991825427342)*z^2 + 681994333183086402768490619461091127342359572258 prod6 = (403224252430258213018808164592556921112551430842*i + 1072501587439756383839928276680242651041055479001)*z^4 + (515192785473442154214411326985372045692209216551*i + 1264811833155190234052567165579767753345088245117)*z^2 + 933753153541436149967745700541145775019625660786*i + 579210675113629004884943196866573303442263067154 Total time: 1.080 seconds, Total memory usage: 3.34MB '200.177' ************** MAGMA ***************** Host 200.177.7.38 (200.177.7.38) Time: Tue Dec 13 21:15:15 2005 Input: p := 1461501624496790265145448589920785493717258890819; n := 1461501624496790265145447380994971188499300027613; Fp := GF(p); b := Fp!0; repeat repeat b := b + 1; until IsSquare(b + 1); y := Root(b + 1, 2); E := EllipticCurve([Fp | 0, b]); G := E![1, y]; until IsZero(n*G); "b =", b; "G =", G; rho := (p - n)^2; "rho =", rho; "rho^3 mod n =", rho^3 mod n; H := rho*G; "H =", H; zeta := H[1]; "zeta =", zeta; "zeta^3 =", zeta^3; lambda := Fp!2; while IsPower(lambda, 3) do lambda +:= 1; end while; "lambda =", lambda; Fp2 := ExtensionField; mu := i + 1; while IsSquare(mu) do mu +:= 1; end while; "mu =", mu; xi := 1/(lambda^2*mu^3); Fp12 := ExtensionField; Et := EllipticCurve([Fp2 | 0, b/xi]); xt := 1; while not IsSquare(xt^3 + b/xi) do xt +:= 1; end while; yt := Sqrt(xt^3 + b/xi); Gt := Et![xt, yt]; "Gt =", Gt; k := 12; assert (p^k - 1) mod n eq 0; chi := (p^k - 1) div n; Ek := EllipticCurve([Fp12 | 0, Fp12!b]); g := function(U, V, Q) if IsZero(U) or IsZero(V) or (U eq -V) or IsZero(Q) then return Fp12!1; end if; /* m := (U eq V) select 3*U[1]^2/(2*U[2]) else (V[2] - U[2])/(V[1] - U[1]); return m*(Q[1] - U[1]) + U[2] - Q[2]; */ if U eq V then m := 3*U[1]^2; s := 2*U[2]; else m := V[2] - U[2]; s := V[1] - U[1]; end if; return m*(Q[1] - U[1]) + s*(U[2] - Q[2]); end function; length := function(r) n := 0; v := 1; while v le r do n +:= 1; v +:= v; end while; return n; end function; bit := function(r, i) return (r div 2^i) mod 2; end function; tate := function(P, Qt) Q := Ek![Qt[1]*z^2, Qt[2]*z^3]; f := 1; A := P; for i := length(n) - 2 to 0 by -1 do f := f^2*g(A, A, Q); A := 2*A; if bit(n, i) eq 1 then f *:= g(A, P, Q); A +:= P; end if; end for; return f^chi; end function; g := tate(G, Gt); "g =", g; /* for j in [1..11] do print "g^(p^" * Sprint(j) * "-1): " * Sprint(g^(p^j-1) ne 1); end for; print "g^(p^12-1): " * Sprint(g^(p^12-1) eq 1); "g^n =", g^n; */ /* for j in [1..100] do "--", j; u := Random(n-1); v := Random(n-1); w := tate(u*G, v*Gt); h := g^((u*v) mod n); if w ne h then "Failure!"; print "u = " * Sprint(u); print "v = " * Sprint(v); print "e(u*P, v*Q) = " * Sprint(w); print "e(P, Q)^(u*v) = " * Sprint(h); quit; end if; end for; "Success!"; */ trace4 := function(g) t := g^p; t ^:= p; t ^:= p; t ^:= p; // g^(p^4) s := t^p; s ^:= p; s ^:= p; s ^:= p; // g^(p^8) return g + t + s; end function; trace6 := function(g) t := g^p; t ^:= p; t ^:= p; t ^:= p; t ^:= p; t ^:= p; // g^(p^6) return g + t; end function; t40 := trace4(g); t41 := t40^p; t42 := t41^p; "trace4(g) =", t40; "trace4(g)' =", t41; "trace4(g)'' =", t42; "sum4 =", t40 + t41 + t42; "prod4 =", t40*t41*t42; t60 := trace6(g); t61 := t60^p; "trace6(g) =", t60; "trace6(g)' =", t61; "sum6 =", t60 + t61; "prod6 =", t60*t61; Output: Magma V2.11-10 Tue Dec 13 2005 21:15:14 on modular [Seed = 2387758467] ------------------------------------- b = 3 G = (1 : 1461501624496790265145448589920785493717258890817 : 1) rho = 1461501624493534334825397658811989710051820598436 rho^3 mod n = 1461501624496790265145447380994971188499300027612 H = (1627965160026674480212199743920457792 : 2 : 1) zeta = 1627965160026674480212199743920457792 zeta^3 = 1 lambda = 2 mu = i + 1 Gt = (8 : 645238442624673913635245604741558906439788126004*i + 1165636379991084560121783183184170319793834310968 : 1) g = (348249054088692360343664384716060307282221240198*i + 632608786394229733147453879629382109816155603996)*z^5 + (78288732951012207123651737359843385303103429616*i + 1421790144970157374993435834657786699116722047068)*z^4 + (126808104546169378741740415470754970160692958373*i + 1395505911017780163560370399495096477368129061128)*z^3 + (1151508433446340267838152948068030563616071711737*i + 1331505104782800437119636772308446788364866291823)*z^2 + (294195016254996862796265848147501809413184337240*i + 648097608752361631735328497390888861121749959677)*z + 800403695404043311146437378782593638877310314601*i + 901249395544166733264846949825665528694219338474 trace4(g) = (380424313638508136225221246412264910482078875119*i + 1263514484059759960390214018643718444669869401746)*z^3 + 939709461715339668293863546426995422914672052984*i + 1242246562135709934649092259556211092365399124603 trace4(g)' = (861131396559103494032251111638107159797161227337*i + 239926059884489090698598604094815138857484117938)*z^3 + 521792162781450596851585043493790070802586837835*i + 1242246562135709934649092259556211092365399124603 trace4(g)'' = (1081077310858282128920227343508520583235180015700*i + 197987140437030304755234571277067049047389489073)*z^3 + 939709461715339668293863546426995422914672052984*i + 1242246562135709934649092259556211092365399124603 sum4 = (861131396559103494032251111638107159797161227337*i + 239926059884489090698598604094815138857484117938)*z^3 + 939709461715339668293863546426995422914672052984*i + 803736437413549273656379598827062289661679592171 prod4 = (1109541557279070038674422371008660670501870047821*i + 825029738971209132845985263278243991593718471901)*z^3 + 1073254876702081306986584990089982603797367449521*i + 1088939299807643406655341764511290451420861288604 trace6(g) = (156577465902024414247303474719686770606206859232*i + 1382078665443524484841423079394787904516185203317)*z^4 + (841515242395890270530857306215275633514884532655*i + 1201508585068810609093824954696108083012473692827)*z^2 + 139305766311296357147426167644401784037361738383*i + 340997166591543201384245309730545563671179786129 trace6(g)' = (39750298548322008993974458110694835814573932369*i + 170269780948936534327757016306807241564764251208)*z^4 + (96267089637067702603160574155571500808560497740*i + 198193976097141083414977909683245959979351734515)*z^2 + 1322195858185493907998022422276383709679897152436*i + 340997166591543201384245309730545563671179786129 sum6 = (196327764450346423241277932830381606420780791601*i + 90846821895670754023731505780809652363690563706)*z^4 + (937782332032957973134017880370847134323445030395*i + 1399702561165951692508802864379354042991825427342)*z^2 + 681994333183086402768490619461091127342359572258 prod6 = (403224252430258213018808164592556921112551430842*i + 1072501587439756383839928276680242651041055479001)*z^4 + (515192785473442154214411326985372045692209216551*i + 1264811833155190234052567165579767753345088245117)*z^2 + 933753153541436149967745700541145775019625660786*i + 579210675113629004884943196866573303442263067154 Total time: 1.070 seconds, Total memory usage: 3.34MB '72.19.1' ************** MAGMA ***************** Host 72.19.126.33 (72.19.126.33) Time: Tue Dec 13 21:14:51 2005 Input: F := FunctionField(GF(5,2)); A := AffineAlgebra; P := PolynomialRing(F); x^-1; > MinimalPolynomial(x-y); Output: Magma V2.11-10 Tue Dec 13 2005 21:14:50 on modular [Seed = 2471448758] ------------------------------------- x^4*y^4 + x^4 + y^4 + 1 z^25 + (t^4 + 1)/t^4*z^5 + 1/t^4*z + (t^9 + t + 4)/t^4 Total time: 0.190 seconds, Total memory usage: 3.34MB '200.177' ************** MAGMA ***************** Host 200.177.7.38 (200.177.7.38) Time: Tue Dec 13 21:14:43 2005 Input: p := 1461501624496790265145448589920785493717258890819; n := 1461501624496790265145447380994971188499300027613; Fp := GF(p); b := Fp!0; repeat repeat b := b + 1; until IsSquare(b + 1); y := Root(b + 1, 2); E := EllipticCurve([Fp | 0, b]); G := E![1, y]; until IsZero(n*G); "b =", b; "G =", G; rho := (p - n)^2; "rho =", rho; "rho^3 mod n =", rho^3 mod n; H := rho*G; "H =", H; zeta := H[1]; "zeta =", zeta; "zeta^3 mod p =", zeta^3 mod p; lambda := Fp!2; while IsPower(lambda, 3) do lambda +:= 1; end while; "lambda =", lambda; Fp2 := ExtensionField; mu := i + 1; while IsSquare(mu) do mu +:= 1; end while; "mu =", mu; xi := 1/(lambda^2*mu^3); Fp12 := ExtensionField; Et := EllipticCurve([Fp2 | 0, b/xi]); xt := 1; while not IsSquare(xt^3 + b/xi) do xt +:= 1; end while; yt := Sqrt(xt^3 + b/xi); Gt := Et![xt, yt]; "Gt =", Gt; k := 12; assert (p^k - 1) mod n eq 0; chi := (p^k - 1) div n; Ek := EllipticCurve([Fp12 | 0, Fp12!b]); g := function(U, V, Q) if IsZero(U) or IsZero(V) or (U eq -V) or IsZero(Q) then return Fp12!1; end if; /* m := (U eq V) select 3*U[1]^2/(2*U[2]) else (V[2] - U[2])/(V[1] - U[1]); return m*(Q[1] - U[1]) + U[2] - Q[2]; */ if U eq V then m := 3*U[1]^2; s := 2*U[2]; else m := V[2] - U[2]; s := V[1] - U[1]; end if; return m*(Q[1] - U[1]) + s*(U[2] - Q[2]); end function; length := function(r) n := 0; v := 1; while v le r do n +:= 1; v +:= v; end while; return n; end function; bit := function(r, i) return (r div 2^i) mod 2; end function; tate := function(P, Qt) Q := Ek![Qt[1]*z^2, Qt[2]*z^3]; f := 1; A := P; for i := length(n) - 2 to 0 by -1 do f := f^2*g(A, A, Q); A := 2*A; if bit(n, i) eq 1 then f *:= g(A, P, Q); A +:= P; end if; end for; return f^chi; end function; g := tate(G, Gt); "g =", g; /* for j in [1..11] do print "g^(p^" * Sprint(j) * "-1): " * Sprint(g^(p^j-1) ne 1); end for; print "g^(p^12-1): " * Sprint(g^(p^12-1) eq 1); "g^n =", g^n; */ /* for j in [1..100] do "--", j; u := Random(n-1); v := Random(n-1); w := tate(u*G, v*Gt); h := g^((u*v) mod n); if w ne h then "Failure!"; print "u = " * Sprint(u); print "v = " * Sprint(v); print "e(u*P, v*Q) = " * Sprint(w); print "e(P, Q)^(u*v) = " * Sprint(h); quit; end if; end for; "Success!"; */ trace4 := function(g) t := g^p; t ^:= p; t ^:= p; t ^:= p; // g^(p^4) s := t^p; s ^:= p; s ^:= p; s ^:= p; // g^(p^8) return g + t + s; end function; trace6 := function(g) t := g^p; t ^:= p; t ^:= p; t ^:= p; t ^:= p; t ^:= p; // g^(p^6) return g + t; end function; t40 := trace4(g); t41 := t40^p; t42 := t41^p; "trace4(g) =", t40; "trace4(g)' =", t41; "trace4(g)'' =", t42; "sum4 =", t40 + t41 + t42; "prod4 =", t40*t41*t42; t60 := trace6(g); t61 := t60^p; "trace6(g) =", t60; "trace6(g)' =", t61; "sum6 =", t60 + t61; "prod6 =", t60*t61; Output: Magma V2.11-10 Tue Dec 13 2005 21:14:42 on modular [Seed = 2554616757] ------------------------------------- b = 3 G = (1 : 1461501624496790265145448589920785493717258890817 : 1) rho = 1461501624493534334825397658811989710051820598436 rho^3 mod n = 1461501624496790265145447380994971188499300027612 H = (1627965160026674480212199743920457792 : 2 : 1) zeta = 1627965160026674480212199743920457792 >> "zeta^3 mod p =", zeta^3 mod p; ^ Runtime error in 'mod': Bad argument types Argument types given: FldFinElt, FldFinElt lambda = 2 mu = i + 1 Gt = (8 : 645238442624673913635245604741558906439788126004*i + 1165636379991084560121783183184170319793834310968 : 1) g = (348249054088692360343664384716060307282221240198*i + 632608786394229733147453879629382109816155603996)*z^5 + (78288732951012207123651737359843385303103429616*i + 1421790144970157374993435834657786699116722047068)*z^4 + (126808104546169378741740415470754970160692958373*i + 1395505911017780163560370399495096477368129061128)*z^3 + (1151508433446340267838152948068030563616071711737*i + 1331505104782800437119636772308446788364866291823)*z^2 + (294195016254996862796265848147501809413184337240*i + 648097608752361631735328497390888861121749959677)*z + 800403695404043311146437378782593638877310314601*i + 901249395544166733264846949825665528694219338474 trace4(g) = (380424313638508136225221246412264910482078875119*i + 1263514484059759960390214018643718444669869401746)*z^3 + 939709461715339668293863546426995422914672052984*i + 1242246562135709934649092259556211092365399124603 trace4(g)' = (861131396559103494032251111638107159797161227337*i + 239926059884489090698598604094815138857484117938)*z^3 + 521792162781450596851585043493790070802586837835*i + 1242246562135709934649092259556211092365399124603 trace4(g)'' = (1081077310858282128920227343508520583235180015700*i + 197987140437030304755234571277067049047389489073)*z^3 + 939709461715339668293863546426995422914672052984*i + 1242246562135709934649092259556211092365399124603 sum4 = (861131396559103494032251111638107159797161227337*i + 239926059884489090698598604094815138857484117938)*z^3 + 939709461715339668293863546426995422914672052984*i + 803736437413549273656379598827062289661679592171 prod4 = (1109541557279070038674422371008660670501870047821*i + 825029738971209132845985263278243991593718471901)*z^3 + 1073254876702081306986584990089982603797367449521*i + 1088939299807643406655341764511290451420861288604 trace6(g) = (156577465902024414247303474719686770606206859232*i + 1382078665443524484841423079394787904516185203317)*z^4 + (841515242395890270530857306215275633514884532655*i + 1201508585068810609093824954696108083012473692827)*z^2 + 139305766311296357147426167644401784037361738383*i + 340997166591543201384245309730545563671179786129 trace6(g)' = (39750298548322008993974458110694835814573932369*i + 170269780948936534327757016306807241564764251208)*z^4 + (96267089637067702603160574155571500808560497740*i + 198193976097141083414977909683245959979351734515)*z^2 + 1322195858185493907998022422276383709679897152436*i + 340997166591543201384245309730545563671179786129 sum6 = (196327764450346423241277932830381606420780791601*i + 90846821895670754023731505780809652363690563706)*z^4 + (937782332032957973134017880370847134323445030395*i + 1399702561165951692508802864379354042991825427342)*z^2 + 681994333183086402768490619461091127342359572258 prod6 = (403224252430258213018808164592556921112551430842*i + 1072501587439756383839928276680242651041055479001)*z^4 + (515192785473442154214411326985372045692209216551*i + 1264811833155190234052567165579767753345088245117)*z^2 + 933753153541436149967745700541145775019625660786*i + 579210675113629004884943196866573303442263067154 Total time: 1.100 seconds, Total memory usage: 3.34MB '200.177' ************** MAGMA ***************** Host 200.177.7.38 (200.177.7.38) Time: Tue Dec 13 21:07:56 2005 Input: p := 1461501624496790265145448589920785493717258890819; n := 1461501624496790265145447380994971188499300027613; Fp := GF(p); b := Fp!0; repeat repeat b := b + 1; until IsSquare(b + 1); y := Root(b + 1, 2); E := EllipticCurve([Fp | 0, b]); G := E![1, y]; until IsZero(n*G); "b =", b; "G =", G; lambda := Fp!2; while IsPower(lambda, 3) do lambda +:= 1; end while; "lambda =", lambda; Fp2 := ExtensionField; mu := i + 1; while IsSquare(mu) do mu +:= 1; end while; "mu =", mu; xi := 1/(lambda^2*mu^3); Fp12 := ExtensionField; Et := EllipticCurve([Fp2 | 0, b/xi]); xt := 1; while not IsSquare(xt^3 + b/xi) do xt +:= 1; end while; yt := Sqrt(xt^3 + b/xi); Gt := Et![xt, yt]; "Gt =", Gt; k := 12; assert (p^k - 1) mod n eq 0; chi := (p^k - 1) div n; Ek := EllipticCurve([Fp12 | 0, Fp12!b]); g := function(U, V, Q) if IsZero(U) or IsZero(V) or (U eq -V) or IsZero(Q) then return Fp12!1; end if; /* m := (U eq V) select 3*U[1]^2/(2*U[2]) else (V[2] - U[2])/(V[1] - U[1]); return m*(Q[1] - U[1]) + U[2] - Q[2]; */ if U eq V then m := 3*U[1]^2; s := 2*U[2]; else m := V[2] - U[2]; s := V[1] - U[1]; end if; return m*(Q[1] - U[1]) + s*(U[2] - Q[2]); end function; length := function(r) n := 0; v := 1; while v le r do n +:= 1; v +:= v; end while; return n; end function; bit := function(r, i) return (r div 2^i) mod 2; end function; tate := function(P, Qt) Q := Ek![Qt[1]*z^2, Qt[2]*z^3]; f := 1; A := P; for i := length(n) - 2 to 0 by -1 do f := f^2*g(A, A, Q); A := 2*A; if bit(n, i) eq 1 then f *:= g(A, P, Q); A +:= P; end if; end for; return f^chi; end function; g := tate(G, Gt); "g =", g; /* for j in [1..11] do print "g^(p^" * Sprint(j) * "-1): " * Sprint(g^(p^j-1) ne 1); end for; print "g^(p^12-1): " * Sprint(g^(p^12-1) eq 1); "g^n =", g^n; */ /* for j in [1..100] do "--", j; u := Random(n-1); v := Random(n-1); w := tate(u*G, v*Gt); h := g^((u*v) mod n); if w ne h then "Failure!"; print "u = " * Sprint(u); print "v = " * Sprint(v); print "e(u*P, v*Q) = " * Sprint(w); print "e(P, Q)^(u*v) = " * Sprint(h); quit; end if; end for; "Success!"; */ trace4 := function(g) t := g^p; t ^:= p; t ^:= p; t ^:= p; // g^(p^4) s := t^p; s ^:= p; s ^:= p; s ^:= p; // g^(p^8) return g + t + s; end function; trace6 := function(g) t := g^p; t ^:= p; t ^:= p; t ^:= p; t ^:= p; t ^:= p; // g^(p^6) return g + t; end function; t40 := trace4(g); t41 := t40^p; t42 := t41^p; "trace4(g) =", t40; "trace4(g)' =", t41; "trace4(g)'' =", t42; "sum4 =", t40 + t41 + t42; "prod4 =", t40*t41*t42; t60 := trace6(g); t61 := t60^p; "trace6(g) =", t60; "trace6(g)' =", t61; "sum6 =", t60 + t61; "prod6 =", t60*t61; Output: Magma V2.11-10 Tue Dec 13 2005 21:07:54 on modular [Seed = 2909901032] ------------------------------------- b = 3 G = (1 : 1461501624496790265145448589920785493717258890817 : 1) lambda = 2 mu = i + 1 Gt = (8 : 645238442624673913635245604741558906439788126004*i + 1165636379991084560121783183184170319793834310968 : 1) g = (348249054088692360343664384716060307282221240198*i + 632608786394229733147453879629382109816155603996)*z^5 + (78288732951012207123651737359843385303103429616*i + 1421790144970157374993435834657786699116722047068)*z^4 + (126808104546169378741740415470754970160692958373*i + 1395505911017780163560370399495096477368129061128)*z^3 + (1151508433446340267838152948068030563616071711737*i + 1331505104782800437119636772308446788364866291823)*z^2 + (294195016254996862796265848147501809413184337240*i + 648097608752361631735328497390888861121749959677)*z + 800403695404043311146437378782593638877310314601*i + 901249395544166733264846949825665528694219338474 trace4(g) = (380424313638508136225221246412264910482078875119*i + 1263514484059759960390214018643718444669869401746)*z^3 + 939709461715339668293863546426995422914672052984*i + 1242246562135709934649092259556211092365399124603 trace4(g)' = (861131396559103494032251111638107159797161227337*i + 239926059884489090698598604094815138857484117938)*z^3 + 521792162781450596851585043493790070802586837835*i + 1242246562135709934649092259556211092365399124603 trace4(g)'' = (1081077310858282128920227343508520583235180015700*i + 197987140437030304755234571277067049047389489073)*z^3 + 939709461715339668293863546426995422914672052984*i + 1242246562135709934649092259556211092365399124603 sum4 = (861131396559103494032251111638107159797161227337*i + 239926059884489090698598604094815138857484117938)*z^3 + 939709461715339668293863546426995422914672052984*i + 803736437413549273656379598827062289661679592171 prod4 = (1109541557279070038674422371008660670501870047821*i + 825029738971209132845985263278243991593718471901)*z^3 + 1073254876702081306986584990089982603797367449521*i + 1088939299807643406655341764511290451420861288604 trace6(g) = (156577465902024414247303474719686770606206859232*i + 1382078665443524484841423079394787904516185203317)*z^4 + (841515242395890270530857306215275633514884532655*i + 1201508585068810609093824954696108083012473692827)*z^2 + 139305766311296357147426167644401784037361738383*i + 340997166591543201384245309730545563671179786129 trace6(g)' = (39750298548322008993974458110694835814573932369*i + 170269780948936534327757016306807241564764251208)*z^4 + (96267089637067702603160574155571500808560497740*i + 198193976097141083414977909683245959979351734515)*z^2 + 1322195858185493907998022422276383709679897152436*i + 340997166591543201384245309730545563671179786129 sum6 = (196327764450346423241277932830381606420780791601*i + 90846821895670754023731505780809652363690563706)*z^4 + (937782332032957973134017880370847134323445030395*i + 1399702561165951692508802864379354042991825427342)*z^2 + 681994333183086402768490619461091127342359572258 prod6 = (403224252430258213018808164592556921112551430842*i + 1072501587439756383839928276680242651041055479001)*z^4 + (515192785473442154214411326985372045692209216551*i + 1264811833155190234052567165579767753345088245117)*z^2 + 933753153541436149967745700541145775019625660786*i + 579210675113629004884943196866573303442263067154 Total time: 1.060 seconds, Total memory usage: 3.34MB '72.19.1' ************** MAGMA ***************** Host 72.19.126.33 (72.19.126.33) Time: Tue Dec 13 21:07:15 2005 Input: F := FunctionField(GF(5,2)); A := AffineAlgebra; P := PolynomialRing(F); x^-1; > MinimalPolynomial(x+y); Output: Magma V2.11-10 Tue Dec 13 2005 21:07:15 on modular [Seed = 2993590238] ------------------------------------- x^4*y^4 + x^4 + y^4 + 1 z^25 + (t^4 + 1)/t^4*z^5 + 1/t^4*z + (4*t^9 + 4*t + 4)/t^4 Total time: 0.190 seconds, Total memory usage: 3.34MB '72.19.1' ************** MAGMA ***************** Host 72.19.126.33 (72.19.126.33) Time: Tue Dec 13 21:06:43 2005 Input: F := FunctionField(GF(5,2)); A := AffineAlgebra; P := PolynomialRing(F); x^-1; > MinimalPolynomial(x+t^(-3)*y); Output: Magma V2.11-10 Tue Dec 13 2005 21:06:43 on modular [Seed = 3077278395] ------------------------------------- x^4*y^4 + x^4 + y^4 + 1 z^25 + (t^60 + t^56 + 4*t^54 + t^52 + 4*t^50 + t^48 + t^42 + 3*t^40 + 3*t^38 + t^36 + t^28 + 2*t^26 + t^24 + t^14 + t^12 + 1)/t^60*z^5 + (t^56 + 4*t^54 + t^52 + 4*t^50 + t^48 + t^42 + 3*t^40 + 3*t^38 + t^36 + t^28 + 2*t^26 + t^24 + t^14 + t^12 + 1)/t^60*z + (4*t^66 + 4*t^56 + 4*t^52 + t^50 + 4*t^48 + t^46 + 4*t^44 + 4*t^38 + 2*t^36 + 2*t^34 + 4*t^32 + 4*t^24 + 3*t^22 + 4*t^20 + 4*t^10 + 4*t^8 + 4)/t^70 Total time: 0.220 seconds, Total memory usage: 3.43MB '72.19.1' ************** MAGMA ***************** Host 72.19.126.33 (72.19.126.33) Time: Tue Dec 13 21:05:48 2005 Input: F := FunctionField(GF(5,2)); A := AffineAlgebra; P := PolynomialRing(F); x^-1; > MinimalPolynomial(x+t^(-3)*y); Output: Magma V2.11-10 Tue Dec 13 2005 21:05:48 on modular [Seed = 3160441471] ------------------------------------- (4*t^2 + 4*t)/(t^3 + 2*t^2 + 3*t + 1)*x*y^2 + 4*t^2/(t^3 + 2*t^2 + 3*t + 1)*x*y + (4*t^3 + 3*t^2 + 4*t)/(t^3 + 2*t^2 + 3*t + 1)*x z^6 + (3*t + 3)/t*z^4 + (4*t^5 + 3)/t^8*z^3 + (3*t^2 + t + 3)/t^2*z^2 + (4*t^6 + 4*t^5 + t + 1)/t^9*z + (t^16 + 2*t^15 + 3*t^14 + t^13 + 3*t^10 + 2*t^5 + 1)/t^16 Total time: 0.190 seconds, Total memory usage: 3.34MB '72.19.1' ************** MAGMA ***************** Host 72.19.126.33 (72.19.126.33) Time: Tue Dec 13 21:04:26 2005 Input: F := FunctionField(IntegerRing()); A := AffineAlgebra; P := PolynomialRing(F); x^-1; > MinimalPolynomial(x+t^(-3)*y); Output: Magma V2.11-10 Tue Dec 13 2005 21:04:26 on modular [Seed = 1100840916] ------------------------------------- (-t^2 - t)/(t^3 + 2*t^2 + 3*t + 1)*x*y^2 - t^2/(t^3 + 2*t^2 + 3*t + 1)*x*y + (-t^3 - 2*t^2 - t)/(t^3 + 2*t^2 + 3*t + 1)*x z^6 + (3*t + 3)/t*z^4 + (-6*t^5 - 2)/t^8*z^3 + (3*t^2 + 6*t + 3)/t^2*z^2 + (-6*t^6 - 6*t^5 + 6*t + 6)/t^9*z + (t^16 + 2*t^15 + 3*t^14 + t^13 + 3*t^10 - 3*t^5 + 1)/t^16 Total time: 0.190 seconds, Total memory usage: 3.34MB '72.19.1' ************** MAGMA ***************** Host 72.19.126.33 (72.19.126.33) Time: Tue Dec 13 21:04:09 2005 Input: F := FunctionField(IntegerRing()); A := AffineAlgebra; P := PolynomialRing(F); x^-1; > MinimalPolynomial(x+t^3*y); Output: Magma V2.11-10 Tue Dec 13 2005 21:04:09 on modular [Seed = 1184528608] ------------------------------------- (-t^2 - t)/(t^3 + 2*t^2 + 3*t + 1)*x*y^2 - t^2/(t^3 + 2*t^2 + 3*t + 1)*x*y + (-t^3 - 2*t^2 - t)/(t^3 + 2*t^2 + 3*t + 1)*x z^6 + (3*t + 3)/t*z^4 + (-2*t^10 - 6*t^3)*z^3 + (3*t^2 + 6*t + 3)/t^2*z^2 + (6*t^10 + 6*t^9 - 6*t^3 - 6*t^2)*z + (t^23 - 3*t^16 + 3*t^9 + t^3 + 2*t^2 + 3*t + 1)/t^3 Total time: 0.190 seconds, Total memory usage: 3.34MB '72.19.1' ************** MAGMA ***************** Host 72.19.126.33 (72.19.126.33) Time: Tue Dec 13 21:03:46 2005 Input: F := FunctionField(IntegerRing()); A := AffineAlgebra; P := PolynomialRing(F); x^-1; > MinimalPolynomial(x+y); Output: Magma V2.11-10 Tue Dec 13 2005 21:03:46 on modular [Seed = 1268217334] ------------------------------------- (-t^2 - t)/(t^3 + 2*t^2 + 3*t + 1)*x*y^2 - t^2/(t^3 + 2*t^2 + 3*t + 1)*x*y + (-t^3 - 2*t^2 - t)/(t^3 + 2*t^2 + 3*t + 1)*x z^6 + (3*t + 3)/t*z^4 + (-2*t - 6)*z^3 + (3*t^2 + 6*t + 3)/t^2*z^2 + (6*t^2 - 6)/t*z + (t^5 - 3*t^4 + 4*t^3 + 2*t^2 + 3*t + 1)/t^3 Total time: 0.190 seconds, Total memory usage: 3.24MB '72.19.1' ************** MAGMA ***************** Host 72.19.126.33 (72.19.126.33) Time: Tue Dec 13 21:01:59 2005 Input: F := FunctionField(IntegerRing()); A := AffineAlgebra; P := PolynomialRing(F); x^-1; > MinimalPolynomial(x); Output: Magma V2.11-10 Tue Dec 13 2005 21:01:59 on modular [Seed = 1368223609] ------------------------------------- (-t^2 - t)/(t^3 + 2*t^2 + 3*t + 1)*x*y^2 - t^2/(t^3 + 2*t^2 + 3*t + 1)*x*y + (-t^3 - 2*t^2 - t)/(t^3 + 2*t^2 + 3*t + 1)*x z^6 + (3*t + 3)/t*z^4 + (3*t^2 + 6*t + 3)/t^2*z^2 + (t^3 + 2*t^2 + 3*t + 1)/t^3 Total time: 0.190 seconds, Total memory usage: 3.24MB '200.177' ************** MAGMA ***************** Host 200.177.7.38 (200.177.7.38) Time: Tue Dec 13 20:57:39 2005 Input: p := 1461501624496790265145448589920785493717258890819; n := 1461501624496790265145447380994971188499300027613; Fp := GF(p); b := Fp!0; repeat repeat b := b + 1; until IsSquare(b + 1); y := Root(b + 1, 2); E := EllipticCurve([Fp | 0, b]); G := E![1, y]; until IsZero(n*G); "b =", b; "G =", G; lambda := Fp!2; while IsPower(lambda, 3) do lambda +:= 1; end while; "lambda =", lambda; Fp2 := ExtensionField; mu := i + 1; while IsSquare(mu) do mu +:= 1; end while; "mu =", mu; xi := 1/(lambda^2*mu^3); Fp12 := ExtensionField; Et := EllipticCurve([Fp2 | 0, b/xi]); xt := 1; while not IsSquare(xt^3 + b/xi) do xt +:= 1; end while; yt := Sqrt(xt^3 + b/xi); Gt := Et![xt, yt]; "Gt =", Gt; k := 12; assert (p^k - 1) mod n eq 0; chi := (p^k - 1) div n; Ek := EllipticCurve([Fp12 | 0, Fp12!b]); g := function(U, V, Q) if IsZero(U) or IsZero(V) or (U eq -V) or IsZero(Q) then return Fp12!1; end if; /* m := (U eq V) select 3*U[1]^2/(2*U[2]) else (V[2] - U[2])/(V[1] - U[1]); return m*(Q[1] - U[1]) + U[2] - Q[2]; */ if U eq V then m := 3*U[1]^2; s := 2*U[2]; else m := V[2] - U[2]; s := V[1] - U[1]; end if; return m*(Q[1] - U[1]) + s*(U[2] - Q[2]); end function; length := function(r) n := 0; v := 1; while v le r do n +:= 1; v +:= v; end while; return n; end function; bit := function(r, i) return (r div 2^i) mod 2; end function; tate := function(P, Qt) Q := Ek![Qt[1]*z^2, Qt[2]*z^3]; f := 1; A := P; for i := length(n) - 2 to 0 by -1 do f := f^2*g(A, A, Q); A := 2*A; if bit(n, i) eq 1 then f *:= g(A, P, Q); A +:= P; end if; end for; return f^chi; end function; g := tate(G, Gt); "g =", g; /* for j in [1..11] do print "g^(p^" * Sprint(j) * "-1): " * Sprint(g^(p^j-1) ne 1); end for; print "g^(p^12-1): " * Sprint(g^(p^12-1) eq 1); "g^n =", g^n; */ /* for j in [1..100] do "--", j; u := Random(n-1); v := Random(n-1); w := tate(u*G, v*Gt); h := g^((u*v) mod n); if w ne h then "Failure!"; print "u = " * Sprint(u); print "v = " * Sprint(v); print "e(u*P, v*Q) = " * Sprint(w); print "e(P, Q)^(u*v) = " * Sprint(h); quit; end if; end for; "Success!"; */ trace4 := function(g) t := g^p; t ^:= p; t ^:= p; t ^:= p; // g^(p^4) s := t^p; s ^:= p; s ^:= p; s ^:= p; // g^(p^8) return g + t + s; end function; trace6 := function(g) t := g^p; t ^:= p; t ^:= p; t ^:= p; t ^:= p; t ^:= p; // g^(p^6) return g + t; end function; t4 := trace4(g); "trace4(g) =", t4; "trace4(g)' =", t4^p; "trace4(g)'' =", t4^(p^2); t6 := trace6(g); "trace6(g) =", t6; "trace6(g)' =", t6^p; Output: Magma V2.11-10 Tue Dec 13 2005 20:57:38 on modular [Seed = 1485072245] ------------------------------------- b = 3 G = (1 : 1461501624496790265145448589920785493717258890817 : 1) lambda = 2 mu = i + 1 Gt = (8 : 645238442624673913635245604741558906439788126004*i + 1165636379991084560121783183184170319793834310968 : 1) g = (348249054088692360343664384716060307282221240198*i + 632608786394229733147453879629382109816155603996)*z^5 + (78288732951012207123651737359843385303103429616*i + 1421790144970157374993435834657786699116722047068)*z^4 + (126808104546169378741740415470754970160692958373*i + 1395505911017780163560370399495096477368129061128)*z^3 + (1151508433446340267838152948068030563616071711737*i + 1331505104782800437119636772308446788364866291823)*z^2 + (294195016254996862796265848147501809413184337240*i + 648097608752361631735328497390888861121749959677)*z + 800403695404043311146437378782593638877310314601*i + 901249395544166733264846949825665528694219338474 trace4(g) = (380424313638508136225221246412264910482078875119*i + 1263514484059759960390214018643718444669869401746)*z^3 + 939709461715339668293863546426995422914672052984*i + 1242246562135709934649092259556211092365399124603 trace4(g)' = (861131396559103494032251111638107159797161227337*i + 239926059884489090698598604094815138857484117938)*z^3 + 521792162781450596851585043493790070802586837835*i + 1242246562135709934649092259556211092365399124603 trace4(g)'' = (1081077310858282128920227343508520583235180015700*i + 197987140437030304755234571277067049047389489073)*z^3 + 939709461715339668293863546426995422914672052984*i + 1242246562135709934649092259556211092365399124603 trace6(g) = (156577465902024414247303474719686770606206859232*i + 1382078665443524484841423079394787904516185203317)*z^4 + (841515242395890270530857306215275633514884532655*i + 1201508585068810609093824954696108083012473692827)*z^2 + 139305766311296357147426167644401784037361738383*i + 340997166591543201384245309730545563671179786129 trace6(g)' = (39750298548322008993974458110694835814573932369*i + 170269780948936534327757016306807241564764251208)*z^4 + (96267089637067702603160574155571500808560497740*i + 198193976097141083414977909683245959979351734515)*z^2 + 1322195858185493907998022422276383709679897152436*i + 340997166591543201384245309730545563671179786129 Total time: 1.080 seconds, Total memory usage: 3.34MB '72.19.1' ************** MAGMA ***************** Host 72.19.126.33 (72.19.126.33) Time: Tue Dec 13 20:42:28 2005 Input: R := FunctionField(GF(5,2)); P := PolynomialRing(R); F := ext< R | y^5+y - x >; F; Q := PolynomialRing(P); Q; G := ext; G; Output: Magma V2.11-10 Tue Dec 13 2005 20:42:27 on modular [Seed = 1673484511] ------------------------------------- Algebraic function field defined over Univariate rational function field over GF(5^2) by y^5 + y + 4*x Univariate Polynomial Ring in z over Univariate Polynomial Ring in y over Univariate rational function field over GF(5^2) Algebraic function field defined over F by $.1^5 + $.1 + 4/x*alpha Total time: 0.310 seconds, Total memory usage: 7.89MB '200.177' ************** MAGMA ***************** Host 200.177.7.38 (200.177.7.38) Time: Tue Dec 13 20:40:50 2005 Input: p := 1461501624496790265145448589920785493717258890819; n := 1461501624496790265145447380994971188499300027613; Fp := GF(p); b := Fp!0; repeat repeat b := b + 1; until IsSquare(b + 1); y := Root(b + 1, 2); E := EllipticCurve([Fp | 0, b]); G := E![1, y]; until IsZero(n*G); "b =", b; "G =", G; lambda := Fp!2; while IsPower(lambda, 3) do lambda +:= 1; end while; "lambda =", lambda; Fp2 := ExtensionField; mu := i + 1; while IsSquare(mu) do mu +:= 1; end while; "mu =", mu; xi := 1/(lambda^2*mu^3); Fp12 := ExtensionField; Et := EllipticCurve([Fp2 | 0, b/xi]); xt := 1; while not IsSquare(xt^3 + b/xi) do xt +:= 1; end while; yt := Sqrt(xt^3 + b/xi); Gt := Et![xt, yt]; "Gt =", Gt; k := 12; assert (p^k - 1) mod n eq 0; chi := (p^k - 1) div n; Ek := EllipticCurve([Fp12 | 0, Fp12!b]); g := function(U, V, Q) if IsZero(U) or IsZero(V) or (U eq -V) or IsZero(Q) then return Fp12!1; end if; /* m := (U eq V) select 3*U[1]^2/(2*U[2]) else (V[2] - U[2])/(V[1] - U[1]); return m*(Q[1] - U[1]) + U[2] - Q[2]; */ if U eq V then m := 3*U[1]^2; s := 2*U[2]; else m := V[2] - U[2]; s := V[1] - U[1]; end if; return m*(Q[1] - U[1]) + s*(U[2] - Q[2]); end function; length := function(r) n := 0; v := 1; while v le r do n +:= 1; v +:= v; end while; return n; end function; bit := function(r, i) return (r div 2^i) mod 2; end function; tate := function(P, Qt) Q := Ek![Qt[1]*z^2, Qt[2]*z^3]; f := 1; A := P; for i := length(n) - 2 to 0 by -1 do f := f^2*g(A, A, Q); A := 2*A; if bit(n, i) eq 1 then f *:= g(A, P, Q); A +:= P; end if; end for; return f^chi; end function; g := tate(G, Gt); "g =", g; /* for j in [1..11] do print "g^(p^" * Sprint(j) * "-1): " * Sprint(g^(p^j-1) ne 1); end for; print "g^(p^12-1): " * Sprint(g^(p^12-1) eq 1); "g^n =", g^n; */ /* for j in [1..100] do "--", j; u := Random(n-1); v := Random(n-1); w := tate(u*G, v*Gt); h := g^((u*v) mod n); if w ne h then "Failure!"; print "u = " * Sprint(u); print "v = " * Sprint(v); print "e(u*P, v*Q) = " * Sprint(w); print "e(P, Q)^(u*v) = " * Sprint(h); quit; end if; end for; "Success!"; */ trace4 := function(g) t := g^p; t ^:= p; t ^:= p; t ^:= p; // g^(p^4) s := t^p; s ^:= p; s ^:= p; s ^:= p; // g^(p^8) return g + t + s; end function; "trace4(g) =", trace4(g); Output: Magma V2.11-10 Tue Dec 13 2005 20:40:49 on modular [Seed = 1756648310] ------------------------------------- b = 3 G = (1 : 1461501624496790265145448589920785493717258890817 : 1) lambda = 2 mu = i + 1 Gt = (8 : 645238442624673913635245604741558906439788126004*i + 1165636379991084560121783183184170319793834310968 : 1) g = (348249054088692360343664384716060307282221240198*i + 632608786394229733147453879629382109816155603996)*z^5 + (78288732951012207123651737359843385303103429616*i + 1421790144970157374993435834657786699116722047068)*z^4 + (126808104546169378741740415470754970160692958373*i + 1395505911017780163560370399495096477368129061128)*z^3 + (1151508433446340267838152948068030563616071711737*i + 1331505104782800437119636772308446788364866291823)*z^2 + (294195016254996862796265848147501809413184337240*i + 648097608752361631735328497390888861121749959677)*z + 800403695404043311146437378782593638877310314601*i + 901249395544166733264846949825665528694219338474 trace4(g) = (380424313638508136225221246412264910482078875119*i + 1263514484059759960390214018643718444669869401746)*z^3 + 939709461715339668293863546426995422914672052984*i + 1242246562135709934649092259556211092365399124603 Total time: 0.900 seconds, Total memory usage: 3.34MB '72.19.1' ************** MAGMA ***************** Host 72.19.126.33 (72.19.126.33) Time: Tue Dec 13 20:40:47 2005 Input: R := FunctionField(GF(5,2)); P := PolynomialRing(R); F := ext< R | y^5+y - x >; F; Q := PolynomialRing(P); G := ext; G; Output: Magma V2.11-10 Tue Dec 13 2005 20:40:47 on modular [Seed = 1823496012] ------------------------------------- Algebraic function field defined over Univariate rational function field over GF(5^2) by y^5 + y + 4*x Algebraic function field defined over F by $.1^5 + $.1 + 4/x*alpha Total time: 0.310 seconds, Total memory usage: 7.89MB '72.19.1' ************** MAGMA ***************** Host 72.19.126.33 (72.19.126.33) Time: Tue Dec 13 20:40:29 2005 Input: R := FunctionField(GF(5,2)); P := PolynomialRing(R); F := ext< R | y^5+y - x >; F; Q := PolynomialRing(P); G := ext; G; Output: Magma V2.11-10 Tue Dec 13 2005 20:40:29 on modular [Seed = 1924024640] ------------------------------------- Algebraic function field defined over Univariate rational function field over GF(5^2) by y^5 + y + 4*x >> G := ext; ^ Runtime error in '-': Bad argument types Argument types given: RngUPolElt[RngUPol[FldFunRat]], FldFunElt >> G;; ^ User error: Identifier 'G' has not been declared or assigned Total time: 0.180 seconds, Total memory usage: 3.34MB '72.19.1' ************** MAGMA ***************** Host 72.19.126.33 (72.19.126.33) Time: Tue Dec 13 20:40:12 2005 Input: R := FunctionField(GF(5,2)); P := PolynomialRing(R); F := ext< R | y^5+y - x >; F; Q := PolynomialRing(P); G := ext; G; Output: Magma V2.11-10 Tue Dec 13 2005 20:40:12 on modular [Seed = 2040876474] ------------------------------------- Algebraic function field defined over Univariate rational function field over GF(5^2) by y^5 + y + 4*x >> G := ext; ^ Runtime error in '-': Bad argument types Argument types given: RngUPolElt[RngUPol[FldFunRat]], FldFunElt >> G;; ^ User error: Identifier 'G' has not been declared or assigned Total time: 0.180 seconds, Total memory usage: 3.34MB '200.177' ************** MAGMA ***************** Host 200.177.7.38 (200.177.7.38) Time: Tue Dec 13 20:40:03 2005 Input: p := 1461501624496790265145448589920785493717258890819; n := 1461501624496790265145447380994971188499300027613; Fp := GF(p); b := Fp!0; repeat repeat b := b + 1; until IsSquare(b + 1); y := Root(b + 1, 2); E := EllipticCurve([Fp | 0, b]); G := E![1, y]; until IsZero(n*G); "b =", b; "G =", G; lambda := Fp!2; while IsPower(lambda, 3) do lambda +:= 1; end while; "lambda =", lambda; Fp2 := ExtensionField; mu := i + 1; while IsSquare(mu) do mu +:= 1; end while; "mu =", mu; xi := 1/(lambda^2*mu^3); Fp12 := ExtensionField; Et := EllipticCurve([Fp2 | 0, b/xi]); xt := 1; while not IsSquare(xt^3 + b/xi) do xt +:= 1; end while; yt := Sqrt(xt^3 + b/xi); Gt := Et![xt, yt]; "Gt =", Gt; k := 12; assert (p^k - 1) mod n eq 0; chi := (p^k - 1) div n; Ek := EllipticCurve([Fp12 | 0, Fp12!b]); g := function(U, V, Q) if IsZero(U) or IsZero(V) or (U eq -V) or IsZero(Q) then return Fp12!1; end if; /* m := (U eq V) select 3*U[1]^2/(2*U[2]) else (V[2] - U[2])/(V[1] - U[1]); return m*(Q[1] - U[1]) + U[2] - Q[2]; */ if U eq V then m := 3*U[1]^2; s := 2*U[2]; else m := V[2] - U[2]; s := V[1] - U[1]; end if; return m*(Q[1] - U[1]) + s*(U[2] - Q[2]); end function; length := function(r) n := 0; v := 1; while v le r do n +:= 1; v +:= v; end while; return n; end function; bit := function(r, i) return (r div 2^i) mod 2; end function; tate := function(P, Qt) Q := Ek![Qt[1]*z^2, Qt[2]*z^3]; f := 1; A := P; for i := length(n) - 2 to 0 by -1 do f := f^2*g(A, A, Q); A := 2*A; if bit(n, i) eq 1 then f *:= g(A, P, Q); A +:= P; end if; end for; return f^chi; end function; g := tate(G, Gt); "g =", g; "g[1] =", g[1]; /* for j in [1..11] do print "g^(p^" * Sprint(j) * "-1): " * Sprint(g^(p^j-1) ne 1); end for; print "g^(p^12-1): " * Sprint(g^(p^12-1) eq 1); "g^n =", g^n; */ /* for j in [1..100] do "--", j; u := Random(n-1); v := Random(n-1); w := tate(u*G, v*Gt); h := g^((u*v) mod n); if w ne h then "Failure!"; print "u = " * Sprint(u); print "v = " * Sprint(v); print "e(u*P, v*Q) = " * Sprint(w); print "e(P, Q)^(u*v) = " * Sprint(h); quit; end if; end for; "Success!"; */ trace4 := function(g) t := g^p; t ^:= p; t ^:= p; t ^:= p; // g^(p^4) s := t^p; s ^:= p; s ^:= p; s ^:= p; // g^(p^8) g + t + s; end function; "trace4(g) =", trace4(g); Output: Magma V2.11-10 Tue Dec 13 2005 20:40:02 on modular [Seed = 2124565171] ------------------------------------- b = 3 G = (1 : 1461501624496790265145448589920785493717258890817 : 1) lambda = 2 mu = i + 1 Gt = (8 : 645238442624673913635245604741558906439788126004*i + 1165636379991084560121783183184170319793834310968 : 1) g = (348249054088692360343664384716060307282221240198*i + 632608786394229733147453879629382109816155603996)*z^5 + (78288732951012207123651737359843385303103429616*i + 1421790144970157374993435834657786699116722047068)*z^4 + (126808104546169378741740415470754970160692958373*i + 1395505911017780163560370399495096477368129061128)*z^3 + (1151508433446340267838152948068030563616071711737*i + 1331505104782800437119636772308446788364866291823)*z^2 + (294195016254996862796265848147501809413184337240*i + 648097608752361631735328497390888861121749959677)*z + 800403695404043311146437378782593638877310314601*i + 901249395544166733264846949825665528694219338474 >> "g[1] =", g[1]; ^ Runtime error in '[]': Bad argument types (380424313638508136225221246412264910482078875119*i + 1263514484059759960390214018643718444669869401746)*z^3 + 939709461715339668293863546426995422914672052984*i + 1242246562135709934649092259556211092365399124603 >> "trace4(g) =", trace4(g); ^ Runtime error: No return statement executed in user-defined function Total time: 0.860 seconds, Total memory usage: 3.34MB '72.19.1' ************** MAGMA ***************** Host 72.19.126.33 (72.19.126.33) Time: Tue Dec 13 20:39:56 2005 Input: R := FunctionField(GF(5,2)); P := PolynomialRing(R); F := ext< R | y^5+y - x >; F; Q := PolynomialRing(P); G := ext; G; Output: Magma V2.11-10 Tue Dec 13 2005 20:39:56 on modular [Seed = 64960464] ------------------------------------- Algebraic function field defined over Univariate rational function field over GF(5^2) by y^5 + y + 4*x >> G := ext; ^ Runtime error in '-': Bad argument types Argument types given: RngUPolElt[RngUPol[FldFunRat]], FldFunElt >> G;; ^ User error: Identifier 'G' has not been declared or assigned Total time: 0.190 seconds, Total memory usage: 3.34MB '72.19.1' ************** MAGMA ***************** Host 72.19.126.33 (72.19.126.33) Time: Tue Dec 13 20:38:46 2005 Input: R := FunctionField(GF(5)); P := PolynomialRing(R); F := ext< R | y^2 - 1/x >; F; Q := PolynomialRing(P); G := ext; G; Output: Magma V2.11-10 Tue Dec 13 2005 20:38:46 on modular [Seed = 348659853] ------------------------------------- Algebraic function field defined over Univariate rational function field over GF(5) by y^2 + 4/x Algebraic function field defined over F by $.1^3 + 4*alpha + 4*x Total time: 0.290 seconds, Total memory usage: 7.85MB '72.19.1' ************** MAGMA ***************** Host 72.19.126.33 (72.19.126.33) Time: Tue Dec 13 20:37:32 2005 Input: R := FunctionField(GF(5)); P := PolynomialRing(R); F := ext< R | y^2 - 1/x >; F; Output: Magma V2.11-10 Tue Dec 13 2005 20:37:32 on modular [Seed = 482355327] ------------------------------------- Algebraic function field defined over Univariate rational function field over GF(5) by y^2 + 4/x Total time: 0.190 seconds, Total memory usage: 3.34MB '72.19.1' ************** MAGMA ***************** Host 72.19.126.33 (72.19.126.33) Time: Tue Dec 13 20:37:26 2005 Input: R := FunctionField(GF(5)); F := ext< R | y^2 - 1/x >; F; Output: Magma V2.11-10 Tue Dec 13 2005 20:37:26 on modular [Seed = 570234184] ------------------------------------- >> F := ext< R | y^2 - 1/x >; ^ User error: Identifier 'y' has not been declared or assigned >> F; ^ User error: Identifier 'F' has not been declared or assigned Total time: 0.190 seconds, Total memory usage: 3.24MB '72.19.1' ************** MAGMA ***************** Host 72.19.126.33 (72.19.126.33) Time: Tue Dec 13 20:37:09 2005 Input: R := FunctionField(GF(5)); P := PolynomialRing(R); F := ext< R | y^2 - 1/x >; F; Output: Magma V2.11-10 Tue Dec 13 2005 20:37:09 on modular [Seed = 653922908] ------------------------------------- Algebraic function field defined over Univariate rational function field over GF(5) by y^2 + 4/x Total time: 0.190 seconds, Total memory usage: 3.34MB '72.19.1' ************** MAGMA ***************** Host 72.19.126.33 (72.19.126.33) Time: Tue Dec 13 20:35:29 2005 Input: R := FunctionField(GF(5,2)); P := PolynomialRing(R); F := FunctionField(y^5+y - x); F; Output: Magma V2.11-10 Tue Dec 13 2005 20:35:28 on modular [Seed = 737615604] ------------------------------------- Algebraic function field defined over Univariate rational function field over GF(5^2) by y^5 + y + 4*x Total time: 0.180 seconds, Total memory usage: 3.34MB '72.19.1' ************** MAGMA ***************** Host 72.19.126.33 (72.19.126.33) Time: Tue Dec 13 20:34:56 2005 Input: R := FunctionField(GF(5,2)); P := PolynomialRing(R); F := FunctionField(y^2 - 1/x); F; Output: Magma V2.11-10 Tue Dec 13 2005 20:34:56 on modular [Seed = 820774351] ------------------------------------- Algebraic function field defined over Univariate rational function field over GF(5^2) by y^2 + 4/x Total time: 0.190 seconds, Total memory usage: 3.34MB '72.19.1' ************** MAGMA ***************** Host 72.19.126.33 (72.19.126.33) Time: Tue Dec 13 20:34:10 2005 Input: R := FunctionField(GF(5)); P := PolynomialRing(R); F := FunctionField(y^2 - 1/x); F; Output: Magma V2.11-10 Tue Dec 13 2005 20:34:09 on modular [Seed = 904462491] ------------------------------------- Algebraic function field defined over Univariate rational function field over GF(5) by y^2 + 4/x Total time: 0.190 seconds, Total memory usage: 3.34MB '200.177' ************** MAGMA ***************** Host 200.177.26.116 (200.177.26.116) Time: Tue Dec 13 20:27:47 2005 Input: p := 1461501624496790265145448589920785493717258890819; n := 1461501624496790265145447380994971188499300027613; Fp := GF(p); b := Fp!0; repeat repeat b := b + 1; until IsSquare(b + 1); y := Root(b + 1, 2); E := EllipticCurve([Fp | 0, b]); G := E![1, y]; until IsZero(n*G); "b =", b; "G =", G; lambda := Fp!2; while IsPower(lambda, 3) do lambda +:= 1; end while; "lambda =", lambda; Fp2 := ExtensionField; mu := i + 1; while IsSquare(mu) do mu +:= 1; end while; "mu =", mu; xi := 1/(lambda^2*mu^3); Fp12 := ExtensionField; Et := EllipticCurve([Fp2 | 0, b/xi]); xt := 1; while not IsSquare(xt^3 + b/xi) do xt +:= 1; end while; yt := Sqrt(xt^3 + b/xi); Gt := Et![xt, yt]; "Gt =", Gt; k := 12; assert (p^k - 1) mod n eq 0; chi := (p^k - 1) div n; Ek := EllipticCurve([Fp12 | 0, Fp12!b]); g := function(U, V, Q) if IsZero(U) or IsZero(V) or (U eq -V) or IsZero(Q) then return Fp12!1; end if; /* m := (U eq V) select 3*U[1]^2/(2*U[2]) else (V[2] - U[2])/(V[1] - U[1]); return m*(Q[1] - U[1]) + U[2] - Q[2]; */ if U eq V then m := 3*U[1]^2; s := 2*U[2]; else m := V[2] - U[2]; s := V[1] - U[1]; end if; return m*(Q[1] - U[1]) + s*(U[2] - Q[2]); end function; length := function(r) n := 0; v := 1; while v le r do n +:= 1; v +:= v; end while; return n; end function; bit := function(r, i) return (r div 2^i) mod 2; end function; tate := function(P, Qt) Q := Ek![Qt[1]*z^2, Qt[2]*z^3]; f := 1; A := P; for i := length(n) - 2 to 0 by -1 do f := f^2*g(A, A, Q); A := 2*A; if bit(n, i) eq 1 then f *:= g(A, P, Q); A +:= P; end if; end for; return f^chi; end function; g := tate(G, Gt); "g =", g; "g[1] =", g[1]; /* for j in [1..11] do print "g^(p^" * Sprint(j) * "-1): " * Sprint(g^(p^j-1) ne 1); end for; print "g^(p^12-1): " * Sprint(g^(p^12-1) eq 1); "g^n =", g^n; */ /* for j in [1..100] do "--", j; u := Random(n-1); v := Random(n-1); w := tate(u*G, v*Gt); h := g^((u*v) mod n); if w ne h then "Failure!"; print "u = " * Sprint(u); print "v = " * Sprint(v); print "e(u*P, v*Q) = " * Sprint(w); print "e(P, Q)^(u*v) = " * Sprint(h); quit; end if; end for; "Success!"; */ Output: Magma V2.11-10 Tue Dec 13 2005 20:27:46 on modular [Seed = 1021315601] ------------------------------------- b = 3 G = (1 : 1461501624496790265145448589920785493717258890817 : 1) lambda = 2 mu = i + 1 Gt = (8 : 816263181872116351510202985179226587277470764815*i + 295865244505705705023665406736615173923424579851 : 1) g = (1113252570408097904801784205204725186435037650621*i + 828892838102560531997994710291403383901103286823)*z^5 + (78288732951012207123651737359843385303103429616*i + 1421790144970157374993435834657786699116722047068)*z^4 + (1334693519950620886403708174450030523556565932446*i + 65995713479010101585078190425689016349129829691)*z^3 + (1151508433446340267838152948068030563616071711737*i + 1331505104782800437119636772308446788364866291823)*z^2 + (1167306608241793402349182741773283684304074553579*i + 813404015744428633410120092529896632595508931142)*z + 800403695404043311146437378782593638877310314601*i + 901249395544166733264846949825665528694219338474 >> "g[1] =", g[1]; ^ Runtime error in '[]': Bad argument types Total time: 0.610 seconds, Total memory usage: 3.34MB '72.19.1' ************** MAGMA ***************** Host 72.19.126.33 (72.19.126.33) Time: Tue Dec 13 20:26:47 2005 Input: FunctionField(FiniteField(11)) Output: Magma V2.11-10 Tue Dec 13 2005 20:26:46 on modular [Seed = 3323654440] ------------------------------------- Univariate rational function field over GF(11) Variables: $.1 Total time: 0.180 seconds, Total memory usage: 3.24MB '72.19.1' ************** MAGMA ***************** Host 72.19.126.33 (72.19.126.33) Time: Tue Dec 13 20:25:00 2005 Input: FunctionField(Integers()) Output: Magma V2.11-10 Tue Dec 13 2005 20:25:00 on modular [Seed = 3239966357] ------------------------------------- Univariate rational function field over Integer Ring Variables: $.1 Total time: 0.190 seconds, Total memory usage: 3.24MB '200.177' ************** MAGMA ***************** Host 200.177.26.116 (200.177.26.116) Time: Tue Dec 13 20:20:26 2005 Input: p := 1461501624496790265145448589920785493717258890819; n := 1461501624496790265145447380994971188499300027613; Fp := GF(p); b := Fp!0; repeat repeat b := b + 1; until IsSquare(b + 1); y := Root(b + 1, 2); E := EllipticCurve([Fp | 0, b]); G := E![1, y]; until IsZero(n*G); "b =", b; "G =", G; lambda := Fp!2; while IsPower(lambda, 3) do lambda +:= 1; end while; "lambda =", lambda; Fp2 := ExtensionField; mu := i + 1; while IsSquare(mu) do mu +:= 1; end while; "mu =", mu; xi := 1/(lambda^2*mu^3); Fp12 := ExtensionField; Et := EllipticCurve([Fp2 | 0, b/xi]); xt := 1; while not IsSquare(xt^3 + b/xi) do xt +:= 1; end while; yt := Sqrt(xt^3 + b/xi); Gt := Et![xt, yt]; "Gt =", Gt; k := 12; assert (p^k - 1) mod n eq 0; chi := (p^k - 1) div n; Ek := EllipticCurve([Fp12 | 0, Fp12!b]); g := function(U, V, Q) if IsZero(U) or IsZero(V) or (U eq -V) or IsZero(Q) then return Fp12!1; end if; /* m := (U eq V) select 3*U[1]^2/(2*U[2]) else (V[2] - U[2])/(V[1] - U[1]); return m*(Q[1] - U[1]) + U[2] - Q[2]; */ if U eq V then m := 3*U[1]^2; s := 2*U[2]; else m := V[2] - U[2]; s := V[1] - U[1]; end if; return m*(Q[1] - U[1]) + s*(U[2] - Q[2]); end function; length := function(r) n := 0; v := 1; while v le r do n +:= 1; v +:= v; end while; return n; end function; bit := function(r, i) return (r div 2^i) mod 2; end function; tate := function(P, Qt) Q := Ek![Qt[1]*z^2, Qt[2]*z^3]; f := 1; A := P; for i := length(n) - 2 to 0 by -1 do f := f^2*g(A, A, Q); A := 2*A; if bit(n, i) eq 1 then f *:= g(A, P, Q); A +:= P; end if; end for; return f^chi; end function; g := tate(G, Gt); "g =", g; /* for j in [1..11] do print "g^(p^" * Sprint(j) * "-1): " * Sprint(g^(p^j-1) ne 1); end for; print "g^(p^12-1): " * Sprint(g^(p^12-1) eq 1); "g^n =", g^n; */ for j in [1..100] do "--", j; u := Random(n-1); v := Random(n-1); w := tate(u*G, v*Gt); h := g^((u*v) mod n); if w ne h then "Failure!"; print "u = " * Sprint(u); print "v = " * Sprint(v); print "e(u*P, v*Q) = " * Sprint(w); print "e(P, Q)^(u*v) = " * Sprint(h); quit; end if; end for; "Success!"; Output: ** WARNING: Computation time exceeded 20 seconds, so computation was terminated after 20 seconds. ** Magma V2.11-10 Tue Dec 13 2005 20:20:06 on modular [Seed = 3373660599] ------------------------------------- b = 3 G = (1 : 1461501624496790265145448589920785493717258890817 : 1) lambda = 2 mu = i + 1 Gt = (8 : 816263181872116351510202985179226587277470764815*i + 295865244505705705023665406736615173923424579851 : 1) g = (1113252570408097904801784205204725186435037650621*i + 828892838102560531997994710291403383901103286823)*z^5 + (78288732951012207123651737359843385303103429616*i + 1421790144970157374993435834657786699116722047068)*z^4 + (1334693519950620886403708174450030523556565932446*i + 65995713479010101585078190425689016349129829691)*z^3 + (1151508433446340267838152948068030563616071711737*i + 1331505104782800437119636772308446788364866291823)*z^2 + (1167306608241793402349182741773283684304074553579*i + 813404015744428633410120092529896632595508931142)*z + 800403695404043311146437378782593638877310314601*i + 901249395544166733264846949825665528694219338474 -- 1 -- 2 -- 3 -- 4 -- 5 -- 6 -- 7 -- 8 -- 9 -- 10 -- 11 -- 12 -- 13 -- 14 -- 15 -- 16 -- 17 -- 18 -- 19 -- 20 -- 21 -- 22 -- 23 -- 24 -- 25 -- 26 -- 27 -- 28 -- 29 -- 30 -- 31 -- 32 -- 33 -- 34 -- 35 -- 36 -- 37 -- 38 -- 39 -- 40 -- 41 -- 42 -- 43 -- 44 -- 45 -- 46 -- 47 -- 48 Errors: /bin/sh: line 1: 31729 Alarm clock nice -n 19 /usr/local/bin/magma '200.177' ************** MAGMA ***************** Host 200.177.26.116 (200.177.26.116) Time: Tue Dec 13 20:15:58 2005 Input: p := 1461501624496790265145448589920785493717258890819; n := 1461501624496790265145447380994971188499300027613; Fp := GF(p); b := Fp!0; repeat repeat b := b + 1; until IsSquare(b + 1); y := Root(b + 1, 2); E := EllipticCurve([Fp | 0, b]); G := E![1, y]; until IsZero(n*G); "b =", b; "G =", G; lambda := Fp!2; while IsPower(lambda, 3) do lambda +:= 1; end while; "lambda =", lambda; Fp2 := ExtensionField; mu := i + 1; while IsSquare(mu) do mu +:= 1; end while; "mu =", mu; xi := 1/(lambda^2*mu^3); Fp12 := ExtensionField; Et := EllipticCurve([Fp2 | 0, b/xi]); xt := 1; while not IsSquare(xt^3 + b/xi) do xt +:= 1; end while; yt := Sqrt(xt^3 + b/xi); Gt := Et![xt, yt]; "Gt =", Gt; k := 12; assert (p^k - 1) mod n eq 0; chi := (p^k - 1) div n; Ek := EllipticCurve([Fp12 | 0, Fp12!b]); g := function(U, V, Q) if IsZero(U) or IsZero(V) or (U eq -V) or IsZero(Q) then return Fp12!1; end if; m := (U eq V) select 3*U[1]^2/(2*U[2]) else (V[2] - U[2])/(V[1] - U[1]); return m*(Q[1] - U[1]) + U[2] - Q[2]; end function; length := function(r) n := 0; v := 1; while v le r do n +:= 1; v +:= v; end while; return n; end function; bit := function(r, i) return (r div 2^i) mod 2; end function; tate := function(P, Qt) Q := Ek![Qt[1]*z^2, Qt[2]*z^3]; f := 1; A := P; for i := length(n) - 2 to 0 by -1 do f := f^2*g(A, A, Q); A := 2*A; if bit(n, i) eq 1 then f *:= g(A, P, Q); A +:= P; end if; end for; return f^chi; end function; g := tate(G, Gt); "g =", g; /* for j in [1..11] do print "g^(p^" * Sprint(j) * "-1): " * Sprint(g^(p^j-1) ne 1); end for; print "g^(p^12-1): " * Sprint(g^(p^12-1) eq 1); "g^n =", g^n; */ for j in [1..100] do "--", j; u := Random(n-1); v := Random(n-1); w := tate(u*G, v*Gt); h := g^((u*v) mod n); if w ne h then "Failure!"; print "u = " * Sprint(u); print "v = " * Sprint(v); print "e(u*P, v*Q) = " * Sprint(w); print "e(P, Q)^(u*v) = " * Sprint(h); quit; end if; end for; "Success!"; Output: ** WARNING: Computation time exceeded 20 seconds, so computation was terminated after 20 seconds. ** Magma V2.11-10 Tue Dec 13 2005 20:15:38 on modular [Seed = 3490505170] ------------------------------------- b = 3 G = (1 : 1461501624496790265145448589920785493717258890817 : 1) lambda = 2 mu = i + 1 Gt = (8 : 645238442624673913635245604741558906439788126004*i + 1165636379991084560121783183184170319793834310968 : 1) g = (348249054088692360343664384716060307282221240198*i + 632608786394229733147453879629382109816155603996)*z^5 + (78288732951012207123651737359843385303103429616*i + 1421790144970157374993435834657786699116722047068)*z^4 + (126808104546169378741740415470754970160692958373*i + 1395505911017780163560370399495096477368129061128)*z^3 + (1151508433446340267838152948068030563616071711737*i + 1331505104782800437119636772308446788364866291823)*z^2 + (294195016254996862796265848147501809413184337240*i + 648097608752361631735328497390888861121749959677)*z + 800403695404043311146437378782593638877310314601*i + 901249395544166733264846949825665528694219338474 -- 1 -- 2 -- 3 -- 4 -- 5 -- 6 -- 7 -- 8 -- 9 -- 10 -- 11 -- 12 -- 13 -- 14 -- 15 -- 16 -- 17 -- 18 -- 19 -- 20 -- 21 -- 22 -- 23 -- 24 -- 25 -- 26 -- 27 -- 28 -- 29 -- 30 -- 31 -- 32 -- 33 -- 34 -- 35 -- 36 -- 37 -- 38 -- 39 -- 40 -- 41 -- 42 -- 43 -- 44 -- 45 -- 46 -- 47 -- 48 Errors: /bin/sh: line 1: 31720 Alarm clock nice -n 19 /usr/local/bin/magma '200.177' ************** MAGMA ***************** Host 200.177.26.116 (200.177.26.116) Time: Tue Dec 13 20:14:45 2005 Input: p := 1461501624496790265145448589920785493717258890819; n := 1461501624496790265145447380994971188499300027613; Fp := GF(p); b := Fp!0; repeat repeat b := b + 1; until IsSquare(b + 1); y := Root(b + 1, 2); E := EllipticCurve([Fp | 0, b]); G := E![1, y]; until IsZero(n*G); "b =", b; "G =", G; lambda := Fp!2; while IsPower(lambda, 3) do lambda +:= 1; end while; "lambda =", lambda; Fp2 := ExtensionField; mu := i + 1; while IsSquare(mu) do mu +:= 1; end while; "mu =", mu; xi := 1/(lambda^2*mu^3); Fp12 := ExtensionField; Et := EllipticCurve([Fp2 | 0, b/xi]); xt := 1; while not IsSquare(xt^3 + b/xi) do xt +:= 1; end while; yt := Sqrt(xt^3 + b/xi); Gt := Et![xt, yt]; "Gt =", Gt; k := 12; assert (p^k - 1) mod n eq 0; chi := (p^k - 1) div n; Ek := EllipticCurve([Fp12 | 0, Fp12!b]); g := function(U, V, Q) if IsZero(U) or IsZero(V) or (U eq -V) or IsZero(Q) then return Fp12!1; end if; m := (U eq V) select 3*U[1]^2/(2*U[2]) else (V[2] - U[2])/(V[1] - U[1]); return m*(Q[1] - U[1]) + U[2] - Q[2]; end function; length := function(r) n := 0; v := 1; while v le r do n +:= 1; v +:= v; end while; return n; end function; bit := function(r, i) return (r div 2^i) mod 2; end function; tate := function(P, Qt) Q := Ek![Qt[1]*z^2, Qt[2]*z^3]; f := 1; A := P; for i := length(n) - 2 to 0 by -1 do f := f^2*g(A, A, Q); A := 2*A; if bit(n, i) eq 1 then f *:= g(A, P, Q); A +:= P; end if; end for; return f^chi; end function; g := tate(G, Gt); "g =", g; /* for j in [1..11] do print "g^(p^" * Sprint(j) * "-1): " * Sprint(g^(p^j-1) ne 1); end for; print "g^(p^12-1): " * Sprint(g^(p^12-1) eq 1); "g^n =", g^n; */ for j in [1..100] do u := Random(n-1); v := Random(n-1); w := tate(u*G, v*Gt); h := g^((u*v) mod n); if w ne h then "Failure!"; print "u = " * Sprint(u); print "v = " * Sprint(v); print "e(u*P, v*Q) = " * Sprint(w); print "e(P, Q)^(u*v) = " * Sprint(h); quit; end if; end for; "Success!"; Output: ** WARNING: Computation time exceeded 20 seconds, so computation was terminated after 20 seconds. ** Magma V2.11-10 Tue Dec 13 2005 20:14:25 on modular [Seed = 3674728722] ------------------------------------- b = 3 G = (1 : 1461501624496790265145448589920785493717258890817 : 1) lambda = 2 mu = i + 1 Gt = (8 : 816263181872116351510202985179226587277470764815*i + 295865244505705705023665406736615173923424579851 : 1) g = (1113252570408097904801784205204725186435037650621*i + 828892838102560531997994710291403383901103286823)*z^5 + (78288732951012207123651737359843385303103429616*i + 1421790144970157374993435834657786699116722047068)*z^4 + (1334693519950620886403708174450030523556565932446*i + 65995713479010101585078190425689016349129829691)*z^3 + (1151508433446340267838152948068030563616071711737*i + 1331505104782800437119636772308446788364866291823)*z^2 + (1167306608241793402349182741773283684304074553579*i + 813404015744428633410120092529896632595508931142)*z + 800403695404043311146437378782593638877310314601*i + 901249395544166733264846949825665528694219338474 Errors: /bin/sh: line 1: 31715 Alarm clock nice -n 19 /usr/local/bin/magma '200.177' ************** MAGMA ***************** Host 200.177.26.116 (200.177.26.116) Time: Tue Dec 13 20:13:26 2005 Input: p := 1461501624496790265145448589920785493717258890819; n := 1461501624496790265145447380994971188499300027613; Fp := GF(p); b := Fp!0; repeat repeat b := b + 1; until IsSquare(b + 1); y := Root(b + 1, 2); E := EllipticCurve([Fp | 0, b]); G := E![1, y]; until IsZero(n*G); "b =", b; "G =", G; lambda := Fp!2; while IsPower(lambda, 3) do lambda +:= 1; end while; "lambda =", lambda; Fp2 := ExtensionField; mu := i + 1; while IsSquare(mu) do mu +:= 1; end while; "mu =", mu; xi := 1/(lambda^2*mu^3); Fp12 := ExtensionField; Et := EllipticCurve([Fp2 | 0, b/xi]); xt := 1; while not IsSquare(xt^3 + b/xi) do xt +:= 1; end while; yt := Sqrt(xt^3 + b/xi); Gt := Et![xt, yt]; "Gt =", Gt; k := 12; assert (p^k - 1) mod n eq 0; chi := (p^k - 1) div n; Ek := EllipticCurve([Fp12 | 0, Fp12!b]); g := function(U, V, Q) if IsZero(U) or IsZero(V) or (U eq -V) or IsZero(Q) then return Fp12!1; end if; m := (U eq V) select 3*U[1]^2/(2*U[2]) else (V[2] - U[2])/(V[1] - U[1]); return m*(Q[1] - U[1]) + U[2] - Q[2]; end function; length := function(r) n := 0; v := 1; while v le r do n +:= 1; v +:= v; end while; return n; end function; bit := function(r, i) return (r div 2^i) mod 2; end function; tate := function(P, Qt) Q := Ek![Qt[1]*z^2, Qt[2]*z^3]; f := 1; A := P; for i := length(n) - 2 to 0 by -1 do f := f^2*g(A, A, Q); A := 2*A; if bit(r, i) eq 1 then f *:= g(A, P, Q); A +:= P; end if; end for; return f^chi; end function; g := tate(G, Gt); "g =", g; /* for j in [1..11] do print "g^(p^" * Sprint(j) * "-1): " * Sprint(g^(p^j-1) ne 1); end for; print "g^(p^12-1): " * Sprint(g^(p^12-1) eq 1); "g^n =", g^n; */ for j in [1..100] do u := Random(n-1); v := Random(n-1); w := tate(u*G, v*Gt); h := g^((u*v) mod n); if w ne h then "Failure!"; print "u = " * Sprint(u); print "v = " * Sprint(v); print "e(u*P, v*Q) = " * Sprint(w); print "e(P, Q)^(u*v) = " * Sprint(h); quit; end if; end for; "Success!"; Output: Magma V2.11-10 Tue Dec 13 2005 20:13:25 on modular [Seed = 3862615147] ------------------------------------- b = 3 G = (1 : 1461501624496790265145448589920785493717258890817 : 1) lambda = 2 mu = i + 1 Gt = (8 : 816263181872116351510202985179226587277470764815*i + 295865244505705705023665406736615173923424579851 : 1) >> if bit(r, i) eq 1 then ^ User error: Identifier 'r' has not been declared or assigned >> g := tate(G, Gt); ^ User error: Identifier 'tate' has not been declared or assigned g = function(U, V, Q) ... end function >> w := tate(u*G, v*Gt); ^ User error: Identifier 'tate' has not been declared or assigned Success! Total time: 0.260 seconds, Total memory usage: 3.34MB '70.172.' ************** MAGMA ***************** Host 70.172.216.115 (70.172.216.115) Time: Tue Dec 13 20:13:13 2005 Input: factor 74037563479561712828046796097429573142593188889231 28908493623263897276503402826627689199641962511784 39958943305021275853701189680982867331732731089309 00552505116877063299072396380786710086096962537934 650563796359 Output: Magma V2.11-10 Tue Dec 13 2005 20:13:13 on modular [Seed = 3778925424] ------------------------------------- >> factor 74037563479561712828046796097429573142593188889231 ^ User error: bad syntax >> 39958943305021275853701189680982867331732731089309 ^ User error: bad syntax >> 650563796359; ^ User error: bad syntax Total time: 0.190 seconds, Total memory usage: 3.24MB '70.172.' ************** MAGMA ***************** Host 70.172.216.115 (70.172.216.115) Time: Tue Dec 13 20:12:56 2005 Input: fac 74037563479561712828046796097429573142593188889231 28908493623263897276503402826627689199641962511784 39958943305021275853701189680982867331732731089309 00552505116877063299072396380786710086096962537934 650563796359 Output: Magma V2.11-10 Tue Dec 13 2005 20:12:56 on modular [Seed = 3946305867] ------------------------------------- >> fac 74037563479561712828046796097429573142593188889231 ^ User error: bad syntax >> 39958943305021275853701189680982867331732731089309 ^ User error: bad syntax >> 650563796359; ^ User error: bad syntax Total time: 0.180 seconds, Total memory usage: 3.24MB '72.19.1' ************** MAGMA ***************** Host 72.19.126.33 (72.19.126.33) Time: Tue Dec 13 20:12:28 2005 Input: K := GF(16) Output: Magma V2.11-10 Tue Dec 13 2005 20:12:28 on modular [Seed = 4096313008] ------------------------------------- Total time: 0.180 seconds, Total memory usage: 3.24MB '200.177' ************** MAGMA ***************** Host 200.177.26.116 (200.177.26.116) Time: Tue Dec 13 20:12:24 2005 Input: p := 1461501624496790265145448589920785493717258890819; n := 1461501624496790265145447380994971188499300027613; Fp := GF(p); b := Fp!0; repeat repeat b := b + 1; until IsSquare(b + 1); y := Root(b + 1, 2); E := EllipticCurve([Fp | 0, b]); G := E![1, y]; until IsZero(n*G); "b =", b; "G =", G; lambda := Fp!2; while IsPower(lambda, 3) do lambda +:= 1; end while; "lambda =", lambda; Fp2 := ExtensionField; mu := i + 1; while IsSquare(mu) do mu +:= 1; end while; "mu =", mu; xi := 1/(lambda^2*mu^3); Fp12 := ExtensionField; Et := EllipticCurve([Fp2 | 0, b/xi]); xt := 1; while not IsSquare(xt^3 + b/xi) do xt +:= 1; end while; yt := Sqrt(xt^3 + b/xi); Gt := Et![xt, yt]; "Gt =", Gt; k := 12; assert (p^k - 1) mod n eq 0; chi := (p^k - 1) div n; Ek := EllipticCurve([Fp12 | 0, Fp12!b]); g := function(U, V, Q) if IsZero(U) or IsZero(V) or (U eq -V) or IsZero(Q) then return Fp12!1; end if; m := (U eq V) select 3*U[1]^2/(2*U[2]) else (V[2] - U[2])/(V[1] - U[1]); return m*(Q[1] - U[1]) + U[2] - Q[2]; end function; length := function(r) n := 0; v := 1; while v le r do n +:= 1; v +:= v; end while; return n; end function; bit := function(r, i) return (r div 2^i) mod 2; end function; tate := function(P, Qt) Q := Ek![Qt[1]*z^2, Qt[2]*z^3]; f := 1; A := P; for i := length(r) - 2 to 0 by -1 do f := f^2*g(A, A, Q); A := 2*A; if bit(r, i) eq 1 then f *:= g(A, P, Q); A +:= P; end if; end for; return f^chi; end function; g := tate(G, Gt); "g =", g; /* for j in [1..11] do print "g^(p^" * Sprint(j) * "-1): " * Sprint(g^(p^j-1) ne 1); end for; print "g^(p^12-1): " * Sprint(g^(p^12-1) eq 1); "g^n =", g^n; */ for j in [1..100] do u := Random(n-1); v := Random(n-1); w := tate(u*G, v*Gt); h := g^((u*v) mod n); if w ne h then "Failure!"; print "u = " * Sprint(u); print "v = " * Sprint(v); print "e(u*P, v*Q) = " * Sprint(w); print "e(P, Q)^(u*v) = " * Sprint(h); quit; end if; end for; "Success!"; Output: Magma V2.11-10 Tue Dec 13 2005 20:12:24 on modular [Seed = 4029465264] ------------------------------------- b = 3 G = (1 : 1461501624496790265145448589920785493717258890817 : 1) lambda = 2 mu = i + 1 Gt = (8 : 816263181872116351510202985179226587277470764815*i + 295865244505705705023665406736615173923424579851 : 1) >> for i := length(r) - 2 to 0 by -1 do ^ User error: Identifier 'r' has not been declared or assigned >> g := tate(G, Gt); ^ User error: Identifier 'tate' has not been declared or assigned g = function(U, V, Q) ... end function >> w := tate(u*G, v*Gt); ^ User error: Identifier 'tate' has not been declared or assigned Success! Total time: 0.230 seconds, Total memory usage: 3.34MB '200.177' ************** MAGMA ***************** Host 200.177.26.116 (200.177.26.116) Time: Tue Dec 13 20:10:50 2005 Input: Zz := PolynomialRing(Integers()); P := 36*z^4 + 36*z^3 + 24*z^2 + 6*z + 1; v := 0; while true do v +:= 1; t := 6*v^2 + 1; u := -v; p := Evaluate(P, u); n := p + 1 - t; if p mod 4 eq 3 and p mod 9 eq 4 and IsProbablePrime(p) and IsProbablePrime(n) then break; end if; u := v; p := Evaluate(P, u); n := p + 1 - t; if p mod 4 eq 3 and p mod 9 eq 4 and IsProbablePrime(p) and IsProbablePrime(n) then break; end if; end while; "u =", u; "p =", p; "n =", n; Fp := GF(p); b := Fp!0; repeat repeat b := b + 1; until IsSquare(b + 1); y := Root(b + 1, 2); E := EllipticCurve([Fp | 0, b]); G := E![1, y]; until IsZero(n*G); "b =", b; "G =", G; lambda := Fp!2; while IsPower(lambda, 3) do lambda +:= 1; end while; "lambda =", lambda; Fp2 := ExtensionField; mu := i + 1; while IsSquare(mu) do mu +:= 1; end while; "mu =", mu; xi := 1/(lambda^2*mu^3); Fp12 := ExtensionField; Et := EllipticCurve([Fp2 | 0, b/xi]); xt := 1; while not IsSquare(xt^3 + b/xi) do xt +:= 1; end while; yt := Sqrt(xt^3 + b/xi); Gt := Et![xt, yt]; "Gt =", Gt; k := 12; assert (p^k - 1) mod n eq 0; chi := (p^k - 1) div n; Ek := EllipticCurve([Fp12 | 0, Fp12!b]); g := function(U, V, Q) if IsZero(U) or IsZero(V) or (U eq -V) or IsZero(Q) then return Fp12!1; end if; assert U[3] eq 1; assert V[3] eq 1; assert Q[3] eq 1; m := (U eq V) select 3*U[1]^2/(2*U[2]) else (V[2] - U[2])/(V[1] - U[1]); return m*(Q[1] - U[1]) + U[2] - Q[2]; end function; length := function(r) n := 0; v := 1; while v le r do n +:= 1; v +:= v; end while; return n; end function; bit := function(r, i) return (r div 2^i) mod 2; end function; miller := function(r, P, Q) f := 1; A := P; for i := length(r) - 2 to 0 by -1 do f := f^2*g(A, A, Q); A := 2*A; if bit(r, i) eq 1 then f *:= g(A, P, Q); A +:= P; end if; end for; return f; end function; tate := function(P, Qt) Q := Ek![Qt[1]*z^2, Qt[2]*z^3]; return miller(n, P, Q)^chi; end function; g := tate(G, Gt); "g =", g; /* for j in [1..11] do print "g^(p^" * Sprint(j) * "-1): " * Sprint(g^(p^j-1) ne 1); end for; print "g^(p^12-1): " * Sprint(g^(p^12-1) eq 1); "g^n =", g^n; */ /* for u in [1..n-1] do w := tate(u*G, Gt); //s := g^((u*v) mod n); s := tate(G, u*Gt); h := g^u; if w ne s or w ne h then "Failure!"; print "u = " * Sprint(u); print "e(u*P, Q) = " * Sprint(w); print "e(P, u*Q) = " * Sprint(s); print "e(P, Q)^u = " * Sprint(h); quit; end if; end for; */ for j in [1..100] do u := Random(n-1); v := Random(n-1); w := tate(u*G, v*Gt); h := g^((u*v) mod n); if w ne h then "Failure!"; print "u = " * Sprint(u); print "v = " * Sprint(v); print "e(u*P, v*Q) = " * Sprint(w); print "e(P, Q)^(u*v) = " * Sprint(h); quit; end if; end for; "Success!"; Output: Magma V2.11-10 Tue Dec 13 2005 20:10:50 on modular [Seed = 4213688793] ------------------------------------- u = 1 p = 103 n = 97 b = 12 G = (1 : 61 : 1) lambda = 2 mu = i + 2 Gt = (4 : 44*i + 6 : 1) g = (76*i + 99)*z^5 + (3*i + 57)*z^4 + (24*i + 88)*z^3 + (78*i + 63)*z^2 + (58*i + 84)*z + 23*i + 29 Success! Total time: 0.220 seconds, Total memory usage: 3.53MB '200.177' ************** MAGMA ***************** Host 200.177.26.116 (200.177.26.116) Time: Tue Dec 13 20:09:33 2005 Input: Zz := PolynomialRing(Integers()); P := 36*z^4 + 36*z^3 + 24*z^2 + 6*z + 1; v := 0; while true do v +:= 1; t := 6*v^2 + 1; u := -v; p := Evaluate(P, u); n := p + 1 - t; if p mod 4 eq 3 and p mod 9 eq 4 and IsProbablePrime(p) and IsProbablePrime(n) then break; end if; u := v; p := Evaluate(P, u); n := p + 1 - t; if p mod 4 eq 3 and p mod 9 eq 4 and IsProbablePrime(p) and IsProbablePrime(n) then break; end if; end while; "u =", u; "p =", p; "n =", n; Fp := GF(p); b := Fp!0; repeat repeat b := b + 1; until IsSquare(b + 1); y := Root(b + 1, 2); E := EllipticCurve([Fp | 0, b]); G := E![1, y]; until IsZero(n*G); "b =", b; "G =", G; lambda := Fp!2; while IsPower(lambda, 3) do lambda +:= 1; end while; "lambda =", lambda; Fp2 := ExtensionField; mu := i + 1; while IsSquare(mu) do mu +:= 1; end while; "mu =", mu; xi := 1/(lambda^2*mu^3); Fp12 := ExtensionField; Et := EllipticCurve([Fp2 | 0, b/xi]); xt := 1; while not IsSquare(xt^3 + b/xi) do xt +:= 1; end while; yt := Sqrt(xt^3 + b/xi); Gt := Et![xt, yt]; "Gt =", Gt; k := 12; assert (p^k - 1) mod n eq 0; chi := (p^k - 1) div n; Ek := EllipticCurve([Fp12 | 0, Fp12!b]); g := function(U, V, Q) if IsZero(U) or IsZero(V) or (U eq -V) or IsZero(Q) then return Fp12!1; end if; assert U[3] eq 1; assert V[3] eq 1; assert Q[3] eq 1; m := (U eq V) select 3*U[1]^2/(2*U[2]) else (V[2] - U[2])/(V[1] - U[1]); return m*(Q[1] - U[1]) + U[2] - Q[2]; end function; length := function(r) n := 0; v := 1; while v le r do n +:= 1; v +:= v; end while; return n; end function; bit := function(r, i) return (r div 2^i) mod 2; end function; miller := function(r, P, Q) f := 1; A := P; for i := length(r) - 2 to 0 by -1 do f := f^2*g(A, A, Q); A := 2*A; if bit(r, i) eq 1 then f *:= g(A, P, Q); A +:= P; end if; end for; return f; end function; tate := function(P, Qt) Q := Ek![Qt[1]*z^2, Qt[2]*z^3]; return miller(n, P, Q)^chi; end function; g := tate(G, Gt); "g =", g; /* for j in [1..11] do print "g^(p^" * Sprint(j) * "-1): " * Sprint(g^(p^j-1) ne 1); end for; print "g^(p^12-1): " * Sprint(g^(p^12-1) eq 1); "g^n =", g^n; */ /* for u in [1..n-1] do w := tate(u*G, Gt); //s := g^((u*v) mod n); s := tate(G, u*Gt); h := g^u; if w ne s or w ne h then "Failure!"; print "u = " * Sprint(u); print "e(u*P, Q) = " * Sprint(w); print "e(P, u*Q) = " * Sprint(s); print "e(P, Q)^u = " * Sprint(h); quit; end if; end for; */ for j in [1..100] do u := Random(n-1); v := Random(n-1); w := tate(u*G, v*Gt); h := g^((u*v) mod n); if w ne s or w ne h then "Failure!"; print "u = " * Sprint(u); print "v = " * Sprint(v); print "e(u*P, v*Q) = " * Sprint(w); print "e(P, Q)^(u*v) = " * Sprint(h); quit; end if; end for; "Success!"; Output: Magma V2.11-10 Tue Dec 13 2005 20:09:32 on modular [Seed = 2254091634] ------------------------------------- u = 1 p = 103 n = 97 b = 12 G = (1 : 61 : 1) lambda = 2 mu = i + 2 Gt = (4 : 44*i + 6 : 1) g = (76*i + 99)*z^5 + (3*i + 57)*z^4 + (24*i + 88)*z^3 + (78*i + 63)*z^2 + (58*i + 84)*z + 23*i + 29 >> if w ne s or w ne h then ^ User error: Identifier 's' has not been declared or assigned Success! Total time: 0.190 seconds, Total memory usage: 3.53MB '200.177' ************** MAGMA ***************** Host 200.177.26.116 (200.177.26.116) Time: Tue Dec 13 20:01:22 2005 Input: Zz := PolynomialRing(Integers()); P := 36*z^4 + 36*z^3 + 24*z^2 + 6*z + 1; v := 0; while true do v +:= 1; t := 6*v^2 + 1; u := -v; p := Evaluate(P, u); n := p + 1 - t; if p mod 4 eq 3 and p mod 9 eq 4 and IsProbablePrime(p) and IsProbablePrime(n) then break; end if; u := v; p := Evaluate(P, u); n := p + 1 - t; if p mod 4 eq 3 and p mod 9 eq 4 and IsProbablePrime(p) and IsProbablePrime(n) then break; end if; end while; "u =", u; "p =", p; "n =", n; Fp := GF(p); b := Fp!0; repeat repeat b := b + 1; until IsSquare(b + 1); y := Root(b + 1, 2); E := EllipticCurve([Fp | 0, b]); G := E![1, y]; until IsZero(n*G); "b =", b; "G =", G; lambda := Fp!2; while IsPower(lambda, 3) do lambda +:= 1; end while; "lambda =", lambda; Fp2 := ExtensionField; mu := i + 1; while IsSquare(mu) do mu +:= 1; end while; "mu =", mu; xi := 1/(lambda^2*mu^3); Fp12 := ExtensionField; Et := EllipticCurve([Fp2 | 0, b/xi]); xt := 1; while not IsSquare(xt^3 + b/xi) do xt +:= 1; end while; yt := Sqrt(xt^3 + b/xi); Gt := Et![xt, yt]; "Gt =", Gt; k := 12; assert (p^k - 1) mod n eq 0; chi := (p^k - 1) div n; Ek := EllipticCurve([Fp12 | 0, Fp12!b]); g := function(U, V, Q) if IsZero(U) or IsZero(V) or (U eq -V) or IsZero(Q) then return Fp12!1; end if; assert U[3] eq 1; assert V[3] eq 1; assert Q[3] eq 1; m := (U eq V) select 3*U[1]^2/(2*U[2]) else (V[2] - U[2])/(V[1] - U[1]); return m*(Q[1] - U[1]) + U[2] - Q[2]; end function; length := function(r) n := 0; v := 1; while v le r do n +:= 1; v +:= v; end while; return n; end function; bit := function(r, i) return (r div 2^i) mod 2; end function; miller := function(r, P, Q) f := 1; A := P; for i := length(r) - 2 to 0 by -1 do f := f^2*g(A, A, Q); A := 2*A; if bit(r, i) eq 1 then f *:= g(A, P, Q); A +:= P; end if; end for; return f; end function; tate := function(P, Qt) Q := Ek![Qt[1]*z^2, Qt[2]*z^3]; return miller(n, P, Q)^chi; end function; g := tate(G, Gt); "g =", g; /* for j in [1..11] do print "g^(p^" * Sprint(j) * "-1): " * Sprint(g^(p^j-1) ne 1); end for; print "g^(p^12-1): " * Sprint(g^(p^12-1) eq 1); "g^n =", g^n; */ for u in [1..n-1] do w := tate(u*G, Gt); //s := g^((u*v) mod n); s := tate(G, u*Gt); h := g^u; if w ne s or w ne h then "Failure!"; print "u = " * Sprint(u); print "e(u*P, Q) = " * Sprint(w); print "e(P, u*Q) = " * Sprint(s); print "e(P, Q)^u = " * Sprint(h); quit; end if; end for; "Success!"; Output: Magma V2.11-10 Tue Dec 13 2005 20:01:22 on modular [Seed = 3077282413] ------------------------------------- u = 1 p = 103 n = 97 b = 12 G = (1 : 61 : 1) lambda = 2 mu = i + 2 Gt = (4 : 44*i + 6 : 1) g = (76*i + 99)*z^5 + (3*i + 57)*z^4 + (24*i + 88)*z^3 + (78*i + 63)*z^2 + (58*i + 84)*z + 23*i + 29 Success! Total time: 0.240 seconds, Total memory usage: 3.53MB '200.177' ************** MAGMA ***************** Host 200.177.26.116 (200.177.26.116) Time: Tue Dec 13 20:00:15 2005 Input: Zz := PolynomialRing(Integers()); P := 36*z^4 + 36*z^3 + 24*z^2 + 6*z + 1; v := 0; while true do v +:= 1; t := 6*v^2 + 1; u := -v; p := Evaluate(P, u); n := p + 1 - t; if p mod 4 eq 3 and p mod 9 eq 4 and IsProbablePrime(p) and IsProbablePrime(n) then break; end if; u := v; p := Evaluate(P, u); n := p + 1 - t; if p mod 4 eq 3 and p mod 9 eq 4 and IsProbablePrime(p) and IsProbablePrime(n) then break; end if; end while; "u =", u; "p =", p; "n =", n; Fp := GF(p); b := Fp!0; repeat repeat b := b + 1; until IsSquare(b + 1); y := Root(b + 1, 2); E := EllipticCurve([Fp | 0, b]); G := E![1, y]; until IsZero(n*G); "b =", b; "G =", G; lambda := Fp!2; while IsPower(lambda, 3) do lambda +:= 1; end while; "lambda =", lambda; Fp2 := ExtensionField; mu := i + 1; while IsSquare(mu) do mu +:= 1; end while; "mu =", mu; xi := 1/(lambda^2*mu^3); Fp12 := ExtensionField; Et := EllipticCurve([Fp2 | 0, b/xi]); xt := 1; while not IsSquare(xt^3 + b/xi) do xt +:= 1; end while; yt := Sqrt(xt^3 + b/xi); Gt := Et![xt, yt]; "Gt =", Gt; k := 12; assert (p^k - 1) mod n eq 0; chi := (p^k - 1) div n; Ek := EllipticCurve([Fp12 | 0, Fp12!b]); g := function(U, V, Q) if IsZero(U) or IsZero(V) or (U eq -V) or IsZero(Q) then return Fp12!1; end if; assert U[3] eq 1; assert V[3] eq 1; assert Q[3] eq 1; m := (U eq V) select 3*U[1]^2/(2*U[2]) else (V[2] - U[2])/(V[1] - U[1]); return m*(Q[1] - U[1]) + U[2] - Q[2]; end function; length := function(r) n := 0; v := 1; while v le r do n +:= 1; v +:= v; end while; return n; end function; bit := function(r, i) return (r div 2^i) mod 2; end function; miller := function(r, P, Q) f := 1; A := P; for i := length(r) - 2 to 0 by -1 do f := f^2*g(A, A, Q); A := 2*A; if bit(r, i) eq 1 then "***", i; f *:= g(A, P, Q); A +:= P; end if; end for; return f; end function; tate := function(P, Qt) Q := Ek![Qt[1]*z^2, Qt[2]*z^3]; return miller(n, P, Q)^chi; end function; g := tate(G, Gt); "g =", g; /* for j in [1..11] do print "g^(p^" * Sprint(j) * "-1): " * Sprint(g^(p^j-1) ne 1); end for; print "g^(p^12-1): " * Sprint(g^(p^12-1) eq 1); "g^n =", g^n; */ for u in [1..n-1] do w := tate(u*G, Gt); //s := g^((u*v) mod n); s := tate(G, u*Gt); h := g^u; if w ne s or w ne h then "Failure!"; print "u = " * Sprint(u); print "e(u*P, Q) = " * Sprint(w); print "e(P, u*Q) = " * Sprint(s); print "e(P, Q)^u = " * Sprint(h); quit; end if; end for; "Success!"; Output: Magma V2.11-10 Tue Dec 13 2005 20:00:15 on modular [Seed = 2993594133] ------------------------------------- u = 1 p = 103 n = 97 b = 12 G = (1 : 61 : 1) lambda = 2 mu = i + 2 Gt = (4 : 44*i + 6 : 1) *** 5 *** 0 g = (76*i + 99)*z^5 + (3*i + 57)*z^4 + (24*i + 88)*z^3 + (78*i + 63)*z^2 + (58*i + 84)*z + 23*i + 29 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 *** 5 *** 0 Success! Total time: 0.250 seconds, Total memory usage: 3.53MB '200.177' ************** MAGMA ***************** Host 200.177.26.116 (200.177.26.116) Time: Tue Dec 13 19:53:53 2005 Input: Zz := PolynomialRing(Integers()); P := 36*z^4 + 36*z^3 + 24*z^2 + 6*z + 1; v := 0; while true do v +:= 1; t := 6*v^2 + 1; u := -v; p := Evaluate(P, u); n := p + 1 - t; if p mod 4 eq 3 and p mod 9 eq 4 and IsProbablePrime(p) and IsProbablePrime(n) then break; end if; u := v; p := Evaluate(P, u); n := p + 1 - t; if p mod 4 eq 3 and p mod 9 eq 4 and IsProbablePrime(p) and IsProbablePrime(n) then break; end if; end while; "u =", u; "p =", p; "n =", n; Fp := GF(p); b := Fp!0; repeat repeat b := b + 1; until IsSquare(b + 1); y := Root(b + 1, 2); E := EllipticCurve([Fp | 0, b]); G := E![1, y]; until IsZero(n*G); "b =", b; "G =", G; lambda := Fp!2; while IsPower(lambda, 3) do lambda +:= 1; end while; "lambda =", lambda; Fp2 := ExtensionField; mu := i + 1; while IsSquare(mu) do mu +:= 1; end while; "mu =", mu; xi := 1/(lambda^2*mu^3); Fp12 := ExtensionField; Et := EllipticCurve([Fp2 | 0, b/xi]); xt := 1; while not IsSquare(xt^3 + b/xi) do xt +:= 1; end while; yt := Sqrt(xt^3 + b/xi); Gt := Et![xt, yt]; "Gt =", Gt; k := 12; assert (p^k - 1) mod n eq 0; chi := (p^k - 1) div n; Ek := EllipticCurve([Fp12 | 0, Fp12!b]); g := function(U, V, Q) if IsZero(U) or IsZero(V) or (U eq -V) or IsZero(Q) then return Fp12!1; end if; assert U[3] eq 1; assert V[3] eq 1; assert Q[3] eq 1; m := (U eq V) select 3*U[1]^2/(2*U[2]) else (V[2] - U[2])/(V[1] - U[1]); return m*(Q[1] - U[1]) + U[2] - Q[2]; end function; length := function(r) n := 0; v := 1; while v le r do n +:= 1; v +:= v; end while; return n; end function; bit := function(r, i) return (r div 2^i) mod 2; end function; miller := function(r, P, Q) f := 1; A := P; for i := length(r) - 1 to 0 by -1 do f := f^2*g(A, A, Q); A := 2*A; if bit(r, i) eq 1 then "***", i; f *:= g(A, P, Q); A +:= P; end if; end for; return f; end function; tate := function(P, Qt) Q := Ek![Qt[1]*z^2, Qt[2]*z^3]; return miller(n, P, Q)^chi; end function; g := tate(G, Gt); "g =", g; /* for j in [1..11] do print "g^(p^" * Sprint(j) * "-1): " * Sprint(g^(p^j-1) ne 1); end for; print "g^(p^12-1): " * Sprint(g^(p^12-1) eq 1); "g^n =", g^n; */ for u in [1..n-1] do w := tate(u*G, Gt); //s := g^((u*v) mod n); s := tate(G, u*Gt); h := g^u; if w ne s or w ne h then "Failure!"; print "u = " * Sprint(u); print "e(u*P, Q) = " * Sprint(w); print "e(P, u*Q) = " * Sprint(s); print "e(P, Q)^u = " * Sprint(h); quit; end if; end for; "Success!"; Output: Magma V2.11-10 Tue Dec 13 2005 19:53:53 on modular [Seed = 2090881637] ------------------------------------- u = 1 p = 103 n = 97 b = 12 G = (1 : 61 : 1) lambda = 2 mu = i + 2 Gt = (4 : 44*i + 6 : 1) *** 6 *** 5 *** 0 g = (4*i + 70)*z^5 + (i + 79)*z^4 + (78*i + 73)*z^3 + (24*i + 89)*z^2 + (95*i + 55)*z + 71*i + 56 *** 6 *** 5 *** 0 *** 6 *** 5 *** 0 *** 6 *** 5 *** 0 *** 6 *** 5 *** 0 Failure! u = 2 e(u*P, Q) = (38*i + 16)*z^5 + (101*i + 58)*z^4 + (43*i + 45)*z^3 + (102*i + 83)*z^2 + (61*i + 73)*z + 20*i + 35 e(P, u*Q) = (100*i + 19)*z^5 + (89*i + 82)*z^4 + (74*i + 97)*z^3 + (74*i + 77)*z^2 + (92*i + 40)*z + 92*i + 48 e(P, Q)^u = (48*i + 96)*z^5 + (98*i + 69)*z^4 + (91*i + 98)*z^3 + (100*i + 11)*z^2 + (48*i + 67)*z + 29*i + 33 Total time: 0.190 seconds, Total memory usage: 3.53MB '65.191.' ************** MAGMA ***************** Host 65.191.73.113 (65.191.73.113) Time: Tue Dec 13 19:16:04 2005 Input: K := FiniteField(2); > C := LinearCode; M:=VerticalJoin(GeneratorMatrix(C), ParityCheckMatrix(C)); M1:=EchelonForm(M); M2:=Submatrix(M1,22,22,14,35); M2; C2:=LinearCode(M2); WeightDistribution(C2); C2; L:=MinimumWords(C2); C3:=LinearCode(L); C3; Output: Magma V2.11-10 Tue Dec 13 2005 19:16:04 on modular [Seed = 587072862] ------------------------------------- [1 0 0 0 0 0 0 0 0 0 1 0 0 1 0 1 0 1 1 1 0 0 0 0 1 1 1 0 0 1 0 0 0 1 0] [0 1 0 0 0 0 0 0 0 0 1 0 1 0 0 1 0 0 1 1 0 0 0 0 0 1 1 1 0 1 1 1 1 0 1] [0 0 1 0 0 0 0 0 0 0 0 1 0 1 0 0 1 0 0 1 1 0 0 0 1 1 0 1 0 0 0 1 1 0 1] [0 0 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 1 1 0 0 0 1 0 0 0 1 0 0 0 1 1 0] [0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 1 1 1 1 0 0 0 0 0 0 1 0 0 1 0 1 1 1 0 0] [0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 1 1 1 1 0 0 0 0 0 0 1 0 0 1 0 1 1 1 0] [0 0 0 0 0 0 1 0 0 0 1 1 1 1 0 1 1 1 1 1 0 0 0 0 0 0 0 1 0 1 1 1 0 1 1] [0 0 0 0 0 0 0 1 0 0 1 1 1 0 0 1 1 0 0 0 1 0 0 0 0 0 0 0 0 1 0 1 1 0 0] [0 0 0 0 0 0 0 0 1 0 0 1 1 1 0 1 0 0 1 1 1 0 0 0 0 0 0 0 1 1 0 1 0 0 1] [0 0 0 0 0 0 0 0 0 1 1 1 0 1 0 0 1 1 1 0 1 0 0 0 0 0 0 0 0 1 0 0 1 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 0 0 0 0 0 0 0 1 1 1 1 1 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 1 0 1 0 1 0 0 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 1 1 1 0 1 0 0 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 1 0 1 0 0 1 1 1] [ <0, 1>, <8, 14>, <10, 182>, <12, 805>, <14, 2200>, <16, 3885>, <18, 4340>, <20, 3066>, <22, 1400>, <24, 420>, <26, 70>, <28, 1> ] [35, 14, 8] Linear Code over GF(2) Generator matrix: [1 0 0 0 0 0 0 0 0 0 1 0 0 1 0 1 0 1 1 1 0 0 0 0 1 1 1 0 0 1 0 0 0 1 0] [0 1 0 0 0 0 0 0 0 0 1 0 1 0 0 1 0 0 1 1 0 0 0 0 0 1 1 1 0 1 1 1 1 0 1] [0 0 1 0 0 0 0 0 0 0 0 1 0 1 0 0 1 0 0 1 1 0 0 0 1 1 0 1 0 0 0 1 1 0 1] [0 0 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 1 1 0 0 0 1 0 0 0 1 0 0 0 1 1 0] [0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 1 1 1 1 0 0 0 0 0 0 1 0 0 1 0 1 1 1 0 0] [0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 1 1 1 1 0 0 0 0 0 0 1 0 0 1 0 1 1 1 0] [0 0 0 0 0 0 1 0 0 0 1 1 1 1 0 1 1 1 1 1 0 0 0 0 0 0 0 1 0 1 1 1 0 1 1] [0 0 0 0 0 0 0 1 0 0 1 1 1 0 0 1 1 0 0 0 1 0 0 0 0 0 0 0 0 1 0 1 1 0 0] [0 0 0 0 0 0 0 0 1 0 0 1 1 1 0 1 0 0 1 1 1 0 0 0 0 0 0 0 1 1 0 1 0 0 1] [0 0 0 0 0 0 0 0 0 1 1 1 0 1 0 0 1 1 1 0 1 0 0 0 0 0 0 0 0 1 0 0 1 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 0 0 0 0 0 0 0 1 1 1 1 1 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 1 0 1 0 1 0 0 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 1 1 1 0 1 0 0 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 1 0 1 0 0 1 1 1] >> C3:=LinearCode(L); ^ Runtime error in 'LinearCode': Bad argument types Argument types given: SetEnum[ModTupFldElt] >> C3;; ^ User error: Identifier 'C3' has not been declared or assigned Total time: 0.190 seconds, Total memory usage: 3.34MB '65.191.' ************** MAGMA ***************** Host 65.191.73.113 (65.191.73.113) Time: Tue Dec 13 19:11:47 2005 Input: K := FiniteField(2); > C := LinearCode; M:=VerticalJoin(GeneratorMatrix(C), ParityCheckMatrix(C)); M1:=EchelonForm(M); M2:=Submatrix(M1,22,22,14,35); M2; C2:=LinearCode(M2); WeightDistribution(C2); C2; MinimumWords(C2); Output: Magma V2.11-10 Tue Dec 13 2005 19:11:47 on modular [Seed = 753929303] ------------------------------------- [1 0 0 0 0 0 0 0 0 0 1 0 0 1 0 1 0 1 1 1 0 0 0 0 1 1 1 0 0 1 0 0 0 1 0] [0 1 0 0 0 0 0 0 0 0 1 0 1 0 0 1 0 0 1 1 0 0 0 0 0 1 1 1 0 1 1 1 1 0 1] [0 0 1 0 0 0 0 0 0 0 0 1 0 1 0 0 1 0 0 1 1 0 0 0 1 1 0 1 0 0 0 1 1 0 1] [0 0 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 1 1 0 0 0 1 0 0 0 1 0 0 0 1 1 0] [0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 1 1 1 1 0 0 0 0 0 0 1 0 0 1 0 1 1 1 0 0] [0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 1 1 1 1 0 0 0 0 0 0 1 0 0 1 0 1 1 1 0] [0 0 0 0 0 0 1 0 0 0 1 1 1 1 0 1 1 1 1 1 0 0 0 0 0 0 0 1 0 1 1 1 0 1 1] [0 0 0 0 0 0 0 1 0 0 1 1 1 0 0 1 1 0 0 0 1 0 0 0 0 0 0 0 0 1 0 1 1 0 0] [0 0 0 0 0 0 0 0 1 0 0 1 1 1 0 1 0 0 1 1 1 0 0 0 0 0 0 0 1 1 0 1 0 0 1] [0 0 0 0 0 0 0 0 0 1 1 1 0 1 0 0 1 1 1 0 1 0 0 0 0 0 0 0 0 1 0 0 1 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 0 0 0 0 0 0 0 1 1 1 1 1 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 1 0 1 0 1 0 0 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 1 1 1 0 1 0 0 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 1 0 1 0 0 1 1 1] [ <0, 1>, <8, 14>, <10, 182>, <12, 805>, <14, 2200>, <16, 3885>, <18, 4340>, <20, 3066>, <22, 1400>, <24, 420>, <26, 70>, <28, 1> ] [35, 14, 8] Linear Code over GF(2) Generator matrix: [1 0 0 0 0 0 0 0 0 0 1 0 0 1 0 1 0 1 1 1 0 0 0 0 1 1 1 0 0 1 0 0 0 1 0] [0 1 0 0 0 0 0 0 0 0 1 0 1 0 0 1 0 0 1 1 0 0 0 0 0 1 1 1 0 1 1 1 1 0 1] [0 0 1 0 0 0 0 0 0 0 0 1 0 1 0 0 1 0 0 1 1 0 0 0 1 1 0 1 0 0 0 1 1 0 1] [0 0 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 1 1 0 0 0 1 0 0 0 1 0 0 0 1 1 0] [0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 1 1 1 1 0 0 0 0 0 0 1 0 0 1 0 1 1 1 0 0] [0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 1 1 1 1 0 0 0 0 0 0 1 0 0 1 0 1 1 1 0] [0 0 0 0 0 0 1 0 0 0 1 1 1 1 0 1 1 1 1 1 0 0 0 0 0 0 0 1 0 1 1 1 0 1 1] [0 0 0 0 0 0 0 1 0 0 1 1 1 0 0 1 1 0 0 0 1 0 0 0 0 0 0 0 0 1 0 1 1 0 0] [0 0 0 0 0 0 0 0 1 0 0 1 1 1 0 1 0 0 1 1 1 0 0 0 0 0 0 0 1 1 0 1 0 0 1] [0 0 0 0 0 0 0 0 0 1 1 1 0 1 0 0 1 1 1 0 1 0 0 0 0 0 0 0 0 1 0 0 1 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 0 0 0 0 0 0 0 1 1 1 1 1 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 1 0 1 0 1 0 0 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 1 1 1 0 1 0 0 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 1 0 1 0 0 1 1 1] { (0 0 1 0 1 0 0 0 1 0 0 1 0 0 0 0 0 1 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 1 0), (0 0 0 1 0 1 0 0 0 1 0 0 1 0 0 0 0 0 1 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 1), (0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 1 1 1 1 1 0 1 0 0), (1 0 1 0 0 0 0 0 0 1 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 1 0 0 0), (0 0 0 0 1 0 1 0 0 0 1 0 0 1 0 0 0 0 0 1 0 0 0 1 1 0 0 0 1 0 0 0 0 0 0), (0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 1 1 1 0 1 0 0 1), (0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 1 0 1 0 1 0 0 1 1), (0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 0 0 1 0 1 1 1 0 1 0), (0 1 0 1 0 0 0 1 0 0 1 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 1 0 0), (0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 1 0 1 0 0 1 1 1), (0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 1 0 0 0 0 1 1 1 0 1), (0 1 0 0 0 0 1 0 1 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 1 0 0 0 0), (0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 1 0 1 0 0 1 1 1 0), (1 0 0 0 0 1 0 1 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 1 1 0 0 0 1 0 0 0 0 0) } Total time: 0.190 seconds, Total memory usage: 3.34MB '65.191.' ************** MAGMA ***************** Host 65.191.73.113 (65.191.73.113) Time: Tue Dec 13 18:52:22 2005 Input: K := FiniteField(2); > C := LinearCode; M:=VerticalJoin(GeneratorMatrix(C), ParityCheckMatrix(C)); M1:=EchelonForm(M); M2:=Submatrix(M1,22,22,14,35); M2; C2:=LinearCode(M2); WeightDistribution(C2); C2; Output: Magma V2.11-10 Tue Dec 13 2005 18:52:22 on modular [Seed = 3256803149] ------------------------------------- [1 0 0 0 0 0 0 0 0 0 1 0 0 1 0 1 0 1 1 1 0 0 0 0 1 1 1 0 0 1 0 0 0 1 0] [0 1 0 0 0 0 0 0 0 0 1 0 1 0 0 1 0 0 1 1 0 0 0 0 0 1 1 1 0 1 1 1 1 0 1] [0 0 1 0 0 0 0 0 0 0 0 1 0 1 0 0 1 0 0 1 1 0 0 0 1 1 0 1 0 0 0 1 1 0 1] [0 0 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 1 1 0 0 0 1 0 0 0 1 0 0 0 1 1 0] [0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 1 1 1 1 0 0 0 0 0 0 1 0 0 1 0 1 1 1 0 0] [0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 1 1 1 1 0 0 0 0 0 0 1 0 0 1 0 1 1 1 0] [0 0 0 0 0 0 1 0 0 0 1 1 1 1 0 1 1 1 1 1 0 0 0 0 0 0 0 1 0 1 1 1 0 1 1] [0 0 0 0 0 0 0 1 0 0 1 1 1 0 0 1 1 0 0 0 1 0 0 0 0 0 0 0 0 1 0 1 1 0 0] [0 0 0 0 0 0 0 0 1 0 0 1 1 1 0 1 0 0 1 1 1 0 0 0 0 0 0 0 1 1 0 1 0 0 1] [0 0 0 0 0 0 0 0 0 1 1 1 0 1 0 0 1 1 1 0 1 0 0 0 0 0 0 0 0 1 0 0 1 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 0 0 0 0 0 0 0 1 1 1 1 1 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 1 0 1 0 1 0 0 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 1 1 1 0 1 0 0 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 1 0 1 0 0 1 1 1] [ <0, 1>, <8, 14>, <10, 182>, <12, 805>, <14, 2200>, <16, 3885>, <18, 4340>, <20, 3066>, <22, 1400>, <24, 420>, <26, 70>, <28, 1> ] [35, 14, 8] Linear Code over GF(2) Generator matrix: [1 0 0 0 0 0 0 0 0 0 1 0 0 1 0 1 0 1 1 1 0 0 0 0 1 1 1 0 0 1 0 0 0 1 0] [0 1 0 0 0 0 0 0 0 0 1 0 1 0 0 1 0 0 1 1 0 0 0 0 0 1 1 1 0 1 1 1 1 0 1] [0 0 1 0 0 0 0 0 0 0 0 1 0 1 0 0 1 0 0 1 1 0 0 0 1 1 0 1 0 0 0 1 1 0 1] [0 0 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 1 1 0 0 0 1 0 0 0 1 0 0 0 1 1 0] [0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 1 1 1 1 0 0 0 0 0 0 1 0 0 1 0 1 1 1 0 0] [0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 1 1 1 1 0 0 0 0 0 0 1 0 0 1 0 1 1 1 0] [0 0 0 0 0 0 1 0 0 0 1 1 1 1 0 1 1 1 1 1 0 0 0 0 0 0 0 1 0 1 1 1 0 1 1] [0 0 0 0 0 0 0 1 0 0 1 1 1 0 0 1 1 0 0 0 1 0 0 0 0 0 0 0 0 1 0 1 1 0 0] [0 0 0 0 0 0 0 0 1 0 0 1 1 1 0 1 0 0 1 1 1 0 0 0 0 0 0 0 1 1 0 1 0 0 1] [0 0 0 0 0 0 0 0 0 1 1 1 0 1 0 0 1 1 1 0 1 0 0 0 0 0 0 0 0 1 0 0 1 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 0 0 0 0 0 0 0 1 1 1 1 1 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 1 0 1 0 1 0 0 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 1 1 1 0 1 0 0 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 1 0 1 0 0 1 1 1] Total time: 0.190 seconds, Total memory usage: 3.24MB '65.191.' ************** MAGMA ***************** Host 65.191.73.113 (65.191.73.113) Time: Tue Dec 13 18:46:56 2005 Input: K := FiniteField(2); > C := LinearCode; M:=VerticalJoin(GeneratorMatrix(C), ParityCheckMatrix(C)); M1:=EchelonForm(M); M2:=Submatrix(M1,22,22,14,35); M2; C2:=LinearCode(M2); WeightDistribution(C2); Output: Magma V2.11-10 Tue Dec 13 2005 18:46:55 on modular [Seed = 3423654554] ------------------------------------- [1 0 0 0 0 0 0 0 0 0 1 0 0 1 0 1 0 1 1 1 0 0 0 0 1 1 1 0 0 1 0 0 0 1 0] [0 1 0 0 0 0 0 0 0 0 1 0 1 0 0 1 0 0 1 1 0 0 0 0 0 1 1 1 0 1 1 1 1 0 1] [0 0 1 0 0 0 0 0 0 0 0 1 0 1 0 0 1 0 0 1 1 0 0 0 1 1 0 1 0 0 0 1 1 0 1] [0 0 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 1 1 0 0 0 1 0 0 0 1 0 0 0 1 1 0] [0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 1 1 1 1 0 0 0 0 0 0 1 0 0 1 0 1 1 1 0 0] [0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 1 1 1 1 0 0 0 0 0 0 1 0 0 1 0 1 1 1 0] [0 0 0 0 0 0 1 0 0 0 1 1 1 1 0 1 1 1 1 1 0 0 0 0 0 0 0 1 0 1 1 1 0 1 1] [0 0 0 0 0 0 0 1 0 0 1 1 1 0 0 1 1 0 0 0 1 0 0 0 0 0 0 0 0 1 0 1 1 0 0] [0 0 0 0 0 0 0 0 1 0 0 1 1 1 0 1 0 0 1 1 1 0 0 0 0 0 0 0 1 1 0 1 0 0 1] [0 0 0 0 0 0 0 0 0 1 1 1 0 1 0 0 1 1 1 0 1 0 0 0 0 0 0 0 0 1 0 0 1 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 0 0 0 0 0 0 0 1 1 1 1 1 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 1 0 1 0 1 0 0 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 1 1 1 0 1 0 0 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 1 0 1 0 0 1 1 1] [ <0, 1>, <8, 14>, <10, 182>, <12, 805>, <14, 2200>, <16, 3885>, <18, 4340>, <20, 3066>, <22, 1400>, <24, 420>, <26, 70>, <28, 1> ] Total time: 0.190 seconds, Total memory usage: 3.24MB '65.191.' ************** MAGMA ***************** Host 65.191.73.113 (65.191.73.113) Time: Tue Dec 13 18:42:13 2005 Input: K := FiniteField(2); > C := LinearCode; M:=VerticalJoin(GeneratorMatrix(C), ParityCheckMatrix(C)); M1:=EchelonForm(M); M2:=Submatrix(M1,22,22,14,35); M2; Output: Magma V2.11-10 Tue Dec 13 2005 18:42:12 on modular [Seed = 3591031227] ------------------------------------- [1 0 0 0 0 0 0 0 0 0 1 0 0 1 0 1 0 1 1 1 0 0 0 0 1 1 1 0 0 1 0 0 0 1 0] [0 1 0 0 0 0 0 0 0 0 1 0 1 0 0 1 0 0 1 1 0 0 0 0 0 1 1 1 0 1 1 1 1 0 1] [0 0 1 0 0 0 0 0 0 0 0 1 0 1 0 0 1 0 0 1 1 0 0 0 1 1 0 1 0 0 0 1 1 0 1] [0 0 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 1 1 0 0 0 1 0 0 0 1 0 0 0 1 1 0] [0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 1 1 1 1 0 0 0 0 0 0 1 0 0 1 0 1 1 1 0 0] [0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 1 1 1 1 0 0 0 0 0 0 1 0 0 1 0 1 1 1 0] [0 0 0 0 0 0 1 0 0 0 1 1 1 1 0 1 1 1 1 1 0 0 0 0 0 0 0 1 0 1 1 1 0 1 1] [0 0 0 0 0 0 0 1 0 0 1 1 1 0 0 1 1 0 0 0 1 0 0 0 0 0 0 0 0 1 0 1 1 0 0] [0 0 0 0 0 0 0 0 1 0 0 1 1 1 0 1 0 0 1 1 1 0 0 0 0 0 0 0 1 1 0 1 0 0 1] [0 0 0 0 0 0 0 0 0 1 1 1 0 1 0 0 1 1 1 0 1 0 0 0 0 0 0 0 0 1 0 0 1 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 0 0 0 0 0 0 0 1 1 1 1 1 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 1 0 1 0 1 0 0 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 1 1 1 0 1 0 0 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 1 0 1 0 0 1 1 1] Total time: 0.180 seconds, Total memory usage: 3.24MB '65.191.' ************** MAGMA ***************** Host 65.191.73.113 (65.191.73.113) Time: Tue Dec 13 18:38:53 2005 Input: K := FiniteField(2); > C := LinearCode; M:=VerticalJoin(GeneratorMatrix(C), ParityCheckMatrix(C)); M:=EchelonForm(M); M2:=Submatrix(M,22,22,14,35); M2; Output: Magma V2.11-10 Tue Dec 13 2005 18:38:53 on modular [Seed = 3879468883] ------------------------------------- [1 0 0 0 0 0 0 0 0 0 1 0 0 1 0 1 0 1 1 1 0 0 0 0 1 1 1 0 0 1 0 0 0 1 0] [0 1 0 0 0 0 0 0 0 0 1 0 1 0 0 1 0 0 1 1 0 0 0 0 0 1 1 1 0 1 1 1 1 0 1] [0 0 1 0 0 0 0 0 0 0 0 1 0 1 0 0 1 0 0 1 1 0 0 0 1 1 0 1 0 0 0 1 1 0 1] [0 0 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 1 1 0 0 0 1 0 0 0 1 0 0 0 1 1 0] [0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 1 1 1 1 0 0 0 0 0 0 1 0 0 1 0 1 1 1 0 0] [0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 1 1 1 1 0 0 0 0 0 0 1 0 0 1 0 1 1 1 0] [0 0 0 0 0 0 1 0 0 0 1 1 1 1 0 1 1 1 1 1 0 0 0 0 0 0 0 1 0 1 1 1 0 1 1] [0 0 0 0 0 0 0 1 0 0 1 1 1 0 0 1 1 0 0 0 1 0 0 0 0 0 0 0 0 1 0 1 1 0 0] [0 0 0 0 0 0 0 0 1 0 0 1 1 1 0 1 0 0 1 1 1 0 0 0 0 0 0 0 1 1 0 1 0 0 1] [0 0 0 0 0 0 0 0 0 1 1 1 0 1 0 0 1 1 1 0 1 0 0 0 0 0 0 0 0 1 0 0 1 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 0 0 0 0 0 0 0 1 1 1 1 1 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 1 0 1 0 1 0 0 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 1 1 1 0 1 0 0 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 1 0 1 0 0 1 1 1] Total time: 0.180 seconds, Total memory usage: 3.24MB '65.191.' ************** MAGMA ***************** Host 65.191.73.113 (65.191.73.113) Time: Tue Dec 13 18:37:30 2005 Input: K := FiniteField(2); > C := LinearCode; M:=VerticalJoin(GeneratorMatrix(C), ParityCheckMatrix(C)); M:=EchelonForm(M); M2:=Submatrix(M,22,22,35,14); M2; Output: Magma V2.11-10 Tue Dec 13 2005 18:37:30 on modular [Seed = 4013160428] ------------------------------------- [1 0 0 0 0 0 0 0 0 0 1 0 0 1] [0 1 0 0 0 0 0 0 0 0 1 0 1 0] [0 0 1 0 0 0 0 0 0 0 0 1 0 1] [0 0 0 1 0 0 0 0 0 0 1 1 0 0] [0 0 0 0 1 0 0 0 0 0 0 1 1 0] [0 0 0 0 0 1 0 0 0 0 0 0 1 1] [0 0 0 0 0 0 1 0 0 0 1 1 1 1] [0 0 0 0 0 0 0 1 0 0 1 1 1 0] [0 0 0 0 0 0 0 0 1 0 0 1 1 1] [0 0 0 0 0 0 0 0 0 1 1 1 0 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0] Total time: 0.180 seconds, Total memory usage: 3.24MB '65.191.' ************** MAGMA ***************** Host 65.191.73.113 (65.191.73.113) Time: Tue Dec 13 18:37:09 2005 Input: K := FiniteField(2); > C := LinearCode; M:=VerticalJoin(GeneratorMatrix(C), ParityCheckMatrix(C)); M:=EchelonForm(M); M2:=Submathrix(M,22,22,35,14); M2; Output: Magma V2.11-10 Tue Dec 13 2005 18:37:08 on modular [Seed = 3895783669] ------------------------------------- >> M2:=Submathrix(M,22,22,35,14); ^ User error: Identifier 'Submathrix' has not been declared or assigned >> M2;; ^ User error: Identifier 'M2' has not been declared or assigned Total time: 0.180 seconds, Total memory usage: 3.24MB '65.191.' ************** MAGMA ***************** Host 65.191.73.113 (65.191.73.113) Time: Tue Dec 13 18:32:27 2005 Input: K := FiniteField(2); > C := LinearCode; M:=VerticalJoin(GeneratorMatrix(C), ParityCheckMatrix(C)); M:=EchelonForm(M); M; Output: Magma V2.11-10 Tue Dec 13 2005 18:32:26 on modular [Seed = 4063161281] ------------------------------------- [1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 1 0 1 0 0 0 0 0 0 0 0 1 1 0 0 0 1 1 0] [0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 1 0 1 0 0 0 0 1 1 1 0 1 1 1 0 0 0 0] [0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 0 1 1 0 1 0 0 0 0 0 0 1 1 1 0 0 1 0 1 0 0] [0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 1 0 1 1 0 0 0 1 1 0 1 1 1 1 0 1 0 1] [0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 1 0 1 0 0 0 0 1 0 0 0 1 0 1 0 1 1 0] [0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 1 0 1 0 0 0 0 1 0 0 0 1 0 1 0 1 1] [0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 0 1 0 1 0 0 0 0 0 1 0 0 1 0 1 0 1 0] [0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 1 0 1 0 0 0 0 0 1 1 1 0 1 0 1 0 0 0 0] [0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 1 0 1 0 0 0 0 0 1 1 1 0 1 0 1 0 0 0] [0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 1 0 1 0 0 0 1 1 0 1 1 0 0 0 1 1 1] [0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 1 0 0 1 0 0 0 1 0 0 0 1 0 0 1 1 1 1] [0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 1 0 0 1 0 0 0 0 0 1 0 0 0 1 1 0 1 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 1 0 0 1 0 0 0 0 0 1 0 0 0 1 1 0 1 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 0 0 0 1 0 1 0 0 0 0 0 0 0 1 1 0 1 1 1 1 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 1 1 0 0 0 0 1 0 1 0 0 0 1 0 0 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 1 1 0 1 1 0 0 0 0 0 1 0 1 1 1 0 0 1 0 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 1 1 0 1 1 0 0 0 1 1 0 0 0 1 0 0 0 0 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 0 0 0 0 0 1 1 0 0 0 0 1 1 0 0 0 0 0 0 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 1 1 0 0 0 0 0 1 1 0 0 1 0 0 1 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 1 1 0 0 0 0 0 0 1 1 1 1 0 1 0 0 1 0 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 1 1 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 1 0 1 0 1 1 1 0 0 0 0 1 1 1 0 0 1 0 0 0 1 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 1 0 0 1 0 0 1 1 0 0 0 0 0 1 1 1 0 1 1 1 1 0 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 1 0 0 1 0 0 1 1 0 0 0 1 1 0 1 0 0 0 1 1 0 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 1 1 0 0 0 1 0 0 0 1 0 0 0 1 1 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 1 1 1 1 0 0 0 0 0 0 1 0 0 1 0 1 1 1 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 1 1 1 1 0 0 0 0 0 0 1 0 0 1 0 1 1 1 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 1 1 1 0 1 1 1 1 1 0 0 0 0 0 0 0 1 0 1 1 1 0 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 1 0 0 1 1 0 0 0 1 0 0 0 0 0 0 0 0 1 0 1 1 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 1 0 1 0 0 1 1 1 0 0 0 0 0 0 0 1 1 0 1 0 0 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 1 0 0 1 1 1 0 1 0 0 0 0 0 0 0 0 1 0 0 1 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 0 0 0 0 0 0 0 1 1 1 1 1 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 1 0 1 0 1 0 0 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 1 1 1 0 1 0 0 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 1 0 1 0 0 1 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0] Total time: 0.190 seconds, Total memory usage: 3.24MB '65.191.' ************** MAGMA ***************** Host 65.191.73.113 (65.191.73.113) Time: Tue Dec 13 18:30:17 2005 Input: K := FiniteField(2); > C := LinearCode; M:=VerticalJoin(GeneratorMatrix(C), ParityCheckMatrix(C)); M:=EchelonForm(M); Output: Magma V2.11-10 Tue Dec 13 2005 18:30:17 on modular [Seed = 4280542543] ------------------------------------- Total time: 0.190 seconds, Total memory usage: 3.24MB '65.191.' ************** MAGMA ***************** Host 65.191.73.113 (65.191.73.113) Time: Tue Dec 13 18:27:47 2005 Input: K := FiniteField(2); > C := LinearCode [1, 0, 0, 0, 0, 1, 1, 1], [0, 1, 0, 0, 1, 0, 1, 1], > [0, 0, 1, 0, 1, 1, 0, 1]>; M:=VerticalJoin(GeneratorMatrix(C), ParityCheckMatrix(C)); EchelonForm(M); Output: Magma V2.11-10 Tue Dec 13 2005 18:27:47 on modular [Seed = 2170395211] ------------------------------------- [1 0 0 0 0 1 1 1] [0 1 0 0 0 1 0 1] [0 0 1 0 0 0 1 1] [0 0 0 1 0 0 0 0] [0 0 0 0 1 1 1 0] [0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0] Total time: 0.190 seconds, Total memory usage: 3.24MB '65.191.' ************** MAGMA ***************** Host 65.191.73.113 (65.191.73.113) Time: Tue Dec 13 18:15:06 2005 Input: K := FiniteField(2); > C := LinearCode [1, 0, 0, 0, 0, 1, 1, 1], [0, 1, 0, 0, 1, 0, 1, 1], > [0, 0, 1, 0, 1, 1, 0, 1]>; M:=VerticalJoin(GeneratorMatrix(C), ParityCheckMatrix(C)); C := LinearCode(M); GeneratorMatrix(C); Output: Magma V2.11-10 Tue Dec 13 2005 18:15:06 on modular [Seed = 2504622934] ------------------------------------- [1 0 0 0 0 1 1 1] [0 1 0 0 0 1 0 1] [0 0 1 0 0 0 1 1] [0 0 0 1 0 0 0 0] [0 0 0 0 1 1 1 0] Total time: 0.190 seconds, Total memory usage: 3.24MB '65.191.' ************** MAGMA ***************** Host 65.191.73.113 (65.191.73.113) Time: Tue Dec 13 18:11:56 2005 Input: K := FiniteField(2); > C := LinearCode [1, 0, 0, 0, 0, 1, 1, 1], [0, 1, 0, 0, 1, 0, 1, 1], > [0, 0, 1, 0, 1, 1, 0, 1], [0, 0, 0, 1, 1, 1, 1, 0]>; M:=VerticalJoin(GeneratorMatrix(C), ParityCheckMatrix(C)); M; Output: Magma V2.11-10 Tue Dec 13 2005 18:11:56 on modular [Seed = 2672003482] ------------------------------------- [1 0 0 0 0 1 1 1] [0 1 0 0 1 0 1 1] [0 0 1 0 1 1 0 1] [0 0 0 1 1 1 1 0] [1 0 0 0 0 1 1 1] [0 1 0 0 1 0 1 1] [0 0 1 0 1 1 0 1] [0 0 0 1 1 1 1 0] Total time: 0.190 seconds, Total memory usage: 3.24MB '65.191.' ************** MAGMA ***************** Host 65.191.73.113 (65.191.73.113) Time: Tue Dec 13 18:11:38 2005 Input: K := FiniteField(2); > C := LinearCode [1, 0, 0, 0, 0, 1, 1, 1], [0, 1, 0, 0, 1, 0, 1, 1], > [0, 0, 1, 0, 1, 1, 0, 1], [0, 0, 0, 1, 1, 1, 1, 0]>; M:=VerticalJoin(GeneratorMatrix(C), ParityCheckMatrix(C)); Output: Magma V2.11-10 Tue Dec 13 2005 18:11:37 on modular [Seed = 2554629768] ------------------------------------- Total time: 0.180 seconds, Total memory usage: 3.24MB '65.191.' ************** MAGMA ***************** Host 65.191.73.113 (65.191.73.113) Time: Tue Dec 13 18:06:45 2005 Input: K := FiniteField(2); > C := LinearCode [1, 0, 0, 0, 0, 1, 1, 1], [0, 1, 0, 0, 1, 0, 1, 1], > [0, 0, 1, 0, 1, 1, 0, 1], [0, 0, 0, 1, 1, 1, 1, 0]>; GeneratorMatrix(C); ParityCheckMatrix(C); Output: Magma V2.11-10 Tue Dec 13 2005 18:06:44 on modular [Seed = 3026756489] ------------------------------------- [1 0 0 0 0 1 1 1] [0 1 0 0 1 0 1 1] [0 0 1 0 1 1 0 1] [0 0 0 1 1 1 1 0] [1 0 0 0 0 1 1 1] [0 1 0 0 1 0 1 1] [0 0 1 0 1 1 0 1] [0 0 0 1 1 1 1 0] Total time: 0.180 seconds, Total memory usage: 3.24MB '65.191.' ************** MAGMA ***************** Host 65.191.73.113 (65.191.73.113) Time: Tue Dec 13 18:06:09 2005 Input: K := FiniteField(2); > C := LinearCode [1, 0, 0, 0, 0, 1, 1, 1], [0, 1, 0, 0, 1, 0, 1, 1], > [0, 0, 1, 0, 1, 1, 0, 1], [0, 0, 0, 1, 1, 1, 1, 0]>; GeneratorMatrix(C); Output: Magma V2.11-10 Tue Dec 13 2005 18:06:09 on modular [Seed = 2976751977] ------------------------------------- [1 0 0 0 0 1 1 1] [0 1 0 0 1 0 1 1] [0 0 1 0 1 1 0 1] [0 0 0 1 1 1 1 0] Total time: 0.190 seconds, Total memory usage: 3.24MB '65.191.' ************** MAGMA ***************** Host 65.191.73.113 (65.191.73.113) Time: Tue Dec 13 18:05:35 2005 Input: K := FiniteField(2); > C := LinearCode [1, 0, 0, 0, 0, 1, 1, 1], [0, 1, 0, 0, 1, 0, 1, 1], > [0, 0, 1, 0, 1, 1, 0, 1], [0, 0, 0, 1, 1, 1, 1, 0]>; C; Output: Magma V2.11-10 Tue Dec 13 2005 18:05:35 on modular [Seed = 3127290428] ------------------------------------- [8, 4, 4] Linear Code over GF(2) Generator matrix: [1 0 0 0 0 1 1 1] [0 1 0 0 1 0 1 1] [0 0 1 0 1 1 0 1] [0 0 0 1 1 1 1 0] Total time: 0.190 seconds, Total memory usage: 3.24MB '65.191.' ************** MAGMA ***************** Host 65.191.73.113 (65.191.73.113) Time: Tue Dec 13 17:53:01 2005 Input: K := FiniteField(2); > C := LinearCode; S, f := StandardForm(C); S; Output: Magma V2.11-10 Tue Dec 13 2005 17:53:01 on modular [Seed = 64950519] ------------------------------------- [56, 21] Linear Code over GF(2) Generator matrix: [1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 0 0 0 0 1 1 1 0 1 0 1 0 0 1 1 0 0 1 1 0 0 0 1 1 0 1 0 0] [0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 1 0 1 0 0 1 0 0 0 1 0 0 1 0 1 1 1 0 0 0 1 1 0 0 1 1 0 1 0] [0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 1 0 0 0 1 1 0 0 0 0 0 0 0 1 1 1 1 1 0 0 0 0 0 0 0 1 1 1 1 0 0] [0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 1 1 1 1 0 1 1 0 0 1 1 0 1] [0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 1 0 1 1 0 0 1 1 1 1 1 1 0 1 1 1 0 0 1 1 0 1 1 1 1 0 0 1 1 0] [0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 0 0 0 0 1 0 1 1 1 0 0 0 1 1 1 0 0 0 1 0 0 1 0 0 1 0 1 0 0 0 1] [0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 1 1 1 0 0 1 0 1 0 1 1 0 1 0 0 1 1 1 1 1 0 1 1 0 0 0 1 0 1 0] [0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 1 1 0 0 1 1 0 0 0 1 0 0 1 0 1 1 0 0 1 1 1 0 1 1 0 1 0 0 1 1] [0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 1 1 1 0 1 0 1 1 1 0 0 1 1 1 0 0 0 1 1 1 1 1 1 1 1 0 1 1 0 0 0] [0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 0 1 1 1 1 0 0 0 0 1 1 0 1 1 0 1 0 1 1 1 0 1 0 0 1 0 1 1 0 0] [0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 1 1 1 0 0 1 0 1 1 1 1 1 0 0 0 0 1 1 1 1 0 1 1 0 0 0 0 0 1 0 1 1 0] [0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 1 0 1 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 0 1 0 1 1 1 0 1 0 0 1] [0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 1 1 1 1 1 0 0 1 1 1 0 1 1 1 0 0 1 1 1 1 1 0 1 0 0 1 1 1 0 0 0 1 0 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 1 0 0 0 1 0 0 0 1 0 1 0 1 0 1 1 0 0 1 1 1 0 1 0 0 0 1 1 0 0 0 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 1 1 0 0 0 0 1 0 0 1 1 1 0 1 1 1 0 1 0 1 0 0 1 1 1 0 1 1 0 1 0 0 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0 1 1 0 0 1 1 0 0 0 1 1 0 1 0 0 1 1 1 0 0 0 1 0 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0 1 1 1 0 1 1 0 0 0 1 1 0 1 0 0 1 1 1 0 0 0 1 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 1 0 1 1 1 0 1 0 0 0 1 0 1 1 0 0 1 0 1 1 0 0 1 1 0 0 0 1 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 1 0 1 0 0 1 1 1 0 0 1 1 1 0 1 0 1 1 1 0 1 0 1 0 1 0 0 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 1 0 1 1 0 1 0 0 1 1 0 1 1 0 0 0 0 1 1 0 0 0 1 0 0 0 1 0 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 1 1 1 1 0 1 0 0 0 1 0 1 1 0 0 1 0 1 1 0 0 0 1 0 0 0 1 0 1] Total time: 0.180 seconds, Total memory usage: 3.34MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Tue Dec 13 14:55:57 2005 Input: Zz := PolynomialRing(Integers()); P := 36*z^4 + 36*z^3 + 24*z^2 + 6*z + 1; v := 0; while true do v +:= 1; t := 6*v^2 + 1; u := -v; p := Evaluate(P, u); n := p + 1 - t; if p mod 4 eq 3 and p mod 9 eq 4 and IsProbablePrime(p) and IsProbablePrime(n) then break; end if; u := v; p := Evaluate(P, u); n := p + 1 - t; if p mod 4 eq 3 and p mod 9 eq 4 and IsProbablePrime(p) and IsProbablePrime(n) then break; end if; end while; "u =", u; "p =", p; "n =", n; Fp := GF(p); b := Fp!0; repeat repeat b := b + 1; until IsSquare(b + 1); y := Root(b + 1, 2); E := EllipticCurve([Fp | 0, b]); G := E![1, y]; until IsZero(n*G); "b =", b; "G =", G; lambda := Fp!2; while IsPower(lambda, 3) do lambda +:= 1; end while; "lambda =", lambda; Fp2 := ExtensionField; mu := i + 1; while IsSquare(mu) do mu +:= 1; end while; "mu =", mu; xi := 1/(lambda^2*mu^3); Fp12 := ExtensionField; Et := EllipticCurve([Fp2 | 0, b/xi]); xt := 1; while not IsSquare(xt^3 + b/xi) do xt +:= 1; end while; yt := Sqrt(xt^3 + b/xi); Gt := Et![xt, yt]; "Gt =", Gt; k := 12; assert (p^k - 1) mod n eq 0; chi := (p^k - 1) div n; Ek := EllipticCurve([Fp12 | 0, Fp12!b]); g := function(U, V, Q) if IsZero(U) or IsZero(V) or (U eq -V) or IsZero(Q) then return Fp12!1; end if; assert U[3] eq 1; assert V[3] eq 1; assert Q[3] eq 1; m := (U eq V) select 3*U[1]^2/(2*U[2]) else (V[2] - U[2])/(V[1] - U[1]); return m*(Q[1] - U[1]) + U[2] - Q[2]; end function; length := function(r) n := 0; v := 1; while v le r do n +:= 1; v +:= v; end while; return n; end function; bit := function(r, i) return (r div 2^i) mod 2; end function; miller := function(r, P, Q) f := 1; A := P; for i := length(r) - 1 to 0 by -1 do f := f^2*g(A, A, Q); A := 2*A; if bit(r, i) eq 1 then f *:= g(A, P, Q); A +:= P; end if; end for; return f; end function; tate := function(P, Qt) Q := Ek![Qt[1]*z^2, Qt[2]*z^3]; /* "P =", P; "Q' =", Qt; "Q =", Q; */ return miller(n, P, Q)^chi; end function; g := tate(G, Gt); "g =", g; /* for j in [1..11] do print "g^(p^" * Sprint(j) * "-1): " * Sprint(g^(p^j-1) ne 1); end for; print "g^(p^12-1): " * Sprint(g^(p^12-1) eq 1); "g^n =", g^n; */ for u in [1..n-1] do w := tate(u*G, Gt); //s := g^((u*v) mod n); s := tate(G, u*Gt); h := g^u; if w ne s or w ne h then "Failure!"; print "u = " * Sprint(u); print "e(u*P, Q) = " * Sprint(w); print "e(P, u*Q) = " * Sprint(s); print "e(P, Q)^u = " * Sprint(h); quit; end if; end for; "Success!"; Output: Magma V2.11-10 Tue Dec 13 2005 14:55:57 on modular [Seed = 1807159056] ------------------------------------- u = 1 p = 103 n = 97 b = 12 G = (1 : 61 : 1) lambda = 2 mu = i + 2 Gt = (4 : 44*i + 6 : 1) g = (4*i + 70)*z^5 + (i + 79)*z^4 + (78*i + 73)*z^3 + (24*i + 89)*z^2 + (95*i + 55)*z + 71*i + 56 Failure! u = 2 e(u*P, Q) = (38*i + 16)*z^5 + (101*i + 58)*z^4 + (43*i + 45)*z^3 + (102*i + 83)*z^2 + (61*i + 73)*z + 20*i + 35 e(P, u*Q) = (100*i + 19)*z^5 + (89*i + 82)*z^4 + (74*i + 97)*z^3 + (74*i + 77)*z^2 + (92*i + 40)*z + 92*i + 48 e(P, Q)^u = (48*i + 96)*z^5 + (98*i + 69)*z^4 + (91*i + 98)*z^3 + (100*i + 11)*z^2 + (48*i + 67)*z + 29*i + 33 Total time: 0.200 seconds, Total memory usage: 3.53MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Tue Dec 13 14:55:01 2005 Input: Zz := PolynomialRing(Integers()); P := 36*z^4 + 36*z^3 + 24*z^2 + 6*z + 1; v := 0; while true do v +:= 1; t := 6*v^2 + 1; u := -v; p := Evaluate(P, u); n := p + 1 - t; if p mod 4 eq 3 and p mod 9 eq 4 and IsProbablePrime(p) and IsProbablePrime(n) then break; end if; u := v; p := Evaluate(P, u); n := p + 1 - t; if p mod 4 eq 3 and p mod 9 eq 4 and IsProbablePrime(p) and IsProbablePrime(n) then break; end if; end while; "u =", u; "p =", p; "n =", n; Fp := GF(p); b := Fp!0; repeat repeat b := b + 1; until IsSquare(b + 1); y := Root(b + 1, 2); E := EllipticCurve([Fp | 0, b]); G := E![1, y]; until IsZero(n*G); "b =", b; "G =", G; lambda := Fp!2; while IsPower(lambda, 3) do lambda +:= 1; end while; "lambda =", lambda; Fp2 := ExtensionField; mu := i + 1; while IsSquare(mu) do mu +:= 1; end while; "mu =", mu; xi := 1/(lambda^2*mu^3); Fp12 := ExtensionField; Et := EllipticCurve([Fp2 | 0, b/xi]); xt := 1; while not IsSquare(xt^3 + b/xi) do xt +:= 1; end while; yt := Sqrt(xt^3 + b/xi); Gt := Et![xt, yt]; "Gt =", Gt; k := 12; assert (p^k - 1) mod n eq 0; chi := (p^k - 1) div n; Ek := EllipticCurve([Fp12 | 0, Fp12!b]); g := function(U, V, Q) if IsZero(U) or IsZero(V) or (U eq -V) or IsZero(Q) then return Fp12!1; end if; assert U[3] eq 1; assert V[3] eq 1; assert Q[3] eq 1; m := (U eq V) select 3*U[1]^2/(2*U[2]) else (V[2] - U[2])/(V[1] - U[1]); return m*(Q[1] - U[1]) + U[2] - Q[2]; end function; length := function(r) n := 0; v := 1; while v le r do n +:= 1; v +:= v; end while; return n; end function; bit := function(r, i) return (r div 2^i) mod 2; end function; miller := function(r, P, Q) f := 1; A := P; for i := length(r) - 1 to 0 by -1 do f := f^2*g(A, A, Q); A := 2*A; if bit(r, i) eq 1 then f *:= g(A, P, Q); A +:= P; end if; end for; return f; end function; tate := function(P, Qt) Q := Ek![Qt[1]*z^2, Qt[2]*z^3]; /* "P =", P; "Q' =", Qt; "Q =", Q; */ return miller(n, P, Q)^chi; end function; g := tate(G, Gt); "g =", g; /* for j in [1..11] do print "g^(p^" * Sprint(j) * "-1): " * Sprint(g^(p^j-1) ne 1); end for; print "g^(p^12-1): " * Sprint(g^(p^12-1) eq 1); "g^n =", g^n; */ for u in [1..n-1] do w := tate(u*G, Gt); //s := g^((u*v) mod n); s := tate(G, u*Gt); if w ne s then "Failure!"; print "u = " * Sprint(u); print "e(u*P, Q) = " * Sprint(w); print "e(P, u*Q) = " * Sprint(s); quit; end if; end for; "Success!"; Output: Magma V2.11-10 Tue Dec 13 2005 14:55:01 on modular [Seed = 1857161805] ------------------------------------- u = 1 p = 103 n = 97 b = 12 G = (1 : 61 : 1) lambda = 2 mu = i + 2 Gt = (4 : 44*i + 6 : 1) g = (4*i + 70)*z^5 + (i + 79)*z^4 + (78*i + 73)*z^3 + (24*i + 89)*z^2 + (95*i + 55)*z + 71*i + 56 Failure! u = 2 e(u*P, Q) = (38*i + 16)*z^5 + (101*i + 58)*z^4 + (43*i + 45)*z^3 + (102*i + 83)*z^2 + (61*i + 73)*z + 20*i + 35 e(P, u*Q) = (100*i + 19)*z^5 + (89*i + 82)*z^4 + (74*i + 97)*z^3 + (74*i + 77)*z^2 + (92*i + 40)*z + 92*i + 48 Total time: 0.190 seconds, Total memory usage: 3.53MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Tue Dec 13 14:52:25 2005 Input: Zz := PolynomialRing(Integers()); P := 36*z^4 + 36*z^3 + 24*z^2 + 6*z + 1; v := 0; while true do v +:= 1; t := 6*v^2 + 1; u := -v; p := Evaluate(P, u); n := p + 1 - t; if p mod 4 eq 3 and p mod 9 eq 4 and IsProbablePrime(p) and IsProbablePrime(n) then break; end if; u := v; p := Evaluate(P, u); n := p + 1 - t; if p mod 4 eq 3 and p mod 9 eq 4 and IsProbablePrime(p) and IsProbablePrime(n) then break; end if; end while; "u =", u; "p =", p; "n =", n; Fp := GF(p); b := Fp!0; repeat repeat b := b + 1; until IsSquare(b + 1); y := Root(b + 1, 2); E := EllipticCurve([Fp | 0, b]); G := E![1, y]; until IsZero(n*G); "b =", b; "G =", G; lambda := Fp!2; while IsPower(lambda, 3) do lambda +:= 1; end while; "lambda =", lambda; Fp2 := ExtensionField; mu := i + 1; while IsSquare(mu) do mu +:= 1; end while; "mu =", mu; xi := 1/(lambda^2*mu^3); Fp12 := ExtensionField; Et := EllipticCurve([Fp2 | 0, b/xi]); xt := 1; while not IsSquare(xt^3 + b/xi) do xt +:= 1; end while; yt := Sqrt(xt^3 + b/xi); Gt := Et![xt, yt]; "Gt =", Gt; k := 12; assert (p^k - 1) mod n eq 0; chi := (p^k - 1) div n; Ek := EllipticCurve([Fp12 | 0, Fp12!b]); g := function(U, V, Q) if IsZero(U) or IsZero(V) or (U eq -V) or IsZero(Q) then return Fp12!1; end if; assert U[3] eq 1; assert V[3] eq 1; assert Q[3] eq 1; m := (U eq V) select 3*U[1]^2/(2*U[2]) else (V[2] - U[2])/(V[1] - U[1]); return m*(Q[1] - U[1]) + U[2] - Q[2]; end function; length := function(r) n := 0; v := 1; while v le r do n +:= 1; v +:= v; end while; return n; end function; bit := function(r, i) return (r div 2^i) mod 2; end function; miller := function(r, P, Q) f := 1; A := P; for i := length(r) - 1 to 0 by -1 do f := f^2*g(A, A, Q); A := 2*A; if bit(r, i) eq 1 then f *:= g(A, P, Q); A +:= P; end if; end for; return f; end function; tate := function(P, Qt) Q := Ek![Qt[1]*z^2, Qt[2]*z^3]; /* "P =", P; "Q' =", Qt; "Q =", Q; */ return miller(n, P, Q)^chi; end function; g := tate(G, Gt); "g =", g; /* for j in [1..11] do print "g^(p^" * Sprint(j) * "-1): " * Sprint(g^(p^j-1) ne 1); end for; print "g^(p^12-1): " * Sprint(g^(p^12-1) eq 1); "g^n =", g^n; */ v := 1; for u in [1..n-1] do w := tate(u*G, v*Gt); s := g^((u*v) mod n); if w ne s then "Failure!"; print "u = " * Sprint(u); print "v = " * Sprint(v); print "e(u*P,v*Q) = " * Sprint(w); print "e(P,Q)^(uv) = " * Sprint(s); quit; end if; end for; "Success!"; Output: Magma V2.11-10 Tue Dec 13 2005 14:52:24 on modular [Seed = 2024021302] ------------------------------------- u = 1 p = 103 n = 97 b = 12 G = (1 : 61 : 1) lambda = 2 mu = i + 2 Gt = (4 : 44*i + 6 : 1) g = (4*i + 70)*z^5 + (i + 79)*z^4 + (78*i + 73)*z^3 + (24*i + 89)*z^2 + (95*i + 55)*z + 71*i + 56 Failure! u = 2 v = 1 e(u*P,v*Q) = (38*i + 16)*z^5 + (101*i + 58)*z^4 + (43*i + 45)*z^3 + (102*i + 83)*z^2 + (61*i + 73)*z + 20*i + 35 e(P,Q)^(uv) = (48*i + 96)*z^5 + (98*i + 69)*z^4 + (91*i + 98)*z^3 + (100*i + 11)*z^2 + (48*i + 67)*z + 29*i + 33 Total time: 0.200 seconds, Total memory usage: 3.53MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Tue Dec 13 14:51:09 2005 Input: Zz := PolynomialRing(Integers()); P := 36*z^4 + 36*z^3 + 24*z^2 + 6*z + 1; v := 0; while true do v +:= 1; t := 6*v^2 + 1; u := -v; p := Evaluate(P, u); n := p + 1 - t; if p mod 4 eq 3 and p mod 9 eq 4 and IsProbablePrime(p) and IsProbablePrime(n) then break; end if; u := v; p := Evaluate(P, u); n := p + 1 - t; if p mod 4 eq 3 and p mod 9 eq 4 and IsProbablePrime(p) and IsProbablePrime(n) then break; end if; end while; "u =", u; "p =", p; "n =", n; Fp := GF(p); b := Fp!0; repeat repeat b := b + 1; until IsSquare(b + 1); y := Root(b + 1, 2); E := EllipticCurve([Fp | 0, b]); G := E![1, y]; until IsZero(n*G); "b =", b; "G =", G; lambda := Fp!2; while IsPower(lambda, 3) do lambda +:= 1; end while; "lambda =", lambda; Fp2 := ExtensionField; mu := i + 1; while IsSquare(mu) do mu +:= 1; end while; "mu =", mu; xi := 1/(lambda^2*mu^3); Fp12 := ExtensionField; Et := EllipticCurve([Fp2 | 0, b/xi]); xt := 1; while not IsSquare(xt^3 + b/xi) do xt +:= 1; end while; yt := Sqrt(xt^3 + b/xi); Gt := Et![xt, yt]; "Gt =", Gt; k := 12; assert (p^k - 1) mod n eq 0; chi := (p^k - 1) div n; Ek := EllipticCurve([Fp12 | 0, Fp12!b]); g := function(U, V, Q) if IsZero(U) or IsZero(V) or (U eq -V) or IsZero(Q) then return Fp12!1; end if; assert U[3] eq 1; assert V[3] eq 1; assert Q[3] eq 1; m := (U eq V) select 3*U[1]^2/(2*U[2]) else (V[2] - U[2])/(V[1] - U[1]); return m*(Q[1] - U[1]) + U[2] - Q[2]; end function; length := function(r) n := 0; v := 1; while v le r do n +:= 1; v +:= v; end while; return n; end function; bit := function(r, i) return (r div 2^i) mod 2; end function; miller := function(r, P, Q) f := 1; A := P; for i := length(r) - 1 to 0 by -1 do f := f^2*g(A, A, Q); A := 2*A; if bit(r, i) eq 1 then f *:= g(A, P, Q); A +:= P; end if; end for; return f; end function; tate := function(P, Qt) Q := Ek![Qt[1]*z^2, Qt[2]*z^3]; /* "P =", P; "Q' =", Qt; "Q =", Q; */ return miller(n, P, Q)^chi; end function; g := tate(G, Gt); "g =", g; /* for j in [1..11] do print "g^(p^" * Sprint(j) * "-1): " * Sprint(g^(p^j-1) ne 1); end for; print "g^(p^12-1): " * Sprint(g^(p^12-1) eq 1); "g^n =", g^n; */ for v, u in [1..n-1] do w := tate(u*G, v*Gt); s := g^((u*v) mod n); if w ne s then "Failure!"; print "u = " * Sprint(u); print "v = " * Sprint(v); print "e(u*P,v*Q) = " * Sprint(w); print "e(P,Q)^(uv) = " * Sprint(s); quit; end if; end for; "Success!"; Output: Magma V2.11-10 Tue Dec 13 2005 14:51:08 on modular [Seed = 1907168363] ------------------------------------- u = 1 p = 103 n = 97 b = 12 G = (1 : 61 : 1) lambda = 2 mu = i + 2 Gt = (4 : 44*i + 6 : 1) g = (4*i + 70)*z^5 + (i + 79)*z^4 + (78*i + 73)*z^3 + (24*i + 89)*z^2 + (95*i + 55)*z + 71*i + 56 Failure! u = 2 v = 1 e(u*P,v*Q) = (38*i + 16)*z^5 + (101*i + 58)*z^4 + (43*i + 45)*z^3 + (102*i + 83)*z^2 + (61*i + 73)*z + 20*i + 35 e(P,Q)^(uv) = (48*i + 96)*z^5 + (98*i + 69)*z^4 + (91*i + 98)*z^3 + (100*i + 11)*z^2 + (48*i + 67)*z + 29*i + 33 Total time: 0.200 seconds, Total memory usage: 3.53MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Tue Dec 13 14:49:25 2005 Input: Zz := PolynomialRing(Integers()); P := 36*z^4 + 36*z^3 + 24*z^2 + 6*z + 1; v := 0; while true do v +:= 1; t := 6*v^2 + 1; u := -v; p := Evaluate(P, u); n := p + 1 - t; if p mod 4 eq 3 and p mod 9 eq 4 and IsProbablePrime(p) and IsProbablePrime(n) then break; end if; u := v; p := Evaluate(P, u); n := p + 1 - t; if p mod 4 eq 3 and p mod 9 eq 4 and IsProbablePrime(p) and IsProbablePrime(n) then break; end if; end while; "u =", u; "p =", p; "n =", n; Fp := GF(p); b := Fp!0; repeat repeat b := b + 1; until IsSquare(b + 1); y := Root(b + 1, 2); E := EllipticCurve([Fp | 0, b]); G := E![1, y]; until IsZero(n*G); "b =", b; "G =", G; lambda := Fp!2; while IsPower(lambda, 3) do lambda +:= 1; end while; "lambda =", lambda; Fp2 := ExtensionField; mu := i + 1; while IsSquare(mu) do mu +:= 1; end while; "mu =", mu; xi := 1/(lambda^2*mu^3); Fp12 := ExtensionField; Et := EllipticCurve([Fp2 | 0, b/xi]); xt := 1; while not IsSquare(xt^3 + b/xi) do xt +:= 1; end while; yt := Sqrt(xt^3 + b/xi); Gt := Et![xt, yt]; "Gt =", Gt; k := 12; assert (p^k - 1) mod n eq 0; chi := (p^k - 1) div n; Ek := EllipticCurve([Fp12 | 0, Fp12!b]); g := function(U, V, Q) if IsZero(U) or IsZero(V) or (U eq -V) or IsZero(Q) then return Fp12!1; end if; assert U[3] eq 1; assert V[3] eq 1; assert Q[3] eq 1; m := (U eq V) select 3*U[1]^2/(2*U[2]) else (V[2] - U[2])/(V[1] - U[1]); return m*(Q[1] - U[1]) + U[2] - Q[2]; end function; length := function(r) n := 0; v := 1; while v le r do n +:= 1; v +:= v; end while; return n; end function; bit := function(r, i) return (r div 2^i) mod 2; end function; miller := function(r, P, Q) f := 1; A := P; for i := length(r) - 1 to 0 by -1 do f := f^2*g(A, A, Q); A := 2*A; if bit(r, i) eq 1 then f *:= g(A, P, Q); A +:= P; end if; end for; return f; end function; tate := function(P, Qt) Q := Ek![Qt[1]*z^2, Qt[2]*z^3]; /* "P =", P; "Q' =", Qt; "Q =", Q; */ return miller(n, P, Q)^chi; end function; g := tate(G, Gt); "g =", g; /* for j in [1..11] do print "g^(p^" * Sprint(j) * "-1): " * Sprint(g^(p^j-1) ne 1); end for; print "g^(p^12-1): " * Sprint(g^(p^12-1) eq 1); "g^n =", g^n; */ for u, v in [1..n-1] do w := tate(u*G, v*Gt); s := g^((u*v) mod n); if w ne s then "Failure!"; print "u = " * Sprint(u); print "v = " * Sprint(v); print "e(u*P,v*Q) = " * Sprint(w); print "e(P,Q)^(uv) = " * Sprint(s); quit; end if; end for; "Success!"; Output: Magma V2.11-10 Tue Dec 13 2005 14:49:25 on modular [Seed = 1957171847] ------------------------------------- u = 1 p = 103 n = 97 b = 12 G = (1 : 61 : 1) lambda = 2 mu = i + 2 Gt = (4 : 44*i + 6 : 1) g = (4*i + 70)*z^5 + (i + 79)*z^4 + (78*i + 73)*z^3 + (24*i + 89)*z^2 + (95*i + 55)*z + 71*i + 56 Failure! u = 1 v = 2 e(u*P,v*Q) = (100*i + 19)*z^5 + (89*i + 82)*z^4 + (74*i + 97)*z^3 + (74*i + 77)*z^2 + (92*i + 40)*z + 92*i + 48 e(P,Q)^(uv) = (48*i + 96)*z^5 + (98*i + 69)*z^4 + (91*i + 98)*z^3 + (100*i + 11)*z^2 + (48*i + 67)*z + 29*i + 33 Total time: 0.200 seconds, Total memory usage: 3.53MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Tue Dec 13 14:48:24 2005 Input: Zz := PolynomialRing(Integers()); P := 36*z^4 + 36*z^3 + 24*z^2 + 6*z + 1; v := 0; while true do v +:= 1; t := 6*v^2 + 1; u := -v; p := Evaluate(P, u); n := p + 1 - t; if p mod 4 eq 3 and p mod 9 eq 4 and IsProbablePrime(p) and IsProbablePrime(n) then break; end if; u := v; p := Evaluate(P, u); n := p + 1 - t; if p mod 4 eq 3 and p mod 9 eq 4 and IsProbablePrime(p) and IsProbablePrime(n) then break; end if; end while; "u =", u; "p =", p; "n =", n; Fp := GF(p); b := Fp!0; repeat repeat b := b + 1; until IsSquare(b + 1); y := Root(b + 1, 2); E := EllipticCurve([Fp | 0, b]); G := E![1, y]; until IsZero(n*G); "b =", b; "G =", G; lambda := Fp!2; while IsPower(lambda, 3) do lambda +:= 1; end while; "lambda =", lambda; Fp2 := ExtensionField; mu := i + 1; while IsSquare(mu) do mu +:= 1; end while; "mu =", mu; xi := 1/(lambda^2*mu^3); Fp12 := ExtensionField; Et := EllipticCurve([Fp2 | 0, b/xi]); xt := 1; while not IsSquare(xt^3 + b/xi) do xt +:= 1; end while; yt := Sqrt(xt^3 + b/xi); Gt := Et![xt, yt]; "Gt =", Gt; k := 12; assert (p^k - 1) mod n eq 0; chi := (p^k - 1) div n; Ek := EllipticCurve([Fp12 | 0, Fp12!b]); g := function(U, V, Q) if IsZero(U) or IsZero(V) or (U eq -V) or IsZero(Q) then return Fp12!1; end if; assert U[3] eq 1; assert V[3] eq 1; assert Q[3] eq 1; m := (U eq V) select 3*U[1]^2/(2*U[2]) else (V[2] - U[2])/(V[1] - U[1]); return m*(Q[1] - U[1]) + U[2] - Q[2]; end function; length := function(r) n := 0; v := 1; while v le r do n +:= 1; v +:= v; end while; return n; end function; bit := function(r, i) return (r div 2^i) mod 2; end function; miller := function(r, P, Q) f := 1; A := P; for i := length(r) - 1 to 0 by -1 do f := f^2*g(A, A, Q); A := 2*A; if bit(r, i) eq 1 then f *:= g(A, P, Q); A +:= P; end if; end for; return f; end function; tate := function(P, Qt) Q := Ek![Qt[1]*z^2, Qt[2]*z^3]; /* "P =", P; "Q' =", Qt; "Q =", Q; */ return miller(n, P, Q)^chi; end function; g := tate(G, Gt); "g =", g; /* for j in [1..11] do print "g^(p^" * Sprint(j) * "-1): " * Sprint(g^(p^j-1) ne 1); end for; print "g^(p^12-1): " * Sprint(g^(p^12-1) eq 1); "g^n =", g^n; */ for u in [1..n-1] do for v in [1..n-1] do w := tate(u*G, v*Gt); s := g^((u*v) mod n); if w ne s then "Failure!"; print "u = " * Sprint(u); print "v = " * Sprint(v); print "e(u*P,v*Q) = " * Sprint(w); print "e(P,Q)^(uv) = " * Sprint(s); quit; end if; end for; end for; "Success!"; Output: Magma V2.11-10 Tue Dec 13 2005 14:48:23 on modular [Seed = 198667706] ------------------------------------- u = 1 p = 103 n = 97 b = 12 G = (1 : 61 : 1) lambda = 2 mu = i + 2 Gt = (4 : 44*i + 6 : 1) g = (4*i + 70)*z^5 + (i + 79)*z^4 + (78*i + 73)*z^3 + (24*i + 89)*z^2 + (95*i + 55)*z + 71*i + 56 Failure! u = 1 v = 2 e(u*P,v*Q) = (100*i + 19)*z^5 + (89*i + 82)*z^4 + (74*i + 97)*z^3 + (74*i + 77)*z^2 + (92*i + 40)*z + 92*i + 48 e(P,Q)^(uv) = (48*i + 96)*z^5 + (98*i + 69)*z^4 + (91*i + 98)*z^3 + (100*i + 11)*z^2 + (48*i + 67)*z + 29*i + 33 Total time: 0.200 seconds, Total memory usage: 3.53MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Tue Dec 13 14:46:16 2005 Input: Zz := PolynomialRing(Integers()); P := 36*z^4 + 36*z^3 + 24*z^2 + 6*z + 1; v := 0; while true do v +:= 1; t := 6*v^2 + 1; u := -v; p := Evaluate(P, u); n := p + 1 - t; if p mod 4 eq 3 and p mod 9 eq 4 and IsProbablePrime(p) and IsProbablePrime(n) then break; end if; u := v; p := Evaluate(P, u); n := p + 1 - t; if p mod 4 eq 3 and p mod 9 eq 4 and IsProbablePrime(p) and IsProbablePrime(n) then break; end if; end while; "u =", u; "p =", p; "n =", n; Fp := GF(p); b := Fp!0; repeat repeat b := b + 1; until IsSquare(b + 1); y := Root(b + 1, 2); E := EllipticCurve([Fp | 0, b]); G := E![1, y]; until IsZero(n*G); "b =", b; "G =", G; lambda := Fp!2; while IsPower(lambda, 3) do lambda +:= 1; end while; "lambda =", lambda; Fp2 := ExtensionField; mu := i + 1; while IsSquare(mu) do mu +:= 1; end while; "mu =", mu; xi := 1/(lambda^2*mu^3); Fp12 := ExtensionField; Et := EllipticCurve([Fp2 | 0, b/xi]); xt := 1; while not IsSquare(xt^3 + b/xi) do xt +:= 1; end while; yt := Sqrt(xt^3 + b/xi); Gt := Et![xt, yt]; "Gt =", Gt; k := 12; assert (p^k - 1) mod n eq 0; chi := (p^k - 1) div n; Ek := EllipticCurve([Fp12 | 0, Fp12!b]); g := function(U, V, Q) if IsZero(U) or IsZero(V) or (U eq -V) or IsZero(Q) then return Fp12!1; end if; assert U[3] eq 1; assert V[3] eq 1; assert Q[3] eq 1; m := (U eq V) select 3*U[1]^2/(2*U[2]) else (V[2] - U[2])/(V[1] - U[1]); return m*(Q[1] - U[1]) + U[2] - Q[2]; end function; length := function(r) n := 0; v := 1; while v le r do n +:= 1; v +:= v; end while; return n; end function; bit := function(r, i) return (r div 2^i) mod 2; end function; miller := function(r, P, Q) f := 1; A := P; for i := length(r) - 1 to 0 by -1 do f := f^2*g(A, A, Q); A := 2*A; if bit(r, i) eq 1 then f *:= g(A, P, Q); A +:= P; end if; end for; return f; end function; tate := function(P, Qt) Q := Ek![Qt[1]*z^2, Qt[2]*z^3]; /* "P =", P; "Q' =", Qt; "Q =", Q; */ return miller(n, P, Q)^chi; end function; g := tate(G, Gt); "g =", g; /* for j in [1..11] do print "g^(p^" * Sprint(j) * "-1): " * Sprint(g^(p^j-1) ne 1); end for; print "g^(p^12-1): " * Sprint(g^(p^12-1) eq 1); "g^n =", g^n; */ for u in [1..n-1] do for v in [1..n-1] do w := tate(u*G, v*Gt); if w ne g^((u*v) mod n) then "Failure: u =", u, ", v =", v, ", e(u*P,v*Q) =", w, ", e(P,Q)^(uv) =", g^((u*v) mod n); quit; end if; end for; end for; "Success!"; Output: Magma V2.11-10 Tue Dec 13 2005 14:46:16 on modular [Seed = 31287103] ------------------------------------- u = 1 p = 103 n = 97 b = 12 G = (1 : 61 : 1) lambda = 2 mu = i + 2 Gt = (4 : 44*i + 6 : 1) g = (4*i + 70)*z^5 + (i + 79)*z^4 + (78*i + 73)*z^3 + (24*i + 89)*z^2 + (95*i + 55)*z + 71*i + 56 Failure: u = 1 , v = 2 , e(u*P,v*Q) = (100*i + 19)*z^5 + (89*i + 82)*z^4 + (74*i + 97)*z^3 + (74*i + 77)*z^2 + (92*i + 40)*z + 92*i + 48 , e(P,Q)^(uv) = (48*i + 96)*z^5 + (98*i + 69)*z^4 + (91*i + 98)*z^3 + (100*i + 11)*z^2 + (48*i + 67)*z + 29*i + 33 Total time: 0.190 seconds, Total memory usage: 3.53MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Tue Dec 13 14:45:31 2005 Input: Zz := PolynomialRing(Integers()); P := 36*z^4 + 36*z^3 + 24*z^2 + 6*z + 1; v := 0; while true do v +:= 1; t := 6*v^2 + 1; u := -v; p := Evaluate(P, u); n := p + 1 - t; if p mod 4 eq 3 and p mod 9 eq 4 and IsProbablePrime(p) and IsProbablePrime(n) then break; end if; u := v; p := Evaluate(P, u); n := p + 1 - t; if p mod 4 eq 3 and p mod 9 eq 4 and IsProbablePrime(p) and IsProbablePrime(n) then break; end if; end while; "u =", u; "p =", p; "n =", n; Fp := GF(p); b := Fp!0; repeat repeat b := b + 1; until IsSquare(b + 1); y := Root(b + 1, 2); E := EllipticCurve([Fp | 0, b]); G := E![1, y]; until IsZero(n*G); "b =", b; "G =", G; lambda := Fp!2; while IsPower(lambda, 3) do lambda +:= 1; end while; "lambda =", lambda; Fp2 := ExtensionField; mu := i + 1; while IsSquare(mu) do mu +:= 1; end while; "mu =", mu; xi := 1/(lambda^2*mu^3); Fp12 := ExtensionField; Et := EllipticCurve([Fp2 | 0, b/xi]); xt := 1; while not IsSquare(xt^3 + b/xi) do xt +:= 1; end while; yt := Sqrt(xt^3 + b/xi); Gt := Et![xt, yt]; "Gt =", Gt; k := 12; assert (p^k - 1) mod n eq 0; chi := (p^k - 1) div n; Ek := EllipticCurve([Fp12 | 0, Fp12!b]); g := function(U, V, Q) if IsZero(U) or IsZero(V) or (U eq -V) or IsZero(Q) then return Fp12!1; end if; assert U[3] eq 1; assert V[3] eq 1; assert Q[3] eq 1; m := (U eq V) select 3*U[1]^2/(2*U[2]) else (V[2] - U[2])/(V[1] - U[1]); return m*(Q[1] - U[1]) + U[2] - Q[2]; end function; length := function(r) n := 0; v := 1; while v le r do n +:= 1; v +:= v; end while; return n; end function; bit := function(r, i) return (r div 2^i) mod 2; end function; miller := function(r, P, Q) f := 1; A := P; for i := length(r) - 1 to 0 by -1 do f := f^2*g(A, A, Q); A := 2*A; if bit(r, i) eq 1 then f *:= g(A, P, Q); A +:= P; end if; end for; return f; end function; tate := function(P, Qt) Q := Ek![Qt[1]*z^2, Qt[2]*z^3]; /* "P =", P; "Q' =", Qt; "Q =", Q; */ return miller(n, P, Q)^chi; end function; g := tate(G, Gt); "g =", g; /* for j in [1..11] do print "g^(p^" * Sprint(j) * "-1): " * Sprint(g^(p^j-1) ne 1); end for; print "g^(p^12-1): " * Sprint(g^(p^12-1) eq 1); "g^n =", g^n; */ for u in [0..n-1] do for v in [0..n-1] do w := tate(u*G, v*Gt); if w ne g^((u*v) mod n) then "Failure: u =", u, ", v =", v, ", e(u*P,v*Q) =", w, ", e(P,Q)^(uv) =", g^((u*v) mod n); quit; end if; end for; end for; "Success!"; Output: Magma V2.11-10 Tue Dec 13 2005 14:45:30 on modular [Seed = 81290216] ------------------------------------- u = 1 p = 103 n = 97 b = 12 G = (1 : 61 : 1) lambda = 2 mu = i + 2 Gt = (4 : 44*i + 6 : 1) g = (4*i + 70)*z^5 + (i + 79)*z^4 + (78*i + 73)*z^3 + (24*i + 89)*z^2 + (95*i + 55)*z + 71*i + 56 tate( P: (0 : 1 : 0), Qt: (0 : 1 : 0) ) >> Q := Ek![Qt[1]*z^2, Qt[2]*z^3]; ^ Runtime error in '!': Illegal coercion Success! Total time: 0.190 seconds, Total memory usage: 3.53MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Tue Dec 13 14:44:16 2005 Input: Zz := PolynomialRing(Integers()); P := 36*z^4 + 36*z^3 + 24*z^2 + 6*z + 1; v := 0; while true do v +:= 1; t := 6*v^2 + 1; u := -v; p := Evaluate(P, u); n := p + 1 - t; if p mod 4 eq 3 and p mod 9 eq 4 and IsProbablePrime(p) and IsProbablePrime(n) then break; end if; u := v; p := Evaluate(P, u); n := p + 1 - t; if p mod 4 eq 3 and p mod 9 eq 4 and IsProbablePrime(p) and IsProbablePrime(n) then break; end if; end while; "u =", u; "p =", p; "n =", n; Fp := GF(p); b := Fp!0; repeat repeat b := b + 1; until IsSquare(b + 1); y := Root(b + 1, 2); E := EllipticCurve([Fp | 0, b]); G := E![1, y]; until IsZero(n*G); "b =", b; "G =", G; lambda := Fp!2; while IsPower(lambda, 3) do lambda +:= 1; end while; "lambda =", lambda; Fp2 := ExtensionField; mu := i + 1; while IsSquare(mu) do mu +:= 1; end while; "mu =", mu; xi := 1/(lambda^2*mu^3); Fp12 := ExtensionField; Et := EllipticCurve([Fp2 | 0, b/xi]); xt := 1; while not IsSquare(xt^3 + b/xi) do xt +:= 1; end while; yt := Sqrt(xt^3 + b/xi); Gt := Et![xt, yt]; "Gt =", Gt; k := 12; assert (p^k - 1) mod n eq 0; chi := (p^k - 1) div n; Ek := EllipticCurve([Fp12 | 0, Fp12!b]); g := function(U, V, Q) if IsZero(U) or IsZero(V) or (U eq -V) or IsZero(Q) then return Fp12!1; end if; assert U[3] eq 1; assert V[3] eq 1; assert Q[3] eq 1; m := (U eq V) select 3*U[1]^2/(2*U[2]) else (V[2] - U[2])/(V[1] - U[1]); return m*(Q[1] - U[1]) + U[2] - Q[2]; end function; length := function(r) n := 0; v := 1; while v le r do n +:= 1; v +:= v; end while; return n; end function; bit := function(r, i) return (r div 2^i) mod 2; end function; miller := function(r, P, Q) f := 1; A := P; for i := length(r) - 1 to 0 by -1 do f := f^2*g(A, A, Q); A := 2*A; if bit(r, i) eq 1 then f *:= g(A, P, Q); A +:= P; end if; end for; return f; end function; tate := function(P, Qt) Q := Ek![Qt[1]*z^2, Qt[2]*z^3]; /* "P =", P; "Q' =", Qt; "Q =", Q; */ return miller(n, P, Q)^chi; end function; g := tate(G, Gt); "g =", g; /* for j in [1..11] do print "g^(p^" * Sprint(j) * "-1): " * Sprint(g^(p^j-1) ne 1); end for; print "g^(p^12-1): " * Sprint(g^(p^12-1) eq 1); "g^n =", g^n; */ for j in [1..100] do u := Random(n - 1); v := Random(n - 1); w := tate(u*G, v*Gt); if w ne g^((u*v) mod n) then "Failure: u =", u, ", v =", v, ", e(u*P,v*Q) =", w, ", e(P,Q)^(uv) =", g^((u*v) mod n); quit; end if; end for; "Success!"; Output: Magma V2.11-10 Tue Dec 13 2005 14:44:16 on modular [Seed = 466058429] ------------------------------------- u = 1 p = 103 n = 97 b = 12 G = (1 : 61 : 1) lambda = 2 mu = i + 2 Gt = (4 : 44*i + 6 : 1) g = (4*i + 70)*z^5 + (i + 79)*z^4 + (78*i + 73)*z^3 + (24*i + 89)*z^2 + (95*i + 55)*z + 71*i + 56 Failure: u = 88 , v = 67 , e(u*P,v*Q) = (100*i + 78)*z^5 + (36*i + 83)*z^4 + (30*i + 95)*z^3 + (93*i + 66)*z^2 + (21*i + 39)*z + 74*i + 33 , e(P,Q)^(uv) = (28*i + 78)*z^5 + (62*i + 21)*z^4 + (82*i + 32)*z^3 + (36*i + 57)*z^2 + (69*i + 26)*z + 95*i + 78 Total time: 0.190 seconds, Total memory usage: 3.53MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Tue Dec 13 14:39:51 2005 Input: p := 103;//1461501624496790265145448589920785493717258890819; n := 97;//1461501624496790265145447380994971188499300027613; Fp := GF(p); Fp2 := ExtensionField; lambda := Fp!2; mu := 2 + i;//1 + i; xi := 1/(lambda^2*mu^3); Fp12 := ExtensionField; b := Fp!12;//Fp!3; y0 := Fp!61;//Fp!2; // -Sqrt(1 + b); E := EllipticCurve([0, b]); G := E![1, y0]; "G =", G; Et := EllipticCurve([0, b/xi]); xt := 1; while not IsSquare(xt^3 + b/xi) do xt +:= 1; end while; yt := Sqrt(xt^3 + b/xi); Gt := Et![xt, yt]; "Gt =", Gt; k := 12; assert (p^k - 1) mod n eq 0; chi := (p^k - 1) div n; Ek := EllipticCurve([0, Fp12!b]); g := function(U, V, Q) if IsZero(U) or IsZero(V) or (U eq -V) or IsZero(Q) then return Fp12!1; end if; assert U[3] eq 1; assert V[3] eq 1; assert Q[3] eq 1; m := (U eq V) select 3*U[1]^2/(2*U[2]) else (V[2] - U[2])/(V[1] - U[1]); return m*(Q[1] - U[1]) + U[2] - Q[2]; end function; length := function(r) n := 0; v := 1; while v le r do n +:= 1; v +:= v; end while; return n; end function; bit := function(r, i) return (r div 2^i) mod 2; end function; miller := function(r, P, Q) f := 1; A := P; for i := length(r) - 1 to 0 by -1 do f := f^2*g(A, A, Q); A := 2*A; if bit(r, i) eq 1 then f *:= g(A, P, Q); A +:= P; end if; end for; return f; end function; tate := function(P, Qt) Q := Ek![Qt[1]*z^2, Qt[2]*z^3]; /* "P =", P; "Q' =", Qt; "Q =", Q; */ return miller(n, P, Q)^chi; end function; g := tate(G, Gt); "g =", g; /* for j in [1..11] do print "g^(p^" * Sprint(j) * "-1): " * Sprint(g^(p^j-1) ne 1); end for; print "g^(p^12-1): " * Sprint(g^(p^12-1) eq 1); "g^n =", g^n; */ for j in [1..100] do u := Random(n - 1); v := Random(n - 1); w := tate(u*G, v*Gt); if w ne g^((u*v) mod n) then "Failure: u =", u, ", v =", v, ", e(u*P,v*Q) =", w, ", e(P,Q)^(uv) =", g^((u*v) mod n); quit; end if; end for; "Success!"; Output: Magma V2.11-10 Tue Dec 13 2005 14:39:51 on modular [Seed = 298676640] ------------------------------------- G = (1 : 61 : 1) Gt = (4 : 44*i + 6 : 1) g = (4*i + 70)*z^5 + (i + 79)*z^4 + (78*i + 73)*z^3 + (24*i + 89)*z^2 + (95*i + 55)*z + 71*i + 56 Failure: u = 2 , v = 18 , e(u*P,v*Q) = (50*i + 27)*z^5 + (94*i + 26)*z^4 + (96*i + 54)*z^3 + (19*i + 50)*z^2 + (61*i + 86)*z + 74*i + 83 , e(P,Q)^(uv) = (85*i + 97)*z^5 + (46*i + 29)*z^4 + (25*i + 30)*z^3 + (5*i + 40)*z^2 + (59*i + 45)*z + 71*i + 56 Total time: 0.190 seconds, Total memory usage: 3.53MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Tue Dec 13 14:39:14 2005 Input: p := 103;//1461501624496790265145448589920785493717258890819; n := 97;//1461501624496790265145447380994971188499300027613; Fp := GF(p); Fp2 := ExtensionField; lambda := Fp!2; mu := 2 + i;//1 + i; xi := 1/(lambda^2*mu^3); Fp12 := ExtensionField; b := Fp!12;//Fp!3; y0 := Fp!61;//Fp!2; // -Sqrt(1 + b); E := EllipticCurve([0, b]); G := E![1, y0]; Et := EllipticCurve([0, b/xi]); xt := 1; while not IsSquare(xt^3 + b/xi) do xt +:= 1; end while; "xt =", xt; yt := Sqrt(xt^3 + b/xi); Gt := Et![xt, yt]; k := 12; assert (p^k - 1) mod n eq 0; chi := (p^k - 1) div n; Ek := EllipticCurve([0, Fp12!b]); g := function(U, V, Q) if IsZero(U) or IsZero(V) or (U eq -V) or IsZero(Q) then return Fp12!1; end if; assert U[3] eq 1; assert V[3] eq 1; assert Q[3] eq 1; m := (U eq V) select 3*U[1]^2/(2*U[2]) else (V[2] - U[2])/(V[1] - U[1]); return m*(Q[1] - U[1]) + U[2] - Q[2]; end function; length := function(r) n := 0; v := 1; while v le r do n +:= 1; v +:= v; end while; return n; end function; bit := function(r, i) return (r div 2^i) mod 2; end function; miller := function(r, P, Q) f := 1; A := P; for i := length(r) - 1 to 0 by -1 do f := f^2*g(A, A, Q); A := 2*A; if bit(r, i) eq 1 then f *:= g(A, P, Q); A +:= P; end if; end for; return f; end function; tate := function(P, Qt) Q := Ek![Qt[1]*z^2, Qt[2]*z^3]; /* "P =", P; "Q' =", Qt; "Q =", Q; */ return miller(n, P, Q)^chi; end function; g := tate(G, Gt); "g =", g; /* for j in [1..11] do print "g^(p^" * Sprint(j) * "-1): " * Sprint(g^(p^j-1) ne 1); end for; print "g^(p^12-1): " * Sprint(g^(p^12-1) eq 1); "g^n =", g^n; */ for j in [1..100] do u := Random(n - 1); v := Random(n - 1); w := tate(u*G, v*Gt); if w ne g^((u*v) mod n) then "Failure: u =", u, ", v =", v, ", e(u*P,v*Q) =", w, ", e(P,Q)^(uv) =", g^((u*v) mod n); quit; end if; end for; "Success!"; Output: Magma V2.11-10 Tue Dec 13 2005 14:39:14 on modular [Seed = 348681280] ------------------------------------- xt = 4 g = (4*i + 70)*z^5 + (i + 79)*z^4 + (78*i + 73)*z^3 + (24*i + 89)*z^2 + (95*i + 55)*z + 71*i + 56 Failure: u = 77 , v = 45 , e(u*P,v*Q) = (99*i + 33)*z^5 + (i + 79)*z^4 + (25*i + 30)*z^3 + (24*i + 89)*z^2 + (8*i + 48)*z + 71*i + 56 , e(P,Q)^(uv) = (45*i + 90)*z^5 + (29*i + 53)*z^4 + (91*i + 98)*z^3 + (68*i + 94)*z^2 + (10*i + 44)*z + 29*i + 33 Total time: 0.200 seconds, Total memory usage: 3.53MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Tue Dec 13 14:37:09 2005 Input: p := 1461501624496790265145448589920785493717258890819; n := 1461501624496790265145447380994971188499300027613; Fp := GF(p); Fp2 := ExtensionField; lambda := Fp!2; mu := 1 + i; xi := 1/(lambda^2*mu^3); Fp12 := ExtensionField; b := Fp!3; y0 := Fp!2; // -Sqrt(1 + b); E := EllipticCurve([0, b]); G := E![1, y0]; Et := EllipticCurve([0, b/xi]); //xt := i + 2; xt := 8; /* xt := i; while not IsSquare(xt^3 + b/xi) do xt +:= 1; end while; "xt =", xt; */ yt := Sqrt(xt^3 + b/xi); Gt := Et![xt, yt]; k := 12; assert (p^k - 1) mod n eq 0; chi := (p^k - 1) div n; Ek := EllipticCurve([0, Fp12!b]); g := function(U, V, Q) if IsZero(U) or IsZero(V) or (U eq -V) or IsZero(Q) then return Fp12!1; end if; assert U[3] eq 1; assert V[3] eq 1; assert Q[3] eq 1; m := (U eq V) select 3*U[1]^2/(2*U[2]) else (V[2] - U[2])/(V[1] - U[1]); return m*(Q[1] - U[1]) + U[2] - Q[2]; end function; length := function(r) n := 0; v := 1; while v le r do n +:= 1; v +:= v; end while; return n; end function; bit := function(r, i) return (r div 2^i) mod 2; end function; miller := function(r, P, Q) f := 1; A := P; for i := length(r) - 1 to 0 by -1 do f := f^2*g(A, A, Q); A := 2*A; if bit(r, i) eq 1 then f *:= g(A, P, Q); A +:= P; end if; end for; return f; end function; tate := function(P, Qt) Q := Ek![Qt[1]*z^2, Qt[2]*z^3]; /* "P =", P; "Q' =", Qt; "Q =", Q; */ return miller(n, P, Q)^chi; end function; g := tate(G, Gt); "g =", g; /* for j in [1..11] do print "g^(p^" * Sprint(j) * "-1): " * Sprint(g^(p^j-1) ne 1); end for; print "g^(p^12-1): " * Sprint(g^(p^12-1) eq 1); "g^n =", g^n; */ for j in [1..100] do u := Random(n - 1); v := Random(n - 1); w := tate(u*G, v*Gt); if w ne g^((u*v) mod n) then "Failure: u =", u, ", v =", v, ", e(u*P,v*Q) =", w, ", e(P,Q)^(uv) =", g^((u*v) mod n); quit; end if; end for; "Success!"; Output: Magma V2.11-10 Tue Dec 13 2005 14:37:08 on modular [Seed = 1005017833] ------------------------------------- g = (788575332791159297943382330888091507234373160984*i + 1058591418264462827430151667682426387205715653646)*z^5 + (1322305885575807535232020102800928943661744848278*i + 1409518745655936467893014453594650508665852052962)*z^4 + (92691491674328017873297655882559873492848083281*i + 833387513557317502960994534404369304220347785370)*z^3 + (494268977073817690135764886501032821621331118299*i + 1229571523737605236254885852230274558531837523170)*z^2 + (808544033252198825980179735588413753558824714098*i + 1078734709554906413376551726137377859976359534003)*z + 901493422496026382770929617713581877034333521597*i + 179130707045747991601837568997307929180024496359 Failure: u = 946857947927091542053549758428905646280859693128 , v = 466000635125302880497731311434509156520464968357 , e(u*P,v*Q) = (1200099635135265526944287342088491944694417283646*i + 1014718590403343121885833198299782572285854256079)*z^5 + (471917906168926683696779282454416809308080942623*i + 812644610572628849110985110390868588547305952867)*z^4 + (1324320698252798975422880297923770527123745467545*i + 1369801539108789273837312698785305468703191114557)*z^3 + (912504363855611966661376277297362740269326191111*i + 1380551513148965958602464026524016649810176191967)*z^2 + (1105893840184059270391354805924659396077920064041*i + 1288971271799831078521824062316836734081548434254)*z + 597889428879138809242066013037790879345607008490*i + 1169462561227871511533990611160523461698537007163 , e(P,Q)^(uv) = (1279762112918260549426510071012241774233201710837*i + 1038569981133317417069888654877080847608690012705)*z^5 + (1209071413845908584505709653556653498685569554252*i + 367978315180861868580674923069718724352829282487)*z^4 + (1092679439520606553629045996698544887309903354323*i + 1432439429241915293719430050078044125231284711989)*z^3 + (720945178705548260866392701230718571277060392582*i + 965355359457859959621310893985759282425870986789)*z^2 + (237204670011952913218038313811081503208054984454*i + 1073500538414130670572005779575000303603195535920)*z + 37244205619087872590547375873385141388703250234*i + 1082282633627808574389817278679504033474868288016 Total time: 1.050 seconds, Total memory usage: 3.34MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Tue Dec 13 14:36:11 2005 Input: Zz := PolynomialRing(Integers()); P := 36*z^4 + 36*z^3 + 24*z^2 + 6*z + 1; v := 0; while true do v +:= 1; t := 6*v^2 + 1; u := -v; p := Evaluate(P, u); n := p + 1 - t; if p mod 4 eq 3 and p mod 9 eq 4 and IsProbablePrime(p) and IsProbablePrime(n) then break; end if; u := v; p := Evaluate(P, u); n := p + 1 - t; if p mod 4 eq 3 and p mod 9 eq 4 and IsProbablePrime(p) and IsProbablePrime(n) then break; end if; end while; "u =", u; "p =", p; "n =", n; K := GF(p); b := K!0; repeat repeat b := b + 1; until IsSquare(b + 1); y := Root(b + 1, 2); E := EllipticCurve([K|0,b]); G := E![1,y]; until IsZero(n*G); "b =", b; "G =", G; lambda := K!2; while IsPower(lambda, 3) do lambda +:= 1; end while; "lambda =", lambda; K2 := ExtensionField; mu := i + 1; while IsSquare(mu) do mu +:= 1; end while; "mu =", mu; Output: Magma V2.11-10 Tue Dec 13 2005 14:36:10 on modular [Seed = 1055023010] ------------------------------------- u = 1 p = 103 n = 97 b = 12 G = (1 : 61 : 1) lambda = 2 mu = i + 2 Total time: 0.190 seconds, Total memory usage: 3.43MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Tue Dec 13 14:32:16 2005 Input: Zz := PolynomialRing(Integers()); P := 36*z^4 + 36*z^3 + 24*z^2 + 6*z + 1; v := 0; while true do v +:= 1; t := 6*v^2 + 1; u := -v; p := Evaluate(P, u); n := p + 1 - t; if p mod 4 eq 3 and p mod 9 eq 4 and IsProbablePrime(p) and IsProbablePrime(n) then break; end if; u := v; p := Evaluate(P, u); n := p + 1 - t; if p mod 4 eq 3 and p mod 9 eq 4 and IsProbablePrime(p) and IsProbablePrime(n) then break; end if; end while; "u =", u; "p =", p; "n =", n; K := GF(p); b := K!0; repeat repeat b := b + 1; until IsSquare(b + 1); y := Root(b + 1, 2); E := EllipticCurve([K|0,b]); G := E![1,y]; until IsZero(n*G); "b =", b; "G =", G; Output: Magma V2.11-10 Tue Dec 13 2005 14:32:15 on modular [Seed = 887641789] ------------------------------------- u = 1 p = 103 n = 97 b = 12 G = (1 : 61 : 1) Total time: 0.180 seconds, Total memory usage: 3.34MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Tue Dec 13 14:32:00 2005 Input: Zz := PolynomialRing(Integers()); P := 36*z^4 + 36*z^3 + 24*z^2 + 6*z + 1; v := 0; while true do v +:= 1; t := 6*v^2 + 1; u := -v; p := Evaluate(P, u); n := p + 1 - t; if p mod 4 eq 3 and p mod 9 eq 4 and IsProbablePrime(p) and IsProbablePrime(n) then break; end if; u := v; p := Evaluate(P, u); n := p + 1 - t; if p mod 4 eq 3 and p mod 9 eq 4 and IsProbablePrime(p) and IsProbablePrime(n) then break; end if; end while; "u =", u; "p =", p; "n =", n; K := GF(p); b := K!0; repeat repeat b := b + 1; until IsSquare(b + 1); y := Root(b + 1, 2); E := EllipticCurve([k|0,b]); G := E![1,y]; until IsZero(n*G); "b =", b; "G =", G; Output: Magma V2.11-10 Tue Dec 13 2005 14:32:00 on modular [Seed = 3390482509] ------------------------------------- u = 1 p = 103 n = 97 >> E := EllipticCurve([k|0,b]); ^ User error: Identifier 'k' has not been declared or assigned b = 0 >> "G =", G; ^ User error: Identifier 'G' has not been declared or assigned Total time: 0.190 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Tue Dec 13 14:30:09 2005 Input: Zz := PolynomialRing(Integers()); P := 36*z^4 + 36*z^3 + 24*z^2 + 6*z + 1; v := 0; while true do v +:= 1; t := 6*v^2 + 1; u := -v; p := Evaluate(P, u); n := p + 1 - t; if p mod 4 eq 3 and p mod 9 eq 4 and IsProbablePrime(p) and IsProbablePrime(n) then break; end if; u := v; p := Evaluate(P, u); n := p + 1 - t; if p mod 4 eq 3 and p mod 9 eq 4 and IsProbablePrime(p) and IsProbablePrime(n) then break; end if; end while; "u =", u; "p =", p; "n =", n; Output: Magma V2.11-10 Tue Dec 13 2005 14:30:08 on modular [Seed = 3474173368] ------------------------------------- u = 1 p = 103 n = 97 Total time: 0.190 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Tue Dec 13 14:28:50 2005 Input: Zz := PolynomialRing(Integers()); P := 36*z^4 + 36*z^3 + 24*z^2 + 6*z + 1; v := 0; while true do v +:= 1; t := 6*v^2 + 1; u := -v; p := Evaluate(P, u); n := p + 1 - t; if IsProbablePrime(p) and IsProbablePrime(n) then break; end if; u := v; p := Evaluate(P, u); n := p + 1 - t; if IsProbablePrime(p) and IsProbablePrime(n) then break; end if; end while; "u =", u; "p =", p; "n =", n; Output: Magma V2.11-10 Tue Dec 13 2005 14:28:49 on modular [Seed = 3289949954] ------------------------------------- u = -1 p = 19 n = 13 Total time: 0.190 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Tue Dec 13 14:25:25 2005 Input: Zz := PolynomialRing(Integers()); P := 36*z^4 + 36*z^3 + 24*z^2 + 6*z + 1; u := 0; while true do u +:= 1; t := 6*u^2 + 1; p := Evaluate(P, -u); if not IsProbablePrime(p) then continue; end if; n := p + 1 - t; if IsProbablePrime(n) then break; end if; p := Evaluate(P, u); if not IsProbablePrime(p) then continue; end if; n := p + 1 - t; if IsProbablePrime(n) then break; end if; end while; "u =", u; "p =", p; "n =", n; Output: Magma V2.11-10 Tue Dec 13 2005 14:25:24 on modular [Seed = 3657873865] ------------------------------------- u = 1 p = 19 n = 13 Total time: 0.190 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Tue Dec 13 14:17:13 2005 Input: p := 1461501624496790265145448589920785493717258890819; n := 1461501624496790265145447380994971188499300027613; Fp := GF(p); Fp2 := ExtensionField; lambda := 2; mu := 1 + i; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; b := Fp!3; y0 := Fp!2; // -Sqrt(1 + b); E := EllipticCurve([0, b]); G := E![1, y0]; Et := EllipticCurve([0, b/xi]); //xt := i + 2; xt := 8; /* xt := i; while not IsSquare(xt^3 + b/xi) do xt +:= 1; end while; "xt =", xt; */ yt := Sqrt(xt^3 + b/xi); Gt := Et![xt, yt]; k := 12; assert (p^k - 1) mod n eq 0; chi := (p^k - 1) div n; Ek := EllipticCurve([0, Fp12!b]); g := function(U, V, Q) if IsZero(U) or IsZero(V) or (U eq -V) or IsZero(Q) then return Fp12!1; end if; assert U[3] eq 1; assert V[3] eq 1; assert Q[3] eq 1; m := (U eq V) select 3*U[1]^2/(2*U[2]) else (V[2] - U[2])/(V[1] - U[1]); return m*(Q[1] - U[1]) + U[2] - Q[2]; end function; length := function(r) n := 0; v := 1; while v le r do n +:= 1; v +:= v; end while; return n; end function; bit := function(r, i) return (r div 2^i) mod 2; end function; miller := function(r, P, Q) f := 1; A := P; for i := length(r) - 1 to 0 by -1 do f := f^2*g(A, A, Q); A := 2*A; if bit(r, i) eq 1 then f *:= g(A, P, Q); A +:= P; end if; end for; return f; end function; tate := function(P, Qt) Q := Ek![Qt[1]*z^2, Qt[2]*z^3]; /* "P =", P; "Q' =", Qt; "Q =", Q; */ return miller(n, P, Q)^chi; end function; g := tate(G, Gt); "g =", g; /* for j in [1..11] do print "g^(p^" * Sprint(j) * "-1): " * Sprint(g^(p^j-1) ne 1); end for; print "g^(p^12-1): " * Sprint(g^(p^12-1) eq 1); "g^n =", g^n; */ for j in [1..100] do u := Random(n - 1); v := Random(n - 1); w := tate(u*G, v*Gt); if w ne g^((u*v) mod n) then "Failure: u =", u, ", v =", v, ", e(u*P,v*Q) =", w, ", e(P,Q)^(uv) =", g^((u*v) mod n); quit; end if; end for; "Success!"; Output: Magma V2.11-10 Tue Dec 13 2005 14:17:12 on modular [Seed = 4113148154] ------------------------------------- g = (672926291705630967202066259032693986482885729835*i + 402910206232327437715296922238359106511543237173)*z^5 + (1322305885575807535232020102800928943661744848278*i + 1409518745655936467893014453594650508665852052962)*z^4 + (1368810132822462247272150934038225620224410807538*i + 628114110939472762184454055516416189496911105449)*z^3 + (494268977073817690135764886501032821621331118299*i + 1229571523737605236254885852230274558531837523170)*z^2 + (652957591244591439165268854332371740158434176721*i + 382766914941883851768896863783407633740899356816)*z + 901493422496026382770929617713581877034333521597*i + 179130707045747991601837568997307929180024496359 Failure: u = 787544600137698273840913483523578412686940791799 , v = 633390288056570917236749546569673252543054620654 , e(u*P,v*Q) = (1230799145370778246341173515929335566649536643006*i + 480750625874954609025504843133359714744867086605)*z^5 + (1042350571388618446487630159957378489367482972903*i + 603997525618261102690174964818300174247773796868)*z^4 + (372555237390881172830113633133348948005361857273*i + 795790796595629198651221834435852032328843762695)*z^3 + (294507650957009594258002783265863173832662354454*i + 608683974417992602158011087154420253986790025605)*z^2 + (7609910714021721107059555931969091666598777225*i + 945198760430716673304112293252204618143489684435)*z + 147738933904891503161234155216849327398150311466*i + 11975084496371106465181782342706929746861740417 , e(P,Q)^(uv) = (436069547787440145692125043594555141196201225212*i + 104917863794017341921655052505855786976213992405)*z^5 + (1273611771612726823696618599863708218222654331128*i + 58016250495606580058214502460360819634176535723)*z^4 + (650807760957229285905208902992656562655973656443*i + 522387035783265890583694065207579752375558876825)*z^3 + (935118843989569600386157522041707999880163319815*i + 769360049790716169376779087226867932064575695394)*z^2 + (1137422255938782465848831089043436888213998499734*i + 255708389714534001533432868391992802173796862328)*z + 61611046909036180490590110705335451925853639750*i + 146857281263717838984053265699458256927236573517 Total time: 1.100 seconds, Total memory usage: 3.34MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Tue Dec 13 14:15:53 2005 Input: p := 1461501624496790265145448589920785493717258890819; n := 1461501624496790265145447380994971188499300027613; Fp := GF(p); Fp2 := ExtensionField; lambda := 2; mu := 1 + i; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; b := Fp!3; y0 := Fp!2; // -Sqrt(1 + b); E := EllipticCurve([0, b]); G := E![1, y0]; Et := EllipticCurve([0, b/xi]); //xt := i + 2; xt := 8; /* xt := i; while not IsSquare(xt^3 + b/xi) do xt +:= 1; end while; "xt =", xt; */ yt := Sqrt(xt^3 + b/xi); Gt := Et![xt, yt]; k := 12; assert (p^k - 1) mod n eq 0; chi := (p^k - 1) div n; Ek := EllipticCurve([0, Fp12!b]); g := function(U, V, Q) if IsZero(U) or IsZero(V) or (U eq -V) or IsZero(Q) then return Fp12!1; end if; x_U := U[0]; assert U[3] eq 1; assert V[3] eq 1; assert Q[3] eq 1; m := (U eq V) select 3*U[1]^2/(2*U[1]) else (V[2] - U[2])/(V[1] - U[1]); return m*(Q[1] - U[1]) + U[2] - Q[2]; end function; length := function(r) n := 0; v := 1; while v le r do n +:= 1; v +:= v; end while; return n; end function; bit := function(r, i) return (r div 2^i) mod 2; end function; miller := function(r, P, Q) f := 1; A := P; for i := length(r) - 1 to 0 by -1 do f := f^2*g(A, A, Q); A := 2*A; if bit(r, i) eq 1 then f *:= g(A, P, Q); A +:= P; end if; end for; return f; end function; tate := function(P, Qt) Q := Ek![Qt[1]*z^2, Qt[2]*z^3]; /* "P =", P; "Q' =", Qt; "Q =", Q; */ return miller(n, P, Q)^chi; end function; g := tate(G, Gt); "g =", g; /* for j in [1..11] do print "g^(p^" * Sprint(j) * "-1): " * Sprint(g^(p^j-1) ne 1); end for; print "g^(p^12-1): " * Sprint(g^(p^12-1) eq 1); "g^n =", g^n; */ for j in [1..100] do u := Random(n - 1); v := Random(n - 1); w := tate(u*G, v*Gt); if w ne g^((u*v) mod n) then "Failure: u =", u, ", v =", v, ", e(u*P,v*Q) =", w, ", e(P,Q)^(uv) =", g^((u*v) mod n); quit; end if; end for; "Success!"; Output: Magma V2.11-10 Tue Dec 13 2005 14:15:53 on modular [Seed = 2370949971] ------------------------------------- tate( P: (1 : 2 : 1), Qt: (8 : 645238442624673913635245604741558906439788126004*i + 11... ) miller( r: 1461501624496790265145447380994971188499300027613, P: (1 : 2 : 1), Q: (8*z^2 : (645238442624673913635245604741558906439788126004*i... ) g( U: (1 : 2 : 1), V: (1 : 2 : 1), Q: (8*z^2 : (645238442624673913635245604741558906439788126004*i... ) >> x_U := U[0]; ^ Runtime error in '[]': Argument 2 (0) should be in the range [1 .. 3] g = function(U, V, Q) ... end function tate( P: (1168716806992048472600575426428280493482442262392 : 9145640..., Qt: (1398335284650143722212829649205089955966390414241*i + 51796... ) miller( r: 1461501624496790265145447380994971188499300027613, P: (1168716806992048472600575426428280493482442262392 : 9145640..., Q: ((1398335284650143722212829649205089955966390414241*i + 5179... ) g( U: (1168716806992048472600575426428280493482442262392 : 9145640..., V: (1168716806992048472600575426428280493482442262392 : 9145640..., Q: ((1398335284650143722212829649205089955966390414241*i + 5179... ) >> x_U := U[0]; ^ Runtime error in '[]': Argument 2 (0) should be in the range [1 .. 3] Success! Total time: 0.230 seconds, Total memory usage: 3.34MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Tue Dec 13 14:14:47 2005 Input: p := 1461501624496790265145448589920785493717258890819; n := 1461501624496790265145447380994971188499300027613; Fp := GF(p); Fp2 := ExtensionField; lambda := 2; mu := 1 + i; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; b := Fp!3; y0 := Fp!2; // -Sqrt(1 + b); E := EllipticCurve([0, b]); G := E![1, y0]; Et := EllipticCurve([0, b/xi]); //xt := i + 2; xt := 8; /* xt := i; while not IsSquare(xt^3 + b/xi) do xt +:= 1; end while; "xt =", xt; */ yt := Sqrt(xt^3 + b/xi); Gt := Et![xt, yt]; k := 12; assert (p^k - 1) mod n eq 0; chi := (p^k - 1) div n; Ek := EllipticCurve([0, Fp12!b]); g := function(U, V, Q) if IsZero(U) or IsZero(V) or (U eq -V) or IsZero(Q) then return Fp12!1; end if; assert U[3] eq 1; assert V[3] eq 1; assert Q[3] eq 1; m := (U eq V) select 3*U[1]^2/(2*U[1]) else (V[2] - U[2])/(V[1] - U[1]); return m*(Q[1] - U[1]) + U[2] - Q[2]; end function; length := function(r) n := 0; v := 1; while v le r do n +:= 1; v +:= v; end while; return n; end function; bit := function(r, i) return (r div 2^i) mod 2; end function; miller := function(r, P, Q) f := 1; A := P; for i := length(r) - 1 to 0 by -1 do f := f^2*g(A, A, Q); A := 2*A; if bit(r, i) eq 1 then f *:= g(A, P, Q); A +:= P; end if; end for; return f; end function; tate := function(P, Qt) Q := Ek![Qt[1]*z^2, Qt[2]*z^3]; /* "P =", P; "Q' =", Qt; "Q =", Q; */ return miller(n, P, Q)^chi; end function; g := tate(G, Gt); "g =", g; /* for j in [1..11] do print "g^(p^" * Sprint(j) * "-1): " * Sprint(g^(p^j-1) ne 1); end for; print "g^(p^12-1): " * Sprint(g^(p^12-1) eq 1); "g^n =", g^n; */ for j in [1..100] do u := Random(n - 1); v := Random(n - 1); w := tate(u*G, v*Gt); if w ne g^((u*v) mod n) then "Failure: u =", u, ", v =", v, ", e(u*P,v*Q) =", w, ", e(P,Q)^(uv) =", g^((u*v) mod n); quit; end if; end for; "Success!"; Output: Magma V2.11-10 Tue Dec 13 2005 14:14:46 on modular [Seed = 2187254803] ------------------------------------- g = (1294412285019402306363363178405640025835588490573*i + 110083851862711115164294604712404064123786212316)*z^5 + (1461000921493363626807176655087143303615070519998*i + 179329319464877725831768652574235563708218294754)*z^4 + (1380159870452351478229999574688122314163279894835*i + 608853380181358438321396844362562256927503461575)*z^3 + (1096834514295262897057490471840451625632598605569*i + 556024354574832251790797618637935038587303588339)*z^2 + (1312769194087813280589806690561007885216498370207*i + 585387117445278704678873601298024025030517986150)*z + 737166616423533515609953322860296363171325419696*i + 327478258348140990171576024019950781712369617012 Failure: u = 1125474438691666154889619531753276137514054616092 , v = 74355561060828237152603481423932850628604339121 , e(u*P,v*Q) = (429751061675843899340369203815082176545408527718*i + 909427573279119657754805669761900114071342873652)*z^5 + (1247734288733857748021692265553075464695937579208*i + 743816016218828928608456547179207548076982979307)*z^4 + (487557256047992095318968005263397437474936254189*i + 37857860423732375651447446890944269006478056096)*z^3 + (1163989855974890736075862642655486087920170035473*i + 203749602481885254344251451474004759409452749236)*z^2 + (1030453876983867035177297133699795873715007249920*i + 660735836157490264065099937197471719042581607777)*z + 81803167227496355034247834271195469737156426272*i + 956207628639205541360074132469925387897212870999 , e(P,Q)^(uv) = (456387365657668154214383519085672897734254356213*i + 901464030134534466212711635931829911205601723280)*z^5 + (1457502911931788790996440618804285616966470935812*i + 167907066249828659721266469744918999176634507737)*z^4 + (1117081125135662063752403398732529743718000625159*i + 1166482808557132720766303763649223676515724544694)*z^3 + (45409092783112768705197353118716020133117497726*i + 1140195580893668606687433084773043246385384621649)*z^2 + (797400106022357958571873126411052707015877860255*i + 1144849651621705114749107513915197993111731626826)*z + 398139448757274963177977906845541386205679308934*i + 1011464188230029570370071800294314999300195638774 Total time: 1.080 seconds, Total memory usage: 3.34MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Tue Dec 13 14:13:45 2005 Input: p := 1461501624496790265145448589920785493717258890819; n := 1461501624496790265145447380994971188499300027613; Fp := GF(p); Fp2 := ExtensionField; lambda := 2; mu := 1 + i; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; b := Fp!3; y0 := Fp!2; // -Sqrt(1 + b); E := EllipticCurve([0, b]); G := E![1, y0]; Et := EllipticCurve([0, b/xi]); //xt := i + 2; xt := 8; /* xt := i; while not IsSquare(xt^3 + b/xi) do xt +:= 1; end while; "xt =", xt; */ yt := Sqrt(xt^3 + b/xi); Gt := Et![xt, yt]; k := 12; assert (p^k - 1) mod n eq 0; chi := (p^k - 1) div n; Ek := EllipticCurve([0, Fp12!b]); g := function(U, V, Q) if IsZero(U) or IsZero(V) or (U eq -V) or IsZero(Q) then return Fp12!1; end if; assert U[3] eq 1; assert V[3] eq 1; assert Q[3] eq 1; m := (U eq V) select 3*U[1]^2/(2*U[1]) else (V[2] - U[2])/(V[1] - U[1]); return m*(Q[1] - U[1]) + U[1] - Q[2]; end function; length := function(r) n := 0; v := 1; while v le r do n +:= 1; v +:= v; end while; return n; end function; bit := function(r, i) return (r div 2^i) mod 2; end function; miller := function(r, P, Q) f := 1; A := P; for i := length(r) - 1 to 0 by -1 do f := f^2*g(A, A, Q); A := 2*A; if bit(r, i) eq 1 then f *:= g(A, P, Q); A +:= P; end if; end for; return f; end function; tate := function(P, Qt) Q := Ek![Qt[1]*z^2, Qt[2]*z^3]; /* "P =", P; "Q' =", Qt; "Q =", Q; */ return miller(n, P, Q)^chi; end function; g := tate(G, Gt); "g =", g; /* for j in [1..11] do print "g^(p^" * Sprint(j) * "-1): " * Sprint(g^(p^j-1) ne 1); end for; print "g^(p^12-1): " * Sprint(g^(p^12-1) eq 1); "g^n =", g^n; */ for j in [1..100] do u := Random(n - 1); v := Random(n - 1); w := tate(u*G, v*Gt); if w ne g^((u*v) mod n) then "Failure: u =", u, ", v =", v, ", e(u*P,v*Q) =", w, ", e(P,Q)^(uv) =", g^((u*v) mod n); quit; end if; end for; "Success!"; Output: Magma V2.11-10 Tue Dec 13 2005 14:13:44 on modular [Seed = 2270943704] ------------------------------------- g = (1414620835193373657646613880233559577407734609545*i + 352489175314961464607671418528001817189092256541)*z^5 + (904802261065103105936668720505067133656667597074*i + 1343248922269291340494538393490943353845819222256)*z^4 + (1025700017505207783921241027028089077317301724291*i + 1267567142236668396047190065512596226428512971416)*z^3 + (283587609846347412879109527640610607003328134432*i + 250487955904041646310509260856358041239663937723)*z^2 + (110054808378578108265445578093874268052578876998*i + 202041545854984388890408940644851972139794936772)*z + 257093848091331658607960702914357017287219708568*i + 652950073647454119068347296682657355787566777234 Failure: u = 658677412291089462095414946131774452312281593714 , v = 1216437558465192851244742711765770083951252389636 , e(u*P,v*Q) = (1399072020171236105190679320673870340346132836857*i + 557339706670292369068837859617582039559121726559)*z^5 + (87632175260928283896022294886624356298791691273*i + 353631789820014829475382783849498669504077254473)*z^4 + (1167181401062038608632643315708112439190409866068*i + 1201319255475829211069929657476285565774671872992)*z^3 + (1314520118761252810746572930401672772734452109507*i + 692190026558869491946574975355715885291753444864)*z^2 + (30302970564993022126258376903270909561081118427*i + 981711504087328555212568427163427828202637031781)*z + 650156321266501086248612693725874390933511156097*i + 86504195952951481401572117229146789328835422166 , e(P,Q)^(uv) = (213093635849167276598922493416793532239780363365*i + 176954146803611932474107371546291585496100314567)*z^5 + (423996100507151792116824541957513700044018239694*i + 83422655284377911844489375762414059048084741713)*z^4 + (1422886585162015559920092658102846713850101330779*i + 458574078442061740914292994408151093044898187067)*z^3 + (273126339780048501849982943960562129337069997055*i + 1303685246506966196047520334746858504911291400698)*z^2 + (1304770219556858465763105537968022378874250386356*i + 416455068376200642044392053732047037130013150572)*z + 398741370625816481193912713678168496900587107922*i + 581927151108935829460509086487928349289055788193 Total time: 1.060 seconds, Total memory usage: 3.34MB '65.191.' ************** MAGMA ***************** Host 65.191.73.113 (65.191.73.113) Time: Tue Dec 13 13:05:57 2005 Input: K := FiniteField(2); > C := LinearCode; S, f := StandardForm(C); Submatrix(S, 1, 22, 21, 35); Output: Magma V2.11-10 Tue Dec 13 2005 13:05:57 on modular [Seed = 804479389] ------------------------------------- >> Submatrix(S, 1, 22, 21, 35); ^ Runtime error in 'Submatrix': Bad argument types Argument types given: CodeLinFld, RngIntElt, RngIntElt, RngIntElt, RngIntElt Total time: 0.200 seconds, Total memory usage: 3.34MB '65.191.' ************** MAGMA ***************** Host 65.191.73.113 (65.191.73.113) Time: Tue Dec 13 13:03:43 2005 Input: K := FiniteField(2); > C := LinearCode; S, f := StandardForm(C); ExtractBlock(S, 1, 22, 21, 35); Output: Magma V2.11-10 Tue Dec 13 2005 13:03:42 on modular [Seed = 653945327] ------------------------------------- >> ExtractBlock(S, 1, 22, 21, 35); ^ Runtime error in 'ExtractBlock': Bad argument types Argument types given: CodeLinFld, RngIntElt, RngIntElt, RngIntElt, RngIntElt Total time: 0.190 seconds, Total memory usage: 3.34MB '65.191.' ************** MAGMA ***************** Host 65.191.73.113 (65.191.73.113) Time: Tue Dec 13 13:01:22 2005 Input: K := FiniteField(2); > C := LinearCode; S, f := StandardForm(C); S; Output: Magma V2.11-10 Tue Dec 13 2005 13:01:22 on modular [Seed = 1021342585] ------------------------------------- [56, 21] Linear Code over GF(2) Generator matrix: [1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 0 0 0 0 1 1 1 0 1 0 1 0 0 1 1 0 0 1 1 0 0 0 1 1 0 1 0 0] [0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 1 0 1 0 0 1 0 0 0 1 0 0 1 0 1 1 1 0 0 0 1 1 0 0 1 1 0 1 0] [0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 1 0 0 0 1 1 0 0 0 0 0 0 0 1 1 1 1 1 0 0 0 0 0 0 0 1 1 1 1 0 0] [0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 1 1 1 1 0 1 1 0 0 1 1 0 1] [0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 1 0 1 1 0 0 1 1 1 1 1 1 0 1 1 1 0 0 1 1 0 1 1 1 1 0 0 1 1 0] [0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 0 0 0 0 1 0 1 1 1 0 0 0 1 1 1 0 0 0 1 0 0 1 0 0 1 0 1 0 0 0 1] [0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 1 1 1 0 0 1 0 1 0 1 1 0 1 0 0 1 1 1 1 1 0 1 1 0 0 0 1 0 1 0] [0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 1 1 0 0 1 1 0 0 0 1 0 0 1 0 1 1 0 0 1 1 1 0 1 1 0 1 0 0 1 1] [0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 1 1 1 0 1 0 1 1 1 0 0 1 1 1 0 0 0 1 1 1 1 1 1 1 1 0 1 1 0 0 0] [0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 0 1 1 1 1 0 0 0 0 1 1 0 1 1 0 1 0 1 1 1 0 1 0 0 1 0 1 1 0 0] [0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 1 1 1 0 0 1 0 1 1 1 1 1 0 0 0 0 1 1 1 1 0 1 1 0 0 0 0 0 1 0 1 1 0] [0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 1 0 1 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 0 1 0 1 1 1 0 1 0 0 1] [0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 1 1 1 1 1 0 0 1 1 1 0 1 1 1 0 0 1 1 1 1 1 0 1 0 0 1 1 1 0 0 0 1 0 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 1 0 0 0 1 0 0 0 1 0 1 0 1 0 1 1 0 0 1 1 1 0 1 0 0 0 1 1 0 0 0 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 1 1 0 0 0 0 1 0 0 1 1 1 0 1 1 1 0 1 0 1 0 0 1 1 1 0 1 1 0 1 0 0 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0 1 1 0 0 1 1 0 0 0 1 1 0 1 0 0 1 1 1 0 0 0 1 0 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0 1 1 1 0 1 1 0 0 0 1 1 0 1 0 0 1 1 1 0 0 0 1 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 1 0 1 1 1 0 1 0 0 0 1 0 1 1 0 0 1 0 1 1 0 0 1 1 0 0 0 1 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 1 0 1 0 0 1 1 1 0 0 1 1 1 0 1 0 1 1 1 0 1 0 1 0 1 0 0 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 1 0 1 1 0 1 0 0 1 1 0 1 1 0 0 0 0 1 1 0 0 0 1 0 0 0 1 0 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 1 1 1 1 0 1 0 0 0 1 0 1 1 0 0 1 0 1 1 0 0 0 1 0 0 0 1 0 1] Total time: 0.190 seconds, Total memory usage: 3.34MB '65.191.' ************** MAGMA ***************** Host 65.191.73.113 (65.191.73.113) Time: Tue Dec 13 13:00:59 2005 Input: K := FiniteField(2); > C := LinearCode; S, f := StandardForm(C); Output: Magma V2.11-10 Tue Dec 13 2005 13:00:59 on modular [Seed = 837647965] ------------------------------------- Total time: 0.190 seconds, Total memory usage: 3.34MB '65.191.' ************** MAGMA ***************** Host 65.191.73.113 (65.191.73.113) Time: Tue Dec 13 12:41:53 2005 Input: M:=[1 1 1 1 1 1 1 0 0 0 0 1 1 1 0 1 0 1 0 0 1 1 0 0 1 1 0 0 0 1 1 0 1 0 0] [1 0 0 0 1 0 1 0 1 0 0 1 0 0 0 1 0 0 1 0 1 1 1 0 0 0 1 1 0 0 1 1 0 1 0] [1 1 1 0 1 0 0 0 1 1 0 0 0 0 0 0 0 1 1 1 1 1 0 0 0 0 0 0 0 1 1 1 1 0 0] [1 1 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 1 1 1 1 0 1 1 0 0 1 1 0 1] [1 0 0 1 1 1 0 1 1 0 0 1 1 1 1 1 1 0 1 1 1 0 0 1 1 0 1 1 1 1 0 0 1 1 0] [1 0 1 1 0 0 0 0 1 0 1 1 1 0 0 0 1 1 1 0 0 0 1 0 0 1 0 0 1 0 1 0 0 0 1] [1 0 1 0 0 1 1 1 0 0 1 0 1 0 1 1 0 1 0 0 1 1 1 1 1 0 1 1 0 0 0 1 0 1 0] [1 1 1 0 0 1 1 0 0 1 1 0 0 0 1 0 0 1 0 1 1 0 0 1 1 1 0 1 1 0 1 0 0 1 1] [1 1 0 1 1 1 1 0 1 0 1 1 1 0 0 1 1 1 0 0 0 1 1 1 1 1 1 1 1 0 1 1 0 0 0] [1 0 0 1 1 0 1 1 1 1 0 0 0 0 1 1 0 1 1 0 1 0 1 1 1 0 1 0 0 1 0 1 1 0 0] [1 0 1 1 1 0 0 1 0 1 1 1 1 1 0 0 0 0 1 1 1 1 0 1 1 0 0 0 0 0 1 0 1 1 0] [1 0 1 0 0 0 1 0 1 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 0 1 0 1 1 1 0 1 0 0 1] [1 1 1 1 1 1 0 0 1 1 1 0 1 1 1 0 0 1 1 1 1 1 0 1 0 0 1 1 1 0 0 0 1 0 1] [0 1 0 1 0 0 0 1 0 0 0 1 0 1 0 1 0 1 1 0 0 1 1 1 0 1 0 0 0 1 1 0 0 0 1] [0 1 1 1 0 0 0 0 1 0 0 1 1 1 0 1 1 1 0 1 0 1 0 0 1 1 1 0 1 1 0 1 0 0 1] [0 1 1 0 0 0 0 0 0 1 0 1 1 0 0 1 1 0 0 0 1 1 0 1 0 0 1 1 1 0 0 0 1 0 1] [0 0 1 1 0 0 0 0 0 0 1 0 1 1 1 0 1 1 0 0 0 1 1 0 1 0 0 1 1 1 0 0 0 1 0] [0 0 0 0 1 0 1 0 1 1 1 0 1 0 0 0 1 0 1 1 0 0 1 0 1 1 0 0 1 1 0 0 0 1 0] [0 0 0 0 1 1 1 0 1 0 0 1 1 1 0 0 1 1 1 0 1 0 1 1 1 0 1 0 1 0 1 0 0 1 1] [0 0 0 0 1 1 0 1 1 0 1 0 0 1 1 0 1 1 0 0 0 0 1 1 0 0 0 1 0 0 0 1 0 1 1] [0 0 0 0 0 1 1 1 1 1 0 1 0 0 0 1 0 1 1 0 0 1 0 1 1 0 0 0 1 0 0 0 1 0 1]; Transpose(M); Output: Magma V2.11-10 Tue Dec 13 2005 12:41:53 on modular [Seed = 3624180745] ------------------------------------- >> M:=[1 1 1 1 1 1 1 0 0 0 0 1 1 1 0 1 0 1 0 0 1 1 0 0 1 1 0 0 0 1 1 0 1 0 0] ^ User error: bad syntax >> [1 0 0 0 1 0 1 0 1 0 0 1 0 0 0 1 0 0 1 0 1 1 1 0 0 0 1 1 0 0 1 1 0 1 0] ^ User error: bad syntax >> [1 1 1 0 1 0 0 0 1 1 0 0 0 0 0 0 0 1 1 1 1 1 0 0 0 0 0 0 0 1 1 1 1 0 0] ^ User error: bad syntax >> [1 1 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 1 1 1 1 0 1 1 0 0 1 1 0 1] ^ User error: bad syntax >> [1 0 0 1 1 1 0 1 1 0 0 1 1 1 1 1 1 0 1 1 1 0 0 1 1 0 1 1 1 1 0 0 1 1 0] ^ User error: bad syntax >> [1 0 1 1 0 0 0 0 1 0 1 1 1 0 0 0 1 1 1 0 0 0 1 0 0 1 0 0 1 0 1 0 0 0 1] ^ User error: bad syntax >> [1 0 1 0 0 1 1 1 0 0 1 0 1 0 1 1 0 1 0 0 1 1 1 1 1 0 1 1 0 0 0 1 0 1 0] ^ User error: bad syntax >> [1 1 1 0 0 1 1 0 0 1 1 0 0 0 1 0 0 1 0 1 1 0 0 1 1 1 0 1 1 0 1 0 0 1 1] ^ User error: bad syntax >> [1 1 0 1 1 1 1 0 1 0 1 1 1 0 0 1 1 1 0 0 0 1 1 1 1 1 1 1 1 0 1 1 0 0 0] ^ User error: bad syntax >> [1 0 0 1 1 0 1 1 1 1 0 0 0 0 1 1 0 1 1 0 1 0 1 1 1 0 1 0 0 1 0 1 1 0 0] ^ User error: bad syntax >> [1 0 1 1 1 0 0 1 0 1 1 1 1 1 0 0 0 0 1 1 1 1 0 1 1 0 0 0 0 0 1 0 1 1 0] ^ User error: bad syntax >> [1 0 1 0 0 0 1 0 1 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 0 1 0 1 1 1 0 1 0 0 1] ^ User error: bad syntax >> [1 1 1 1 1 1 0 0 1 1 1 0 1 1 1 0 0 1 1 1 1 1 0 1 0 0 1 1 1 0 0 0 1 0 1] ^ User error: bad syntax >> [0 1 0 1 0 0 0 1 0 0 0 1 0 1 0 1 0 1 1 0 0 1 1 1 0 1 0 0 0 1 1 0 0 0 1] ^ User error: bad syntax >> [0 1 1 1 0 0 0 0 1 0 0 1 1 1 0 1 1 1 0 1 0 1 0 0 1 1 1 0 1 1 0 1 0 0 1] ^ User error: bad syntax >> [0 1 1 0 0 0 0 0 0 1 0 1 1 0 0 1 1 0 0 0 1 1 0 1 0 0 1 1 1 0 0 0 1 0 1] ^ User error: bad syntax >> [0 0 1 1 0 0 0 0 0 0 1 0 1 1 1 0 1 1 0 0 0 1 1 0 1 0 0 1 1 1 0 0 0 1 0] ^ User error: bad syntax >> [0 0 0 0 1 0 1 0 1 1 1 0 1 0 0 0 1 0 1 1 0 0 1 0 1 1 0 0 1 1 0 0 0 1 0] ^ User error: bad syntax >> [0 0 0 0 1 1 1 0 1 0 0 1 1 1 0 0 1 1 1 0 1 0 1 1 1 0 1 0 1 0 1 0 0 1 1] ^ User error: bad syntax >> [0 0 0 0 1 1 0 1 1 0 1 0 0 1 1 0 1 1 0 0 0 0 1 1 0 0 0 1 0 0 0 1 0 1 1] ^ User error: bad syntax >> [0 0 0 0 0 1 1 1 1 1 0 1 0 0 0 1 0 1 1 0 0 1 0 1 1 0 0 0 1 0 0 0 1 0 1]; ^ User error: bad syntax >> Transpose(M);; ^ User error: Identifier 'M' has not been declared or assigned Total time: 0.190 seconds, Total memory usage: 3.24MB '146.6.1' ************** MAGMA ***************** Host 146.6.139.217 (146.6.139.217) Time: Tue Dec 13 12:10:52 2005 Input: count:=0; m:=5; q:=2^m; F1:=GF(q); F2:=GF(q^2); Trace0:={@0@}; for i:=1 to q-1 do dd:=F1.1^i; if Trace(dd) eq 0 then Trace0:=Trace0 join {@dd@}; end if; end for; d:=0; for i:=1 to q-1 do a:=F1.1^i; for j:=1 to #Trace0 do b:=Trace0[j]; P:=PolynomialRing(F1); C:=HyperellipticCurve(a*x^5+b*x^3+x,x); N1:=#C; //over Fq R:=PolynomialRing(F2); CC:=HyperellipticCurve(a*z^5+b*z^3+z,z); N2:=#CC; //over Fq^2; a1:=N1-q-1; a2:=(N2-1-q^2+a1^2)/2; if not IsSquare(a1^2-4*a2+8*q) then if IsDivisibleBy(a2, 2^Ceiling(m/2)) then delta:=(a2+2*q)^2-4*q*a1^2; V:=0; if delta ne 0 then V:=Valuation(delta,2); end if; B:=delta/(2^V); if IsOdd(V) or (B-1 mod 8 ne 0) then // count:=count+1; // print C,"(a1,a2)= (", a1 ,",", a2,")"; //print "(",a1, a2,")"; end if; end if; // end if; J:=q^2+a1*q+a1+a2+1; Rat:=#RationalPoints(C); Jac:=Jacobian(C); R:=RingOfIntegers(); if R!J mod 8 eq 0 then for ii:=1 to Floor(J) do DP:=Points(Jac)[ii]; if HasOrder(DP,8) then for jj:=1 to Floor(J) do DQ:=Points(Jac)[jj]; if HasOrder(DQ,8) then WeilPairing(DP,DQ,8); end if; end for; //print DD, "has order 8, a=", a," b= ", b; end if; end for; count; end if; count:=0; end if; end for; end for; print "done"; Output: ** WARNING: Computation time exceeded 20 seconds, so computation was terminated after 20 seconds. ** Magma V2.11-10 Tue Dec 13 2005 12:10:32 on modular [Seed = 2893082388] ------------------------------------- 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 Errors: /bin/sh: line 1: 29596 Alarm clock nice -n 19 /usr/local/bin/magma '146.6.1' ************** MAGMA ***************** Host 146.6.139.217 (146.6.139.217) Time: Tue Dec 13 12:08:28 2005 Input: count:=0; m:=5; q:=2^m; F1:=GF(q); F2:=GF(q^2); Trace0:={@0@}; for i:=1 to q-1 do dd:=F1.1^i; if Trace(dd) eq 0 then Trace0:=Trace0 join {@dd@}; end if; end for; d:=0; for i:=1 to q-1 do a:=F1.1^i; for j:=1 to #Trace0 do b:=Trace0[j]; P:=PolynomialRing(F1); C:=HyperellipticCurve(a*x^5+b*x^3+x,x); N1:=#C; //over Fq R:=PolynomialRing(F2); CC:=HyperellipticCurve(a*z^5+b*z^3+z,z); N2:=#CC; //over Fq^2; a1:=N1-q-1; a2:=(N2-1-q^2+a1^2)/2; if not IsSquare(a1^2-4*a2+8*q) then if IsDivisibleBy(a2, 2^Ceiling(m/2)) then delta:=(a2+2*q)^2-4*q*a1^2; V:=0; if delta ne 0 then V:=Valuation(delta,2); end if; B:=delta/(2^V); if IsOdd(V) or (B-1 mod 8 ne 0) then // count:=count+1; // print C,"(a1,a2)= (", a1 ,",", a2,")"; //print "(",a1, a2,")"; end if; end if; // end if; J:=q^2+a1*q+a1+a2+1; Rat:=#RationalPoints(C); Jac:=Jacobian(C); R:=RingOfIntegers(); if R!J mod 8 eq 0 then for ii:=1 to Floor(J) do DP:=Points(Jac)[ii]; if HasOrder(DP,8) then for jj:=1 to Floor(J) do DQ:=Points(Jac)[jj]; if HasOrder(DQ,8) then WeilPairing(DP,DQ,3); end if; end for; //print DD, "has order 8, a=", a," b= ", b; end if; end for; count; end if; count:=0; end if; end for; end for; print "done"; Output: Magma V2.11-10 Tue Dec 13 2005 12:08:27 on modular [Seed = 3127310721] ------------------------------------- >> WeilPairing(DP,DQ,3); ^ Runtime error in 'WeilPairing': divisors must be m-torsion elements done Total time: 0.780 seconds, Total memory usage: 3.72MB '146.6.1' ************** MAGMA ***************** Host 146.6.139.217 (146.6.139.217) Time: Tue Dec 13 12:08:14 2005 Input: m:=5; q:=2^m; F1:=GF(q); F2:=GF(q^2); Trace0:={@0@}; for i:=1 to q-1 do dd:=F1.1^i; if Trace(dd) eq 0 then Trace0:=Trace0 join {@dd@}; end if; end for; d:=0; for i:=1 to q-1 do a:=F1.1^i; for j:=1 to #Trace0 do b:=Trace0[j]; P:=PolynomialRing(F1); C:=HyperellipticCurve(a*x^5+b*x^3+x,x); N1:=#C; //over Fq R:=PolynomialRing(F2); CC:=HyperellipticCurve(a*z^5+b*z^3+z,z); N2:=#CC; //over Fq^2; a1:=N1-q-1; a2:=(N2-1-q^2+a1^2)/2; if not IsSquare(a1^2-4*a2+8*q) then if IsDivisibleBy(a2, 2^Ceiling(m/2)) then delta:=(a2+2*q)^2-4*q*a1^2; V:=0; if delta ne 0 then V:=Valuation(delta,2); end if; B:=delta/(2^V); if IsOdd(V) or (B-1 mod 8 ne 0) then // count:=count+1; // print C,"(a1,a2)= (", a1 ,",", a2,")"; //print "(",a1, a2,")"; end if; end if; // end if; J:=q^2+a1*q+a1+a2+1; Rat:=#RationalPoints(C); Jac:=Jacobian(C); R:=RingOfIntegers(); if R!J mod 8 eq 0 then for ii:=1 to Floor(J) do DP:=Points(Jac)[ii]; if HasOrder(DP,8) then for jj:=1 to Floor(J) do DQ:=Points(Jac)[jj]; if HasOrder(DQ,8) then WeilPairing(DP,DQ,3); end if; end for; //print DD, "has order 8, a=", a," b= ", b; end if; end for; count; end if; count:=0; end if; end for; end for; print "done"; Output: Magma V2.11-10 Tue Dec 13 2005 12:08:14 on modular [Seed = 3043494267] ------------------------------------- >> count; ^ User error: Identifier 'count' has not been declared or assigned done Total time: 0.190 seconds, Total memory usage: 3.34MB '146.6.1' ************** MAGMA ***************** Host 146.6.139.217 (146.6.139.217) Time: Tue Dec 13 12:07:59 2005 Input: m:=5; q:=2^m; F1:=GF(q); F2:=GF(q^2); Trace0:={@0@}; for i:=1 to q-1 do dd:=F1.1^i; if Trace(dd) eq 0 then Trace0:=Trace0 join {@dd@}; end if; end for; d:=0; for i:=1 to q-1 do a:=F1.1^i; for j:=1 to #Trace0 do b:=Trace0[j]; P:=PolynomialRing(F1); C:=HyperellipticCurve(a*x^5+b*x^3+x,x); N1:=#C; //over Fq R:=PolynomialRing(F2); CC:=HyperellipticCurve(a*z^5+b*z^3+z,z); N2:=#CC; //over Fq^2; a1:=N1-q-1; a2:=(N2-1-q^2+a1^2)/2; if not IsSquare(a1^2-4*a2+8*q) then if IsDivisibleBy(a2, 2^Ceiling(m/2)) then delta:=(a2+2*q)^2-4*q*a1^2; V:=0; if delta ne 0 then V:=Valuation(delta,2); end if; B:=delta/(2^V); if IsOdd(V) or (B-1 mod 8 ne 0) then // count:=count+1; // print C,"(a1,a2)= (", a1 ,",", a2,")"; //print "(",a1, a2,")"; end if; end if; // end if; J:=q^2+a1*q+a1+a2+1; Rat:=#RationalPoints(C); Jac:=Jacobian(C); R:=RingOfIntegers(); if R!J mod 8 eq 0 then for ii:=1 to Floor(J) do DP:=Points(Jac)[ii]; if HasOrder(DP,8) then for jj:=1 to Floor(J) do DQ:=Points(Jac)[jj]; if HasOrder(DQ,8) then WeilPairing(DP,DQ,3); end if end for //print DD, "has order 8, a=", a," b= ", b; end if; end for; count; end if; count:=0; end if; end for; end for; print "done"; Output: Magma V2.11-10 Tue Dec 13 2005 12:07:58 on modular [Seed = 1318318736] ------------------------------------- >> end for ^ User error: bad syntax >> end if; ^ User error: bad syntax >> end for; ^ User error: bad syntax >> count; ^ User error: Identifier 'count' has not been declared or assigned >> end if; ^ User error: bad syntax >> end if; ^ User error: bad syntax >> end for; ^ User error: bad syntax >> end for; ^ User error: bad syntax done Total time: 0.180 seconds, Total memory usage: 3.34MB '65.191.' ************** MAGMA ***************** Host 65.191.73.113 (65.191.73.113) Time: Tue Dec 13 11:58:23 2005 Input: K := FiniteField(2); > C := LinearCode; S, f := StandardForm(C); S; Output: Magma V2.11-10 Tue Dec 13 2005 11:58:22 on modular [Seed = 449291396] ------------------------------------- [56, 21] Linear Code over GF(2) Generator matrix: [1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 0 0 0 0 1 1 1 0 1 0 1 0 0 1 1 0 0 1 1 0 0 0 1 1 0 1 0 0] [0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 1 0 1 0 0 1 0 0 0 1 0 0 1 0 1 1 1 0 0 0 1 1 0 0 1 1 0 1 0] [0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 1 0 0 0 1 1 0 0 0 0 0 0 0 1 1 1 1 1 0 0 0 0 0 0 0 1 1 1 1 0 0] [0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 1 1 1 1 0 1 1 0 0 1 1 0 1] [0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 1 0 1 1 0 0 1 1 1 1 1 1 0 1 1 1 0 0 1 1 0 1 1 1 1 0 0 1 1 0] [0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 0 0 0 0 1 0 1 1 1 0 0 0 1 1 1 0 0 0 1 0 0 1 0 0 1 0 1 0 0 0 1] [0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 1 1 1 0 0 1 0 1 0 1 1 0 1 0 0 1 1 1 1 1 0 1 1 0 0 0 1 0 1 0] [0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 1 1 0 0 1 1 0 0 0 1 0 0 1 0 1 1 0 0 1 1 1 0 1 1 0 1 0 0 1 1] [0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 1 1 1 0 1 0 1 1 1 0 0 1 1 1 0 0 0 1 1 1 1 1 1 1 1 0 1 1 0 0 0] [0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 0 1 1 1 1 0 0 0 0 1 1 0 1 1 0 1 0 1 1 1 0 1 0 0 1 0 1 1 0 0] [0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 1 1 1 0 0 1 0 1 1 1 1 1 0 0 0 0 1 1 1 1 0 1 1 0 0 0 0 0 1 0 1 1 0] [0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 1 0 1 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 0 1 0 1 1 1 0 1 0 0 1] [0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 1 1 1 1 1 0 0 1 1 1 0 1 1 1 0 0 1 1 1 1 1 0 1 0 0 1 1 1 0 0 0 1 0 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 1 0 0 0 1 0 0 0 1 0 1 0 1 0 1 1 0 0 1 1 1 0 1 0 0 0 1 1 0 0 0 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 1 1 0 0 0 0 1 0 0 1 1 1 0 1 1 1 0 1 0 1 0 0 1 1 1 0 1 1 0 1 0 0 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0 1 1 0 0 1 1 0 0 0 1 1 0 1 0 0 1 1 1 0 0 0 1 0 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0 1 1 1 0 1 1 0 0 0 1 1 0 1 0 0 1 1 1 0 0 0 1 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 1 0 1 1 1 0 1 0 0 0 1 0 1 1 0 0 1 0 1 1 0 0 1 1 0 0 0 1 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 1 0 1 0 0 1 1 1 0 0 1 1 1 0 1 0 1 1 1 0 1 0 1 0 1 0 0 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 1 0 1 1 0 1 0 0 1 1 0 1 1 0 0 0 0 1 1 0 0 0 1 0 0 0 1 0 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 1 1 1 1 0 1 0 0 0 1 0 1 1 0 0 1 0 1 1 0 0 0 1 0 0 0 1 0 1] Total time: 0.190 seconds, Total memory usage: 3.34MB '84.56.2' ************** MAGMA ***************** Host 84.56.209.34 (84.56.209.34) Time: Tue Dec 13 11:39:59 2005 Input: R:=PolynomialRing(GF(2)); f:=x^155+x^62+1; K:=ext; I:=ideal; Output: Magma V2.11-10 Tue Dec 13 2005 11:39:59 on modular [Seed = 804427072] ------------------------------------- >> I:=ideal;; ^ Runtime error: No constructor provided for this type of object Total time: 0.190 seconds, Total memory usage: 3.24MB '84.56.2' ************** MAGMA ***************** Host 84.56.209.34 (84.56.209.34) Time: Tue Dec 13 11:38:59 2005 Input: R:=PolynomialRing(GF(2)); f:=x^155+x^62+1; K:=ext; I:=Ideal; Output: Magma V2.11-10 Tue Dec 13 2005 11:38:59 on modular [Seed = 1055097901] ------------------------------------- >> I:=Ideal;; ^ User error: bad syntax Total time: 0.190 seconds, Total memory usage: 3.24MB '84.56.2' ************** MAGMA ***************** Host 84.56.209.34 (84.56.209.34) Time: Tue Dec 13 11:37:27 2005 Input: R:=PolynomialRing(GF(2)); f:=x^155+x^62+1; K:=ext; Decomposition(K); Output: Magma V2.11-10 Tue Dec 13 2005 11:37:27 on modular [Seed = 921406913] ------------------------------------- >> Decomposition(K);; ^ Runtime error in 'Decomposition': Bad argument types Argument types given: FldFin Total time: 0.180 seconds, Total memory usage: 3.24MB '84.56.2' ************** MAGMA ***************** Host 84.56.209.34 (84.56.209.34) Time: Tue Dec 13 11:36:34 2005 Input: R:=PolynomialRing(GF(2)); f:=x^155+x^62+1; K:=ext; Decomposition(K,f); Output: Magma V2.11-10 Tue Dec 13 2005 11:36:33 on modular [Seed = 837585552] ------------------------------------- >> Decomposition(K,f);; ^ Runtime error in 'Decomposition': Bad argument types Argument types given: FldFin, RngUPolElt[FldFin] Total time: 0.180 seconds, Total memory usage: 3.24MB '84.56.2' ************** MAGMA ***************** Host 84.56.209.34 (84.56.209.34) Time: Tue Dec 13 11:34:06 2005 Input: R:=PolynomialRing(GF(2)); f:=x^155+x^62+1; K:=ext; Decomposition(K,f); Output: Magma V2.11-10 Tue Dec 13 2005 11:34:05 on modular [Seed = 3440408833] ------------------------------------- >> Decomposition(K,f);; ^ Runtime error in 'Decomposition': Bad argument types Argument types given: FldFin, RngUPolElt[FldFin] Total time: 0.190 seconds, Total memory usage: 3.24MB '84.56.2' ************** MAGMA ***************** Host 84.56.209.34 (84.56.209.34) Time: Tue Dec 13 11:32:13 2005 Input: R:=PolynomialRing(GF(2)); f:=x^155+x^62+1; K:=ext; Decomposition(f); Output: Magma V2.11-10 Tue Dec 13 2005 11:32:13 on modular [Seed = 3340399483] ------------------------------------- >> Decomposition(f);; ^ Runtime error in 'Decomposition': Bad argument types Argument types given: RngUPolElt[FldFin] Total time: 0.190 seconds, Total memory usage: 3.24MB '84.56.2' ************** MAGMA ***************** Host 84.56.209.34 (84.56.209.34) Time: Tue Dec 13 11:31:11 2005 Input: R:=PolynomialRing(GF(2)); f:=x^155+x^62+1; K:=ext; Decomposition(f); Output: Magma V2.11-10 Tue Dec 13 2005 11:31:11 on modular [Seed = 3223155902] ------------------------------------- >> K:=ext; ^ User error: Identifier 'definingPolynomial' has not been declared or assigned >> Decomposition(f);; ^ Runtime error in 'Decomposition': Bad argument types Argument types given: RngUPolElt[FldFin] Total time: 0.180 seconds, Total memory usage: 3.24MB '84.56.2' ************** MAGMA ***************** Host 84.56.209.34 (84.56.209.34) Time: Tue Dec 13 11:29:51 2005 Input: R:=PolynomialRing(GF(2)); definingPolynomial:=x^155+x^62+1; K:=ext; Decomposition(f); Output: Magma V2.11-10 Tue Dec 13 2005 11:29:51 on modular [Seed = 3540936221] ------------------------------------- >> Decomposition(f);; ^ User error: Identifier 'f' has not been declared or assigned Total time: 0.180 seconds, Total memory usage: 3.24MB '65.191.' ************** MAGMA ***************** Host 65.191.73.113 (65.191.73.113) Time: Tue Dec 13 11:26:17 2005 Input: > K := FiniteField(2); > E := LinearCode; > WeightDistribution(E); Output: Magma V2.11-10 Tue Dec 13 2005 11:26:17 on modular [Seed = 3979368693] ------------------------------------- [ <0, 1>, <2, 105>, <4, 1365>, <6, 5005>, <8, 6435>, <10, 3003>, <12, 455>, <14, 15> ] Total time: 0.180 seconds, Total memory usage: 3.24MB '65.191.' ************** MAGMA ***************** Host 65.191.73.113 (65.191.73.113) Time: Tue Dec 13 10:54:53 2005 Input: > K := FiniteField(2); > C := LinearCode; > D:=Dual(C); > (D meet C) eq C; Output: Magma V2.11-10 Tue Dec 13 2005 10:54:53 on modular [Seed = 2354127216] ------------------------------------- true Total time: 0.190 seconds, Total memory usage: 3.24MB '65.191.' ************** MAGMA ***************** Host 65.191.73.113 (65.191.73.113) Time: Tue Dec 13 10:52:23 2005 Input: > K := FiniteField(2); > C := LinearCode; > D:=Dual(C); > (d meet C) eq C; Output: Magma V2.11-10 Tue Dec 13 2005 10:52:22 on modular [Seed = 2220435843] ------------------------------------- >> (d meet C) eq C; ^ User error: Identifier 'd' has not been declared or assigned Total time: 0.180 seconds, Total memory usage: 3.24MB '65.191.' ************** MAGMA ***************** Host 65.191.73.113 (65.191.73.113) Time: Tue Dec 13 10:40:50 2005 Input: > K := FiniteField(2); > K56 := VectorSpace(K, 56); > C := LinearCode(sub); > D:=Hull(C); > WeightDistribution(D); Output: Magma V2.11-10 Tue Dec 13 2005 10:40:50 on modular [Seed = 2671916844] ------------------------------------- >> D:=Hull(C); ^ User error: Identifier 'Hull' has not been declared or assigned >> WeightDistribution(D); ^ User error: Identifier 'D' has not been declared or assigned Total time: 0.190 seconds, Total memory usage: 3.24MB '65.191.' ************** MAGMA ***************** Host 65.191.73.113 (65.191.73.113) Time: Tue Dec 13 10:35:53 2005 Input: > K := FiniteField(2); > K56 := VectorSpace(K, 56); > C := LinearCode(sub; > D:=Hull(C); > WeightDistribution(D); Output: Magma V2.11-10 Tue Dec 13 2005 10:35:53 on modular [Seed = 2876637891] ------------------------------------- >> 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]>; ^ User error: bad syntax >> D:=Hull(C); ^ User error: Identifier 'C' has not been declared or assigned >> WeightDistribution(D); ^ User error: Identifier 'D' has not been declared or assigned Total time: 0.180 seconds, Total memory usage: 3.24MB '65.191.' ************** MAGMA ***************** Host 65.191.73.113 (65.191.73.113) Time: Tue Dec 13 10:25:19 2005 Input: K := FiniteField(2); > C := LinearCode; > D:=Hull(C); > WeightDistribution(D); Output: Magma V2.11-10 Tue Dec 13 2005 10:25:19 on modular [Seed = 2726233676] ------------------------------------- >> D:=Hull(C); ^ User error: Identifier 'Hull' has not been declared or assigned >> WeightDistribution(D); ^ User error: Identifier 'D' has not been declared or assigned Total time: 0.190 seconds, Total memory usage: 3.24MB '65.191.' ************** MAGMA ***************** Host 65.191.73.113 (65.191.73.113) Time: Tue Dec 13 10:23:39 2005 Input: K := FiniteField(2); > C := LinearCode; D:=Hull(C); WeightDistribution(D); Output: Magma V2.11-10 Tue Dec 13 2005 10:23:39 on modular [Seed = 3210877941] ------------------------------------- >> D:=Hull(C); ^ User error: Identifier 'Hull' has not been declared or assigned >> WeightDistribution(D); ^ User error: Identifier 'D' has not been declared or assigned Total time: 0.210 seconds, Total memory usage: 3.24MB '65.191.' ************** MAGMA ***************** Host 65.191.73.113 (65.191.73.113) Time: Tue Dec 13 10:20:00 2005 Input: K := FiniteField(2); > C := LinearCode; C; WeightDistribution(C); Output: Magma V2.11-10 Tue Dec 13 2005 10:19:57 on modular [Seed = 820876674] ------------------------------------- [56, 21] Linear Code over GF(2) Generator matrix: [1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 1 1 0 0 0 0 1 1 1 0 0 0 0 1 1 1 0 1 0 1 0 0 1 1 0 0 1 1 0 0 0 1 1 0 1 0 0] [0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 1 0 0 1 0 0 0 1 0 0 1 0 1 1 1 0 0 0 1 1 0 0 1 1 0 1 0] [0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 1 0 0 0 0 0 1 0 0 0 1 1 0 0 0 0 0 0 0 1 1 1 1 1 0 0 0 0 0 0 0 1 1 1 1 0 0] [0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 1 1 1 1 0 1 1 0 0 1 1 0 1] [0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 1 1 0 1 1 0 0 1 1 1 1 1 1 0 1 1 1 0 0 1 1 0 1 1 1 1 0 0 1 1 0] [0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 0 1 1 1 0 0 0 1 1 1 0 0 0 1 0 0 1 0 0 1 0 1 0 0 0 1] [0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 1 1 1 0 0 1 0 1 0 1 1 0 1 0 0 1 1 1 1 1 0 1 1 0 0 0 1 0 1 0] [0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 1 1 0 0 0 0 0 0 1 1 0 0 1 1 0 0 0 1 0 0 1 0 1 1 0 0 1 1 1 0 1 1 0 1 0 0 1 1] [0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 1 0 0 0 0 1 1 1 0 1 0 1 1 1 0 0 1 1 1 0 0 0 1 1 1 1 1 1 1 1 0 1 1 0 0 0] [0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 1 0 1 1 1 1 0 0 0 0 1 1 0 1 1 0 1 0 1 1 1 0 1 0 0 1 0 1 1 0 0] [0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 1 1 0 0 0 0 1 0 0 1 0 1 1 1 1 1 0 0 0 0 1 1 1 1 0 1 1 0 0 0 0 0 1 0 1 1 0] [0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 1 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 0 1 0 1 1 1 0 1 0 0 1] [0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 1 1 1 0 0 0 0 1 1 0 0 1 1 1 0 1 1 1 0 0 1 1 1 1 1 0 1 0 0 1 1 1 0 0 0 1 0 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 1 0 0 0 0 0 0 0 1 0 0 0 1 0 1 0 1 0 1 1 0 0 1 1 1 0 1 0 0 0 1 1 0 0 0 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 1 0 0 0 0 0 0 0 0 1 0 0 1 1 1 0 1 1 1 0 1 0 1 0 0 1 1 1 0 1 1 0 1 0 0 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 0 0 0 0 0 0 0 0 0 0 1 0 1 1 0 0 1 1 0 0 0 1 1 0 1 0 0 1 1 1 0 0 0 1 0 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 0 0 0 0 0 0 0 0 0 0 1 0 1 1 1 0 1 1 0 0 0 1 1 0 1 0 0 1 1 1 0 0 0 1 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 1 0 1 1 1 0 1 0 0 0 1 0 1 1 0 0 1 0 1 1 0 0 1 1 0 0 0 1 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 1 0 1 0 0 1 1 1 0 0 1 1 1 0 1 0 1 1 1 0 1 0 1 0 1 0 0 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 0 1 1 0 1 0 0 1 1 0 1 1 0 0 0 0 1 1 0 0 0 1 0 0 0 1 0 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 1 1 1 0 1 0 0 0 1 0 1 1 0 0 1 0 1 1 0 0 0 1 0 0 0 1 0 1] [ <0, 1>, <16, 5082>, <20, 91168>, <24, 507045>, <28, 890560>, <32, 507045>, <36, 91168>, <40, 5082>, <56, 1> ] Total time: 0.240 seconds, Total memory usage: 3.34MB '65.191.' ************** MAGMA ***************** Host 65.191.73.113 (65.191.73.113) Time: Tue Dec 13 10:16:57 2005 Input: K := FiniteField(2); > C := LinearCode; E := Hull(C); E; WeightDistribution(E); Output: Magma V2.11-10 Tue Dec 13 2005 10:16:56 on modular [Seed = 2671917382] ------------------------------------- >> E := Hull(C); ^ User error: Identifier 'Hull' has not been declared or assigned >> E; ^ User error: Identifier 'E' has not been declared or assigned >> WeightDistribution(E); ^ User error: Identifier 'E' has not been declared or assigned Total time: 0.220 seconds, Total memory usage: 3.24MB '65.191.' ************** MAGMA ***************** Host 65.191.73.113 (65.191.73.113) Time: Tue Dec 13 10:01:43 2005 Input: K := FiniteField(2); > C := LinearCode; D := Dual(C); WeightDistribution(C); time WeightDistribution(D); Output: Magma V2.11-10 Tue Dec 13 2005 10:01:42 on modular [Seed = 4096349653] ------------------------------------- [ <0, 1>, <16, 5082>, <20, 91168>, <24, 507045>, <28, 890560>, <32, 507045>, <36, 91168>, <40, 5082>, <56, 1> ] [ <0, 1>, <8, 1155>, <10, 34496>, <12, 539280>, <14, 5480640>, <16, 40029297>, <18, 200963840>, <20, 755023808>, <22, 2027262720>, <24, 4185896715>, <26, 6289084032>, <28, 7351106400>, <30, 6289084032>, <32, 4185896715>, <34, 2027262720>, <36, 755023808>, <38, 200963840>, <40, 40029297>, <42, 5480640>, <44, 539280>, <46, 34496>, <48, 1155>, <56, 1> ] Time: 0.020 Total time: 0.250 seconds, Total memory usage: 3.34MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Tue Dec 13 10:00:16 2005 Input: p := 1461501624496790265145448589920785493717258890819; n := 1461501624496790265145447380994971188499300027613; Fp := GF(p); Fp2 := ExtensionField; lambda := 2; mu := 1 + i; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; b := Fp!3; y0 := Fp!2; // -Sqrt(1 + b); E := EllipticCurve([0, b]); G := E![1, y0]; Et := EllipticCurve([0, b/xi]); //xt := i + 2; xt := 8; /* xt := i; while not IsSquare(xt^3 + b/xi) do xt +:= 1; end while; "xt =", xt; */ yt := Sqrt(xt^3 + b/xi); Gt := Et![xt, yt]; k := 12; assert (p^k - 1) mod n eq 0; chi := (p^k - 1) div n; Ek := EllipticCurve([0, Fp12!b]); g := function(U, V, Q) if IsZero(U) or IsZero(V) or (U eq -V) or IsZero(Q) then return Fp12!1; end if; m := (U eq V) select 3*U[1]^2/(2*U[1]) else (V[2] - U[2])/(V[1] - U[1]); return m*(Q[1] - U[1]) + U[1] - Q[2]; end function; length := function(r) n := 0; v := 1; while v le r do n +:= 1; v +:= v; end while; return n; end function; bit := function(r, i) return (r div 2^i) mod 2; end function; miller := function(r, P, Q) f := 1; A := P; for i := length(r) - 1 to 0 by -1 do f := f^2*g(A, A, Q); A := 2*A; if bit(r, i) eq 1 then f *:= g(A, P, Q); A +:= P; end if; end for; return f; end function; tate := function(P, Qt) Q := Ek![Qt[1]*z^2, Qt[2]*z^3]; /* "P =", P; "Q' =", Qt; "Q =", Q; */ return miller(n, P, Q)^chi; end function; g := tate(G, Gt); "g =", g; /* for j in [1..11] do print "g^(p^" * Sprint(j) * "-1): " * Sprint(g^(p^j-1) ne 1); end for; print "g^(p^12-1): " * Sprint(g^(p^12-1) eq 1); "g^n =", g^n; */ for j in [1..100] do u := Random(n - 1); v := Random(n - 1); w := tate(u*G, v*Gt); if w ne g^((u*v) mod n) then "Failure: u =", u, ", v =", v, "e(u*P,v*Q) =", w, ", e(P,Q)^(uv) =", g^((u*v) mod n); quit; end if; end for; "Success!"; Output: Magma V2.11-10 Tue Dec 13 2005 10:00:15 on modular [Seed = 164932481] ------------------------------------- g = (1414620835193373657646613880233559577407734609545*i + 352489175314961464607671418528001817189092256541)*z^5 + (904802261065103105936668720505067133656667597074*i + 1343248922269291340494538393490943353845819222256)*z^4 + (1025700017505207783921241027028089077317301724291*i + 1267567142236668396047190065512596226428512971416)*z^3 + (283587609846347412879109527640610607003328134432*i + 250487955904041646310509260856358041239663937723)*z^2 + (110054808378578108265445578093874268052578876998*i + 202041545854984388890408940644851972139794936772)*z + 257093848091331658607960702914357017287219708568*i + 652950073647454119068347296682657355787566777234 Failure: u = 505653466152672793617695632599288515920022369214 , v = 813309314074995504442616266927015985129671509525 e(u*P,v*Q) = (229450318520897930832122824950759136162186600899*i + 171458666541069865322322784204702213676643928600)*z^5 + (1123336210201860974010594496403076935746616156201*i + 382509869157557354897123314935302020523067652238)*z^4 + (195956015357740925670293015392589736976005190419*i + 1201060680146586880867752998079193262658782520318)*z^3 + (1400183652328377308210652428671451994834327508147*i + 885783581485769151127555190406908658009200060139)*z^2 + (241990578305321134124934647372233092586143696750*i + 353283743758609339844557914979526193600469875967)*z + 69301698278701722581645298580459933625029213350*i + 522130179428839937361552509967162010658032702262 , e(P,Q)^(uv) = (1233480993554786834008498621906536758921898377565*i + 1190015354691216118850119495146059439731677746380)*z^5 + (921756594327110983814290309172804473768320708717*i + 738420790495346685113640877223616173148157182434)*z^4 + (1448167631698704692933635090700677473693607566307*i + 467911266345108570245537574526004995346892189176)*z^3 + (819222557435172257255808600604347664799809662153*i + 691163176033377731593142083640280484981447850039)*z^2 + (1178974401966800441343583985193986104164302821005*i + 276511259554730579432106796424120788700936371162)*z + 646813776638959146887930847368281873288578357077*i + 1043047602020859885512585427844085790318024216544 Total time: 1.080 seconds, Total memory usage: 3.34MB '65.191.' ************** MAGMA ***************** Host 65.191.73.113 (65.191.73.113) Time: Tue Dec 13 09:47:58 2005 Input: K := FiniteField(2); > D := LinearCode; > aut := AutomorphismGroup(D); Order(aut); > FactoredOrder(aut); > CompositionFactors(aut); Generators(aut); IsSelfOrthogonal(D); Output: Magma V2.11-10 Tue Dec 13 2005 09:47:57 on modular [Seed = 1552007222] ------------------------------------- 7 [ <7, 1> ] G | Cyclic(7) 1 { (1, 2, 3, 4, 5, 6, 7)(8, 9, 10, 11, 12, 13, 14)(15, 16, 17, 18, 19, 20, 21)(22, 23, 24, 25, 26, 27, 28)(29, 30, 31, 32, 33, 34, 35)(36, 37, 38, 39, 40, 41, 42)(43, 44, 45, 46, 47, 48, 49)(50, 51, 52, 53, 54, 55, 56) } false Total time: 0.280 seconds, Total memory usage: 5.51MB '65.191.' ************** MAGMA ***************** Host 65.191.73.113 (65.191.73.113) Time: Tue Dec 13 09:47:01 2005 Input: K := FiniteField(2); > C := LinearCode; > aut := AutomorphismGroup(C); Order(aut); > FactoredOrder(aut); > CompositionFactors(aut); Generators(aut); IsSelfOrthogonal(C); Output: Magma V2.11-10 Tue Dec 13 2005 09:47:01 on modular [Seed = 1385156544] ------------------------------------- 7 [ <7, 1> ] G | Cyclic(7) 1 { (1, 2, 3, 4, 5, 6, 7)(8, 9, 10, 11, 12, 13, 14)(15, 16, 17, 18, 19, 20, 21)(22, 23, 24, 25, 26, 27, 28)(29, 30, 31, 32, 33, 34, 35)(36, 37, 38, 39, 40, 41, 42)(43, 44, 45, 46, 47, 48, 49)(50, 51, 52, 53, 54, 55, 56) } false Total time: 0.250 seconds, Total memory usage: 5.51MB '65.191.' ************** MAGMA ***************** Host 65.191.73.113 (65.191.73.113) Time: Tue Dec 13 09:43:19 2005 Input: K := FiniteField(2); > C := LinearCode [1, 0, 0, 0, 0, 1, 1, 1], [0, 1, 0, 0, 1, 0, 1, 1], > [0, 0, 1, 0, 1, 1, 0, 1], [0, 0, 0, 1, 1, 1, 1, 0]>; > aut := AutomorphismGroup(C); Order(aut); > FactoredOrder(aut); > CompositionFactors(aut); Generators(aut); IsSelfOrthogonal(C); Output: Magma V2.11-10 Tue Dec 13 2005 09:43:18 on modular [Seed = 1468713510] ------------------------------------- 1344 [ <2, 6>, <3, 1>, <7, 1> ] G | A(1, 7) = L(2, 7) * | Cyclic(2) * | Cyclic(2) * | Cyclic(2) 1 { (3, 4)(7, 8), (4, 6)(5, 7), (4, 7)(5, 6), (1, 2)(5, 6), (2, 4, 3)(6, 8, 7) } true Total time: 0.200 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Tue Dec 13 09:41:20 2005 Input: p := 1461501624496790265145448589920785493717258890819; n := 1461501624496790265145447380994971188499300027613; Fp := GF(p); Fp2 := ExtensionField; lambda := 2; mu := 1 + i; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; b := Fp!3; y0 := Fp!2; // -Sqrt(1 + b); E := EllipticCurve([0, b]); G := E![1, y0]; Et := EllipticCurve([0, b/xi]); //xt := i + 2; xt := 8; /* xt := i; while not IsSquare(xt^3 + b/xi) do xt +:= 1; end while; "xt =", xt; */ yt := Sqrt(xt^3 + b/xi); Gt := Et![xt, yt]; k := 12; assert (p^k - 1) mod n eq 0; chi := (p^k - 1) div n; Ek := EllipticCurve([0, Fp12!b]); g := function(U, V, Q) if IsZero(U) or IsZero(V) or (U eq -V) or IsZero(Q) then return Fp12!1; end if; m := (U eq V) select 3*U[1]^2/(2*U[1]) else (V[2] - U[2])/(V[1] - U[1]); return m*(Q[1] - U[1]) + U[1] - Q[2]; end function; length := function(r) n := 0; v := 1; while v le r do n +:= 1; v +:= v; end while; return n; end function; bit := function(r, i) return (r div 2^i) mod 2; end function; miller := function(r, P, Q) f := 1; A := P; for i := length(r) - 1 to 0 by -1 do f := f^2*g(A, A, Q); A := 2*A; if bit(r, i) eq 1 then f *:= g(A, P, Q); A +:= P; end if; end for; return f; end function; tate := function(P, Qt) Q := Ek![Qt[1]*z^2, Qt[2]*z^3]; /* "P =", P; "Q' =", Qt; "Q =", Q; */ return miller(n, P, Q)^chi; end function; g := tate(G, Gt); "g =", g; /* for j in [1..11] do print "g^(p^" * Sprint(j) * "-1): " * Sprint(g^(p^j-1) ne 1); end for; print "g^(p^12-1): " * Sprint(g^(p^12-1) eq 1); "g^n =", g^n; */ for j in [1..100] do u := Random(n - 1); v := Random(n - 1); if tate(u*G, v*Gt) ne g^((u*v) mod n) then "Failure: u =", u, ", v =", v; quit; end if; end for; "Success!"; Output: Magma V2.11-10 Tue Dec 13 2005 09:41:19 on modular [Seed = 1840313237] ------------------------------------- g = (46880789303416607498834709687225916309524281274*i + 1109012449181828800537777171392783676528166634278)*z^5 + (904802261065103105936668720505067133656667597074*i + 1343248922269291340494538393490943353845819222256)*z^4 + (435801606991582481224207562892696416399957166528*i + 193934482260121869098258524408189267288745919403)*z^3 + (283587609846347412879109527640610607003328134432*i + 250487955904041646310509260856358041239663937723)*z^2 + (1351446816118212156880003011826911225664680013821*i + 1259460078641805876255039649275933521577463954047)*z + 257093848091331658607960702914357017287219708568*i + 652950073647454119068347296682657355787566777234 Failure: u = 1026132414533535312271167998032452174431467642015 , v = 607317891842996534038173466812558720729177303044 Total time: 1.060 seconds, Total memory usage: 3.34MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Tue Dec 13 09:30:18 2005 Input: p := 1461501624496790265145448589920785493717258890819; n := 1461501624496790265145447380994971188499300027613; Fp := GF(p); Fp2 := ExtensionField; lambda := 2; mu := 1 + i; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; b := Fp!3; y0 := Fp!2; // -Sqrt(1 + b); E := EllipticCurve([0, b]); G := E![1, y0]; Et := EllipticCurve([0, b/xi]); //xt := i + 2; xt := 8; /* xt := i; while not IsSquare(xt^3 + b/xi) do xt +:= 1; end while; "xt =", xt; */ yt := Sqrt(xt^3 + b/xi); Gt := Et![xt, yt]; k := 12; assert (p^k - 1) mod n eq 0; chi := (p^k - 1) div n; Ek := EllipticCurve([0, Fp12!b]); g := function(U, V, Q) if IsZero(U) or IsZero(V) or (U eq -V) or IsZero(Q) then return Fp12!1; end if; m := (U eq V) select 3*U[1]^2/(2*U[1]) else (V[2] - U[2])/(V[1] - U[1]); return m*(Q[1] - U[1]) + U[1] - Q[2]; end function; length := function(r) n := 0; v := 1; while v le r do n +:= 1; v +:= v; end while; return n; end function; bit := function(r, i) return (r div 2^i) mod 2; end function; miller := function(r, P, Q) f := 1; A := P; for i := length(r) - 1 to 0 by -1 do f := f^2*g(A, A, Q); A := 2*A; if bit(r, i) eq 1 then f *:= g(A, P, Q); A +:= P; end if; end for; return f; end function; tate := function(P, Qt) Q := Ek![Qt[1]*z^2, Qt[2]*z^3]; /* "P =", P; "Q' =", Qt; "Q =", Q; */ return miller(n, P, Q)^chi; end function; g := tate(G, Gt); "g =", g; for j in [1..11] do print "g^(p^" * Sprint(j) * "-1): " * Sprint(g^(p^j-1) ne 1); end for; print "g^(p^12-1): " * Sprint(g^(p^12-1) eq 1); "g^n =", g^n; Output: Magma V2.11-10 Tue Dec 13 2005 09:30:15 on modular [Seed = 1924131826] ------------------------------------- g = (1414620835193373657646613880233559577407734609545*i + 352489175314961464607671418528001817189092256541)*z^5 + (904802261065103105936668720505067133656667597074*i + 1343248922269291340494538393490943353845819222256)*z^4 + (1025700017505207783921241027028089077317301724291*i + 1267567142236668396047190065512596226428512971416)*z^3 + (283587609846347412879109527640610607003328134432*i + 250487955904041646310509260856358041239663937723)*z^2 + (110054808378578108265445578093874268052578876998*i + 202041545854984388890408940644851972139794936772)*z + 257093848091331658607960702914357017287219708568*i + 652950073647454119068347296682657355787566777234 g^(p^1-1): true g^(p^2-1): true g^(p^3-1): true g^(p^4-1): true g^(p^5-1): true g^(p^6-1): true g^(p^7-1): true g^(p^8-1): true g^(p^9-1): true g^(p^10-1): true g^(p^11-1): true g^(p^12-1): true g^n = 1 Total time: 2.600 seconds, Total memory usage: 3.34MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Tue Dec 13 09:28:11 2005 Input: p := 1461501624496790265145448589920785493717258890819; n := 1461501624496790265145447380994971188499300027613; Fp := GF(p); Fp2 := ExtensionField; lambda := 2; mu := 1 + i; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; b := Fp!3; y0 := Fp!2; // -Sqrt(1 + b); E := EllipticCurve([0, b]); G := E![1, y0]; Et := EllipticCurve([0, b/xi]); xt := i + 2; /* xt := i; while not IsSquare(xt^3 + b/xi) do xt +:= 1; end while; "xt =", xt; */ yt := Sqrt(xt^3 + b/xi); Gt := Et![xt, yt]; k := 12; assert (p^k - 1) mod n eq 0; chi := (p^k - 1) div n; Ek := EllipticCurve([0, Fp12!b]); g := function(U, V, Q) if IsZero(U) or IsZero(V) or (U eq -V) or IsZero(Q) then return Fp12!1; end if; m := (U eq V) select 3*U[1]^2/(2*U[1]) else (V[2] - U[2])/(V[1] - U[1]); return m*(Q[1] - U[1]) + U[1] - Q[2]; end function; length := function(r) n := 0; v := 1; while v le r do n +:= 1; v +:= v; end while; return n; end function; bit := function(r, i) return (r div 2^i) mod 2; end function; miller := function(r, P, Q) f := 1; A := P; for i := length(r) - 1 to 0 by -1 do f := f^2*g(A, A, Q); A := 2*A; if bit(r, i) eq 1 then f *:= g(A, P, Q); A +:= P; end if; end for; return f; end function; tate := function(P, Qt) Q := Ek![Qt[1]*z^2, Qt[2]*z^3]; "P =", P; "Q' =", Qt; "Q =", Q; return miller(n, P, Q)^chi; end function; g := tate(G, Gt); "g =", g; for j in [1..12] do print "g^(p^" * Sprint(j) * "-1) = " * Sprint(g^(p^j-1)); end for; "g^n =", g^n; Output: Magma V2.11-10 Tue Dec 13 2005 09:28:08 on modular [Seed = 2655192982] ------------------------------------- P = (1 : 2 : 1) Q' = (i + 2 : 1097950022558348869462494891919937327680507168109*i + 1374283222983568961963002763601919595127481124512 : 1) Q = ((i + 2)*z^2 : (1097950022558348869462494891919937327680507168109*i + 1374283222983568961963002763601919595127481124512)*z^3 : 1) g = (123909007522715632304234432926990441159731420430*i + 637508010256500182107280699368993348477341084485)*z^5 + (688703507104756931070001853914995131005373541312*i + 1002621352581475249329708955843984496240073153982)*z^4 + (608846424657090324998119234825851616974125548764*i + 771482020725020581050629871627202529613821639823)*z^3 + (1392434018234511176428044389335844610478592711217*i + 473331107538064635840791020785366158595476919127)*z^2 + (73970378918706542820629598671516884756426271661*i + 365943832704743787495421878796345795274953421973)*z + 616166846676061100095127430332774495441755394581*i + 889255287756560160849573452496040306572455585395 g^(p^1-1) = (748100618927567348520741622873551546451532782022*i + 1428699130224630511049471936323405187567302837671)*z^5 + (1202680812184493569478300721391313577308119778263*i + 313447253330080409810020430322639687095227202514)*z^4 + (896962745306077156991764130250048502701811544250*i + 75030114724933747382754664337864814318468370719)*z^3 + (333682831282116263631143017468030399561367002231*i + 518405572876237815324490177632651602018332790765)*z^2 + (1112317314850298844232882970368687655422664826708*i + 560782544915172117676888817418082341837354667733)*z + 1202838306203487510142528951450976199443662125144*i + 795069427093050539880893267448617311116978407546 g^(p^2-1) = (1122685762559073106420486640172303029900241318667*i + 1206761843969459490842516022375418025481873864298)*z^5 + (3817024364748341053470651752544249413211110060*i + 92074379894423550543412574098909660792084933016)*z^4 + (608846424657090324998119234825851616974125548764*i + 771482020725020581050629871627202529613821639823)*z^3 + (477186809706685400548284839616627303398814604581*i + 1353293254655497653064288440912729947389627707084)*z^2 + (708592972296574786248902668360725858769675156568*i + 566475776092548476468859447971519767492233615991)*z + 616166846676061100095127430332774495441755394581*i + 889255287756560160849573452496040306572455585395 g^(p^3-1) = (1357106999709464496564779036298599057498303412530*i + 1345682320959266667240129620417040749623418914188)*z^5 + (936139532752018606210117538341071166608350863727*i + 1112461501477852426997296904293113682769958766774)*z^4 + (1445405172445096850777244515807849246142141374340*i + 396466674385106777111606626855996450744375790171)*z^3 + (1461250712521757914096163977919390965791567384854*i + 244970246429598896709919226014995935882868586820)*z^2 + (542551411651261070353031322827794497138348015296*i + 429807684156175367374279713695611598069895859239)*z + 70531838254713320311910681453696149021611168678*i + 1398834529239549175114023695374662216300177982250 g^(p^4-1) = (1221509504491967374311557208672867933832795980673*i + 200467472176772056987212698380766546792219162971)*z^5 + (208862746409985936264596906005246300650626057668*i + 565357618624448102009812399714319141941210262495)*z^4 + (865756177282313545985213216550036708550476280949*i + 475475525215527373084617709122508974760880884335)*z^3 + (102466015727909809069678183573961098275241875184*i + 940151235303014027110245043307168921216332322768)*z^2 + (928922738858698562216100850689154274539784497877*i + 382002999584106301024558712410303674095683347609)*z + 1220974658126123565670019751888593761044993991185*i + 211811947018249923223576976718524137383767937965 g^(p^5-1) = (594480147410955283734684832159447099240412938228*i + 1007447468906095875580426526654019572773339214368)*z^5 + (1152066755303591353030630678326466707035079216340*i + 651594961867943713451573799740600135483141099859)*z^4 + (49501392213291562027375381267081647404991522810*i + 820710015151236425244004586118888126082585441078)*z^3 + (42188653754860119450135480418886642353674596443*i + 562408702302581793941475002165818938687417258094)*z^2 + (1091572228871275931758790486093603710299273846570*i + 577436592715763274817514347647367029829638669007)*z + 1168565180281449525586662304102209555431570678308*i + 503582600681258840386198926408111711693060441073 g^(p^6-1) = (205768684348313624077451572794465353568914781421*i + 613950226393615708182994901900063381875511405659)*z^5 + (1428633672987200370786940114820930632812523417902*i + 395314511348759603485431448487808080038157695517)*z^4 + (881979103651631189249974992863662825116960349537*i + 1044381106242866386024323569275011144406191731131)*z^3 + (377475555156278737030694625815440794829353520650*i + 60549325767510300874857944491779135450365424950)*z^2 + (904476028311783023806480875092919429317592811440*i + 654508681633842315960851796564179216185709460867)*z + 826543425518806376820411130172467469135354246707*i + 546674685375929320723432354108401487302836191956 g^(p^7-1) = (334999190429734450302032977239633601705995714949*i + 285499081478684180346094108130235958500726170807)*z^5 + (1357641212310828158490838523101409841505272979404*i + 678284707222532143244512307089503927268688656266)*z^4 + (447713965858424633050352197649395842462816234715*i + 523942895907349573254753909344390113077006213603)*z^3 + (395448289533461064020784297720245760316432679996*i + 256449527939203012159253238775693631557658125017)*z^2 + (825189319832069956822612513847208165005287290855*i + 333752408118167725005777483986738208826843204644)*z + 292936444215340739558786285818575938285688212511*i + 503582600681258840386198926408111711693060441073 g^(p^8-1) = (128794472520144390825537958245889556576532668457*i + 682817146795319322173041394863892996788054910205)*z^5 + (712683874015386917422759238642315687393680373361*i + 1227924079489556276578945161248737876531470377844)*z^4 + (595745447214476719160235373370748785166782609870*i + 986026099281262892060830880798276518956378006484)*z^3 + (375673708125247397107051660343678979770619117775*i + 114357758639702456761704170918133809768845643944)*z^2 + (1411889287093050769181126017526789790323252735684*i + 1105315081259072298561198519568055599119405178147)*z + 1220974658126123565670019751888593761044993991185*i + 211811947018249923223576976718524137383767937965 g^(p^9-1) = (1223692038621048430153872436913389348975101235031*i + 474621643755870221635939468034787852720049039479)*z^5 + (936139532752018606210117538341071166608350863727*i + 349040123018937838148151685627671810947300124045)*z^4 + (1376144652516737512570670115379753945711985528623*i + 1373311158685001552683394130170484212465308688527)*z^3 + (244970246429598896709919226014995935882868586820*i + 1461250712521757914096163977919390965791567384854)*z^2 + (51496949206574234593005076323843447627546727734*i + 22372145650816023655116449432656061648752517842)*z + 1390969786242076944833537908467089344695647722141*i + 1398834529239549175114023695374662216300177982250 g^(p^10-1) = (214906854415001526420727516821492022657286151722*i + 1078733394767620857341100458097159613475302832855)*z^5 + (768981093027284993021976084253246113298674239447*i + 366805892020891465272327059977891336685100803821)*z^4 + (608846424657090324998119234825851616974125548764*i + 771482020725020581050629871627202529613821639823)*z^3 + (1053382421052383953314567950889099073557110465840*i + 1096378886800018241385817718143474881449413155427)*z^2 + (678938273281508936075916322888542750191157462590*i + 529082015699498001181167263152919930950071852855)*z + 616166846676061100095127430332774495441755394581*i + 889255287756560160849573452496040306572455585395 g^(p^11-1) = (548580783601665930616001142054617430470218498646*i + 1451628029083941079148995842838738809076287723525)*z^5 + (2545939521711331370283095426719602351824833799*i + 1344046085896870808486244472445559740184244945852)*z^4 + (837618043175865668846772504097006035270479350279*i + 786494693152924360802801900817858926731509706423)*z^3 + (1402153388011088316650991466733555001741554598260*i + 247983074468381684089738348912296989265022486477)*z^2 + (1185906070039435994832468343291972442612922376685*i + 305123309751637633980829646230021828644012617207)*z + 258663318293302755002919638469809294273596765675*i + 795069427093050539880893267448617311116978407546 g^(p^12-1) = 1 g^n = 1 Total time: 2.589 seconds, Total memory usage: 3.34MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Tue Dec 13 09:25:43 2005 Input: p := 1461501624496790265145448589920785493717258890819; n := 1461501624496790265145447380994971188499300027613; Fp := GF(p); Fp2 := ExtensionField; lambda := 2; mu := 1 + i; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; b := Fp!3; y0 := Fp!2; // -Sqrt(1 + b); E := EllipticCurve([0, b]); G := E![1, y0]; Et := EllipticCurve([0, b/xi]); //xt := i + 2; xt := 8; /* xt := i; while not IsSquare(xt^3 + b/xi) do xt +:= 1; end while; "xt =", xt; */ yt := Sqrt(xt^3 + b/xi); Gt := Et![xt, yt]; k := 12; assert (p^k - 1) mod n eq 0; chi := (p^k - 1) div n; Ek := EllipticCurve([0, Fp12!b]); g := function(U, V, Q) if IsZero(U) or IsZero(V) or (U eq -V) or IsZero(Q) then return Fp12!1; end if; m := (U eq V) select 3*U[1]^2/(2*U[1]) else (V[2] - U[2])/(V[1] - U[1]); return m*(Q[1] - U[1]) + U[1] - Q[2]; end function; length := function(r) n := 0; v := 1; while v le r do n +:= 1; v +:= v; end while; return n; end function; bit := function(r, i) return (r div 2^i) mod 2; end function; miller := function(r, P, Q) f := 1; A := P; for i := length(r) - 1 to 0 by -1 do f := f^2*g(A, A, Q); A := 2*A; if bit(r, i) eq 1 then f *:= g(A, P, Q); A +:= P; end if; end for; return f; end function; tate := function(P, Qt) Q := Ek![Qt[1]*z^2, Qt[2]*z^3]; //"P =", P; "Q' =", Qt; "Q =", Q; return miller(n, P, Q)^chi; end function; g := tate(G, Gt); "g =", g; for j in [1..11] do print "g^(p^" * Sprint(j) * "-1): " * Sprint(g^(p^j-1) ne 1); end for; print "g^(p^12-1): " * Sprint(g^(p^j-1) eq 1); "g^n =", g^n; Output: Magma V2.11-10 Tue Dec 13 2005 09:25:40 on modular [Seed = 2538212611] ------------------------------------- g = (46880789303416607498834709687225916309524281274*i + 1109012449181828800537777171392783676528166634278)*z^5 + (904802261065103105936668720505067133656667597074*i + 1343248922269291340494538393490943353845819222256)*z^4 + (435801606991582481224207562892696416399957166528*i + 193934482260121869098258524408189267288745919403)*z^3 + (283587609846347412879109527640610607003328134432*i + 250487955904041646310509260856358041239663937723)*z^2 + (1351446816118212156880003011826911225664680013821*i + 1259460078641805876255039649275933521577463954047)*z + 257093848091331658607960702914357017287219708568*i + 652950073647454119068347296682657355787566777234 g^(p^1-1): true g^(p^2-1): true g^(p^3-1): true g^(p^4-1): true g^(p^5-1): true g^(p^6-1): true g^(p^7-1): true g^(p^8-1): true g^(p^9-1): true g^(p^10-1): true g^(p^11-1): true >> print "g^(p^12-1): " * Sprint(g^(p^j-1) eq 1); ^ User error: Identifier 'j' has not been declared or assigned g^n = 1 Total time: 2.640 seconds, Total memory usage: 3.34MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Tue Dec 13 09:21:25 2005 Input: p := 1461501624496790265145448589920785493717258890819; n := 1461501624496790265145447380994971188499300027613; Fp := GF(p); Fp2 := ExtensionField; lambda := 2; mu := 1 + i; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; b := Fp!3; y0 := Fp!2; // -Sqrt(1 + b); E := EllipticCurve([0, b]); G := E![1, y0]; Et := EllipticCurve([0, b/xi]); xt := i + 2; /* xt := i; while not IsSquare(xt^3 + b/xi) do xt +:= 1; end while; "xt =", xt; */ yt := Sqrt(xt^3 + b/xi); Gt := Et![xt, yt]; k := 12; assert (p^k - 1) mod n eq 0; chi := (p^k - 1) div n; Ek := EllipticCurve([0, Fp12!b]); g := function(U, V, Q) if IsZero(U) or IsZero(V) or (U eq -V) or IsZero(Q) then return Fp12!1; end if; m := (U eq V) select 3*U[1]^2/(2*U[1]) else (V[2] - U[2])/(V[1] - U[1]); return m*(Q[1] - U[1]) + U[1] - Q[2]; end function; length := function(r) n := 0; v := 1; while v le r do n +:= 1; v +:= v; end while; return n; end function; bit := function(r, i) return (r div 2^i) mod 2; end function; miller := function(r, P, Q) f := 1; A := P; for i := length(r) - 1 to 0 by -1 do f := f^2*g(A, A, Q); A := 2*A; if bit(r, i) eq 1 then f *:= g(A, P, Q); A +:= P; end if; end for; return f; end function; tate := function(P, Qt) Q := Ek![Qt[1]*z^2, Qt[2]*z^3]; "P =", P; "Q' =", Qt; "Q =", Q; return miller(n, P, Q)^chi; end function; g := tate(G, Gt); "g =", g; for j in [1..12] do print "g^(p^" * Sprint(j) * "-1) = " * Sprint(g^(p^j-1)); end for; "g^n =", g^n; Output: Magma V2.11-10 Tue Dec 13 2005 09:21:17 on modular [Seed = 2943497218] ------------------------------------- P = (1 : 2 : 1) Q' = (i + 2 : 363551601938441395682953698000848166036751722710*i + 87218401513221303182445826318865898589777766307 : 1) Q = ((i + 2)*z^2 : (363551601938441395682953698000848166036751722710*i + 87218401513221303182445826318865898589777766307)*z^3 : 1) g = (1337592616974074632841214156993795052557527470389*i + 823993614240290083038167890551792145239917806334)*z^5 + (688703507104756931070001853914995131005373541312*i + 1002621352581475249329708955843984496240073153982)*z^4 + (852655199839699940147329355094933876743133342055*i + 690019603771769684094818718293582964103437250996)*z^3 + (1392434018234511176428044389335844610478592711217*i + 473331107538064635840791020785366158595476919127)*z^2 + (1387531245578083722324818991249268608960832619158*i + 1095557791792046477650026711124439698442305468846)*z + 616166846676061100095127430332774495441755394581*i + 889255287756560160849573452496040306572455585395 g^(p^1-1) = (713401005569222916624706967047233947265726108797*i + 32802494272159754095976653597380306149956053148)*z^5 + (1202680812184493569478300721391313577308119778263*i + 313447253330080409810020430322639687095227202514)*z^4 + (564538879190713108153684459670736991015447346569*i + 1386471509771856517762693925582920679398790520100)*z^3 + (333682831282116263631143017468030399561367002231*i + 518405572876237815324490177632651602018332790765)*z^2 + (349184309646491420912565619552097838294594064111*i + 900719079581618147468559772502703151879904223086)*z + 1202838306203487510142528951450976199443662125144*i + 795069427093050539880893267448617311116978407546 g^(p^2-1) = (338815861937717158724961949748482463817017572152*i + 254739780527330774302932567545367468235385026521)*z^5 + (3817024364748341053470651752544249413211110060*i + 92074379894423550543412574098909660792084933016)*z^4 + (852655199839699940147329355094933876743133342055*i + 690019603771769684094818718293582964103437250996)*z^3 + (477186809706685400548284839616627303398814604581*i + 1353293254655497653064288440912729947389627707084)*z^2 + (752908652200215478896545921560059634947583734251*i + 895025848404241788676589141949265726225025274828)*z + 616166846676061100095127430332774495441755394581*i + 889255287756560160849573452496040306572455585395 g^(p^3-1) = (104394624787325768580669553622186436218955478289*i + 115819303537523597905318969503744744093839976631)*z^5 + (936139532752018606210117538341071166608350863727*i + 1112461501477852426997296904293113682769958766774)*z^4 + (16096452051693414368204074112936247575117516479*i + 1065034950111683488033841963064789042972883100648)*z^3 + (1461250712521757914096163977919390965791567384854*i + 244970246429598896709919226014995935882868586820)*z^2 + (918950212845529194792417267092990996578910875523*i + 1031693940340614897771168876225173895647363031580)*z + 70531838254713320311910681453696149021611168678*i + 1398834529239549175114023695374662216300177982250 g^(p^4-1) = (239992120004822890833891381247917559884462910146*i + 1261034152320018208158235891540018946925039727848)*z^5 + (208862746409985936264596906005246300650626057668*i + 565357618624448102009812399714319141941210262495)*z^4 + (595745447214476719160235373370748785166782609870*i + 986026099281262892060830880798276518956378006484)*z^3 + (102466015727909809069678183573961098275241875184*i + 940151235303014027110245043307168921216332322768)*z^2 + (532578885638091702929347739231631219177474392942*i + 1079498624912683964120889877510481819621575543210)*z + 1220974658126123565670019751888593761044993991185*i + 211811947018249923223576976718524137383767937965 g^(p^5-1) = (867021477085834981410763757761338394476845952591*i + 454054155590694389565022063266765920943919676451)*z^5 + (1152066755303591353030630678326466707035079216340*i + 651594961867943713451573799740600135483141099859)*z^4 + (1412000232283498703118073208653703846312267368009*i + 640791609345553839901444003801897367634673449741)*z^3 + (42188653754860119450135480418886642353674596443*i + 562408702302581793941475002165818938687417258094)*z^2 + (369929395625514333386658103827181783417985044249*i + 884065031781026990327934242273418463887620221812)*z + 1168565180281449525586662304102209555431570678308*i + 503582600681258840386198926408111711693060441073 g^(p^6-1) = (1255732940148476641067997017126320140148344109398*i + 847551398103174556962453688020722111841747485160)*z^5 + (1428633672987200370786940114820930632812523417902*i + 395314511348759603485431448487808080038157695517)*z^4 + (579522520845159075895473597057122668600298541282*i + 417120518253923879121125020645774349311067159688)*z^3 + (377475555156278737030694625815440794829353520650*i + 60549325767510300874857944491779135450365424950)*z^2 + (557025596185007241338967714827866064399666079379*i + 806992942862947949184596793356606277531549429952)*z + 826543425518806376820411130172467469135354246707*i + 546674685375929320723432354108401487302836191956 g^(p^7-1) = (1126502434067055814843415612681151892011263175870*i + 1176002543018106084799354481790549535216532720012)*z^5 + (1357641212310828158490838523101409841505272979404*i + 678284707222532143244512307089503927268688656266)*z^4 + (1013787658638365632095096392271389651254442656104*i + 937558728589440691890694680576395380640252677216)*z^3 + (395448289533461064020784297720245760316432679996*i + 256449527939203012159253238775693631557658125017)*z^2 + (636312304664720308322836076073577328711971599964*i + 1127749216378622540139671105934047284890415686175)*z + 292936444215340739558786285818575938285688212511*i + 503582600681258840386198926408111711693060441073 g^(p^8-1) = (1332707151976645874319910631674895937140726222362*i + 778684477701470942972407195056892496929203980614)*z^5 + (712683874015386917422759238642315687393680373361*i + 1227924079489556276578945161248737876531470377844)*z^4 + (865756177282313545985213216550036708550476280949*i + 475475525215527373084617709122508974760880884335)*z^3 + (375673708125247397107051660343678979770619117775*i + 114357758639702456761704170918133809768845643944)*z^2 + (49612337403739495964322572393995703394006155135*i + 356186543237717966584250070352729894597853712672)*z + 1220974658126123565670019751888593761044993991185*i + 211811947018249923223576976718524137383767937965 g^(p^9-1) = (237809585875741834991576153007396144742157655788*i + 986879980740920043509509121885997640997209851340)*z^5 + (936139532752018606210117538341071166608350863727*i + 349040123018937838148151685627671810947300124045)*z^4 + (85356971980052752574778474541031548005273362196*i + 88190465811788712462054459750301281251950202292)*z^3 + (244970246429598896709919226014995935882868586820*i + 1461250712521757914096163977919390965791567384854)*z^2 + (1410004675290216030552443513596942046089712163085*i + 1439129478845974241490332140488129432068506372977)*z + 1390969786242076944833537908467089344695647722141*i + 1398834529239549175114023695374662216300177982250 g^(p^10-1) = (1246594770081788738724721073099293471059972739097*i + 382768229729169407804348131823625880241956057964)*z^5 + (768981093027284993021976084253246113298674239447*i + 366805892020891465272327059977891336685100803821)*z^4 + (852655199839699940147329355094933876743133342055*i + 690019603771769684094818718293582964103437250996)*z^3 + (1053382421052383953314567950889099073557110465840*i + 1096378886800018241385817718143474881449413155427)*z^2 + (782563351215281329069532267032242743526101428229*i + 932419608797292263964281326767865562767187037964)*z + 616166846676061100095127430332774495441755394581*i + 889255287756560160849573452496040306572455585395 g^(p^11-1) = (912920840895124334529447447866168063247040392173*i + 9873595412849185996452747082046684640971167294)*z^5 + (2545939521711331370283095426719602351824833799*i + 1344046085896870808486244472445559740184244945852)*z^4 + (623883581320924596298676085823779458446779540540*i + 675006931343865904342646689102926566985749184396)*z^3 + (1402153388011088316650991466733555001741554598260*i + 247983074468381684089738348912296989265022486477)*z^2 + (275595554457354270312980246628813051104336514134*i + 1156378314745152631164618943690763665073246273612)*z + 258663318293302755002919638469809294273596765675*i + 795069427093050539880893267448617311116978407546 g^(p^12-1) = 1 g^n = 1 Total time: 2.680 seconds, Total memory usage: 3.34MB '84.56.2' ************** MAGMA ***************** Host 84.56.209.34 (84.56.209.34) Time: Tue Dec 13 09:09:16 2005 Input: R:=PolynomialRing(GF(2)); definingPolynomial:=x^155+x^62+1; K:=ext; CompositionFactors(155); Output: Magma V2.11-10 Tue Dec 13 2005 09:09:15 on modular [Seed = 3010340552] ------------------------------------- >> CompositionFactors(155);; ^ Runtime error in 'CompositionFactors': Bad argument types Argument types given: RngIntElt Total time: 0.190 seconds, Total memory usage: 3.24MB '84.56.2' ************** MAGMA ***************** Host 84.56.209.34 (84.56.209.34) Time: Tue Dec 13 09:07:43 2005 Input: R:=PolynomialRing(GF(2)); definingPolynomial:=x^155+x^62+1; K:=ext; CompositionFactors(definingPolynomial); Output: Magma V2.11-10 Tue Dec 13 2005 09:07:43 on modular [Seed = 3741468928] ------------------------------------- >> CompositionFactors(definingPolynomial);; ^ Runtime error in 'CompositionFactors': Bad argument types Argument types given: RngUPolElt[FldFin] Total time: 0.180 seconds, Total memory usage: 3.24MB '84.56.2' ************** MAGMA ***************** Host 84.56.209.34 (84.56.209.34) Time: Tue Dec 13 09:04:55 2005 Input: R:=PolynomialRing(GF(2)); definingPolynomial:=x^155+x^62+1; K:=ext; Output: Magma V2.11-10 Tue Dec 13 2005 09:04:55 on modular [Seed = 3657911469] ------------------------------------- Total time: 0.190 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Tue Dec 13 09:03:36 2005 Input: p := 1461501624496790265145448589920785493717258890819; n := 1461501624496790265145447380994971188499300027613; Fp := GF(p); Fp2 := ExtensionField; lambda := 2; mu := 1 + i; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; b := Fp!3; y0 := Fp!2; // -Sqrt(1 + b); E := EllipticCurve([0, b]); G := E![1, y0]; Et := EllipticCurve([0, b/xi]); xt := i + 2; /* xt := i; while not IsSquare(xt^3 + b/xi) do xt +:= 1; end while; "xt =", xt; */ yt := Sqrt(xt^3 + b/xi); Gt := Et![xt, yt]; k := 12; assert (p^k - 1) mod n eq 0; chi := (p^k - 1) div n; Ek := EllipticCurve([0, Fp12!b]); g := function(U, V, Q) if IsZero(U) or IsZero(V) or (U eq -V) or IsZero(Q) then return Fp12!1; end if; m := (U eq V) select 3*U[1]^2/(2*U[1]) else (V[2] - U[2])/(V[1] - U[1]); return m*(Q[1] - U[1]) + U[1] - Q[2]; end function; length := function(r) n := 0; v := 1; while v le r do n +:= 1; v +:= v; end while; return n; end function; bit := function(r, i) return (r div 2^i) mod 2; end function; miller := function(r, P, Q) f := 1; A := P; for i := length(r) - 1 to 0 by -1 do f := f^2*g(A, A, Q); A := 2*A; if bit(r, i) eq 1 then f *:= g(A, P, Q); A +:= P; end if; end for; return f; end function; tate := function(P, Qt) Q := Ek![Qt[1]*z^2, Qt[2]*z^3]; "P =", P; "Q' =", Qt; "Q =", Q; return miller(n, P, Q)^chi; end function; "miller =", miller(n, G, Gt); "tate =", tate(G, Gt); Output: Magma V2.11-10 Tue Dec 13 2005 09:03:36 on modular [Seed = 3574088221] ------------------------------------- miller = 162471042542776450109546911262159626540194874282*i + 1254993595675210729223411613343751283964425868707 P = (1 : 2 : 1) Q' = (i + 2 : 363551601938441395682953698000848166036751722710*i + 87218401513221303182445826318865898589777766307 : 1) Q = ((i + 2)*z^2 : (363551601938441395682953698000848166036751722710*i + 87218401513221303182445826318865898589777766307)*z^3 : 1) tate = (1337592616974074632841214156993795052557527470389*i + 823993614240290083038167890551792145239917806334)*z^5 + (688703507104756931070001853914995131005373541312*i + 1002621352581475249329708955843984496240073153982)*z^4 + (852655199839699940147329355094933876743133342055*i + 690019603771769684094818718293582964103437250996)*z^3 + (1392434018234511176428044389335844610478592711217*i + 473331107538064635840791020785366158595476919127)*z^2 + (1387531245578083722324818991249268608960832619158*i + 1095557791792046477650026711124439698442305468846)*z + 616166846676061100095127430332774495441755394581*i + 889255287756560160849573452496040306572455585395 Total time: 0.630 seconds, Total memory usage: 3.34MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Tue Dec 13 08:59:32 2005 Input: p := 1461501624496790265145448589920785493717258890819; n := 1461501624496790265145447380994971188499300027613; Fp := GF(p); Fp2 := ExtensionField; lambda := 2; mu := 1 + i; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; b := Fp!3; y0 := Fp!2; // -Sqrt(1 + b); E := EllipticCurve([0, b]); G := E![1, y0]; Et := EllipticCurve([0, b/xi]); xt := i + 2; /* xt := i; while not IsSquare(xt^3 + b/xi) do xt +:= 1; end while; "xt =", xt; */ yt := Sqrt(xt^3 + b/xi); Gt := Et![xt, yt]; k := 12; assert (p^k - 1) mod n eq 0; z := (p^k - 1) div n; g := function(U, V, Q) if IsZero(U) or IsZero(V) or (U eq -V) or IsZero(Q) then return Fp12!1; end if; m := (U eq V) select 3*U[1]^2/(2*U[1]) else (V[2] - U[2])/(V[1] - U[1]); return m*(Q[1] - U[1]) + U[1] - Q[2]; end function; length := function(r) n := 0; v := 1; while v le r do n +:= 1; v +:= v; end while; return n; end function; bit := function(r, i) return (r div 2^i) mod 2; end function; miller := function(r, P, Q) f := 1; A := P; for i := length(r) - 1 to 0 by -1 do f := f^2*g(A, A, Q); A := 2*A; if bit(r, i) eq 1 then f *:= g(A, P, Q); A +:= P; end if; end for; return f; end function; tate := function(P, Q) return miller(n, P, Q)^z; end function; "P =", G; "Q =", Gt; "miller =", miller(n, G, Gt); "tate =", tate(G, Gt); Output: Magma V2.11-10 Tue Dec 13 2005 08:59:31 on modular [Seed = 4013064028] ------------------------------------- P = (1 : 2 : 1) Q = (i + 2 : 363551601938441395682953698000848166036751722710*i + 87218401513221303182445826318865898589777766307 : 1) miller = 162471042542776450109546911262159626540194874282*i + 1254993595675210729223411613343751283964425868707 tate = 1 Total time: 0.260 seconds, Total memory usage: 3.34MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Tue Dec 13 08:58:31 2005 Input: p := 1461501624496790265145448589920785493717258890819; n := 1461501624496790265145447380994971188499300027613; Fp := GF(p); Fp2 := ExtensionField; lambda := 2; mu := 1 + i; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; b := Fp!3; y0 := Fp!2; // -Sqrt(1 + b); E := EllipticCurve([0, b]); G := E![1, y0]; Et := EllipticCurve([0, b/xi]); xt := i + 2; /* xt := i; while not IsSquare(xt^3 + b/xi) do xt +:= 1; end while; "xt =", xt; */ yt := Sqrt(xt^3 + b/xi); Gt := Et![xt, yt]; k := 12; assert (p^k - 1) mod n eq 0; z := (p^k - 1) div n; g := function(U, V, Q) if IsZero(U) or IsZero(V) or (U eq -V) or IsZero(Q) then return Fp12!1; end if; m := (U eq V) select 3*U[1]^2/(2*U[1]) else (V[2] - U[2])/(V[1] - U[1]); return m*(Q[1] - U[1]) + U[1] - Q[2]; end function; length := function(r) n := 0; v := 1; while v le r do n +:= 1; v +:= v; end while; return n; end function; bit := function(r, i) return (r div 2^i) mod 2; end function; miller := function(r, P, Q) f := 1; A := P; for i := length(r) - 1 to 0 by -1 do f := f^2*g(A, A, Q); A := 2*A; if bit(r, i) eq 1 then f *:= g(A, P, Q); A +:= P; end if; end for; return f; end function; tate := function(P, Q) return miller(n, P, Q)^z; end function; "miller =", miller(n, G, Gt); "tate =", tate(G, Gt); Output: Magma V2.11-10 Tue Dec 13 2005 08:58:31 on modular [Seed = 3929504285] ------------------------------------- miller = 694976144959185627384757159853755106109815530714*i + 1311958513305109301215342597778813598163540723157 tate = 1 Total time: 0.250 seconds, Total memory usage: 3.34MB '84.56.2' ************** MAGMA ***************** Host 84.56.209.34 (84.56.209.34) Time: Tue Dec 13 08:58:30 2005 Input: R:=PolynomialRing(GF(2)); Output: Magma V2.11-10 Tue Dec 13 2005 08:58:23 on modular [Seed = 3845682666] ------------------------------------- Total time: 0.210 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Tue Dec 13 08:50:40 2005 Input: p := 1461501624496790265145448589920785493717258890819; n := 1461501624496790265145447380994971188499300027613; Fp := GF(p); Fp2 := ExtensionField; lambda := 2; mu := 1 + i; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; b := Fp!3; y0 := Fp!2; // -Sqrt(1 + b); E := EllipticCurve([0, b]); G := E![1, y0]; Et := EllipticCurve([0, b/xi]); xt := lambda^3; /* xt := 1; while not IsSquare(xt^3 + b/xi) do xt +:= 1; end while; "xt =", xt; */ yt := Sqrt(xt^3 + b/xi); Gt := Et![xt, yt]; k := 12; assert (p^k - 1) mod n eq 0; z := (p^k - 1) div n; g := function(U, V, Q) if IsZero(U) or IsZero(V) or (U eq -V) or IsZero(Q) then return Fp12!1; end if; m := (U eq V) select 3*U[1]^2/(2*U[1]) else (V[2] - U[2])/(V[1] - U[1]); return m*(Q[1] - U[1]) + U[1] - Q[2]; end function; length := function(r) n := 0; v := 1; while v le r do n +:= 1; v +:= v; end while; return n; end function; bit := function(r, i) return (r div 2^i) mod 2; end function; miller := function(r, P, Q) f := 1; A := P; for i := length(r) - 1 to 0 by -1 do f := f^2*g(A, A, Q); A := 2*A; if bit(r, i) eq 1 then f *:= g(A, P, Q); A +:= P; end if; end for; return f; end function; tate := function(P, Q) return miller(n, P, Q)^z; end function; "miller =", miller(n, G, Gt); "tate =", tate(G, Gt); Roots(24+25*i, 3); xt := i; while not IsSquare(xt^3 + b/xi) do xt +:= 1; end while; "xt =", xt; Output: Magma V2.11-10 Tue Dec 13 2005 08:50:18 on modular [Seed = 4179913490] ------------------------------------- miller = 34978940859506703873739429950455443570933508707*i + 411686203016070577518067565293188378259628682113 tate = 1 >> Roots(24+25*i, 3); ^ Runtime error in 'Roots': Bad argument types Argument types given: FldFinElt, RngIntElt xt = i + 2 Total time: 0.300 seconds, Total memory usage: 3.34MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Tue Dec 13 08:42:17 2005 Input: p := 1461501624496790265145448589920785493717258890819; n := 1461501624496790265145447380994971188499300027613; Fp := GF(p); Fp2 := ExtensionField; lambda := 2; mu := 1 + i; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; b := Fp!3; y0 := Fp!2; // -Sqrt(1 + b); E := EllipticCurve([0, b]); G := E![1, y0]; Et := EllipticCurve([0, b/xi]); xt := lambda^3; /* xt := 1; while not IsSquare(xt^3 + b/xi) do xt +:= 1; end while; "xt =", xt; */ yt := Sqrt(xt^3 + b/xi); Gt := Et![xt, yt]; k := 12; assert (p^k - 1) mod n eq 0; z := (p^k - 1) div n; g := function(U, V, Q) if IsZero(U) or IsZero(V) or (U eq -V) or IsZero(Q) then return Fp12!1; end if; m := (U eq V) select 3*U[1]^2/(2*U[1]) else (V[2] - U[2])/(V[1] - U[1]); return m*(Q[1] - U[1]) + U[1] - Q[2]; end function; length := function(r) n := 0; v := 1; while v le r do n +:= 1; v +:= v; end while; return n; end function; bit := function(r, i) return (r div 2^i) mod 2; end function; miller := function(r, P, Q) f := 1; A := P; for i := length(r) - 1 to 0 by -1 do f := f^2*g(A, A, Q); A := 2*A; if bit(r, i) eq 1 then f *:= g(A, P, Q); A +:= P; end if; end for; return f; end function; tate := function(P, Q) return miller(n, P, Q)^z; end function; "miller =", miller(n, P, Q); "tate =", tate(G, Gt); Output: Magma V2.11-10 Tue Dec 13 2005 08:42:01 on modular [Seed = 4046224683] ------------------------------------- >> "miller =", miller(n, P, Q); ^ User error: Identifier 'P' has not been declared or assigned tate = 1 Total time: 0.270 seconds, Total memory usage: 3.34MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Tue Dec 13 08:37:54 2005 Input: p := 1461501624496790265145448589920785493717258890819; n := 1461501624496790265145447380994971188499300027613; Fp := GF(p); Fp2 := ExtensionField; lambda := 2; mu := 1 + i; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; b := Fp!3; y0 := Fp!2; // -Sqrt(1 + b); E := EllipticCurve([0, b]); G := E![1, y0]; Et := EllipticCurve([0, b/xi]); xt := lambda^3; /* xt := 1; while not IsSquare(xt^3 + b/xi) do xt +:= 1; end while; "xt =", xt; */ yt := Sqrt(xt^3 + b/xi); Gt := Et![xt, yt]; g := function(U, V, Q) if IsZero(U) or IsZero(V) or (U eq -V) or IsZero(Q) then return Fp12!1; end if; m := (U eq V) select 3*U[1]^2/(2*U[1]) else (V[2] - U[2])/(V[1] - U[1]); return m*(Q[1] - U[1]) + U[1] - Q[2]; end function; length := function(r) n := 0; v := 1; while v le r do n +:= 1; v +:= v; end while; return n; end function; bit := function(r, i) return (r div 2^i) mod 2; end function; miller := function(r, P, Q) f := 1; A := P; for i := length(r) - 1 to 0 by -1 do f := f^2*g(A, A, Q); A := 2*A; if bit(r, i) eq 1 then f *:= g(A, P, Q); A +:= P; end if; end for; return f; end function; tate := function(r, q, k, P, Q) // assert (q^k - 1) mod r eq 0; z := (q^k - 1) div r; return miller(r, P, Q)^z; end function; tate(n, p, 12, G, Gt); Output: Magma V2.11-10 Tue Dec 13 2005 08:37:53 on modular [Seed = 516133855] ------------------------------------- 1 Total time: 0.240 seconds, Total memory usage: 3.34MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Tue Dec 13 08:36:36 2005 Input: p := 1461501624496790265145448589920785493717258890819; n := 1461501624496790265145447380994971188499300027613; Fp := GF(p); Fp2 := ExtensionField; lambda := 2; mu := 1 + i; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; b := Fp!3; y0 := Fp!2; // -Sqrt(1 + b); E := EllipticCurve([0, b]); G := E![1, y0]; Et := EllipticCurve([0, b/xi]); xt := lambda^3; /* xt := 1; while not IsSquare(xt^3 + b/xi) do xt +:= 1; end while; "xt =", xt; */ yt := Sqrt(xt^3 + b/xi); Gt := Et![xt, yt]; g := function(U, V, Q) if IsZero(U) or IsZero(V) or (U eq -V) or IsZero(Q) then return Fp12!1; end if; m := (U eq V) select 3*U[1]^2/(2*U[1]) else (V[2] - U[2])/(V[1] - U[1]); return m*(Q[1] - U[1]) + U[1] - Q[2]; end function; length := function(r) n := 0; v := 1; while v le r do n +:= 1; v +:= v; end while; return n; end function; bit := function(r, i) return (r div 2^i) mod 2; end function; miller := function(r, P, Q) f := 1; A := P; for i := length(r) - 1 to 0 by -1 do f := f^2*g(A, A, Q); A := 2*A; if bit(r, i) eq 1 then f *:= g(A, P, Q); A +:= P; end if; end for; return f; end function; tate := function(r, q, k, P, Q) m := miller(r, P, Q); assert (q^k - 1) mod r eq 0; z := (q^k - 1) div r; return m^z; end function; tate(n, p, 12, G, Gt); Output: Magma V2.11-10 Tue Dec 13 2005 08:36:33 on modular [Seed = 415601929] ------------------------------------- 1 Total time: 0.260 seconds, Total memory usage: 3.34MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Tue Dec 13 08:34:42 2005 Input: p := 1461501624496790265145448589920785493717258890819; n := 1461501624496790265145447380994971188499300027613; Fp := GF(p); Fp2 := ExtensionField; lambda := 2; mu := 1 + i; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; b := Fp!3; y0 := Fp!2; // -Sqrt(1 + b); E := EllipticCurve([0, b]); G := E![1, y0]; Et := EllipticCurve([0, b/xi]); xt := lambda^3; /* xt := 1; while not IsSquare(xt^3 + b/xi) do xt +:= 1; end while; "xt =", xt; */ yt := Sqrt(xt^3 + b/xi); Gt := Et![xt, yt]; g := function(U, V, Q) if IsZero(U) or IsZero(V) or (U eq -V) or IsZero(Q) then return Fp12!1; end if; m := (U eq V) select 3*U[1]^2/(2*U[1]) else (V[2] - U[2])/(V[1] - U[1]); return m*(Q[1] - U[1]) + U[1] - Q[2]; end function; length := function(r) n := 0; v := 1; while v le r do n +:= 1; v +:= v; end while; return n; end function; bit := function(r, i) return (r div 2^i) mod 2; end function; miller := function(r, P, Q) f := 1; A := P; for i := length(r) - 1 to 0 by -1 do f := f^2*g(A, A, Q); A := 2*A; if bit(r, i) eq 1 then f *:= g(A, P, Q); A +:= P; end if; end for; return f; end function; tate := function(r, q, k, P, Q) return miller(r, P, Q)^((q^k - 1)/r); end function; tate(n, p, 12, G, Gt); Output: Magma V2.11-10 Tue Dec 13 2005 08:34:23 on modular [Seed = 332307842] ------------------------------------- tate( r: 1461501624496790265145447380994971188499300027613, q: 1461501624496790265145448589920785493717258890819, k: 12, P: (1 : 2 : 1), Q: (8 : 645238442624673913635245604741558906439788126004*i + 11... ) >> return miller(r, P, Q)^((q^k - 1)/r); ^ Runtime error in '^': Bad argument types Argument types given: FldFinElt, FldFinElt Total time: 0.290 seconds, Total memory usage: 3.34MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Tue Dec 13 08:32:15 2005 Input: p := 1461501624496790265145448589920785493717258890819; n := 1461501624496790265145447380994971188499300027613; Fp := GF(p); Fp2 := ExtensionField; lambda := 2; mu := 1 + i; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; b := Fp!3; y0 := Fp!2; // -Sqrt(1 + b); E := EllipticCurve([0, b]); G := E![1, y0]; Et := EllipticCurve([0, b/xi]); xt := lambda^3; /* xt := 1; while not IsSquare(xt^3 + b/xi) do xt +:= 1; end while; "xt =", xt; */ yt := Sqrt(xt^3 + b/xi); Gt := Et![xt, yt]; g := function(U, V, Q) if IsZero(U) or IsZero(V) or (U eq -V) or IsZero(Q) then return Fp12!1; end if; m := (U eq V) select 3*U[1]^2/(2*U[1]) else (V[2] - U[2])/(V[1] - U[1]); return m*(Q[1] - U[1]) + U[1] - Q[2]; end function; length := function(r) n := 0; v := 1; while v le r do n +:= 1; v +:= v; end while; return n; end function; bit := function(r, i) return (r div 2^i) mod 2; end function; miller := function(r, P, Q) f := 1; A := P; for i := length(r) - 1 to 0 by -1 do f := f^2*g(A, A, Q); A := 2*A; if bit(r, i) = 1 then f *:= g(A, P, Q); A +:= P; end if; end for; return f; end function; tate := function(r, q, k, P, Q) return miller(r, P, Q)^((q^k - 1)/r); end function; tate(n, p, 12, G, Gt); Output: Magma V2.11-10 Tue Dec 13 2005 08:31:57 on modular [Seed = 770745107] ------------------------------------- tate( r: 1461501624496790265145447380994971188499300027613, q: 1461501624496790265145448589920785493717258890819, k: 12, P: (1 : 2 : 1), Q: (8 : 816263181872116351510202985179226587277470764815*i + 29... ) miller( r: 1461501624496790265145447380994971188499300027613, P: (1 : 2 : 1), Q: (8 : 816263181872116351510202985179226587277470764815*i + 29... ) >> if bit(r, i) = 1 then ^ Runtime error in if: Logical expected Total time: 0.319 seconds, Total memory usage: 3.34MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Tue Dec 13 08:19:36 2005 Input: p := 1461501624496790265145448589920785493717258890819; Fp := GF(p); Fp2 := ExtensionField; lambda := 2; mu := 1 + i; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; b := Fp!3; y0 := Fp!2; // -Sqrt(1 + b); E := EllipticCurve([0, b]); G := E![1, y0]; Et := EllipticCurve([0, b/xi]); xt := 2; while not IsSquare(xt^3 + b/xi) do xt +:= 1; end while; "xt =", xt; Gt := Et![xt, Sqrt(xt^3 + b/xi)]; Gt; g := function(U, V, Q) if IsZero(U) or IsZero(V) or (U eq -V) or IsZero(Q) then return Fp12!1; end if; m := (U eq V) select 3*U[1]^2/(2*U[1]) else (V[2] - U[2])/(V[1] - U[1]); return m*(Q[1] - U[1]) + U[1] - Q[2]; end function; length := function(r) n := 0; v := 1; while v le r do n +:= 1; v +:= v; end while; return n; end function; bit := function(r, i) return (r div 2^i) mod 2; end function; miller := function(r, P, Q) f := 1; A := P; for i := length(r) - 1 to 0 by -1 do f := f^2*g(A, A, Q); A := 2*A; if bit(r, i) = 1 then f *:= g(A, P, Q); A +:= P; end if; end for; return f; end function; tate := function(r, q, k, P, Q) return miller(r, P, Q)^((q^k - 1)/r); end function; Output: Magma V2.11-10 Tue Dec 13 2005 08:19:24 on modular [Seed = 620342536] ------------------------------------- xt = 8 (8 : 816263181872116351510202985179226587277470764815*i + 295865244505705705023665406736615173923424579851 : 1) Total time: 0.260 seconds, Total memory usage: 3.34MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Tue Dec 13 08:17:14 2005 Input: p := 1461501624496790265145448589920785493717258890819; Fp := GF(p); Fp2 := ExtensionField; lambda := 2; mu := 1 + i; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; b := Fp!3; y0 := Fp!2; // -Sqrt(1 + b); E := EllipticCurve([0, b]); G := E![1, y0]; Et := EllipticCurve([0, b/xi]); Gt := Et![1, Sqrt(1 + b/xi)]; Gt; g := function(U, V, Q) if IsZero(U) or IsZero(V) or (U eq -V) or IsZero(Q) then return Fp12!1; end if; m := (U eq V) select 3*U[1]^2/(2*U[1]) else (V[2] - U[2])/(V[1] - U[1]); return m*(Q[1] - U[1]) + U[1] - Q[2]; end function; length := function(r) n := 0; v := 1; while v le r do n +:= 1; v +:= v; end while; return n; end function; bit := function(r, i) return (r div 2^i) mod 2; end function; miller := function(r, P, Q) f := 1; A := P; for i := length(r) - 1 to 0 by -1 do f := f^2*g(A, A, Q); A := 2*A; if bit(r, i) = 1 then f *:= g(A, P, Q); A +:= P; end if; end for; return f; end function; tate := function(r, q, k, P, Q) return miller(r, P, Q)^((q^k - 1)/r); end function; Output: Magma V2.11-10 Tue Dec 13 2005 08:17:11 on modular [Seed = 1038137242] ------------------------------------- >> Gt := Et![1, Sqrt(1 + b/xi)]; ^ Runtime error in 'Sqrt': Argument has no square root >> Gt; ^ User error: Identifier 'Gt' has not been declared or assigned Total time: 0.240 seconds, Total memory usage: 3.34MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Tue Dec 13 08:15:08 2005 Input: p := 1461501624496790265145448589920785493717258890819; Fp := GF(p); Fp2 := ExtensionField; lambda := 2; mu := 1 + i; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; b := Fp!3; y0 := Fp!2; // -Sqrt(1 + b); E := EllipticCurve([0, b]); G := E![1, y0]; Et := EllipticCurve([0, b/xi]); Et; g := function(U, V, Q) if IsZero(U) or IsZero(V) or (U eq -V) or IsZero(Q) then return Fp12!1; end if; m := (U eq V) select 3*U[1]^2/(2*U[1]) else (V[2] - U[2])/(V[1] - U[1]); return m*(Q[1] - U[1]) + U[1] - Q[2]; end function; length := function(r) n := 0; v := 1; while v le r do n +:= 1; v +:= v; end while; return n; end function; bit := function(r, i) return (r div 2^i) mod 2; end function; miller := function(r, P, Q) f := 1; A := P; for i := length(r) - 1 to 0 by -1 do f := f^2*g(A, A, Q); A := 2*A; if bit(r, i) = 1 then f *:= g(A, P, Q); A +:= P; end if; end for; return f; end function; tate := function(r, q, k, P, Q) return miller(r, P, Q)^((q^k - 1)/r); end function; Output: Magma V2.11-10 Tue Dec 13 2005 08:15:05 on modular [Seed = 871282970] ------------------------------------- Elliptic Curve defined by y^2 = x^3 + (24*i + 1461501624496790265145448589920785493717258890795) over GF(1461501624496790265145448589920785493717258890819^2) Total time: 0.230 seconds, Total memory usage: 3.34MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Tue Dec 13 08:14:46 2005 Input: p := 1461501624496790265145448589920785493717258890819; Fp := GF(p); Fp2 := ExtensionField; lambda := 2; mu := 1 + i; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; b := Fp!3; y0 := Fp!2; // -Sqrt(1 + b); E := EllipticCurve([0, b]); G := E![1, y0]; G; g := function(U, V, Q) if IsZero(U) or IsZero(V) or (U eq -V) or IsZero(Q) then return Fp12!1; end if; m := (U eq V) select 3*U[1]^2/(2*U[1]) else (V[2] - U[2])/(V[1] - U[1]); return m*(Q[1] - U[1]) + U[1] - Q[2]; end function; length := function(r) n := 0; v := 1; while v le r do n +:= 1; v +:= v; end while; return n; end function; bit := function(r, i) return (r div 2^i) mod 2; end function; miller := function(r, P, Q) f := 1; A := P; for i := length(r) - 1 to 0 by -1 do f := f^2*g(A, A, Q); A := 2*A; if bit(r, i) = 1 then f *:= g(A, P, Q); A +:= P; end if; end for; return f; end function; tate := function(r, q, k, P, Q) return miller(r, P, Q)^((q^k - 1)/r); end function; Output: Magma V2.11-10 Tue Dec 13 2005 08:14:43 on modular [Seed = 1318306941] ------------------------------------- (1 : 2 : 1) Total time: 0.230 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Tue Dec 13 08:13:39 2005 Input: p := 1461501624496790265145448589920785493717258890819; Fp := GF(p); Fp2 := ExtensionField; lambda := 2; mu := 1 + i; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; b := Fp!3; y0 := Fp!2; // -Sqrt(1 + b); E := EllipticCurve([0, b]); G := E![1, y0]; G; g := function(U, V, Q) if IsZero(U) or IsZero(V) or (U eq -V) or IsZero(Q) then return Fp12!1; end if; m := (U eq V) select 3*U[1]^2/(2*U[1]) else (V[2] - U[2])/(V[1] - U[1]); return m*(Q[1] - U[1]) + U[1] - Q[2]; end function; length := function(r) n := 0; v := 1; while v le r do n +:= 1; v +:= v; end while; return n; end function; bit := function(r, i) return (r div 2^i) mod 2; end function; miller := function(r, P, Q) f := 1; A := P; for i := length(r) - 1 to 0 by -1 do f := f^2*g(A, A, Q); A := 2*A; if bit(r, i) = 1 then f *:= g(A, P, Q); A +:= P; end if; end for; return f; end function; tate := function(r, q, k, P, Q) return miller(r, P, Q)^((q^k - 1)/r); end function; Output: Magma V2.11-10 Tue Dec 13 2005 08:13:35 on modular [Seed = 1234485958] ------------------------------------- (1 : 2 : 1) Total time: 0.220 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Tue Dec 13 08:12:47 2005 Input: p := 1461501624496790265145448589920785493717258890819; Fp := GF(p); Fp2 := ExtensionField; lambda := 2; mu := 1 + i; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; b := Fp!3; E := EllipticCurve([0, b]); y0 := -Sqrt(1 + b); G := E![1, y0]; G; g := function(U, V, Q) if IsZero(U) or IsZero(V) or (U eq -V) or IsZero(Q) then return Fp12!1; end if; m := (U eq V) select 3*U[1]^2/(2*U[1]) else (V[2] - U[2])/(V[1] - U[1]); return m*(Q[1] - U[1]) + U[1] - Q[2]; end function; length := function(r) n := 0; v := 1; while v le r do n +:= 1; v +:= v; end while; return n; end function; bit := function(r, i) return (r div 2^i) mod 2; end function; miller := function(r, P, Q) f := 1; A := P; for i := length(r) - 1 to 0 by -1 do f := f^2*g(A, A, Q); A := 2*A; if bit(r, i) = 1 then f *:= g(A, P, Q); A +:= P; end if; end for; return f; end function; tate := function(r, q, k, P, Q) return miller(r, P, Q)^((q^k - 1)/r); end function; Output: Magma V2.11-10 Tue Dec 13 2005 08:12:44 on modular [Seed = 1552013280] ------------------------------------- (1 : 2 : 1) Total time: 0.220 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Tue Dec 13 08:12:00 2005 Input: p := 1461501624496790265145448589920785493717258890819; Fp := GF(p); Fp2 := ExtensionField; lambda := 2; mu := 1 + i; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; b := Fp!3; E := EllipticCurve([0, b]); y0 := -Sqrt(1 + b); E; y0; g := function(U, V, Q) if IsZero(U) or IsZero(V) or (U eq -V) or IsZero(Q) then return Fp12!1; end if; m := (U eq V) select 3*U[1]^2/(2*U[1]) else (V[2] - U[2])/(V[1] - U[1]); return m*(Q[1] - U[1]) + U[1] - Q[2]; end function; length := function(r) n := 0; v := 1; while v le r do n +:= 1; v +:= v; end while; return n; end function; bit := function(r, i) return (r div 2^i) mod 2; end function; miller := function(r, P, Q) f := 1; A := P; for i := length(r) - 1 to 0 by -1 do f := f^2*g(A, A, Q); A := 2*A; if bit(r, i) = 1 then f *:= g(A, P, Q); A +:= P; end if; end for; return f; end function; tate := function(r, q, k, P, Q) return miller(r, P, Q)^((q^k - 1)/r); end function; Output: Magma V2.11-10 Tue Dec 13 2005 08:11:57 on modular [Seed = 1468718266] ------------------------------------- Elliptic Curve defined by y^2 = x^3 + 3 over GF(1461501624496790265145448589920785493717258890819) 2 Total time: 0.240 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Tue Dec 13 08:11:47 2005 Input: p := 1461501624496790265145448589920785493717258890819; Fp := GF(p); Fp2 := ExtensionField; lambda := 2; mu := 1 + i; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; b := Fp!3; E := EllipticCurve([0, b]); y0 := p - Sqrt(1 + b); E; y0; g := function(U, V, Q) if IsZero(U) or IsZero(V) or (U eq -V) or IsZero(Q) then return Fp12!1; end if; m := (U eq V) select 3*U[1]^2/(2*U[1]) else (V[2] - U[2])/(V[1] - U[1]); return m*(Q[1] - U[1]) + U[1] - Q[2]; end function; length := function(r) n := 0; v := 1; while v le r do n +:= 1; v +:= v; end while; return n; end function; bit := function(r, i) return (r div 2^i) mod 2; end function; miller := function(r, P, Q) f := 1; A := P; for i := length(r) - 1 to 0 by -1 do f := f^2*g(A, A, Q); A := 2*A; if bit(r, i) = 1 then f *:= g(A, P, Q); A +:= P; end if; end for; return f; end function; tate := function(r, q, k, P, Q) return miller(r, P, Q)^((q^k - 1)/r); end function; Output: Magma V2.11-10 Tue Dec 13 2005 08:11:44 on modular [Seed = 1385159090] ------------------------------------- Elliptic Curve defined by y^2 = x^3 + 3 over GF(1461501624496790265145448589920785493717258890819) 2 Total time: 0.230 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Tue Dec 13 08:11:19 2005 Input: p := 1461501624496790265145448589920785493717258890819; Fp := GF(p); Fp2 := ExtensionField; lambda := 2; mu := 1 + i; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; b := Fp!3; E := EllipticCurve([0, b]); y0 := Sqrt(1 + b); E; y0; g := function(U, V, Q) if IsZero(U) or IsZero(V) or (U eq -V) or IsZero(Q) then return Fp12!1; end if; m := (U eq V) select 3*U[1]^2/(2*U[1]) else (V[2] - U[2])/(V[1] - U[1]); return m*(Q[1] - U[1]) + U[1] - Q[2]; end function; length := function(r) n := 0; v := 1; while v le r do n +:= 1; v +:= v; end while; return n; end function; bit := function(r, i) return (r div 2^i) mod 2; end function; miller := function(r, P, Q) f := 1; A := P; for i := length(r) - 1 to 0 by -1 do f := f^2*g(A, A, Q); A := 2*A; if bit(r, i) = 1 then f *:= g(A, P, Q); A +:= P; end if; end for; return f; end function; tate := function(r, q, k, P, Q) return miller(r, P, Q)^((q^k - 1)/r); end function; Output: Magma V2.11-10 Tue Dec 13 2005 08:11:16 on modular [Seed = 1840309865] ------------------------------------- Elliptic Curve defined by y^2 = x^3 + 3 over GF(1461501624496790265145448589920785493717258890819) 1461501624496790265145448589920785493717258890817 Total time: 0.240 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Tue Dec 13 08:10:59 2005 Input: p := 1461501624496790265145448589920785493717258890819; Fp := GF(p); Fp2 := ExtensionField; lambda := 2; mu := 1 + i; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; b := 3; E := EllipticCurve([0, b]); y0 := Sqrt(1 + b); E; y0; g := function(U, V, Q) if IsZero(U) or IsZero(V) or (U eq -V) or IsZero(Q) then return Fp12!1; end if; m := (U eq V) select 3*U[1]^2/(2*U[1]) else (V[2] - U[2])/(V[1] - U[1]); return m*(Q[1] - U[1]) + U[1] - Q[2]; end function; length := function(r) n := 0; v := 1; while v le r do n +:= 1; v +:= v; end while; return n; end function; bit := function(r, i) return (r div 2^i) mod 2; end function; miller := function(r, P, Q) f := 1; A := P; for i := length(r) - 1 to 0 by -1 do f := f^2*g(A, A, Q); A := 2*A; if bit(r, i) = 1 then f *:= g(A, P, Q); A +:= P; end if; end for; return f; end function; tate := function(r, q, k, P, Q) return miller(r, P, Q)^((q^k - 1)/r); end function; Output: Magma V2.11-10 Tue Dec 13 2005 08:10:55 on modular [Seed = 1756750663] ------------------------------------- Elliptic Curve defined by y^2 = x^3 + 3 over Rational Field 2.00000000000000000000000000000000000000 Total time: 0.250 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Tue Dec 13 08:08:18 2005 Input: p := 1461501624496790265145448589920785493717258890819; Fp := GF(p); Fp2 := ExtensionField; lambda := 2; mu := 1 + i; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; g := function(U, V, Q) if IsZero(U) or IsZero(V) or (U eq -V) or IsZero(Q) then return Fp12!1; end if; m := (U eq V) select 3*U[1]^2/(2*U[1]) else (V[2] - U[2])/(V[1] - U[1]); return m*(Q[1] - U[1]) + U[1] - Q[2]; end function; length := function(r) n := 0; v := 1; while v le r do n +:= 1; v +:= v; end while; return n; end function; bit := function(r, i) return (r div 2^i) mod 2; end function; miller := function(r, P, Q) f := 1; A := P; for i := length(r) - 1 to 0 by -1 do f := f^2*g(A, A, Q); A := 2*A; if bit(r, i) = 1 then f *:= g(A, P, Q); A +:= P; end if; end for; return f; end function; tate := function(r, q, k, P, Q) return miller(r, P, Q)^((q^k - 1)/r); end function; for r in [0..64] do r, ":", length(r); end for; Output: Magma V2.11-10 Tue Dec 13 2005 08:08:11 on modular [Seed = 2007692220] ------------------------------------- 0 : 0 1 : 1 2 : 2 3 : 2 4 : 3 5 : 3 6 : 3 7 : 3 8 : 4 9 : 4 10 : 4 11 : 4 12 : 4 13 : 4 14 : 4 15 : 4 16 : 5 17 : 5 18 : 5 19 : 5 20 : 5 21 : 5 22 : 5 23 : 5 24 : 5 25 : 5 26 : 5 27 : 5 28 : 5 29 : 5 30 : 5 31 : 5 32 : 6 33 : 6 34 : 6 35 : 6 36 : 6 37 : 6 38 : 6 39 : 6 40 : 6 41 : 6 42 : 6 43 : 6 44 : 6 45 : 6 46 : 6 47 : 6 48 : 6 49 : 6 50 : 6 51 : 6 52 : 6 53 : 6 54 : 6 55 : 6 56 : 6 57 : 6 58 : 6 59 : 6 60 : 6 61 : 6 62 : 6 63 : 6 64 : 7 Total time: 0.220 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Tue Dec 13 08:06:55 2005 Input: p := 1461501624496790265145448589920785493717258890819; Fp := GF(p); Fp2 := ExtensionField; lambda := 2; mu := 1 + i; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; g := function(U, V, Q) if IsZero(U) or IsZero(V) or (U eq -V) or IsZero(Q) then return Fp12!1; end if; m := (U eq V) select 3*U[1]^2/(2*U[1]) else (V[2] - U[2])/(V[1] - U[1]); return m*(Q[1] - U[1]) + U[1] - Q[2]; end function; length := function(r) n := 0; v := 1; while v le r do n +:= 1; v +:= v; end while; return n; end function; bit := function(r, i) return (r div 2^i) mod 2; end function; miller := function(r, P, Q) f := 1; A := P; for i := t - 1 to 0 by -1 do f := f^2*g(A, A, Q); A := 2*A; if bit(r, i) = 1 then f *:= g(A, P, Q); A +:= P; end if; end for; return f; end function; tate := function(r, q, k, P, Q) return miller(r, P, Q)^((q^k - 1)/r); end function; for r in [0..64] do r, ":", length(r); end for; Output: Magma V2.11-10 Tue Dec 13 2005 08:06:48 on modular [Seed = 2387806870] ------------------------------------- >> for i := t - 1 to 0 by -1 do ^ User error: Identifier 't' has not been declared or assigned >> return miller(r, P, Q)^((q^k - 1)/r); ^ User error: Identifier 'miller' has not been declared or assigned 0 : 0 1 : 1 2 : 2 3 : 2 4 : 3 5 : 3 6 : 3 7 : 3 8 : 4 9 : 4 10 : 4 11 : 4 12 : 4 13 : 4 14 : 4 15 : 4 16 : 5 17 : 5 18 : 5 19 : 5 20 : 5 21 : 5 22 : 5 23 : 5 24 : 5 25 : 5 26 : 5 27 : 5 28 : 5 29 : 5 30 : 5 31 : 5 32 : 6 33 : 6 34 : 6 35 : 6 36 : 6 37 : 6 38 : 6 39 : 6 40 : 6 41 : 6 42 : 6 43 : 6 44 : 6 45 : 6 46 : 6 47 : 6 48 : 6 49 : 6 50 : 6 51 : 6 52 : 6 53 : 6 54 : 6 55 : 6 56 : 6 57 : 6 58 : 6 59 : 6 60 : 6 61 : 6 62 : 6 63 : 6 64 : 7 Total time: 0.240 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Tue Dec 13 08:03:03 2005 Input: p := 1461501624496790265145448589920785493717258890819; Fp := GF(p); Fp2 := ExtensionField; lambda := 2; mu := 1 + i; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; g := function(U, V, Q) if IsZero(U) or IsZero(V) or (U eq -V) or IsZero(Q) then return Fp12!1; end if; m := (U eq V) select 3*U[1]^2/(2*U[1]) else (V[2] - U[2])/(V[1] - U[1]); return m*(Q[1] - U[1]) + U[1] - Q[2]; end function; bit := function(r, i) return (r div 2^i) mod 2; end function; miller := function(r, P, Q) f := 1; A := P; for i := t – 1 to 0 by –1 do f := f^2*g(A, A, Q); A := 2*A; if bit(r, i) = 1 then f *:= g(A, P, Q); A +:= P; end if; end for; return f; end function; Output: Magma V2.11-10 Tue Dec 13 2005 08:02:46 on modular [Seed = 2287275406] ------------------------------------- >> for i := t ^Ö 1 to 0 by ^Ö1 do ^ User error: Unknown character (ASCII value 150) >> f := f^2*g(A, A, Q); A := 2*A; ^ User error: Identifier 'f' has not been declared or assigned >> f := f^2*g(A, A, Q); A := 2*A; ^ User error: Identifier 'A' has not been declared or assigned >> if bit(r, i) = 1 then ^ User error: Identifier 'r' has not been declared or assigned >> end for; ^ User error: bad syntax >> return f; ^ User error: A 'return' can only be used inside a procedure or function >> end function; ^ User error: bad syntax Total time: 0.270 seconds, Total memory usage: 3.24MB '213.78.' ************** MAGMA ***************** Host 213.78.42.15 (213.78.42.15) Time: Tue Dec 13 06:18:14 2005 Input: 10! Output: Magma V2.11-10 Tue Dec 13 2005 06:18:13 on modular [Seed = 298647435] ------------------------------------- >> 10!; ^ User error: bad syntax Total time: 0.220 seconds, Total memory usage: 3.24MB '130.83.' ************** MAGMA ***************** Host 130.83.244.131 (130.83.244.131) Time: Tue Dec 13 06:14:16 2005 Input: F2 := FiniteField(2); P := PolynomialRing(F2); p := x^230 + x^50 + x^33 + x^32 + 1; F := ext< F2 | p >; a := z; E := EllipticCurve([1, 0, 0, 0, a]); time #E; FactoredOrder(E); Output: ** WARNING: Computation time exceeded 20 seconds, so computation was terminated after 20 seconds. ** Magma V2.11-10 Tue Dec 13 2005 06:13:32 on modular [Seed = 1234478843] ------------------------------------- 1725436586697640946858688965569256371777165413914956878562009594085376 Time: 0.170 Errors: /bin/sh: line 1: 11302 Alarm clock nice -n 19 /usr/local/bin/magma '159.149' ************** MAGMA ***************** Host 159.149.2.252 (159.149.2.252) Time: Tue Dec 13 04:03:09 2005 Input: "Replace this by some code, then click [PARI] or [MAGMA]!" Output: Magma V2.11-10 Tue Dec 13 2005 04:02:39 on modular [Seed = 2420989386] ------------------------------------- Replace this by some code, then click [PARI] or [MAGMA]! Total time: 0.370 seconds, Total memory usage: 3.24MB '60.225.' ************** MAGMA ***************** Host 60.225.131.213 (60.225.131.213) Time: Mon Dec 12 18:17:50 2005 Input: Factorization(100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001); Output: ** WARNING: Computation time exceeded 20 seconds, so computation was terminated after 20 seconds. ** Magma V2.11-10 Mon Dec 12 2005 18:17:30 on modular [Seed = 1890399814] ------------------------------------- Errors: /bin/sh: line 1: 32588 Alarm clock nice -n 19 /usr/local/bin/magma '60.225.' ************** MAGMA ***************** Host 60.225.131.213 (60.225.131.213) Time: Mon Dec 12 18:16:29 2005 Input: Factorization(10000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000); Output: Magma V2.11-10 Mon Dec 12 2005 18:16:28 on modular [Seed = 2141337743] ------------------------------------- [ <2, 709>, <5, 709> ] Total time: 0.190 seconds, Total memory usage: 3.24MB '80.201.' ************** MAGMA ***************** Host 80.201.74.36 (80.201.74.36) Time: Mon Dec 12 15:19:29 2005 Input: factor(111111111111111111111111111111111) Output: Magma V2.11-10 Mon Dec 12 2005 15:19:29 on modular [Seed = 2170493422] ------------------------------------- >> factor(111111111111111111111111111111111) ^ User error: Identifier 'factor' has not been declared or assigned Total time: 0.190 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Mon Dec 12 14:29:29 2005 Input: p := 1461501624496790265145448589920785493717258890819; Fp := GF(p); Fp2 := ExtensionField; lambda := 2; mu := 1 + i; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; g := function(U, V, Q) if IsZero(U) or IsZero(V) or (U eq -V) or IsZero(Q) then return Fp12!1; end if; m := (U eq V) select 3*U[1]^2/(2*U[1]) else (V[2] - U[2])/(V[1] - U[1]); return m*(Q[1] - U[1]) + U[1] - Q[2]; end function; Output: Magma V2.11-10 Mon Dec 12 2005 14:29:29 on modular [Seed = 1418385651] ------------------------------------- Total time: 0.200 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Mon Dec 12 14:29:20 2005 Input: p := 1461501624496790265145448589920785493717258890819; Fp := GF(p); Fp2 := ExtensionField; lambda := 2; mu := 1 + i; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; g := function(U, V, Q) if IsZero(U) or IsZero(V) or (u eq -V) or IsZero(Q) then return Fp12!1; end if; m := (U eq V) select 3*U[1]^2/(2*U[1]) else (V[2] - U[2])/(V[1] - U[1]); return m*(Q[1] - U[1]) + U[1] - Q[2]; end function; Output: Magma V2.11-10 Mon Dec 12 2005 14:29:20 on modular [Seed = 1535498229] ------------------------------------- >> if IsZero(U) or IsZero(V) or (u eq -V) or IsZero(Q) then ^ User error: Identifier 'u' has not been declared or assigned Total time: 0.200 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Mon Dec 12 14:06:22 2005 Input: p := 1461501624496790265145448589920785493717258890819; Fp := GF(p); Fp2 := ExtensionField; lambda := 2; mu := 1 + i; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; Output: Magma V2.11-10 Mon Dec 12 2005 14:06:21 on modular [Seed = 2057630569] ------------------------------------- Total time: 0.200 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Mon Dec 12 14:04:05 2005 Input: p := 1461501624496790265145448589920785493717258890819; Fp := GF(p); Fp2 := ext; lambda := 2; mu := 1 + i; xi := 1/(-8 + 8*i); Fp12 := ext; Output: Magma V2.11-10 Mon Dec 12 2005 14:04:05 on modular [Seed = 14584475] ------------------------------------- >> Fp2 := ext; ^ User error: bad syntax >> mu := 1 + i; ^ User error: Identifier 'i' has not been declared or assigned >> xi := 1/(-8 + 8*i); ^ User error: Identifier 'i' has not been declared or assigned >> Fp12 := ext; ^ User error: bad syntax Total time: 0.200 seconds, Total memory usage: 3.24MB '131.188' ************** MAGMA ***************** Host 131.188.166.152 (131.188.166.152) Time: Mon Dec 12 11:58:49 2005 Input: 1+2; Output: Magma V2.11-10 Mon Dec 12 2005 11:58:49 on modular [Seed = 248570239] ------------------------------------- 3 Total time: 0.190 seconds, Total memory usage: 3.24MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Mon Dec 12 07:14:00 2005 Input: G :=DirichletGroup(15); G; X :=Elements(G); X; Y :=X[4]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := SortDecomposition(NewformDecomposition(NewSubspace(CuspidalSubspace(M)))); D; qEigenform(D[1],80);Parent($1); Output: Magma V2.11-10 Mon Dec 12 2005 07:14:00 on modular [Seed = 586942663] ------------------------------------- Group of Dirichlet characters of modulus 15 over Rational Field [ 1, $.1, $.2, $.1*$.2 ] 15 2 [ Modular symbols space of level 15, weight 3, character $.1*$.2, and dimension 1 over Rational Field, Modular symbols space of level 15, weight 3, character $.1*$.2, and dimension 1 over Rational Field ] q + q^2 - 3*q^3 - 3*q^4 + 5*q^5 - 3*q^6 - 7*q^8 + 9*q^9 + 5*q^10 + 9*q^12 - 15*q^15 + 5*q^16 - 14*q^17 + 9*q^18 - 22*q^19 - 15*q^20 + 34*q^23 + 21*q^24 + 25*q^25 - 27*q^27 - 15*q^30 + 2*q^31 + 33*q^32 - 14*q^34 - 27*q^36 - 22*q^38 - 35*q^40 + 45*q^45 + 34*q^46 - 14*q^47 - 15*q^48 + 49*q^49 + 25*q^50 + 42*q^51 - 86*q^53 - 27*q^54 + 66*q^57 + 45*q^60 - 118*q^61 + 2*q^62 + 13*q^64 + 42*q^68 - 102*q^69 - 63*q^72 - 75*q^75 + 66*q^76 + 98*q^79 + O(q^80) Power series ring in q over Rational Field Total time: 0.280 seconds, Total memory usage: 4.81MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Mon Dec 12 07:13:46 2005 Input: G :=DirichletGroup(15); G; X :=Elements(G); X; Y :=X[4]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := SortDecomposition(NewformDecomposition(NewSubspace(CuspidalSubspace(M)))); D; qEigenform(D[12],80);Parent($1); Output: Magma V2.11-10 Mon Dec 12 2005 07:13:42 on modular [Seed = 670633276] ------------------------------------- Group of Dirichlet characters of modulus 15 over Rational Field [ 1, $.1, $.2, $.1*$.2 ] 15 2 [ Modular symbols space of level 15, weight 3, character $.1*$.2, and dimension 1 over Rational Field, Modular symbols space of level 15, weight 3, character $.1*$.2, and dimension 1 over Rational Field ] >> qEigenform(D[12],80);Parent($1);; ^ Runtime error in '[]': Sequence element 12 not defined Set of sequences over Power Structure of ModSym Total time: 0.300 seconds, Total memory usage: 4.42MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Mon Dec 12 06:27:07 2005 Input: G :=DirichletGroup(75,CyclotomicField(4)); G; X :=Elements(G); X; Y :=X[8]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 2, 1); D := SortDecomposition(NewformDecomposition(NewSubspace(CuspidalSubspace(M)))); D; qEigenform(D[2],80);Parent($1); Output: Magma V2.11-10 Mon Dec 12 2005 06:27:06 on modular [Seed = 1054876634] ------------------------------------- Group of Dirichlet characters of modulus 75 over Cyclotomic Field of order 4 and degree 2 [ 1, $.1, $.2, $.1*$.2, $.2^2, $.1*$.2^2, $.2^3, $.1*$.2^3 ] 15 4 [ Modular symbols space of level 75, weight 2, character $.1*$.2^3, and dimension 2 over Cyclotomic Field of order 4 and degree 2, Modular symbols space of level 75, weight 2, character $.1*$.2^3, and dimension 2 over Cyclotomic Field of order 4 and degree 2 ] q + a*q^2 + zeta_4*a*q^3 + zeta_4*q^4 - 3*q^6 - zeta_4*a*q^8 - 3*zeta_4*q^9 - a*q^12 + 5*q^16 - 4*a*q^17 - 3*zeta_4*a*q^18 + 4*zeta_4*q^19 + 2*zeta_4*a*q^23 + 3*zeta_4*q^24 + 3*a*q^27 - 8*q^31 + 3*a*q^32 - 12*zeta_4*q^34 + 3*q^36 + 4*zeta_4*a*q^38 - 6*q^46 + 6*a*q^47 + 5*zeta_4*a*q^48 + 7*zeta_4*q^49 + 12*q^51 - 8*zeta_4*a*q^53 + 9*zeta_4*q^54 - 4*a*q^57 + 2*q^61 - 8*a*q^62 - zeta_4*q^64 - 4*zeta_4*a*q^68 - 6*zeta_4*q^69 - 3*a*q^72 - 4*q^76 - 16*zeta_4*q^79 + O(q^80) Power series ring in q over Univariate Quotient Polynomial Algebra in a over Cyclotomic Field of order 4 and degree 2 with modulus a^2 - 3*zeta_4 Total time: 0.380 seconds, Total memory usage: 5.04MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Mon Dec 12 06:26:16 2005 Input: G :=DirichletGroup(75,CyclotomicField(4)); G; X :=Elements(G); X; Y :=X[8]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 2, 1); D := SortDecomposition(NewformDecomposition(NewSubspace(CuspidalSubspace(M)))); D; qEigenform(D[1],80);Parent($1); Output: Magma V2.11-10 Mon Dec 12 2005 06:26:16 on modular [Seed = 620378386] ------------------------------------- Group of Dirichlet characters of modulus 75 over Cyclotomic Field of order 4 and degree 2 [ 1, $.1, $.2, $.1*$.2, $.2^2, $.1*$.2^2, $.2^3, $.1*$.2^3 ] 15 4 [ Modular symbols space of level 75, weight 2, character $.1*$.2^3, and dimension 2 over Cyclotomic Field of order 4 and degree 2, Modular symbols space of level 75, weight 2, character $.1*$.2^3, and dimension 2 over Cyclotomic Field of order 4 and degree 2 ] q - zeta_4*a*q^3 - 2*zeta_4*q^4 + a*q^7 + 3*zeta_4*q^9 - 2*a*q^12 + 3*zeta_4*a*q^13 - 4*q^16 + zeta_4*q^19 - 3*q^21 + 3*a*q^27 - 2*zeta_4*a*q^28 + 7*q^31 + 6*q^36 - 4*a*q^37 - 9*zeta_4*q^39 - 7*zeta_4*a*q^43 + 4*zeta_4*a*q^48 + 4*zeta_4*q^49 + 6*a*q^52 + a*q^57 - 13*q^61 + 3*zeta_4*a*q^63 + 8*zeta_4*q^64 - 9*a*q^67 + 8*zeta_4*a*q^73 + 2*q^76 - 4*zeta_4*q^79 + O(q^80) Power series ring in q over Univariate Quotient Polynomial Algebra in a over Cyclotomic Field of order 4 and degree 2 with modulus a^2 + 3*zeta_4 Total time: 0.400 seconds, Total memory usage: 5.04MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Mon Dec 12 06:23:10 2005 Input: G :=DirichletGroup(45,CyclotomicField(4)); G; X :=Elements(G); X; Y :=X[8]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 2, 1); D := SortDecomposition(NewformDecomposition(NewSubspace(CuspidalSubspace(M)))); D; qEigenform(D[1],80);Parent($1); Output: Magma V2.11-10 Mon Dec 12 2005 06:23:10 on modular [Seed = 1940745603] ------------------------------------- Group of Dirichlet characters of modulus 45 over Cyclotomic Field of order 4 and degree 2 [ 1, $.1, $.2, $.1*$.2, $.2^2, $.1*$.2^2, $.2^3, $.1*$.2^3 ] 15 4 [ Modular symbols space of level 45, weight 2, character $.1*$.2^3, and dimension 2 over Cyclotomic Field of order 4 and degree 2 ] q + a*q^2 - zeta_4*q^4 + (2*zeta_4 - 1)*a*q^5 + (2*zeta_4 - 2)*q^7 - 3*zeta_4*a*q^8 + (-zeta_4 - 2)*q^10 + (-2*zeta_4 - 2)*a*q^11 + (zeta_4 + 1)*q^13 + (2*zeta_4 - 2)*a*q^14 + q^16 + 4*a*q^17 + (zeta_4 + 2)*a*q^20 + (-2*zeta_4 + 2)*q^22 + 4*zeta_4*a*q^23 + (-3*zeta_4 + 4)*q^25 + (zeta_4 + 1)*a*q^26 + (2*zeta_4 + 2)*q^28 + (-3*zeta_4 + 3)*a*q^29 - 4*q^31 - 5*a*q^32 + 4*zeta_4*q^34 + (-6*zeta_4 - 2)*a*q^35 + (-zeta_4 + 1)*q^37 + (6*zeta_4 - 3)*q^40 + (zeta_4 + 1)*a*q^41 + (-8*zeta_4 - 8)*q^43 + (2*zeta_4 - 2)*a*q^44 - 4*q^46 - 8*a*q^47 - zeta_4*q^49 + (-3*zeta_4 + 4)*a*q^50 + (-zeta_4 + 1)*q^52 + 4*zeta_4*a*q^53 + (6*zeta_4 + 2)*q^55 + (6*zeta_4 + 6)*a*q^56 + (3*zeta_4 + 3)*q^58 + (-6*zeta_4 + 6)*a*q^59 + 8*q^61 - 4*a*q^62 - 7*zeta_4*q^64 + (zeta_4 - 3)*a*q^65 + (-4*zeta_4 + 4)*q^67 - 4*zeta_4*a*q^68 + (-2*zeta_4 + 6)*q^70 + (4*zeta_4 + 4)*a*q^71 + (zeta_4 + 1)*q^73 + (-zeta_4 + 1)*a*q^74 + 8*a*q^77 + 12*zeta_4*q^79 + O(q^80) Power series ring in q over Univariate Quotient Polynomial Algebra in a over Cyclotomic Field of order 4 and degree 2 with modulus a^2 - zeta_4 Total time: 0.340 seconds, Total memory usage: 4.98MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Mon Dec 12 06:21:51 2005 Input: G :=DirichletGroup(45,CyclotomicField(4)); G; X :=Elements(G); X; Y :=X[8]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 2, 1); D := SortDecomposition(NewformDecomposition(NewSubspace(CuspidalSubspace(M)))); D; qEigenform(D[1],300); Output: Magma V2.11-10 Mon Dec 12 2005 06:21:50 on modular [Seed = 2024171096] ------------------------------------- Group of Dirichlet characters of modulus 45 over Cyclotomic Field of order 4 and degree 2 [ 1, $.1, $.2, $.1*$.2, $.2^2, $.1*$.2^2, $.2^3, $.1*$.2^3 ] 15 4 [ Modular symbols space of level 45, weight 2, character $.1*$.2^3, and dimension 2 over Cyclotomic Field of order 4 and degree 2 ] q + a*q^2 - zeta_4*q^4 + (2*zeta_4 - 1)*a*q^5 + (2*zeta_4 - 2)*q^7 - 3*zeta_4*a*q^8 + (-zeta_4 - 2)*q^10 + (-2*zeta_4 - 2)*a*q^11 + (zeta_4 + 1)*q^13 + (2*zeta_4 - 2)*a*q^14 + q^16 + 4*a*q^17 + (zeta_4 + 2)*a*q^20 + (-2*zeta_4 + 2)*q^22 + 4*zeta_4*a*q^23 + (-3*zeta_4 + 4)*q^25 + (zeta_4 + 1)*a*q^26 + (2*zeta_4 + 2)*q^28 + (-3*zeta_4 + 3)*a*q^29 - 4*q^31 - 5*a*q^32 + 4*zeta_4*q^34 + (-6*zeta_4 - 2)*a*q^35 + (-zeta_4 + 1)*q^37 + (6*zeta_4 - 3)*q^40 + (zeta_4 + 1)*a*q^41 + (-8*zeta_4 - 8)*q^43 + (2*zeta_4 - 2)*a*q^44 - 4*q^46 - 8*a*q^47 - zeta_4*q^49 + (-3*zeta_4 + 4)*a*q^50 + (-zeta_4 + 1)*q^52 + 4*zeta_4*a*q^53 + (6*zeta_4 + 2)*q^55 + (6*zeta_4 + 6)*a*q^56 + (3*zeta_4 + 3)*q^58 + (-6*zeta_4 + 6)*a*q^59 + 8*q^61 - 4*a*q^62 - 7*zeta_4*q^64 + (zeta_4 - 3)*a*q^65 + (-4*zeta_4 + 4)*q^67 - 4*zeta_4*a*q^68 + (-2*zeta_4 + 6)*q^70 + (4*zeta_4 + 4)*a*q^71 + (zeta_4 + 1)*q^73 + (-zeta_4 + 1)*a*q^74 + 8*a*q^77 + 12*zeta_4*q^79 + (2*zeta_4 - 1)*a*q^80 + (zeta_4 - 1)*q^82 + 4*zeta_4*a*q^83 + (-4*zeta_4 - 8)*q^85 + (-8*zeta_4 - 8)*a*q^86 + (-6*zeta_4 - 6)*q^88 + (9*zeta_4 - 9)*a*q^89 - 4*q^91 + 4*a*q^92 - 8*zeta_4*q^94 + (11*zeta_4 - 11)*q^97 - zeta_4*a*q^98 + (-4*zeta_4 - 3)*q^100 + (-11*zeta_4 - 11)*a*q^101 + (10*zeta_4 + 10)*q^103 + (-3*zeta_4 + 3)*a*q^104 - 4*q^106 + 4*a*q^107 + (6*zeta_4 + 2)*a*q^110 + (2*zeta_4 - 2)*q^112 - 14*zeta_4*a*q^113 + (-8*zeta_4 + 4)*q^115 + (-3*zeta_4 - 3)*a*q^116 + (6*zeta_4 + 6)*q^118 + (8*zeta_4 - 8)*a*q^119 + 3*q^121 + 8*a*q^122 + 4*zeta_4*q^124 + (11*zeta_4 + 2)*a*q^125 + (-10*zeta_4 + 10)*q^127 + 3*zeta_4*a*q^128 + (-3*zeta_4 - 1)*q^130 + (10*zeta_4 + 10)*a*q^131 + (-4*zeta_4 + 4)*a*q^134 + 12*q^136 + 10*a*q^137 - 12*zeta_4*q^139 + (2*zeta_4 - 6)*a*q^140 + (4*zeta_4 - 4)*q^142 - 4*zeta_4*a*q^143 + (3*zeta_4 - 9)*q^145 + (zeta_4 + 1)*a*q^146 + (-zeta_4 - 1)*q^148 + (3*zeta_4 - 3)*a*q^149 + 8*q^151 + 8*zeta_4*q^154 + (-8*zeta_4 + 4)*a*q^155 + (5*zeta_4 - 5)*q^157 + 12*zeta_4*a*q^158 + (5*zeta_4 + 10)*q^160 + (-8*zeta_4 - 8)*a*q^161 + (-8*zeta_4 - 8)*q^163 + (-zeta_4 + 1)*a*q^164 - 4*q^166 - 20*a*q^167 - 11*zeta_4*q^169 + (-4*zeta_4 - 8)*a*q^170 + (8*zeta_4 - 8)*q^172 - 14*zeta_4*a*q^173 + (14*zeta_4 - 2)*q^175 + (-2*zeta_4 - 2)*a*q^176 + (-9*zeta_4 - 9)*q^178 + (-18*zeta_4 + 18)*a*q^179 - 16*q^181 - 4*a*q^182 + 12*zeta_4*q^184 + (3*zeta_4 + 1)*a*q^185 + (-8*zeta_4 + 8)*q^187 + 8*zeta_4*a*q^188 + (16*zeta_4 + 16)*a*q^191 + (zeta_4 + 1)*q^193 + (11*zeta_4 - 11)*a*q^194 - q^196 - 14*a*q^197 + (-12*zeta_4 - 9)*a*q^200 + (-11*zeta_4 + 11)*q^202 + 12*zeta_4*a*q^203 + (-3*zeta_4 - 1)*q^205 + (10*zeta_4 + 10)*a*q^206 + (zeta_4 + 1)*q^208 - 4*q^211 + 4*a*q^212 + 4*zeta_4*q^214 + (-8*zeta_4 + 24)*a*q^215 + (-8*zeta_4 + 8)*q^217 + (-2*zeta_4 + 6)*q^220 + (4*zeta_4 + 4)*a*q^221 + (10*zeta_4 + 10)*q^223 + (-10*zeta_4 + 10)*a*q^224 + 14*q^226 - 8*a*q^227 + 6*zeta_4*q^229 + (-8*zeta_4 + 4)*a*q^230 + (-9*zeta_4 + 9)*q^232 + 4*zeta_4*a*q^233 + (8*zeta_4 + 16)*q^235 + (-6*zeta_4 - 6)*a*q^236 + (-8*zeta_4 - 8)*q^238 + (12*zeta_4 - 12)*a*q^239 - 10*q^241 + 3*a*q^242 - 8*zeta_4*q^244 + (zeta_4 + 2)*a*q^245 + 12*zeta_4*a*q^248 + (2*zeta_4 - 11)*q^250 + (-14*zeta_4 - 14)*a*q^251 + (8*zeta_4 + 8)*q^253 + (-10*zeta_4 + 10)*a*q^254 - 17*q^256 - 2*a*q^257 + 4*zeta_4*q^259 + (3*zeta_4 + 1)*a*q^260 + (10*zeta_4 - 10)*q^262 + 4*zeta_4*a*q^263 + (-8*zeta_4 + 4)*q^265 + (-4*zeta_4 - 4)*q^268 + (9*zeta_4 - 9)*a*q^269 - 16*q^271 + 4*a*q^272 + 10*zeta_4*q^274 + (-2*zeta_4 - 14)*a*q^275 + (11*zeta_4 - 11)*q^277 - 12*zeta_4*a*q^278 + (-18*zeta_4 - 6)*q^280 + (7*zeta_4 + 7)*a*q^281 + (-8*zeta_4 - 8)*q^283 + (-4*zeta_4 + 4)*a*q^284 + 4*q^286 - 4*a*q^287 - zeta_4*q^289 + (3*zeta_4 - 9)*a*q^290 + (-zeta_4 + 1)*q^292 - 14*zeta_4*a*q^293 + (6*zeta_4 - 18)*q^295 + (-3*zeta_4 - 3)*a*q^296 + (-3*zeta_4 - 3)*q^298 + (4*zeta_4 - 4)*a*q^299 + O(q^300) Total time: 0.550 seconds, Total memory usage: 7.41MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Mon Dec 12 06:21:37 2005 Input: G :=DirichletGroup(45,CyclotomicField(4)); G; X :=Elements(G); X; Y :=X[16]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 2, 1); D := SortDecomposition(NewformDecomposition(NewSubspace(CuspidalSubspace(M)))); D; qEigenform(D[1],300); Output: Magma V2.11-10 Mon Dec 12 2005 06:21:36 on modular [Seed = 2141282641] ------------------------------------- Group of Dirichlet characters of modulus 45 over Cyclotomic Field of order 4 and degree 2 [ 1, $.1, $.2, $.1*$.2, $.2^2, $.1*$.2^2, $.2^3, $.1*$.2^3 ] >> Y :=X[16]; Conductor(Y); Order(Y); ^ Runtime error in '[]': Sequence element 16 not defined >> Y :=X[16]; Conductor(Y); Order(Y); ^ User error: Identifier 'Y' has not been declared or assigned >> Y :=X[16]; Conductor(Y); Order(Y); ^ User error: Identifier 'Y' has not been declared or assigned >> M := ModularSymbols(Y, 2, 1); ^ User error: Identifier 'Y' has not been declared or assigned >> SortDecomposition(NewformDecomposition(NewSubspace(CuspidalSubspace(M)))); ^ User error: Identifier 'M' has not been declared or assigned >> D; ^ User error: Identifier 'D' has not been declared or assigned >> qEigenform(D[1],300);; ^ User error: Identifier 'D' has not been declared or assigned Total time: 0.190 seconds, Total memory usage: 3.34MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Mon Dec 12 06:02:56 2005 Input: G :=DirichletGroup(15); G; X :=Elements(G); X; Y :=X[4]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := SortDecomposition(NewformDecomposition(NewSubspace(CuspidalSubspace(M)))); D; qEigenform(D[1],300); Output: Magma V2.11-10 Mon Dec 12 2005 06:02:55 on modular [Seed = 3026890500] ------------------------------------- Group of Dirichlet characters of modulus 15 over Rational Field [ 1, $.1, $.2, $.1*$.2 ] 15 2 [ Modular symbols space of level 15, weight 3, character $.1*$.2, and dimension 1 over Rational Field, Modular symbols space of level 15, weight 3, character $.1*$.2, and dimension 1 over Rational Field ] q + q^2 - 3*q^3 - 3*q^4 + 5*q^5 - 3*q^6 - 7*q^8 + 9*q^9 + 5*q^10 + 9*q^12 - 15*q^15 + 5*q^16 - 14*q^17 + 9*q^18 - 22*q^19 - 15*q^20 + 34*q^23 + 21*q^24 + 25*q^25 - 27*q^27 - 15*q^30 + 2*q^31 + 33*q^32 - 14*q^34 - 27*q^36 - 22*q^38 - 35*q^40 + 45*q^45 + 34*q^46 - 14*q^47 - 15*q^48 + 49*q^49 + 25*q^50 + 42*q^51 - 86*q^53 - 27*q^54 + 66*q^57 + 45*q^60 - 118*q^61 + 2*q^62 + 13*q^64 + 42*q^68 - 102*q^69 - 63*q^72 - 75*q^75 + 66*q^76 + 98*q^79 + 25*q^80 + 81*q^81 + 154*q^83 - 70*q^85 + 45*q^90 - 102*q^92 - 6*q^93 - 14*q^94 - 110*q^95 - 99*q^96 + 49*q^98 - 75*q^100 + 42*q^102 - 86*q^106 + 106*q^107 + 81*q^108 - 22*q^109 - 206*q^113 + 66*q^114 + 170*q^115 + 105*q^120 + 121*q^121 - 118*q^122 - 6*q^124 + 125*q^125 - 119*q^128 - 135*q^135 + 98*q^136 + 226*q^137 - 102*q^138 - 262*q^139 + 42*q^141 + 45*q^144 - 147*q^147 - 75*q^150 - 238*q^151 + 154*q^152 - 126*q^153 + 10*q^155 + 98*q^158 + 258*q^159 + 165*q^160 + 81*q^162 + 154*q^166 - 254*q^167 + 169*q^169 - 70*q^170 - 198*q^171 + 154*q^173 - 135*q^180 + 122*q^181 + 354*q^183 - 238*q^184 - 6*q^186 + 42*q^188 - 110*q^190 - 39*q^192 - 147*q^196 - 374*q^197 - 142*q^199 - 175*q^200 - 126*q^204 + 306*q^207 + 362*q^211 + 258*q^212 + 106*q^214 + 189*q^216 - 22*q^218 + 225*q^225 - 206*q^226 - 134*q^227 - 198*q^228 + 218*q^229 + 170*q^230 + 34*q^233 - 70*q^235 - 294*q^237 - 75*q^240 - 478*q^241 + 121*q^242 - 243*q^243 + 354*q^244 + 245*q^245 - 14*q^248 - 462*q^249 + 125*q^250 + 210*q^255 - 171*q^256 + 466*q^257 - 446*q^263 - 430*q^265 - 135*q^270 + 482*q^271 - 70*q^272 + 226*q^274 + 306*q^276 - 262*q^278 + 18*q^279 + 42*q^282 + 330*q^285 + 297*q^288 - 93*q^289 + 394*q^293 - 147*q^294 + O(q^300) Total time: 0.500 seconds, Total memory usage: 7.23MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Mon Dec 12 06:02:39 2005 Input: G :=DirichletGroup(15); G; X :=Elements(G); X; Y :=X[1]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := SortDecomposition(NewformDecomposition(NewSubspace(CuspidalSubspace(M)))); D; qEigenform(D[1],300); Output: Magma V2.11-10 Mon Dec 12 2005 06:02:39 on modular [Seed = 3110576665] ------------------------------------- Group of Dirichlet characters of modulus 15 over Rational Field [ 1, $.1, $.2, $.1*$.2 ] 1 1 [] >> qEigenform(D[1],300);; ^ Runtime error in '[]': Sequence element 1 not defined Total time: 0.230 seconds, Total memory usage: 4.59MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Mon Dec 12 06:02:30 2005 Input: G :=DirichletGroup(15); G; X :=Elements(G); X; Y :=X[16]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := SortDecomposition(NewformDecomposition(NewSubspace(CuspidalSubspace(M)))); D; qEigenform(D[1],300); Output: Magma V2.11-10 Mon Dec 12 2005 06:02:30 on modular [Seed = 3194004759] ------------------------------------- Group of Dirichlet characters of modulus 15 over Rational Field [ 1, $.1, $.2, $.1*$.2 ] >> Y :=X[16]; Conductor(Y); Order(Y); ^ Runtime error in '[]': Sequence element 16 not defined >> Y :=X[16]; Conductor(Y); Order(Y); ^ User error: Identifier 'Y' has not been declared or assigned >> Y :=X[16]; Conductor(Y); Order(Y); ^ User error: Identifier 'Y' has not been declared or assigned >> M := ModularSymbols(Y, 3, 1); ^ User error: Identifier 'Y' has not been declared or assigned >> SortDecomposition(NewformDecomposition(NewSubspace(CuspidalSubspace(M)))); ^ User error: Identifier 'M' has not been declared or assigned >> D; ^ User error: Identifier 'D' has not been declared or assigned >> qEigenform(D[1],300);; ^ User error: Identifier 'D' has not been declared or assigned Total time: 0.200 seconds, Total memory usage: 3.34MB '64.229.' ************** MAGMA ***************** Host 64.229.200.59 (64.229.200.59) Time: Mon Dec 12 04:20:10 2005 Input: Factorization(40^2+40+41); Output: Magma V2.11-10 Mon Dec 12 2005 04:20:10 on modular [Seed = 1401770230] ------------------------------------- [ <41, 2> ] Total time: 0.190 seconds, Total memory usage: 3.24MB '64.229.' ************** MAGMA ***************** Host 64.229.200.59 (64.229.200.59) Time: Mon Dec 12 04:19:59 2005 Input: Factorization(41^2+41+41); Output: Magma V2.11-10 Mon Dec 12 2005 04:19:59 on modular [Seed = 1485200264] ------------------------------------- [ <41, 1>, <43, 1> ] Total time: 0.180 seconds, Total memory usage: 3.24MB '64.229.' ************** MAGMA ***************** Host 64.229.200.59 (64.229.200.59) Time: Mon Dec 12 04:19:51 2005 Input: Factorization(40^2+40+41); Output: Magma V2.11-10 Mon Dec 12 2005 04:19:50 on modular [Seed = 1568890500] ------------------------------------- [ <41, 2> ] Total time: 0.180 seconds, Total memory usage: 3.24MB '64.229.' ************** MAGMA ***************** Host 64.229.200.59 (64.229.200.59) Time: Mon Dec 12 04:19:42 2005 Input: Factorization(39^2+39+41); Output: Magma V2.11-10 Mon Dec 12 2005 04:19:42 on modular [Seed = 1117536643] ------------------------------------- [ <1601, 1> ] Total time: 0.190 seconds, Total memory usage: 3.24MB '64.229.' ************** MAGMA ***************** Host 64.229.200.59 (64.229.200.59) Time: Mon Dec 12 04:19:34 2005 Input: Factorization(31^2+31+41); Output: Magma V2.11-10 Mon Dec 12 2005 04:19:34 on modular [Seed = 1201226894] ------------------------------------- [ <1033, 1> ] Total time: 0.190 seconds, Total memory usage: 3.24MB '64.229.' ************** MAGMA ***************** Host 64.229.200.59 (64.229.200.59) Time: Mon Dec 12 04:19:23 2005 Input: Factorization(26^2+26+41); Output: Magma V2.11-10 Mon Dec 12 2005 04:19:23 on modular [Seed = 1284657080] ------------------------------------- [ <743, 1> ] Total time: 0.190 seconds, Total memory usage: 3.24MB '64.229.' ************** MAGMA ***************** Host 64.229.200.59 (64.229.200.59) Time: Mon Dec 12 04:19:15 2005 Input: Factorization(19^2+19+41); Output: Magma V2.11-10 Mon Dec 12 2005 04:19:15 on modular [Seed = 1907323533] ------------------------------------- [ <421, 1> ] Total time: 0.190 seconds, Total memory usage: 3.24MB '64.229.' ************** MAGMA ***************** Host 64.229.200.59 (64.229.200.59) Time: Mon Dec 12 04:19:10 2005 Input: Factorization(9^2+9+41); Output: Magma V2.11-10 Mon Dec 12 2005 04:19:09 on modular [Seed = 1990751636] ------------------------------------- [ <131, 1> ] Total time: 0.190 seconds, Total memory usage: 3.24MB '64.229.' ************** MAGMA ***************** Host 64.229.200.59 (64.229.200.59) Time: Mon Dec 12 04:19:03 2005 Input: Factorization(4^2+4+41); Output: Magma V2.11-10 Mon Dec 12 2005 04:19:03 on modular [Seed = 2074441875] ------------------------------------- [ <61, 1> ] Total time: 0.190 seconds, Total memory usage: 3.24MB '64.229.' ************** MAGMA ***************** Host 64.229.200.59 (64.229.200.59) Time: Mon Dec 12 04:18:55 2005 Input: Factorization(3^2+3+41); Output: Magma V2.11-10 Mon Dec 12 2005 04:18:55 on modular [Seed = 1623090032] ------------------------------------- [ <53, 1> ] Total time: 0.190 seconds, Total memory usage: 3.24MB '64.229.' ************** MAGMA ***************** Host 64.229.200.59 (64.229.200.59) Time: Mon Dec 12 04:18:47 2005 Input: Factorization(2^2+2+41); Output: Magma V2.11-10 Mon Dec 12 2005 04:18:47 on modular [Seed = 1706780285] ------------------------------------- [ <47, 1> ] Total time: 0.190 seconds, Total memory usage: 3.24MB '64.229.' ************** MAGMA ***************** Host 64.229.200.59 (64.229.200.59) Time: Mon Dec 12 04:18:40 2005 Input: Factorization(1^2+1+41); Output: Magma V2.11-10 Mon Dec 12 2005 04:18:40 on modular [Seed = 1790208373] ------------------------------------- [ <43, 1> ] Total time: 0.190 seconds, Total memory usage: 3.24MB '64.229.' ************** MAGMA ***************** Host 64.229.200.59 (64.229.200.59) Time: Mon Dec 12 04:18:31 2005 Input: 1^2+1+41 Output: Magma V2.11-10 Mon Dec 12 2005 04:18:31 on modular [Seed = 1873898599] ------------------------------------- 43 Total time: 0.190 seconds, Total memory usage: 3.24MB '64.229.' ************** MAGMA ***************** Host 64.229.200.59 (64.229.200.59) Time: Mon Dec 12 04:12:46 2005 Input: e^(Pi 163^(1/2)) Output: Magma V2.11-10 Mon Dec 12 2005 04:12:46 on modular [Seed = 2671895545] ------------------------------------- >> e^(Pi 163^(1/2)); ^ User error: bad syntax Total time: 0.190 seconds, Total memory usage: 3.24MB '64.229.' ************** MAGMA ***************** Host 64.229.200.59 (64.229.200.59) Time: Mon Dec 12 04:12:36 2005 Input: e^(pi 163^(1/2)) Output: Magma V2.11-10 Mon Dec 12 2005 04:12:35 on modular [Seed = 2220554472] ------------------------------------- >> e^(pi 163^(1/2)); ^ User error: bad syntax Total time: 0.190 seconds, Total memory usage: 3.24MB '70.228.' ************** MAGMA ***************** Host 70.228.67.16 (70.228.67.16) Time: Mon Dec 12 02:46:25 2005 Input: A := Matrix(2,2,[1,2,3,4]) SmithForm(A) Output: Magma V2.11-10 Mon Dec 12 2005 02:46:25 on modular [Seed = 1150955005] ------------------------------------- >> SmithForm(A); ^ User error: bad syntax Total time: 0.180 seconds, Total memory usage: 3.24MB '70.228.' ************** MAGMA ***************** Host 70.228.67.16 (70.228.67.16) Time: Mon Dec 12 02:45:59 2005 Input: A := Matrix(2,2,[1,2,3,4]) Output: Magma V2.11-10 Mon Dec 12 2005 02:45:59 on modular [Seed = 1334912656] ------------------------------------- Total time: 0.180 seconds, Total memory usage: 3.24MB '70.228.' ************** MAGMA ***************** Host 70.228.67.16 (70.228.67.16) Time: Mon Dec 12 02:45:28 2005 Input: A := Matrix(2,2,[1,2,3,4]) Output: Magma V2.11-10 Mon Dec 12 2005 02:45:28 on modular [Seed = 1251222458] ------------------------------------- Total time: 0.180 seconds, Total memory usage: 3.24MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Sun Dec 11 15:31:08 2005 Input: G :=DirichletGroup(1155); G; X :=Elements(G); X; Output: Magma V2.11-10 Sun Dec 11 2005 15:31:07 on modular [Seed = 3026942808] ------------------------------------- Group of Dirichlet characters of modulus 1155 over Rational Field [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3, $.4, $.1*$.4, $.2*$.4, $.1*$.2*$.4, $.3*$.4, $.1*$.3*$.4, $.2*$.3*$.4, $.1*$.2*$.3*$.4 ] Total time: 0.200 seconds, Total memory usage: 3.34MB '155.207' ************** MAGMA ***************** Host 155.207.209.204 (155.207.209.204) Time: Sun Dec 11 13:10:04 2005 Input: R := PolynomialRing(Integers()); > > f :=x^4-2*x^2+2; > > K := NumberField(f); > > O := MaximalOrder(K); > > I := IntegralBasis(K); > > print I; > >NormEquation(O, 4*5^4); Output: Magma V2.11-10 Sun Dec 11 2005 13:10:03 on modular [Seed = 1251304016] ------------------------------------- [ 1, y, y^2, y^3 ] true [ [8, 0, -9, 4], [4, -8, -1, 2], [-8, 0, 7, 0], [4, 8, -1, -2], [8, 0, -9, -4], [4, -8, -3, 6], [0, 0, 5, 0], [4, 8, -3, -6], [8, 0, -1, 0] ] Total time: 0.400 seconds, Total memory usage: 3.72MB '155.207' ************** MAGMA ***************** Host 155.207.209.204 (155.207.209.204) Time: Sun Dec 11 13:09:51 2005 Input: R := PolynomialRing(Integers()); > > f :=x^4-2*x^2+2; > > K := NumberField(f); > > O := MaximalOrder(K); > > I := IntegralBasis(K); > > print I; > >NormEquation(O, 4); Output: Magma V2.11-10 Sun Dec 11 2005 13:09:51 on modular [Seed = 2007523193] ------------------------------------- [ 1, y, y^2, y^3 ] true [ [0, 0, 1, 0] ] Total time: 0.340 seconds, Total memory usage: 3.72MB '155.207' ************** MAGMA ***************** Host 155.207.209.204 (155.207.209.204) Time: Sun Dec 11 13:08:48 2005 Input: R := PolynomialRing(Integers()); > > f :=x^4-2*x^2+2; > > K := NumberField(f); > > O := MaximalOrder(K); > > I := IntegralBasis(K); > > print I; > >NormEquation(O, 4*5^4); Output: Magma V2.11-10 Sun Dec 11 2005 13:08:48 on modular [Seed = 2554835146] ------------------------------------- [ 1, y, y^2, y^3 ] true [ [8, 0, -9, 4], [4, -8, -1, 2], [-8, 0, 7, 0], [4, 8, -1, -2], [8, 0, -9, -4], [4, -8, -3, 6], [0, 0, 5, 0], [4, 8, -3, -6], [8, 0, -1, 0] ] Total time: 0.400 seconds, Total memory usage: 3.72MB '155.207' ************** MAGMA ***************** Host 155.207.209.204 (155.207.209.204) Time: Sun Dec 11 13:08:33 2005 Input: R := PolynomialRing(Integers()); > > f :=x^4-2*x^2+2; > > K := NumberField(f); > > O := MaximalOrder(K); > > I := IntegralBasis(K); > > print I; > >NormEquation(O, 3); Output: Magma V2.11-10 Sun Dec 11 2005 13:08:33 on modular [Seed = 2521415381] ------------------------------------- [ 1, y, y^2, y^3 ] false Total time: 0.310 seconds, Total memory usage: 3.72MB '155.207' ************** MAGMA ***************** Host 155.207.209.204 (155.207.209.204) Time: Sun Dec 11 13:08:13 2005 Input: R := PolynomialRing(Integers()); > > f :=x^8 - 20*x^6 + 98*x^4 - 76*x^2 + 1; > > K := NumberField(f); > > O := MaximalOrder(K); > > I := IntegralBasis(K); > > print I; > >NormEquation(O, 3); Output: Magma V2.11-10 Sun Dec 11 2005 13:08:12 on modular [Seed = 2371003875] ------------------------------------- [ 1, y, y^2, y^3, 1/8*(y^4 + 4*y^3 + 6*y^2 + 4*y + 7), 1/8*(y^5 + 6*y^3 + 4*y^2 + 7*y + 4), 1/56*(y^6 + 28*y^3 + 35*y^2 + 28*y + 22), 1/112*(y^7 + y^6 + 7*y^5 + 7*y^4 + 21*y^3 + 77*y^2 + 71*y + 15) ] false Total time: 0.520 seconds, Total memory usage: 3.90MB '84.167.' ************** MAGMA ***************** Host 84.167.225.156 (84.167.225.156) Time: Sun Dec 11 10:16:20 2005 Input: F2 := FiniteField(2); P := PolynomialRing(F2); p := x^120 + x^41 + x^35 + x^32 + 1; F := ext< F2 | p >; a := z; E := EllipticCurve([1, 0, 0, 0, a]); time #E; FactoredOrder(E); Output: Magma V2.11-10 Sun Dec 11 2005 10:16:19 on modular [Seed = 3707952194] ------------------------------------- 1329227995784915872847313096830369792 Time: 0.290 [ <2, 14>, <26113, 1>, <368539681, 1>, <8430212384348387921, 1> ] Total time: 0.620 seconds, Total memory usage: 3.82MB '85.226.' ************** MAGMA ***************** Host 85.226.120.223 (85.226.120.223) Time: Sun Dec 11 09:41:05 2005 Input: R := PolynomialRing(RationalField(),1); I := ideal< R | (x - 1)^2*x>; S := R/I; S; IsNilpotent((x-1)^2); Output: Magma V2.11-10 Sun Dec 11 2005 09:41:04 on modular [Seed = 1468542603] ------------------------------------- Affine Algebra of rank 1 over Rational Field Lexicographical Order Variables: x Quotient relations: [ x^3 - 2*x^2 + x ] Magma: Internal error Please mail this entire run [**WITH THE FOLLOWING LINES**] to magma-bugs@maths.usyd.edu.au Version: 2.11-10 Link date: Thu Nov 4 20:39:55 EST 2004 Machine type: x86_64-linux Initial seed: 1468542603 Time to this point: 0.19 Memory usage: 3.24MB Internal error in glue_ring_elt_is_nilpotent() at ring/elt_glue.c, line 367 Total time: 0.190 seconds, Total memory usage: 3.24MB '85.226.' ************** MAGMA ***************** Host 85.226.120.223 (85.226.120.223) Time: Sun Dec 11 09:40:50 2005 Input: R := PolynomialRing(RationalField(),1); I := ideal< R | (x - 1)^2*x>; S := R/I; S; Output: Magma V2.11-10 Sun Dec 11 2005 09:40:49 on modular [Seed = 1518674807] ------------------------------------- Affine Algebra of rank 1 over Rational Field Lexicographical Order Variables: x Quotient relations: [ x^3 - 2*x^2 + x ] Total time: 0.190 seconds, Total memory usage: 3.24MB '85.226.' ************** MAGMA ***************** Host 85.226.120.223 (85.226.120.223) Time: Sun Dec 11 09:40:43 2005 Input: R := PolynomialRing(RationalField(),1); I := ideal< R | (x - 1)^2*x>; S := R/I; Output: Magma V2.11-10 Sun Dec 11 2005 09:40:42 on modular [Seed = 1568810091] ------------------------------------- Total time: 0.190 seconds, Total memory usage: 3.24MB '85.226.' ************** MAGMA ***************** Host 85.226.120.223 (85.226.120.223) Time: Sun Dec 11 09:40:14 2005 Input: R := PolyonialRing(RationalField(),1); I := ideal< R | (x - 1)^2*x>; S := R/I; Output: Magma V2.11-10 Sun Dec 11 2005 09:40:13 on modular [Seed = 2793004982] ------------------------------------- >> R := PolyonialRing(RationalField(),1); ^ User error: Identifier 'PolyonialRing' has not been declared or assigned >> I := ideal< R | (x - 1)^2*x>; ^ User error: Identifier 'R' has not been declared or assigned >> S := R/I;; ^ User error: Identifier 'R' has not been declared or assigned Total time: 0.200 seconds, Total memory usage: 3.24MB '67.62.1' ************** MAGMA ***************** Host 67.62.112.123 (67.62.112.123) Time: Sun Dec 11 07:14:56 2005 Input: c:=10; e:=180; r:=c*e; r; Output: Magma V2.11-10 Sun Dec 11 2005 07:14:55 on modular [Seed = 3390411179] ------------------------------------- 1800 Total time: 0.190 seconds, Total memory usage: 3.24MB '67.62.1' ************** MAGMA ***************** Host 67.62.112.123 (67.62.112.123) Time: Sun Dec 11 06:40:38 2005 Input: t:=11111111111; y:=11111111111; g:=t*y; g; Output: Magma V2.11-10 Sun Dec 11 2005 06:40:37 on modular [Seed = 2220611561] ------------------------------------- 123456790120987654321 Total time: 0.240 seconds, Total memory usage: 3.24MB '67.62.1' ************** MAGMA ***************** Host 67.62.112.123 (67.62.112.123) Time: Sun Dec 11 06:34:58 2005 Input: c:=11; e:=18; r:=c*e; r; Output: Magma V2.11-10 Sun Dec 11 2005 06:34:58 on modular [Seed = 2421153066] ------------------------------------- 198 Total time: 0.200 seconds, Total memory usage: 3.24MB '67.62.1' ************** MAGMA ***************** Host 67.62.112.123 (67.62.112.123) Time: Sun Dec 11 06:34:47 2005 Input: c:=11; e:=11; r:=c*e; r; Output: Magma V2.11-10 Sun Dec 11 2005 06:34:47 on modular [Seed = 2604979732] ------------------------------------- 121 Total time: 0.200 seconds, Total memory usage: 3.24MB '217.24.' ************** MAGMA ***************** Host 217.24.144.35 (217.24.144.35) Time: Sun Dec 11 06:33:09 2005 Input: c:=11; e:=11; r:=c*e; r; Output: Magma V2.11-10 Sun Dec 11 2005 06:33:09 on modular [Seed = 3778981068] ------------------------------------- 121 Total time: 0.200 seconds, Total memory usage: 3.24MB '217.24.' ************** MAGMA ***************** Host 217.24.144.35 (217.24.144.35) Time: Sun Dec 11 06:27:13 2005 Input: c:=123; e:=11; r:=c*e; r; Output: Magma V2.11-10 Sun Dec 11 2005 06:27:11 on modular [Seed = 3657798797] ------------------------------------- 1353 Total time: 0.190 seconds, Total memory usage: 3.24MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Sun Dec 11 04:38:47 2005 Input: G :=DirichletGroup(175,CyclotomicField(4));G; X :=Elements(G);X; Y :=X[6]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 2, 1); D := SortDecomposition(NewformDecomposition(NewSubspace(CuspidalSubspace(M)))); D; qEigenform(D[1],104);Parent($1); Output: Magma V2.11-10 Sun Dec 11 2005 04:38:47 on modular [Seed = 921601736] ------------------------------------- Group of Dirichlet characters of modulus 175 over Cyclotomic Field of order 4 and degree 2 [ 1, $.1, $.1^2, $.1^3, $.2, $.1*$.2, $.1^2*$.2, $.1^3*$.2 ] 35 4 [ Modular symbols space of level 175, weight 2, character $.1*$.2, and dimension 2 over Cyclotomic Field of order 4 and degree 2, Modular symbols space of level 175, weight 2, character $.1*$.2, and dimension 2 over Cyclotomic Field of order 4 and degree 2, Modular symbols space of level 175, weight 2, character $.1*$.2, and dimension 2 over Cyclotomic Field of order 4 and degree 2, Modular symbols space of level 175, weight 2, character $.1*$.2, and dimension 4 over Cyclotomic Field of order 4 and degree 2 ] q - 1/3*a*q^3 - 2*zeta_4*q^4 + 1/3*zeta_4*a*q^7 + 4*zeta_4*q^9 - 3*q^11 + 2/3*zeta_4*a*q^12 - 1/3*a*q^13 - 4*q^16 - zeta_4*a*q^17 + 7*q^21 - 1/3*zeta_4*a*q^27 + 2/3*a*q^28 - 9*zeta_4*q^29 + a*q^33 + 8*q^36 + 7*zeta_4*q^39 + 6*zeta_4*q^44 - zeta_4*a*q^47 + 4/3*a*q^48 - 7*zeta_4*q^49 - 21*q^51 + 2/3*zeta_4*a*q^52 - 4/3*a*q^63 + 8*zeta_4*q^64 - 2*a*q^68 + 12*q^71 + 4/3*a*q^73 - zeta_4*a*q^77 + zeta_4*q^79 + 5*q^81 - 2*a*q^83 - 14*zeta_4*q^84 + 3*zeta_4*a*q^87 + 7*q^91 + 7/3*zeta_4*a*q^97 - 12*zeta_4*q^99 - 1/3*a*q^103 + O(q^104) Power series ring in q over Univariate Quotient Polynomial Algebra in a over Cyclotomic Field of order 4 and degree 2 with modulus a^2 - 63*zeta_4 Total time: 0.530 seconds, Total memory usage: 5.46MB '83.194.' ************** MAGMA ***************** Host 83.194.162.38 (83.194.162.38) Time: Sun Dec 11 03:35:56 2005 Input: "Replace this by some code, then click [PARI] or [MAGMA]!" Output: Magma V2.11-10 Sun Dec 11 2005 03:35:55 on modular [Seed = 4147073094] ------------------------------------- Replace this by some code, then click [PARI] or [MAGMA]! Total time: 0.190 seconds, Total memory usage: 3.24MB '72.19.1' ************** MAGMA ***************** Host 72.19.126.33 (72.19.126.33) Time: Sat Dec 10 16:55:59 2005 Input: Q:=GaloisField(35098201); P:=PolynomialRing(Q,2); I:=ideal; Groebner(I); I; Output: Magma V2.11-10 Sat Dec 10 2005 16:55:56 on modular [Seed = 3290220920] ------------------------------------- Ideal of Polynomial ring of rank 2 over GF(35098201) Lexicographical Order Variables: x, y Dimension 0 Groebner basis: [ x + 33784728*y^33 + 15744019*y^32 + 14466235*y^31 + 14937582*y^30 + 9988153*y^29 + 24849537*y^28 + 13827463*y^27 + 10851940*y^26 + 25333828*y^25 + 29238403*y^24 + 35087366*y^23 + 3185785*y^22 + 12125255*y^21 + 11305600*y^20 + 713800*y^19 + 11882241*y^18 + 23388419*y^17 + 12677392*y^16 + 20159861*y^15 + 31143912*y^14 + 33062327*y^13 + 11580434*y^12 + 10629964*y^11 + 14094725*y^10 + 30606411*y^9 + 20913610*y^8 + 23355486*y^7 + 32139384*y^6 + 35026862*y^5 + 11038274*y^4 + 26690476*y^3 + 752845*y^2 + 9514380*y + 16409093, y^34 + 34*y^33 + 561*y^32 + 5984*y^31 + 46376*y^30 + 278256*y^29 + 1344904*y^28 + 5379616*y^27 + 18156204*y^26 + 17353055*y^25 + 25833537*y^24 + 5312152*y^23 + 21881025*y^22 + 15430534*y^21 + 23145801*y^20 + 30861068*y^19 + 27872968*y^18 + 17124952*y^17 + 27873223*y^16 + 30858484*y^15 + 23140973*y^14 + 15452362*y^13 + 21951303*y^12 + 5362540*y^11 + 25762851*y^10 + 17184126*y^9 + 18014271*y^8 + 5332492*y^7 + 1358334*y^6 + 297330*y^5 + 54824*y^4 + 7259*y^3 + 714*y^2 + 34*y + 1 ] Total time: 0.240 seconds, Total memory usage: 3.43MB '84.59.7' ************** MAGMA ***************** Host 84.59.79.231 (84.59.79.231) Time: Sat Dec 10 11:35:50 2005 Input: factor(1111111111111) Output: Magma V2.11-10 Sat Dec 10 2005 11:35:50 on modular [Seed = 3895502914] ------------------------------------- >> factor(1111111111111); ^ User error: Identifier 'factor' has not been declared or assigned Total time: 0.190 seconds, Total memory usage: 3.24MB '86.138.' ************** MAGMA ***************** Host 86.138.172.89 (86.138.172.89) Time: Sat Dec 10 11:04:37 2005 Input: "Replace this by some code, then click [PARI] or [MAGMA]!" Output: Magma V2.11-10 Sat Dec 10 2005 11:04:37 on modular [Seed = 654285311] ------------------------------------- Replace this by some code, then click [PARI] or [MAGMA]! Total time: 0.200 seconds, Total memory usage: 3.24MB '200.177' ************** MAGMA ***************** Host 200.177.7.181 (200.177.7.181) Time: Sat Dec 10 10:31:44 2005 Input: BitLength := function(k) if k lt 0 then return -1; end if; v := k; n := 0; while v ne 0 do v div:= 2; n +:= 1; end while; return n; end function; TrailingZeroes := function(k) if k lt 0 then return -1; end if; v := k; s := 0; while v ne 0 and v mod 2 eq 0 do v div:= 2; s +:= 1; end while; return s; end function; GetBit := function(k, j) return (k div 2^j) mod 2; end function; Lucas := function(Q, k) n := BitLength(k); s := TrailingZeroes(k); Uh := 1; Vl := 2; Vh := 0; Ql := 1; Qh := 1; for j := n - 1 to s + 1 by -1 do assert Vl*Vh eq 0; Ql *:= Qh; if GetBit(k, j) eq 1 then Qh := Ql*Q; Uh *:= Vh; Vl *:= Vh; Vh *:= Vh; Vh -:= 2*Qh; assert (Vh ne 0) or (Uh eq 0 and Vh eq 0 and Vh eq -2*Qh); else Qh := Ql; Uh *:= Vl; Uh -:= Ql; Vh *:= Vl; Vl *:= Vl; Vl -:= 2*Ql; assert (Vl ne 0) or (Uh eq -Ql and Vh eq 0 and Vl eq -2*Ql); end if; assert (Vl eq 0 and Uh eq 0) or (Vh eq 0); end for; assert Vl*Vh eq 0; //assert (Vl eq 0 and Uh eq 0 and Vh eq -2*Qh) or (Vh eq 0 and Uh eq -Ql and Vl eq -2*Ql); Ql *:= Qh; Qh := Ql*Q; Uh *:= Vl; Uh -:= Ql; Vl *:= Vh; Ql *:= Qh; for j := 1 to s do Uh *:= Vl; Vl *:= Vl; Vl -:= 2*Ql; Ql *:= Ql; end for; assert Uh*Vl eq 0; return Uh, Vl; end function; for i, j in [1..10] do i, ":", Lucas(j, i); end for; Output: Magma V2.11-10 Sat Dec 10 2005 10:31:43 on modular [Seed = 1940571360] ------------------------------------- 1 : 1 0 1 : 1 0 1 : 1 0 1 : 1 0 1 : 1 0 1 : 1 0 1 : 1 0 1 : 1 0 1 : 1 0 1 : 1 0 2 : 0 -2 2 : 0 -4 2 : 0 -6 2 : 0 -8 2 : 0 -10 2 : 0 -12 2 : 0 -14 2 : 0 -16 2 : 0 -18 2 : 0 -20 3 : -1 0 3 : -2 0 3 : -3 0 3 : -4 0 3 : -5 0 3 : -6 0 3 : -7 0 3 : -8 0 3 : -9 0 3 : -10 0 4 : 0 2 4 : 0 8 4 : 0 18 4 : 0 32 4 : 0 50 4 : 0 72 4 : 0 98 4 : 0 128 4 : 0 162 4 : 0 200 5 : 1 0 5 : 4 0 5 : 9 0 5 : 16 0 5 : 25 0 5 : 36 0 5 : 49 0 5 : 64 0 5 : 81 0 5 : 100 0 6 : 0 -2 6 : 0 -16 6 : 0 -54 6 : 0 -128 6 : 0 -250 6 : 0 -432 6 : 0 -686 6 : 0 -1024 6 : 0 -1458 6 : 0 -2000 7 : -1 0 7 : -8 0 7 : -27 0 7 : -64 0 7 : -125 0 7 : -216 0 7 : -343 0 7 : -512 0 7 : -729 0 7 : -1000 0 8 : 0 2 8 : 0 32 8 : 0 162 8 : 0 512 8 : 0 1250 8 : 0 2592 8 : 0 4802 8 : 0 8192 8 : 0 13122 8 : 0 20000 9 : 1 0 9 : 16 0 9 : 81 0 9 : 256 0 9 : 625 0 9 : 1296 0 9 : 2401 0 9 : 4096 0 9 : 6561 0 9 : 10000 0 10 : 0 -2 10 : 0 -64 10 : 0 -486 10 : 0 -2048 10 : 0 -6250 10 : 0 -15552 10 : 0 -33614 10 : 0 -65536 10 : 0 -118098 10 : 0 -200000 Total time: 0.200 seconds, Total memory usage: 3.34MB '200.177' ************** MAGMA ***************** Host 200.177.7.181 (200.177.7.181) Time: Sat Dec 10 10:28:35 2005 Input: BitLength := function(k) if k lt 0 then return -1; end if; v := k; n := 0; while v ne 0 do v div:= 2; n +:= 1; end while; return n; end function; TrailingZeroes := function(k) if k lt 0 then return -1; end if; v := k; s := 0; while v ne 0 and v mod 2 eq 0 do v div:= 2; s +:= 1; end while; return s; end function; GetBit := function(k, j) return (k div 2^j) mod 2; end function; Lucas := function(Q, k) n := BitLength(k); s := TrailingZeroes(k); Uh := 1; Vl := 2; Vh := 0; Ql := 1; Qh := 1; for j := n - 1 to s + 1 by -1 do assert Vl*Vh eq 0; Ql *:= Qh; if GetBit(k, j) eq 1 then Qh := Ql*Q; Uh *:= Vh; Vl *:= Vh; Vh *:= Vh; Vh -:= 2*Qh; else Qh := Ql; Uh *:= Vl; Uh -:= Ql; Vh *:= Vl; Vl *:= Vl; Vl -:= 2*Ql; end if; assert (Vl eq 0 and Uh eq 0) or (Vh eq 0); end for; assert Vl*Vh eq 0; //assert (Vl eq 0 and Uh eq 0 and Vh eq -2*Qh) or (Vh eq 0 and Uh eq -Ql and Vl eq -2*Ql); Ql *:= Qh; Qh := Ql*Q; Uh *:= Vl; Uh -:= Ql; Vl *:= Vh; Ql *:= Qh; for j := 1 to s do Uh *:= Vl; Vl *:= Vl; Vl -:= 2*Ql; Ql *:= Ql; end for; assert Uh*Vl eq 0; return Uh, Vl; end function; for i, j in [1..10] do i, ":", Lucas(j, i); end for; Output: Magma V2.11-10 Sat Dec 10 2005 10:28:34 on modular [Seed = 2141636625] ------------------------------------- 1 : 1 0 1 : 1 0 1 : 1 0 1 : 1 0 1 : 1 0 1 : 1 0 1 : 1 0 1 : 1 0 1 : 1 0 1 : 1 0 2 : 0 -2 2 : 0 -4 2 : 0 -6 2 : 0 -8 2 : 0 -10 2 : 0 -12 2 : 0 -14 2 : 0 -16 2 : 0 -18 2 : 0 -20 3 : -1 0 3 : -2 0 3 : -3 0 3 : -4 0 3 : -5 0 3 : -6 0 3 : -7 0 3 : -8 0 3 : -9 0 3 : -10 0 4 : 0 2 4 : 0 8 4 : 0 18 4 : 0 32 4 : 0 50 4 : 0 72 4 : 0 98 4 : 0 128 4 : 0 162 4 : 0 200 5 : 1 0 5 : 4 0 5 : 9 0 5 : 16 0 5 : 25 0 5 : 36 0 5 : 49 0 5 : 64 0 5 : 81 0 5 : 100 0 6 : 0 -2 6 : 0 -16 6 : 0 -54 6 : 0 -128 6 : 0 -250 6 : 0 -432 6 : 0 -686 6 : 0 -1024 6 : 0 -1458 6 : 0 -2000 7 : -1 0 7 : -8 0 7 : -27 0 7 : -64 0 7 : -125 0 7 : -216 0 7 : -343 0 7 : -512 0 7 : -729 0 7 : -1000 0 8 : 0 2 8 : 0 32 8 : 0 162 8 : 0 512 8 : 0 1250 8 : 0 2592 8 : 0 4802 8 : 0 8192 8 : 0 13122 8 : 0 20000 9 : 1 0 9 : 16 0 9 : 81 0 9 : 256 0 9 : 625 0 9 : 1296 0 9 : 2401 0 9 : 4096 0 9 : 6561 0 9 : 10000 0 10 : 0 -2 10 : 0 -64 10 : 0 -486 10 : 0 -2048 10 : 0 -6250 10 : 0 -15552 10 : 0 -33614 10 : 0 -65536 10 : 0 -118098 10 : 0 -200000 Total time: 0.190 seconds, Total memory usage: 3.24MB '200.177' ************** MAGMA ***************** Host 200.177.7.181 (200.177.7.181) Time: Sat Dec 10 10:27:19 2005 Input: BitLength := function(k) if k lt 0 then return -1; end if; v := k; n := 0; while v ne 0 do v div:= 2; n +:= 1; end while; return n; end function; TrailingZeroes := function(k) if k lt 0 then return -1; end if; v := k; s := 0; while v ne 0 and v mod 2 eq 0 do v div:= 2; s +:= 1; end while; return s; end function; GetBit := function(k, j) return (k div 2^j) mod 2; end function; Lucas := function(Q, k) n := BitLength(k); s := TrailingZeroes(k); Uh := 1; Vl := 2; Vh := 0; Ql := 1; Qh := 1; for j := n - 1 to s + 1 by -1 do assert Vl*Vh eq 0; Ql *:= Qh; if GetBit(k, j) eq 1 then Qh := Ql*Q; Uh *:= Vh; Vl *:= Vh; Vh *:= Vh; Vh -:= 2*Qh; else Qh := Ql; Uh *:= Vl; Uh -:= Ql; Vh *:= Vl; Vl *:= Vl; Vl -:= 2*Ql; end if; assert (Vl eq 0 and Uh eq 0) or (Vh eq 0); end for; assert Vl*Vh eq 0; //assert (Vl eq 0 and Uh eq 0 and Vh eq -2*Qh) or (Vh eq 0 and Uh eq -Ql and Vl eq -2*Ql); Ql *:= Qh; Qh := Ql*Q; Uh *:= Vl; Uh -:= Ql; Vl *:= Vh; Ql *:= Qh; for j := 1 to s do Uh *:= Vl; Vl *:= Vl; Vl -:= 2*Ql; Ql *:= Ql; end for; assert Uh*Vl eq 0; return Uh, Vl; end function; for i in [1..100] do i, ":", Lucas(5, i); end for; Output: Magma V2.11-10 Sat Dec 10 2005 10:27:18 on modular [Seed = 1639502331] ------------------------------------- 1 : 1 0 2 : 0 -10 3 : -5 0 4 : 0 50 5 : 25 0 6 : 0 -250 7 : -125 0 8 : 0 1250 9 : 625 0 10 : 0 -6250 11 : -3125 0 12 : 0 31250 13 : 15625 0 14 : 0 -156250 15 : -78125 0 16 : 0 781250 17 : 390625 0 18 : 0 -3906250 19 : -1953125 0 20 : 0 19531250 21 : 9765625 0 22 : 0 -97656250 23 : -48828125 0 24 : 0 488281250 25 : 244140625 0 26 : 0 -2441406250 27 : -1220703125 0 28 : 0 12207031250 29 : 6103515625 0 30 : 0 -61035156250 31 : -30517578125 0 32 : 0 305175781250 33 : 152587890625 0 34 : 0 -1525878906250 35 : -762939453125 0 36 : 0 7629394531250 37 : 3814697265625 0 38 : 0 -38146972656250 39 : -19073486328125 0 40 : 0 190734863281250 41 : 95367431640625 0 42 : 0 -953674316406250 43 : -476837158203125 0 44 : 0 4768371582031250 45 : 2384185791015625 0 46 : 0 -23841857910156250 47 : -11920928955078125 0 48 : 0 119209289550781250 49 : 59604644775390625 0 50 : 0 -596046447753906250 51 : -298023223876953125 0 52 : 0 2980232238769531250 53 : 1490116119384765625 0 54 : 0 -14901161193847656250 55 : -7450580596923828125 0 56 : 0 74505805969238281250 57 : 37252902984619140625 0 58 : 0 -372529029846191406250 59 : -186264514923095703125 0 60 : 0 1862645149230957031250 61 : 931322574615478515625 0 62 : 0 -9313225746154785156250 63 : -4656612873077392578125 0 64 : 0 46566128730773925781250 65 : 23283064365386962890625 0 66 : 0 -232830643653869628906250 67 : -116415321826934814453125 0 68 : 0 1164153218269348144531250 69 : 582076609134674072265625 0 70 : 0 -5820766091346740722656250 71 : -2910383045673370361328125 0 72 : 0 29103830456733703613281250 73 : 14551915228366851806640625 0 74 : 0 -145519152283668518066406250 75 : -72759576141834259033203125 0 76 : 0 727595761418342590332031250 77 : 363797880709171295166015625 0 78 : 0 -3637978807091712951660156250 79 : -1818989403545856475830078125 0 80 : 0 18189894035458564758300781250 81 : 9094947017729282379150390625 0 82 : 0 -90949470177292823791503906250 83 : -45474735088646411895751953125 0 84 : 0 454747350886464118957519531250 85 : 227373675443232059478759765625 0 86 : 0 -2273736754432320594787597656250 87 : -1136868377216160297393798828125 0 88 : 0 11368683772161602973937988281250 89 : 5684341886080801486968994140625 0 90 : 0 -56843418860808014869689941406250 91 : -28421709430404007434844970703125 0 92 : 0 284217094304040074348449707031250 93 : 142108547152020037174224853515625 0 94 : 0 -1421085471520200371742248535156250 95 : -710542735760100185871124267578125 0 96 : 0 7105427357601001858711242675781250 97 : 3552713678800500929355621337890625 0 98 : 0 -35527136788005009293556213378906250 99 : -17763568394002504646778106689453125 0 100 : 0 177635683940025046467781066894531250 Total time: 0.190 seconds, Total memory usage: 3.24MB '200.177' ************** MAGMA ***************** Host 200.177.7.181 (200.177.7.181) Time: Sat Dec 10 10:27:02 2005 Input: BitLength := function(k) if k lt 0 then return -1; end if; v := k; n := 0; while v ne 0 do v div:= 2; n +:= 1; end while; return n; end function; TrailingZeroes := function(k) if k lt 0 then return -1; end if; v := k; s := 0; while v ne 0 and v mod 2 eq 0 do v div:= 2; s +:= 1; end while; return s; end function; GetBit := function(k, j) return (k div 2^j) mod 2; end function; Lucas := function(Q, k) n := BitLength(k); s := TrailingZeroes(k); Uh := 1; Vl := 2; Vh := 0; Ql := 1; Qh := 1; for j := n - 1 to s + 1 by -1 do assert Vl*Vh eq 0; Ql *:= Qh; if GetBit(k, j) eq 1 then Qh := Ql*Q; Uh *:= Vh; Vl *:= Vh; Vh *:= Vh; Vh -:= 2*Qh; else Qh := Ql; Uh *:= Vl; Uh -:= Ql; Vh *:= Vl; Vl *:= Vl; Vl -:= 2*Ql; end if; assert (Vl eq 0 and Uh eq 0) or (Vh eq 0); end for; assert Vl*Vh eq 0; //assert (Vl eq 0 and Uh eq 0 and Vh eq -2*Qh) or (Vh eq 0 and Uh eq -Ql and Vl eq -2*Ql); Ql *:= Qh; Qh := Ql*Q; Uh *:= Vl; Uh -:= Ql; Vl *:= Vh; Ql *:= Qh; for j := 1 to s do Uh *:= Vl; Vl *:= Vl; Vl -:= 2*Ql; Ql *:= Ql; end for; assert Uh*Vl eq 0; return Uh, Vl; end function; for i in [1, 100] do i, ":", Lucas(5, i); end for; Output: Magma V2.11-10 Sat Dec 10 2005 10:27:01 on modular [Seed = 1690162413] ------------------------------------- 1 : 1 0 100 : 0 177635683940025046467781066894531250 Total time: 0.190 seconds, Total memory usage: 3.24MB '200.177' ************** MAGMA ***************** Host 200.177.7.181 (200.177.7.181) Time: Sat Dec 10 10:26:01 2005 Input: BitLength := function(k) if k lt 0 then return -1; end if; v := k; n := 0; while v ne 0 do v div:= 2; n +:= 1; end while; return n; end function; TrailingZeroes := function(k) if k lt 0 then return -1; end if; v := k; s := 0; while v ne 0 and v mod 2 eq 0 do v div:= 2; s +:= 1; end while; return s; end function; GetBit := function(k, j) return (k div 2^j) mod 2; end function; Lucas := function(Q, k) n := BitLength(k); s := TrailingZeroes(k); Uh := 1; Vl := 2; Vh := 0; Ql := 1; Qh := 1; for j := n - 1 to s + 1 by -1 do assert Vl*Vh eq 0; Ql *:= Qh; if GetBit(k, j) eq 1 then Qh := Ql*Q; Uh *:= Vh; Vl *:= Vh; Vh *:= Vh; Vh -:= 2*Qh; else Qh := Ql; Uh *:= Vl; Uh -:= Ql; Vh *:= Vl; Vl *:= Vl; Vl -:= 2*Ql; end if; assert (Vl eq 0 and Uh eq 0) or (Vh eq 0 and Uh eq -Ql); end for; assert Vl*Vh eq 0; //assert (Vl eq 0 and Uh eq 0 and Vh eq -2*Qh) or (Vh eq 0 and Uh eq -Ql and Vl eq -2*Ql); Ql *:= Qh; Qh := Ql*Q; Uh *:= Vl; Uh -:= Ql; Vl *:= Vh; Ql *:= Qh; for j := 1 to s do Uh *:= Vl; Vl *:= Vl; Vl -:= 2*Ql; Ql *:= Ql; end for; assert Uh*Vl eq 0; return Uh, Vl; end function; for i in [1, 100] do i, ":", Lucas(5, i); end for; Output: Magma V2.11-10 Sat Dec 10 2005 10:26:01 on modular [Seed = 1874255786] ------------------------------------- 1 : 1 0 Lucas( Q: 5, k: 100 ) >> assert (Vl eq 0 and Uh eq 0) or (Vh eq 0 and Uh eq -Ql); ^ Runtime error in assert: Assertion failed Total time: 0.190 seconds, Total memory usage: 3.24MB '200.177' ************** MAGMA ***************** Host 200.177.7.181 (200.177.7.181) Time: Sat Dec 10 10:24:16 2005 Input: BitLength := function(k) if k lt 0 then return -1; end if; v := k; n := 0; while v ne 0 do v div:= 2; n +:= 1; end while; return n; end function; TrailingZeroes := function(k) if k lt 0 then return -1; end if; v := k; s := 0; while v ne 0 and v mod 2 eq 0 do v div:= 2; s +:= 1; end while; return s; end function; GetBit := function(k, j) return (k div 2^j) mod 2; end function; Lucas := function(Q, k) n := BitLength(k); s := TrailingZeroes(k); Uh := 1; Vl := 2; Vh := 0; Ql := 1; Qh := 1; for j := n - 1 to s + 1 by -1 do assert Vl*Vh eq 0; Ql *:= Qh; if GetBit(k, j) eq 1 then Qh := Ql*Q; Uh *:= Vh; Vl *:= Vh; Vh *:= Vh; Vh -:= 2*Qh; else Qh := Ql; Uh *:= Vl; Uh -:= Ql; Vh *:= Vl; Vl *:= Vl; Vl -:= 2*Ql; end if; assert (Vl eq 0 and Uh eq 0) or (Vh eq 0 and Uh eq -Ql); end for; assert Vl*Vh eq 0; //assert (Vl eq 0 and Uh eq 0 and Vh eq -2*Qh) or (Vh eq 0 and Uh eq -Ql and Vl eq -2*Ql); Ql *:= Qh; Qh := Ql*Q; Uh *:= Vl; Uh -:= Ql; Vl *:= Vh; Ql *:= Qh; for j := 1 to s do Uh *:= Vl; Vl *:= Vl; Vl -:= 2*Ql; Ql *:= Ql; end for; return Uh, Vl; end function; Lucas(5, 1); Lucas(5, 2); Lucas(5, 3); Lucas(5, 4); Lucas(5, 5); Lucas(5, 6); Lucas(5, 7); Lucas(5, 8); Output: Magma V2.11-10 Sat Dec 10 2005 10:24:16 on modular [Seed = 1384899134] ------------------------------------- 1 0 0 -10 -5 0 0 50 25 0 0 -250 -125 0 0 1250 Total time: 0.190 seconds, Total memory usage: 3.24MB '200.177' ************** MAGMA ***************** Host 200.177.7.181 (200.177.7.181) Time: Sat Dec 10 10:23:11 2005 Input: BitLength := function(k) if k lt 0 then return -1; end if; v := k; n := 0; while v ne 0 do v div:= 2; n +:= 1; end while; return n; end function; TrailingZeroes := function(k) if k lt 0 then return -1; end if; v := k; s := 0; while v ne 0 and v mod 2 eq 0 do v div:= 2; s +:= 1; end while; return s; end function; GetBit := function(k, j) return (k div 2^j) mod 2; end function; Lucas := function(Q, k) n := BitLength(k); s := TrailingZeroes(k); Uh := 1; Vl := 2; Vh := 0; Ql := 1; Qh := 1; for j := n - 1 to s + 1 by -1 do assert Vl*Vh eq 0; Ql *:= Qh; if GetBit(k, j) eq 1 then Qh := Ql*Q; Uh *:= Vh; Vl *:= Vh; Vh *:= Vh; Vh -:= 2*Qh; else Qh := Ql; Uh *:= Vl; Uh -:= Ql; Vh *:= Vl; Vl *:= Vl; Vl -:= 2*Ql; end if; assert (Vl eq 0 and Uh eq 0 and Vh eq -2*Qh) or (Vh eq 0 and Uh eq -Ql and Vl eq -2*Ql); end for; assert Vl*Vh eq 0; //assert (Vl eq 0 and Uh eq 0 and Vh eq -2*Qh) or (Vh eq 0 and Uh eq -Ql and Vl eq -2*Ql); Ql *:= Qh; Qh := Ql*Q; Uh *:= Vl; Uh -:= Ql; Vl *:= Vh; Ql *:= Qh; for j := 1 to s do Uh *:= Vl; Vl *:= Vl; Vl -:= 2*Ql; Ql *:= Ql; end for; return Uh, Vl; end function; Lucas(5, 1); Lucas(5, 2); Lucas(5, 3); Lucas(5, 4); Lucas(5, 5); Lucas(5, 6); Lucas(5, 7); Lucas(5, 8); Output: Magma V2.11-10 Sat Dec 10 2005 10:23:10 on modular [Seed = 1535306441] ------------------------------------- 1 0 0 -10 -5 0 0 50 25 0 0 -250 Lucas( Q: 5, k: 7 ) >> assert (Vl eq 0 and Uh eq 0 and Vh eq -2*Qh) or (Vh eq 0 and Uh eq ^ Runtime error in assert: Assertion failed 0 1250 Total time: 0.190 seconds, Total memory usage: 3.24MB '200.177' ************** MAGMA ***************** Host 200.177.7.181 (200.177.7.181) Time: Sat Dec 10 10:22:25 2005 Input: BitLength := function(k) if k lt 0 then return -1; end if; v := k; n := 0; while v ne 0 do v div:= 2; n +:= 1; end while; return n; end function; TrailingZeroes := function(k) if k lt 0 then return -1; end if; v := k; s := 0; while v ne 0 and v mod 2 eq 0 do v div:= 2; s +:= 1; end while; return s; end function; GetBit := function(k, j) return (k div 2^j) mod 2; end function; Lucas := function(Q, k) n := BitLength(k); s := TrailingZeroes(k); Uh := 1; Vl := 2; Vh := 0; Ql := 1; Qh := 1; for j := n - 1 to s + 1 by -1 do assert Vl*Vh eq 0; Ql *:= Qh; if GetBit(k, j) eq 1 then Qh := Ql*Q; Uh *:= Vh; Vl *:= Vh; Vh *:= Vh; Vh -:= 2*Qh; else Qh := Ql; Uh *:= Vl; Uh -:= Ql; Vh *:= Vl; Vl *:= Vl; Vl -:= 2*Ql; end if; assert (Vl eq 0 and Uh eq 0 and Vh eq -2*Qh) or (Vh eq 0 and Uh eq -Ql and Vl eq -2*Ql); end for; assert Vl*Vh eq 0; assert (Vl eq 0 and Uh eq 0 and Vh eq -2*Qh) or (Vh eq 0 and Uh eq -Ql and Vl eq -2*Ql); Ql *:= Qh; Qh := Ql*Q; Uh *:= Vl; Uh -:= Ql; Vl *:= Vh; Ql *:= Qh; for j := 1 to s do Uh *:= Vl; Vl *:= Vl; Vl -:= 2*Ql; Ql *:= Ql; end for; return Uh, Vl; end function; //Lucas(5, 1); //Lucas(5, 2); Lucas(5, 3); Lucas(5, 4); Lucas(5, 5); Lucas(5, 6); Lucas(5, 7); Lucas(5, 8); Output: Magma V2.11-10 Sat Dec 10 2005 10:22:25 on modular [Seed = 1552280993] ------------------------------------- -5 0 Lucas( Q: 5, k: 4 ) >> assert (Vl eq 0 and Uh eq 0 and Vh eq -2*Qh) or (Vh eq 0 and Uh eq -Ql ^ Runtime error in assert: Assertion failed 25 0 0 -250 Lucas( Q: 5, k: 7 ) >> assert (Vl eq 0 and Uh eq 0 and Vh eq -2*Qh) or (Vh eq 0 and Uh eq ^ Runtime error in assert: Assertion failed Lucas( Q: 5, k: 8 ) >> assert (Vl eq 0 and Uh eq 0 and Vh eq -2*Qh) or (Vh eq 0 and Uh eq -Ql ^ Runtime error in assert: Assertion failed Total time: 0.200 seconds, Total memory usage: 3.34MB '200.177' ************** MAGMA ***************** Host 200.177.7.181 (200.177.7.181) Time: Sat Dec 10 10:21:29 2005 Input: BitLength := function(k) if k lt 0 then return -1; end if; v := k; n := 0; while v ne 0 do v div:= 2; n +:= 1; end while; return n; end function; TrailingZeroes := function(k) if k lt 0 then return -1; end if; v := k; s := 0; while v ne 0 and v mod 2 eq 0 do v div:= 2; s +:= 1; end while; return s; end function; GetBit := function(k, j) return (k div 2^j) mod 2; end function; Lucas := function(Q, k) n := BitLength(k); s := TrailingZeroes(k); Uh := 1; Vl := 2; Vh := 0; Ql := 1; Qh := 1; for j := n - 1 to s + 1 by -1 do assert Vl*Vh eq 0; Ql *:= Qh; if GetBit(k, j) eq 1 then Qh := Ql*Q; Uh *:= Vh; Vl *:= Vh; Vh *:= Vh; Vh -:= 2*Qh; else Qh := Ql; Uh *:= Vl; Uh -:= Ql; Vh *:= Vl; Vl *:= Vl; Vl -:= 2*Ql; end if; assert (Vl eq 0 and Uh eq 0 and Vh eq -2*Qh) or (Vh eq 0 and Uh eq -Ql and Vl eq -2*Ql); end for; assert Vl*Vh eq 0; assert (Vl eq 0 and Uh eq 0 and Vh eq -2*Qh) or (Vh eq 0 and Uh eq -Ql and Vl eq -2*Ql); Ql *:= Qh; Qh := Ql*Q; Uh *:= Vl; Uh -:= Ql; Vl *:= Vh; Ql *:= Qh; for j := 1 to s do Uh *:= Vl; Vl *:= Vl; Vl -:= 2*Ql; Ql *:= Ql; end for; return Uh, Vl; end function; Lucas(5, 1); Lucas(5, 2); Lucas(5, 3); Lucas(5, 4); Lucas(5, 5); Output: Magma V2.11-10 Sat Dec 10 2005 10:21:29 on modular [Seed = 3010069127] ------------------------------------- Lucas( Q: 5, k: 1 ) >> assert (Vl eq 0 and Uh eq 0 and Vh eq -2*Qh) or (Vh eq 0 and Uh eq -Ql ^ Runtime error in assert: Assertion failed Lucas( Q: 5, k: 2 ) >> assert (Vl eq 0 and Uh eq 0 and Vh eq -2*Qh) or (Vh eq 0 and Uh eq -Ql ^ Runtime error in assert: Assertion failed -5 0 Lucas( Q: 5, k: 4 ) >> assert (Vl eq 0 and Uh eq 0 and Vh eq -2*Qh) or (Vh eq 0 and Uh eq -Ql ^ Runtime error in assert: Assertion failed 25 0 Total time: 0.190 seconds, Total memory usage: 3.34MB '80.184.' ************** MAGMA ***************** Host 80.184.165.238 (80.184.165.238) Time: Sat Dec 10 10:21:23 2005 Input: gcd(105;384) Output: Magma V2.11-10 Sat Dec 10 2005 10:21:23 on modular [Seed = 3027043722] ------------------------------------- >> gcd(105;384); ^ User error: bad syntax Total time: 0.180 seconds, Total memory usage: 3.24MB '200.177' ************** MAGMA ***************** Host 200.177.7.181 (200.177.7.181) Time: Sat Dec 10 10:14:32 2005 Input: BitLength := function(k) if k lt 0 then return -1; end if; v := k; n := 0; while v ne 0 do v div:= 2; n +:= 1; end while; return n; end function; TrailingZeroes := function(k) if k lt 0 then return -1; end if; v := k; s := 0; while v ne 0 and v mod 2 eq 0 do v div:= 2; s +:= 1; end while; return s; end function; GetBit := function(k, j) return (k div 2^j) mod 2; end function; Lucas := function(Q, k) n := BitLength(k); s := TrailingZeroes(k); Uh := 1; Vl := 2; Vh := 0; Ql := 1; Qh := 1; for j := n - 1 to s + 1 by -1 do assert Vl eq 0 or Vh eq 0; Ql *:= Qh; if GetBit(k, j) eq 1 then Qh := Ql*Q; Uh *:= Vh; Vl *:= Vh; Vh *:= Vh; Vh -:= 2*Qh; else Qh := Ql; Uh *:= Vl; Uh -:= Ql; Vh *:= Vl; Vl *:= Vl; Vl -:= 2*Ql; end if; assert (Vl eq 0 and Uh eq 0 and Vh eq -2*Qh) or (Vh eq 0 and Uh eq -Ql and Vl eq -2*Ql); end for; Ql *:= Qh; Qh := Ql*Q; Uh *:= Vl; Uh -:= Ql; Vl *:= Vh; Ql *:= Qh; for j := 1 to s do Uh *:= Vl; Vl *:= Vl; Vl -:= 2*Ql; Ql *:= Ql; end for; return Uh, Vl; end function; Lucas(5, 3); Output: Magma V2.11-10 Sat Dec 10 2005 10:14:31 on modular [Seed = 2521769228] ------------------------------------- -5 0 Total time: 0.190 seconds, Total memory usage: 3.24MB '200.177' ************** MAGMA ***************** Host 200.177.7.181 (200.177.7.181) Time: Sat Dec 10 10:10:40 2005 Input: BitLength := function(k) if k lt 0 then return -1; end if; v := k; n := 0; while v ne 0 do v div:= 2; n +:= 1; end while; return n; end function; TrailingZeroes := function(k) if k lt 0 then return -1; end if; v := k; s := 0; while v ne 0 and v mod 2 eq 0 do v div:= 2; s +:= 1; end while; return s; end function; GetBit := function(k, j) return (k div 2^j) mod 2; end function; Lucas := function(Q, k) n := BitLength(k); s := TrailingZeroes(k); Uh := 1; Vl := 2; Vh := 0; Ql := 1; Qh := 1; for j := n - 1 to s + 1 by -1 do assert Vl eq 0 or Vh eq 0; Ql *:= Qh; if GetBit(k, j) eq 1 then Qh := Ql*Q; Uh *:= Vh; Vl *:= Vh; Vh *:= Vh; Vh -:= 2*Qh; else Qh := Ql; Uh *:= Vl; Uh -:= Ql; Vh *:= Vl; Vl *:= Vl; Vl -:= 2*Ql; end if; end for; Ql *:= Qh; Qh := Ql*Q; Uh *:= Vl; Uh -:= Ql; Vl *:= Vh; Ql *:= Qh; for j := 1 to s do Uh *:= Vl; Vl *:= Vl; Vl -:= 2*Ql; Ql *:= Ql; end for; return Uh, Vl; end function; Lucas(5, 3); Output: Magma V2.11-10 Sat Dec 10 2005 10:10:40 on modular [Seed = 2554410021] ------------------------------------- -5 0 Total time: 0.210 seconds, Total memory usage: 3.24MB '200.177' ************** MAGMA ***************** Host 200.177.7.181 (200.177.7.181) Time: Sat Dec 10 10:09:26 2005 Input: BitLength := function(k) if k lt 0 then return -1; end if; v := k; n := 0; while v ne 0 do v div:= 2; n +:= 1; end while; return n; end function; TrailingZeroes := function(k) if k lt 0 then return -1; end if; v := k; s := 0; while v ne 0 and v mod 2 eq 0 do v div:= 2; s +:= 1; end while; return s; end function; GetBit := function(k, j) return (k div 2^j) mod 2; end function; Lucas := function(Q, k) n := BitLength(k); s := TrailingZeroes(k); Uh := 1; Vl := 2; Vh := 0; Ql := 1; Qh := 1; for j := n - 1 to s + 1 by -1 do Ql *:= Qh; if GetBit(k, j) eq 1 then Qh := Ql*Q; Uh *:= Vh; Vl *:= Vh; Vh *:= Vh; Vh -:= 2*Qh; else Qh := Ql; Uh *:= Vl; Uh -:= Ql; Vh *:= Vl; Vl *:= Vl; Vl -:= 2*Ql; end if; end for; Ql *:= Qh; Qh := Ql*Q; Uh *:= Vl; Uh -:= Ql; Vl *:= Vh; Ql *:= Qh; for j := 1 to s do Uh *:= Vl; Vl *:= Vl; Vl -:= 2*Ql; Ql *:= Ql; end for; return Uh, Vl; end function; Lucas(5, 3); Output: Magma V2.11-10 Sat Dec 10 2005 10:09:25 on modular [Seed = 2638492021] ------------------------------------- -5 0 Total time: 0.200 seconds, Total memory usage: 3.24MB '200.177' ************** MAGMA ***************** Host 200.177.7.181 (200.177.7.181) Time: Sat Dec 10 10:07:32 2005 Input: BitLength := function(k) if k lt 0 then return -1; end if; v := k; n := 0; while v ne 0 do v div:= 2; n +:= 1; end while; return n; end function; TrailingZeroes := function(k) if k lt 0 then return -1; end if; v := k; s := 0; while v ne 0 and v mod 2 eq 0 do v div:= 2; s +:= 1; end while; return s; end function; TestBit := function(k, j) return (2^j and k) ne 0; end function; Lucas := function(Q, k) n := BitLength(k); s := TrailingZeroes(k); Uh := 1; Vl := 2; Vh := 0; Ql := 1; Qh := 1; for j := n - 1 to s + 1 by -1 do Ql *:= Qh; if TestBit(k, j) then Qh := Ql*Q; Uh *:= Vh; Vl *:= Vh; Vh *:= Vh; Vh -:= 2*Qh; else Qh := Ql; Uh *:= Vl; Uh -:= Ql; Vh *:= Vl; Vl *:= Vl; Vl -:= 2*Ql; end if; end for; Ql *:= Qh; Qh := Ql*Q; Uh *:= Vl; Uh -:= Ql; Vl *:= Vh; Ql *:= Qh; for j := 1 to s do Uh *:= Vl; Vl *:= Vl; Vl -:= 2*Ql; Ql *:= Ql; end for; return Uh, Vl; end function; Lucas(5, 3); Output: Magma V2.11-10 Sat Dec 10 2005 10:07:31 on modular [Seed = 4045954447] ------------------------------------- Lucas( Q: 5, k: 3 ) TestBit( k: 3, j: 1 ) >> return (2^j and k) ne 0; ^ Runtime error: Expected a logical for the 'and' operator Total time: 0.190 seconds, Total memory usage: 3.24MB '65.191.' ************** MAGMA ***************** Host 65.191.73.113 (65.191.73.113) Time: Sat Dec 10 08:40:00 2005 Input: K := FiniteField(2); > C := LinearCode; IsSelfOrthogonal(C); WeightDistribution(C); Output: Magma V2.11-10 Sat Dec 10 2005 08:38:02 on modular [Seed = 2023874932] ------------------------------------- false [ <0, 1>, <16, 5082>, <20, 91168>, <24, 507045>, <28, 890560>, <32, 507045>, <36, 91168>, <40, 5082>, <56, 1> ] Total time: 0.100 seconds, Total memory usage: 3.34MB '219.108' ************** MAGMA ***************** Host 219.108.73.204 (219.108.73.204) Time: Sat Dec 10 06:55:26 2005 Input: V:=EvenWeightCode(27); e1:=CharacteristicVector(VectorSpace(GF(2),27),{1}); // for v1 in V do v1:=Random(V); x:=v1+e1; v:=Vector(GF(2),56,[0: i in [1..28]] cat [1] cat Eltseq(x)); v; Weight(v); Output: Magma V2.11-10 Sat Dec 10 2005 06:55:10 on modular [Seed = 1267911207] ------------------------------------- (0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 1 0 1 0 0 1 0 1 1 1 1 0 0 1 0 1 0 1 1 1 0 1 0) 16 Total time: 0.180 seconds, Total memory usage: 3.24MB '219.108' ************** MAGMA ***************** Host 219.108.73.204 (219.108.73.204) Time: Sat Dec 10 06:54:52 2005 Input: V:=EvenWeightCode(27); e1:=CharacteristicVector(VectorSpace(GF(2),27),{1}); // for v1 in V do v1:=Random(V); x:=v1+e1; v:=Vector(GF(2),56,[0: i in [1..28]] cat [1] cat Eltseq(x)); v; Output: Magma V2.11-10 Sat Dec 10 2005 06:54:42 on modular [Seed = 2959684093] ------------------------------------- (0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 0 0 0 0 1 1 1 1 1 1 0 0 0 1 0 0 1 0 1 0 0 0 0) Total time: 0.200 seconds, Total memory usage: 3.24MB '219.108' ************** MAGMA ***************** Host 219.108.73.204 (219.108.73.204) Time: Sat Dec 10 06:54:47 2005 Input: V:=EvenWeightCode(27); e1:=CharacteristicVector(VectorSpace(GF(2),27),{1}); // for v1 in V do v1:=Random(V); x:=v1+e1; v:=Vector(GF(2),56,[0: i in [1..28]] cat [1] cat Eltseq(x)); v; Output: Magma V2.11-10 Sat Dec 10 2005 06:54:08 on modular [Seed = 3010081052] ------------------------------------- (0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 0 0 0 0 1 1 0 1 1 0 1 1 0 0 0 0 0 0 1 1 0 0) Total time: 0.180 seconds, Total memory usage: 3.24MB '219.108' ************** MAGMA ***************** Host 219.108.73.204 (219.108.73.204) Time: Sat Dec 10 06:53:02 2005 Input: V:=EvenWeightCode(27); e1:=CharacteristicVector(VectorSpace(GF(2),27),{1}); // for v1 in V do v1:=Randam(V); x:=v1+e1; v:=Vector(GF(2),56,[0: i in [1..28]] cat [1] cat Eltseq(x)); v; Output: Magma V2.11-10 Sat Dec 10 2005 06:52:59 on modular [Seed = 3143776623] ------------------------------------- >> v1:=Randam(V); ^ User error: Identifier 'Randam' has not been declared or assigned >> x:=v1+e1; ^ User error: Identifier 'v1' has not been declared or assigned >> v:=Vector(GF(2),56,[0: i in [1..28]] cat [1] cat Eltseq(x)); ^ User error: Identifier 'x' has not been declared or assigned >> v; ^ User error: Identifier 'v' has not been declared or assigned Total time: 0.220 seconds, Total memory usage: 3.24MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Sat Dec 10 03:43:11 2005 Input: G :=DirichletGroup(210); G; X :=Elements(G); X; Y :=X[1]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 6, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; Output: ** WARNING: Computation time exceeded 20 seconds, so computation was terminated after 20 seconds. ** Magma V2.11-10 Sat Dec 10 2005 03:42:50 on modular [Seed = 3177475070] ------------------------------------- Group of Dirichlet characters of modulus 210 over Rational Field [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3 ] 1 1 Errors: /bin/sh: line 1: 2804 Alarm clock nice -n 19 /usr/local/bin/magma '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Sat Dec 10 03:21:17 2005 Input: G :=DirichletGroup(610); G; X :=Elements(G); X; Y :=X[1]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 6, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; Output: ** WARNING: Computation used more memory than allowed. ** Magma V2.11-10 Sat Dec 10 2005 03:21:10 on modular [Seed = 3256553084] ------------------------------------- Group of Dirichlet characters of modulus 610 over Rational Field [ 1, $.1, $.2, $.1*$.2 ] 1 1 Current total memory usage: 68.2MB, failed memory request: 29.7MB System Error: User memory limit has been reached >> D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); ^ Runtime error: Variable 'M' has not been initialized >> D; ^ User error: Identifier 'D' has not been declared or assigned Total time: 3.600 seconds, Total memory usage: 68.18MB '69.175.' ************** MAGMA ***************** Host 69.175.68.230 (69.175.68.230) Time: Fri Dec 9 12:20:25 2005 Input: x = 1 + 2 Output: Magma V2.11-10 Fri Dec 9 2005 12:20:25 on modular [Seed = 1201518809] ------------------------------------- >> x = 1 + 2; ^ User error: Identifier 'x' has not been declared or assigned Total time: 0.190 seconds, Total memory usage: 3.24MB '87.69.5' ************** MAGMA ***************** Host 87.69.58.225 (87.69.58.225) Time: Fri Dec 9 12:12:21 2005 Input: "Replace this by some code, then click [PARI] or [MAGMA]!" Output: Magma V2.11-10 Fri Dec 9 2005 12:12:20 on modular [Seed = 2621859112] ------------------------------------- Replace this by some code, then click [PARI] or [MAGMA]! Total time: 0.200 seconds, Total memory usage: 3.24MB '152.6.1' ************** MAGMA ***************** Host 152.6.19.192 (152.6.19.192) Time: Fri Dec 9 11:40:12 2005 Input: K := FiniteField(2); > C := LinearCode; > aut := AutomorphismGroup(C); Order(aut); > FactoredOrder(aut); > CompositionFactors(aut); Generators(aut); IsSelfOrthogonal(C); Output: Magma V2.11-10 Fri Dec 9 2005 11:40:12 on modular [Seed = 2170253740] ------------------------------------- 1344 [ <2, 6>, <3, 1>, <7, 1> ] G | A(1, 7) = L(2, 7) * | Cyclic(2) * | Cyclic(2) * | Cyclic(2) 1 { (3, 4)(7, 8), (4, 6)(5, 7), (4, 7)(5, 6), (1, 2)(5, 6), (2, 4, 3)(6, 8, 7) } true Total time: 0.200 seconds, Total memory usage: 3.24MB '152.6.1' ************** MAGMA ***************** Host 152.6.19.192 (152.6.19.192) Time: Fri Dec 9 11:39:02 2005 Input: K := FiniteField(2); > C := LinearCode; > aut := AutomorphismGroup(C); Order(aut); > FactoredOrder(aut); > CompositionFactors(aut); Generators(aut); IsSelfOrthogonal(C); Output: Magma V2.11-10 Fri Dec 9 2005 11:39:01 on modular [Seed = 4196403870] ------------------------------------- 1344 [ <2, 6>, <3, 1>, <7, 1> ] G | A(1, 7) = L(2, 7) * | Cyclic(2) * | Cyclic(2) * | Cyclic(2) 1 { (3, 4)(7, 8), (4, 6)(5, 7), (4, 7)(5, 6), (1, 2)(5, 6), (2, 4, 3)(6, 8, 7) } true Total time: 0.200 seconds, Total memory usage: 3.24MB '152.6.1' ************** MAGMA ***************** Host 152.6.19.192 (152.6.19.192) Time: Fri Dec 9 10:46:16 2005 Input: K := FiniteField(2); > C := LinearCode; > aut := AutomorphismGroup(C); Order(aut); > FactoredOrder(aut); > CompositionFactors(aut); Generators(aut); IsSelfOrthogonal(C); WeightDistribution(C); Output: Magma V2.11-10 Fri Dec 9 2005 10:46:15 on modular [Seed = 804637353] ------------------------------------- 7 [ <7, 1> ] G | Cyclic(7) 1 { (1, 2, 3, 4, 5, 6, 7)(8, 9, 10, 11, 12, 13, 14)(15, 16, 17, 18, 19, 20, 21)(22, 23, 24, 25, 26, 27, 28)(29, 30, 31, 32, 33, 34, 35)(36, 37, 38, 39, 40, 41, 42)(43, 44, 45, 46, 47, 48, 49)(50, 51, 52, 53, 54, 55, 56) } false [ <0, 1>, <16, 5082>, <20, 91168>, <24, 507045>, <28, 890560>, <32, 507045>, <36, 91168>, <40, 5082>, <56, 1> ] Total time: 0.310 seconds, Total memory usage: 5.51MB '65.191.' ************** MAGMA ***************** Host 65.191.73.113 (65.191.73.113) Time: Fri Dec 9 09:21:10 2005 Input: K := FiniteField(2); > C := LinearCode; > aut := AutomorphismGroup(C); Order(aut); > FactoredOrder(aut); > CompositionFactors(aut); Generators(aut); IsSelfOrthogonal(C); WeightDistribution(C); Output: Magma V2.11-10 Fri Dec 9 2005 09:21:09 on modular [Seed = 2554480911] ------------------------------------- 7 [ <7, 1> ] G | Cyclic(7) 1 { (1, 2, 3, 4, 5, 6, 7)(8, 9, 10, 11, 12, 13, 14)(15, 16, 17, 18, 19, 20, 21)(22, 23, 24, 25, 26, 27, 28)(29, 30, 31, 32, 33, 34, 35)(36, 37, 38, 39, 40, 41, 42)(43, 44, 45, 46, 47, 48, 49)(50, 51, 52, 53, 54, 55, 56) } false [ <0, 1>, <16, 5082>, <20, 91168>, <24, 507045>, <28, 890560>, <32, 507045>, <36, 91168>, <40, 5082>, <56, 1> ] Total time: 0.300 seconds, Total memory usage: 5.51MB '65.191.' ************** MAGMA ***************** Host 65.191.73.113 (65.191.73.113) Time: Fri Dec 9 09:16:23 2005 Input: K := FiniteField(2); > C := LinearCode; > aut := AutomorphismGroup(C); Order(aut); > FactoredOrder(aut); > CompositionFactors(aut); Generators(aut); IsSelfOrthogonal(C); Output: Magma V2.11-10 Fri Dec 9 2005 09:16:23 on modular [Seed = 2404736378] ------------------------------------- 7 [ <7, 1> ] G | Cyclic(7) 1 { (1, 2, 3, 4, 5, 6, 7)(8, 9, 10, 11, 12, 13, 14)(15, 16, 17, 18, 19, 20, 21)(22, 23, 24, 25, 26, 27, 28)(29, 30, 31, 32, 33, 34, 35)(36, 37, 38, 39, 40, 41, 42)(43, 44, 45, 46, 47, 48, 49)(50, 51, 52, 53, 54, 55, 56) } false Total time: 0.270 seconds, Total memory usage: 5.51MB '65.191.' ************** MAGMA ***************** Host 65.191.73.113 (65.191.73.113) Time: Fri Dec 9 08:49:20 2005 Input: K := FiniteField(2); > C := LinearCode [1, 0, 0, 0, 0, 1, 1, 1], [0, 1, 0, 0, 1, 0, 1, 1], > [0, 0, 1, 0, 1, 1, 0, 1], [0, 0, 0, 1, 1, 1, 1, 0]>; > aut := AutomorphismGroup(C); Order(aut); > FactoredOrder(aut); > CompositionFactors(aut); Generators(aut); IsSelfOrthogonal(C); Output: Magma V2.11-10 Fri Dec 9 2005 08:49:19 on modular [Seed = 3996395251] ------------------------------------- 1344 [ <2, 6>, <3, 1>, <7, 1> ] G | A(1, 7) = L(2, 7) * | Cyclic(2) * | Cyclic(2) * | Cyclic(2) 1 { (3, 4)(7, 8), (4, 6)(5, 7), (4, 7)(5, 6), (1, 2)(5, 6), (2, 4, 3)(6, 8, 7) } true Total time: 0.200 seconds, Total memory usage: 3.24MB '65.191.' ************** MAGMA ***************** Host 65.191.73.113 (65.191.73.113) Time: Fri Dec 9 08:48:35 2005 Input: K := FiniteField(2); > C := LinearCode [1 0 0 0 0 1 1 1] [0 1 0 0 1 0 1 1] > [0 0 1 0 1 1 0 1] [0 0 0 1 1 1 1 0]>; > aut := AutomorphismGroup(C); Order(aut); > FactoredOrder(aut); > CompositionFactors(aut); Generators(aut); IsSelfOrthogonal(C); Output: Magma V2.11-10 Fri Dec 9 2005 08:48:35 on modular [Seed = 3829279893] ------------------------------------- >> [1 0 0 0 0 1 1 1] [0 1 0 0 1 0 1 1] ^ User error: bad syntax >> [0 0 1 0 1 1 0 1] [0 0 0 1 1 1 1 0]>; ^ User error: bad syntax >> aut := AutomorphismGroup(C); ^ User error: Identifier 'C' has not been declared or assigned >> Order(aut); ^ User error: Identifier 'aut' has not been declared or assigned >> FactoredOrder(aut); ^ User error: Identifier 'aut' has not been declared or assigned >> CompositionFactors(aut); ^ User error: Identifier 'aut' has not been declared or assigned >> Generators(aut); ^ User error: Identifier 'aut' has not been declared or assigned >> IsSelfOrthogonal(C);; ^ User error: Identifier 'C' has not been declared or assigned Total time: 0.190 seconds, Total memory usage: 3.24MB '65.191.' ************** MAGMA ***************** Host 65.191.73.113 (65.191.73.113) Time: Fri Dec 9 08:46:50 2005 Input: K := FiniteField(2); > C := LinearCode [1, 0, 0, 0, 0, 1, 1, 1], [0, 1, 0, 0, 1, 0, 1, 1], > [0, 0, 1, 0, 1, 1, 0, 1], [0, 0, 0, 1, 1, 1, 1, 0]>; > aut := AutomorphismGroup(C); Order(aut); > FactoredOrder(aut); > CompositionFactors(aut); Generators(aut); IsSelfOrthogonal(C); Output: Magma V2.11-10 Fri Dec 9 2005 08:46:50 on modular [Seed = 3812174443] ------------------------------------- 1344 [ <2, 6>, <3, 1>, <7, 1> ] G | A(1, 7) = L(2, 7) * | Cyclic(2) * | Cyclic(2) * | Cyclic(2) 1 { (3, 4)(7, 8), (4, 6)(5, 7), (4, 7)(5, 6), (1, 2)(5, 6), (2, 4, 3)(6, 8, 7) } true Total time: 0.190 seconds, Total memory usage: 3.24MB '141.20.' ************** MAGMA ***************** Host 141.20.57.85 (141.20.57.85) Time: Fri Dec 9 08:35:26 2005 Input: H := PermutationGroup< 5 | (1,2,3,5),(1,2,3,4) >; H; FPGroup(H); Output: Magma V2.11-10 Fri Dec 9 2005 08:35:26 on modular [Seed = 3474261139] ------------------------------------- Permutation group H acting on a set of cardinality 5 (1, 2, 3, 5) (1, 2, 3, 4) Finitely presented group on 2 generators Relations $.1^4 = Id($) $.2^4 = Id($) ($.2^-1 * $.1)^3 = Id($) ($.1^-1, $.2^-1)^2 = Id($) $.2^-1 * $.1^-2 * $.2^-1 * $.1^2 * $.2^2 * $.1^-1 * $.2^-1 = Id($) Total time: 0.190 seconds, Total memory usage: 3.34MB '141.20.' ************** MAGMA ***************** Host 141.20.57.85 (141.20.57.85) Time: Fri Dec 9 08:31:58 2005 Input: H := PermutationGroup< 9 | (1,2,4)(5,6,8)(3,9,7), (4,5,6)(7,9,8) >; H; FPGroup(H); Output: Magma V2.11-10 Fri Dec 9 2005 08:31:57 on modular [Seed = 3390049989] ------------------------------------- Permutation group H acting on a set of cardinality 9 (1, 2, 4)(3, 9, 7)(5, 6, 8) (4, 5, 6)(7, 9, 8) Finitely presented group on 2 generators Relations $.1^-3 = Id($) $.2^-3 = Id($) ($.1^-1 * $.2^-1)^4 = Id($) $.1^-1 * $.2^-1 * $.1 * $.2^-1 * $.1^-1 * $.2^-1 * $.1 * $.2 * $.1^-1 * $.2 * $.1 * $.2 = Id($) Total time: 0.190 seconds, Total memory usage: 3.34MB '141.20.' ************** MAGMA ***************** Host 141.20.57.85 (141.20.57.85) Time: Fri Dec 9 08:31:32 2005 Input: H := PermutationGroup< 9 | (1,2,4)(5,6,8)(3,9,7), (4,5,6)(7,9,8) >; H; Output: Magma V2.11-10 Fri Dec 9 2005 08:31:32 on modular [Seed = 3323724464] ------------------------------------- Permutation group H acting on a set of cardinality 9 (1, 2, 4)(3, 9, 7)(5, 6, 8) (4, 5, 6)(7, 9, 8) Total time: 0.200 seconds, Total memory usage: 3.24MB '141.20.' ************** MAGMA ***************** Host 141.20.57.85 (141.20.57.85) Time: Fri Dec 9 08:30:49 2005 Input: H := PermutationGroup< 9 | (1,2,4)(5,6,8)(3,9,7), (4,5,6)(7,9,8) >; Output: Magma V2.11-10 Fri Dec 9 2005 08:30:49 on modular [Seed = 1038612854] ------------------------------------- Total time: 0.200 seconds, Total memory usage: 3.24MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Fri Dec 9 08:00:24 2005 Input: K:=PolynomialRing(RationalField(),3); s:=(1-2*b)*(a+b+c-1); print Factorization(s^2+(a+b+c-1)*(4*a*b*c-a-b-c+1)); print Factorization(4*b*(b+c-1)*(a+b-1)-(4*a*b*c-a-b-c+1)); Output: Magma V2.11-10 Fri Dec 9 2005 08:00:24 on modular [Seed = 2254223954] ------------------------------------- [ , , , ] [ , ] Total time: 0.190 seconds, Total memory usage: 3.24MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Fri Dec 9 07:59:55 2005 Input: K:=PolynomialRing(RationalField(),3); s:=(1-2*b)*(a+b+c-1); print Factorization(s^2+(a+b+c-1)*(4*a*b*c-a-b-c+1)); print Factorization(4*a*(a+c-1)*(a+b-1)-(4*a*b*c-a-b-c+1)); Output: Magma V2.11-10 Fri Dec 9 2005 07:59:54 on modular [Seed = 1267824573] ------------------------------------- [ , , , ] [ , ] Total time: 0.190 seconds, Total memory usage: 3.24MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Fri Dec 9 07:57:36 2005 Input: K:=PolynomialRing(RationalField(),3); s:=(1-2*a)*(a+b+c-1); print Factorization(s^2+(a+b+c-1)*(4*a*b*c-a-b-c+1)); print Factorization(4*a*(a+c-1)*(a+b-1)-(4*a*b*c-a-b-c+1)); Output: Magma V2.11-10 Fri Dec 9 2005 07:57:35 on modular [Seed = 1622826083] ------------------------------------- [ , , , ] [ , ] Total time: 0.200 seconds, Total memory usage: 3.24MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Fri Dec 9 07:57:26 2005 Input: K:=PolynomialRing(RationalField(),3); s:=(1-2*a)*(a+b+c-1); print Factorization(s^2+(a+b+c-1)*(4*a*b*c-a-b-c+1)); print Factorization(4*a*(a+c-1)*(a+b-1)*(a+b+c-1)-(4*a*b*c-a-b-c+1)); Output: Magma V2.11-10 Fri Dec 9 2005 07:57:26 on modular [Seed = 1789941344] ------------------------------------- [ , , , ] [ ] Total time: 0.190 seconds, Total memory usage: 3.24MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Fri Dec 9 07:56:29 2005 Input: K:=PolynomialRing(RationalField(),3); s:=(1-2*a)*(a+b+c-1); print Factorization(s^2+(a+b+c-1)*(4*a*b*c-a-b-c+1)); print Factorization(4*a*(a+c-1)*(a+b-1)*(a+b+c-1)-(a+b+c-1)*(4*a*b*c-a-b-c+1)); Output: Magma V2.11-10 Fri Dec 9 2005 07:56:28 on modular [Seed = 2057324474] ------------------------------------- [ , , , ] [ , ] Total time: 0.190 seconds, Total memory usage: 3.24MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Fri Dec 9 07:56:13 2005 Input: K:=PolynomialRing(RationalField(),3); s:=(1-2*a)*(a+b+c-1); print Factorization(s^2+(a+b+c-1)*(4*a*b*c-a-b-c+1)); print Factorization(a*(a+c-1)*(a+b-1)*(a+b+c-1)-(a+b+c-1)*(4*a*b*c-a-b-c+1)); Output: Magma V2.11-10 Fri Dec 9 2005 07:56:13 on modular [Seed = 64569042] ------------------------------------- [ , , , ] [ , ] Total time: 0.200 seconds, Total memory usage: 3.34MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Fri Dec 9 07:55:35 2005 Input: K:=PolynomialRing(RationalField(),3); s:=(1-2*a)*(a+b+c-1); print Factorization(s^2+(a+b+c-1)*(4*a*b*c-a-b-c+1)); print Factorization(a*(a+c-1)*(a*(a+b-1)-(4*a*b*c-a-b-c+1)); Output: Magma V2.11-10 Fri Dec 9 2005 07:55:34 on modular [Seed = 181417968] ------------------------------------- [ , , , ] >> print Factorization(a*(a+c-1)*(a*(a+b-1)-(4*a*b*c-a-b-c+1));; ^ User error: bad syntax Total time: 0.200 seconds, Total memory usage: 3.24MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Fri Dec 9 07:55:18 2005 Input: K:=PolynomialRing(RationalField(),3); //s:=(1-2*a)*(a+b+c-1); //print Factorization(s^2+(a+b+c-1)*(4*a*b*c-a-b-c+1)); print Factorization(a*(a+c-1)*(a+b+c-1)*(a+b-1)-(4*a*b*c-a-b-c+1)); Output: Magma V2.11-10 Fri Dec 9 2005 07:55:17 on modular [Seed = 331951770] ------------------------------------- [ ] Total time: 0.200 seconds, Total memory usage: 3.24MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Fri Dec 9 07:55:08 2005 Input: K:=PolynomialRing(RationalField(),3); //s:=(1-2*a)*(a+b+c-1); //print Factorization(s^2+(a+b+c-1)*(4*a*b*c-a-b-c+1)); print Factorization(a*(a+c-1)*(a+b-1)-(4*a*b*c-a-b-c+1)); Output: Magma V2.11-10 Fri Dec 9 2005 07:55:08 on modular [Seed = 365897861] ------------------------------------- [ ] Total time: 0.190 seconds, Total memory usage: 3.24MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Fri Dec 9 07:54:40 2005 Input: K:=PolynomialRing(RationalField(),3); //s:=(1-2*a)*(a+b+c-1); //print Factorization(s^2+(a+b+c-1)*(4*a*b*c-a-b-c+1)); print Factorization(a^2*b^2*c^3*(a+b-1)-(4*a*b*c-a-b-c+1)); Output: Magma V2.11-10 Fri Dec 9 2005 07:54:40 on modular [Seed = 482746800] ------------------------------------- [ ] Total time: 0.200 seconds, Total memory usage: 3.24MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Fri Dec 9 07:54:23 2005 Input: K:=PolynomialRing(RationalField(),3); //s:=(1-2*a)*(a+b+c-1); //print Factorization(s^2+(a+b+c-1)*(4*a*b*c-a-b-c+1)); print Factorization(a^3*b^3*c^3*(a+b-1)-(4*a*b*c-a-b-c+1)); Output: Magma V2.11-10 Fri Dec 9 2005 07:54:23 on modular [Seed = 637474918] ------------------------------------- [ ] Total time: 0.190 seconds, Total memory usage: 3.24MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Fri Dec 9 07:54:03 2005 Input: K:=PolynomialRing(RationalField(),3); //s:=(1-2*a)*(a+b+c-1); //print Factorization(s^2+(a+b+c-1)*(4*a*b*c-a-b-c+1)); print Factorization(a*b*(a+b-1)-(4*a*b*c-a-b-c+1)); Output: Magma V2.11-10 Fri Dec 9 2005 07:54:02 on modular [Seed = 771166331] ------------------------------------- [ ] Total time: 0.190 seconds, Total memory usage: 3.24MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Fri Dec 9 07:52:46 2005 Input: K:=PolynomialRing(RationalField(),3); //s:=(1-2*a)*(a+b+c-1); //print Factorization(s^2+(a+b+c-1)*(4*a*b*c-a-b-c+1)); print Factorization(a*b*c*(a+b-1)-(4*a*b*c-a-b-c+1)); Output: Magma V2.11-10 Fri Dec 9 2005 07:52:45 on modular [Seed = 1021706525] ------------------------------------- [ ] Total time: 0.200 seconds, Total memory usage: 3.24MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Fri Dec 9 07:49:19 2005 Input: K:=PolynomialRing(RationalField(),3); //s:=(1-2*a)*(a+b+c-1); //print Factorization(s^2+(a+b+c-1)*(4*a*b*c-a-b-c+1)); print Factorization(2*a^2*b^2*c*(a+b-1)*(a+b+c-1)-(a+b+c-1)*(4*a*b*c-a-b-c+1)); Output: Magma V2.11-10 Fri Dec 9 2005 07:49:19 on modular [Seed = 3608013221] ------------------------------------- [ , ] Total time: 0.200 seconds, Total memory usage: 3.34MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Fri Dec 9 07:48:58 2005 Input: K:=PolynomialRing(RationalField(),3); //s:=(1-2*a)*(a+b+c-1); //print Factorization(s^2+(a+b+c-1)*(4*a*b*c-a-b-c+1)); print Factorization(a^2*b^2*c*(a+b-1)*(a+b+c-1)-(a+b+c-1)*(4*a*b*c-a-b-c+1)); Output: Magma V2.11-10 Fri Dec 9 2005 07:48:57 on modular [Seed = 3724861658] ------------------------------------- [ , ] Total time: 0.200 seconds, Total memory usage: 3.34MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Fri Dec 9 07:48:40 2005 Input: K:=PolynomialRing(RationalField(),3); //s:=(1-2*a)*(a+b+c-1); //print Factorization(s^2+(a+b+c-1)*(4*a*b*c-a-b-c+1)); print Factorization(1/2*a*b*(a+b-1)*(a+b+c-1)-(a+b+c-1)*(4*a*b*c-a-b-c+1)); Output: Magma V2.11-10 Fri Dec 9 2005 07:48:40 on modular [Seed = 3778797556] ------------------------------------- [ , ] Total time: 0.190 seconds, Total memory usage: 3.34MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Fri Dec 9 07:48:33 2005 Input: K:=PolynomialRing(RationalField(),3); //s:=(1-2*a)*(a+b+c-1); //print Factorization(s^2+(a+b+c-1)*(4*a*b*c-a-b-c+1)); print Factorization(-1/2*a*b*(a+b-1)*(a+b+c-1)-(a+b+c-1)*(4*a*b*c-a-b-c+1)); Output: Magma V2.11-10 Fri Dec 9 2005 07:48:33 on modular [Seed = 3945911799] ------------------------------------- [ , ] Total time: 0.190 seconds, Total memory usage: 3.43MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Fri Dec 9 07:48:26 2005 Input: K:=PolynomialRing(RationalField(),3); //s:=(1-2*a)*(a+b+c-1); //print Factorization(s^2+(a+b+c-1)*(4*a*b*c-a-b-c+1)); print Factorization(-1/4*a*b*(a+b-1)*(a+b+c-1)-(a+b+c-1)*(4*a*b*c-a-b-c+1)); Output: Magma V2.11-10 Fri Dec 9 2005 07:48:26 on modular [Seed = 3979858890] ------------------------------------- [ , ] Total time: 0.190 seconds, Total memory usage: 3.43MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Fri Dec 9 07:48:20 2005 Input: K:=PolynomialRing(RationalField(),3); //s:=(1-2*a)*(a+b+c-1); //print Factorization(s^2+(a+b+c-1)*(4*a*b*c-a-b-c+1)); print Factorization(1/4*a*b*(a+b-1)*(a+b+c-1)-(a+b+c-1)*(4*a*b*c-a-b-c+1)); Output: Magma V2.11-10 Fri Dec 9 2005 07:48:20 on modular [Seed = 4146973150] ------------------------------------- [ , ] Total time: 0.200 seconds, Total memory usage: 3.53MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Fri Dec 9 07:48:13 2005 Input: K:=PolynomialRing(RationalField(),3); //s:=(1-2*a)*(a+b+c-1); //print Factorization(s^2+(a+b+c-1)*(4*a*b*c-a-b-c+1)); print Factorization(1/4*a*b*c*(a+b-1)*(a+b+c-1)-(a+b+c-1)*(4*a*b*c-a-b-c+1)); Output: Magma V2.11-10 Fri Dec 9 2005 07:48:13 on modular [Seed = 4163028680] ------------------------------------- [ , ] Total time: 0.190 seconds, Total memory usage: 3.43MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Fri Dec 9 07:48:07 2005 Input: K:=PolynomialRing(RationalField(),3); //s:=(1-2*a)*(a+b+c-1); //print Factorization(s^2+(a+b+c-1)*(4*a*b*c-a-b-c+1)); print Factorization(-1/4*a*b*c*(a+b-1)*(a+b+c-1)-(a+b+c-1)*(4*a*b*c-a-b-c+1)); Output: Magma V2.11-10 Fri Dec 9 2005 07:48:06 on modular [Seed = 2186853560] ------------------------------------- [ , ] Total time: 0.200 seconds, Total memory usage: 3.43MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Fri Dec 9 07:48:01 2005 Input: K:=PolynomialRing(RationalField(),3); //s:=(1-2*a)*(a+b+c-1); //print Factorization(s^2+(a+b+c-1)*(4*a*b*c-a-b-c+1)); print Factorization(-1/2*a*b*c*(a+b-1)*(a+b+c-1)-(a+b+c-1)*(4*a*b*c-a-b-c+1)); Output: Magma V2.11-10 Fri Dec 9 2005 07:48:01 on modular [Seed = 2271066555] ------------------------------------- [ , ] Total time: 0.200 seconds, Total memory usage: 3.43MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Fri Dec 9 07:47:55 2005 Input: K:=PolynomialRing(RationalField(),3); //s:=(1-2*a)*(a+b+c-1); //print Factorization(s^2+(a+b+c-1)*(4*a*b*c-a-b-c+1)); print Factorization(1/2*a*b*c*(a+b-1)*(a+b+c-1)-(a+b+c-1)*(4*a*b*c-a-b-c+1)); Output: Magma V2.11-10 Fri Dec 9 2005 07:47:54 on modular [Seed = 2303964750] ------------------------------------- [ , ] Total time: 0.200 seconds, Total memory usage: 3.34MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Fri Dec 9 07:47:36 2005 Input: K:=PolynomialRing(RationalField(),3); //s:=(1-2*a)*(a+b+c-1); //print Factorization(s^2+(a+b+c-1)*(4*a*b*c-a-b-c+1)); print Factorization(16*a^2*b^2*c^2*(a+b-1)*(a+b+c-1)-(a+b+c-1)*(4*a*b*c-a-b-c+1)); Output: Magma V2.11-10 Fri Dec 9 2005 07:47:35 on modular [Seed = 2471078985] ------------------------------------- [ , ] Total time: 0.190 seconds, Total memory usage: 3.24MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Fri Dec 9 07:47:27 2005 Input: K:=PolynomialRing(RationalField(),3); //s:=(1-2*a)*(a+b+c-1); //print Factorization(s^2+(a+b+c-1)*(4*a*b*c-a-b-c+1)); print Factorization(8*a^2*b^2*c^2*(a+b-1)*(a+b+c-1)-(a+b+c-1)*(4*a*b*c-a-b-c+1)); Output: Magma V2.11-10 Fri Dec 9 2005 07:47:27 on modular [Seed = 2505026167] ------------------------------------- [ , ] Total time: 0.200 seconds, Total memory usage: 3.43MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Fri Dec 9 07:47:13 2005 Input: K:=PolynomialRing(RationalField(),3); //s:=(1-2*a)*(a+b+c-1); //print Factorization(s^2+(a+b+c-1)*(4*a*b*c-a-b-c+1)); print Factorization(6*a^2*b^2*c^2*(a+b-1)*(a+b+c-1)-(a+b+c-1)*(4*a*b*c-a-b-c+1)); Output: Magma V2.11-10 Fri Dec 9 2005 07:47:13 on modular [Seed = 2672140403] ------------------------------------- [ , ] Total time: 0.200 seconds, Total memory usage: 3.24MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Fri Dec 9 07:47:05 2005 Input: K:=PolynomialRing(RationalField(),3); //s:=(1-2*a)*(a+b+c-1); //print Factorization(s^2+(a+b+c-1)*(4*a*b*c-a-b-c+1)); print Factorization(6*a*b*c*(a+b-1)*(a+b+c-1)-(a+b+c-1)*(4*a*b*c-a-b-c+1)); Output: Magma V2.11-10 Fri Dec 9 2005 07:47:05 on modular [Seed = 2709233237] ------------------------------------- [ , ] Total time: 0.190 seconds, Total memory usage: 3.43MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Fri Dec 9 07:46:59 2005 Input: K:=PolynomialRing(RationalField(),3); //s:=(1-2*a)*(a+b+c-1); //print Factorization(s^2+(a+b+c-1)*(4*a*b*c-a-b-c+1)); print Factorization(8*a*b*c*(a+b-1)*(a+b+c-1)-(a+b+c-1)*(4*a*b*c-a-b-c+1)); Output: Magma V2.11-10 Fri Dec 9 2005 07:46:58 on modular [Seed = 2876347476] ------------------------------------- [ , ] Total time: 0.200 seconds, Total memory usage: 3.43MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Fri Dec 9 07:46:42 2005 Input: K:=PolynomialRing(RationalField(),3); //s:=(1-2*a)*(a+b+c-1); //print Factorization(s^2+(a+b+c-1)*(4*a*b*c-a-b-c+1)); print Factorization(-2*a*b*c*(a+b-1)*(a+b+c-1)-(a+b+c-1)*(4*a*b*c-a-b-c+1)); Output: Magma V2.11-10 Fri Dec 9 2005 07:46:41 on modular [Seed = 2910294561] ------------------------------------- [ , ] Total time: 0.190 seconds, Total memory usage: 3.34MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Fri Dec 9 07:46:34 2005 Input: K:=PolynomialRing(RationalField(),3); //s:=(1-2*a)*(a+b+c-1); //print Factorization(s^2+(a+b+c-1)*(4*a*b*c-a-b-c+1)); print Factorization(-2*b*c*(a+b-1)*(a+b+c-1)-(a+b+c-1)*(4*a*b*c-a-b-c+1)); Output: Magma V2.11-10 Fri Dec 9 2005 07:46:34 on modular [Seed = 3077408800] ------------------------------------- [ , ] Total time: 0.190 seconds, Total memory usage: 3.43MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Fri Dec 9 07:46:29 2005 Input: K:=PolynomialRing(RationalField(),3); //s:=(1-2*a)*(a+b+c-1); //print Factorization(s^2+(a+b+c-1)*(4*a*b*c-a-b-c+1)); print Factorization(-b*c*(a+b-1)*(a+b+c-1)-(a+b+c-1)*(4*a*b*c-a-b-c+1)); Output: Magma V2.11-10 Fri Dec 9 2005 07:46:28 on modular [Seed = 3093464865] ------------------------------------- [ , ] Total time: 0.200 seconds, Total memory usage: 3.53MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Fri Dec 9 07:46:20 2005 Input: K:=PolynomialRing(RationalField(),3); //s:=(1-2*a)*(a+b+c-1); //print Factorization(s^2+(a+b+c-1)*(4*a*b*c-a-b-c+1)); print Factorization(-a*b*(a+b-1)*(a+b+c-1)-(a+b+c-1)*(4*a*b*c-a-b-c+1)); Output: Magma V2.11-10 Fri Dec 9 2005 07:46:19 on modular [Seed = 1117289956] ------------------------------------- [ , ] Total time: 0.200 seconds, Total memory usage: 3.34MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Fri Dec 9 07:46:12 2005 Input: K:=PolynomialRing(RationalField(),3); //s:=(1-2*a)*(a+b+c-1); //print Factorization(s^2+(a+b+c-1)*(4*a*b*c-a-b-c+1)); print Factorization(-a*c*(a+b-1)*(a+b+c-1)-(a+b+c-1)*(4*a*b*c-a-b-c+1)); Output: Magma V2.11-10 Fri Dec 9 2005 07:46:12 on modular [Seed = 1201502446] ------------------------------------- [ , ] Total time: 0.190 seconds, Total memory usage: 3.34MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Fri Dec 9 07:46:05 2005 Input: K:=PolynomialRing(RationalField(),3); //s:=(1-2*a)*(a+b+c-1); //print Factorization(s^2+(a+b+c-1)*(4*a*b*c-a-b-c+1)); print Factorization(-a*b*c*(a+b-1)*(a+b+c-1)-(a+b+c-1)*(4*a*b*c-a-b-c+1)); Output: Magma V2.11-10 Fri Dec 9 2005 07:46:04 on modular [Seed = 1234401016] ------------------------------------- [ , ] Total time: 0.190 seconds, Total memory usage: 3.24MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Fri Dec 9 07:45:41 2005 Input: K:=PolynomialRing(RationalField(),3); //s:=(1-2*a)*(a+b+c-1); //print Factorization(s^2+(a+b+c-1)*(4*a*b*c-a-b-c+1)); print Factorization(-4*a*b*c*(a+b-1)*(a+b+c-1)-(a+b+c-1)*(4*a*b*c-a-b-c+1)); Output: Magma V2.11-10 Fri Dec 9 2005 07:45:41 on modular [Seed = 1351249688] ------------------------------------- [ , ] Total time: 0.190 seconds, Total memory usage: 3.24MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Fri Dec 9 07:45:33 2005 Input: K:=PolynomialRing(RationalField(),3); //s:=(1-2*a)*(a+b+c-1); //print Factorization(s^2+(a+b+c-1)*(4*a*b*c-a-b-c+1)); print Factorization(-4*a^2*b^2*c^2*(a+b-1)*(a+b+c-1)-(a+b+c-1)*(4*a*b*c-a-b-c+1)); Output: Magma V2.11-10 Fri Dec 9 2005 07:45:32 on modular [Seed = 1435462148] ------------------------------------- [ , ] Total time: 0.190 seconds, Total memory usage: 3.43MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Fri Dec 9 07:45:24 2005 Input: K:=PolynomialRing(RationalField(),3); //s:=(1-2*a)*(a+b+c-1); //print Factorization(s^2+(a+b+c-1)*(4*a*b*c-a-b-c+1)); print Factorization(-2*a^2*b^2*c^2*(a+b-1)*(a+b+c-1)-(a+b+c-1)*(4*a*b*c-a-b-c+1)); Output: Magma V2.11-10 Fri Dec 9 2005 07:45:23 on modular [Seed = 1602576385] ------------------------------------- [ , ] Total time: 0.200 seconds, Total memory usage: 3.43MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Fri Dec 9 07:45:18 2005 Input: K:=PolynomialRing(RationalField(),3); //s:=(1-2*a)*(a+b+c-1); //print Factorization(s^2+(a+b+c-1)*(4*a*b*c-a-b-c+1)); print Factorization(2*a^2*b^2*c^2*(a+b-1)*(a+b+c-1)-(a+b+c-1)*(4*a*b*c-a-b-c+1)); Output: Magma V2.11-10 Fri Dec 9 2005 07:45:18 on modular [Seed = 1639669250] ------------------------------------- [ , ] Total time: 0.190 seconds, Total memory usage: 3.43MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Fri Dec 9 07:45:05 2005 Input: K:=PolynomialRing(RationalField(),3); //s:=(1-2*a)*(a+b+c-1); //print Factorization(s^2+(a+b+c-1)*(4*a*b*c-a-b-c+1)); print Factorization(4*a^2*b^2*c^2*(a+b-1)*(a+b+c-1)-(a+b+c-1)*(4*a*b*c-a-b-c+1)); Output: Magma V2.11-10 Fri Dec 9 2005 07:45:05 on modular [Seed = 1806783495] ------------------------------------- [ , ] Total time: 0.200 seconds, Total memory usage: 3.34MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Fri Dec 9 07:44:58 2005 Input: K:=PolynomialRing(RationalField(),3); //s:=(1-2*a)*(a+b+c-1); //print Factorization(s^2+(a+b+c-1)*(4*a*b*c-a-b-c+1)); print Factorization(4*a^2*b^2*(a+b-1)*(a+b+c-1)-(a+b+c-1)*(4*a*b*c-a-b-c+1)); Output: Magma V2.11-10 Fri Dec 9 2005 07:44:58 on modular [Seed = 1823888155] ------------------------------------- [ , ] Total time: 0.200 seconds, Total memory usage: 3.24MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Fri Dec 9 07:44:51 2005 Input: K:=PolynomialRing(RationalField(),3); //s:=(1-2*a)*(a+b+c-1); //print Factorization(s^2+(a+b+c-1)*(4*a*b*c-a-b-c+1)); print Factorization(2*a^2*b^2*(a+b-1)*(a+b+c-1)-(a+b+c-1)*(4*a*b*c-a-b-c+1)); Output: Magma V2.11-10 Fri Dec 9 2005 07:44:51 on modular [Seed = 1991002376] ------------------------------------- [ , ] Total time: 0.200 seconds, Total memory usage: 3.34MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Fri Dec 9 07:44:45 2005 Input: K:=PolynomialRing(RationalField(),3); //s:=(1-2*a)*(a+b+c-1); //print Factorization(s^2+(a+b+c-1)*(4*a*b*c-a-b-c+1)); print Factorization(a^2*b^2*(a+b-1)*(a+b+c-1)-(a+b+c-1)*(4*a*b*c-a-b-c+1)); Output: Magma V2.11-10 Fri Dec 9 2005 07:44:44 on modular [Seed = 2074166395] ------------------------------------- [ , ] Total time: 0.190 seconds, Total memory usage: 3.34MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Fri Dec 9 07:44:37 2005 Input: K:=PolynomialRing(RationalField(),3); //s:=(1-2*a)*(a+b+c-1); //print Factorization(s^2+(a+b+c-1)*(4*a*b*c-a-b-c+1)); print Factorization(a^2*b*(a+b-1)*(a+b+c-1)-(a+b+c-1)*(4*a*b*c-a-b-c+1)); Output: Magma V2.11-10 Fri Dec 9 2005 07:44:36 on modular [Seed = 2108113524] ------------------------------------- [ , ] Total time: 0.190 seconds, Total memory usage: 3.34MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Fri Dec 9 07:44:12 2005 Input: K:=PolynomialRing(RationalField(),3); //s:=(1-2*a)*(a+b+c-1); //print Factorization(s^2+(a+b+c-1)*(4*a*b*c-a-b-c+1)); print Factorization(a*b*(a+b-1)*(a+b+c-1)-(a+b+c-1)*(4*a*b*c-a-b-c+1)); Output: Magma V2.11-10 Fri Dec 9 2005 07:44:12 on modular [Seed = 131938342] ------------------------------------- [ , ] Total time: 0.190 seconds, Total memory usage: 3.34MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Fri Dec 9 07:44:06 2005 Input: K:=PolynomialRing(RationalField(),3); //s:=(1-2*a)*(a+b+c-1); //print Factorization(s^2+(a+b+c-1)*(4*a*b*c-a-b-c+1)); print Factorization(a*c*(a+b-1)*(a+b+c-1)-(a+b+c-1)*(4*a*b*c-a-b-c+1)); Output: Magma V2.11-10 Fri Dec 9 2005 07:44:06 on modular [Seed = 164836898] ------------------------------------- [ , ] Total time: 0.190 seconds, Total memory usage: 3.24MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Fri Dec 9 07:43:59 2005 Input: K:=PolynomialRing(RationalField(),3); //s:=(1-2*a)*(a+b+c-1); //print Factorization(s^2+(a+b+c-1)*(4*a*b*c-a-b-c+1)); print Factorization(b*c*(a+b-1)*(a+b+c-1)-(a+b+c-1)*(4*a*b*c-a-b-c+1)); Output: Magma V2.11-10 Fri Dec 9 2005 07:43:59 on modular [Seed = 315108640] ------------------------------------- [ , ] Total time: 0.190 seconds, Total memory usage: 3.43MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Fri Dec 9 07:43:53 2005 Input: K:=PolynomialRing(RationalField(),3); //s:=(1-2*a)*(a+b+c-1); //print Factorization(s^2+(a+b+c-1)*(4*a*b*c-a-b-c+1)); print Factorization(2*b*c*(a+b-1)*(a+b+c-1)-(a+b+c-1)*(4*a*b*c-a-b-c+1)); Output: Magma V2.11-10 Fri Dec 9 2005 07:43:53 on modular [Seed = 399321123] ------------------------------------- [ , ] Total time: 0.190 seconds, Total memory usage: 3.43MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Fri Dec 9 07:43:47 2005 Input: K:=PolynomialRing(RationalField(),3); //s:=(1-2*a)*(a+b+c-1); //print Factorization(s^2+(a+b+c-1)*(4*a*b*c-a-b-c+1)); print Factorization(2*a*c*(a+b-1)*(a+b+c-1)-(a+b+c-1)*(4*a*b*c-a-b-c+1)); Output: Magma V2.11-10 Fri Dec 9 2005 07:43:47 on modular [Seed = 415377190] ------------------------------------- [ , ] Total time: 0.190 seconds, Total memory usage: 3.43MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Fri Dec 9 07:43:41 2005 Input: K:=PolynomialRing(RationalField(),3); //s:=(1-2*a)*(a+b+c-1); //print Factorization(s^2+(a+b+c-1)*(4*a*b*c-a-b-c+1)); print Factorization(2*a*b*(a+b-1)*(a+b+c-1)-(a+b+c-1)*(4*a*b*c-a-b-c+1)); Output: Magma V2.11-10 Fri Dec 9 2005 07:43:40 on modular [Seed = 499590620] ------------------------------------- [ , ] Total time: 0.200 seconds, Total memory usage: 3.34MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Fri Dec 9 07:43:18 2005 Input: K:=PolynomialRing(RationalField(),3); //s:=(1-2*a)*(a+b+c-1); //print Factorization(s^2+(a+b+c-1)*(4*a*b*c-a-b-c+1)); print Factorization(4*a*b*c*(a+b-1)*(a+b+c-1)-(a+b+c-1)*(4*a*b*c-a-b-c+1)); Output: Magma V2.11-10 Fri Dec 9 2005 07:43:17 on modular [Seed = 670900223] ------------------------------------- [ , ] Total time: 0.190 seconds, Total memory usage: 3.34MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Fri Dec 9 07:43:11 2005 Input: K:=PolynomialRing(RationalField(),3); //s:=(1-2*a)*(a+b+c-1); //print Factorization(s^2+(a+b+c-1)*(4*a*b*c-a-b-c+1)); print Factorization(2*a*b*c*(a+b-1)*(a+b+c-1)-(a+b+c-1)*(4*a*b*c-a-b-c+1)); Output: Magma V2.11-10 Fri Dec 9 2005 07:43:11 on modular [Seed = 686952650] ------------------------------------- [ , ] Total time: 0.190 seconds, Total memory usage: 3.53MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Fri Dec 9 07:42:47 2005 Input: K:=PolynomialRing(RationalField(),3); //s:=(1-2*a)*(a+b+c-1); //print Factorization(s^2+(a+b+c-1)*(4*a*b*c-a-b-c+1)); print Factorization(a*b*c*(a+b-1)*(a+b+c-1)-(a+b+c-1)*(4*a*b*c-a-b-c+1)); Output: Magma V2.11-10 Fri Dec 9 2005 07:42:46 on modular [Seed = 854067913] ------------------------------------- [ , ] Total time: 0.190 seconds, Total memory usage: 3.34MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Fri Dec 9 07:39:45 2005 Input: K:=PolynomialRing(RationalField(),3); s:=(1-2*a)*(a+b+c-1); print Factorization(s^2+(a+b+c-1)*(4*a*b*c-a-b-c+1)); Output: Magma V2.11-10 Fri Dec 9 2005 07:39:44 on modular [Seed = 3239836143] ------------------------------------- [ , , , ] Total time: 0.190 seconds, Total memory usage: 3.24MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Fri Dec 9 07:39:28 2005 Input: K:=PolynomialRing(RationalField(),3); s:=(2*a-1)*(a+b+c-1); print Factorization(s^2-(a+b+c-1)*(4*a*b*c-a-b-c+1)); Output: Magma V2.11-10 Fri Dec 9 2005 07:39:28 on modular [Seed = 3340632057] ------------------------------------- [ , ] Total time: 0.190 seconds, Total memory usage: 3.34MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Fri Dec 9 07:39:22 2005 Input: K:=PolynomialRing(RationalField(),3); s:=(2*a-1)*(a+b+c-1); print Factorization(s^2-(a+b+c-1)*(4*a*b*c-a-b-c+1)); Output: Magma V2.11-10 Fri Dec 9 2005 07:39:21 on modular [Seed = 3440899082] ------------------------------------- [ , ] Total time: 0.190 seconds, Total memory usage: 3.34MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Fri Dec 9 07:39:13 2005 Input: K:=PolynomialRing(RationalField(),3); s:=(a-b)*(a+b+c-1); print Factorization(s^2-(a+b+c-1)*(4*a*b*c-a-b-c+1)); Output: Magma V2.11-10 Fri Dec 9 2005 07:39:13 on modular [Seed = 3490375967] ------------------------------------- [ , ] Total time: 0.200 seconds, Total memory usage: 3.34MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Fri Dec 9 07:38:58 2005 Input: K:=PolynomialRing(RationalField(),3); s:=a-b; print Factorization(s^2-(a+b+c-1)*(4*a*b*c-a-b-c+1)); Output: Magma V2.11-10 Fri Dec 9 2005 07:38:54 on modular [Seed = 3812219395] ------------------------------------- [ ] Total time: 0.240 seconds, Total memory usage: 3.24MB '192.122' ************** MAGMA ***************** Host 192.122.134.249 (192.122.134.249) Time: Fri Dec 9 00:30:06 2005 Input: F := GF(8); a :=PrimitiveElement(F); (1+a)^(-1); Output: Magma V2.11-10 Fri Dec 9 2005 00:30:05 on modular [Seed = 4213288604] ------------------------------------- F.1^4 Total time: 0.190 seconds, Total memory usage: 3.24MB '192.122' ************** MAGMA ***************** Host 192.122.134.249 (192.122.134.249) Time: Fri Dec 9 00:29:43 2005 Input: F := GF(8); a :=PrimitiveElement(F); a^(-1); Output: Magma V2.11-10 Fri Dec 9 2005 00:29:43 on modular [Seed = 2254230523] ------------------------------------- F.1^6 Total time: 0.190 seconds, Total memory usage: 3.24MB '192.122' ************** MAGMA ***************** Host 192.122.134.249 (192.122.134.249) Time: Fri Dec 9 00:29:07 2005 Input: F := GF(8); a :=PrimitiveElement(F); Inverse(a); Output: Magma V2.11-10 Fri Dec 9 2005 00:29:07 on modular [Seed = 2371344532] ------------------------------------- >> Inverse(a);; ^ Runtime error in 'Inverse': Argument 1 must be a variable reference (use ~) Argument types given: FldFinElt Total time: 0.190 seconds, Total memory usage: 3.24MB '192.122' ************** MAGMA ***************** Host 192.122.134.249 (192.122.134.249) Time: Fri Dec 9 00:28:02 2005 Input: F := GF(8); a :=PrimitiveElement(F); Trace(a+1); Output: Magma V2.11-10 Fri Dec 9 2005 00:28:02 on modular [Seed = 2337398099] ------------------------------------- 1 Total time: 0.180 seconds, Total memory usage: 3.24MB '192.122' ************** MAGMA ***************** Host 192.122.134.249 (192.122.134.249) Time: Fri Dec 9 00:26:31 2005 Input: F := GF(8); a :=PrimitiveElement(F); Trace(a); Output: Magma V2.11-10 Fri Dec 9 2005 00:26:31 on modular [Seed = 2521612331] ------------------------------------- 0 Total time: 0.190 seconds, Total memory usage: 3.24MB '192.122' ************** MAGMA ***************** Host 192.122.134.249 (192.122.134.249) Time: Fri Dec 9 00:25:50 2005 Input: F := GF(8); a :=PrimitiveElement(F); a; Output: Magma V2.11-10 Fri Dec 9 2005 00:25:50 on modular [Seed = 2638728185] ------------------------------------- F.1 Total time: 0.180 seconds, Total memory usage: 3.24MB '192.122' ************** MAGMA ***************** Host 192.122.134.249 (192.122.134.249) Time: Fri Dec 9 00:25:13 2005 Input: F := GF(8); F; Output: Magma V2.11-10 Fri Dec 9 2005 00:25:12 on modular [Seed = 2554515098] ------------------------------------- Finite field of size 2^3 Total time: 0.180 seconds, Total memory usage: 3.24MB '192.122' ************** MAGMA ***************** Host 192.122.134.249 (192.122.134.249) Time: Fri Dec 9 00:24:46 2005 Input: F := GF(8) Output: Magma V2.11-10 Fri Dec 9 2005 00:24:46 on modular [Seed = 2810048899] ------------------------------------- Total time: 0.200 seconds, Total memory usage: 3.24MB '65.191.' ************** MAGMA ***************** Host 65.191.73.113 (65.191.73.113) Time: Thu Dec 8 23:56:47 2005 Input: K := FiniteField(2); > C := LinearCode [1, 0, 0, 0, 0, 1, 1, 1], [0, 1, 0, 0, 1, 0, 1, 1], > [0, 0, 1, 0, 1, 1, 0, 1], [0, 0, 0, 1, 1, 1, 1, 0]>; > aut := AutomorphismGroup(C); Order(aut); > FactoredOrder(aut); > CompositionFactors(aut); Generators(aut); IsSelfDual(C); Output: Magma V2.11-10 Thu Dec 8 2005 23:56:47 on modular [Seed = 131943090] ------------------------------------- 1344 [ <2, 6>, <3, 1>, <7, 1> ] G | A(1, 7) = L(2, 7) * | Cyclic(2) * | Cyclic(2) * | Cyclic(2) 1 { (3, 4)(7, 8), (4, 6)(5, 7), (4, 7)(5, 6), (1, 2)(5, 6), (2, 4, 3)(6, 8, 7) } true Total time: 0.190 seconds, Total memory usage: 3.24MB '65.191.' ************** MAGMA ***************** Host 65.191.73.113 (65.191.73.113) Time: Thu Dec 8 23:56:22 2005 Input: K := FiniteField(2); > C := LinearCode [1, 0, 0, 0, 0, 1, 1, 1], [0, 1, 0, 0, 1, 0, 1, 1], > [0, 0, 1, 0, 1, 1, 0, 1], [0, 0, 0, 1, 1, 1, 1, 0]>; > aut := AutomorphismGroup(C); Order(aut); > FactoredOrder(aut); > CompositionFactors(aut); Generators(aut); IsSelfOrthogonal(C); Output: Magma V2.11-10 Thu Dec 8 2005 23:56:22 on modular [Seed = 47730510] ------------------------------------- 1344 [ <2, 6>, <3, 1>, <7, 1> ] G | A(1, 7) = L(2, 7) * | Cyclic(2) * | Cyclic(2) * | Cyclic(2) 1 { (3, 4)(7, 8), (4, 6)(5, 7), (4, 7)(5, 6), (1, 2)(5, 6), (2, 4, 3)(6, 8, 7) } true Total time: 0.190 seconds, Total memory usage: 3.24MB '65.191.' ************** MAGMA ***************** Host 65.191.73.113 (65.191.73.113) Time: Thu Dec 8 23:55:57 2005 Input: K := FiniteField(2); > C := LinearCode [1, 0, 0, 0, 0, 1, 1, 1], [0, 1, 0, 0, 1, 0, 1, 1], > [0, 0, 1, 0, 1, 1, 0, 1], [0, 0, 0, 1, 1, 1, 1, 0]>; > aut := AutomorphismGroup(C); Order(aut); > FactoredOrder(aut); > CompositionFactors(aut); Generators(aut); IsSelfOrthogonal(C) : Code -> BoolElt Output: Magma V2.11-10 Thu Dec 8 2005 23:55:56 on modular [Seed = 164840559] ------------------------------------- 1344 [ <2, 6>, <3, 1>, <7, 1> ] G | A(1, 7) = L(2, 7) * | Cyclic(2) * | Cyclic(2) * | Cyclic(2) 1 { (3, 4)(7, 8), (4, 6)(5, 7), (4, 7)(5, 6), (1, 2)(5, 6), (2, 4, 3)(6, 8, 7) } >> IsSelfOrthogonal(C) : Code -> BoolElt ^ User error: bad syntax Total time: 0.180 seconds, Total memory usage: 3.24MB '65.191.' ************** MAGMA ***************** Host 65.191.73.113 (65.191.73.113) Time: Thu Dec 8 23:52:06 2005 Input: K := FiniteField(2); > C := LinearCode [1, 0, 0, 0, 0, 1, 1, 1], [0, 1, 0, 0, 1, 0, 1, 1], > [0, 0, 1, 0, 1, 1, 0, 1], [0, 0, 0, 1, 1, 1, 1, 0]>; > aut := AutomorphismGroup(C); Order(aut); > FactoredOrder(aut); > CompositionFactors(aut); Generators(aut); Output: Magma V2.11-10 Thu Dec 8 2005 23:52:06 on modular [Seed = 298540893] ------------------------------------- 1344 [ <2, 6>, <3, 1>, <7, 1> ] G | A(1, 7) = L(2, 7) * | Cyclic(2) * | Cyclic(2) * | Cyclic(2) 1 { (3, 4)(7, 8), (4, 6)(5, 7), (4, 7)(5, 6), (1, 2)(5, 6), (2, 4, 3)(6, 8, 7) } Total time: 0.190 seconds, Total memory usage: 3.24MB '65.191.' ************** MAGMA ***************** Host 65.191.73.113 (65.191.73.113) Time: Thu Dec 8 23:50:33 2005 Input: K := FiniteField(2); > C := LinearCode [1, 0, 0, 0, 0, 1, 1, 1], [0, 1, 0, 0, 1, 0, 1, 1], > [0, 0, 1, 0, 1, 1, 0, 1], [0, 0, 0, 1, 1, 1, 1, 0]>; > aut := AutomorphismGroup(C); Order(aut); > FactoredOrder(aut); > CompositionFactors(aut); Output: Magma V2.11-10 Thu Dec 8 2005 23:50:33 on modular [Seed = 482759231] ------------------------------------- 1344 [ <2, 6>, <3, 1>, <7, 1> ] G | A(1, 7) = L(2, 7) * | Cyclic(2) * | Cyclic(2) * | Cyclic(2) 1 Total time: 0.190 seconds, Total memory usage: 3.24MB '65.191.' ************** MAGMA ***************** Host 65.191.73.113 (65.191.73.113) Time: Thu Dec 8 23:49:30 2005 Input: K := FiniteField(2); > C := LinearCode [1, 0, 0, 0, 0, 1, 1, 1], [0, 1, 0, 0, 1, 0, 1, 1], > [0, 0, 1, 0, 1, 1, 0, 1], [0, 0, 0, 1, 1, 1, 1, 0]>; > aut := AutomorphismGroup(C); > FactoredOrder(aut); > CompositionFactors(aut); Output: Magma V2.11-10 Thu Dec 8 2005 23:49:30 on modular [Seed = 465655035] ------------------------------------- [ <2, 6>, <3, 1>, <7, 1> ] G | A(1, 7) = L(2, 7) * | Cyclic(2) * | Cyclic(2) * | Cyclic(2) 1 Total time: 0.190 seconds, Total memory usage: 3.24MB '65.191.' ************** MAGMA ***************** Host 65.191.73.113 (65.191.73.113) Time: Thu Dec 8 23:43:43 2005 Input: K := FiniteField(2); > C := LinearCode [1, 0, 0, 0, 0, 1, 1, 1], [0, 1, 0, 0, 1, 0, 1, 1], > [0, 0, 1, 0, 1, 1, 0, 1], [0, 0, 0, 1, 1, 1, 1, 0]>; AutomorphismGroup(C: parameters) : Code -> GrpPerm, PowMap, Map Output: Magma V2.11-10 Thu Dec 8 2005 23:43:42 on modular [Seed = 754345064] ------------------------------------- >> AutomorphismGroup(C: parameters) : Code -> GrpPerm, PowMap, Map ^ User error: bad syntax Total time: 0.190 seconds, Total memory usage: 3.24MB '65.191.' ************** MAGMA ***************** Host 65.191.73.113 (65.191.73.113) Time: Thu Dec 8 23:40:08 2005 Input: K := FiniteField(2); > C := LinearCode [1, 0, 0, 0, 0, 1, 1, 1], [0, 1, 0, 0, 1, 0, 1, 1], > [0, 0, 1, 0, 1, 1, 0, 1], [0, 0, 0, 1, 1, 1, 1, 0]>; IsSelfOrthogonal(C) : Code -> BoolElt Output: Magma V2.11-10 Thu Dec 8 2005 23:40:08 on modular [Seed = 904887463] ------------------------------------- >> IsSelfOrthogonal(C) : Code -> BoolElt ^ User error: bad syntax Total time: 0.190 seconds, Total memory usage: 3.24MB '65.191.' ************** MAGMA ***************** Host 65.191.73.113 (65.191.73.113) Time: Thu Dec 8 23:39:47 2005 Input: K := FiniteField(2); > C := LinearCode [1, 0, 0, 0, 0, 1, 1, 1], [0, 1, 0, 0, 1, 0, 1, 1], > [0, 0, 1, 0, 1, 1, 0, 1], [0, 0, 0, 1, 1, 1, 1, 0]>; IsSelfOrthogonal(C) : Code -> BoolElt; Output: Magma V2.11-10 Thu Dec 8 2005 23:39:47 on modular [Seed = 820672921] ------------------------------------- >> IsSelfOrthogonal(C) : Code -> BoolElt; ^ User error: bad syntax Total time: 0.180 seconds, Total memory usage: 3.24MB '65.191.' ************** MAGMA ***************** Host 65.191.73.113 (65.191.73.113) Time: Thu Dec 8 23:39:10 2005 Input: K := FiniteField(2); > C := LinearCode [1, 0, 0, 0, 0, 1, 1, 1], [0, 1, 0, 0, 1, 0, 1, 1], > [0, 0, 1, 0, 1, 1, 0, 1], [0, 0, 0, 1, 1, 1, 1, 0]>; Output: Magma V2.11-10 Thu Dec 8 2005 23:39:10 on modular [Seed = 1072002667] ------------------------------------- Total time: 0.190 seconds, Total memory usage: 3.24MB '65.191.' ************** MAGMA ***************** Host 65.191.73.113 (65.191.73.113) Time: Thu Dec 8 23:38:44 2005 Input: K := FiniteField(2); > C := LinearCode [1, 0, 0, 0, 0, 1, 1, 1], [0, 1, 0, 0, 1, 0, 1, 1], > [0, 0, 1, 0, 1, 1, 0, 1], [0, 0, 0, 1, 1, 1, 1, 0]; Output: Magma V2.11-10 Thu Dec 8 2005 23:38:44 on modular [Seed = 3474294432] ------------------------------------- >> [0, 0, 1, 0, 1, 1, 0, 1], [0, 0, 0, 1, 1, 1, 1, 0]; ^ User error: bad syntax Total time: 0.190 seconds, Total memory usage: 3.24MB '65.191.' ************** MAGMA ***************** Host 65.191.73.113 (65.191.73.113) Time: Thu Dec 8 23:35:40 2005 Input: C := BCHCode(GF(2),23,2); C; Output: Magma V2.11-10 Thu Dec 8 2005 23:35:40 on modular [Seed = 3607993712] ------------------------------------- [23, 12, 7] "BCH code (d = 2, b = 1)" Cyclic Linear Code over GF(2) Generator matrix: [1 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 1 1 1 0 1 0] [0 1 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 1 1 1 0 1] [0 0 1 0 0 0 0 0 0 0 0 0 1 1 1 1 0 1 1 0 1 0 0] [0 0 0 1 0 0 0 0 0 0 0 0 0 1 1 1 1 0 1 1 0 1 0] [0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 1 1 1 0 1 1 0 1] [0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 1 1 0 0 1 1 0 0] [0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 1 1 0 0 1 1 0] [0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 1 1 0 0 1 1] [0 0 0 0 0 0 0 0 1 0 0 0 1 1 0 1 1 1 0 0 0 1 1] [0 0 0 0 0 0 0 0 0 1 0 0 1 0 1 0 1 0 0 1 0 1 1] [0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 1 0 0 1 1 1 1 1] [0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 1 1 1 0 1 0 1] Total time: 0.190 seconds, Total memory usage: 3.24MB '65.191.' ************** MAGMA ***************** Host 65.191.73.113 (65.191.73.113) Time: Thu Dec 8 23:34:17 2005 Input: > C := BCHCode(GF(2),23,2); > C; Output: Magma V2.11-10 Thu Dec 8 2005 23:34:17 on modular [Seed = 3691419310] ------------------------------------- [23, 12, 7] "BCH code (d = 2, b = 1)" Cyclic Linear Code over GF(2) Generator matrix: [1 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 1 1 1 0 1 0] [0 1 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 1 1 1 0 1] [0 0 1 0 0 0 0 0 0 0 0 0 1 1 1 1 0 1 1 0 1 0 0] [0 0 0 1 0 0 0 0 0 0 0 0 0 1 1 1 1 0 1 1 0 1 0] [0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 1 1 1 0 1 1 0 1] [0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 1 1 0 0 1 1 0 0] [0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 1 1 0 0 1 1 0] [0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 1 1 0 0 1 1] [0 0 0 0 0 0 0 0 1 0 0 0 1 1 0 1 1 1 0 0 0 1 1] [0 0 0 0 0 0 0 0 0 1 0 0 1 0 1 0 1 0 0 1 0 1 1] [0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 1 0 0 1 1 1 1 1] [0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 1 1 1 0 1 0 1] Total time: 0.190 seconds, Total memory usage: 3.24MB '65.191.' ************** MAGMA ***************** Host 65.191.73.113 (65.191.73.113) Time: Thu Dec 8 23:10:54 2005 Input: 2+3 Output: Magma V2.11-10 Thu Dec 8 2005 23:10:54 on modular [Seed = 3812205334] ------------------------------------- 5 Total time: 0.190 seconds, Total memory usage: 3.24MB '60.225.' ************** MAGMA ***************** Host 60.225.131.213 (60.225.131.213) Time: Thu Dec 8 22:13:05 2005 Input: Factorization(1111111111111111111111111111111111112222222222222222222222222222222222222222222222222222222222222222223); Output: Magma V2.11-10 Thu Dec 8 2005 22:13:04 on modular [Seed = 1569122342] ------------------------------------- [ <3, 2>, <41, 1>, <191, 1>, <6246943, 1>, <252365633729939524730246496421381170326644302337345845070768633297938261556952\ 8675426211359, 1> ] Total time: 0.210 seconds, Total memory usage: 3.24MB '60.225.' ************** MAGMA ***************** Host 60.225.131.213 (60.225.131.213) Time: Thu Dec 8 22:12:31 2005 Input: Factorization(11111111111111111111111111111111111122222222222222222222222222222222222222222222222222222222222222222233333333333333333333333333333333333333333333333333333333333333333333333333333333333333); Output: ** WARNING: Computation time exceeded 20 seconds, so computation was terminated after 20 seconds. ** Magma V2.11-10 Thu Dec 8 2005 22:12:11 on modular [Seed = 1518464874] ------------------------------------- Errors: /bin/sh: line 1: 30817 Alarm clock nice -n 19 /usr/local/bin/magma '60.225.' ************** MAGMA ***************** Host 60.225.131.213 (60.225.131.213) Time: Thu Dec 8 22:10:50 2005 Input: Factorization(1111111111111111111111111111111111112222222222222222222222222222222222222222222222222222222222222222223333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444); Output: ** WARNING: Computation time exceeded 20 seconds, so computation was terminated after 20 seconds. ** Magma V2.11-10 Thu Dec 8 2005 22:10:29 on modular [Seed = 1740427769] ------------------------------------- Errors: /bin/sh: line 1: 30812 Alarm clock nice -n 19 /usr/local/bin/magma '60.225.' ************** MAGMA ***************** Host 60.225.131.213 (60.225.131.213) Time: Thu Dec 8 22:09:25 2005 Input: Factorization(1111111111111111111111111111111111112222222222222222222222222222222222222222222222222222222222222222223333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444455555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555566666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666); Output: ** WARNING: Computation time exceeded 20 seconds, so computation was terminated after 20 seconds. ** Magma V2.11-10 Thu Dec 8 2005 22:09:05 on modular [Seed = 1823988230] ------------------------------------- Errors: /bin/sh: line 1: 30807 Alarm clock nice -n 19 /usr/local/bin/magma '60.225.' ************** MAGMA ***************** Host 60.225.131.213 (60.225.131.213) Time: Thu Dec 8 22:08:02 2005 Input: Factorization(11111111111111111111111111111111111122222222222222222222222222222222222222222222222222222222222222222233333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333334444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666667777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999990); Output: ** WARNING: Computation time exceeded 20 seconds, so computation was terminated after 20 seconds. ** Magma V2.11-10 Thu Dec 8 2005 22:07:41 on modular [Seed = 2024001470] ------------------------------------- Errors: /bin/sh: line 1: 30787 Alarm clock nice -n 19 /usr/local/bin/magma '60.225.' ************** MAGMA ***************** Host 60.225.131.213 (60.225.131.213) Time: Thu Dec 8 22:06:11 2005 Input: Factorization(111111111111111111111111111111111111222222222222222222222222222222222222222222222222222222222222222222333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333344444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444445555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555556666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666677777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777778888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888889999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999900000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000); Output: ** WARNING: Computation time exceeded 20 seconds, so computation was terminated after 20 seconds. ** Magma V2.11-10 Thu Dec 8 2005 22:05:50 on modular [Seed = 115192243] ------------------------------------- Errors: /bin/sh: line 1: 30781 Alarm clock nice -n 19 /usr/local/bin/magma '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Thu Dec 8 18:52:57 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); A1:=Matrix(K,3,3,[[b,1,1],[0,a,1],[0,0,c]]); S:=Transpose(A)*A^(-1); F,T:=PrimaryRationalForm(S); B:=T*A*Transpose(T); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*B*Transpose(P)-B)); G:=GroebnerBasis(I); Q1:=Matrix(K,3,3,[(2*b-1)/(a+b-1),(a-b)/(a+b-1),0,(b-a)/(a+b-1),(2*a-1)/(a+b-1),0, 0,0,1]); Kt:=FunctionField(K); D:=(2*a*b*c-a-b-c+1)/(a*b*c); s:=t; px:=-(2*s+D); py:=1-s^2; q:=s^2+s*D+1; Q:=Matrix(Kt,3,3,[1,0,0,0,D*px/q+py/q,-px/q,0,px/q,py/q]); Q:=Q1*T^(-1)*Q*T; print (Q*A*Transpose(Q)-A1); print Factorization(Numerator(Evaluate(q,-D/2+(1-2*a)*(a+b+c-1)/(2*a*b*c)))); for i in [1..3] do for j in [1..3] do print Evaluate(Q[i,j],-D/2+(2*a-1)*(a+b+c-1)/(2*a*b*c)); end for; end for; Output: Magma V2.11-10 Thu Dec 8 2005 18:52:57 on modular [Seed = 432184689] ------------------------------------- [0 0 0] [0 0 0] [0 0 0] [ , , ] 1/a -1 0 (-a^2 + a*c + 2*a - 1)/(a^2 + a*c - a) 0 (-2*a + 1)/(a + c - 1) (2*a*c - c)/(a^2 + a*c - a) 0 (-a + c)/(a + c - 1) Total time: 0.370 seconds, Total memory usage: 3.63MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Thu Dec 8 18:51:58 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); A1:=Matrix(K,3,3,[[b,1,1],[0,a,1],[0,0,c]]); S:=Transpose(A)*A^(-1); F,T:=PrimaryRationalForm(S); B:=T*A*Transpose(T); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*B*Transpose(P)-B)); G:=GroebnerBasis(I); Q1:=Matrix(K,3,3,[(2*b-1)/(a+b-1),(a-b)/(a+b-1),0,(b-a)/(a+b-1),(2*a-1)/(a+b-1),0, 0,0,1]); Kt:=FunctionField(K); D:=(2*a*b*c-a-b-c+1)/(a*b*c); s:=t; px:=-(2*s+D); py:=1-s^2; q:=s^2+s*D+1; Q:=Matrix(Kt,3,3,[1,0,0,0,D*px/q+py/q,-px/q,0,px/q,py/q]); Q:=Q1*T^(-1)*Q*T; print (Q*A*Transpose(Q)-A1); print Factorization(Numerator(Evaluate(q,-D/2+(1-2*a)*(a+b+c-1)/(2*a*b*c)))); for i in [1..3] do for j in [1..3] do print Evaluate(Q[i,j],-D/2+(1-2*a)*(a+b+c-1)/(2*a*b*c)); end for; end for; Output: Magma V2.11-10 Thu Dec 8 2005 18:51:58 on modular [Seed = 315201085] ------------------------------------- [0 0 0] [0 0 0] [0 0 0] [ , , ] (a^3 - 4*a^2*b*c + a^2*c - a^2 + 4*a*b^2*c - a*b^2 + a*b - b^2*c)/(a^4 + 2*a^3*b + a^3*c - 3*a^3 + a^2*b^2 + 2*a^2*b*c - 4*a^2*b - 2*a^2*c + 3*a^2 + a*b^2*c - a*b^2 - 2*a*b*c + 2*a*b + a*c - a) (-a^4 - 2*a^3*b - a^3*c + a^3 - a^2*b^2 + 6*a^2*b*c + 2*a^2*b - a*b^2*c + a*b^2 - 4*a*b*c - a*b + b*c)/(a^4 + 2*a^3*b + a^3*c - 3*a^3 + a^2*b^2 + 2*a^2*b*c - 4*a^2*b - 2*a^2*c + 3*a^2 + a*b^2*c - a*b^2 - 2*a*b*c + 2*a*b + a*c - a) (4*a^2*b - 4*a*b + b)/(a^3 + a^2*b + a^2*c - 2*a^2 + a*b*c - a*b - a*c + a) (-a^4 - 2*a^3*b + a^3*c + 4*a^3 - a^2*b^2 - 6*a^2*b*c + 4*a^2*b - 4*a^2 + a*b^2*c + 4*a*b*c - a*b + a - b*c)/(a^4 + 2*a^3*b + a^3*c - 3*a^3 + a^2*b^2 + 2*a^2*b*c - 4*a^2*b - 2*a^2*c + 3*a^2 + a*b^2*c - a*b^2 - 2*a*b*c + 2*a*b + a*c - a) (-4*a^3*c - 2*a^3 + 4*a^2*b*c - 2*a^2*b + 6*a^2*c + 3*a^2 - 2*a*b*c + a*b - 4*a*c - a + c)/(a^4 + 2*a^3*b + a^3*c - 3*a^3 + a^2*b^2 + 2*a^2*b*c - 4*a^2*b - 2*a^2*c + 3*a^2 + a*b^2*c - a*b^2 - 2*a*b*c + 2*a*b + a*c - a) (-2*a^3 + 2*a^2*b + 5*a^2 - a*b - 4*a + 1)/(a^3 + a^2*b + a^2*c - 2*a^2 + a*b*c - a*b - a*c + a) (2*a^2*c - 2*a*b*c - a*c + b*c)/(a^3 + a^2*b + a^2*c - 2*a^2 + a*b*c - a*b - a*c + a) (-4*a^2*c + 4*a*c - c)/(a^3 + a^2*b + a^2*c - 2*a^2 + a*b*c - a*b - a*c + a) (-a^2 + a*c + 2*a - 1)/(a^2 + a*c - a) Total time: 0.370 seconds, Total memory usage: 3.63MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Thu Dec 8 18:49:04 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); A1:=Matrix(K,3,3,[[b,1,1],[0,a,1],[0,0,c]]); S:=Transpose(A)*A^(-1); F,T:=PrimaryRationalForm(S); B:=T*A*Transpose(T); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*B*Transpose(P)-B)); G:=GroebnerBasis(I); Q1:=Matrix(K,3,3,[(2*b-1)/(a+b-1),(a-b)/(a+b-1),0,(b-a)/(a+b-1),(2*a-1)/(a+b-1),0, 0,0,1]); Kt:=FunctionField(K); D:=(2*a*b*c-a-b-c+1)/(a*b*c); s:=t; px:=-(2*s+D); py:=1-s^2; q:=s^2+s*D+1; Q:=Matrix(Kt,3,3,[1,0,0,0,D*px/q+py/q,-px/q,0,px/q,py/q]); Q:=Q1*T^(-1)*Q*T; print (Q*A*Transpose(Q)-A1); print Factorization(Numerator(Evaluate(q,-D/2+(1-2*a)*(a+b+c-1)/(2*a*b*c)))); for i in [1..3] do for j in [1..3] do print Evaluate(Q[i,j],-D/2+(0)*(a+b+c-1)/(2*a*b*c)); end for; end for; print Factorization(Numerator(2*(a-b)*(a+b+c-1)*(a+b-1)*(b+c-1)*(c+a-1)-(a+b+c-1)*(4*a*b*c-a-b-c+1))); Output: Magma V2.11-10 Thu Dec 8 2005 18:49:04 on modular [Seed = 720482155] ------------------------------------- [0 0 0] [0 0 0] [0 0 0] [ , , ] (2*b - 1)/(a + b - 1) (a - b)/(a + b - 1) 0 (-a + b)/(a + b - 1) (2*a - 1)/(a + b - 1) 0 0 0 1 [ , ] Total time: 0.370 seconds, Total memory usage: 3.72MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Thu Dec 8 18:48:41 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); A1:=Matrix(K,3,3,[[b,1,1],[0,a,1],[0,0,c]]); S:=Transpose(A)*A^(-1); F,T:=PrimaryRationalForm(S); B:=T*A*Transpose(T); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*B*Transpose(P)-B)); G:=GroebnerBasis(I); Q1:=Matrix(K,3,3,[(2*b-1)/(a+b-1),(a-b)/(a+b-1),0,(b-a)/(a+b-1),(2*a-1)/(a+b-1),0, 0,0,1]); Kt:=FunctionField(K); D:=(2*a*b*c-a-b-c+1)/(a*b*c); s:=t; px:=-(2*s+D); py:=1-s^2; q:=s^2+s*D+1; Q:=Matrix(Kt,3,3,[1,0,0,0,D*px/q+py/q,-px/q,0,px/q,py/q]); Q:=Q1*T^(-1)*Q*T; print (Q*A*Transpose(Q)-A1); print Factorization(Numerator(Evaluate(q,-D/2+(1-2*a)*(a+b+c-1)/(2*a*b*c)))); for i in [1..3] do for j in [1..3] do print Evaluate(Q[i,j],-D/2+(0)*(a+b+c-1)/(2*a*b*c)); end for; end for; print Factorization(Numerator(2*a*b*(a+b+c-1)*(a+b-1)*(b+c-1)*(c+a-1)-(a+b+c-1)*(4*a*b*c-a-b-c+1))); Output: Magma V2.11-10 Thu Dec 8 2005 18:48:41 on modular [Seed = 670868047] ------------------------------------- [0 0 0] [0 0 0] [0 0 0] [ , , ] (2*b - 1)/(a + b - 1) (a - b)/(a + b - 1) 0 (-a + b)/(a + b - 1) (2*a - 1)/(a + b - 1) 0 0 0 1 [ , ] Total time: 0.370 seconds, Total memory usage: 3.82MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Thu Dec 8 18:48:11 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); A1:=Matrix(K,3,3,[[b,1,1],[0,a,1],[0,0,c]]); S:=Transpose(A)*A^(-1); F,T:=PrimaryRationalForm(S); B:=T*A*Transpose(T); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*B*Transpose(P)-B)); G:=GroebnerBasis(I); Q1:=Matrix(K,3,3,[(2*b-1)/(a+b-1),(a-b)/(a+b-1),0,(b-a)/(a+b-1),(2*a-1)/(a+b-1),0, 0,0,1]); Kt:=FunctionField(K); D:=(2*a*b*c-a-b-c+1)/(a*b*c); s:=t; px:=-(2*s+D); py:=1-s^2; q:=s^2+s*D+1; Q:=Matrix(Kt,3,3,[1,0,0,0,D*px/q+py/q,-px/q,0,px/q,py/q]); Q:=Q1*T^(-1)*Q*T; print (Q*A*Transpose(Q)-A1); print Factorization(Numerator(Evaluate(q,-D/2+(1-2*a)*(a+b+c-1)/(2*a*b*c)))); for i in [1..3] do for j in [1..3] do print Evaluate(Q[i,j],-D/2+(0)*(a+b+c-1)/(2*a*b*c)); end for; end for; print Factorization(Numerator(2*a*b*c*(a+b+c-1)*(a+b-1)*(b+c-1)*(c+a-1)-(a+b+c-1)*(4*a*b*c-a-b-c+1))); Output: Magma V2.11-10 Thu Dec 8 2005 18:48:10 on modular [Seed = 1055243061] ------------------------------------- [0 0 0] [0 0 0] [0 0 0] [ , , ] (2*b - 1)/(a + b - 1) (a - b)/(a + b - 1) 0 (-a + b)/(a + b - 1) (2*a - 1)/(a + b - 1) 0 0 0 1 [ , ] Total time: 0.380 seconds, Total memory usage: 3.82MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Thu Dec 8 18:47:45 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); A1:=Matrix(K,3,3,[[b,1,1],[0,a,1],[0,0,c]]); S:=Transpose(A)*A^(-1); F,T:=PrimaryRationalForm(S); B:=T*A*Transpose(T); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*B*Transpose(P)-B)); G:=GroebnerBasis(I); Q1:=Matrix(K,3,3,[(2*b-1)/(a+b-1),(a-b)/(a+b-1),0,(b-a)/(a+b-1),(2*a-1)/(a+b-1),0, 0,0,1]); Kt:=FunctionField(K); D:=(2*a*b*c-a-b-c+1)/(a*b*c); s:=t; px:=-(2*s+D); py:=1-s^2; q:=s^2+s*D+1; Q:=Matrix(Kt,3,3,[1,0,0,0,D*px/q+py/q,-px/q,0,px/q,py/q]); Q:=Q1*T^(-1)*Q*T; print (Q*A*Transpose(Q)-A1); print Factorization(Numerator(Evaluate(q,-D/2+(1-2*a)*(a+b+c-1)/(2*a*b*c)))); for i in [1..3] do for j in [1..3] do print Evaluate(Q[i,j],-D/2+(0)*(a+b+c-1)/(2*a*b*c)); end for; end for; print Factorization(Numerator((a+b+c-1)*(a+b-1)*(b+c-1)*(c+a-1)-(a+b+c-1)*(4*a*b*c-a-b-c+1))); Output: Magma V2.11-10 Thu Dec 8 2005 18:47:45 on modular [Seed = 1004584491] ------------------------------------- [0 0 0] [0 0 0] [0 0 0] [ , , ] (2*b - 1)/(a + b - 1) (a - b)/(a + b - 1) 0 (-a + b)/(a + b - 1) (2*a - 1)/(a + b - 1) 0 0 0 1 [ , ] Total time: 0.380 seconds, Total memory usage: 3.63MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Thu Dec 8 18:39:28 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); A1:=Matrix(K,3,3,[[b,1,1],[0,a,1],[0,0,c]]); S:=Transpose(A)*A^(-1); F,T:=PrimaryRationalForm(S); B:=T*A*Transpose(T); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*B*Transpose(P)-B)); G:=GroebnerBasis(I); Q1:=Matrix(K,3,3,[(2*b-1)/(a+b-1),(a-b)/(a+b-1),0,(b-a)/(a+b-1),(2*a-1)/(a+b-1),0, 0,0,1]); Kt:=FunctionField(K); D:=(2*a*b*c-a-b-c+1)/(a*b*c); s:=t; px:=-(2*s+D); py:=1-s^2; q:=s^2+s*D+1; Q:=Matrix(Kt,3,3,[1,0,0,0,D*px/q+py/q,-px/q,0,px/q,py/q]); Q:=Q1*T^(-1)*Q*T; print (Q*A*Transpose(Q)-A1); print Factorization(Numerator(Evaluate(q,-D/2+(1-2*a)*(a+b+c-1)/(2*a*b*c)))); for i in [1..3] do for j in [1..3] do print Evaluate(Q[i,j],-D/2+(0)*(a+b+c-1)/(2*a*b*c)); end for; end for; print (2*(a+b-1)*(b+c-1)*(a+c-1)-4*a*b*c+a+b+c-1); print Factorization(Numerator(2*a^2*b + 2*a^2*c - 2*a^2 + 2*a*b^2 - 6*a*b + 2*a*c^2 - 6*a*c + 5*a + 2*b^2*c - 2*b^2 + 2*b*c^2 - 6*b*c + 5*b - 2*c^2 + 5*c - 3 )); Output: Magma V2.11-10 Thu Dec 8 2005 18:39:28 on modular [Seed = 3474213069] ------------------------------------- [0 0 0] [0 0 0] [0 0 0] [ , , ] (2*b - 1)/(a + b - 1) (a - b)/(a + b - 1) 0 (-a + b)/(a + b - 1) (2*a - 1)/(a + b - 1) 0 0 0 1 2*a^2*b + 2*a^2*c - 2*a^2 + 2*a*b^2 - 6*a*b + 2*a*c^2 - 6*a*c + 5*a + 2*b^2*c - 2*b^2 + 2*b*c^2 - 6*b*c + 5*b - 2*c^2 + 5*c - 3 [ ] Total time: 0.370 seconds, Total memory usage: 3.63MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Thu Dec 8 18:39:09 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); A1:=Matrix(K,3,3,[[b,1,1],[0,a,1],[0,0,c]]); S:=Transpose(A)*A^(-1); F,T:=PrimaryRationalForm(S); B:=T*A*Transpose(T); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*B*Transpose(P)-B)); G:=GroebnerBasis(I); Q1:=Matrix(K,3,3,[(2*b-1)/(a+b-1),(a-b)/(a+b-1),0,(b-a)/(a+b-1),(2*a-1)/(a+b-1),0, 0,0,1]); Kt:=FunctionField(K); D:=(2*a*b*c-a-b-c+1)/(a*b*c); s:=t; px:=-(2*s+D); py:=1-s^2; q:=s^2+s*D+1; Q:=Matrix(Kt,3,3,[1,0,0,0,D*px/q+py/q,-px/q,0,px/q,py/q]); Q:=Q1*T^(-1)*Q*T; print (Q*A*Transpose(Q)-A1); print Factorization(Numerator(Evaluate(q,-D/2+(1-2*a)*(a+b+c-1)/(2*a*b*c)))); for i in [1..3] do for j in [1..3] do print Evaluate(Q[i,j],-D/2+(0)*(a+b+c-1)/(2*a*b*c)); end for; end for; print (2*(a+b-1)*(b+c-1)*(a+c-1)-4*a*b*c+a+b+c-1); print Factorization(2*a^2*b + 2*a^2*c - 2*a^2 + 2*a*b^2 - 6*a*b + 2*a*c^2 - 6*a*c + 5*a + 2*b^2*c - 2*b^2 + 2*b*c^2 - 6*b*c + 5*b - 2*c^2 + 5*c - 3 ); Output: Magma V2.11-10 Thu Dec 8 2005 18:39:08 on modular [Seed = 3307094763] ------------------------------------- [0 0 0] [0 0 0] [0 0 0] [ , , ] (2*b - 1)/(a + b - 1) (a - b)/(a + b - 1) 0 (-a + b)/(a + b - 1) (2*a - 1)/(a + b - 1) 0 0 0 1 2*a^2*b + 2*a^2*c - 2*a^2 + 2*a*b^2 - 6*a*b + 2*a*c^2 - 6*a*c + 5*a + 2*b^2*c - 2*b^2 + 2*b*c^2 - 6*b*c + 5*b - 2*c^2 + 5*c - 3 >> print Factorization(2*a^2*b + 2*a^2*c - 2*a^2 + 2*a*b^2 - 6*a*b + 2*a*c^2 - ^ Runtime error in 'Factorization': Bad argument types Argument types given: FldFunRatMElt Total time: 0.370 seconds, Total memory usage: 3.63MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Thu Dec 8 18:38:30 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); A1:=Matrix(K,3,3,[[b,1,1],[0,a,1],[0,0,c]]); S:=Transpose(A)*A^(-1); F,T:=PrimaryRationalForm(S); B:=T*A*Transpose(T); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*B*Transpose(P)-B)); G:=GroebnerBasis(I); Q1:=Matrix(K,3,3,[(2*b-1)/(a+b-1),(a-b)/(a+b-1),0,(b-a)/(a+b-1),(2*a-1)/(a+b-1),0, 0,0,1]); Kt:=FunctionField(K); D:=(2*a*b*c-a-b-c+1)/(a*b*c); s:=t; px:=-(2*s+D); py:=1-s^2; q:=s^2+s*D+1; Q:=Matrix(Kt,3,3,[1,0,0,0,D*px/q+py/q,-px/q,0,px/q,py/q]); Q:=Q1*T^(-1)*Q*T; print (Q*A*Transpose(Q)-A1); print Factorization(Numerator(Evaluate(q,-D/2+(1-2*a)*(a+b+c-1)/(2*a*b*c)))); for i in [1..3] do for j in [1..3] do print Evaluate(Q[i,j],-D/2+(0)*(a+b+c-1)/(2*a*b*c)); end for; end for; print (2*(a+b-1)*(b+c-1)*(a+c-1)-4*a*b*c+a+b+c-1); Output: Magma V2.11-10 Thu Dec 8 2005 18:38:30 on modular [Seed = 3607910163] ------------------------------------- [0 0 0] [0 0 0] [0 0 0] [ , , ] (2*b - 1)/(a + b - 1) (a - b)/(a + b - 1) 0 (-a + b)/(a + b - 1) (2*a - 1)/(a + b - 1) 0 0 0 1 2*a^2*b + 2*a^2*c - 2*a^2 + 2*a*b^2 - 6*a*b + 2*a*c^2 - 6*a*c + 5*a + 2*b^2*c - 2*b^2 + 2*b*c^2 - 6*b*c + 5*b - 2*c^2 + 5*c - 3 Total time: 0.370 seconds, Total memory usage: 3.63MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Thu Dec 8 18:37:25 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); A1:=Matrix(K,3,3,[[b,1,1],[0,a,1],[0,0,c]]); S:=Transpose(A)*A^(-1); F,T:=PrimaryRationalForm(S); B:=T*A*Transpose(T); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*B*Transpose(P)-B)); G:=GroebnerBasis(I); Q1:=Matrix(K,3,3,[(2*b-1)/(a+b-1),(a-b)/(a+b-1),0,(b-a)/(a+b-1),(2*a-1)/(a+b-1),0, 0,0,1]); Kt:=FunctionField(K); D:=(2*a*b*c-a-b-c+1)/(a*b*c); s:=t; px:=-(2*s+D); py:=1-s^2; q:=s^2+s*D+1; Q:=Matrix(Kt,3,3,[1,0,0,0,D*px/q+py/q,-px/q,0,px/q,py/q]); Q:=Q1*T^(-1)*Q*T; print (Q*A*Transpose(Q)-A1); print Factorization(Numerator(Evaluate(q,-D/2+(1-2*a)*(a+b+c-1)/(2*a*b*c)))); for i in [1..3] do for j in [1..3] do print Evaluate(Q[i,j],-D/2+(0)*(a+b+c-1)/(2*a*b*c)); end for; end for; print ((a+b-1)*(b+c-1)*(a+c-1)-4*a*b*c+a+b+c-1); print Factorization(Numerator(a^2*b + a^2*c - a^2 + a*b^2 - 2*a*b*c - 3*a*b + a*c^2 - 3*a*c + 3*a + b^2*c - b^2 + b*c^2 - 3*b*c + 3*b - c^2 + 3*c - 2)); Output: Magma V2.11-10 Thu Dec 8 2005 18:37:25 on modular [Seed = 3996479081] ------------------------------------- [0 0 0] [0 0 0] [0 0 0] [ , , ] (2*b - 1)/(a + b - 1) (a - b)/(a + b - 1) 0 (-a + b)/(a + b - 1) (2*a - 1)/(a + b - 1) 0 0 0 1 a^2*b + a^2*c - a^2 + a*b^2 - 2*a*b*c - 3*a*b + a*c^2 - 3*a*c + 3*a + b^2*c - b^2 + b*c^2 - 3*b*c + 3*b - c^2 + 3*c - 2 [ ] Total time: 0.370 seconds, Total memory usage: 3.63MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Thu Dec 8 18:36:58 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); A1:=Matrix(K,3,3,[[b,1,1],[0,a,1],[0,0,c]]); S:=Transpose(A)*A^(-1); F,T:=PrimaryRationalForm(S); B:=T*A*Transpose(T); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*B*Transpose(P)-B)); G:=GroebnerBasis(I); Q1:=Matrix(K,3,3,[(2*b-1)/(a+b-1),(a-b)/(a+b-1),0,(b-a)/(a+b-1),(2*a-1)/(a+b-1),0, 0,0,1]); Kt:=FunctionField(K); D:=(2*a*b*c-a-b-c+1)/(a*b*c); s:=t; px:=-(2*s+D); py:=1-s^2; q:=s^2+s*D+1; Q:=Matrix(Kt,3,3,[1,0,0,0,D*px/q+py/q,-px/q,0,px/q,py/q]); Q:=Q1*T^(-1)*Q*T; print (Q*A*Transpose(Q)-A1); print Factorization(Numerator(Evaluate(q,-D/2+(1-2*a)*(a+b+c-1)/(2*a*b*c)))); for i in [1..3] do for j in [1..3] do print Evaluate(Q[i,j],-D/2+(0)*(a+b+c-1)/(2*a*b*c)); end for; end for; print ((a+b-1)*(b+c-1)*(a+c-1)-4*a*b*c+a+b+c-1); Output: Magma V2.11-10 Thu Dec 8 2005 18:36:57 on modular [Seed = 3846071630] ------------------------------------- [0 0 0] [0 0 0] [0 0 0] [ , , ] (2*b - 1)/(a + b - 1) (a - b)/(a + b - 1) 0 (-a + b)/(a + b - 1) (2*a - 1)/(a + b - 1) 0 0 0 1 a^2*b + a^2*c - a^2 + a*b^2 - 2*a*b*c - 3*a*b + a*c^2 - 3*a*c + 3*a + b^2*c - b^2 + b*c^2 - 3*b*c + 3*b - c^2 + 3*c - 2 Total time: 0.360 seconds, Total memory usage: 3.63MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Thu Dec 8 18:34:34 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); A1:=Matrix(K,3,3,[[b,1,1],[0,a,1],[0,0,c]]); S:=Transpose(A)*A^(-1); F,T:=PrimaryRationalForm(S); B:=T*A*Transpose(T); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*B*Transpose(P)-B)); G:=GroebnerBasis(I); Q1:=Matrix(K,3,3,[(2*b-1)/(a+b-1),(a-b)/(a+b-1),0,(b-a)/(a+b-1),(2*a-1)/(a+b-1),0, 0,0,1]); Kt:=FunctionField(K); D:=(2*a*b*c-a-b-c+1)/(a*b*c); s:=t; px:=-(2*s+D); py:=1-s^2; q:=s^2+s*D+1; Q:=Matrix(Kt,3,3,[1,0,0,0,D*px/q+py/q,-px/q,0,px/q,py/q]); Q:=Q1*T^(-1)*Q*T; print (Q*A*Transpose(Q)-A1); print Factorization(Numerator(Evaluate(q,-D/2+(1-2*a)*(a+b+c-1)/(2*a*b*c)))); for i in [1..3] do for j in [1..3] do print Evaluate(Q[i,j],-D/2+(0)*(a+b+c-1)/(2*a*b*c)); end for; end for; Output: Magma V2.11-10 Thu Dec 8 2005 18:34:33 on modular [Seed = 3812123608] ------------------------------------- [0 0 0] [0 0 0] [0 0 0] [ , , ] (2*b - 1)/(a + b - 1) (a - b)/(a + b - 1) 0 (-a + b)/(a + b - 1) (2*a - 1)/(a + b - 1) 0 0 0 1 Total time: 0.370 seconds, Total memory usage: 3.63MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Thu Dec 8 18:34:18 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); A1:=Matrix(K,3,3,[[b,1,1],[0,a,1],[0,0,c]]); S:=Transpose(A)*A^(-1); F,T:=PrimaryRationalForm(S); B:=T*A*Transpose(T); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*B*Transpose(P)-B)); G:=GroebnerBasis(I); Q1:=Matrix(K,3,3,[(2*b-1)/(a+b-1),(a-b)/(a+b-1),0,(b-a)/(a+b-1),(2*a-1)/(a+b-1),0, 0,0,1]); Kt:=FunctionField(K); D:=(2*a*b*c-a-b-c+1)/(a*b*c); s:=t; px:=-(2*s+D); py:=1-s^2; q:=s^2+s*D+1; Q:=Matrix(Kt,3,3,[1,0,0,0,D*px/q+py/q,-px/q,0,px/q,py/q]); Q:=Q1*T^(-1)*Q*T; print (Q*A*Transpose(Q)-A1); print Factorization(Denominator(Evaluate(q,-D/2+(1-2*a)*(a+b+c-1)/(2*a*b*c)))); for i in [1..3] do for j in [1..3] do print Evaluate(Q[i,j],-D/2+(0)*(a+b+c-1)/(2*a*b*c)); end for; end for; Output: Magma V2.11-10 Thu Dec 8 2005 18:34:17 on modular [Seed = 4263870261] ------------------------------------- [0 0 0] [0 0 0] [0 0 0] [ , , ] (2*b - 1)/(a + b - 1) (a - b)/(a + b - 1) 0 (-a + b)/(a + b - 1) (2*a - 1)/(a + b - 1) 0 0 0 1 Total time: 0.370 seconds, Total memory usage: 3.63MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Thu Dec 8 18:34:01 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); A1:=Matrix(K,3,3,[[b,1,1],[0,a,1],[0,0,c]]); S:=Transpose(A)*A^(-1); F,T:=PrimaryRationalForm(S); B:=T*A*Transpose(T); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*B*Transpose(P)-B)); G:=GroebnerBasis(I); Q1:=Matrix(K,3,3,[(2*b-1)/(a+b-1),(a-b)/(a+b-1),0,(b-a)/(a+b-1),(2*a-1)/(a+b-1),0, 0,0,1]); Kt:=FunctionField(K); D:=(2*a*b*c-a-b-c+1)/(a*b*c); s:=t; px:=-(2*s+D); py:=1-s^2; q:=s^2+s*D+1; Q:=Matrix(Kt,3,3,[1,0,0,0,D*px/q+py/q,-px/q,0,px/q,py/q]); Q:=Q1*T^(-1)*Q*T; print (Q*A*Transpose(Q)-A1); print Factorization(Denominator(Evaluate(q,-D/2+(0)*(a+b+c-1)/(2*a*b*c)))); for i in [1..3] do for j in [1..3] do print Evaluate(Q[i,j],-D/2+(0)*(a+b+c-1)/(2*a*b*c)); end for; end for; Output: Magma V2.11-10 Thu Dec 8 2005 18:34:01 on modular [Seed = 4113462318] ------------------------------------- [0 0 0] [0 0 0] [0 0 0] [ , , ] (2*b - 1)/(a + b - 1) (a - b)/(a + b - 1) 0 (-a + b)/(a + b - 1) (2*a - 1)/(a + b - 1) 0 0 0 1 Total time: 0.360 seconds, Total memory usage: 3.63MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Thu Dec 8 18:33:47 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); A1:=Matrix(K,3,3,[[b,1,1],[0,a,1],[0,0,c]]); S:=Transpose(A)*A^(-1); F,T:=PrimaryRationalForm(S); B:=T*A*Transpose(T); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*B*Transpose(P)-B)); G:=GroebnerBasis(I); Q1:=Matrix(K,3,3,[(2*b-1)/(a+b-1),(a-b)/(a+b-1),0,(b-a)/(a+b-1),(2*a-1)/(a+b-1),0, 0,0,1]); Kt:=FunctionField(K); D:=(2*a*b*c-a-b-c+1)/(a*b*c); s:=t; px:=-(2*s+D); py:=1-s^2; q:=s^2+s*D+1; Q:=Matrix(Kt,3,3,[1,0,0,0,D*px/q+py/q,-px/q,0,px/q,py/q]); Q:=Q1*T^(-1)*Q*T; print (Q*A*Transpose(Q)-A1); print Factorization(Denominator(Evaluate(q,-D/2+(0)*(a+b+c-1)/(2*a*b*c)))); for i in [1..3] do for j in [1..3] do print Evaluate(Q[i,j],-D/2+(2)*(a+b+c-1)/(2*a*b*c)); end for; end for; Output: Magma V2.11-10 Thu Dec 8 2005 18:33:46 on modular [Seed = 4029378840] ------------------------------------- [0 0 0] [0 0 0] [0 0 0] [ , , ] (2*a*b^2*c + a*b*c + 1/2*a*b + 1/4*a - 2*b^2*c + 1/2*b^2 - 1/2*b*c - 1/4*b + 1/4*c - 1/4)/(a^2*b*c + 3/4*a^2 + a*b^2*c - a*b*c + 3/2*a*b + 3/4*a*c - 3/2*a + 3/4*b^2 + 3/4*b*c - 3/2*b - 3/4*c + 3/4) (a^2*b*c - 1/4*a^2 - a*b^2*c - 4*a*b*c - a*b - 1/4*a*c + 1/4*a - 3/4*b^2 + 9/4*b*c + 3/4*b)/(a^2*b*c + 3/4*a^2 + a*b^2*c - a*b*c + 3/2*a*b + 3/4*a*c - 3/2*a + 3/4*b^2 + 3/4*b*c - 3/2*b - 3/4*c + 3/4) (-2*a*b + 3*b)/(a*b*c + 3/4*a + 3/4*b + 3/4*c - 3/4) (-a^2*b*c - 3/4*a^2 + a*b^2*c + 4*a*b*c - a*b + 1/4*a*c + 7/4*a - 1/4*b^2 - 9/4*b*c + 5/4*b - 1)/(a^2*b*c + 3/4*a^2 + a*b^2*c - a*b*c + 3/2*a*b + 3/4*a*c - 3/2*a + 3/4*b^2 + 3/4*b*c - 3/2*b - 3/4*c + 3/4) (2*a^2*b*c + 2*a^2*c + 1/2*a^2 - 3*a*b*c + 1/2*a*b - 9/2*a*c - 5/4*a - 3/4*b + 9/4*c + 3/4)/(a^2*b*c + 3/4*a^2 + a*b^2*c - a*b*c + 3/2*a*b + 3/4*a*c - 3/2*a + 3/4*b^2 + 3/4*b*c - 3/2*b - 3/4*c + 3/4) (-2*a*b - 3*a + 3)/(a*b*c + 3/4*a + 3/4*b + 3/4*c - 3/4) (2*b*c + c)/(a*b*c + 3/4*a + 3/4*b + 3/4*c - 3/4) (2*a*c - 3*c)/(a*b*c + 3/4*a + 3/4*b + 3/4*c - 3/4) (a*b*c - 9/4*a - 9/4*b + 3/4*c + 9/4)/(a*b*c + 3/4*a + 3/4*b + 3/4*c - 3/4) Total time: 0.370 seconds, Total memory usage: 3.63MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Thu Dec 8 18:32:23 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); A1:=Matrix(K,3,3,[[b,1,1],[0,a,1],[0,0,c]]); S:=Transpose(A)*A^(-1); F,T:=PrimaryRationalForm(S); B:=T*A*Transpose(T); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*B*Transpose(P)-B)); G:=GroebnerBasis(I); Q1:=Matrix(K,3,3,[(2*b-1)/(a+b-1),(a-b)/(a+b-1),0,(b-a)/(a+b-1),(2*a-1)/(a+b-1),0, 0,0,1]); Kt:=FunctionField(K); D:=(2*a*b*c-a-b-c+1)/(a*b*c); s:=t; px:=-(2*s+D); py:=1-s^2; q:=s^2+s*D+1; Q:=Matrix(Kt,3,3,[1,0,0,0,D*px/q+py/q,-px/q,0,px/q,py/q]); Q:=Q1*T^(-1)*Q*T; print (Q*A*Transpose(Q)-A1); print Factorization(Denominator(Evaluate(Q[3,3],-D/2+(0)*(a+b+c-1)/(2*a*b*c)))); for i in [1..3] do for j in [1..3] do print Evaluate(Q[i,j],-D/2+(2)*(a+b+c-1)/(2*a*b*c)); end for; end for; Output: Magma V2.11-10 Thu Dec 8 2005 18:32:23 on modular [Seed = 2320567103] ------------------------------------- [0 0 0] [0 0 0] [0 0 0] [] (2*a*b^2*c + a*b*c + 1/2*a*b + 1/4*a - 2*b^2*c + 1/2*b^2 - 1/2*b*c - 1/4*b + 1/4*c - 1/4)/(a^2*b*c + 3/4*a^2 + a*b^2*c - a*b*c + 3/2*a*b + 3/4*a*c - 3/2*a + 3/4*b^2 + 3/4*b*c - 3/2*b - 3/4*c + 3/4) (a^2*b*c - 1/4*a^2 - a*b^2*c - 4*a*b*c - a*b - 1/4*a*c + 1/4*a - 3/4*b^2 + 9/4*b*c + 3/4*b)/(a^2*b*c + 3/4*a^2 + a*b^2*c - a*b*c + 3/2*a*b + 3/4*a*c - 3/2*a + 3/4*b^2 + 3/4*b*c - 3/2*b - 3/4*c + 3/4) (-2*a*b + 3*b)/(a*b*c + 3/4*a + 3/4*b + 3/4*c - 3/4) (-a^2*b*c - 3/4*a^2 + a*b^2*c + 4*a*b*c - a*b + 1/4*a*c + 7/4*a - 1/4*b^2 - 9/4*b*c + 5/4*b - 1)/(a^2*b*c + 3/4*a^2 + a*b^2*c - a*b*c + 3/2*a*b + 3/4*a*c - 3/2*a + 3/4*b^2 + 3/4*b*c - 3/2*b - 3/4*c + 3/4) (2*a^2*b*c + 2*a^2*c + 1/2*a^2 - 3*a*b*c + 1/2*a*b - 9/2*a*c - 5/4*a - 3/4*b + 9/4*c + 3/4)/(a^2*b*c + 3/4*a^2 + a*b^2*c - a*b*c + 3/2*a*b + 3/4*a*c - 3/2*a + 3/4*b^2 + 3/4*b*c - 3/2*b - 3/4*c + 3/4) (-2*a*b - 3*a + 3)/(a*b*c + 3/4*a + 3/4*b + 3/4*c - 3/4) (2*b*c + c)/(a*b*c + 3/4*a + 3/4*b + 3/4*c - 3/4) (2*a*c - 3*c)/(a*b*c + 3/4*a + 3/4*b + 3/4*c - 3/4) (a*b*c - 9/4*a - 9/4*b + 3/4*c + 9/4)/(a*b*c + 3/4*a + 3/4*b + 3/4*c - 3/4) Total time: 0.370 seconds, Total memory usage: 3.63MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Thu Dec 8 18:31:56 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); A1:=Matrix(K,3,3,[[b,1,1],[0,a,1],[0,0,c]]); S:=Transpose(A)*A^(-1); F,T:=PrimaryRationalForm(S); B:=T*A*Transpose(T); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*B*Transpose(P)-B)); G:=GroebnerBasis(I); Q1:=Matrix(K,3,3,[(2*b-1)/(a+b-1),(a-b)/(a+b-1),0,(b-a)/(a+b-1),(2*a-1)/(a+b-1),0, 0,0,1]); Kt:=FunctionField(K); D:=(2*a*b*c-a-b-c+1)/(a*b*c); s:=t; px:=-(2*s+D); py:=1-s^2; q:=s^2+s*D+1; Q:=Matrix(Kt,3,3,[1,0,0,0,D*px/q+py/q,-px/q,0,px/q,py/q]); Q:=Q1*T^(-1)*Q*T; print (Q*A*Transpose(Q)-A1); print Factorization(Denominator(Evaluate(Q[3,3],-D/2+(2)*(a+b+c-1)/(2*a*b*c)))); for i in [1..3] do for j in [1..3] do print Evaluate(Q[i,j],-D/2+(2)*(a+b+c-1)/(2*a*b*c)); end for; end for; Output: Magma V2.11-10 Thu Dec 8 2005 18:31:55 on modular [Seed = 2170159320] ------------------------------------- [0 0 0] [0 0 0] [0 0 0] [ ] (2*a*b^2*c + a*b*c + 1/2*a*b + 1/4*a - 2*b^2*c + 1/2*b^2 - 1/2*b*c - 1/4*b + 1/4*c - 1/4)/(a^2*b*c + 3/4*a^2 + a*b^2*c - a*b*c + 3/2*a*b + 3/4*a*c - 3/2*a + 3/4*b^2 + 3/4*b*c - 3/2*b - 3/4*c + 3/4) (a^2*b*c - 1/4*a^2 - a*b^2*c - 4*a*b*c - a*b - 1/4*a*c + 1/4*a - 3/4*b^2 + 9/4*b*c + 3/4*b)/(a^2*b*c + 3/4*a^2 + a*b^2*c - a*b*c + 3/2*a*b + 3/4*a*c - 3/2*a + 3/4*b^2 + 3/4*b*c - 3/2*b - 3/4*c + 3/4) (-2*a*b + 3*b)/(a*b*c + 3/4*a + 3/4*b + 3/4*c - 3/4) (-a^2*b*c - 3/4*a^2 + a*b^2*c + 4*a*b*c - a*b + 1/4*a*c + 7/4*a - 1/4*b^2 - 9/4*b*c + 5/4*b - 1)/(a^2*b*c + 3/4*a^2 + a*b^2*c - a*b*c + 3/2*a*b + 3/4*a*c - 3/2*a + 3/4*b^2 + 3/4*b*c - 3/2*b - 3/4*c + 3/4) (2*a^2*b*c + 2*a^2*c + 1/2*a^2 - 3*a*b*c + 1/2*a*b - 9/2*a*c - 5/4*a - 3/4*b + 9/4*c + 3/4)/(a^2*b*c + 3/4*a^2 + a*b^2*c - a*b*c + 3/2*a*b + 3/4*a*c - 3/2*a + 3/4*b^2 + 3/4*b*c - 3/2*b - 3/4*c + 3/4) (-2*a*b - 3*a + 3)/(a*b*c + 3/4*a + 3/4*b + 3/4*c - 3/4) (2*b*c + c)/(a*b*c + 3/4*a + 3/4*b + 3/4*c - 3/4) (2*a*c - 3*c)/(a*b*c + 3/4*a + 3/4*b + 3/4*c - 3/4) (a*b*c - 9/4*a - 9/4*b + 3/4*c + 9/4)/(a*b*c + 3/4*a + 3/4*b + 3/4*c - 3/4) Total time: 0.370 seconds, Total memory usage: 3.63MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Thu Dec 8 18:31:46 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); A1:=Matrix(K,3,3,[[b,1,1],[0,a,1],[0,0,c]]); S:=Transpose(A)*A^(-1); F,T:=PrimaryRationalForm(S); B:=T*A*Transpose(T); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*B*Transpose(P)-B)); G:=GroebnerBasis(I); Q1:=Matrix(K,3,3,[(2*b-1)/(a+b-1),(a-b)/(a+b-1),0,(b-a)/(a+b-1),(2*a-1)/(a+b-1),0, 0,0,1]); Kt:=FunctionField(K); D:=(2*a*b*c-a-b-c+1)/(a*b*c); s:=t; px:=-(2*s+D); py:=1-s^2; q:=s^2+s*D+1; Q:=Matrix(Kt,3,3,[1,0,0,0,D*px/q+py/q,-px/q,0,px/q,py/q]); Q:=Q1*T^(-1)*Q*T; print (Q*A*Transpose(Q)-A1); print Factorization(Denominator(Evaluate(Q[3,3],-D/2+(2)*(a+b+c-1)/(2*a*b*c)))); for i in [1..3] do for j in [1..3] do print Evaluate(Q[i,j],-D/2+(1)*(a+b+c-1)/(2*a*b*c)); end for; end for; Output: Magma V2.11-10 Thu Dec 8 2005 18:31:45 on modular [Seed = 2571245770] ------------------------------------- [0 0 0] [0 0 0] [0 0 0] [ ] (2*a*b - b)/(a^2 + a*b - a) (a^2 - a*b - 2*a + 1)/(a^2 + a*b - a) (-a + 1)/(a*c) (-a^2 + a*b + 2*a - 1)/(a^2 + a*b - a) (2*a^2*b + a^2 - 2*a*b - 2*a + 1)/(a^2*b + a*b^2 - a*b) (-a*b - a + 1)/(a*b*c) 1/a (a - 1)/(a*b) (a*b*c - a - b + 1)/(a*b*c) Total time: 0.370 seconds, Total memory usage: 3.63MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Thu Dec 8 18:30:11 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); A1:=Matrix(K,3,3,[[b,1,1],[0,a,1],[0,0,c]]); S:=Transpose(A)*A^(-1); F,T:=PrimaryRationalForm(S); B:=T*A*Transpose(T); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*B*Transpose(P)-B)); G:=GroebnerBasis(I); Q1:=Matrix(K,3,3,[(2*b-1)/(a+b-1),(a-b)/(a+b-1),0,(b-a)/(a+b-1),(2*a-1)/(a+b-1),0, 0,0,1]); Kt:=FunctionField(K); D:=(2*a*b*c-a-b-c+1)/(a*b*c); s:=t; px:=-(2*s+D); py:=1-s^2; q:=s^2+s*D+1; Q:=Matrix(Kt,3,3,[1,0,0,0,D*px/q+py/q,-px/q,0,px/q,py/q]); Q:=Q1*T^(-1)*Q*T; print (Q*A*Transpose(Q)-A1); print Factorization(Denominator(Evaluate(Q[3,3],-D/2+(1)*(a+b+c-1)/(2*a*b*c)))); for i in [1..3] do for j in [1..3] do print Evaluate(Q[i,j],-D/2+(1)*(a+b+c-1)/(2*a*b*c)); end for; end for; Output: Magma V2.11-10 Thu Dec 8 2005 18:30:10 on modular [Seed = 2488210852] ------------------------------------- [0 0 0] [0 0 0] [0 0 0] [ , , ] (2*a*b - b)/(a^2 + a*b - a) (a^2 - a*b - 2*a + 1)/(a^2 + a*b - a) (-a + 1)/(a*c) (-a^2 + a*b + 2*a - 1)/(a^2 + a*b - a) (2*a^2*b + a^2 - 2*a*b - 2*a + 1)/(a^2*b + a*b^2 - a*b) (-a*b - a + 1)/(a*b*c) 1/a (a - 1)/(a*b) (a*b*c - a - b + 1)/(a*b*c) Total time: 0.370 seconds, Total memory usage: 3.63MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Thu Dec 8 18:29:33 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); A1:=Matrix(K,3,3,[[b,1,1],[0,a,1],[0,0,c]]); S:=Transpose(A)*A^(-1); F,T:=PrimaryRationalForm(S); B:=T*A*Transpose(T); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*B*Transpose(P)-B)); G:=GroebnerBasis(I); Q1:=Matrix(K,3,3,[(2*b-1)/(a+b-1),(a-b)/(a+b-1),0,(b-a)/(a+b-1),(2*a-1)/(a+b-1),0, 0,0,1]); Kt:=FunctionField(K); D:=(2*a*b*c-a-b-c+1)/(a*b*c); s:=t; px:=-(2*s+D); py:=1-s^2; q:=s^2+s*D+1; Q:=Matrix(Kt,3,3,[1,0,0,0,D*px/q+py/q,-px/q,0,px/q,py/q]); Q:=Q1*T^(-1)*Q*T; print (Q*A*Transpose(Q)-A1); print Factorization(Denominator(Evaluate(Q[3,3],-D/2+(1)*(a+b+c-1)/(2*a*b*c)))); for i in [1..3] do for j in [1..3] do print Evaluate(Q[i,j],-D/2+(2*a-1)*(a+b+c-1)/(2*a*b*c)); end for; end for; Output: Magma V2.11-10 Thu Dec 8 2005 18:29:33 on modular [Seed = 2826119751] ------------------------------------- [0 0 0] [0 0 0] [0 0 0] [ , , ] 1/a -1 0 (-a^2 + a*c + 2*a - 1)/(a^2 + a*c - a) 0 (-2*a + 1)/(a + c - 1) (2*a*c - c)/(a^2 + a*c - a) 0 (-a + c)/(a + c - 1) Total time: 0.370 seconds, Total memory usage: 3.63MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Thu Dec 8 18:28:31 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); A1:=Matrix(K,3,3,[[b,1,1],[0,a,1],[0,0,c]]); S:=Transpose(A)*A^(-1); F,T:=PrimaryRationalForm(S); B:=T*A*Transpose(T); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*B*Transpose(P)-B)); G:=GroebnerBasis(I); Q1:=Matrix(K,3,3,[(2*b-1)/(a+b-1),(a-b)/(a+b-1),0,(b-a)/(a+b-1),(2*a-1)/(a+b-1),0, 0,0,1]); Kt:=FunctionField(K); D:=(2*a*b*c-a-b-c+1)/(a*b*c); s:=t; px:=-(2*s+D); py:=1-s^2; q:=s^2+s*D+1; Q:=Matrix(Kt,3,3,[1,0,0,0,D*px/q+py/q,-px/q,0,px/q,py/q]); Q:=Q1*T^(-1)*Q*T; print (Q*A*Transpose(Q)-A1); print Factorization(Denominator(Evaluate(Q[3,3],-D/2+(a^2+a*b-b)*(a+b+c-1)/(2*a*b*c)))); for i in [1..3] do for j in [1..3] do print Evaluate(Q[i,j],-D/2+(2*a-1)*(a+b+c-1)/(2*a*b*c)); end for; end for; Output: Magma V2.11-10 Thu Dec 8 2005 18:28:31 on modular [Seed = 2709136140] ------------------------------------- [0 0 0] [0 0 0] [0 0 0] [ ] 1/a -1 0 (-a^2 + a*c + 2*a - 1)/(a^2 + a*c - a) 0 (-2*a + 1)/(a + c - 1) (2*a*c - c)/(a^2 + a*c - a) 0 (-a + c)/(a + c - 1) Total time: 0.370 seconds, Total memory usage: 3.63MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Thu Dec 8 18:27:49 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); A1:=Matrix(K,3,3,[[b,1,1],[0,a,1],[0,0,c]]); S:=Transpose(A)*A^(-1); F,T:=PrimaryRationalForm(S); B:=T*A*Transpose(T); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*B*Transpose(P)-B)); G:=GroebnerBasis(I); Q1:=Matrix(K,3,3,[(2*b-1)/(a+b-1),(a-b)/(a+b-1),0,(b-a)/(a+b-1),(2*a-1)/(a+b-1),0, 0,0,1]); Kt:=FunctionField(K); D:=(2*a*b*c-a-b-c+1)/(a*b*c); s:=t; px:=-(2*s+D); py:=1-s^2; q:=s^2+s*D+1; Q:=Matrix(Kt,3,3,[1,0,0,0,D*px/q+py/q,-px/q,0,px/q,py/q]); Q:=Q1*T^(-1)*Q*T; print (Q*A*Transpose(Q)-A1); print Factorization(Denominator(Evaluate(Q[3,3],-D/2+(a^2+a*b-b)*(a+b+c-1)/(2*a*b*c))); for i in [1..3] do for j in [1..3] do print Evaluate(Q[i,j],-D/2+(2*a-1)*(a+b+c-1)/(2*a*b*c)); end for; end for; Output: Magma V2.11-10 Thu Dec 8 2005 18:27:49 on modular [Seed = 3211017720] ------------------------------------- [0 0 0] [0 0 0] [0 0 0] >> ^2+a*b-b)*(a+b+c-1)/(2*a*b*c))); ^ User error: bad syntax 1/a -1 0 (-a^2 + a*c + 2*a - 1)/(a^2 + a*c - a) 0 (-2*a + 1)/(a + c - 1) (2*a*c - c)/(a^2 + a*c - a) 0 (-a + c)/(a + c - 1) Total time: 0.370 seconds, Total memory usage: 3.63MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Thu Dec 8 18:27:19 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); A1:=Matrix(K,3,3,[[b,1,1],[0,a,1],[0,0,c]]); S:=Transpose(A)*A^(-1); F,T:=PrimaryRationalForm(S); B:=T*A*Transpose(T); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*B*Transpose(P)-B)); G:=GroebnerBasis(I); Q1:=Matrix(K,3,3,[(2*b-1)/(a+b-1),(a-b)/(a+b-1),0,(b-a)/(a+b-1),(2*a-1)/(a+b-1),0, 0,0,1]); Kt:=FunctionField(K); D:=(2*a*b*c-a-b-c+1)/(a*b*c); s:=t; px:=-(2*s+D); py:=1-s^2; q:=s^2+s*D+1; Q:=Matrix(Kt,3,3,[1,0,0,0,D*px/q+py/q,-px/q,0,px/q,py/q]); Q:=Q1*T^(-1)*Q*T; print (Q*A*Transpose(Q)-A1); print Factorization(Evaluate(Denominator(Q[3,3])),-D/2+(a^2+a*b-b)*(a+b+c-1)/(2*a*b*c)); for i in [1..3] do for j in [1..3] do print Evaluate(Q[i,j],-D/2+(2*a-1)*(a+b+c-1)/(2*a*b*c)); end for; end for; Output: Magma V2.11-10 Thu Dec 8 2005 18:27:18 on modular [Seed = 3126933726] ------------------------------------- [0 0 0] [0 0 0] [0 0 0] >> print Factorization(Evaluate(Denominator(Q[3,3])),-D/2+(a^2+a*b-b)*(a+b+c-1 ^ Runtime error in 'Evaluate': Bad argument types Argument types given: RngUPolElt[FldFunRat] 1/a -1 0 (-a^2 + a*c + 2*a - 1)/(a^2 + a*c - a) 0 (-2*a + 1)/(a + c - 1) (2*a*c - c)/(a^2 + a*c - a) 0 (-a + c)/(a + c - 1) Total time: 0.370 seconds, Total memory usage: 3.63MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Thu Dec 8 18:26:12 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); A1:=Matrix(K,3,3,[[b,1,1],[0,a,1],[0,0,c]]); S:=Transpose(A)*A^(-1); F,T:=PrimaryRationalForm(S); B:=T*A*Transpose(T); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*B*Transpose(P)-B)); G:=GroebnerBasis(I); Q1:=Matrix(K,3,3,[(2*b-1)/(a+b-1),(a-b)/(a+b-1),0,(b-a)/(a+b-1),(2*a-1)/(a+b-1),0, 0,0,1]); Kt:=FunctionField(K); D:=(2*a*b*c-a-b-c+1)/(a*b*c); s:=t; px:=-(2*s+D); py:=1-s^2; q:=s^2+s*D+1; Q:=Matrix(Kt,3,3,[1,0,0,0,D*px/q+py/q,-px/q,0,px/q,py/q]); Q:=Q1*T^(-1)*Q*T; print (Q*A*Transpose(Q)-A1); for i in [1..3] do for j in [1..3] do print Evaluate(Q[i,j],-D/2+(a^2+a*b-b)*(a+b+c-1)/(2*a*b*c)); end for; end for; Output: Magma V2.11-10 Thu Dec 8 2005 18:26:12 on modular [Seed = 2976527255] ------------------------------------- [0 0 0] [0 0 0] [0 0 0] (a^3 - 2*a^2*b*c + a^2*b + a^2*c + a^2 + 2*a*c - a + 2*b*c - b + c - 1)/(a^4 + 2*a^3*b + a^3*c + a^2*b^2 + a^2*b*c - 2*a^2*b + a^2*c - 2*a*b^2 - 2*a*b*c + a*c + b^2 + b*c + c - 1) (-a^4 - 2*a^3*b - a^3*c - a^3 - a^2*b^2 + a^2*b*c + a^2*b - 2*a^2*c + a^2 + 2*a*b^2 - 2*a*b*c + 2*a*b - a*c + a - b^2 + b*c - b)/(a^4 + 2*a^3*b + a^3*c + a^2*b^2 + a^2*b*c - 2*a^2*b + a^2*c - 2*a*b^2 - 2*a*b*c + a*c + b^2 + b*c + c - 1) (2*a^3*b + 2*a^2*b^2 - 2*a^2*b - 4*a*b^2 + 2*b^2)/(a^4 + 2*a^3*b + a^3*c + a^2*b^2 + a^2*b*c - 2*a^2*b + a^2*c - 2*a*b^2 - 2*a*b*c + a*c + b^2 + b*c + c - 1) (-a^4 - 2*a^3*b + a^3*c + a^3 - a^2*b^2 - a^2*b*c + 3*a^2*b + 2*a^2*c + a^2 + 2*a*b^2 + 2*a*b*c + a*c - a - b^2 - b*c - b)/(a^4 + 2*a^3*b + a^3*c + a^2*b^2 + a^2*b*c - 2*a^2*b + a^2*c - 2*a*b^2 - 2*a*b*c + a*c + b^2 + b*c + c - 1) (-2*a^3*c - a^3 - a^2*b + a^2*c + a^2 + 2*a*b + a - b + c - 1)/(a^4 + 2*a^3*b + a^3*c + a^2*b^2 + a^2*b*c - 2*a^2*b + a^2*c - 2*a*b^2 - 2*a*b*c + a*c + b^2 + b*c + c - 1) (-2*a^4 - 2*a^3*b + 2*a^2*b - 2*a^2 - 2*a*b + 2*b)/(a^4 + 2*a^3*b + a^3*c + a^2*b^2 + a^2*b*c - 2*a^2*b + a^2*c - 2*a*b^2 - 2*a*b*c + a*c + b^2 + b*c + c - 1) (2*a^3*c + 2*a^2*b*c + 2*a^2*c - 2*b*c)/(a^4 + 2*a^3*b + a^3*c + a^2*b^2 + a^2*b*c - 2*a^2*b + a^2*c - 2*a*b^2 - 2*a*b*c + a*c + b^2 + b*c + c - 1) (-2*a^3*c - 2*a^2*b*c + 2*a^2*c + 4*a*b*c - 2*b*c)/(a^4 + 2*a^3*b + a^3*c + a^2*b^2 + a^2*b*c - 2*a^2*b + a^2*c - 2*a*b^2 - 2*a*b*c + a*c + b^2 + b*c + c - 1) (-a^4 - 2*a^3*b + a^3*c - a^2*b^2 + a^2*b*c + 2*a^2*b + a^2*c - 2*a^2 + 2*a*b^2 - 2*a*b*c - 2*a*b + a*c - b^2 + b*c + 2*b + c - 1)/(a^4 + 2*a^3*b + a^3*c + a^2*b^2 + a^2*b*c - 2*a^2*b + a^2*c - 2*a*b^2 - 2*a*b*c + a*c + b^2 + b*c + c - 1) Total time: 0.380 seconds, Total memory usage: 3.63MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Thu Dec 8 18:25:18 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); A1:=Matrix(K,3,3,[[b,1,1],[0,a,1],[0,0,c]]); S:=Transpose(A)*A^(-1); F,T:=PrimaryRationalForm(S); B:=T*A*Transpose(T); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*B*Transpose(P)-B)); G:=GroebnerBasis(I); Q1:=Matrix(K,3,3,[(2*b-1)/(a+b-1),(a-b)/(a+b-1),0,(b-a)/(a+b-1),(2*a-1)/(a+b-1),0, 0,0,1]); Kt:=FunctionField(K); D:=(2*a*b*c-a-b-c+1)/(a*b*c); s:=t; px:=-(2*s+D); py:=1-s^2; q:=s^2+s*D+1; Q:=Matrix(Kt,3,3,[1,0,0,0,D*px/q+py/q,-px/q,0,px/q,py/q]); Q:=Q1*T^(-1)*Q*T; print (Q*A*Transpose(Q)-A1); for i in [1..3] do for j in [1..3] do print Evaluate(Q[i,j],-D/2+(a-b)*(a+b+c-1)/(2*a*b*c)); end for; end for; Output: Magma V2.11-10 Thu Dec 8 2005 18:25:18 on modular [Seed = 1285017504] ------------------------------------- [0 0 0] [0 0 0] [0 0 0] (a + 2*b*c - b + c - 1)/(a^2 - 2*a*b + a*c + b^2 + b*c + c - 1) (-a^2 + 2*a*b - a*c + a - b^2 + b*c - b)/(a^2 - 2*a*b + a*c + b^2 + b*c + c - 1) (-2*a*b + 2*b^2)/(a^2 - 2*a*b + a*c + b^2 + b*c + c - 1) (-a^2 + 2*a*b + a*c + a - b^2 - b*c - b)/(a^2 - 2*a*b + a*c + b^2 + b*c + c - 1) (2*a*c + a - b + c - 1)/(a^2 - 2*a*b + a*c + b^2 + b*c + c - 1) (-2*a^2 + 2*a*b - 2*a + 2*b)/(a^2 - 2*a*b + a*c + b^2 + b*c + c - 1) (2*a*c - 2*b*c)/(a^2 - 2*a*b + a*c + b^2 + b*c + c - 1) (2*a*c - 2*b*c)/(a^2 - 2*a*b + a*c + b^2 + b*c + c - 1) (-a^2 + 2*a*b + a*c - 2*a - b^2 + b*c + 2*b + c - 1)/(a^2 - 2*a*b + a*c + b^2 + b*c + c - 1) Total time: 0.370 seconds, Total memory usage: 3.63MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Thu Dec 8 18:25:02 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); A1:=Matrix(K,3,3,[[b,1,1],[0,a,1],[0,0,c]]); S:=Transpose(A)*A^(-1); F,T:=PrimaryRationalForm(S); B:=T*A*Transpose(T); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*B*Transpose(P)-B)); G:=GroebnerBasis(I); Q1:=Matrix(K,3,3,[(2*b-1)/(a+b-1),(a-b)/(a+b-1),0,(b-a)/(a+b-1),(2*a-1)/(a+b-1),0, 0,0,1]); Kt:=FunctionField(K); D:=(2*a*b*c-a-b-c+1)/(a*b*c); s:=t; px:=-(2*s+D); py:=1-s^2; q:=s^2+s*D+1; Q:=Matrix(Kt,3,3,[1,0,0,0,D*px/q+py/q,-px/q,0,px/q,py/q]); Q:=Q1*T^(-1)*Q*T; print (Q*A*Transpose(Q)-A1); for i in [1..3] do for j in [1..3] do print Evaluate(Q[i,j],-D/2+(a-b)*(a+b+c-1)/(2*a*b*c))); end for; end for; Output: Magma V2.11-10 Thu Dec 8 2005 18:25:02 on modular [Seed = 1267779767] ------------------------------------- [0 0 0] [0 0 0] [0 0 0] >> print Evaluate(Q[i,j],-D/2+(a-b)*(a+b+c-1)/(2*a*b*c))); ^ User error: bad syntax >> end for; ^ User error: bad syntax >> end for; ^ User error: bad syntax Total time: 0.370 seconds, Total memory usage: 3.63MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Thu Dec 8 18:24:30 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); A1:=Matrix(K,3,3,[[b,1,1],[0,a,1],[0,0,c]]); S:=Transpose(A)*A^(-1); F,T:=PrimaryRationalForm(S); B:=T*A*Transpose(T); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*B*Transpose(P)-B)); G:=GroebnerBasis(I); Q1:=Matrix(K,3,3,[(2*b-1)/(a+b-1),(a-b)/(a+b-1),0,(b-a)/(a+b-1),(2*a-1)/(a+b-1),0, 0,0,1]); Kt:=FunctionField(K); D:=(2*a*b*c-a-b-c+1)/(a*b*c); s:=t; px:=-(2*s+D); py:=1-s^2; q:=s^2+s*D+1; Q:=Matrix(Kt,3,3,[1,0,0,0,D*px/q+py/q,-px/q,0,px/q,py/q]); Q:=Q1*T^(-1)*Q*T; print (Q*A*Transpose(Q)-A1); for i in [1..3] do for j in [1..3] do print Evaluate(Q[i,j],-D/2+(a-b)*(2*b-1)*(a+b+c-1)/((2*a*b*c)*(1-2*a))); end for; end for; Output: Magma V2.11-10 Thu Dec 8 2005 18:24:29 on modular [Seed = 1100661470] ------------------------------------- [0 0 0] [0 0 0] [0 0 0] (4*a*b^2*c - a*b^2 - 2*a*b*c + 1/4*a - 2*b^3*c + b^3 + b^2*c - b^2 - 1/2*b*c + 3/4*b + 1/4*c - 1/4)/(a^2*b^2 + 4*a^2*b*c - a^2*b - 3/4*a^2 - 2*a*b^3 - 3*a*b^2*c + 2*a*b^2 - a*b*c - 1/2*a*b - 3/4*a*c + a + b^4 + b^3*c - b^3 + 1/4*b^2 + 1/4*b*c + 1/4*c - 1/4) (-a^2*b^2 + 4*a^2*b*c - a^2*b - 1/4*a^2 + 2*a*b^3 - 5*a*b^2*c + a*b^2 - a*b*c + 1/2*a*b - 1/4*a*c + 1/4*a - b^4 + b^3*c + b^2*c - 1/4*b^2 + 1/4*b*c - 1/4*b)/(a^2*b^2 + 4*a^2*b*c - a^2*b - 3/4*a^2 - 2*a*b^3 - 3*a*b^2*c + 2*a*b^2 - a*b*c - 1/2*a*b - 3/4*a*c + a + b^4 + b^3*c - b^3 + 1/4*b^2 + 1/4*b*c + 1/4*c - 1/4) (4*a^2*b^2 - 2*a^2*b - 6*a*b^3 + 2*a*b^2 + 1/2*a*b + 2*b^4 - 1/2*b^2)/(a^2*b^2 + 4*a^2*b*c - a^2*b - 3/4*a^2 - 2*a*b^3 - 3*a*b^2*c + 2*a*b^2 - a*b*c - 1/2*a*b - 3/4*a*c + a + b^4 + b^3*c - b^3 + 1/4*b^2 + 1/4*b*c + 1/4*c - 1/4) (-a^2*b^2 - 4*a^2*b*c + a^2*b + 3/4*a^2 + 2*a*b^3 + 5*a*b^2*c - 3*a*b^2 + a*b*c + 1/2*a*b + 1/4*a*c - 3/4*a - b^4 - b^3*c + 2*b^3 - b^2*c - 5/4*b^2 - 1/4*b*c + 3/4*b)/(a^2*b^2 + 4*a^2*b*c - a^2*b - 3/4*a^2 - 2*a*b^3 - 3*a*b^2*c + 2*a*b^2 - a*b*c - 1/2*a*b - 3/4*a*c + a + b^4 + b^3*c - b^3 + 1/4*b^2 + 1/4*b*c + 1/4*c - 1/4) (4*a^2*b*c - 2*a^2*b + 2*a^2*c - a^2 - 2*a*b^2*c + 3*a*b^2 - 4*a*b*c - 3/2*a*c + 5/4*a - b^3 + b^2*c + b*c - 1/4*b + 1/4*c - 1/4)/(a^2*b^2 + 4*a^2*b*c - a^2*b - 3/4*a^2 - 2*a*b^3 - 3*a*b^2*c + 2*a*b^2 - a*b*c - 1/2*a*b - 3/4*a*c + a + b^4 + b^3*c - b^3 + 1/4*b^2 + 1/4*b*c + 1/4*c - 1/4) (2*a^2*b^2 + 2*a^2*b - 3/2*a^2 - 2*a*b^3 - 4*a*b^2 + 3/2*a*b + 1/2*a + 2*b^3 - 1/2*b)/(a^2*b^2 + 4*a^2*b*c - a^2*b - 3/4*a^2 - 2*a*b^3 - 3*a*b^2*c + 2*a*b^2 - a*b*c - 1/2*a*b - 3/4*a*c + a + b^4 + b^3*c - b^3 + 1/4*b^2 + 1/4*b*c + 1/4*c - 1/4) (-2*a*b^2*c + 2*a*b*c - 1/2*a*c + 2*b^3*c - 2*b^2*c + 1/2*b*c)/(a^2*b^2 + 4*a^2*b*c - a^2*b - 3/4*a^2 - 2*a*b^3 - 3*a*b^2*c + 2*a*b^2 - a*b*c - 1/2*a*b - 3/4*a*c + a + b^4 + b^3*c - b^3 + 1/4*b^2 + 1/4*b*c + 1/4*c - 1/4) (-4*a^2*b*c + 2*a^2*c + 6*a*b^2*c - 2*a*b*c - 1/2*a*c - 2*b^3*c + 1/2*b*c)/(a^2*b^2 + 4*a^2*b*c - a^2*b - 3/4*a^2 - 2*a*b^3 - 3*a*b^2*c + 2*a*b^2 - a*b*c - 1/2*a*b - 3/4*a*c + a + b^4 + b^3*c - b^3 + 1/4*b^2 + 1/4*b*c + 1/4*c - 1/4) (-a^2*b^2 + 4*a^2*b*c + 3*a^2*b - 9/4*a^2 + 2*a*b^3 - 3*a*b^2*c - 4*a*b^2 - a*b*c + 1/2*a*b - 3/4*a*c + 3/2*a - b^4 + b^3*c + b^3 + 3/4*b^2 + 1/4*b*c - 1/2*b + 1/4*c - 1/4)/(a^2*b^2 + 4*a^2*b*c - a^2*b - 3/4*a^2 - 2*a*b^3 - 3*a*b^2*c + 2*a*b^2 - a*b*c - 1/2*a*b - 3/4*a*c + a + b^4 + b^3*c - b^3 + 1/4*b^2 + 1/4*b*c + 1/4*c - 1/4) Total time: 0.380 seconds, Total memory usage: 3.63MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Thu Dec 8 18:23:16 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); A1:=Matrix(K,3,3,[[b,1,1],[0,a,1],[0,0,c]]); S:=Transpose(A)*A^(-1); F,T:=PrimaryRationalForm(S); B:=T*A*Transpose(T); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*B*Transpose(P)-B)); G:=GroebnerBasis(I); Q1:=Matrix(K,3,3,[(2*b-1)/(a+b-1),(a-b)/(a+b-1),0,(b-a)/(a+b-1),(2*a-1)/(a+b-1),0, 0,0,1]); Kt:=FunctionField(K); D:=(2*a*b*c-a-b-c+1)/(a*b*c); s:=t; px:=-(2*s+D); py:=1-s^2; q:=s^2+s*D+1; Q:=Matrix(Kt,3,3,[1,0,0,0,D*px/q+py/q,-px/q,0,px/q,py/q]); Q:=Q1*T^(-1)*Q*T; print (Q*A*Transpose(Q)-A1); for i in [1..3] do for j in [1..3] do print Evaluate(Q[i,j],-D/2+(a+3*b-2)*(a+b+c-1)/(2*a*b*c)); end for; end for; Output: Magma V2.11-10 Thu Dec 8 2005 18:23:16 on modular [Seed = 1518459148] ------------------------------------- [0 0 0] [0 0 0] [0 0 0] (a^3 + 2*a^2*b*c + 9*a^2*b + a^2*c - 7*a^2 + 4*a*b^2*c + 23*a*b^2 + 6*a*b*c - 38*a*b - 6*a*c + 15*a - 30*b^3*c + 15*b^3 + 53*b^2*c - 39*b^2 - 36*b*c + 33*b + 9*c - 9)/(a^4 + 8*a^3*b + a^3*c - 6*a^3 + 22*a^2*b^2 + 11*a^2*b*c - 34*a^2*b - 5*a^2*c + 12*a^2 + 24*a*b^3 + 19*a*b^2*c - 58*a*b^2 - 26*a*b*c + 44*a*b + 7*a*c - 10*a + 9*b^4 + 9*b^3*c - 30*b^3 - 21*b^2*c + 36*b^2 + 15*b*c - 18*b - 3*c + 3) (-a^4 - 8*a^3*b - a^3*c + 7*a^3 - 22*a^2*b^2 - 9*a^2*b*c + 35*a^2*b + 6*a^2*c - 15*a^2 - 24*a*b^3 - 31*a*b^2*c + 49*a*b^2 + 32*a*b*c - 34*a*b - 9*a*c + 9*a - 9*b^4 + 9*b^3*c + 21*b^3 - 6*b^2*c - 15*b^2 + b*c + 3*b)/(a^4 + 8*a^3*b + a^3*c - 6*a^3 + 22*a^2*b^2 + 11*a^2*b*c - 34*a^2*b - 5*a^2*c + 12*a^2 + 24*a*b^3 + 19*a*b^2*c - 58*a*b^2 - 26*a*b*c + 44*a*b + 7*a*c - 10*a + 9*b^4 + 9*b^3*c - 30*b^3 - 21*b^2*c + 36*b^2 + 15*b*c - 18*b - 3*c + 3) (-2*a^2*b + 2*a*b + 18*b^3 - 18*b^2 + 4*b)/(a^3 + 7*a^2*b + a^2*c - 5*a^2 + 15*a*b^2 + 10*a*b*c - 22*a*b - 4*a*c + 7*a + 9*b^3 + 9*b^2*c - 21*b^2 - 12*b*c + 15*b + 3*c - 3) (-a^4 - 8*a^3*b + a^3*c + 7*a^3 - 22*a^2*b^2 + 9*a^2*b*c + 43*a^2*b - 6*a^2*c - 19*a^2 - 24*a*b^3 + 31*a*b^2*c + 81*a*b^2 - 32*a*b*c - 82*a*b + 9*a*c + 25*a - 9*b^4 - 9*b^3*c + 45*b^3 + 6*b^2*c - 75*b^2 - b*c + 51*b - 12)/(a^4 + 8*a^3*b + a^3*c - 6*a^3 + 22*a^2*b^2 + 11*a^2*b*c - 34*a^2*b - 5*a^2*c + 12*a^2 + 24*a*b^3 + 19*a*b^2*c - 58*a*b^2 - 26*a*b*c + 44*a*b + 7*a*c - 10*a + 9*b^4 + 9*b^3*c - 30*b^3 - 21*b^2*c + 36*b^2 + 15*b*c - 18*b - 3*c + 3) (2*a^3*c + a^3 + 4*a^2*b*c + a^2*b - 3*a^2*c - 3*a^2 - 30*a*b^2*c - 9*a*b^2 + 22*a*b*c + 10*a*b - 4*a*c - a - 9*b^3 + 9*b^2*c + 21*b^2 - 6*b*c - 15*b + c + 3)/(a^4 + 8*a^3*b + a^3*c - 6*a^3 + 22*a^2*b^2 + 11*a^2*b*c - 34*a^2*b - 5*a^2*c + 12*a^2 + 24*a*b^3 + 19*a*b^2*c - 58*a*b^2 - 26*a*b*c + 44*a*b + 7*a*c - 10*a + 9*b^4 + 9*b^3*c - 30*b^3 - 21*b^2*c + 36*b^2 + 15*b*c - 18*b - 3*c + 3) (-2*a^3 - 16*a^2*b + 8*a^2 - 30*a*b^2 + 38*a*b - 10*a + 18*b^2 - 18*b + 4)/(a^3 + 7*a^2*b + a^2*c - 5*a^2 + 15*a*b^2 + 10*a*b*c - 22*a*b - 4*a*c + 7*a + 9*b^3 + 9*b^2*c - 21*b^2 - 12*b*c + 15*b + 3*c - 3) (2*a^2*c + 16*a*b*c - 10*a*c + 30*b^2*c - 38*b*c + 12*c)/(a^3 + 7*a^2*b + a^2*c - 5*a^2 + 15*a*b^2 + 10*a*b*c - 22*a*b - 4*a*c + 7*a + 9*b^3 + 9*b^2*c - 21*b^2 - 12*b*c + 15*b + 3*c - 3) (2*a^2*c - 2*a*c - 18*b^2*c + 18*b*c - 4*c)/(a^3 + 7*a^2*b + a^2*c - 5*a^2 + 15*a*b^2 + 10*a*b*c - 22*a*b - 4*a*c + 7*a + 9*b^3 + 9*b^2*c - 21*b^2 - 12*b*c + 15*b + 3*c - 3) (-a^2 - 6*a*b + a*c + 2*a - 9*b^2 + 9*b*c + 6*b - 3*c - 1)/(a^2 + 6*a*b + a*c - 4*a + 9*b^2 + 9*b*c - 12*b - 3*c + 3) Total time: 0.380 seconds, Total memory usage: 3.63MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Thu Dec 8 18:17:06 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); A1:=Matrix(K,3,3,[[b,1,1],[0,a,1],[0,0,c]]); S:=Transpose(A)*A^(-1); F,T:=PrimaryRationalForm(S); B:=T*A*Transpose(T); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*B*Transpose(P)-B)); G:=GroebnerBasis(I); Q1:=Matrix(K,3,3,[(2*b-1)/(a+b-1),(a-b)/(a+b-1),0,(b-a)/(a+b-1),(2*a-1)/(a+b-1),0, 0,0,1]); Kt:=FunctionField(K); D:=(2*a*b*c-a-b-c+1)/(a*b*c); s:=t; px:=-(2*s+D); py:=1-s^2; q:=s^2+s*D+1; Q:=Matrix(Kt,3,3,[1,0,0,0,D*px/q+py/q,-px/q,0,px/q,py/q]); Q:=Q1*T^(-1)*Q*T; print (Q*A*Transpose(Q)-A1); for i in [1..3] do for j in [1..3] do print Evaluate(Q[i,j],-D/2+(-1+2*a)*(a+b+c-1)/(2*a*b*c)); end for; end for; Output: Magma V2.11-10 Thu Dec 8 2005 18:17:06 on modular [Seed = 2057406394] ------------------------------------- [0 0 0] [0 0 0] [0 0 0] 1/a -1 0 (-a^2 + a*c + 2*a - 1)/(a^2 + a*c - a) 0 (-2*a + 1)/(a + c - 1) (2*a*c - c)/(a^2 + a*c - a) 0 (-a + c)/(a + c - 1) Total time: 0.370 seconds, Total memory usage: 3.63MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Thu Dec 8 18:16:00 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); A1:=Matrix(K,3,3,[[b,1,1],[0,a,1],[0,0,c]]); S:=Transpose(A)*A^(-1); F,T:=PrimaryRationalForm(S); B:=T*A*Transpose(T); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*B*Transpose(P)-B)); G:=GroebnerBasis(I); Q1:=Matrix(K,3,3,[(2*b-1)/(a+b-1),(a-b)/(a+b-1),0,(b-a)/(a+b-1),(2*a-1)/(a+b-1),0, 0,0,1]); Kt:=FunctionField(K); D:=(2*a*b*c-a-b-c+1)/(a*b*c); s:=t; px:=-(2*s+D); py:=1-s^2; q:=s^2+s*D+1; Q:=Matrix(Kt,3,3,[1,0,0,0,D*px/q+py/q,-px/q,0,px/q,py/q]); Q:=Q1*T^(-1)*Q*T; print (Q*A*Transpose(Q)-A1); for i in [1..3] do for j in [1..3] do print Evaluate(Q[i,j],-D/2+(1-2*b)*(a+b+c-1)/(2*a*b*c)); end for; end for; Output: Magma V2.11-10 Thu Dec 8 2005 18:15:59 on modular [Seed = 115211163] ------------------------------------- [0 0 0] [0 0 0] [0 0 0] 0 (-b + c)/(b + c - 1) (2*b - 1)/(b + c - 1) -1 (-b + c)/(b^2 + b*c - b) (2*b - 1)/(b^2 + b*c - b) 0 (-2*b*c + c)/(b^2 + b*c - b) (-b^2 + b*c + 2*b - 1)/(b^2 + b*c - b) Total time: 0.370 seconds, Total memory usage: 3.63MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Thu Dec 8 18:15:21 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); A1:=Matrix(K,3,3,[[b,1,1],[0,a,1],[0,0,c]]); S:=Transpose(A)*A^(-1); F,T:=PrimaryRationalForm(S); B:=T*A*Transpose(T); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*B*Transpose(P)-B)); G:=GroebnerBasis(I); Q1:=Matrix(K,3,3,[(2*b-1)/(a+b-1),(a-b)/(a+b-1),0,(b-a)/(a+b-1),(2*a-1)/(a+b-1),0, 0,0,1]); Kt:=FunctionField(K); D:=(2*a*b*c-a-b-c+1)/(a*b*c); s:=t; px:=-(2*s+D); py:=1-s^2; q:=s^2+s*D+1; Q:=Matrix(Kt,3,3,[1,0,0,0,D*px/q+py/q,-px/q,0,px/q,py/q]); Q:=Q1*T^(-1)*Q*T; print (Q*A*Transpose(Q)-A1); for i in [1..3] do for j in [1..3] do print Evaluate(Denominator(Q[i,j]),-D/2+(1-2*b)*(a+b+c-1)/(2*a*b*c)); end for; end for; print Factorization(Numerator(a^2*b + a^2*c - a^2 + 2*a*b^2 + 3*a*b*c - 4*a*b + a*c^2 - 3*a*c + 2*a + b^3 + 2*b^2*c - 3*b^2 + b*c^2 - 4*b*c + 3*b - c^2 + 2*c - 1)); Output: Magma V2.11-10 Thu Dec 8 2005 18:15:21 on modular [Seed = 349170095] ------------------------------------- [0 0 0] [0 0 0] [0 0 0] (a^2*b + a^2*c - a^2 + 2*a*b^2 + 3*a*b*c - 4*a*b + a*c^2 - 3*a*c + 2*a + b^3 + 2*b^2*c - 3*b^2 + b*c^2 - 4*b*c + 3*b - c^2 + 2*c - 1)/(a^2*b*c^2) (a^2*b + a^2*c - a^2 + 2*a*b^2 + 3*a*b*c - 4*a*b + a*c^2 - 3*a*c + 2*a + b^3 + 2*b^2*c - 3*b^2 + b*c^2 - 4*b*c + 3*b - c^2 + 2*c - 1)/(a^2*b*c^2) (a^2*b + a^2*c - a^2 + 2*a*b^2 + 3*a*b*c - 4*a*b + a*c^2 - 3*a*c + 2*a + b^3 + 2*b^2*c - 3*b^2 + b*c^2 - 4*b*c + 3*b - c^2 + 2*c - 1)/(a^2*b*c^2) (a^2*b + a^2*c - a^2 + 2*a*b^2 + 3*a*b*c - 4*a*b + a*c^2 - 3*a*c + 2*a + b^3 + 2*b^2*c - 3*b^2 + b*c^2 - 4*b*c + 3*b - c^2 + 2*c - 1)/(a^2*b*c^2) (a^2*b + a^2*c - a^2 + 2*a*b^2 + 3*a*b*c - 4*a*b + a*c^2 - 3*a*c + 2*a + b^3 + 2*b^2*c - 3*b^2 + b*c^2 - 4*b*c + 3*b - c^2 + 2*c - 1)/(a^2*b*c^2) (a^2*b + a^2*c - a^2 + 2*a*b^2 + 3*a*b*c - 4*a*b + a*c^2 - 3*a*c + 2*a + b^3 + 2*b^2*c - 3*b^2 + b*c^2 - 4*b*c + 3*b - c^2 + 2*c - 1)/(a^2*b*c^2) (a^2*b + a^2*c - a^2 + 2*a*b^2 + 3*a*b*c - 4*a*b + a*c^2 - 3*a*c + 2*a + b^3 + 2*b^2*c - 3*b^2 + b*c^2 - 4*b*c + 3*b - c^2 + 2*c - 1)/(a^2*b*c^2) (a^2*b + a^2*c - a^2 + 2*a*b^2 + 3*a*b*c - 4*a*b + a*c^2 - 3*a*c + 2*a + b^3 + 2*b^2*c - 3*b^2 + b*c^2 - 4*b*c + 3*b - c^2 + 2*c - 1)/(a^2*b*c^2) (a^2*b + a^2*c - a^2 + 2*a*b^2 + 3*a*b*c - 4*a*b + a*c^2 - 3*a*c + 2*a + b^3 + 2*b^2*c - 3*b^2 + b*c^2 - 4*b*c + 3*b - c^2 + 2*c - 1)/(a^2*b*c^2) [ , , ] Total time: 0.370 seconds, Total memory usage: 3.63MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Thu Dec 8 18:14:46 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); A1:=Matrix(K,3,3,[[b,1,1],[0,a,1],[0,0,c]]); S:=Transpose(A)*A^(-1); F,T:=PrimaryRationalForm(S); B:=T*A*Transpose(T); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*B*Transpose(P)-B)); G:=GroebnerBasis(I); Q1:=Matrix(K,3,3,[(2*b-1)/(a+b-1),(a-b)/(a+b-1),0,(b-a)/(a+b-1),(2*a-1)/(a+b-1),0, 0,0,1]); Kt:=FunctionField(K); D:=(2*a*b*c-a-b-c+1)/(a*b*c); s:=t; px:=-(2*s+D); py:=1-s^2; q:=s^2+s*D+1; Q:=Matrix(Kt,3,3,[1,0,0,0,D*px/q+py/q,-px/q,0,px/q,py/q]); Q:=Q1*T^(-1)*Q*T; print (Q*A*Transpose(Q)-A1); for i in [1..3] do for j in [1..3] do print Evaluate(Denominator(Q[i,j]),-D/2+(1-2*b)*(a+b+c-1)/(2*a*b*c)); end for; end for; print Factorization(Numerator(1/4*a^4 + 1/2*a^3*c - 1/2*a^3 - 1/2*a^2*b^2 + 1/2*a^2*b*c + 1/2*a^2*b + 1/4*a^2*c^2 - 1/2*a^2*c + 1/2*a*b^2*c + 1/2*a*b^2 + 1/2*a*b*c^2 - a*b - 1/2*a*c + 1/2*a + 1/4*b^4 + 1/2*b^3*c - 1/2*b^3 + 1/4*b^2*c^2 - 1/2*b^2*c - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)); Output: Magma V2.11-10 Thu Dec 8 2005 18:14:45 on modular [Seed = 331935300] ------------------------------------- [0 0 0] [0 0 0] [0 0 0] (a^2*b + a^2*c - a^2 + 2*a*b^2 + 3*a*b*c - 4*a*b + a*c^2 - 3*a*c + 2*a + b^3 + 2*b^2*c - 3*b^2 + b*c^2 - 4*b*c + 3*b - c^2 + 2*c - 1)/(a^2*b*c^2) (a^2*b + a^2*c - a^2 + 2*a*b^2 + 3*a*b*c - 4*a*b + a*c^2 - 3*a*c + 2*a + b^3 + 2*b^2*c - 3*b^2 + b*c^2 - 4*b*c + 3*b - c^2 + 2*c - 1)/(a^2*b*c^2) (a^2*b + a^2*c - a^2 + 2*a*b^2 + 3*a*b*c - 4*a*b + a*c^2 - 3*a*c + 2*a + b^3 + 2*b^2*c - 3*b^2 + b*c^2 - 4*b*c + 3*b - c^2 + 2*c - 1)/(a^2*b*c^2) (a^2*b + a^2*c - a^2 + 2*a*b^2 + 3*a*b*c - 4*a*b + a*c^2 - 3*a*c + 2*a + b^3 + 2*b^2*c - 3*b^2 + b*c^2 - 4*b*c + 3*b - c^2 + 2*c - 1)/(a^2*b*c^2) (a^2*b + a^2*c - a^2 + 2*a*b^2 + 3*a*b*c - 4*a*b + a*c^2 - 3*a*c + 2*a + b^3 + 2*b^2*c - 3*b^2 + b*c^2 - 4*b*c + 3*b - c^2 + 2*c - 1)/(a^2*b*c^2) (a^2*b + a^2*c - a^2 + 2*a*b^2 + 3*a*b*c - 4*a*b + a*c^2 - 3*a*c + 2*a + b^3 + 2*b^2*c - 3*b^2 + b*c^2 - 4*b*c + 3*b - c^2 + 2*c - 1)/(a^2*b*c^2) (a^2*b + a^2*c - a^2 + 2*a*b^2 + 3*a*b*c - 4*a*b + a*c^2 - 3*a*c + 2*a + b^3 + 2*b^2*c - 3*b^2 + b*c^2 - 4*b*c + 3*b - c^2 + 2*c - 1)/(a^2*b*c^2) (a^2*b + a^2*c - a^2 + 2*a*b^2 + 3*a*b*c - 4*a*b + a*c^2 - 3*a*c + 2*a + b^3 + 2*b^2*c - 3*b^2 + b*c^2 - 4*b*c + 3*b - c^2 + 2*c - 1)/(a^2*b*c^2) (a^2*b + a^2*c - a^2 + 2*a*b^2 + 3*a*b*c - 4*a*b + a*c^2 - 3*a*c + 2*a + b^3 + 2*b^2*c - 3*b^2 + b*c^2 - 4*b*c + 3*b - c^2 + 2*c - 1)/(a^2*b*c^2) [ , , ] Total time: 0.370 seconds, Total memory usage: 3.63MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Thu Dec 8 18:12:48 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); A1:=Matrix(K,3,3,[[b,1,1],[0,a,1],[0,0,c]]); S:=Transpose(A)*A^(-1); F,T:=PrimaryRationalForm(S); B:=T*A*Transpose(T); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*B*Transpose(P)-B)); G:=GroebnerBasis(I); Q1:=Matrix(K,3,3,[(2*b-1)/(a+b-1),(a-b)/(a+b-1),0,(b-a)/(a+b-1),(2*a-1)/(a+b-1),0, 0,0,1]); Kt:=FunctionField(K); D:=(2*a*b*c-a-b-c+1)/(a*b*c); s:=t; px:=-(2*s+D); py:=1-s^2; q:=s^2+s*D+1; Q:=Matrix(Kt,3,3,[1,0,0,0,D*px/q+py/q,-px/q,0,px/q,py/q]); Q:=Q1*T^(-1)*Q*T; print (Q*A*Transpose(Q)-A1); for i in [1..3] do for j in [1..3] do print Evaluate(Denominator(Q[i,j]),-D/2+(a-b)*(a+b+c-1)/(2*a*b*c)); end for; end for; print Factorization(Numerator(1/4*a^4 + 1/2*a^3*c - 1/2*a^3 - 1/2*a^2*b^2 + 1/2*a^2*b*c + 1/2*a^2*b + 1/4*a^2*c^2 - 1/2*a^2*c + 1/2*a*b^2*c + 1/2*a*b^2 + 1/2*a*b*c^2 - a*b - 1/2*a*c + 1/2*a + 1/4*b^4 + 1/2*b^3*c - 1/2*b^3 + 1/4*b^2*c^2 - 1/2*b^2*c - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)); Output: Magma V2.11-10 Thu Dec 8 2005 18:12:48 on modular [Seed = 703780843] ------------------------------------- [0 0 0] [0 0 0] [0 0 0] (1/4*a^4 + 1/2*a^3*c - 1/2*a^3 - 1/2*a^2*b^2 + 1/2*a^2*b*c + 1/2*a^2*b + 1/4*a^2*c^2 - 1/2*a^2*c + 1/2*a*b^2*c + 1/2*a*b^2 + 1/2*a*b*c^2 - a*b - 1/2*a*c + 1/2*a + 1/4*b^4 + 1/2*b^3*c - 1/2*b^3 + 1/4*b^2*c^2 - 1/2*b^2*c - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2) (1/4*a^4 + 1/2*a^3*c - 1/2*a^3 - 1/2*a^2*b^2 + 1/2*a^2*b*c + 1/2*a^2*b + 1/4*a^2*c^2 - 1/2*a^2*c + 1/2*a*b^2*c + 1/2*a*b^2 + 1/2*a*b*c^2 - a*b - 1/2*a*c + 1/2*a + 1/4*b^4 + 1/2*b^3*c - 1/2*b^3 + 1/4*b^2*c^2 - 1/2*b^2*c - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2) (1/4*a^4 + 1/2*a^3*c - 1/2*a^3 - 1/2*a^2*b^2 + 1/2*a^2*b*c + 1/2*a^2*b + 1/4*a^2*c^2 - 1/2*a^2*c + 1/2*a*b^2*c + 1/2*a*b^2 + 1/2*a*b*c^2 - a*b - 1/2*a*c + 1/2*a + 1/4*b^4 + 1/2*b^3*c - 1/2*b^3 + 1/4*b^2*c^2 - 1/2*b^2*c - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2) (1/4*a^4 + 1/2*a^3*c - 1/2*a^3 - 1/2*a^2*b^2 + 1/2*a^2*b*c + 1/2*a^2*b + 1/4*a^2*c^2 - 1/2*a^2*c + 1/2*a*b^2*c + 1/2*a*b^2 + 1/2*a*b*c^2 - a*b - 1/2*a*c + 1/2*a + 1/4*b^4 + 1/2*b^3*c - 1/2*b^3 + 1/4*b^2*c^2 - 1/2*b^2*c - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2) (1/4*a^4 + 1/2*a^3*c - 1/2*a^3 - 1/2*a^2*b^2 + 1/2*a^2*b*c + 1/2*a^2*b + 1/4*a^2*c^2 - 1/2*a^2*c + 1/2*a*b^2*c + 1/2*a*b^2 + 1/2*a*b*c^2 - a*b - 1/2*a*c + 1/2*a + 1/4*b^4 + 1/2*b^3*c - 1/2*b^3 + 1/4*b^2*c^2 - 1/2*b^2*c - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2) (1/4*a^4 + 1/2*a^3*c - 1/2*a^3 - 1/2*a^2*b^2 + 1/2*a^2*b*c + 1/2*a^2*b + 1/4*a^2*c^2 - 1/2*a^2*c + 1/2*a*b^2*c + 1/2*a*b^2 + 1/2*a*b*c^2 - a*b - 1/2*a*c + 1/2*a + 1/4*b^4 + 1/2*b^3*c - 1/2*b^3 + 1/4*b^2*c^2 - 1/2*b^2*c - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2) (1/4*a^4 + 1/2*a^3*c - 1/2*a^3 - 1/2*a^2*b^2 + 1/2*a^2*b*c + 1/2*a^2*b + 1/4*a^2*c^2 - 1/2*a^2*c + 1/2*a*b^2*c + 1/2*a*b^2 + 1/2*a*b*c^2 - a*b - 1/2*a*c + 1/2*a + 1/4*b^4 + 1/2*b^3*c - 1/2*b^3 + 1/4*b^2*c^2 - 1/2*b^2*c - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2) (1/4*a^4 + 1/2*a^3*c - 1/2*a^3 - 1/2*a^2*b^2 + 1/2*a^2*b*c + 1/2*a^2*b + 1/4*a^2*c^2 - 1/2*a^2*c + 1/2*a*b^2*c + 1/2*a*b^2 + 1/2*a*b*c^2 - a*b - 1/2*a*c + 1/2*a + 1/4*b^4 + 1/2*b^3*c - 1/2*b^3 + 1/4*b^2*c^2 - 1/2*b^2*c - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2) (1/4*a^4 + 1/2*a^3*c - 1/2*a^3 - 1/2*a^2*b^2 + 1/2*a^2*b*c + 1/2*a^2*b + 1/4*a^2*c^2 - 1/2*a^2*c + 1/2*a*b^2*c + 1/2*a*b^2 + 1/2*a*b*c^2 - a*b - 1/2*a*c + 1/2*a + 1/4*b^4 + 1/2*b^3*c - 1/2*b^3 + 1/4*b^2*c^2 - 1/2*b^2*c - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2) [ , , ] Total time: 0.370 seconds, Total memory usage: 3.63MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Thu Dec 8 18:12:25 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); A1:=Matrix(K,3,3,[[b,1,1],[0,a,1],[0,0,c]]); S:=Transpose(A)*A^(-1); F,T:=PrimaryRationalForm(S); B:=T*A*Transpose(T); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*B*Transpose(P)-B)); G:=GroebnerBasis(I); Q1:=Matrix(K,3,3,[(2*b-1)/(a+b-1),(a-b)/(a+b-1),0,(b-a)/(a+b-1),(2*a-1)/(a+b-1),0, 0,0,1]); Kt:=FunctionField(K); D:=(2*a*b*c-a-b-c+1)/(a*b*c); s:=t; px:=-(2*s+D); py:=1-s^2; q:=s^2+s*D+1; Q:=Matrix(Kt,3,3,[1,0,0,0,D*px/q+py/q,-px/q,0,px/q,py/q]); Q:=Q1*T^(-1)*Q*T; print (Q*A*Transpose(Q)-A1); for i in [1..3] do for j in [1..3] do print Evaluate(Denominator(Q[i,j]),-D/2+(a-b)*(a+b+c-1)/(2*a*b*c)); end for; end for; print Factorization(Numerator(a^2 - 2*a*b + a*c + b^2 + b*c + c - 1)); Output: Magma V2.11-10 Thu Dec 8 2005 18:12:24 on modular [Seed = 620747470] ------------------------------------- [0 0 0] [0 0 0] [0 0 0] (1/4*a^4 + 1/2*a^3*c - 1/2*a^3 - 1/2*a^2*b^2 + 1/2*a^2*b*c + 1/2*a^2*b + 1/4*a^2*c^2 - 1/2*a^2*c + 1/2*a*b^2*c + 1/2*a*b^2 + 1/2*a*b*c^2 - a*b - 1/2*a*c + 1/2*a + 1/4*b^4 + 1/2*b^3*c - 1/2*b^3 + 1/4*b^2*c^2 - 1/2*b^2*c - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2) (1/4*a^4 + 1/2*a^3*c - 1/2*a^3 - 1/2*a^2*b^2 + 1/2*a^2*b*c + 1/2*a^2*b + 1/4*a^2*c^2 - 1/2*a^2*c + 1/2*a*b^2*c + 1/2*a*b^2 + 1/2*a*b*c^2 - a*b - 1/2*a*c + 1/2*a + 1/4*b^4 + 1/2*b^3*c - 1/2*b^3 + 1/4*b^2*c^2 - 1/2*b^2*c - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2) (1/4*a^4 + 1/2*a^3*c - 1/2*a^3 - 1/2*a^2*b^2 + 1/2*a^2*b*c + 1/2*a^2*b + 1/4*a^2*c^2 - 1/2*a^2*c + 1/2*a*b^2*c + 1/2*a*b^2 + 1/2*a*b*c^2 - a*b - 1/2*a*c + 1/2*a + 1/4*b^4 + 1/2*b^3*c - 1/2*b^3 + 1/4*b^2*c^2 - 1/2*b^2*c - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2) (1/4*a^4 + 1/2*a^3*c - 1/2*a^3 - 1/2*a^2*b^2 + 1/2*a^2*b*c + 1/2*a^2*b + 1/4*a^2*c^2 - 1/2*a^2*c + 1/2*a*b^2*c + 1/2*a*b^2 + 1/2*a*b*c^2 - a*b - 1/2*a*c + 1/2*a + 1/4*b^4 + 1/2*b^3*c - 1/2*b^3 + 1/4*b^2*c^2 - 1/2*b^2*c - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2) (1/4*a^4 + 1/2*a^3*c - 1/2*a^3 - 1/2*a^2*b^2 + 1/2*a^2*b*c + 1/2*a^2*b + 1/4*a^2*c^2 - 1/2*a^2*c + 1/2*a*b^2*c + 1/2*a*b^2 + 1/2*a*b*c^2 - a*b - 1/2*a*c + 1/2*a + 1/4*b^4 + 1/2*b^3*c - 1/2*b^3 + 1/4*b^2*c^2 - 1/2*b^2*c - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2) (1/4*a^4 + 1/2*a^3*c - 1/2*a^3 - 1/2*a^2*b^2 + 1/2*a^2*b*c + 1/2*a^2*b + 1/4*a^2*c^2 - 1/2*a^2*c + 1/2*a*b^2*c + 1/2*a*b^2 + 1/2*a*b*c^2 - a*b - 1/2*a*c + 1/2*a + 1/4*b^4 + 1/2*b^3*c - 1/2*b^3 + 1/4*b^2*c^2 - 1/2*b^2*c - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2) (1/4*a^4 + 1/2*a^3*c - 1/2*a^3 - 1/2*a^2*b^2 + 1/2*a^2*b*c + 1/2*a^2*b + 1/4*a^2*c^2 - 1/2*a^2*c + 1/2*a*b^2*c + 1/2*a*b^2 + 1/2*a*b*c^2 - a*b - 1/2*a*c + 1/2*a + 1/4*b^4 + 1/2*b^3*c - 1/2*b^3 + 1/4*b^2*c^2 - 1/2*b^2*c - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2) (1/4*a^4 + 1/2*a^3*c - 1/2*a^3 - 1/2*a^2*b^2 + 1/2*a^2*b*c + 1/2*a^2*b + 1/4*a^2*c^2 - 1/2*a^2*c + 1/2*a*b^2*c + 1/2*a*b^2 + 1/2*a*b*c^2 - a*b - 1/2*a*c + 1/2*a + 1/4*b^4 + 1/2*b^3*c - 1/2*b^3 + 1/4*b^2*c^2 - 1/2*b^2*c - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2) (1/4*a^4 + 1/2*a^3*c - 1/2*a^3 - 1/2*a^2*b^2 + 1/2*a^2*b*c + 1/2*a^2*b + 1/4*a^2*c^2 - 1/2*a^2*c + 1/2*a*b^2*c + 1/2*a*b^2 + 1/2*a*b*c^2 - a*b - 1/2*a*c + 1/2*a + 1/4*b^4 + 1/2*b^3*c - 1/2*b^3 + 1/4*b^2*c^2 - 1/2*b^2*c - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2) [ ] Total time: 0.370 seconds, Total memory usage: 3.63MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Thu Dec 8 18:11:51 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); A1:=Matrix(K,3,3,[[b,1,1],[0,a,1],[0,0,c]]); S:=Transpose(A)*A^(-1); F,T:=PrimaryRationalForm(S); B:=T*A*Transpose(T); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*B*Transpose(P)-B)); G:=GroebnerBasis(I); Q1:=Matrix(K,3,3,[(2*b-1)/(a+b-1),(a-b)/(a+b-1),0,(b-a)/(a+b-1),(2*a-1)/(a+b-1),0, 0,0,1]); Kt:=FunctionField(K); D:=(2*a*b*c-a-b-c+1)/(a*b*c); s:=t; px:=-(2*s+D); py:=1-s^2; q:=s^2+s*D+1; Q:=Matrix(Kt,3,3,[1,0,0,0,D*px/q+py/q,-px/q,0,px/q,py/q]); Q:=Q1*T^(-1)*Q*T; print (Q*A*Transpose(Q)-A1); for i in [1..3] do for j in [1..3] do print Evaluate(Q[i,j],-D/2+(a-b)*(a+b+c-1)/(2*a*b*c)); end for; end for; print Factorization(Numerator(a^2 - 2*a*b + a*c + b^2 + b*c + c - 1)); Output: Magma V2.11-10 Thu Dec 8 2005 18:11:51 on modular [Seed = 603512230] ------------------------------------- [0 0 0] [0 0 0] [0 0 0] (a + 2*b*c - b + c - 1)/(a^2 - 2*a*b + a*c + b^2 + b*c + c - 1) (-a^2 + 2*a*b - a*c + a - b^2 + b*c - b)/(a^2 - 2*a*b + a*c + b^2 + b*c + c - 1) (-2*a*b + 2*b^2)/(a^2 - 2*a*b + a*c + b^2 + b*c + c - 1) (-a^2 + 2*a*b + a*c + a - b^2 - b*c - b)/(a^2 - 2*a*b + a*c + b^2 + b*c + c - 1) (2*a*c + a - b + c - 1)/(a^2 - 2*a*b + a*c + b^2 + b*c + c - 1) (-2*a^2 + 2*a*b - 2*a + 2*b)/(a^2 - 2*a*b + a*c + b^2 + b*c + c - 1) (2*a*c - 2*b*c)/(a^2 - 2*a*b + a*c + b^2 + b*c + c - 1) (2*a*c - 2*b*c)/(a^2 - 2*a*b + a*c + b^2 + b*c + c - 1) (-a^2 + 2*a*b + a*c - 2*a - b^2 + b*c + 2*b + c - 1)/(a^2 - 2*a*b + a*c + b^2 + b*c + c - 1) [ ] Total time: 0.370 seconds, Total memory usage: 3.63MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Thu Dec 8 18:11:06 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); A1:=Matrix(K,3,3,[[b,1,1],[0,a,1],[0,0,c]]); S:=Transpose(A)*A^(-1); F,T:=PrimaryRationalForm(S); B:=T*A*Transpose(T); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*B*Transpose(P)-B)); G:=GroebnerBasis(I); Q1:=Matrix(K,3,3,[(2*b-1)/(a+b-1),(a-b)/(a+b-1),0,(b-a)/(a+b-1),(2*a-1)/(a+b-1),0, 0,0,1]); Kt:=FunctionField(K); D:=(2*a*b*c-a-b-c+1)/(a*b*c); s:=t; px:=-(2*s+D); py:=1-s^2; q:=s^2+s*D+1; Q:=Matrix(Kt,3,3,[1,0,0,0,D*px/q+py/q,-px/q,0,px/q,py/q]); Q:=Q1*T^(-1)*Q*T; print (Q*A*Transpose(Q)-A1); for i in [1..3] do for j in [1..3] do print Evaluate(Q[i,j],-D/2-(a-b)*(a+b+c-1)/(2*a*b*c)); end for; end for; print Factorization(Numerator(a^2 - 2*a*b + a*c + b^2 + b*c + c - 1)); Output: Magma V2.11-10 Thu Dec 8 2005 18:11:06 on modular [Seed = 1055244422] ------------------------------------- [0 0 0] [0 0 0] [0 0 0] (a^3 - 6*a^2*b*c - 3*a^2*b + a^2*c + a^2 + 20*a*b^2*c - a*b^2 - 10*a*b*c + 2*a*b + 2*a*c - a - 6*b^3*c + 3*b^3 + 5*b^2*c - 7*b^2 - 4*b*c + 5*b + c - 1)/(a^4 + a^3*c - 2*a^3 - 2*a^2*b^2 + 3*a^2*b*c + 2*a^2*b - a^2*c + 3*a*b^2*c + 2*a*b^2 - 2*a*b*c - 4*a*b - a*c + 2*a + b^4 + b^3*c - 2*b^3 - b^2*c - b*c + 2*b + c - 1) (-a^4 - a^3*c - a^3 + 2*a^2*b^2 + 15*a^2*b*c - a^2*b - 2*a^2*c + a^2 - 15*a*b^2*c + a*b^2 - 2*a*b - a*c + a - b^4 + b^3*c + b^3 + 2*b^2*c + b^2 + b*c - b)/(a^4 + a^3*c - 2*a^3 - 2*a^2*b^2 + 3*a^2*b*c + 2*a^2*b - a^2*c + 3*a*b^2*c + 2*a*b^2 - 2*a*b*c - 4*a*b - a*c + 2*a + b^4 + b^3*c - 2*b^3 - b^2*c - b*c + 2*b + c - 1) (6*a^2*b - 8*a*b^2 - 2*a*b + 2*b^3 + 2*b^2)/(a^3 - a^2*b + a^2*c - a^2 - a*b^2 + 2*a*b*c + 2*a*b - a + b^3 + b^2*c - b^2 - b - c + 1) (-a^4 + a^3*c + 3*a^3 + 2*a^2*b^2 - 15*a^2*b*c - 5*a^2*b + 2*a^2*c + a^2 + 15*a*b^2*c - 3*a*b^2 + 6*a*b + a*c - 3*a - b^4 - b^3*c + 5*b^3 - 2*b^2*c - 7*b^2 - b*c + 3*b)/(a^4 + a^3*c - 2*a^3 - 2*a^2*b^2 + 3*a^2*b*c + 2*a^2*b - a^2*c + 3*a*b^2*c + 2*a*b^2 - 2*a*b*c - 4*a*b - a*c + 2*a + b^4 + b^3*c - 2*b^3 - b^2*c - b*c + 2*b + c - 1) (-6*a^3*c - 3*a^3 + 20*a^2*b*c + a^2*b + 5*a^2*c + a^2 - 6*a*b^2*c + 3*a*b^2 - 10*a*b*c - 6*a*b - 4*a*c + 3*a - b^3 + b^2*c + b^2 + 2*b*c + b + c - 1)/(a^4 + a^3*c - 2*a^3 - 2*a^2*b^2 + 3*a^2*b*c + 2*a^2*b - a^2*c + 3*a*b^2*c + 2*a*b^2 - 2*a*b*c - 4*a*b - a*c + 2*a + b^4 + b^3*c - 2*b^3 - b^2*c - b*c + 2*b + c - 1) (-2*a^3 + 8*a^2*b + 4*a^2 - 6*a*b^2 - 6*a*b - 2*a + 2*b^2 + 2*b)/(a^3 - a^2*b + a^2*c - a^2 - a*b^2 + 2*a*b*c + 2*a*b - a + b^3 + b^2*c - b^2 - b - c + 1) (2*a^2*c - 8*a*b*c + 2*a*c + 6*b^2*c - 2*b*c)/(a^3 - a^2*b + a^2*c - a^2 - a*b^2 + 2*a*b*c + 2*a*b - a + b^3 + b^2*c - b^2 - b - c + 1) (-6*a^2*c + 8*a*b*c + 2*a*c - 2*b^2*c - 2*b*c)/(a^3 - a^2*b + a^2*c - a^2 - a*b^2 + 2*a*b*c + 2*a*b - a + b^3 + b^2*c - b^2 - b - c + 1) (-a^2 + 2*a*b + a*c + 2*a - b^2 + b*c - 2*b + c - 1)/(a^2 - 2*a*b + a*c + b^2 + b*c + c - 1) [ ] Total time: 0.380 seconds, Total memory usage: 3.63MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Thu Dec 8 18:10:33 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); A1:=Matrix(K,3,3,[[b,1,1],[0,a,1],[0,0,c]]); S:=Transpose(A)*A^(-1); F,T:=PrimaryRationalForm(S); B:=T*A*Transpose(T); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*B*Transpose(P)-B)); G:=GroebnerBasis(I); Q1:=Matrix(K,3,3,[(2*b-1)/(a+b-1),(a-b)/(a+b-1),0,(b-a)/(a+b-1),(2*a-1)/(a+b-1),0, 0,0,1]); Kt:=FunctionField(K); D:=(2*a*b*c-a-b-c+1)/(a*b*c); s:=t; px:=-(2*s+D); py:=1-s^2; q:=s^2+s*D+1; Q:=Matrix(Kt,3,3,[1,0,0,0,D*px/q+py/q,-px/q,0,px/q,py/q]); Q:=Q1*T^(-1)*Q*T; print (Q*A*Transpose(Q)-A1); for i in [1..3] do for j in [1..3] do print Evaluate(Q[i,j],-D/2+(a-b)*(a+b+c-1)/(2*a*b*c)); end for; end for; print Factorization(Numerator(a^2 - 2*a*b + a*c + b^2 + b*c + c - 1)); Output: Magma V2.11-10 Thu Dec 8 2005 18:10:32 on modular [Seed = 904842095] ------------------------------------- [0 0 0] [0 0 0] [0 0 0] (a + 2*b*c - b + c - 1)/(a^2 - 2*a*b + a*c + b^2 + b*c + c - 1) (-a^2 + 2*a*b - a*c + a - b^2 + b*c - b)/(a^2 - 2*a*b + a*c + b^2 + b*c + c - 1) (-2*a*b + 2*b^2)/(a^2 - 2*a*b + a*c + b^2 + b*c + c - 1) (-a^2 + 2*a*b + a*c + a - b^2 - b*c - b)/(a^2 - 2*a*b + a*c + b^2 + b*c + c - 1) (2*a*c + a - b + c - 1)/(a^2 - 2*a*b + a*c + b^2 + b*c + c - 1) (-2*a^2 + 2*a*b - 2*a + 2*b)/(a^2 - 2*a*b + a*c + b^2 + b*c + c - 1) (2*a*c - 2*b*c)/(a^2 - 2*a*b + a*c + b^2 + b*c + c - 1) (2*a*c - 2*b*c)/(a^2 - 2*a*b + a*c + b^2 + b*c + c - 1) (-a^2 + 2*a*b + a*c - 2*a - b^2 + b*c + 2*b + c - 1)/(a^2 - 2*a*b + a*c + b^2 + b*c + c - 1) [ ] Total time: 0.370 seconds, Total memory usage: 3.63MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Thu Dec 8 18:09:23 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); A1:=Matrix(K,3,3,[[b,1,1],[0,a,1],[0,0,c]]); S:=Transpose(A)*A^(-1); F,T:=PrimaryRationalForm(S); B:=T*A*Transpose(T); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*B*Transpose(P)-B)); G:=GroebnerBasis(I); Q1:=Matrix(K,3,3,[(2*b-1)/(a+b-1),(a-b)/(a+b-1),0,(b-a)/(a+b-1),(2*a-1)/(a+b-1),0, 0,0,1]); Kt:=FunctionField(K); D:=(2*a*b*c-a-b-c+1)/(a*b*c); s:=t; px:=-(2*s+D); py:=1-s^2; q:=s^2+s*D+1; Q:=Matrix(Kt,3,3,[1,0,0,0,D*px/q+py/q,-px/q,0,px/q,py/q]); Q:=Q1*T^(-1)*Q*T; print (Q*A*Transpose(Q)-A1); for i in [1..3] do for j in [1..3] do print Evaluate(Q[i,j],-D/2+(a-b)*(a+b+c-1)/(2*a*b*c)); end for; end for; Output: Magma V2.11-10 Thu Dec 8 2005 18:09:22 on modular [Seed = 820760096] ------------------------------------- [0 0 0] [0 0 0] [0 0 0] (a + 2*b*c - b + c - 1)/(a^2 - 2*a*b + a*c + b^2 + b*c + c - 1) (-a^2 + 2*a*b - a*c + a - b^2 + b*c - b)/(a^2 - 2*a*b + a*c + b^2 + b*c + c - 1) (-2*a*b + 2*b^2)/(a^2 - 2*a*b + a*c + b^2 + b*c + c - 1) (-a^2 + 2*a*b + a*c + a - b^2 - b*c - b)/(a^2 - 2*a*b + a*c + b^2 + b*c + c - 1) (2*a*c + a - b + c - 1)/(a^2 - 2*a*b + a*c + b^2 + b*c + c - 1) (-2*a^2 + 2*a*b - 2*a + 2*b)/(a^2 - 2*a*b + a*c + b^2 + b*c + c - 1) (2*a*c - 2*b*c)/(a^2 - 2*a*b + a*c + b^2 + b*c + c - 1) (2*a*c - 2*b*c)/(a^2 - 2*a*b + a*c + b^2 + b*c + c - 1) (-a^2 + 2*a*b + a*c - 2*a - b^2 + b*c + 2*b + c - 1)/(a^2 - 2*a*b + a*c + b^2 + b*c + c - 1) Total time: 0.370 seconds, Total memory usage: 3.63MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Thu Dec 8 18:09:07 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); A1:=Matrix(K,3,3,[[b,1,1],[0,a,1],[0,0,c]]); S:=Transpose(A)*A^(-1); F,T:=PrimaryRationalForm(S); B:=T*A*Transpose(T); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*B*Transpose(P)-B)); G:=GroebnerBasis(I); Q1:=Matrix(K,3,3,[(2*b-1)/(a+b-1),(a-b)/(a+b-1),0,(b-a)/(a+b-1),(2*a-1)/(a+b-1),0, 0,0,1]); Kt:=FunctionField(K); D:=(2*a*b*c-a-b-c+1)/(a*b*c); s:=t; px:=-(2*s+D); py:=1-s^2; q:=s^2+s*D+1; Q:=Matrix(Kt,3,3,[1,0,0,0,D*px/q+py/q,-px/q,0,px/q,py/q]); Q:=Q1*T^(-1)*Q*T; print (Q*A*Transpose(Q)-A1); for i in [1..3] do for j in [1..3] do print Evaluate(Q[i,j]),-D/2+(a-b)*(a+b+c-1)/(2*a*b*c)); end for; end for; Output: Magma V2.11-10 Thu Dec 8 2005 18:09:07 on modular [Seed = 3457495278] ------------------------------------- [0 0 0] [0 0 0] [0 0 0] >> print Evaluate(Q[i,j]),-D/2+(a-b)*(a+b+c-1)/(2*a*b*c)); ^ User error: bad syntax >> end for; ^ User error: bad syntax >> end for; ^ User error: bad syntax Total time: 0.360 seconds, Total memory usage: 3.63MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Thu Dec 8 18:05:30 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); A1:=Matrix(K,3,3,[[b,1,1],[0,a,1],[0,0,c]]); S:=Transpose(A)*A^(-1); F,T:=PrimaryRationalForm(S); B:=T*A*Transpose(T); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*B*Transpose(P)-B)); G:=GroebnerBasis(I); Q1:=Matrix(K,3,3,[(2*b-1)/(a+b-1),(a-b)/(a+b-1),0,(b-a)/(a+b-1),(2*a-1)/(a+b-1),0, 0,0,1]); Kt:=FunctionField(K); D:=(2*a*b*c-a-b-c+1)/(a*b*c); s:=t; px:=-(2*s+D); py:=1-s^2; q:=s^2+s*D+1; Q:=Matrix(Kt,3,3,[1,0,0,0,D*px/q+py/q,-px/q,0,px/q,py/q]); Q:=Q1*T^(-1)*Q*T; print (Q*A*Transpose(Q)-A1); for i in [1..3] do for j in [1..3] do print Evaluate(Denominator(Q[i,j]),-D/2+(a-b)/(2*a*b)); end for; end for; print Factorization(Numerator(a^2*b*c + 1/4*a^2*c^2 - 1/4*a^2 + a*b^2*c + 1/2*a*b*c^2 - a*b*c - 1/2*a*b - 1/2*a*c + 1/2*a + 1/4*b^2*c^2 - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)); Output: Magma V2.11-10 Thu Dec 8 2005 18:05:29 on modular [Seed = 3657508304] ------------------------------------- [0 0 0] [0 0 0] [0 0 0] (a^2*b*c + 1/4*a^2*c^2 - 1/4*a^2 + a*b^2*c + 1/2*a*b*c^2 - a*b*c - 1/2*a*b - 1/2*a*c + 1/2*a + 1/4*b^2*c^2 - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2) (a^2*b*c + 1/4*a^2*c^2 - 1/4*a^2 + a*b^2*c + 1/2*a*b*c^2 - a*b*c - 1/2*a*b - 1/2*a*c + 1/2*a + 1/4*b^2*c^2 - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2) (a^2*b*c + 1/4*a^2*c^2 - 1/4*a^2 + a*b^2*c + 1/2*a*b*c^2 - a*b*c - 1/2*a*b - 1/2*a*c + 1/2*a + 1/4*b^2*c^2 - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2) (a^2*b*c + 1/4*a^2*c^2 - 1/4*a^2 + a*b^2*c + 1/2*a*b*c^2 - a*b*c - 1/2*a*b - 1/2*a*c + 1/2*a + 1/4*b^2*c^2 - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2) (a^2*b*c + 1/4*a^2*c^2 - 1/4*a^2 + a*b^2*c + 1/2*a*b*c^2 - a*b*c - 1/2*a*b - 1/2*a*c + 1/2*a + 1/4*b^2*c^2 - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2) (a^2*b*c + 1/4*a^2*c^2 - 1/4*a^2 + a*b^2*c + 1/2*a*b*c^2 - a*b*c - 1/2*a*b - 1/2*a*c + 1/2*a + 1/4*b^2*c^2 - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2) (a^2*b*c + 1/4*a^2*c^2 - 1/4*a^2 + a*b^2*c + 1/2*a*b*c^2 - a*b*c - 1/2*a*b - 1/2*a*c + 1/2*a + 1/4*b^2*c^2 - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2) (a^2*b*c + 1/4*a^2*c^2 - 1/4*a^2 + a*b^2*c + 1/2*a*b*c^2 - a*b*c - 1/2*a*b - 1/2*a*c + 1/2*a + 1/4*b^2*c^2 - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2) (a^2*b*c + 1/4*a^2*c^2 - 1/4*a^2 + a*b^2*c + 1/2*a*b*c^2 - a*b*c - 1/2*a*b - 1/2*a*c + 1/2*a + 1/4*b^2*c^2 - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2) [ , ] Total time: 0.370 seconds, Total memory usage: 3.63MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Thu Dec 8 18:04:49 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); A1:=Matrix(K,3,3,[[b,1,1],[0,a,1],[0,0,c]]); S:=Transpose(A)*A^(-1); F,T:=PrimaryRationalForm(S); B:=T*A*Transpose(T); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*B*Transpose(P)-B)); G:=GroebnerBasis(I); Q1:=Matrix(K,3,3,[(2*b-1)/(a+b-1),(a-b)/(a+b-1),0,(b-a)/(a+b-1),(2*a-1)/(a+b-1),0, 0,0,1]); Kt:=FunctionField(K); D:=(2*a*b*c-a-b-c+1)/(a*b*c); s:=t; px:=-(2*s+D); py:=1-s^2; q:=s^2+s*D+1; Q:=Matrix(Kt,3,3,[1,0,0,0,D*px/q+py/q,-px/q,0,px/q,py/q]); Q:=Q1*T^(-1)*Q*T; print (Q*A*Transpose(Q)-A1); for i in [1..3] do for j in [1..3] do print Evaluate(Denominator(Q[i,j]),-D/2+(a-b)/(2*a*b)); end for; end for; Output: Magma V2.11-10 Thu Dec 8 2005 18:04:49 on modular [Seed = 3607898357] ------------------------------------- [0 0 0] [0 0 0] [0 0 0] (a^2*b*c + 1/4*a^2*c^2 - 1/4*a^2 + a*b^2*c + 1/2*a*b*c^2 - a*b*c - 1/2*a*b - 1/2*a*c + 1/2*a + 1/4*b^2*c^2 - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2) (a^2*b*c + 1/4*a^2*c^2 - 1/4*a^2 + a*b^2*c + 1/2*a*b*c^2 - a*b*c - 1/2*a*b - 1/2*a*c + 1/2*a + 1/4*b^2*c^2 - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2) (a^2*b*c + 1/4*a^2*c^2 - 1/4*a^2 + a*b^2*c + 1/2*a*b*c^2 - a*b*c - 1/2*a*b - 1/2*a*c + 1/2*a + 1/4*b^2*c^2 - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2) (a^2*b*c + 1/4*a^2*c^2 - 1/4*a^2 + a*b^2*c + 1/2*a*b*c^2 - a*b*c - 1/2*a*b - 1/2*a*c + 1/2*a + 1/4*b^2*c^2 - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2) (a^2*b*c + 1/4*a^2*c^2 - 1/4*a^2 + a*b^2*c + 1/2*a*b*c^2 - a*b*c - 1/2*a*b - 1/2*a*c + 1/2*a + 1/4*b^2*c^2 - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2) (a^2*b*c + 1/4*a^2*c^2 - 1/4*a^2 + a*b^2*c + 1/2*a*b*c^2 - a*b*c - 1/2*a*b - 1/2*a*c + 1/2*a + 1/4*b^2*c^2 - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2) (a^2*b*c + 1/4*a^2*c^2 - 1/4*a^2 + a*b^2*c + 1/2*a*b*c^2 - a*b*c - 1/2*a*b - 1/2*a*c + 1/2*a + 1/4*b^2*c^2 - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2) (a^2*b*c + 1/4*a^2*c^2 - 1/4*a^2 + a*b^2*c + 1/2*a*b*c^2 - a*b*c - 1/2*a*b - 1/2*a*c + 1/2*a + 1/4*b^2*c^2 - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2) (a^2*b*c + 1/4*a^2*c^2 - 1/4*a^2 + a*b^2*c + 1/2*a*b*c^2 - a*b*c - 1/2*a*b - 1/2*a*c + 1/2*a + 1/4*b^2*c^2 - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2) Total time: 0.360 seconds, Total memory usage: 3.63MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Thu Dec 8 18:04:16 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); A1:=Matrix(K,3,3,[[b,1,1],[0,a,1],[0,0,c]]); S:=Transpose(A)*A^(-1); F,T:=PrimaryRationalForm(S); B:=T*A*Transpose(T); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*B*Transpose(P)-B)); G:=GroebnerBasis(I); Q1:=Matrix(K,3,3,[(2*b-1)/(a+b-1),(a-b)/(a+b-1),0,(b-a)/(a+b-1),(2*a-1)/(a+b-1),0, 0,0,1]); Kt:=FunctionField(K); D:=(2*a*b*c-a-b-c+1)/(a*b*c); s:=t; px:=-(2*s+D); py:=1-s^2; q:=s^2+s*D+1; Q:=Matrix(Kt,3,3,[1,0,0,0,D*px/q+py/q,-px/q,0,px/q,py/q]); Q:=Q1*T^(-1)*Q*T; print (Q*A*Transpose(Q)-A1); for i in [1..3] do for j in [1..3] do print Factorization(Evaluate(Denominator(Q[i,j]),-D/2+(a-b)/(2*a*b))); end for; end for; Output: Magma V2.11-10 Thu Dec 8 2005 18:04:15 on modular [Seed = 3963032624] ------------------------------------- [0 0 0] [0 0 0] [0 0 0] >> print Factorization(Evaluate(Denominator(Q[i,j]),-D/2+(a-b)/(2*a*b))); ^ Runtime error in 'Factorization': Bad argument types Argument types given: FldFunRatMElt Total time: 0.360 seconds, Total memory usage: 3.63MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Thu Dec 8 18:03:59 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); A1:=Matrix(K,3,3,[[b,1,1],[0,a,1],[0,0,c]]); S:=Transpose(A)*A^(-1); F,T:=PrimaryRationalForm(S); B:=T*A*Transpose(T); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*B*Transpose(P)-B)); G:=GroebnerBasis(I); Q1:=Matrix(K,3,3,[(2*b-1)/(a+b-1),(a-b)/(a+b-1),0,(b-a)/(a+b-1),(2*a-1)/(a+b-1),0, 0,0,1]); Kt:=FunctionField(K); D:=(2*a*b*c-a-b-c+1)/(a*b*c); s:=t; px:=-(2*s+D); py:=1-s^2; q:=s^2+s*D+1; Q:=Matrix(Kt,3,3,[1,0,0,0,D*px/q+py/q,-px/q,0,px/q,py/q]); Q:=Q1*T^(-1)*Q*T; print (Q*A*Transpose(Q)-A1); for i in [1..3] do for j in [1..3] do print Evaluate(Denominator(Q[i,j]),-D/2+(a-b)/(2*a*b)); end for; end for; Output: Magma V2.11-10 Thu Dec 8 2005 18:03:58 on modular [Seed = 3912372484] ------------------------------------- [0 0 0] [0 0 0] [0 0 0] (a^2*b*c + 1/4*a^2*c^2 - 1/4*a^2 + a*b^2*c + 1/2*a*b*c^2 - a*b*c - 1/2*a*b - 1/2*a*c + 1/2*a + 1/4*b^2*c^2 - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2) (a^2*b*c + 1/4*a^2*c^2 - 1/4*a^2 + a*b^2*c + 1/2*a*b*c^2 - a*b*c - 1/2*a*b - 1/2*a*c + 1/2*a + 1/4*b^2*c^2 - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2) (a^2*b*c + 1/4*a^2*c^2 - 1/4*a^2 + a*b^2*c + 1/2*a*b*c^2 - a*b*c - 1/2*a*b - 1/2*a*c + 1/2*a + 1/4*b^2*c^2 - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2) (a^2*b*c + 1/4*a^2*c^2 - 1/4*a^2 + a*b^2*c + 1/2*a*b*c^2 - a*b*c - 1/2*a*b - 1/2*a*c + 1/2*a + 1/4*b^2*c^2 - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2) (a^2*b*c + 1/4*a^2*c^2 - 1/4*a^2 + a*b^2*c + 1/2*a*b*c^2 - a*b*c - 1/2*a*b - 1/2*a*c + 1/2*a + 1/4*b^2*c^2 - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2) (a^2*b*c + 1/4*a^2*c^2 - 1/4*a^2 + a*b^2*c + 1/2*a*b*c^2 - a*b*c - 1/2*a*b - 1/2*a*c + 1/2*a + 1/4*b^2*c^2 - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2) (a^2*b*c + 1/4*a^2*c^2 - 1/4*a^2 + a*b^2*c + 1/2*a*b*c^2 - a*b*c - 1/2*a*b - 1/2*a*c + 1/2*a + 1/4*b^2*c^2 - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2) (a^2*b*c + 1/4*a^2*c^2 - 1/4*a^2 + a*b^2*c + 1/2*a*b*c^2 - a*b*c - 1/2*a*b - 1/2*a*c + 1/2*a + 1/4*b^2*c^2 - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2) (a^2*b*c + 1/4*a^2*c^2 - 1/4*a^2 + a*b^2*c + 1/2*a*b*c^2 - a*b*c - 1/2*a*b - 1/2*a*c + 1/2*a + 1/4*b^2*c^2 - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2) Total time: 0.370 seconds, Total memory usage: 3.63MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Thu Dec 8 18:03:31 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); A1:=Matrix(K,3,3,[[b,1,1],[0,a,1],[0,0,c]]); S:=Transpose(A)*A^(-1); F,T:=PrimaryRationalForm(S); B:=T*A*Transpose(T); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*B*Transpose(P)-B)); G:=GroebnerBasis(I); Q1:=Matrix(K,3,3,[(2*b-1)/(a+b-1),(a-b)/(a+b-1),0,(b-a)/(a+b-1),(2*a-1)/(a+b-1),0, 0,0,1]); Kt:=FunctionField(K); D:=(2*a*b*c-a-b-c+1)/(a*b*c); s:=t; px:=-(2*s+D); py:=1-s^2; q:=s^2+s*D+1; Q:=Matrix(Kt,3,3,[1,0,0,0,D*px/q+py/q,-px/q,0,px/q,py/q]); Q:=Q1*T^(-1)*Q*T; print (Q*A*Transpose(Q)-A1); for i in [1..3] do for j in [1..3] do print Evaluate(Q[i,j],-D/2-(a-b)/(2*a*b)); end for; end for; Output: Magma V2.11-10 Thu Dec 8 2005 18:03:30 on modular [Seed = 3862765079] ------------------------------------- [0 0 0] [0 0 0] [0 0 0] (2*a^3*b^2*c - a^3*b*c^2 - 3/2*a^3*b*c - 1/2*a^3*b + 1/4*a^3*c^2 + 1/2*a^3*c + 1/4*a^3 + 4*a^2*b^3*c + 5*a^2*b^2*c^2 - 13/2*a^2*b^2*c - 3/2*a^2*b^2 - 3/2*a^2*b*c^3 - 11/4*a^2*b*c^2 + 2*a^2*b*c + 9/4*a^2*b + 1/4*a^2*c^3 + 3/4*a^2*c^2 - 1/4*a^2*c - 3/4*a^2 + 2*a*b^4*c + 5*a*b^3*c^2 - 9/2*a*b^3*c - 3/2*a*b^3 + 5*a*b^2*c^3 - 33/4*a*b^2*c^2 + 1/2*a*b^2*c + 15/4*a*b^2 - 5/2*a*b*c^3 + 5/2*a*b*c^2 + 3*a*b*c - 3*a*b + 1/2*a*c^3 - 1/4*a*c^2 - a*c + 3/4*a - b^4*c^2 + 1/2*b^4*c - 1/2*b^4 - 3/2*b^3*c^3 + 11/4*b^3*c^2 - 3*b^3*c + 7/4*b^3 + 5/4*b^2*c^3 - 17/4*b^2*c^2 + 21/4*b^2*c - 9/4*b^2 - b*c^3 + 13/4*b*c^2 - 7/2*b*c + 5/4*b + 1/4*c^3 - 3/4*c^2 + 3/4*c - 1/4)/(a^4*b*c + 1/4*a^4*c^2 - 1/4*a^4 + 3*a^3*b^2*c + 2*a^3*b*c^2 - 3*a^3*b*c - a^3*b + 1/4*a^3*c^3 - 1/2*a^3*c^2 - 3/4*a^3*c + a^3 + 3*a^2*b^3*c + 7/2*a^2*b^2*c^2 - 6*a^2*b^2*c - 3/2*a^2*b^2 + 3/4*a^2*b*c^3 - 7/2*a^2*b*c^2 + 3/4*a^2*b*c + 3*a^2*b - 1/4*a^2*c^3 - 1/2*a^2*c^2 + 9/4*a^2*c - 3/2*a^2 + a*b^4*c + 2*a*b^3*c^2 - 3*a*b^3*c - a*b^3 + 3/4*a*b^2*c^3 - 7/2*a*b^2*c^2 + 3/4*a*b^2*c + 3*a*b^2 - 1/2*a*b*c^3 + 7/2*a*b*c - 3*a*b - 1/4*a*c^3 + 3/2*a*c^2 - 9/4*a*c + a + 1/4*b^4*c^2 - 1/4*b^4 + 1/4*b^3*c^3 - 1/2*b^3*c^2 - 3/4*b^3*c + b^3 - 1/4*b^2*c^3 - 1/2*b^2*c^2 + 9/4*b^2*c - 3/2*b^2 - 1/4*b*c^3 + 3/2*b*c^2 - 9/4*b*c + b + 1/4*c^3 - 3/4*c^2 + 3/4*c - 1/4) (a^4*b*c - 1/4*a^4*c^2 - 1/2*a^4*c - 1/4*a^4 + a^3*b^2*c + 4*a^3*b*c^2 - 5/2*a^3*b*c - 1/2*a^3*b - 1/4*a^3*c^3 - 3/4*a^3*c^2 + 1/4*a^3*c + 3/4*a^3 - a^2*b^3*c + 1/2*a^2*b^2*c^2 + 1/2*a^2*b^2*c + 15/4*a^2*b*c^3 - 19/4*a^2*b*c^2 + 1/4*a^2*b*c + 3/4*a^2*b - 1/2*a^2*c^3 + 1/4*a^2*c^2 + a^2*c - 3/4*a^2 - a*b^4*c - 4*a*b^3*c^2 + 5/2*a*b^3*c + 1/2*a*b^3 - 15/4*a*b^2*c^3 + 19/4*a*b^2*c^2 - 5/4*a*b^2*c - 3/4*a*b^2 - 1/2*a*b*c^2 + 1/2*a*b*c - 1/4*a*c^3 + 3/4*a*c^2 - 3/4*a*c + 1/4*a - 1/4*b^4*c^2 + 1/4*b^4 + 1/4*b^3*c^3 + 3/4*b^3*c^2 + 3/4*b^3*c - 3/4*b^3 + 1/2*b^2*c^3 + 1/4*b^2*c^2 - 3/2*b^2*c + 3/4*b^2 + 1/4*b*c^3 - 3/4*b*c^2 + 3/4*b*c - 1/4*b)/(a^4*b*c + 1/4*a^4*c^2 - 1/4*a^4 + 3*a^3*b^2*c + 2*a^3*b*c^2 - 3*a^3*b*c - a^3*b + 1/4*a^3*c^3 - 1/2*a^3*c^2 - 3/4*a^3*c + a^3 + 3*a^2*b^3*c + 7/2*a^2*b^2*c^2 - 6*a^2*b^2*c - 3/2*a^2*b^2 + 3/4*a^2*b*c^3 - 7/2*a^2*b*c^2 + 3/4*a^2*b*c + 3*a^2*b - 1/4*a^2*c^3 - 1/2*a^2*c^2 + 9/4*a^2*c - 3/2*a^2 + a*b^4*c + 2*a*b^3*c^2 - 3*a*b^3*c - a*b^3 + 3/4*a*b^2*c^3 - 7/2*a*b^2*c^2 + 3/4*a*b^2*c + 3*a*b^2 - 1/2*a*b*c^3 + 7/2*a*b*c - 3*a*b - 1/4*a*c^3 + 3/2*a*c^2 - 9/4*a*c + a + 1/4*b^4*c^2 - 1/4*b^4 + 1/4*b^3*c^3 - 1/2*b^3*c^2 - 3/4*b^3*c + b^3 - 1/4*b^2*c^3 - 1/2*b^2*c^2 + 9/4*b^2*c - 3/2*b^2 - 1/4*b*c^3 + 3/2*b*c^2 - 9/4*b*c + b + 1/4*c^3 - 3/4*c^2 + 3/4*c - 1/4) (a^3*b*c + 3/2*a^2*b*c^2 - 3/2*a^2*b*c - a*b^3*c - 2*a*b^2*c^2 + a*b^2*c - 1/2*a*b*c^2 + 1/2*a*b*c + 1/2*b^3*c^2 + 1/2*b^3*c + 1/2*b^2*c^2 - 1/2*b^2*c)/(a^3*b*c + 1/4*a^3*c^2 - 1/4*a^3 + 2*a^2*b^2*c + 7/4*a^2*b*c^2 - 2*a^2*b*c - 3/4*a^2*b + 1/4*a^2*c^3 - 1/4*a^2*c^2 - 3/4*a^2*c + 3/4*a^2 + a*b^3*c + 7/4*a*b^2*c^2 - 2*a*b^2*c - 3/4*a*b^2 + 1/2*a*b*c^3 - 3/2*a*b*c^2 - 1/2*a*b*c + 3/2*a*b - 3/4*a*c^2 + 3/2*a*c - 3/4*a + 1/4*b^3*c^2 - 1/4*b^3 + 1/4*b^2*c^3 - 1/4*b^2*c^2 - 3/4*b^2*c + 3/4*b^2 - 3/4*b*c^2 + 3/2*b*c - 3/4*b - 1/4*c^3 + 3/4*c^2 - 3/4*c + 1/4) (-a^4*b*c - 1/4*a^4*c^2 + 1/4*a^4 - a^3*b^2*c - 4*a^3*b*c^2 + 3/2*a^3*b*c + 1/2*a^3*b + 1/4*a^3*c^3 + 5/4*a^3*c^2 + 5/4*a^3*c - 3/4*a^3 + a^2*b^3*c + 1/2*a^2*b^2*c^2 - 1/2*a^2*b^2*c - 15/4*a^2*b*c^3 + 13/4*a^2*b*c^2 + 5/4*a^2*b*c - 3/4*a^2*b + 1/2*a^2*c^3 + 1/4*a^2*c^2 - 5/2*a^2*c + 3/4*a^2 + a*b^4*c + 4*a*b^3*c^2 - 3/2*a*b^3*c - 1/2*a*b^3 + 15/4*a*b^2*c^3 - 21/4*a*b^2*c^2 - 1/4*a*b^2*c + 3/4*a*b^2 + 3/2*a*b*c^2 - 1/2*a*b*c + 1/4*a*c^3 - 5/4*a*c^2 + 5/4*a*c - 1/4*a - 1/4*b^4*c^2 + 1/2*b^4*c - 1/4*b^4 - 1/4*b^3*c^3 + 3/4*b^3*c^2 - 9/4*b^3*c + 3/4*b^3 - 1/2*b^2*c^3 - 7/4*b^2*c^2 + 3*b^2*c - 3/4*b^2 - 1/4*b*c^3 + 5/4*b*c^2 - 5/4*b*c + 1/4*b)/(a^4*b*c + 1/4*a^4*c^2 - 1/4*a^4 + 3*a^3*b^2*c + 2*a^3*b*c^2 - 3*a^3*b*c - a^3*b + 1/4*a^3*c^3 - 1/2*a^3*c^2 - 3/4*a^3*c + a^3 + 3*a^2*b^3*c + 7/2*a^2*b^2*c^2 - 6*a^2*b^2*c - 3/2*a^2*b^2 + 3/4*a^2*b*c^3 - 7/2*a^2*b*c^2 + 3/4*a^2*b*c + 3*a^2*b - 1/4*a^2*c^3 - 1/2*a^2*c^2 + 9/4*a^2*c - 3/2*a^2 + a*b^4*c + 2*a*b^3*c^2 - 3*a*b^3*c - a*b^3 + 3/4*a*b^2*c^3 - 7/2*a*b^2*c^2 + 3/4*a*b^2*c + 3*a*b^2 - 1/2*a*b*c^3 + 7/2*a*b*c - 3*a*b - 1/4*a*c^3 + 3/2*a*c^2 - 9/4*a*c + a + 1/4*b^4*c^2 - 1/4*b^4 + 1/4*b^3*c^3 - 1/2*b^3*c^2 - 3/4*b^3*c + b^3 - 1/4*b^2*c^3 - 1/2*b^2*c^2 + 9/4*b^2*c - 3/2*b^2 - 1/4*b*c^3 + 3/2*b*c^2 - 9/4*b*c + b + 1/4*c^3 - 3/4*c^2 + 3/4*c - 1/4) (2*a^4*b*c - a^4*c^2 - 1/2*a^4*c - 1/2*a^4 + 4*a^3*b^2*c + 5*a^3*b*c^2 - 11/2*a^3*b*c - 3/2*a^3*b - 3/2*a^3*c^3 + 5/4*a^3*c^2 - 1/2*a^3*c + 7/4*a^3 + 2*a^2*b^3*c + 5*a^2*b^2*c^2 - 11/2*a^2*b^2*c - 3/2*a^2*b^2 + 5*a^2*b*c^3 - 31/4*a^2*b*c^2 + a^2*b*c + 15/4*a^2*b + 5/4*a^2*c^3 - 9/4*a^2*c^2 + 13/4*a^2*c - 9/4*a^2 - a*b^3*c^2 - 1/2*a*b^3*c - 1/2*a*b^3 - 3/2*a*b^2*c^3 - 5/4*a*b^2*c^2 - 1/2*a*b^2*c + 9/4*a*b^2 - 5/2*a*b*c^3 + 1/2*a*b*c^2 + 4*a*b*c - 3*a*b - a*c^3 + 11/4*a*c^2 - 3*a*c + 5/4*a - 1/4*b^3*c^2 + 1/4*b^3 + 1/4*b^2*c^3 + 3/4*b^2*c^2 + 3/4*b^2*c - 3/4*b^2 + 1/2*b*c^3 + 1/4*b*c^2 - 3/2*b*c + 3/4*b + 1/4*c^3 - 3/4*c^2 + 3/4*c - 1/4)/(a^4*b*c + 1/4*a^4*c^2 - 1/4*a^4 + 3*a^3*b^2*c + 2*a^3*b*c^2 - 3*a^3*b*c - a^3*b + 1/4*a^3*c^3 - 1/2*a^3*c^2 - 3/4*a^3*c + a^3 + 3*a^2*b^3*c + 7/2*a^2*b^2*c^2 - 6*a^2*b^2*c - 3/2*a^2*b^2 + 3/4*a^2*b*c^3 - 7/2*a^2*b*c^2 + 3/4*a^2*b*c + 3*a^2*b - 1/4*a^2*c^3 - 1/2*a^2*c^2 + 9/4*a^2*c - 3/2*a^2 + a*b^4*c + 2*a*b^3*c^2 - 3*a*b^3*c - a*b^3 + 3/4*a*b^2*c^3 - 7/2*a*b^2*c^2 + 3/4*a*b^2*c + 3*a*b^2 - 1/2*a*b*c^3 + 7/2*a*b*c - 3*a*b - 1/4*a*c^3 + 3/2*a*c^2 - 9/4*a*c + a + 1/4*b^4*c^2 - 1/4*b^4 + 1/4*b^3*c^3 - 1/2*b^3*c^2 - 3/4*b^3*c + b^3 - 1/4*b^2*c^3 - 1/2*b^2*c^2 + 9/4*b^2*c - 3/2*b^2 - 1/4*b*c^3 + 3/2*b*c^2 - 9/4*b*c + b + 1/4*c^3 - 3/4*c^2 + 3/4*c - 1/4) (a^3*b*c - 1/2*a^3*c^2 + 1/2*a^3*c + 2*a^2*b*c^2 - a^2*b*c + a^2*c^2 - a^2*c - a*b^3*c - 3/2*a*b^2*c^2 + 1/2*a*b^2*c - 3/2*a*b*c^2 + 1/2*a*b*c - 1/2*a*c^2 + 1/2*a*c + 1/2*b^2*c^2 + 1/2*b^2*c + 1/2*b*c^2 - 1/2*b*c)/(a^3*b*c + 1/4*a^3*c^2 - 1/4*a^3 + 2*a^2*b^2*c + 7/4*a^2*b*c^2 - 2*a^2*b*c - 3/4*a^2*b + 1/4*a^2*c^3 - 1/4*a^2*c^2 - 3/4*a^2*c + 3/4*a^2 + a*b^3*c + 7/4*a*b^2*c^2 - 2*a*b^2*c - 3/4*a*b^2 + 1/2*a*b*c^3 - 3/2*a*b*c^2 - 1/2*a*b*c + 3/2*a*b - 3/4*a*c^2 + 3/2*a*c - 3/4*a + 1/4*b^3*c^2 - 1/4*b^3 + 1/4*b^2*c^3 - 1/4*b^2*c^2 - 3/4*b^2*c + 3/4*b^2 - 3/4*b*c^2 + 3/2*b*c - 3/4*b - 1/4*c^3 + 3/4*c^2 - 3/4*c + 1/4) (-a^2*b*c^2 + 1/2*a^2*c^3 + 1/2*a^2*c^2 - 2*a*b*c^3 + a*b*c^2 + 1/2*a*c^3 - 1/2*a*c^2 + b^3*c^2 + 3/2*b^2*c^3 - 3/2*b^2*c^2 - 1/2*b*c^3 + 1/2*b*c^2)/(a^3*b*c + 1/4*a^3*c^2 - 1/4*a^3 + 2*a^2*b^2*c + 7/4*a^2*b*c^2 - 2*a^2*b*c - 3/4*a^2*b + 1/4*a^2*c^3 - 1/4*a^2*c^2 - 3/4*a^2*c + 3/4*a^2 + a*b^3*c + 7/4*a*b^2*c^2 - 2*a*b^2*c - 3/4*a*b^2 + 1/2*a*b*c^3 - 3/2*a*b*c^2 - 1/2*a*b*c + 3/2*a*b - 3/4*a*c^2 + 3/2*a*c - 3/4*a + 1/4*b^3*c^2 - 1/4*b^3 + 1/4*b^2*c^3 - 1/4*b^2*c^2 - 3/4*b^2*c + 3/4*b^2 - 3/4*b*c^2 + 3/2*b*c - 3/4*b - 1/4*c^3 + 3/4*c^2 - 3/4*c + 1/4) (-a^3*c^2 - 3/2*a^2*c^3 + 3/2*a^2*c^2 + a*b^2*c^2 + 2*a*b*c^3 - a*b*c^2 + 1/2*a*c^3 - 1/2*a*c^2 - 1/2*b^2*c^3 - 1/2*b^2*c^2 - 1/2*b*c^3 + 1/2*b*c^2)/(a^3*b*c + 1/4*a^3*c^2 - 1/4*a^3 + 2*a^2*b^2*c + 7/4*a^2*b*c^2 - 2*a^2*b*c - 3/4*a^2*b + 1/4*a^2*c^3 - 1/4*a^2*c^2 - 3/4*a^2*c + 3/4*a^2 + a*b^3*c + 7/4*a*b^2*c^2 - 2*a*b^2*c - 3/4*a*b^2 + 1/2*a*b*c^3 - 3/2*a*b*c^2 - 1/2*a*b*c + 3/2*a*b - 3/4*a*c^2 + 3/2*a*c - 3/4*a + 1/4*b^3*c^2 - 1/4*b^3 + 1/4*b^2*c^3 - 1/4*b^2*c^2 - 3/4*b^2*c + 3/4*b^2 - 3/4*b*c^2 + 3/2*b*c - 3/4*b - 1/4*c^3 + 3/4*c^2 - 3/4*c + 1/4) (a^2*b*c - 1/4*a^2*c^2 + 1/2*a^2*c - 1/4*a^2 + a*b^2*c + 5/2*a*b*c^2 - a*b*c - 1/2*a*b + 1/4*a*c^3 + 1/2*a*c^2 - 5/4*a*c + 1/2*a - 1/4*b^2*c^2 - 1/2*b^2*c - 1/4*b^2 + 1/4*b*c^3 - 1/2*b*c^2 - 1/4*b*c + 1/2*b + 1/4*c^3 - 3/4*c^2 + 3/4*c - 1/4)/(a^2*b*c + 1/4*a^2*c^2 - 1/4*a^2 + a*b^2*c + 3/2*a*b*c^2 - a*b*c - 1/2*a*b + 1/4*a*c^3 - 3/4*a*c + 1/2*a + 1/4*b^2*c^2 - 1/4*b^2 + 1/4*b*c^3 - 3/4*b*c + 1/2*b + 1/4*c^3 - 3/4*c^2 + 3/4*c - 1/4) Total time: 0.380 seconds, Total memory usage: 3.63MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Thu Dec 8 18:01:41 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); A1:=Matrix(K,3,3,[[b,1,1],[0,a,1],[0,0,c]]); S:=Transpose(A)*A^(-1); F,T:=PrimaryRationalForm(S); B:=T*A*Transpose(T); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*B*Transpose(P)-B)); G:=GroebnerBasis(I); Q1:=Matrix(K,3,3,[(2*b-1)/(a+b-1),(a-b)/(a+b-1),0,(b-a)/(a+b-1),(2*a-1)/(a+b-1),0, 0,0,1]); Kt:=FunctionField(K); D:=(2*a*b*c-a-b-c+1)/(a*b*c); s:=t; px:=-(2*s+D); py:=1-s^2; q:=s^2+s*D+1; Q:=Matrix(Kt,3,3,[1,0,0,0,D*px/q+py/q,-px/q,0,px/q,py/q]); Q:=Q1*T^(-1)*Q*T; print (Q*A*Transpose(Q)-A1); for i in [1..3] do for j in [1..3] do print Evaluate(Q[i,j],-D/2+(a-b)/(2*a*b)); end for; end for; print (1-D/2), (1+D/2); Output: Magma V2.11-10 Thu Dec 8 2005 18:01:41 on modular [Seed = 4230415514] ------------------------------------- [0 0 0] [0 0 0] [0 0 0] (2*a*b^2*c + a*b*c^2 - 1/2*a*b*c - 1/2*a*b + 1/4*a*c^2 - 1/2*a*c + 1/4*a + b^2*c^2 - 1/2*b^2*c - 1/2*b^2 + 1/2*b*c^3 - 1/4*b*c^2 - b*c + 3/4*b + 1/4*c^3 - 3/4*c^2 + 3/4*c - 1/4)/(a^2*b*c + 1/4*a^2*c^2 - 1/4*a^2 + a*b^2*c + 3/2*a*b*c^2 - a*b*c - 1/2*a*b + 1/4*a*c^3 - 3/4*a*c + 1/2*a + 1/4*b^2*c^2 - 1/4*b^2 + 1/4*b*c^3 - 3/4*b*c + 1/2*b + 1/4*c^3 - 3/4*c^2 + 3/4*c - 1/4) (a^2*b*c - 1/4*a^2*c^2 + 1/2*a^2*c - 1/4*a^2 - a*b^2*c + 1/2*a*b*c^2 - 1/2*a*b*c - 1/4*a*c^3 + 3/4*a*c^2 - 3/4*a*c + 1/4*a - 1/4*b^2*c^2 + 1/4*b^2 + 1/4*b*c^3 - 3/4*b*c^2 + 3/4*b*c - 1/4*b)/(a^2*b*c + 1/4*a^2*c^2 - 1/4*a^2 + a*b^2*c + 3/2*a*b*c^2 - a*b*c - 1/2*a*b + 1/4*a*c^3 - 3/4*a*c + 1/2*a + 1/4*b^2*c^2 - 1/4*b^2 + 1/4*b*c^3 - 3/4*b*c + 1/2*b + 1/4*c^3 - 3/4*c^2 + 3/4*c - 1/4) (-a^2*b*c + a*b^2*c - 1/2*a*b*c^2 + 1/2*a*b*c + 1/2*b^2*c^2 - 1/2*b^2*c)/(a^2*b*c + 1/4*a^2*c^2 - 1/4*a^2 + a*b^2*c + 3/2*a*b*c^2 - a*b*c - 1/2*a*b + 1/4*a*c^3 - 3/4*a*c + 1/2*a + 1/4*b^2*c^2 - 1/4*b^2 + 1/4*b*c^3 - 3/4*b*c + 1/2*b + 1/4*c^3 - 3/4*c^2 + 3/4*c - 1/4) (-a^2*b*c - 1/4*a^2*c^2 + 1/4*a^2 + a*b^2*c + 1/2*a*b*c^2 + 1/2*a*b*c + 1/4*a*c^3 - 1/4*a*c^2 + 1/4*a*c - 1/4*a - 1/4*b^2*c^2 - 1/2*b^2*c - 1/4*b^2 - 1/4*b*c^3 + 1/4*b*c^2 - 1/4*b*c + 1/4*b)/(a^2*b*c + 1/4*a^2*c^2 - 1/4*a^2 + a*b^2*c + 3/2*a*b*c^2 - a*b*c - 1/2*a*b + 1/4*a*c^3 - 3/4*a*c + 1/2*a + 1/4*b^2*c^2 - 1/4*b^2 + 1/4*b*c^3 - 3/4*b*c + 1/2*b + 1/4*c^3 - 3/4*c^2 + 3/4*c - 1/4) (2*a^2*b*c + a^2*c^2 + 1/2*a^2*c - 1/2*a^2 + a*b*c^2 - 3/2*a*b*c - 1/2*a*b + 1/2*a*c^3 + 1/4*a*c^2 - 3/2*a*c + 3/4*a - 1/4*b*c^2 + 1/4*b + 1/4*c^3 - 3/4*c^2 + 3/4*c - 1/4)/(a^2*b*c + 1/4*a^2*c^2 - 1/4*a^2 + a*b^2*c + 3/2*a*b*c^2 - a*b*c - 1/2*a*b + 1/4*a*c^3 - 3/4*a*c + 1/2*a + 1/4*b^2*c^2 - 1/4*b^2 + 1/4*b*c^3 - 3/4*b*c + 1/2*b + 1/4*c^3 - 3/4*c^2 + 3/4*c - 1/4) (-a^2*b*c - 1/2*a^2*c^2 - 1/2*a^2*c + a*b^2*c + 1/2*a*b*c^2 + 1/2*a*b*c - 1/2*a*c^2 + 1/2*a*c + 1/2*b*c^2 - 1/2*b*c)/(a^2*b*c + 1/4*a^2*c^2 - 1/4*a^2 + a*b^2*c + 3/2*a*b*c^2 - a*b*c - 1/2*a*b + 1/4*a*c^3 - 3/4*a*c + 1/2*a + 1/4*b^2*c^2 - 1/4*b^2 + 1/4*b*c^3 - 3/4*b*c + 1/2*b + 1/4*c^3 - 3/4*c^2 + 3/4*c - 1/4) (a*b*c^2 + 1/2*a*c^3 - 1/2*a*c^2 - b^2*c^2 - 1/2*b*c^3 + 1/2*b*c^2)/(a^2*b*c + 1/4*a^2*c^2 - 1/4*a^2 + a*b^2*c + 3/2*a*b*c^2 - a*b*c - 1/2*a*b + 1/4*a*c^3 - 3/4*a*c + 1/2*a + 1/4*b^2*c^2 - 1/4*b^2 + 1/4*b*c^3 - 3/4*b*c + 1/2*b + 1/4*c^3 - 3/4*c^2 + 3/4*c - 1/4) (a^2*c^2 - a*b*c^2 + 1/2*a*c^3 - 1/2*a*c^2 - 1/2*b*c^3 + 1/2*b*c^2)/(a^2*b*c + 1/4*a^2*c^2 - 1/4*a^2 + a*b^2*c + 3/2*a*b*c^2 - a*b*c - 1/2*a*b + 1/4*a*c^3 - 3/4*a*c + 1/2*a + 1/4*b^2*c^2 - 1/4*b^2 + 1/4*b*c^3 - 3/4*b*c + 1/2*b + 1/4*c^3 - 3/4*c^2 + 3/4*c - 1/4) (a^2*b*c - 1/4*a^2*c^2 - 1/2*a^2*c - 1/4*a^2 + a*b^2*c + 5/2*a*b*c^2 - a*b*c - 1/2*a*b + 1/4*a*c^3 - 1/2*a*c^2 - 1/4*a*c + 1/2*a - 1/4*b^2*c^2 + 1/2*b^2*c - 1/4*b^2 + 1/4*b*c^3 + 1/2*b*c^2 - 5/4*b*c + 1/2*b + 1/4*c^3 - 3/4*c^2 + 3/4*c - 1/4)/(a^2*b*c + 1/4*a^2*c^2 - 1/4*a^2 + a*b^2*c + 3/2*a*b*c^2 - a*b*c - 1/2*a*b + 1/4*a*c^3 - 3/4*a*c + 1/2*a + 1/4*b^2*c^2 - 1/4*b^2 + 1/4*b*c^3 - 3/4*b*c + 1/2*b + 1/4*c^3 - 3/4*c^2 + 3/4*c - 1/4) (1/2*a + 1/2*b + 1/2*c - 1/2)/(a*b*c) (2*a*b*c - 1/2*a - 1/2*b - 1/2*c + 1/2)/(a*b*c) Total time: 0.370 seconds, Total memory usage: 3.63MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Thu Dec 8 17:50:10 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); A1:=Matrix(K,3,3,[[b,1,1],[0,a,1],[0,0,c]]); S:=Transpose(A)*A^(-1); F,T:=PrimaryRationalForm(S); B:=T*A*Transpose(T); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*B*Transpose(P)-B)); G:=GroebnerBasis(I); Q1:=Matrix(K,3,3,[(2*b-1)/(a+b-1),(a-b)/(a+b-1),0,(b-a)/(a+b-1),(2*a-1)/(a+b-1),0, 0,0,1]); Kt:=FunctionField(K); D:=(2*a*b*c-a-b-c+1)/(a*b*c); s:=t; px:=-(2*s+D); py:=1-s^2; q:=s^2+s*D+1; Q:=Matrix(Kt,3,3,[1,0,0,0,D*px/q+py/q,-px/q,0,px/q,py/q]); Q:=Q1*T^(-1)*Q*T; print (Q*A*Transpose(Q)-A1); for i in [1..3] do for j in [1..3] do print Evaluate(Q[i,j],-D/2); end for; end for; print (1-D/2), (1+D/2); Output: Magma V2.11-10 Thu Dec 8 2005 17:50:10 on modular [Seed = 1117370835] ------------------------------------- [0 0 0] [0 0 0] [0 0 0] (2*b - 1)/(a + b - 1) (a - b)/(a + b - 1) 0 (-a + b)/(a + b - 1) (2*a - 1)/(a + b - 1) 0 0 0 1 (1/2*a + 1/2*b + 1/2*c - 1/2)/(a*b*c) (2*a*b*c - 1/2*a - 1/2*b - 1/2*c + 1/2)/(a*b*c) Total time: 0.370 seconds, Total memory usage: 3.63MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Thu Dec 8 17:46:34 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); A1:=Matrix(K,3,3,[[b,1,1],[0,a,1],[0,0,c]]); S:=Transpose(A)*A^(-1); F,T:=PrimaryRationalForm(S); B:=T*A*Transpose(T); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*B*Transpose(P)-B)); G:=GroebnerBasis(I); Q1:=Matrix(K,3,3,[(2*b-1)/(a+b-1),(a-b)/(a+b-1),0,(b-a)/(a+b-1),(2*a-1)/(a+b-1),0, 0,0,1]); Kt:=FunctionField(K); D:=(2*a*b*c-a-b-c+1)/(a*b*c); s:=t; px:=-(2*s+D); py:=1-s^2; q:=s^2+s*D+1; Q:=Matrix(Kt,3,3,[1,0,0,0,D*px/q+py/q,-px/q,0,px/q,py/q]); Q:=Q1*T^(-1)*Q*T; print (Q*A*Transpose(Q)-A1); for i in [1..3] do for j in [1..3] do print Evaluate(Q[i,j],-D); end for; end for; Output: Magma V2.11-10 Thu Dec 8 2005 17:46:33 on modular [Seed = 1501736355] ------------------------------------- [0 0 0] [0 0 0] [0 0 0] (-a^3*c + a^3 - 2*a^2*b*c^2 + a^2*b*c + 2*a^2*b - a^2*c^2 + 4*a^2*c - 3*a^2 + 2*a*b^2*c + a*b^2 + 2*a*b*c^2 + a*b*c - 4*a*b + 2*a*c^2 - 5*a*c + 3*a - b^2 - 2*b*c + 2*b - c^2 + 2*c - 1)/(a^4*c + 2*a^3*b*c + a^3*c^2 - 2*a^3*c + a^2*b^2*c + a^2*b*c^2 - 2*a^2*b*c - a^2*c^2 + a^2*c) (-a^4*b*c + a^4 - 2*a^3*b^2*c - a^3*b*c^2 + 5*a^3*b*c + 2*a^3*b + 2*a^3*c - 4*a^3 - a^2*b^3*c + a^2*b^2*c^2 + 5*a^2*b^2*c + a^2*b^2 + 4*a^2*b*c^2 - 5*a^2*b*c - 6*a^2*b + a^2*c^2 - 6*a^2*c + 6*a^2 - 3*a*b^2*c - 2*a*b^2 - 3*a*b*c^2 - a*b*c + 6*a*b - 2*a*c^2 + 6*a*c - 4*a + b^2 + 2*b*c - 2*b + c^2 - 2*c + 1)/(a^4*b*c + 2*a^3*b^2*c + a^3*b*c^2 - 2*a^3*b*c + a^2*b^3*c + a^2*b^2*c^2 - 2*a^2*b^2*c - a^2*b*c^2 + a^2*b*c) (2*a^3*b*c - a^3 + 2*a^2*b^2*c^2 + 2*a^2*b^2*c + 2*a^2*b*c^2 - 5*a^2*b*c - 2*a^2*b - 2*a^2*c + 3*a^2 - 3*a*b^2*c - a*b^2 - 3*a*b*c^2 + a*b*c + 4*a*b - a*c^2 + 4*a*c - 3*a + b^2 + 2*b*c - 2*b + c^2 - 2*c + 1)/(a^3*b*c^2 + a^2*b^2*c^2 + a^2*b*c^3 - a^2*b*c^2) (-a^4*b*c - 2*a^3*b^2*c + a^3*b*c^2 - a^3*b*c + a^3*b - a^3*c + a^3 - a^2*b^3*c - a^2*b^2*c^2 - a^2*b^2*c + 2*a^2*b^2 - 4*a^2*b*c^2 + 5*a^2*b*c - a^2*c^2 + 4*a^2*c - 3*a^2 + a*b^3 + 4*a*b^2*c - a*b^2 + 3*a*b*c^2 - a*b*c - 3*a*b + 2*a*c^2 - 5*a*c + 3*a - b^2 - 2*b*c + 2*b - c^2 + 2*c - 1)/(a^4*b*c + 2*a^3*b^2*c + a^3*b*c^2 - 2*a^3*b*c + a^2*b^3*c + a^2*b^2*c^2 - 2*a^2*b^2*c - a^2*b*c^2 + a^2*b*c) (-2*a^4*b*c + a^4*b + a^4 - 2*a^3*b^2*c^2 - a^3*b^2*c + 2*a^3*b^2 - 2*a^3*b*c^2 + 9*a^3*b*c - a^3*b + 2*a^3*c - 4*a^3 + a^2*b^3*c + a^2*b^3 + 3*a^2*b^2*c^2 + 6*a^2*b^2*c - 3*a^2*b^2 + 6*a^2*b*c^2 - 10*a^2*b*c - 3*a^2*b + a^2*c^2 - 6*a^2*c + 6*a^2 - a*b^3 - 5*a*b^2*c - 4*a*b*c^2 + a*b*c + 5*a*b - 2*a*c^2 + 6*a*c - 4*a + b^2 + 2*b*c - 2*b + c^2 - 2*c + 1)/(a^4*b^2*c + 2*a^3*b^3*c + a^3*b^2*c^2 - 2*a^3*b^2*c + a^2*b^4*c + a^2*b^3*c^2 - 2*a^2*b^3*c - a^2*b^2*c^2 + a^2*b^2*c) (-2*a^3*b^2*c^2 + 2*a^3*b^2*c + 3*a^3*b*c - a^3*b - a^3 + 2*a^2*b^3*c + 4*a^2*b^2*c^2 + a^2*b^2*c - 2*a^2*b^2 + 3*a^2*b*c^2 - 8*a^2*b*c - 2*a^2*c + 3*a^2 - a*b^3 - 5*a*b^2*c + a*b^2 - 4*a*b*c^2 + 3*a*b*c + 3*a*b - a*c^2 + 4*a*c - 3*a + b^2 + 2*b*c - 2*b + c^2 - 2*c + 1)/(a^3*b^2*c^2 + a^2*b^3*c^2 + a^2*b^2*c^3 - a^2*b^2*c^2) (2*a^2*b*c^2 - 2*a^2*b*c - a^2*c + a^2 - 2*a*b^2*c - 2*a*b*c^2 + a*b*c + 2*a*b - a*c^2 + 3*a*c - 2*a + b^2 + 2*b*c - 2*b + c^2 - 2*c + 1)/(a^3*b*c + a^2*b^2*c + a^2*b*c^2 - a^2*b*c) (-2*a^3*b*c + a^3 - 2*a^2*b^2*c^2 - 2*a^2*b^2*c - 2*a^2*b*c^2 + 5*a^2*b*c + 2*a^2*b + 2*a^2*c - 3*a^2 + 3*a*b^2*c + a*b^2 + 3*a*b*c^2 - a*b*c - 4*a*b + a*c^2 - 4*a*c + 3*a - b^2 - 2*b*c + 2*b - c^2 + 2*c - 1)/(a^3*b^2*c + a^2*b^3*c + a^2*b^2*c^2 - a^2*b^2*c) (-a^3*b^2*c^2 + 3*a^3*b*c - a^3 - a^2*b^3*c^2 + a^2*b^2*c^3 + a^2*b^2*c^2 + 6*a^2*b^2*c + 3*a^2*b*c^2 - 6*a^2*b*c - 3*a^2*b - 2*a^2*c + 3*a^2 + 3*a*b^3*c + 3*a*b^2*c^2 - 6*a*b^2*c - 3*a*b^2 - 3*a*b*c^2 - a*b*c + 6*a*b - a*c^2 + 4*a*c - 3*a - b^3 - 2*b^2*c + 3*b^2 - b*c^2 + 4*b*c - 3*b + c^2 - 2*c + 1)/(a^3*b^2*c^2 + a^2*b^3*c^2 + a^2*b^2*c^3 - a^2*b^2*c^2) Total time: 0.360 seconds, Total memory usage: 3.63MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Thu Dec 8 17:19:27 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); A1:=Matrix(K,3,3,[[b,1,1],[0,a,1],[0,0,c]]); S:=Transpose(A)*A^(-1); F,T:=PrimaryRationalForm(S); B:=T*A*Transpose(T); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*B*Transpose(P)-B)); G:=GroebnerBasis(I); Q1:=Matrix(K,3,3,[(2*b-1)/(a+b-1),(a-b)/(a+b-1),0,(b-a)/(a+b-1),(2*a-1)/(a+b-1),0, 0,0,1]); Kt:=FunctionField(K); D:=(2*a*b*c-a-b-c+1)/(a*b*c); s:=t; px:=-(2*s+D); py:=1-s^2; q:=s^2+s*D+1; Q:=Matrix(Kt,3,3,[1,0,0,0,D*px/q+py/q,-px/q,0,px/q,py/q]); Q:=Q1*T^(-1)*Q*T; print (Q*A*Transpose(Q)-A1); for i in [1..3] do for j in [1..3] do print Evaluate(Q[i,j],-D/2); end for; end for; print Factorization(Numerator((1-(D/2))^2-(a+b+c-1)/(a*b*c))); Output: Magma V2.11-10 Thu Dec 8 2005 17:19:26 on modular [Seed = 2124778531] ------------------------------------- [0 0 0] [0 0 0] [0 0 0] (2*b - 1)/(a + b - 1) (a - b)/(a + b - 1) 0 (-a + b)/(a + b - 1) (2*a - 1)/(a + b - 1) 0 0 0 1 [ , ] Total time: 0.360 seconds, Total memory usage: 3.72MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Thu Dec 8 17:19:02 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); A1:=Matrix(K,3,3,[[b,1,1],[0,a,1],[0,0,c]]); S:=Transpose(A)*A^(-1); F,T:=PrimaryRationalForm(S); B:=T*A*Transpose(T); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*B*Transpose(P)-B)); G:=GroebnerBasis(I); Q1:=Matrix(K,3,3,[(2*b-1)/(a+b-1),(a-b)/(a+b-1),0,(b-a)/(a+b-1),(2*a-1)/(a+b-1),0, 0,0,1]); Kt:=FunctionField(K); D:=(2*a*b*c-a-b-c+1)/(a*b*c); s:=t; px:=-(2*s+D); py:=1-s^2; q:=s^2+s*D+1; Q:=Matrix(Kt,3,3,[1,0,0,0,D*px/q+py/q,-px/q,0,px/q,py/q]); Q:=Q1*T^(-1)*Q*T; print (Q*A*Transpose(Q)-A1); for i in [1..3] do for j in [1..3] do print Evaluate(Q[i,j],-D/2); end for; end for; print (1-(D/2))^2-(a+b+c-1)/(a*b*c); Output: Magma V2.11-10 Thu Dec 8 2005 17:19:02 on modular [Seed = 2074118430] ------------------------------------- [0 0 0] [0 0 0] [0 0 0] (2*b - 1)/(a + b - 1) (a - b)/(a + b - 1) 0 (-a + b)/(a + b - 1) (2*a - 1)/(a + b - 1) 0 0 0 1 (-a^2*b*c + 1/4*a^2 - a*b^2*c - a*b*c^2 + a*b*c + 1/2*a*b + 1/2*a*c - 1/2*a + 1/4*b^2 + 1/2*b*c - 1/2*b + 1/4*c^2 - 1/2*c + 1/4)/(a^2*b^2*c^2) Total time: 0.360 seconds, Total memory usage: 3.63MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Thu Dec 8 17:18:17 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); A1:=Matrix(K,3,3,[[b,1,1],[0,a,1],[0,0,c]]); S:=Transpose(A)*A^(-1); F,T:=PrimaryRationalForm(S); B:=T*A*Transpose(T); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*B*Transpose(P)-B)); G:=GroebnerBasis(I); Q1:=Matrix(K,3,3,[(2*b-1)/(a+b-1),(a-b)/(a+b-1),0,(b-a)/(a+b-1),(2*a-1)/(a+b-1),0, 0,0,1]); Kt:=FunctionField(K); D:=(2*a*b*c-a-b-c+1)/(a*b*c); s:=t; px:=-(2*s+D); py:=1-s^2; q:=s^2+s*D+1; Q:=Matrix(Kt,3,3,[1,0,0,0,D*px/q+py/q,-px/q,0,px/q,py/q]); Q:=Q1*T^(-1)*Q*T; print (Q*A*Transpose(Q)-A1); for i in [1..3] do for j in [1..3] do print Evaluate(Q[i,j],-D/2); end for; end for; print (1-(D/2))^2; Output: Magma V2.11-10 Thu Dec 8 2005 17:18:16 on modular [Seed = 1907000170] ------------------------------------- [0 0 0] [0 0 0] [0 0 0] (2*b - 1)/(a + b - 1) (a - b)/(a + b - 1) 0 (-a + b)/(a + b - 1) (2*a - 1)/(a + b - 1) 0 0 0 1 (1/4*a^2 + 1/2*a*b + 1/2*a*c - 1/2*a + 1/4*b^2 + 1/2*b*c - 1/2*b + 1/4*c^2 - 1/2*c + 1/4)/(a^2*b^2*c^2) Total time: 0.360 seconds, Total memory usage: 3.63MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Thu Dec 8 17:10:34 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); A1:=Matrix(K,3,3,[[b,1,1],[0,a,1],[0,0,c]]); S:=Transpose(A)*A^(-1); F,T:=PrimaryRationalForm(S); B:=T*A*Transpose(T); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*B*Transpose(P)-B)); G:=GroebnerBasis(I); Q1:=Matrix(K,3,3,[(2*b-1)/(a+b-1),(a-b)/(a+b-1),0,(b-a)/(a+b-1),(2*a-1)/(a+b-1),0, 0,0,1]); Kt:=FunctionField(K); D:=(2*a*b*c-a-b-c+1)/(a*b*c); s:=t; px:=-(2*s+D); py:=1-s^2; q:=s^2+s*D+1; Q:=Matrix(Kt,3,3,[1,0,0,0,D*px/q+py/q,-px/q,0,px/q,py/q]); Q:=Q1*T^(-1)*Q*T; print (Q*A*Transpose(Q)-A1); for i in [1..3] do for j in [1..3] do print Evaluate(Q[i,j],-(2*a*b*c-a-b+1)/(2*a*b*c)); end for; end for; Output: Magma V2.11-10 Thu Dec 8 2005 17:10:33 on modular [Seed = 131918701] ------------------------------------- [0 0 0] [0 0 0] [0 0 0] (2*a^3*b^2*c - a^3*b*c - 1/2*a^3*b + 1/4*a^3 + 4*a^2*b^3*c + 4*a^2*b^2*c^2 - 6*a^2*b^2*c - 3/2*a^2*b^2 - 3*a^2*b*c^2 + 9/4*a^2*b + 5/4*a^2*c - 3/4*a^2 + 2*a*b^4*c + 4*a*b^3*c^2 - 5*a*b^3*c - 3/2*a*b^3 + 2*a*b^2*c^3 - 6*a*b^2*c^2 + 15/4*a*b^2 - 2*a*b*c^3 + 11/2*a*b*c - 3*a*b + 2*a*c^2 - 5/2*a*c + 3/4*a - 1/2*b^4 + b^3*c^2 - 2*b^3*c + 7/4*b^3 + b^2*c^3 - 4*b^2*c^2 + 21/4*b^2*c - 9/4*b^2 - 2*b*c^3 + 5*b*c^2 - 9/2*b*c + 5/4*b + c^3 - 2*c^2 + 5/4*c - 1/4)/(a^4*b*c - 1/4*a^4 + 3*a^3*b^2*c + 2*a^3*b*c^2 - 3*a^3*b*c - a^3*b - 3/4*a^3*c + a^3 + 3*a^2*b^3*c + 4*a^2*b^2*c^2 - 6*a^2*b^2*c - 3/2*a^2*b^2 + a^2*b*c^3 - 4*a^2*b*c^2 + 3/4*a^2*b*c + 3*a^2*b - 1/2*a^2*c^2 + 9/4*a^2*c - 3/2*a^2 + a*b^4*c + 2*a*b^3*c^2 - 3*a*b^3*c - a*b^3 + a*b^2*c^3 - 4*a*b^2*c^2 + 3/4*a*b^2*c + 3*a*b^2 - a*b*c^3 + a*b*c^2 + 7/2*a*b*c - 3*a*b + a*c^2 - 9/4*a*c + a - 1/4*b^4 - 3/4*b^3*c + b^3 - 1/2*b^2*c^2 + 9/4*b^2*c - 3/2*b^2 + b*c^2 - 9/4*b*c + b - 1/2*c^2 + 3/4*c - 1/4) (a^4*b*c - 1/4*a^4 + a^3*b^2*c + 2*a^3*b*c^2 - 2*a^3*b*c - 1/2*a^3*b - 5/4*a^3*c + 3/4*a^3 - a^2*b^3*c + a^2*b*c^3 - 3/4*a^2*b*c + 3/4*a^2*b - 2*a^2*c^2 + 5/2*a^2*c - 3/4*a^2 - a*b^4*c - 2*a*b^3*c^2 + 2*a*b^3*c + 1/2*a*b^3 - a*b^2*c^3 + 4*a*b^2*c^2 - 3/4*a*b^2*c - 3/4*a*b^2 + 2*a*b*c^3 - 4*a*b*c^2 + a*b*c - a*c^3 + 2*a*c^2 - 5/4*a*c + 1/4*a + 1/4*b^4 + 3/4*b^3*c - 3/4*b^3 - 3/2*b^2*c + 3/4*b^2 + 3/4*b*c - 1/4*b)/(a^4*b*c - 1/4*a^4 + 3*a^3*b^2*c + 2*a^3*b*c^2 - 3*a^3*b*c - a^3*b - 3/4*a^3*c + a^3 + 3*a^2*b^3*c + 4*a^2*b^2*c^2 - 6*a^2*b^2*c - 3/2*a^2*b^2 + a^2*b*c^3 - 4*a^2*b*c^2 + 3/4*a^2*b*c + 3*a^2*b - 1/2*a^2*c^2 + 9/4*a^2*c - 3/2*a^2 + a*b^4*c + 2*a*b^3*c^2 - 3*a*b^3*c - a*b^3 + a*b^2*c^3 - 4*a*b^2*c^2 + 3/4*a*b^2*c + 3*a*b^2 - a*b*c^3 + a*b*c^2 + 7/2*a*b*c - 3*a*b + a*c^2 - 9/4*a*c + a - 1/4*b^4 - 3/4*b^3*c + b^3 - 1/2*b^2*c^2 + 9/4*b^2*c - 3/2*b^2 + b*c^2 - 9/4*b*c + b - 1/2*c^2 + 3/4*c - 1/4) (a^2*b*c + a*b^2*c + a*b*c^2 - 3/2*a*b*c - 1/2*b^2*c + 1/2*b*c)/(a^3*b*c - 1/4*a^3 + 2*a^2*b^2*c + 2*a^2*b*c^2 - 2*a^2*b*c - 3/4*a^2*b - 3/4*a^2*c + 3/4*a^2 + a*b^3*c + 2*a*b^2*c^2 - 2*a*b^2*c - 3/4*a*b^2 + a*b*c^3 - 2*a*b*c^2 - 1/2*a*b*c + 3/2*a*b - 1/2*a*c^2 + 3/2*a*c - 3/4*a - 1/4*b^3 - 3/4*b^2*c + 3/4*b^2 - 1/2*b*c^2 + 3/2*b*c - 3/4*b + 1/2*c^2 - 3/4*c + 1/4) (-a^4*b*c + 1/4*a^4 - a^3*b^2*c - 2*a^3*b*c^2 + 2*a^3*b*c + 1/2*a^3*b + 3/4*a^3*c - 3/4*a^3 + a^2*b^3*c - a^2*b*c^3 - 3/4*a^2*b*c - 3/4*a^2*b + a^2*c^2 - a^2*c + 3/4*a^2 + a*b^4*c + 2*a*b^3*c^2 - 2*a*b^3*c - 1/2*a*b^3 + a*b^2*c^3 - 4*a*b^2*c^2 - 3/4*a*b^2*c + 3/4*a*b^2 - 2*a*b*c^3 + 2*a*b*c^2 + 2*a*b*c + a*c^3 - 1/4*a*c - 1/4*a - 1/4*b^4 - 5/4*b^3*c + 3/4*b^3 - b^2*c^2 + 3*b^2*c - 3/4*b^2 + 2*b*c^2 - 9/4*b*c + 1/4*b - c^2 + 1/2*c)/(a^4*b*c - 1/4*a^4 + 3*a^3*b^2*c + 2*a^3*b*c^2 - 3*a^3*b*c - a^3*b - 3/4*a^3*c + a^3 + 3*a^2*b^3*c + 4*a^2*b^2*c^2 - 6*a^2*b^2*c - 3/2*a^2*b^2 + a^2*b*c^3 - 4*a^2*b*c^2 + 3/4*a^2*b*c + 3*a^2*b - 1/2*a^2*c^2 + 9/4*a^2*c - 3/2*a^2 + a*b^4*c + 2*a*b^3*c^2 - 3*a*b^3*c - a*b^3 + a*b^2*c^3 - 4*a*b^2*c^2 + 3/4*a*b^2*c + 3*a*b^2 - a*b*c^3 + a*b*c^2 + 7/2*a*b*c - 3*a*b + a*c^2 - 9/4*a*c + a - 1/4*b^4 - 3/4*b^3*c + b^3 - 1/2*b^2*c^2 + 9/4*b^2*c - 3/2*b^2 + b*c^2 - 9/4*b*c + b - 1/2*c^2 + 3/4*c - 1/4) (2*a^4*b*c - 1/2*a^4 + 4*a^3*b^2*c + 4*a^3*b*c^2 - 5*a^3*b*c - 3/2*a^3*b - a^3*c^2 - 2*a^3*c + 7/4*a^3 + 2*a^2*b^3*c + 4*a^2*b^2*c^2 - 6*a^2*b^2*c - 3/2*a^2*b^2 + 2*a^2*b*c^3 - 6*a^2*b*c^2 + 15/4*a^2*b - a^2*c^3 + 19/4*a^2*c - 9/4*a^2 - a*b^3*c - 1/2*a*b^3 - a*b^2*c^2 + 9/4*a*b^2 + 9/2*a*b*c - 3*a*b + a*c^2 - 7/2*a*c + 5/4*a + 1/4*b^3 + 3/4*b^2*c - 3/4*b^2 - 3/2*b*c + 3/4*b + 3/4*c - 1/4)/(a^4*b*c - 1/4*a^4 + 3*a^3*b^2*c + 2*a^3*b*c^2 - 3*a^3*b*c - a^3*b - 3/4*a^3*c + a^3 + 3*a^2*b^3*c + 4*a^2*b^2*c^2 - 6*a^2*b^2*c - 3/2*a^2*b^2 + a^2*b*c^3 - 4*a^2*b*c^2 + 3/4*a^2*b*c + 3*a^2*b - 1/2*a^2*c^2 + 9/4*a^2*c - 3/2*a^2 + a*b^4*c + 2*a*b^3*c^2 - 3*a*b^3*c - a*b^3 + a*b^2*c^3 - 4*a*b^2*c^2 + 3/4*a*b^2*c + 3*a*b^2 - a*b*c^3 + a*b*c^2 + 7/2*a*b*c - 3*a*b + a*c^2 - 9/4*a*c + a - 1/4*b^4 - 3/4*b^3*c + b^3 - 1/2*b^2*c^2 + 9/4*b^2*c - 3/2*b^2 + b*c^2 - 9/4*b*c + b - 1/2*c^2 + 3/4*c - 1/4) (a^2*b*c + 1/2*a^2*c + a*b^2*c + a*b*c^2 - 1/2*a*b*c - a*c - 1/2*b*c + 1/2*c)/(a^3*b*c - 1/4*a^3 + 2*a^2*b^2*c + 2*a^2*b*c^2 - 2*a^2*b*c - 3/4*a^2*b - 3/4*a^2*c + 3/4*a^2 + a*b^3*c + 2*a*b^2*c^2 - 2*a*b^2*c - 3/4*a*b^2 + a*b*c^3 - 2*a*b*c^2 - 1/2*a*b*c + 3/2*a*b - 1/2*a*c^2 + 3/2*a*c - 3/4*a - 1/4*b^3 - 3/4*b^2*c + 3/4*b^2 - 1/2*b*c^2 + 3/2*b*c - 3/4*b + 1/2*c^2 - 3/4*c + 1/4) (-a*b*c^2 + 1/2*a*c^2 - b^2*c^2 - b*c^3 + 3/2*b*c^2 + c^3 - 1/2*c^2)/(a^3*b*c - 1/4*a^3 + 2*a^2*b^2*c + 2*a^2*b*c^2 - 2*a^2*b*c - 3/4*a^2*b - 3/4*a^2*c + 3/4*a^2 + a*b^3*c + 2*a*b^2*c^2 - 2*a*b^2*c - 3/4*a*b^2 + a*b*c^3 - 2*a*b*c^2 - 1/2*a*b*c + 3/2*a*b - 1/2*a*c^2 + 3/2*a*c - 3/4*a - 1/4*b^3 - 3/4*b^2*c + 3/4*b^2 - 1/2*b*c^2 + 3/2*b*c - 3/4*b + 1/2*c^2 - 3/4*c + 1/4) (-a^2*c^2 - a*b*c^2 - a*c^3 + 3/2*a*c^2 + 1/2*b*c^2 - 1/2*c^2)/(a^3*b*c - 1/4*a^3 + 2*a^2*b^2*c + 2*a^2*b*c^2 - 2*a^2*b*c - 3/4*a^2*b - 3/4*a^2*c + 3/4*a^2 + a*b^3*c + 2*a*b^2*c^2 - 2*a*b^2*c - 3/4*a*b^2 + a*b*c^3 - 2*a*b*c^2 - 1/2*a*b*c + 3/2*a*b - 1/2*a*c^2 + 3/2*a*c - 3/4*a - 1/4*b^3 - 3/4*b^2*c + 3/4*b^2 - 1/2*b*c^2 + 3/2*b*c - 3/4*b + 1/2*c^2 - 3/4*c + 1/4) (a^3*b*c - 1/4*a^3 + 2*a^2*b^2*c + 2*a^2*b*c^2 - 2*a^2*b*c - 3/4*a^2*b - 1/4*a^2*c + 3/4*a^2 + a*b^3*c + 2*a*b^2*c^2 - 2*a*b^2*c - 3/4*a*b^2 + a*b*c^3 - 2*a*b*c^2 + 1/2*a*b*c + 3/2*a*b - 1/2*a*c^2 + 1/2*a*c - 3/4*a - 1/4*b^3 - 1/4*b^2*c + 3/4*b^2 - 1/2*b*c^2 + 1/2*b*c - 3/4*b + 1/2*c^2 - 1/4*c + 1/4)/(a^3*b*c - 1/4*a^3 + 2*a^2*b^2*c + 2*a^2*b*c^2 - 2*a^2*b*c - 3/4*a^2*b - 3/4*a^2*c + 3/4*a^2 + a*b^3*c + 2*a*b^2*c^2 - 2*a*b^2*c - 3/4*a*b^2 + a*b*c^3 - 2*a*b*c^2 - 1/2*a*b*c + 3/2*a*b - 1/2*a*c^2 + 3/2*a*c - 3/4*a - 1/4*b^3 - 3/4*b^2*c + 3/4*b^2 - 1/2*b*c^2 + 3/2*b*c - 3/4*b + 1/2*c^2 - 3/4*c + 1/4) Total time: 0.370 seconds, Total memory usage: 3.63MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Thu Dec 8 17:09:51 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); A1:=Matrix(K,3,3,[[b,1,1],[0,a,1],[0,0,c]]); S:=Transpose(A)*A^(-1); F,T:=PrimaryRationalForm(S); B:=T*A*Transpose(T); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*B*Transpose(P)-B)); G:=GroebnerBasis(I); Q1:=Matrix(K,3,3,[(2*b-1)/(a+b-1),(a-b)/(a+b-1),0,(b-a)/(a+b-1),(2*a-1)/(a+b-1),0, 0,0,1]); Kt:=FunctionField(K); D:=(2*a*b*c-a-b-c+1)/(a*b*c); s:=t; px:=-(2*s+D); py:=1-s^2; q:=s^2+s*D+1; Q:=Matrix(Kt,3,3,[1,0,0,0,D*px/q+py/q,-px/q,0,px/q,py/q]); Q:=Q1*T^(-1)*Q*T; print (Q*A*Transpose(Q)-A1); for i in [1..3] do for j in [1..3] do print Evaluate(Q[i,j],c); end for; end for; Output: Magma V2.11-10 Thu Dec 8 2005 17:09:51 on modular [Seed = 499577089] ------------------------------------- [0 0 0] [0 0 0] [0 0 0] (a^3*b^2*c^4 + 4*a^3*b^2*c^3 + 4*a^3*b^2*c^2 + 3*a^3*b^2*c - 2*a^3*b*c^2 - 4*a^3*b*c - a^3*b + a^3 - 2*a^2*b^3*c^5 - 3*a^2*b^3*c^4 + 3*a^2*b^3*c^2 + 6*a^2*b^3*c + a^2*b^2*c^5 + 3*a^2*b^2*c^4 + 4*a^2*b^2*c^3 - 13*a^2*b^2*c - 3*a^2*b^2 - 4*a^2*b*c^3 - 4*a^2*b*c^2 + 4*a^2*b*c + 6*a^2*b + 3*a^2*c - 3*a^2 - 2*a*b^4*c^3 - a*b^4*c^2 + 3*a*b^4*c - 2*a*b^3*c^4 + 5*a*b^3*c^3 + 7*a*b^3*c^2 - 8*a*b^3*c - 3*a*b^3 + 4*a*b^2*c^4 - 3*a*b^2*c^3 - 10*a*b^2*c^2 + a*b^2*c + 9*a*b^2 - 2*a*b*c^4 + a*b*c^2 + 10*a*b*c - 9*a*b + 3*a*c^2 - 6*a*c + 3*a + b^4*c - b^4 + 2*b^3*c^2 - 6*b^3*c + 4*b^3 + b^2*c^3 - 7*b^2*c^2 + 12*b^2*c - 6*b^2 - 2*b*c^3 + 8*b*c^2 - 10*b*c + 4*b + c^3 - 3*c^2 + 3*c - 1)/(a^4*b^2*c^4 + 2*a^4*b^2*c^3 + a^4*b^2*c^2 - a^4*b*c^2 + 2*a^3*b^3*c^4 + 4*a^3*b^3*c^3 + 2*a^3*b^3*c^2 + a^3*b^2*c^5 - 3*a^3*b^2*c^3 - 5*a^3*b^2*c^2 - 2*a^3*b*c^3 + 3*a^3*b*c^2 + a^2*b^4*c^4 + 2*a^2*b^4*c^3 + a^2*b^4*c^2 + a^2*b^3*c^5 - 3*a^2*b^3*c^3 - 5*a^2*b^3*c^2 - a^2*b^2*c^5 - a^2*b^2*c^4 - 3*a^2*b^2*c^3 + 7*a^2*b^2*c^2 - a^2*b*c^4 + 4*a^2*b*c^3 - 3*a^2*b*c^2 - a*b^4*c^2 - 2*a*b^3*c^3 + 3*a*b^3*c^2 - a*b^2*c^4 + 4*a*b^2*c^3 - 3*a*b^2*c^2 + a*b*c^4 - 2*a*b*c^3 + a*b*c^2) (-a^3*b^2*c^4 - 2*a^3*b^2*c^3 - a^3*b^2*c^2 + 2*a^3*b*c^2 + 3*a^3*b*c - a^3 - 2*a^2*b^3*c^4 - 4*a^2*b^3*c^3 - 2*a^2*b^3*c^2 - a^2*b^2*c^5 - a^2*b^2*c^4 - 3*a^2*b^2*c^3 + 2*a^2*b^2*c^2 + 6*a^2*b^2*c + 4*a^2*b*c^3 + 2*a^2*b*c^2 - 4*a^2*b*c - 3*a^2*b - 3*a^2*c + 3*a^2 - a*b^4*c^4 - 2*a*b^4*c^3 - a*b^4*c^2 + a*b^3*c^5 + 3*a*b^3*c^4 - a*b^3*c^3 + a*b^3*c^2 + 3*a*b^3*c - 4*a*b^2*c^4 + 5*a*b^2*c^3 + 3*a*b^2*c^2 - 2*a*b^2*c - 3*a*b^2 + 2*a*b*c^4 - a*b*c^3 - 7*a*b*c + 6*a*b - 3*a*c^2 + 6*a*c - 3*a + b^4*c^2 + b^3*c^3 - 3*b^3*c^2 + 2*b^3*c - b^3 - 2*b^2*c^3 + 6*b^2*c^2 - 7*b^2*c + 3*b^2 + 2*b*c^3 - 7*b*c^2 + 8*b*c - 3*b - c^3 + 3*c^2 - 3*c + 1)/(a^3*b^2*c^4 + 2*a^3*b^2*c^3 + a^3*b^2*c^2 - a^3*b*c^2 + 2*a^2*b^3*c^4 + 4*a^2*b^3*c^3 + 2*a^2*b^3*c^2 + a^2*b^2*c^5 - 3*a^2*b^2*c^3 - 5*a^2*b^2*c^2 - 2*a^2*b*c^3 + 3*a^2*b*c^2 + a*b^4*c^4 + 2*a*b^4*c^3 + a*b^4*c^2 + a*b^3*c^5 - 3*a*b^3*c^3 - 5*a*b^3*c^2 - a*b^2*c^5 - a*b^2*c^4 - 3*a*b^2*c^3 + 7*a*b^2*c^2 - a*b*c^4 + 4*a*b*c^3 - 3*a*b*c^2 - b^4*c^2 - 2*b^3*c^3 + 3*b^3*c^2 - b^2*c^4 + 4*b^2*c^3 - 3*b^2*c^2 + b*c^4 - 2*b*c^3 + b*c^2) (-2*a^2*b*c^2 - 2*a^2*b*c + a^2 + 2*a*b^2*c^4 + 4*a*b^2*c^3 - 2*a*b^2*c - 2*a*b*c^3 - a*b*c^2 + a*b*c + 2*a*b + 2*a*c - 2*a - b^2*c^2 - b^2*c + b^2 - b*c^3 + 3*b*c - 2*b + c^2 - 2*c + 1)/(a^2*b*c^4 + 2*a^2*b*c^3 + a^2*b*c^2 - a^2*c^2 + a*b^2*c^4 + 2*a*b^2*c^3 + a*b^2*c^2 + a*b*c^5 + a*b*c^4 - a*b*c^3 - 3*a*b*c^2 - 2*a*c^3 + 2*a*c^2 - b^2*c^2 - 2*b*c^3 + 2*b*c^2 - c^4 + 2*c^3 - c^2) (-a^4*b^2*c^4 - 2*a^4*b^2*c^3 - a^4*b^2*c^2 + a^4*b*c^2 - 2*a^3*b^3*c^4 - 4*a^3*b^3*c^3 - 2*a^3*b^3*c^2 + a^3*b^2*c^5 + 5*a^3*b^2*c^4 + 11*a^3*b^2*c^3 + 11*a^3*b^2*c^2 + 3*a^3*b^2*c - a^3*b*c^3 - 8*a^3*b*c^2 - 5*a^3*b*c - a^3*b + a^3*c + a^3 - a^2*b^4*c^4 - 2*a^2*b^4*c^3 - a^2*b^4*c^2 - a^2*b^3*c^5 + a^2*b^3*c^4 + 9*a^2*b^3*c^3 + 12*a^2*b^3*c^2 + 6*a^2*b^3*c + 2*a^2*b^2*c^4 - 3*a^2*b^2*c^3 - 17*a^2*b^2*c^2 - 16*a^2*b^2*c - 3*a^2*b^2 - 2*a^2*b*c^4 - 5*a^2*b*c^3 + 3*a^2*b*c^2 + 10*a^2*b*c + 6*a^2*b + 2*a^2*c^2 - 3*a^2 + 2*a*b^4*c^2 + 3*a*b^4*c + 2*a*b^3*c^3 - 3*a*b^3*c^2 - 11*a*b^3*c - 3*a*b^3 - 4*a*b^2*c^3 - 3*a*b^2*c^2 + 10*a*b^2*c + 9*a*b^2 + a*b*c^3 + 5*a*b*c^2 + a*b*c - 9*a*b + a*c^3 - a*c^2 - 3*a*c + 3*a - b^4 - 2*b^3*c + 4*b^3 - b^2*c^2 + 6*b^2*c - 6*b^2 + 2*b*c^2 - 6*b*c + 4*b - c^2 + 2*c - 1)/(a^4*b^2*c^4 + 2*a^4*b^2*c^3 + a^4*b^2*c^2 - a^4*b*c^2 + 2*a^3*b^3*c^4 + 4*a^3*b^3*c^3 + 2*a^3*b^3*c^2 + a^3*b^2*c^5 - 3*a^3*b^2*c^3 - 5*a^3*b^2*c^2 - 2*a^3*b*c^3 + 3*a^3*b*c^2 + a^2*b^4*c^4 + 2*a^2*b^4*c^3 + a^2*b^4*c^2 + a^2*b^3*c^5 - 3*a^2*b^3*c^3 - 5*a^2*b^3*c^2 - a^2*b^2*c^5 - a^2*b^2*c^4 - 3*a^2*b^2*c^3 + 7*a^2*b^2*c^2 - a^2*b*c^4 + 4*a^2*b*c^3 - 3*a^2*b*c^2 - a*b^4*c^2 - 2*a*b^3*c^3 + 3*a*b^3*c^2 - a*b^2*c^4 + 4*a*b^2*c^3 - 3*a*b^2*c^2 + a*b*c^4 - 2*a*b*c^3 + a*b*c^2) (2*a^3*b*c^3 + 3*a^3*b*c^2 + 3*a^3*b*c - a^3*c - a^3 - 2*a^2*b^2*c^5 - 5*a^2*b^2*c^4 - 4*a^2*b^2*c^3 + a^2*b^2*c^2 + 6*a^2*b^2*c + 2*a^2*b*c^4 + a^2*b*c^3 + 2*a^2*b*c^2 - 7*a^2*b*c - 3*a^2*b - 2*a^2*c^2 + 3*a^2 - a*b^3*c^4 - 4*a*b^3*c^3 - 2*a*b^3*c^2 + 3*a*b^3*c + a*b^2*c^5 + a*b^2*c^4 + 4*a*b^2*c^3 + 8*a*b^2*c^2 - 5*a*b^2*c - 3*a*b^2 + 2*a*b*c^3 - 7*a*b*c^2 - a*b*c + 6*a*b - a*c^3 + a*c^2 + 3*a*c - 3*a + b^3*c^2 + b^3*c - b^3 + b^2*c^3 - b^2*c^2 - 4*b^2*c + 3*b^2 - b*c^3 - b*c^2 + 5*b*c - 3*b + c^2 - 2*c + 1)/(a^3*b^2*c^4 + 2*a^3*b^2*c^3 + a^3*b^2*c^2 - a^3*b*c^2 + 2*a^2*b^3*c^4 + 4*a^2*b^3*c^3 + 2*a^2*b^3*c^2 + a^2*b^2*c^5 - 3*a^2*b^2*c^3 - 5*a^2*b^2*c^2 - 2*a^2*b*c^3 + 3*a^2*b*c^2 + a*b^4*c^4 + 2*a*b^4*c^3 + a*b^4*c^2 + a*b^3*c^5 - 3*a*b^3*c^3 - 5*a*b^3*c^2 - a*b^2*c^5 - a*b^2*c^4 - 3*a*b^2*c^3 + 7*a*b^2*c^2 - a*b*c^4 + 4*a*b*c^3 - 3*a*b*c^2 - b^4*c^2 - 2*b^3*c^3 + 3*b^3*c^2 - b^2*c^4 + 4*b^2*c^3 - 3*b^2*c^2 + b*c^4 - 2*b*c^3 + b*c^2) (-2*a^2*b*c^4 - 4*a^2*b*c^3 - 4*a^2*b*c^2 - 2*a^2*b*c + a^2*c^2 + a^2*c + a^2 - 2*a*b^2*c^2 - 2*a*b^2*c + 2*a*b*c^4 + 2*a*b*c^3 + 3*a*b*c^2 + 3*a*b*c + 2*a*b + a*c^3 - a*c^2 - 2*a + b^2 - b*c^2 + b*c - 2*b - c^3 + c^2 - c + 1)/(a^2*b*c^4 + 2*a^2*b*c^3 + a^2*b*c^2 - a^2*c^2 + a*b^2*c^4 + 2*a*b^2*c^3 + a*b^2*c^2 + a*b*c^5 + a*b*c^4 - a*b*c^3 - 3*a*b*c^2 - 2*a*c^3 + 2*a*c^2 - b^2*c^2 - 2*b*c^3 + 2*b*c^2 - c^4 + 2*c^3 - c^2) (2*a^2*b^2*c^4 + 4*a^2*b^2*c^3 + 4*a^2*b^2*c^2 + 2*a^2*b^2*c - 3*a^2*b*c^2 - 3*a^2*b*c - a^2*b + a^2 + 2*a*b^3*c^2 + 2*a*b^3*c + 2*a*b^2*c^3 - 3*a*b^2*c^2 - 5*a*b^2*c - 2*a*b^2 - 3*a*b*c^3 + a*b*c + 4*a*b + 2*a*c - 2*a - b^3 - 2*b^2*c + 3*b^2 - b*c^2 + 4*b*c - 3*b + c^2 - 2*c + 1)/(a^3*b^2*c^3 + 2*a^3*b^2*c^2 + a^3*b^2*c - a^3*b*c + a^2*b^3*c^3 + 2*a^2*b^3*c^2 + a^2*b^3*c + a^2*b^2*c^4 + a^2*b^2*c^3 - a^2*b^2*c^2 - 3*a^2*b^2*c - 2*a^2*b*c^2 + 2*a^2*b*c - a*b^3*c - 2*a*b^2*c^2 + 2*a*b^2*c - a*b*c^3 + 2*a*b*c^2 - a*b*c) (2*a^2*b*c^2 + 2*a^2*b*c - a^2 - 2*a*b^2*c^4 - 4*a*b^2*c^3 + 2*a*b^2*c + 2*a*b*c^3 + a*b*c^2 - a*b*c - 2*a*b - 2*a*c + 2*a + b^2*c^2 + b^2*c - b^2 + b*c^3 - 3*b*c + 2*b - c^2 + 2*c - 1)/(a^2*b^2*c^3 + 2*a^2*b^2*c^2 + a^2*b^2*c - a^2*b*c + a*b^3*c^3 + 2*a*b^3*c^2 + a*b^3*c + a*b^2*c^4 + a*b^2*c^3 - a*b^2*c^2 - 3*a*b^2*c - 2*a*b*c^2 + 2*a*b*c - b^3*c - 2*b^2*c^2 + 2*b^2*c - b*c^3 + 2*b*c^2 - b*c) (-a^2*b*c^3 - 2*a^2*b*c^2 - a^2*b*c + a^2 - a*b^2*c^3 - 2*a*b^2*c^2 - a*b^2*c + a*b*c^4 + 3*a*b*c^3 + 3*a*b*c^2 + a*b*c + 2*a*b - a*c^2 + a*c - 2*a + b^2 - b*c^2 + b*c - 2*b - c^3 + c^2 - c + 1)/(a^2*b*c^3 + 2*a^2*b*c^2 + a^2*b*c - a^2*c + a*b^2*c^3 + 2*a*b^2*c^2 + a*b^2*c + a*b*c^4 + a*b*c^3 - a*b*c^2 - 3*a*b*c - 2*a*c^2 + 2*a*c - b^2*c - 2*b*c^2 + 2*b*c - c^3 + 2*c^2 - c) Total time: 0.370 seconds, Total memory usage: 3.63MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Thu Dec 8 17:03:08 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); A1:=Matrix(K,3,3,[[b,1,1],[0,a,1],[0,0,c]]); S:=Transpose(A)*A^(-1); F,T:=PrimaryRationalForm(S); B:=T*A*Transpose(T); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*B*Transpose(P)-B)); G:=GroebnerBasis(I); Q1:=Matrix(K,3,3,[(2*b-1)/(a+b-1),(a-b)/(a+b-1),0,(b-a)/(a+b-1),(2*a-1)/(a+b-1),0, 0,0,1]); Kt:=FunctionField(K); D:=(2*a*b*c-a-b-c+1)/(a*b*c); s:=t; px:=-(2*s+D); py:=1-s^2; q:=s^2+s*D+1; Q:=Matrix(Kt,3,3,[1,0,0,0,D*px/q+py/q,-px/q,0,px/q,py/q]); Q:=Q1*T^(-1)*Q*T; print (Q*A*Transpose(Q)-A1); for i in [1..3] do for j in [1..3] do print Evaluate(Q[i,j],-D/2); end for; end for; print Factorization(Numerator(1-D^2/4)); Output: Magma V2.11-10 Thu Dec 8 2005 17:03:07 on modular [Seed = 1038537621] ------------------------------------- [0 0 0] [0 0 0] [0 0 0] (2*b - 1)/(a + b - 1) (a - b)/(a + b - 1) 0 (-a + b)/(a + b - 1) (2*a - 1)/(a + b - 1) 0 0 0 1 [ , ] Total time: 0.370 seconds, Total memory usage: 3.63MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Thu Dec 8 17:02:01 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); A1:=Matrix(K,3,3,[[b,1,1],[0,a,1],[0,0,c]]); S:=Transpose(A)*A^(-1); F,T:=PrimaryRationalForm(S); B:=T*A*Transpose(T); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*B*Transpose(P)-B)); G:=GroebnerBasis(I); Q1:=Matrix(K,3,3,[(2*b-1)/(a+b-1),(a-b)/(a+b-1),0,(b-a)/(a+b-1),(2*a-1)/(a+b-1),0, 0,0,1]); Kt:=FunctionField(K); D:=(2*a*b*c-a-b-c+1)/(a*b*c); s:=t; px:=-(2*s+D); py:=1-s^2; q:=s^2+s*D+1; Q:=Matrix(Kt,3,3,[1,0,0,0,D*px/q+py/q,-px/q,0,px/q,py/q]); Q:=Q1*T^(-1)*Q*T; print (Q*A*Transpose(Q)-A1); for i in [1..3] do for j in [1..3] do print Evaluate(Q[i,j],-D/2); end for; end for; Output: Magma V2.11-10 Thu Dec 8 2005 17:02:01 on modular [Seed = 987878480] ------------------------------------- [0 0 0] [0 0 0] [0 0 0] (2*b - 1)/(a + b - 1) (a - b)/(a + b - 1) 0 (-a + b)/(a + b - 1) (2*a - 1)/(a + b - 1) 0 0 0 1 Total time: 0.370 seconds, Total memory usage: 3.63MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Thu Dec 8 17:01:41 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); A1:=Matrix(K,3,3,[[b,1,1],[0,a,1],[0,0,c]]); S:=Transpose(A)*A^(-1); F,T:=PrimaryRationalForm(S); B:=T*A*Transpose(T); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*B*Transpose(P)-B)); G:=GroebnerBasis(I); Q1:=Matrix(K,3,3,[(2*b-1)/(a+b-1),(a-b)/(a+b-1),0,(b-a)/(a+b-1),(2*a-1)/(a+b-1),0, 0,0,1]); Kt:=FunctionField(K); D:=(2*a*b*c-a-b-c+1)/(a*b*c); s:=t; px:=-(2*s+D); py:=1-s^2; q:=s^2+s*D+1; Q:=Matrix(Kt,3,3,[1,0,0,0,D*px/q+py/q,-px/q,0,px/q,py/q]); Q:=Q1*T^(-1)*Q*T; print (Q*A*Transpose(Q)-A1); for i in [1..3] do for j in [1..3] do print Evaluate(Q[i,j],+D/2); end for; end for; Output: Magma V2.11-10 Thu Dec 8 2005 17:01:40 on modular [Seed = 938268992] ------------------------------------- [0 0 0] [0 0 0] [0 0 0] (2*a^3*b^2*c^2 + a^3*b^2*c - 9/4*a^3*b*c - 3/8*a^3*b + 9/16*a^3 - 2*a^2*b^3*c^3 + a^2*b^3*c^2 + 2*a^2*b^3*c + 2*a^2*b^2*c^3 + 5/2*a^2*b^2*c^2 - 6*a^2*b^2*c - 9/8*a^2*b^2 - 9/2*a^2*b*c^2 + 21/8*a^2*b*c + 45/16*a^2*b + 27/16*a^2*c - 27/16*a^2 - a*b^4*c^2 + a*b^4*c - a*b^3*c^3 + 11/2*a*b^3*c^2 - 13/4*a*b^3*c - 9/8*a*b^3 + 7/2*a*b^2*c^3 - 8*a*b^2*c^2 + 3/4*a*b^2*c + 63/16*a*b^2 - 9/4*a*b*c^3 + 15/8*a*b*c^2 + 39/8*a*b*c - 9/2*a*b + 27/16*a*c^2 - 27/8*a*c + 27/16*a + 1/2*b^4*c - 3/8*b^4 + b^3*c^2 - 23/8*b^3*c + 27/16*b^3 + 1/2*b^2*c^3 - 29/8*b^2*c^2 + 95/16*b^2*c - 45/16*b^2 - 9/8*b*c^3 + 69/16*b*c^2 - 21/4*b*c + 33/16*b + 9/16*c^3 - 27/16*c^2 + 27/16*c - 9/16)/(a^4*b^2*c^2 - 3/4*a^4*b*c + 3/16*a^4 + 2*a^3*b^3*c^2 + a^3*b^2*c^3 - 2*a^3*b^2*c^2 - 9/4*a^3*b^2*c - 3/2*a^3*b*c^2 + 9/4*a^3*b*c + 3/4*a^3*b + 9/16*a^3*c - 3/4*a^3 + a^2*b^4*c^2 + a^2*b^3*c^3 - 2*a^2*b^3*c^2 - 9/4*a^2*b^3*c - a^2*b^2*c^3 - 2*a^2*b^2*c^2 + 9/2*a^2*b^2*c + 9/8*a^2*b^2 - 3/4*a^2*b*c^3 + 3*a^2*b*c^2 - 9/16*a^2*b*c - 9/4*a^2*b + 9/16*a^2*c^2 - 27/16*a^2*c + 9/8*a^2 - 3/4*a*b^4*c - 3/2*a*b^3*c^2 + 9/4*a*b^3*c + 3/4*a*b^3 - 3/4*a*b^2*c^3 + 3*a*b^2*c^2 - 9/16*a*b^2*c - 9/4*a*b^2 + 3/4*a*b*c^3 - 3/8*a*b*c^2 - 21/8*a*b*c + 9/4*a*b + 3/16*a*c^3 - 9/8*a*c^2 + 27/16*a*c - 3/4*a + 3/16*b^4 + 9/16*b^3*c - 3/4*b^3 + 9/16*b^2*c^2 - 27/16*b^2*c + 9/8*b^2 + 3/16*b*c^3 - 9/8*b*c^2 + 27/16*b*c - 3/4*b - 3/16*c^3 + 9/16*c^2 - 9/16*c + 3/16) (-a^4*b^2*c^2 + 7/4*a^4*b*c - 9/16*a^4 - 2*a^3*b^3*c^2 - a^3*b^2*c^3 - a^3*b^2*c^2 + 17/4*a^3*b^2*c + 7/2*a^3*b*c^2 - 5/2*a^3*b*c - 15/8*a^3*b - 27/16*a^3*c + 27/16*a^3 - a^2*b^4*c^2 + a^2*b^3*c^3 - a^2*b^3*c^2 + 13/4*a^2*b^3*c - 2*a^2*b^2*c^3 + 11/2*a^2*b^2*c^2 - 3*a^2*b^2*c - 9/4*a^2*b^2 + 7/4*a^2*b*c^3 - 3/2*a^2*b*c^2 - 63/16*a^2*b*c + 63/16*a^2*b - 27/16*a^2*c^2 + 27/8*a^2*c - 27/16*a^2 + 3/4*a*b^4*c - 1/2*a*b^3*c - 9/8*a*b^3 - 3/4*a*b^2*c^3 + 2*a*b^2*c^2 - 57/16*a*b^2*c + 45/16*a*b^2 + a*b*c^3 - 15/4*a*b*c^2 + 5*a*b*c - 9/4*a*b - 9/16*a*c^3 + 27/16*a*c^2 - 27/16*a*c + 9/16*a - 3/16*b^4 - 5/16*b^3*c + 9/16*b^3 - 1/16*b^2*c^2 + 5/8*b^2*c - 9/16*b^2 + 1/16*b*c^3 + 1/16*b*c^2 - 5/16*b*c + 3/16*b)/(a^4*b^2*c^2 - 3/4*a^4*b*c + 3/16*a^4 + 2*a^3*b^3*c^2 + a^3*b^2*c^3 - 2*a^3*b^2*c^2 - 9/4*a^3*b^2*c - 3/2*a^3*b*c^2 + 9/4*a^3*b*c + 3/4*a^3*b + 9/16*a^3*c - 3/4*a^3 + a^2*b^4*c^2 + a^2*b^3*c^3 - 2*a^2*b^3*c^2 - 9/4*a^2*b^3*c - a^2*b^2*c^3 - 2*a^2*b^2*c^2 + 9/2*a^2*b^2*c + 9/8*a^2*b^2 - 3/4*a^2*b*c^3 + 3*a^2*b*c^2 - 9/16*a^2*b*c - 9/4*a^2*b + 9/16*a^2*c^2 - 27/16*a^2*c + 9/8*a^2 - 3/4*a*b^4*c - 3/2*a*b^3*c^2 + 9/4*a*b^3*c + 3/4*a*b^3 - 3/4*a*b^2*c^3 + 3*a*b^2*c^2 - 9/16*a*b^2*c - 9/4*a*b^2 + 3/4*a*b*c^3 - 3/8*a*b*c^2 - 21/8*a*b*c + 9/4*a*b + 3/16*a*c^3 - 9/8*a*c^2 + 27/16*a*c - 3/4*a + 3/16*b^4 + 9/16*b^3*c - 3/4*b^3 + 9/16*b^2*c^2 - 27/16*b^2*c + 9/8*b^2 + 3/16*b*c^3 - 9/8*b*c^2 + 27/16*b*c - 3/4*b - 3/16*c^3 + 9/16*c^2 - 9/16*c + 3/16) (-a^3*b^2*c + 1/2*a^3*b + 2*a^2*b^3*c^2 - a^2*b^3*c - a^2*b^2*c^2 - 1/2*a^2*b^2*c + a^2*b^2 + a^2*b*c - 3/4*a^2*b - 3/2*a*b^3*c + 1/2*a*b^3 - 3/2*a*b^2*c^2 + 5/2*a*b^2*c - 1/2*a*b^2 + 1/2*a*b*c^2 - 1/2*a*b*c + 1/4*b^3 + 1/2*b^2*c - 1/2*b^2 + 1/4*b*c^2 - 1/2*b*c + 1/4*b)/(a^3*b^2*c^2 - 3/4*a^3*b*c + 3/16*a^3 + a^2*b^3*c^2 + a^2*b^2*c^3 - a^2*b^2*c^2 - 3/2*a^2*b^2*c - 3/2*a^2*b*c^2 + 3/2*a^2*b*c + 9/16*a^2*b + 9/16*a^2*c - 9/16*a^2 - 3/4*a*b^3*c - 3/2*a*b^2*c^2 + 3/2*a*b^2*c + 9/16*a*b^2 - 3/4*a*b*c^3 + 3/2*a*b*c^2 + 3/8*a*b*c - 9/8*a*b + 9/16*a*c^2 - 9/8*a*c + 9/16*a + 3/16*b^3 + 9/16*b^2*c - 9/16*b^2 + 9/16*b*c^2 - 9/8*b*c + 9/16*b + 3/16*c^3 - 9/16*c^2 + 9/16*c - 3/16) (-a^4*b^2*c^2 + 3/4*a^4*b*c - 3/16*a^4 - 2*a^3*b^3*c^2 + a^3*b^2*c^3 + 5*a^3*b^2*c^2 + 13/4*a^3*b^2*c - a^3*b*c^2 - 5*a^3*b*c - 9/8*a^3*b + 3/16*a^3*c + 21/16*a^3 - a^2*b^4*c^2 - a^2*b^3*c^3 + 5*a^2*b^3*c^2 + 17/4*a^2*b^3*c + 2*a^2*b^2*c^3 - 5/2*a^2*b^2*c^2 - 12*a^2*b^2*c - 9/4*a^2*b^2 - 7/4*a^2*b*c^3 - 7/2*a^2*b*c^2 + 111/16*a^2*b*c + 81/16*a^2*b + 15/16*a^2*c^2 + 9/8*a^2*c - 45/16*a^2 + 7/4*a*b^4*c + 5/2*a*b^3*c^2 - 7*a*b^3*c - 15/8*a*b^3 + 3/4*a*b^2*c^3 - 7*a*b^2*c^2 + 105/16*a*b^2*c + 99/16*a*b^2 - a*b*c^3 + 19/4*a*b*c^2 + 3/2*a*b*c - 27/4*a*b + 9/16*a*c^3 - 3/16*a*c^2 - 45/16*a*c + 39/16*a - 9/16*b^4 - 19/16*b^3*c + 39/16*b^3 - 11/16*b^2*c^2 + 31/8*b^2*c - 63/16*b^2 - 1/16*b*c^3 + 23/16*b*c^2 - 67/16*b*c + 45/16*b - 3/4*c^2 + 3/2*c - 3/4)/(a^4*b^2*c^2 - 3/4*a^4*b*c + 3/16*a^4 + 2*a^3*b^3*c^2 + a^3*b^2*c^3 - 2*a^3*b^2*c^2 - 9/4*a^3*b^2*c - 3/2*a^3*b*c^2 + 9/4*a^3*b*c + 3/4*a^3*b + 9/16*a^3*c - 3/4*a^3 + a^2*b^4*c^2 + a^2*b^3*c^3 - 2*a^2*b^3*c^2 - 9/4*a^2*b^3*c - a^2*b^2*c^3 - 2*a^2*b^2*c^2 + 9/2*a^2*b^2*c + 9/8*a^2*b^2 - 3/4*a^2*b*c^3 + 3*a^2*b*c^2 - 9/16*a^2*b*c - 9/4*a^2*b + 9/16*a^2*c^2 - 27/16*a^2*c + 9/8*a^2 - 3/4*a*b^4*c - 3/2*a*b^3*c^2 + 9/4*a*b^3*c + 3/4*a*b^3 - 3/4*a*b^2*c^3 + 3*a*b^2*c^2 - 9/16*a*b^2*c - 9/4*a*b^2 + 3/4*a*b*c^3 - 3/8*a*b*c^2 - 21/8*a*b*c + 9/4*a*b + 3/16*a*c^3 - 9/8*a*c^2 + 27/16*a*c - 3/4*a + 3/16*b^4 + 9/16*b^3*c - 3/4*b^3 + 9/16*b^2*c^2 - 27/16*b^2*c + 9/8*b^2 + 3/16*b*c^3 - 9/8*b*c^2 + 27/16*b*c - 3/4*b - 3/16*c^3 + 9/16*c^2 - 9/16*c + 3/16) (a^4*b*c^2 + a^4*b*c - 1/2*a^4*c - 3/8*a^4 - 2*a^3*b^2*c^3 - a^3*b^2*c^2 + 2*a^3*b^2*c + a^3*b*c^3 + 3/2*a^3*b*c^2 - 7/4*a^3*b*c - 9/8*a^3*b - a^3*c^2 + 1/8*a^3*c + 15/16*a^3 - 2*a^2*b^3*c^2 + a^2*b^3*c + 9/2*a^2*b^2*c^2 - 9/8*a^2*b^2 + 3/2*a^2*b*c^3 - 5/2*a^2*b*c^2 - 9/4*a^2*b*c + 27/16*a^2*b - 1/2*a^2*c^3 + 3/8*a^2*c^2 + 15/16*a^2*c - 9/16*a^2 + 5/4*a*b^3*c - 3/8*a*b^3 + a*b^2*c^2 - 27/8*a*b^2*c + 9/16*a*b^2 - 1/4*a*b*c^3 - 13/8*a*b*c^2 + 19/8*a*b*c - 1/8*a*c^3 + 9/16*a*c^2 - 1/4*a*c - 3/16*a - 3/16*b^3 - 5/16*b^2*c + 9/16*b^2 - 1/16*b*c^2 + 5/8*b*c - 9/16*b + 1/16*c^3 + 1/16*c^2 - 5/16*c + 3/16)/(a^4*b^2*c^2 - 3/4*a^4*b*c + 3/16*a^4 + 2*a^3*b^3*c^2 + a^3*b^2*c^3 - 2*a^3*b^2*c^2 - 9/4*a^3*b^2*c - 3/2*a^3*b*c^2 + 9/4*a^3*b*c + 3/4*a^3*b + 9/16*a^3*c - 3/4*a^3 + a^2*b^4*c^2 + a^2*b^3*c^3 - 2*a^2*b^3*c^2 - 9/4*a^2*b^3*c - a^2*b^2*c^3 - 2*a^2*b^2*c^2 + 9/2*a^2*b^2*c + 9/8*a^2*b^2 - 3/4*a^2*b*c^3 + 3*a^2*b*c^2 - 9/16*a^2*b*c - 9/4*a^2*b + 9/16*a^2*c^2 - 27/16*a^2*c + 9/8*a^2 - 3/4*a*b^4*c - 3/2*a*b^3*c^2 + 9/4*a*b^3*c + 3/4*a*b^3 - 3/4*a*b^2*c^3 + 3*a*b^2*c^2 - 9/16*a*b^2*c - 9/4*a*b^2 + 3/4*a*b*c^3 - 3/8*a*b*c^2 - 21/8*a*b*c + 9/4*a*b + 3/16*a*c^3 - 9/8*a*c^2 + 27/16*a*c - 3/4*a + 3/16*b^4 + 9/16*b^3*c - 3/4*b^3 + 9/16*b^2*c^2 - 27/16*b^2*c + 9/8*b^2 + 3/16*b*c^3 - 9/8*b*c^2 + 27/16*b*c - 3/4*b - 3/16*c^3 + 9/16*c^2 - 9/16*c + 3/16) (-2*a^3*b^2*c^2 - a^3*b^2*c + 3/2*a^3*b*c + 1/2*a^3*b - 1/4*a^3 - a^2*b^3*c + a^2*b^2*c^2 + 5/2*a^2*b^2*c + a^2*b^2 + 3/2*a^2*b*c^2 - 2*a^2*b*c - 3/2*a^2*b - 1/2*a^2*c + 3/4*a^2 + 1/2*a*b^3 - 1/2*a*b^2*c - 5/4*a*b^2 - a*b*c^2 + 3/2*a*b - 1/4*a*c^2 + a*c - 3/4*a + 1/4*b^2 + 1/2*b*c - 1/2*b + 1/4*c^2 - 1/2*c + 1/4)/(a^3*b^2*c^2 - 3/4*a^3*b*c + 3/16*a^3 + a^2*b^3*c^2 + a^2*b^2*c^3 - a^2*b^2*c^2 - 3/2*a^2*b^2*c - 3/2*a^2*b*c^2 + 3/2*a^2*b*c + 9/16*a^2*b + 9/16*a^2*c - 9/16*a^2 - 3/4*a*b^3*c - 3/2*a*b^2*c^2 + 3/2*a*b^2*c + 9/16*a*b^2 - 3/4*a*b*c^3 + 3/2*a*b*c^2 + 3/8*a*b*c - 9/8*a*b + 9/16*a*c^2 - 9/8*a*c + 9/16*a + 3/16*b^3 + 9/16*b^2*c - 9/16*b^2 + 9/16*b*c^2 - 9/8*b*c + 9/16*b + 3/16*c^3 - 9/16*c^2 + 9/16*c - 3/16) (2*a^2*b^2*c^3 + a^2*b^2*c^2 - 5/2*a^2*b*c^2 - 1/2*a^2*b*c + 3/4*a^2*c + a*b^3*c^2 + a*b^2*c^3 - 7/2*a*b^2*c^2 - a*b^2*c - 5/2*a*b*c^3 + 3/2*a*b*c^2 + 5/2*a*b*c + 3/2*a*c^2 - 3/2*a*c - 1/2*b^3*c - b^2*c^2 + 7/4*b^2*c - 1/2*b*c^3 + 5/2*b*c^2 - 2*b*c + 3/4*c^3 - 3/2*c^2 + 3/4*c)/(a^3*b^2*c^2 - 3/4*a^3*b*c + 3/16*a^3 + a^2*b^3*c^2 + a^2*b^2*c^3 - a^2*b^2*c^2 - 3/2*a^2*b^2*c - 3/2*a^2*b*c^2 + 3/2*a^2*b*c + 9/16*a^2*b + 9/16*a^2*c - 9/16*a^2 - 3/4*a*b^3*c - 3/2*a*b^2*c^2 + 3/2*a*b^2*c + 9/16*a*b^2 - 3/4*a*b*c^3 + 3/2*a*b*c^2 + 3/8*a*b*c - 9/8*a*b + 9/16*a*c^2 - 9/8*a*c + 9/16*a + 3/16*b^3 + 9/16*b^2*c - 9/16*b^2 + 9/16*b*c^2 - 9/8*b*c + 9/16*b + 3/16*c^3 - 9/16*c^2 + 9/16*c - 3/16) (a^3*b*c^2 - 1/2*a^3*c - 2*a^2*b^2*c^3 + a^2*b^2*c^2 + a^2*b*c^3 + 1/2*a^2*b*c^2 - a^2*b*c - a^2*c^2 + 3/4*a^2*c + 3/2*a*b^2*c^2 - 1/2*a*b^2*c + 3/2*a*b*c^3 - 5/2*a*b*c^2 + 1/2*a*b*c - 1/2*a*c^3 + 1/2*a*c^2 - 1/4*b^2*c - 1/2*b*c^2 + 1/2*b*c - 1/4*c^3 + 1/2*c^2 - 1/4*c)/(a^3*b^2*c^2 - 3/4*a^3*b*c + 3/16*a^3 + a^2*b^3*c^2 + a^2*b^2*c^3 - a^2*b^2*c^2 - 3/2*a^2*b^2*c - 3/2*a^2*b*c^2 + 3/2*a^2*b*c + 9/16*a^2*b + 9/16*a^2*c - 9/16*a^2 - 3/4*a*b^3*c - 3/2*a*b^2*c^2 + 3/2*a*b^2*c + 9/16*a*b^2 - 3/4*a*b*c^3 + 3/2*a*b*c^2 + 3/8*a*b*c - 9/8*a*b + 9/16*a*c^2 - 9/8*a*c + 9/16*a + 3/16*b^3 + 9/16*b^2*c - 9/16*b^2 + 9/16*b*c^2 - 9/8*b*c + 9/16*b + 3/16*c^3 - 9/16*c^2 + 9/16*c - 3/16) (-a^3*b^2*c^2 + 3/4*a^3*b*c - 1/16*a^3 - a^2*b^3*c^2 + a^2*b^2*c^3 + a^2*b^2*c^2 + 3/2*a^2*b^2*c - 3/2*a^2*b*c - 3/16*a^2*b + 1/16*a^2*c + 3/16*a^2 + 3/4*a*b^3*c - 3/2*a*b^2*c - 3/16*a*b^2 - 3/4*a*b*c^3 + 7/8*a*b*c + 3/8*a*b + 5/16*a*c^2 - 1/8*a*c - 3/16*a - 1/16*b^3 + 1/16*b^2*c + 3/16*b^2 + 5/16*b*c^2 - 1/8*b*c - 3/16*b + 3/16*c^3 - 5/16*c^2 + 1/16*c + 1/16)/(a^3*b^2*c^2 - 3/4*a^3*b*c + 3/16*a^3 + a^2*b^3*c^2 + a^2*b^2*c^3 - a^2*b^2*c^2 - 3/2*a^2*b^2*c - 3/2*a^2*b*c^2 + 3/2*a^2*b*c + 9/16*a^2*b + 9/16*a^2*c - 9/16*a^2 - 3/4*a*b^3*c - 3/2*a*b^2*c^2 + 3/2*a*b^2*c + 9/16*a*b^2 - 3/4*a*b*c^3 + 3/2*a*b*c^2 + 3/8*a*b*c - 9/8*a*b + 9/16*a*c^2 - 9/8*a*c + 9/16*a + 3/16*b^3 + 9/16*b^2*c - 9/16*b^2 + 9/16*b*c^2 - 9/8*b*c + 9/16*b + 3/16*c^3 - 9/16*c^2 + 9/16*c - 3/16) T ** WARNING: Output too long, hence truncated. '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Thu Dec 8 17:01:30 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); A1:=Matrix(K,3,3,[[b,1,1],[0,a,1],[0,0,c]]); S:=Transpose(A)*A^(-1); F,T:=PrimaryRationalForm(S); B:=T*A*Transpose(T); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*B*Transpose(P)-B)); G:=GroebnerBasis(I); Q1:=Matrix(K,3,3,[(2*b-1)/(a+b-1),(a-b)/(a+b-1),0,(b-a)/(a+b-1),(2*a-1)/(a+b-1),0, 0,0,1]); Kt:=FunctionField(K); D:=(2*a*b*c-a-b-c+1)/(a*b*c); s:=t; px:=-(2*s+D); py:=1-s^2; q:=s^2+s*D+1; Q:=Matrix(Kt,3,3,[1,0,0,0,D*px/q+py/q,-px/q,0,px/q,py/q]); Q:=Q1*T^(-1)*Q*T; print (Q*A*Transpose(Q)-A1); for i in [1..3] do for j in [1..3] do print Evaluate(Q[i,j],-D/2); end for; end for; Output: Magma V2.11-10 Thu Dec 8 2005 17:01:30 on modular [Seed = 3540532677] ------------------------------------- [0 0 0] [0 0 0] [0 0 0] (2*b - 1)/(a + b - 1) (a - b)/(a + b - 1) 0 (-a + b)/(a + b - 1) (2*a - 1)/(a + b - 1) 0 0 0 1 Total time: 0.370 seconds, Total memory usage: 3.63MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Thu Dec 8 17:01:20 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); A1:=Matrix(K,3,3,[[b,1,1],[0,a,1],[0,0,c]]); S:=Transpose(A)*A^(-1); F,T:=PrimaryRationalForm(S); B:=T*A*Transpose(T); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*B*Transpose(P)-B)); G:=GroebnerBasis(I); Q1:=Matrix(K,3,3,[(2*b-1)/(a+b-1),(a-b)/(a+b-1),0,(b-a)/(a+b-1),(2*a-1)/(a+b-1),0, 0,0,1]); Kt:=FunctionField(K); D:=(2*a*b*c-a-b-c+1)/(a*b*c); s:=t; px:=-(2*s+D); py:=1-s^2; q:=s^2+s*D+1; Q:=Matrix(Kt,3,3,[1,0,0,0,D*px/q+py/q,-px/q,0,px/q,py/q]); Q:=Q1*T^(-1)*Q*T; print (Q*A*Transpose(Q)-A1); for i in [1..3] do for j in [1..3] do print Evaluate(Q[i,j]),-D/2); end for; end for; Output: Magma V2.11-10 Thu Dec 8 2005 17:01:19 on modular [Seed = 3591190741] ------------------------------------- [0 0 0] [0 0 0] [0 0 0] >> print Evaluate(Q[i,j]),-D/2); ^ User error: bad syntax >> end for; ^ User error: bad syntax >> end for; ^ User error: bad syntax Total time: 0.360 seconds, Total memory usage: 3.63MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Thu Dec 8 16:58:15 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); A1:=Matrix(K,3,3,[[b,1,1],[0,a,1],[0,0,c]]); S:=Transpose(A)*A^(-1); F,T:=PrimaryRationalForm(S); B:=T*A*Transpose(T); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*B*Transpose(P)-B)); G:=GroebnerBasis(I); Q1:=Matrix(K,3,3,[(2*b-1)/(a+b-1),(a-b)/(a+b-1),0,(b-a)/(a+b-1),(2*a-1)/(a+b-1),0, 0,0,1]); Kt:=FunctionField(K); D:=(2*a*b*c-a-b-c+1)/(a*b*c); s:=t; px:=-(2*s+D); py:=1-s^2; q:=s^2+s*D+1; Q:=Matrix(Kt,3,3,[1,0,0,0,D*px/q+py/q,-px/q,0,px/q,py/q]); Q:=Q1*T^(-1)*Q*T; print (Q*A*Transpose(Q)-A1); for i in [1..3] do for j in [1..3] do print Factorization(Denominator(Evaluate(Numerator(Q[i,j]),0))); end for; end for; Output: Magma V2.11-10 Thu Dec 8 2005 16:58:14 on modular [Seed = 3624089474] ------------------------------------- [0 0 0] [0 0 0] [0 0 0] [ , , , , ] [ , , , , ] [ , , , ] [ , , , , ] [ , , , , ] [ , , , ] [ , , , ] [ , , , ] [ , , , ] Total time: 0.370 seconds, Total memory usage: 3.63MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Thu Dec 8 16:57:34 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); A1:=Matrix(K,3,3,[[b,1,1],[0,a,1],[0,0,c]]); S:=Transpose(A)*A^(-1); F,T:=PrimaryRationalForm(S); B:=T*A*Transpose(T); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*B*Transpose(P)-B)); G:=GroebnerBasis(I); Q1:=Matrix(K,3,3,[(2*b-1)/(a+b-1),(a-b)/(a+b-1),0,(b-a)/(a+b-1),(2*a-1)/(a+b-1),0, 0,0,1]); Kt:=FunctionField(K); D:=(2*a*b*c-a-b-c+1)/(a*b*c); s:=t; px:=-(2*s+D); py:=1-s^2; q:=s^2+s*D+1; Q:=Matrix(Kt,3,3,[1,0,0,0,D*px/q+py/q,-px/q,0,px/q,py/q]); Q:=Q1*T^(-1)*Q*T; print (Q*A*Transpose(Q)-A1); for i in [1..3] do for j in [1..3] do print Evaluate(Numerator(Q[i,j]),0); end for; end for; Output: Magma V2.11-10 Thu Dec 8 2005 16:57:34 on modular [Seed = 3256429984] ------------------------------------- [0 0 0] [0 0 0] [0 0 0] (3*a^3*b^2*c^2 + 3*a^3*b^2*c - 4*a^3*b*c - a^3*b + a^3 - 2*a^2*b^3*c^3 + a^2*b^3*c^2 + 6*a^2*b^3*c + 3*a^2*b^2*c^3 + 6*a^2*b^2*c^2 - 13*a^2*b^2*c - 3*a^2*b^2 - 8*a^2*b*c^2 + 4*a^2*b*c + 6*a^2*b + 3*a^2*c - 3*a^2 - 2*a*b^4*c^2 + 3*a*b^4*c - 2*a*b^3*c^3 + 11*a*b^3*c^2 - 8*a*b^3*c - 3*a*b^3 + 6*a*b^2*c^3 - 15*a*b^2*c^2 + a*b^2*c + 9*a*b^2 - 4*a*b*c^3 + 3*a*b*c^2 + 10*a*b*c - 9*a*b + 3*a*c^2 - 6*a*c + 3*a + b^4*c - b^4 + 2*b^3*c^2 - 6*b^3*c + 4*b^3 + b^2*c^3 - 7*b^2*c^2 + 12*b^2*c - 6*b^2 - 2*b*c^3 + 8*b*c^2 - 10*b*c + 4*b + c^3 - 3*c^2 + 3*c - 1)/(a^4*b^2*c^2 + 2*a^3*b^3*c^2 + a^3*b^2*c^3 - 2*a^3*b^2*c^2 + a^2*b^4*c^2 + a^2*b^3*c^3 - 2*a^2*b^3*c^2 - a^2*b^2*c^3 + a^2*b^2*c^2) (-a^3*b^2*c^2 + 3*a^3*b*c - a^3 - 2*a^2*b^3*c^2 - a^2*b^2*c^3 - 3*a^2*b^2*c^2 + 6*a^2*b^2*c + 6*a^2*b*c^2 - 4*a^2*b*c - 3*a^2*b - 3*a^2*c + 3*a^2 - a*b^4*c^2 + a*b^3*c^3 - 3*a*b^3*c^2 + 3*a*b^3*c - 4*a*b^2*c^3 + 9*a*b^2*c^2 - 2*a*b^2*c - 3*a*b^2 + 3*a*b*c^3 - 2*a*b*c^2 - 7*a*b*c + 6*a*b - 3*a*c^2 + 6*a*c - 3*a - b^3*c^2 + 2*b^3*c - b^3 - b^2*c^3 + 5*b^2*c^2 - 7*b^2*c + 3*b^2 + 2*b*c^3 - 7*b*c^2 + 8*b*c - 3*b - c^3 + 3*c^2 - 3*c + 1)/(a^3*b^2*c^2 + 2*a^2*b^3*c^2 + a^2*b^2*c^3 - 2*a^2*b^2*c^2 + a*b^4*c^2 + a*b^3*c^3 - 2*a*b^3*c^2 - a*b^2*c^3 + a*b^2*c^2) (-2*a^2*b*c + a^2 + 2*a*b^2*c^2 - 2*a*b^2*c - 2*a*b*c^2 + a*b*c + 2*a*b + 2*a*c - 2*a - b^2*c + b^2 - b*c^2 + 3*b*c - 2*b + c^2 - 2*c + 1)/(a^2*b*c^2 + a*b^2*c^2 + a*b*c^3 - a*b*c^2) (-a^4*b^2*c^2 - 2*a^3*b^3*c^2 + a^3*b^2*c^3 + 7*a^3*b^2*c^2 + 3*a^3*b^2*c - 3*a^3*b*c^2 - 5*a^3*b*c - a^3*b + a^3*c + a^3 - a^2*b^4*c^2 - a^2*b^3*c^3 + 7*a^2*b^3*c^2 + 6*a^2*b^3*c + 4*a^2*b^2*c^3 - 5*a^2*b^2*c^2 - 16*a^2*b^2*c - 3*a^2*b^2 - 3*a^2*b*c^3 - 4*a^2*b*c^2 + 10*a^2*b*c + 6*a^2*b + 2*a^2*c^2 - 3*a^2 + 3*a*b^4*c + 4*a*b^3*c^2 - 11*a*b^3*c - 3*a*b^3 + a*b^2*c^3 - 11*a*b^2*c^2 + 10*a*b^2*c + 9*a*b^2 - 2*a*b*c^3 + 8*a*b*c^2 + a*b*c - 9*a*b + a*c^3 - a*c^2 - 3*a*c + 3*a - b^4 - 2*b^3*c + 4*b^3 - b^2*c^2 + 6*b^2*c - 6*b^2 + 2*b*c^2 - 6*b*c + 4*b - c^2 + 2*c - 1)/(a^4*b^2*c^2 + 2*a^3*b^3*c^2 + a^3*b^2*c^3 - 2*a^3*b^2*c^2 + a^2*b^4*c^2 + a^2*b^3*c^3 - 2*a^2*b^3*c^2 - a^2*b^2*c^3 + a^2*b^2*c^2) (2*a^3*b*c^2 + 3*a^3*b*c - a^3*c - a^3 - 2*a^2*b^2*c^3 - a^2*b^2*c^2 + 6*a^2*b^2*c + 2*a^2*b*c^3 + 3*a^2*b*c^2 - 7*a^2*b*c - 3*a^2*b - 2*a^2*c^2 + 3*a^2 - 3*a*b^3*c^2 + 3*a*b^3*c - a*b^2*c^3 + 8*a*b^2*c^2 - 5*a*b^2*c - 3*a*b^2 + 2*a*b*c^3 - 6*a*b*c^2 - a*b*c + 6*a*b - a*c^3 + a*c^2 + 3*a*c - 3*a + b^3*c - b^3 + b^2*c^2 - 4*b^2*c + 3*b^2 - 2*b*c^2 + 5*b*c - 3*b + c^2 - 2*c + 1)/(a^3*b^2*c^2 + 2*a^2*b^3*c^2 + a^2*b^2*c^3 - 2*a^2*b^2*c^2 + a*b^4*c^2 + a*b^3*c^3 - 2*a*b^3*c^2 - a*b^2*c^3 + a*b^2*c^2) (-2*a^2*b*c^2 - 2*a^2*b*c + a^2*c + a^2 - 2*a*b^2*c + 3*a*b*c + 2*a*b + a*c^2 - 2*a + b^2 + b*c - 2*b - c + 1)/(a^2*b*c^2 + a*b^2*c^2 + a*b*c^3 - a*b*c^2) (2*a^2*b^2*c^2 + 2*a^2*b^2*c - 3*a^2*b*c - a^2*b + a^2 + 2*a*b^3*c + 2*a*b^2*c^2 - 5*a*b^2*c - 2*a*b^2 - 3*a*b*c^2 + a*b*c + 4*a*b + 2*a*c - 2*a - b^3 - 2*b^2*c + 3*b^2 - b*c^2 + 4*b*c - 3*b + c^2 - 2*c + 1)/(a^3*b^2*c + a^2*b^3*c + a^2*b^2*c^2 - a^2*b^2*c) (2*a^2*b*c - a^2 - 2*a*b^2*c^2 + 2*a*b^2*c + 2*a*b*c^2 - a*b*c - 2*a*b - 2*a*c + 2*a + b^2*c - b^2 + b*c^2 - 3*b*c + 2*b - c^2 + 2*c - 1)/(a^2*b^2*c + a*b^3*c + a*b^2*c^2 - a*b^2*c) (-a^2*b*c + a^2 - a*b^2*c + a*b*c^2 + a*b*c + 2*a*b + a*c - 2*a + b^2 + b*c - 2*b - c + 1)/(a^2*b*c + a*b^2*c + a*b*c^2 - a*b*c) Total time: 0.370 seconds, Total memory usage: 3.63MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Thu Dec 8 16:55:47 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); A1:=Matrix(K,3,3,[[b,1,1],[0,a,1],[0,0,c]]); S:=Transpose(A)*A^(-1); F,T:=PrimaryRationalForm(S); B:=T*A*Transpose(T); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*B*Transpose(P)-B)); G:=GroebnerBasis(I); Q1:=Matrix(K,3,3,[(2*b-1)/(a+b-1),(a-b)/(a+b-1),0,(b-a)/(a+b-1),(2*a-1)/(a+b-1),0, 0,0,1]); Kt:=FunctionField(K); D:=(2*a*b*c-a-b-c+1)/(a*b*c); s:=t; px:=-(2*s+D); py:=1-s^2; q:=s^2+s*D+1; Q:=Matrix(Kt,3,3,[1,0,0,0,D*px/q+py/q,-px/q,0,px/q,py/q]); Q:=Q1*T^(-1)*Q*T; print (Q*A*Transpose(Q)-A1); for i in [1..3] do for j in [1..3] do print Denominator(Q[i,j])*(a+b+c-1)*(a+b-1); end for; end for; Output: Magma V2.11-10 Thu Dec 8 2005 16:55:47 on modular [Seed = 3307089162] ------------------------------------- [0 0 0] [0 0 0] [0 0 0] (a^2 + 2*a*b + a*c - 2*a + b^2 + b*c - 2*b - c + 1)*$.1^2 + (2*a^3*b*c - a^3 + 4*a^2*b^2*c + 2*a^2*b*c^2 - 4*a^2*b*c - 3*a^2*b - 2*a^2*c + 3*a^2 + 2*a*b^3*c + 2*a*b^2*c^2 - 4*a*b^2*c - 3*a*b^2 - 2*a*b*c^2 - 2*a*b*c + 6*a*b - a*c^2 + 4*a*c - 3*a - b^3 - 2*b^2*c + 3*b^2 - b*c^2 + 4*b*c - 3*b + c^2 - 2*c + 1)/(a*b*c)*$.1 + a^2 + 2*a*b + a*c - 2*a + b^2 + b*c - 2*b - c + 1 (a^2 + 2*a*b + a*c - 2*a + b^2 + b*c - 2*b - c + 1)*$.1^2 + (2*a^3*b*c - a^3 + 4*a^2*b^2*c + 2*a^2*b*c^2 - 4*a^2*b*c - 3*a^2*b - 2*a^2*c + 3*a^2 + 2*a*b^3*c + 2*a*b^2*c^2 - 4*a*b^2*c - 3*a*b^2 - 2*a*b*c^2 - 2*a*b*c + 6*a*b - a*c^2 + 4*a*c - 3*a - b^3 - 2*b^2*c + 3*b^2 - b*c^2 + 4*b*c - 3*b + c^2 - 2*c + 1)/(a*b*c)*$.1 + a^2 + 2*a*b + a*c - 2*a + b^2 + b*c - 2*b - c + 1 (a^2 + 2*a*b + a*c - 2*a + b^2 + b*c - 2*b - c + 1)*$.1^2 + (2*a^3*b*c - a^3 + 4*a^2*b^2*c + 2*a^2*b*c^2 - 4*a^2*b*c - 3*a^2*b - 2*a^2*c + 3*a^2 + 2*a*b^3*c + 2*a*b^2*c^2 - 4*a*b^2*c - 3*a*b^2 - 2*a*b*c^2 - 2*a*b*c + 6*a*b - a*c^2 + 4*a*c - 3*a - b^3 - 2*b^2*c + 3*b^2 - b*c^2 + 4*b*c - 3*b + c^2 - 2*c + 1)/(a*b*c)*$.1 + a^2 + 2*a*b + a*c - 2*a + b^2 + b*c - 2*b - c + 1 (a^2 + 2*a*b + a*c - 2*a + b^2 + b*c - 2*b - c + 1)*$.1^2 + (2*a^3*b*c - a^3 + 4*a^2*b^2*c + 2*a^2*b*c^2 - 4*a^2*b*c - 3*a^2*b - 2*a^2*c + 3*a^2 + 2*a*b^3*c + 2*a*b^2*c^2 - 4*a*b^2*c - 3*a*b^2 - 2*a*b*c^2 - 2*a*b*c + 6*a*b - a*c^2 + 4*a*c - 3*a - b^3 - 2*b^2*c + 3*b^2 - b*c^2 + 4*b*c - 3*b + c^2 - 2*c + 1)/(a*b*c)*$.1 + a^2 + 2*a*b + a*c - 2*a + b^2 + b*c - 2*b - c + 1 (a^2 + 2*a*b + a*c - 2*a + b^2 + b*c - 2*b - c + 1)*$.1^2 + (2*a^3*b*c - a^3 + 4*a^2*b^2*c + 2*a^2*b*c^2 - 4*a^2*b*c - 3*a^2*b - 2*a^2*c + 3*a^2 + 2*a*b^3*c + 2*a*b^2*c^2 - 4*a*b^2*c - 3*a*b^2 - 2*a*b*c^2 - 2*a*b*c + 6*a*b - a*c^2 + 4*a*c - 3*a - b^3 - 2*b^2*c + 3*b^2 - b*c^2 + 4*b*c - 3*b + c^2 - 2*c + 1)/(a*b*c)*$.1 + a^2 + 2*a*b + a*c - 2*a + b^2 + b*c - 2*b - c + 1 (a^2 + 2*a*b + a*c - 2*a + b^2 + b*c - 2*b - c + 1)*$.1^2 + (2*a^3*b*c - a^3 + 4*a^2*b^2*c + 2*a^2*b*c^2 - 4*a^2*b*c - 3*a^2*b - 2*a^2*c + 3*a^2 + 2*a*b^3*c + 2*a*b^2*c^2 - 4*a*b^2*c - 3*a*b^2 - 2*a*b*c^2 - 2*a*b*c + 6*a*b - a*c^2 + 4*a*c - 3*a - b^3 - 2*b^2*c + 3*b^2 - b*c^2 + 4*b*c - 3*b + c^2 - 2*c + 1)/(a*b*c)*$.1 + a^2 + 2*a*b + a*c - 2*a + b^2 + b*c - 2*b - c + 1 (a^2 + 2*a*b + a*c - 2*a + b^2 + b*c - 2*b - c + 1)*$.1^2 + (2*a^3*b*c - a^3 + 4*a^2*b^2*c + 2*a^2*b*c^2 - 4*a^2*b*c - 3*a^2*b - 2*a^2*c + 3*a^2 + 2*a*b^3*c + 2*a*b^2*c^2 - 4*a*b^2*c - 3*a*b^2 - 2*a*b*c^2 - 2*a*b*c + 6*a*b - a*c^2 + 4*a*c - 3*a - b^3 - 2*b^2*c + 3*b^2 - b*c^2 + 4*b*c - 3*b + c^2 - 2*c + 1)/(a*b*c)*$.1 + a^2 + 2*a*b + a*c - 2*a + b^2 + b*c - 2*b - c + 1 (a^2 + 2*a*b + a*c - 2*a + b^2 + b*c - 2*b - c + 1)*$.1^2 + (2*a^3*b*c - a^3 + 4*a^2*b^2*c + 2*a^2*b*c^2 - 4*a^2*b*c - 3*a^2*b - 2*a^2*c + 3*a^2 + 2*a*b^3*c + 2*a*b^2*c^2 - 4*a*b^2*c - 3*a*b^2 - 2*a*b*c^2 - 2*a*b*c + 6*a*b - a*c^2 + 4*a*c - 3*a - b^3 - 2*b^2*c + 3*b^2 - b*c^2 + 4*b*c - 3*b + c^2 - 2*c + 1)/(a*b*c)*$.1 + a^2 + 2*a*b + a*c - 2*a + b^2 + b*c - 2*b - c + 1 (a^2 + 2*a*b + a*c - 2*a + b^2 + b*c - 2*b - c + 1)*$.1^2 + (2*a^3*b*c - a^3 + 4*a^2*b^2*c + 2*a^2*b*c^2 - 4*a^2*b*c - 3*a^2*b - 2*a^2*c + 3*a^2 + 2*a*b^3*c + 2*a*b^2*c^2 - 4*a*b^2*c - 3*a*b^2 - 2*a*b*c^2 - 2*a*b*c + 6*a*b - a*c^2 + 4*a*c - 3*a - b^3 - 2*b^2*c + 3*b^2 - b*c^2 + 4*b*c - 3*b + c^2 - 2*c + 1)/(a*b*c)*$.1 + a^2 + 2*a*b + a*c - 2*a + b^2 + b*c - 2*b - c + 1 Total time: 0.370 seconds, Total memory usage: 3.63MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Thu Dec 8 16:52:11 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); A1:=Matrix(K,3,3,[[b,1,1],[0,a,1],[0,0,c]]); S:=Transpose(A)*A^(-1); F,T:=PrimaryRationalForm(S); B:=T*A*Transpose(T); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*B*Transpose(P)-B)); G:=GroebnerBasis(I); Q1:=Matrix(K,3,3,[(2*b-1)/(a+b-1),(a-b)/(a+b-1),0,(b-a)/(a+b-1),(2*a-1)/(a+b-1),0, 0,0,1]); Kt:=FunctionField(K); D:=(2*a*b*c-a-b-c+1)/(a*b*c); s:=t; px:=-(2*s+D); py:=1-s^2; q:=s^2+s*D+1; Q:=Matrix(Kt,3,3,[1,0,0,0,D*px/q+py/q,-px/q,0,px/q,py/q]); Q:=Q1*T^(-1)*Q*T; print (Q*A*Transpose(Q)-A1); for i in [1..1] do for j in [1..3] do print Numerator(Q[i,j])*(a+b+c-1)*(a+b-1); end for; end for; Output: Magma V2.11-10 Thu Dec 8 2005 16:52:11 on modular [Seed = 3424068843] ------------------------------------- [0 0 0] [0 0 0] [0 0 0] (a - 2*b*c + b + c - 1)*$.1^2 + (4*a^2*b*c + a^2*b - 2*a^2 - 4*a*b^2*c^2 + 2*a*b^2*c + 2*a*b^2 + 4*a*b*c^2 + a*b*c - 6*a*b - 4*a*c + 4*a - 2*b^3*c + b^3 - 2*b^2*c^2 + 7*b^2*c - 4*b^2 + 4*b*c^2 - 9*b*c + 5*b - 2*c^2 + 4*c - 2)/(a*b*c)*$.1 + (3*a^3*b^2*c^2 + 3*a^3*b^2*c - 4*a^3*b*c - a^3*b + a^3 - 2*a^2*b^3*c^3 + a^2*b^3*c^2 + 6*a^2*b^3*c + 3*a^2*b^2*c^3 + 6*a^2*b^2*c^2 - 13*a^2*b^2*c - 3*a^2*b^2 - 8*a^2*b*c^2 + 4*a^2*b*c + 6*a^2*b + 3*a^2*c - 3*a^2 - 2*a*b^4*c^2 + 3*a*b^4*c - 2*a*b^3*c^3 + 11*a*b^3*c^2 - 8*a*b^3*c - 3*a*b^3 + 6*a*b^2*c^3 - 15*a*b^2*c^2 + a*b^2*c + 9*a*b^2 - 4*a*b*c^3 + 3*a*b*c^2 + 10*a*b*c - 9*a*b + 3*a*c^2 - 6*a*c + 3*a + b^4*c - b^4 + 2*b^3*c^2 - 6*b^3*c + 4*b^3 + b^2*c^3 - 7*b^2*c^2 + 12*b^2*c - 6*b^2 - 2*b*c^3 + 8*b*c^2 - 10*b*c + 4*b + c^3 - 3*c^2 + 3*c - 1)/(a^2*b^2*c^2) (-a^2 - 2*a*b - a*c + a - b^2 + b*c + b)*$.1^2 + (-2*a^3*b*c + 2*a^3 - 4*a^2*b^2*c - 2*a^2*b*c^2 - 2*a^2*b*c + 5*a^2*b + 4*a^2*c - 4*a^2 - 2*a*b^3*c + 2*a*b^2*c^2 - 2*a*b^2*c + 4*a*b^2 - 4*a*b*c^2 + 9*a*b*c - 6*a*b + 2*a*c^2 - 4*a*c + 2*a + b^3 + b^2*c - 2*b^2 - b*c + b)/(a*b*c)*$.1 + (-a^3*b^2*c^2 + 3*a^3*b*c - a^3 - 2*a^2*b^3*c^2 - a^2*b^2*c^3 - 3*a^2*b^2*c^2 + 6*a^2*b^2*c + 6*a^2*b*c^2 - 4*a^2*b*c - 3*a^2*b - 3*a^2*c + 3*a^2 - a*b^4*c^2 + a*b^3*c^3 - 3*a*b^3*c^2 + 3*a*b^3*c - 4*a*b^2*c^3 + 9*a*b^2*c^2 - 2*a*b^2*c - 3*a*b^2 + 3*a*b*c^3 - 2*a*b*c^2 - 7*a*b*c + 6*a*b - 3*a*c^2 + 6*a*c - 3*a - b^3*c^2 + 2*b^3*c - b^3 - b^2*c^3 + 5*b^2*c^2 - 7*b^2*c + 3*b^2 + 2*b*c^3 - 7*b*c^2 + 8*b*c - 3*b - c^3 + 3*c^2 - 3*c + 1)/(a*b^2*c^2) (2*a*b + 2*b^2 - 2*b)*$.1^2 + (-2*a^3 + 4*a^2*b*c - 4*a^2*b - 2*a^2*c + 3*a^2 + 4*a*b^2*c - 2*a*b^2 - 6*a*b*c + 2*a*b + a*c - b^2 - b*c + 2*b + c - 1)/(a*c)*$.1 + (-2*a^3*b*c + a^3 + 2*a^2*b^2*c^2 - 4*a^2*b^2*c - 2*a^2*b*c^2 + 3*a^2*b*c + 3*a^2*b + 2*a^2*c - 3*a^2 + 2*a*b^3*c^2 - 2*a*b^3*c - 4*a*b^2*c^2 + 2*a*b^2*c + 3*a*b^2 + a*b*c^2 + 4*a*b*c - 6*a*b + a*c^2 - 4*a*c + 3*a - b^3*c + b^3 - b^2*c^2 + 4*b^2*c - 3*b^2 + 2*b*c^2 - 5*b*c + 3*b - c^2 + 2*c - 1)/(a*b*c^2) Total time: 0.360 seconds, Total memory usage: 3.63MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Thu Dec 8 16:51:18 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); A1:=Matrix(K,3,3,[[b,1,1],[0,a,1],[0,0,c]]); S:=Transpose(A)*A^(-1); F,T:=PrimaryRationalForm(S); B:=T*A*Transpose(T); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*B*Transpose(P)-B)); G:=GroebnerBasis(I); Q1:=Matrix(K,3,3,[(2*b-1)/(a+b-1),(a-b)/(a+b-1),0,(b-a)/(a+b-1),(2*a-1)/(a+b-1),0, 0,0,1]); Kt:=FunctionField(K); D:=(2*a*b*c-a-b-c+1)/(a*b*c); s:=t; px:=-(2*s+D); py:=1-s^2; q:=s^2+s*D+1; Q:=Matrix(Kt,3,3,[1,0,0,0,D*px/q+py/q,-px/q,0,px/q,py/q]); Q:=Q1*T^(-1)*Q*T; print (Q*A*Transpose(Q)-A1); for i in [1..1] do for j in [1..3] do print Evaluate(Numerator(Q[i,j]),(a+b+c-1)*(a+b-1)); end for; end for; Output: Magma V2.11-10 Thu Dec 8 2005 16:51:17 on modular [Seed = 4146863630] ------------------------------------- [0 0 0] [0 0 0] [0 0 0] (a^7*b^2*c^2 - 2*a^6*b^3*c^3 + 5*a^6*b^3*c^2 + 3*a^6*b^2*c^3 - 5*a^6*b^2*c^2 - 8*a^5*b^4*c^3 + 10*a^5*b^4*c^2 - 4*a^5*b^3*c^4 + 20*a^5*b^3*c^3 - 20*a^5*b^3*c^2 + 3*a^5*b^2*c^4 - 12*a^5*b^2*c^3 + 14*a^5*b^2*c^2 + a^5*b^2*c - 2*a^5*b*c - 12*a^4*b^5*c^3 + 10*a^4*b^5*c^2 - 12*a^4*b^4*c^4 + 42*a^4*b^4*c^3 - 30*a^4*b^4*c^2 - 2*a^4*b^3*c^5 + 21*a^4*b^3*c^4 - 52*a^4*b^3*c^3 + 40*a^4*b^3*c^2 + 4*a^4*b^3*c + a^4*b^2*c^5 - 9*a^4*b^2*c^4 + 26*a^4*b^2*c^3 - 16*a^4*b^2*c^2 - 12*a^4*b^2*c - 6*a^4*b*c^2 + 8*a^4*b*c - 8*a^3*b^6*c^3 + 5*a^3*b^6*c^2 - 12*a^3*b^5*c^4 + 36*a^3*b^5*c^3 - 20*a^3*b^5*c^2 - 4*a^3*b^4*c^5 + 33*a^3*b^4*c^4 - 68*a^3*b^4*c^3 + 36*a^3*b^4*c^2 + 6*a^3*b^4*c + 6*a^3*b^3*c^5 - 34*a^3*b^3*c^4 + 64*a^3*b^3*c^3 - 20*a^3*b^3*c^2 - 24*a^3*b^3*c - 2*a^3*b^2*c^5 + 13*a^3*b^2*c^4 - 19*a^3*b^2*c^3 - 16*a^3*b^2*c^2 + 33*a^3*b^2*c - 6*a^3*b*c^3 + 18*a^3*b*c^2 - 16*a^3*b*c - a^3*b + a^3 - 2*a^2*b^7*c^3 + a^2*b^7*c^2 - 4*a^2*b^6*c^4 + 11*a^2*b^6*c^3 - 5*a^2*b^6*c^2 - 2*a^2*b^5*c^5 + 15*a^2*b^5*c^4 - 28*a^2*b^5*c^3 + 8*a^2*b^5*c^2 + 4*a^2*b^5*c + 5*a^2*b^4*c^5 - 25*a^2*b^4*c^4 + 34*a^2*b^4*c^3 + 8*a^2*b^4*c^2 - 20*a^2*b^4*c - 4*a^2*b^3*c^5 + 19*a^2*b^3*c^4 - 10*a^2*b^3*c^3 - 42*a^2*b^3*c^2 + 42*a^2*b^3*c + a^2*b^2*c^5 - 3*a^2*b^2*c^4 - 16*a^2*b^2*c^3 + 55*a^2*b^2*c^2 - 41*a^2*b^2*c - 3*a^2*b^2 - 2*a^2*b*c^4 + 12*a^2*b*c^3 - 26*a^2*b*c^2 + 12*a^2*b*c + 6*a^2*b + 3*a^2*c - 3*a^2 - 2*a*b^6*c^2 + a*b^6*c - 4*a*b^5*c^3 + 12*a*b^5*c^2 - 6*a*b^5*c - 2*a*b^4*c^4 + 17*a*b^4*c^3 - 32*a*b^4*c^2 + 17*a*b^4*c + 6*a*b^3*c^4 - 30*a*b^3*c^3 + 49*a*b^3*c^2 - 24*a*b^3*c - 3*a*b^3 - 6*a*b^2*c^4 + 27*a*b^2*c^3 - 39*a*b^2*c^2 + 10*a*b^2*c + 9*a*b^2 + 2*a*b*c^4 - 10*a*b*c^3 + 9*a*b*c^2 + 8*a*b*c - 9*a*b + 3*a*c^2 - 6*a*c + 3*a + b^4*c - b^4 + 2*b^3*c^2 - 6*b^3*c + 4*b^3 + b^2*c^3 - 7*b^2*c^2 + 12*b^2*c - 6*b^2 - 2*b*c^3 + 8*b*c^2 - 10*b*c + 4*b + c^3 - 3*c^2 + 3*c - 1)/(a^4*b^2*c^2 + 2*a^3*b^3*c^2 + a^3*b^2*c^3 - 2*a^3*b^2*c^2 + a^2*b^4*c^2 + a^2*b^3*c^3 - 2*a^2*b^3*c^2 - a^2*b^2*c^3 + a^2*b^2*c^2) (-a^7*b^2*c^2 - 6*a^6*b^3*c^2 - 3*a^6*b^2*c^3 + 5*a^6*b^2*c^2 - 15*a^5*b^4*c^2 - 13*a^5*b^3*c^3 + 25*a^5*b^3*c^2 - 3*a^5*b^2*c^4 + 12*a^5*b^2*c^3 - 12*a^5*b^2*c^2 + 2*a^5*b*c - 20*a^4*b^5*c^2 - 22*a^4*b^4*c^3 + 50*a^4*b^4*c^2 - 8*a^4*b^3*c^4 + 40*a^4*b^3*c^3 - 48*a^4*b^3*c^2 - a^4*b^2*c^5 + 9*a^4*b^2*c^4 - 22*a^4*b^2*c^3 + 12*a^4*b^2*c^2 + 9*a^4*b^2*c + 6*a^4*b*c^2 - 8*a^4*b*c - 15*a^3*b^6*c^2 - 18*a^3*b^5*c^3 + 50*a^3*b^5*c^2 - 6*a^3*b^4*c^4 + 48*a^3*b^4*c^3 - 72*a^3*b^4*c^2 - a^3*b^3*c^5 + 15*a^3*b^3*c^4 - 50*a^3*b^3*c^3 + 36*a^3*b^3*c^2 + 16*a^3*b^3*c + 2*a^3*b^2*c^5 - 11*a^3*b^2*c^4 + 12*a^3*b^2*c^3 + 20*a^3*b^2*c^2 - 28*a^3*b^2*c + 6*a^3*b*c^3 - 18*a^3*b*c^2 + 15*a^3*b*c - a^3 - 6*a^2*b^7*c^2 - 7*a^2*b^6*c^3 + 25*a^2*b^6*c^2 + 24*a^2*b^5*c^3 - 48*a^2*b^5*c^2 + a^2*b^4*c^5 + 3*a^2*b^4*c^4 - 34*a^2*b^4*c^3 + 36*a^2*b^4*c^2 + 14*a^2*b^4*c - 6*a^2*b^3*c^4 + 8*a^2*b^3*c^3 + 24*a^2*b^3*c^2 - 36*a^2*b^3*c - a^2*b^2*c^5 + a^2*b^2*c^4 + 21*a^2*b^2*c^3 - 54*a^2*b^2*c^2 + 36*a^2*b^2*c + 2*a^2*b*c^4 - 12*a^2*b*c^3 + 24*a^2*b*c^2 - 12*a^2*b*c - 3*a^2*b - 3*a^2*c + 3*a^2 - a*b^8*c^2 - a*b^7*c^3 + 5*a*b^7*c^2 + a*b^6*c^4 + 4*a*b^6*c^3 - 12*a*b^6*c^2 + a*b^5*c^5 - 3*a*b^5*c^4 - 6*a*b^5*c^3 + 12*a*b^5*c^2 + 6*a*b^5*c - 2*a*b^4*c^5 + 5*a*b^4*c^4 - 4*a*b^4*c^3 + 12*a*b^4*c^2 - 20*a*b^4*c + a*b^3*c^5 - 7*a*b^3*c^4 + 24*a*b^3*c^3 - 42*a*b^3*c^2 + 27*a*b^3*c + 6*a*b^2*c^4 - 26*a*b^2*c^3 + 37*a*b^2*c^2 - 14*a*b^2*c - 3*a*b^2 - 2*a*b*c^4 + 9*a*b*c^3 - 8*a*b*c^2 - 5*a*b*c + 6*a*b - 3*a*c^2 + 6*a*c - 3*a + b^6*c + 2*b^5*c^2 - 4*b^5*c + b^4*c^3 - 6*b^4*c^2 + 6*b^4*c - 2*b^3*c^3 + 5*b^3*c^2 - 2*b^3*c - b^3 + 3*b^2*c^2 - 6*b^2*c + 3*b^2 + 2*b*c^3 - 7*b*c^2 + 8*b*c - 3*b - c^3 + 3*c^2 - 3*c + 1)/(a^3*b^2*c^2 + 2*a^2*b^3*c^2 + a^2*b^2*c^3 - 2*a^2*b^2*c^2 + a*b^4*c^2 + a*b^3*c^3 - 2*a*b^3*c^2 - a*b^2*c^3 + a*b^2*c^2) (2*a^5*b^2*c^2 + 8*a^4*b^3*c^2 + 4*a^4*b^2*c^3 - 8*a^4*b^2*c^2 - 2*a^4*b*c + 12*a^3*b^4*c^2 + 12*a^3*b^3*c^3 - 24*a^3*b^3*c^2 + 2*a^3*b^2*c^4 - 12*a^3*b^2*c^3 + 16*a^3*b^2*c^2 - 6*a^3*b^2*c - 4*a^3*b*c^2 + 5*a^3*b*c + 8*a^2*b^5*c^2 + 12*a^2*b^4*c^3 - 24*a^2*b^4*c^2 + 4*a^2*b^3*c^4 - 24*a^2*b^3*c^3 + 32*a^2*b^3*c^2 - 6*a^2*b^3*c - 4*a^2*b^2*c^4 + 16*a^2*b^2*c^3 - 24*a^2*b^2*c^2 + 9*a^2*b^2*c - 2*a^2*b*c^3 + 6*a^2*b*c^2 - 5*a^2*b*c + a^2 + 2*a*b^6*c^2 + 4*a*b^5*c^3 - 8*a*b^5*c^2 + 2*a*b^4*c^4 - 12*a*b^4*c^3 + 16*a*b^4*c^2 - 2*a*b^4*c - 4*a*b^3*c^4 + 16*a*b^3*c^3 - 20*a*b^3*c^2 + 3*a*b^3*c + 2*a*b^2*c^4 - 10*a*b^2*c^3 + 12*a*b^2*c^2 - 2*a*b^2*c + a*b*c^3 - 2*a*b*c^2 + 2*a*b + 2*a*c - 2*a - b^4*c - 2*b^3*c^2 + 3*b^3*c - b^2*c^3 + 4*b^2*c^2 - 4*b^2*c + b^2 + b*c^3 - 3*b*c^2 + 4*b*c - 2*b + c^2 - 2*c + 1)/(a^2*b*c^2 + a*b^2*c^2 + a*b*c^3 - a*b*c^2) Total time: 0.370 seconds, Total memory usage: 3.63MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Thu Dec 8 16:47:53 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); A1:=Matrix(K,3,3,[[b,1,1],[0,a,1],[0,0,c]]); S:=Transpose(A)*A^(-1); F,T:=PrimaryRationalForm(S); B:=T*A*Transpose(T); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*B*Transpose(P)-B)); G:=GroebnerBasis(I); Q1:=Matrix(K,3,3,[(2*b-1)/(a+b-1),(a-b)/(a+b-1),0,(b-a)/(a+b-1),(2*a-1)/(a+b-1),0, 0,0,1]); Kt:=FunctionField(K); D:=(2*a*b*c-a-b-c+1)/(a*b*c); s:=t; px:=-(2*s+D); py:=1-s^2; q:=s^2+s*D+1; Q:=Matrix(Kt,3,3,[1,0,0,0,D*px/q+py/q,-px/q,0,px/q,py/q]); Q:=Q1*T^(-1)*Q*T; print (Q*A*Transpose(Q)-A1); for i in [1..3] do for j in [1..3] do print Numerator(Q[i,j])*(a+b+c-1)*(a+b-1); end for; end for; Output: Magma V2.11-10 Thu Dec 8 2005 16:47:52 on modular [Seed = 2538372787] ------------------------------------- [0 0 0] [0 0 0] [0 0 0] (a - 2*b*c + b + c - 1)*$.1^2 + (4*a^2*b*c + a^2*b - 2*a^2 - 4*a*b^2*c^2 + 2*a*b^2*c + 2*a*b^2 + 4*a*b*c^2 + a*b*c - 6*a*b - 4*a*c + 4*a - 2*b^3*c + b^3 - 2*b^2*c^2 + 7*b^2*c - 4*b^2 + 4*b*c^2 - 9*b*c + 5*b - 2*c^2 + 4*c - 2)/(a*b*c)*$.1 + (3*a^3*b^2*c^2 + 3*a^3*b^2*c - 4*a^3*b*c - a^3*b + a^3 - 2*a^2*b^3*c^3 + a^2*b^3*c^2 + 6*a^2*b^3*c + 3*a^2*b^2*c^3 + 6*a^2*b^2*c^2 - 13*a^2*b^2*c - 3*a^2*b^2 - 8*a^2*b*c^2 + 4*a^2*b*c + 6*a^2*b + 3*a^2*c - 3*a^2 - 2*a*b^4*c^2 + 3*a*b^4*c - 2*a*b^3*c^3 + 11*a*b^3*c^2 - 8*a*b^3*c - 3*a*b^3 + 6*a*b^2*c^3 - 15*a*b^2*c^2 + a*b^2*c + 9*a*b^2 - 4*a*b*c^3 + 3*a*b*c^2 + 10*a*b*c - 9*a*b + 3*a*c^2 - 6*a*c + 3*a + b^4*c - b^4 + 2*b^3*c^2 - 6*b^3*c + 4*b^3 + b^2*c^3 - 7*b^2*c^2 + 12*b^2*c - 6*b^2 - 2*b*c^3 + 8*b*c^2 - 10*b*c + 4*b + c^3 - 3*c^2 + 3*c - 1)/(a^2*b^2*c^2) (-a^2 - 2*a*b - a*c + a - b^2 + b*c + b)*$.1^2 + (-2*a^3*b*c + 2*a^3 - 4*a^2*b^2*c - 2*a^2*b*c^2 - 2*a^2*b*c + 5*a^2*b + 4*a^2*c - 4*a^2 - 2*a*b^3*c + 2*a*b^2*c^2 - 2*a*b^2*c + 4*a*b^2 - 4*a*b*c^2 + 9*a*b*c - 6*a*b + 2*a*c^2 - 4*a*c + 2*a + b^3 + b^2*c - 2*b^2 - b*c + b)/(a*b*c)*$.1 + (-a^3*b^2*c^2 + 3*a^3*b*c - a^3 - 2*a^2*b^3*c^2 - a^2*b^2*c^3 - 3*a^2*b^2*c^2 + 6*a^2*b^2*c + 6*a^2*b*c^2 - 4*a^2*b*c - 3*a^2*b - 3*a^2*c + 3*a^2 - a*b^4*c^2 + a*b^3*c^3 - 3*a*b^3*c^2 + 3*a*b^3*c - 4*a*b^2*c^3 + 9*a*b^2*c^2 - 2*a*b^2*c - 3*a*b^2 + 3*a*b*c^3 - 2*a*b*c^2 - 7*a*b*c + 6*a*b - 3*a*c^2 + 6*a*c - 3*a - b^3*c^2 + 2*b^3*c - b^3 - b^2*c^3 + 5*b^2*c^2 - 7*b^2*c + 3*b^2 + 2*b*c^3 - 7*b*c^2 + 8*b*c - 3*b - c^3 + 3*c^2 - 3*c + 1)/(a*b^2*c^2) (2*a*b + 2*b^2 - 2*b)*$.1^2 + (-2*a^3 + 4*a^2*b*c - 4*a^2*b - 2*a^2*c + 3*a^2 + 4*a*b^2*c - 2*a*b^2 - 6*a*b*c + 2*a*b + a*c - b^2 - b*c + 2*b + c - 1)/(a*c)*$.1 + (-2*a^3*b*c + a^3 + 2*a^2*b^2*c^2 - 4*a^2*b^2*c - 2*a^2*b*c^2 + 3*a^2*b*c + 3*a^2*b + 2*a^2*c - 3*a^2 + 2*a*b^3*c^2 - 2*a*b^3*c - 4*a*b^2*c^2 + 2*a*b^2*c + 3*a*b^2 + a*b*c^2 + 4*a*b*c - 6*a*b + a*c^2 - 4*a*c + 3*a - b^3*c + b^3 - b^2*c^2 + 4*b^2*c - 3*b^2 + 2*b*c^2 - 5*b*c + 3*b - c^2 + 2*c - 1)/(a*b*c^2) (-a^2 - 2*a*b + a*c + 3*a - b^2 - b*c + 3*b - 2)*$.1^2 + (-2*a^3*b*c + a^3 - 4*a^2*b^2*c + 2*a^2*b*c^2 + 10*a^2*b*c + 4*a^2*b - a^2*c - 5*a^2 - 2*a*b^3*c - 2*a*b^2*c^2 + 10*a*b^2*c + 5*a*b^2 + 4*a*b*c^2 - 7*a*b*c - 12*a*b - 2*a*c^2 - 2*a*c + 7*a + 2*b^3 + 2*b^2*c - 7*b^2 - 5*b*c + 8*b + 3*c - 3)/(a*b*c)*$.1 + (-a^4*b^2*c^2 - 2*a^3*b^3*c^2 + a^3*b^2*c^3 + 7*a^3*b^2*c^2 + 3*a^3*b^2*c - 3*a^3*b*c^2 - 5*a^3*b*c - a^3*b + a^3*c + a^3 - a^2*b^4*c^2 - a^2*b^3*c^3 + 7*a^2*b^3*c^2 + 6*a^2*b^3*c + 4*a^2*b^2*c^3 - 5*a^2*b^2*c^2 - 16*a^2*b^2*c - 3*a^2*b^2 - 3*a^2*b*c^3 - 4*a^2*b*c^2 + 10*a^2*b*c + 6*a^2*b + 2*a^2*c^2 - 3*a^2 + 3*a*b^4*c + 4*a*b^3*c^2 - 11*a*b^3*c - 3*a*b^3 + a*b^2*c^3 - 11*a*b^2*c^2 + 10*a*b^2*c + 9*a*b^2 - 2*a*b*c^3 + 8*a*b*c^2 + a*b*c - 9*a*b + a*c^3 - a*c^2 - 3*a*c + 3*a - b^4 - 2*b^3*c + 4*b^3 - b^2*c^2 + 6*b^2*c - 6*b^2 + 2*b*c^2 - 6*b*c + 4*b - c^2 + 2*c - 1)/(a^2*b^2*c^2) (-2*a*c - a - b + c + 1)*$.1^2 + (2*a^3*c + a^3 - 4*a^2*b*c^2 - 2*a^2*b*c + 2*a^2*b + 2*a^2*c^2 - a^2*c - a^2 - 4*a*b^2*c + a*b^2 + 5*a*b*c - a + b^2 + b*c - 2*b - c + 1)/(a*b*c)*$.1 + (2*a^3*b*c^2 + 3*a^3*b*c - a^3*c - a^3 - 2*a^2*b^2*c^3 - a^2*b^2*c^2 + 6*a^2*b^2*c + 2*a^2*b*c^3 + 3*a^2*b*c^2 - 7*a^2*b*c - 3*a^2*b - 2*a^2*c^2 + 3*a^2 - 3*a*b^3*c^2 + 3*a*b^3*c - a*b^2*c^3 + 8*a*b^2*c^2 - 5*a*b^2*c - 3*a*b^2 + 2*a*b*c^3 - 6*a*b*c^2 - a*b*c + 6*a*b - a*c^3 + a*c^2 + 3*a*c - 3*a + b^3*c - b^3 + b^2*c^2 - 4*b^2*c + 3*b^2 - 2*b*c^2 + 5*b*c - 3*b + c^2 - 2*c + 1)/(a*b^2*c^2) (-2*a^2 - 2*a*b + 4*a + 2*b - 2)*$.1^2 + (-4*a^3*b*c - 2*a^3*b + a^3 - 4*a^2*b^2*c - 4*a^2*b^2 + 6*a^2*b*c + 6*a^2*b + a^2*c - 3*a^2 - 2*a*b^3 + 2*a*b^2*c + 5*a*b^2 - a*b*c - 6*a*b - 2*a*c + 3*a - b^2 - b*c + 2*b + c - 1)/(a*b*c)*$.1 + (-2*a^3*b*c^2 - 2*a^3*b*c + a^3*c + a^3 - 2*a^2*b^2*c^2 - 4*a^2*b^2*c + 2*a^2*b*c^2 + 6*a^2*b*c + 3*a^2*b + a^2*c^2 - a^2*c - 3*a^2 - 2*a*b^3*c + 5*a*b^2*c + 3*a*b^2 + a*b*c^2 - 2*a*b*c - 6*a*b - a*c^2 - a*c + 3*a + b^3 + b^2*c - 3*b^2 - 2*b*c + 3*b + c - 1)/(a*b*c^2) (2*a*c + 2*b*c - 2*c)*$.1^2 + (4*a^2*b*c + 2*a^2*b - 3*a^2 + 4*a*b^2*c + 4*a*b^2 - 2*a*b*c - 10*a*b - 3*a*c + 6*a + 2*b^3 + 2*b^2*c - 7*b^2 - 5*b*c + 8*b + 3*c - 3)/(a*b)*$.1 + (2*a^3*b^2*c^2 + 2*a^3*b^2*c - 3*a^3*b*c - a^3*b + a^3 + 2*a^2*b^3*c^2 + 4*a^2*b^3*c - 10*a^2*b^2*c - 3*a^2*b^2 - 3*a^2*b*c^2 + 4*a^2*b*c + 6*a^2*b + 2*a^2*c - 3*a^2 + 2*a*b^4*c + 2*a*b^3*c^2 - 7*a*b^3*c - 3*a*b^3 - 5*a*b^2*c^2 + 4*a*b^2*c + 9*a*b^2 + 2*a*b*c^2 + 5*a*b*c - 9*a*b + a*c^2 - 4*a*c + 3*a - b^4 - 2*b^3*c + 4*b^3 - b^2*c^2 + 6*b^2*c - 6*b^2 + 2*b*c^2 - 6*b*c + 4*b - c^2 + 2*c - 1)/(a^2*b^2*c) (-2*a*c - 2*b*c + 2*c)*$.1^2 + (2*a^3 - 4*a^2*b*c + 4*a^2*b + 2*a^2*c - 3*a^2 - 4*a*b^2*c + 2*a*b^2 + 6*a*b*c - 2*a*b - a*c + b^2 + b*c - 2*b - c + 1)/(a*b)*$.1 + (2*a^3*b*c - a^3 - 2*a^2*b^2*c^2 + 4*a^2*b^2*c + 2*a^2*b*c^2 - 3*a^2*b*c - 3*a^2*b - 2*a^2*c + 3*a^2 - 2*a*b^3*c^2 + 2*a*b^3*c + 4*a*b^2*c^2 - 2*a*b^2*c - 3*a*b^2 - a*b*c^2 - 4*a*b*c + 6*a*b - a*c^2 + 4*a*c - 3*a + b^3*c - b^3 + b^2*c^2 - 4*b^2*c + 3*b^2 - 2*b*c^2 + 5*b*c - 3*b + c^2 - 2*c + 1)/(a*b^2*c) (-a^2 - 2*a*b + a*c + 2*a - b^2 + b*c + 2*b - c - 1)*$.1^2 + (-2*a^3*b - 4*a^2*b^2 + 2*a^2*b*c + 4*a^2*b - a^2 - 2*a*b^3 + 2*a*b^2*c + 4*a*b^2 - 2*a*b*c - 4*a*b - a*c + 2*a - b^2 - b*c + 2*b + c - 1)/(a*b)*$.1 + (-a^3*b*c + a^3 - 2*a^2*b^2*c + a^2*b*c^2 + 2*a^2*b*c + 3*a^2*b + a^2*c - 3*a^2 - a*b^3*c + a*b^2*c^2 + 2*a*b^2*c + 3*a*b^2 - a*b*c^2 + a*b*c - 6*a*b - 2*a*c + 3*a + b^3 + b^2*c - 3*b^2 - 2*b*c + 3*b + c - 1)/(a*b*c) Total time: 0.370 seconds, Total memory usage: 3.63MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Thu Dec 8 16:46:52 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); A1:=Matrix(K,3,3,[[b,1,1],[0,a,1],[0,0,c]]); S:=Transpose(A)*A^(-1); F,T:=PrimaryRationalForm(S); B:=T*A*Transpose(T); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*B*Transpose(P)-B)); G:=GroebnerBasis(I); Q1:=Matrix(K,3,3,[(2*b-1)/(a+b-1),(a-b)/(a+b-1),0,(b-a)/(a+b-1),(2*a-1)/(a+b-1),0, 0,0,1]); Kt:=FunctionField(K); D:=(2*a*b*c-a-b-c+1)/(a*b*c); s:=t; px:=-(2*s+D); py:=1-s^2; q:=s^2+s*D+1; Q:=Matrix(Kt,3,3,[1,0,0,0,D*px/q+py/q,-px/q,0,px/q,py/q]); Q:=Q1*T^(-1)*Q*T; print (Q*A*Transpose(Q)-A1); for i in [1..3] do for j in [1..3] do print Numerator(Q[i,j])*(a+b+c-1)^2*(a+b-1); end for; end for; Output: Magma V2.11-10 Thu Dec 8 2005 16:46:52 on modular [Seed = 2587979898] ------------------------------------- [0 0 0] [0 0 0] [0 0 0] (a^2 - 2*a*b*c + 2*a*b + 2*a*c - 2*a - 2*b^2*c + b^2 - 2*b*c^2 + 4*b*c - 2*b + c^2 - 2*c + 1)*$.1^2 + (4*a^3*b*c + a^3*b - 2*a^3 - 4*a^2*b^2*c^2 + 6*a^2*b^2*c + 3*a^2*b^2 + 8*a^2*b*c^2 - 2*a^2*b*c - 9*a^2*b - 6*a^2*c + 6*a^2 - 4*a*b^3*c^2 + 3*a*b^3 - 4*a*b^2*c^3 + 8*a*b^2*c^2 + 8*a*b^2*c - 12*a*b^2 + 4*a*b*c^3 + a*b*c^2 - 20*a*b*c + 15*a*b - 6*a*c^2 + 12*a*c - 6*a - 2*b^4*c + b^4 - 4*b^3*c^2 + 10*b^3*c - 5*b^3 - 2*b^2*c^3 + 13*b^2*c^2 - 20*b^2*c + 9*b^2 + 4*b*c^3 - 15*b*c^2 + 18*b*c - 7*b - 2*c^3 + 6*c^2 - 6*c + 2)/(a*b*c)*$.1 + (3*a^4*b^2*c^2 + 3*a^4*b^2*c - 4*a^4*b*c - a^4*b + a^4 - 2*a^3*b^3*c^3 + 4*a^3*b^3*c^2 + 9*a^3*b^3*c + 6*a^3*b^2*c^3 + 6*a^3*b^2*c^2 - 20*a^3*b^2*c - 4*a^3*b^2 - 12*a^3*b*c^2 + 7*a^3*b*c + 8*a^3*b + 4*a^3*c - 4*a^3 - 2*a^2*b^4*c^3 - a^2*b^4*c^2 + 9*a^2*b^4*c - 2*a^2*b^3*c^4 + 4*a^2*b^3*c^3 + 22*a^2*b^3*c^2 - 27*a^2*b^3*c - 6*a^2*b^3 + 3*a^2*b^2*c^4 + 9*a^2*b^2*c^3 - 42*a^2*b^2*c^2 + 15*a^2*b^2*c + 18*a^2*b^2 - 12*a^2*b*c^3 + 15*a^2*b*c^2 + 15*a^2*b*c - 18*a^2*b + 6*a^2*c^2 - 12*a^2*c + 6*a^2 - 2*a*b^5*c^2 + 3*a*b^5*c - 4*a*b^4*c^3 + 16*a*b^4*c^2 - 10*a*b^4*c - 4*a*b^4 - 2*a*b^3*c^4 + 19*a*b^3*c^3 - 32*a*b^3*c^2 + 16*a*b^3 + 6*a*b^2*c^4 - 24*a*b^2*c^3 + 12*a*b^2*c^2 + 30*a*b^2*c - 24*a*b^2 - 4*a*b*c^4 + 5*a*b*c^3 + 18*a*b*c^2 - 35*a*b*c + 16*a*b + 4*a*c^3 - 12*a*c^2 + 12*a*c - 4*a + b^5*c - b^5 + 3*b^4*c^2 - 8*b^4*c + 5*b^4 + 3*b^3*c^3 - 15*b^3*c^2 + 22*b^3*c - 10*b^3 + b^2*c^4 - 10*b^2*c^3 + 27*b^2*c^2 - 28*b^2*c + 10*b^2 - 2*b*c^4 + 11*b*c^3 - 21*b*c^2 + 17*b*c - 5*b + c^4 - 4*c^3 + 6*c^2 - 4*c + 1)/(a^2*b^2*c^2) (-a^3 - 3*a^2*b - 2*a^2*c + 2*a^2 - 3*a*b^2 - 2*a*b*c + 4*a*b - a*c^2 + 2*a*c - a - b^3 + 2*b^2 + b*c^2 - b)*$.1^2 + (-2*a^4*b*c + 2*a^4 - 6*a^3*b^2*c - 4*a^3*b*c^2 + 7*a^3*b + 6*a^3*c - 6*a^3 - 6*a^2*b^3*c - 4*a^2*b^2*c^2 + 9*a^2*b^2 - 2*a^2*b*c^3 - 4*a^2*b*c^2 + 20*a^2*b*c - 15*a^2*b + 6*a^2*c^2 - 12*a^2*c + 6*a^2 - 2*a*b^4*c + 5*a*b^3 + 2*a*b^2*c^3 - 8*a*b^2*c^2 + 16*a*b^2*c - 12*a*b^2 - 4*a*b*c^3 + 15*a*b*c^2 - 20*a*b*c + 9*a*b + 2*a*c^3 - 6*a*c^2 + 6*a*c - 2*a + b^4 + 2*b^3*c - 3*b^3 + b^2*c^2 - 4*b^2*c + 3*b^2 - b*c^2 + 2*b*c - b)/(a*b*c)*$.1 + (-a^4*b^2*c^2 + 3*a^4*b*c - a^4 - 3*a^3*b^3*c^2 - 2*a^3*b^2*c^3 - 2*a^3*b^2*c^2 + 9*a^3*b^2*c + 9*a^3*b*c^2 - 7*a^3*b*c - 4*a^3*b - 4*a^3*c + 4*a^3 - 3*a^2*b^4*c^2 - 2*a^2*b^3*c^3 - 4*a^2*b^3*c^2 + 9*a^2*b^3*c - a^2*b^2*c^4 - 6*a^2*b^2*c^3 + 24*a^2*b^2*c^2 - 12*a^2*b^2*c - 6*a^2*b^2 + 9*a^2*b*c^3 - 12*a^2*b*c^2 - 9*a^2*b*c + 12*a^2*b - 6*a^2*c^2 + 12*a^2*c - 6*a^2 - a*b^5*c^2 - 2*a*b^4*c^2 + 3*a*b^4*c + a*b^3*c^4 - 8*a*b^3*c^3 + 14*a*b^3*c^2 - 3*a*b^3*c - 4*a*b^3 - 4*a*b^2*c^4 + 15*a*b^2*c^3 - 8*a*b^2*c^2 - 15*a*b^2*c + 12*a*b^2 + 3*a*b*c^4 - 3*a*b*c^3 - 15*a*b*c^2 + 27*a*b*c - 12*a*b - 4*a*c^3 + 12*a*c^2 - 12*a*c + 4*a - b^4*c^2 + 2*b^4*c - b^4 - 2*b^3*c^3 + 8*b^3*c^2 - 10*b^3*c + 4*b^3 - b^2*c^4 + 8*b^2*c^3 - 19*b^2*c^2 + 18*b^2*c - 6*b^2 + 2*b*c^4 - 10*b*c^3 + 18*b*c^2 - 14*b*c + 4*b - c^4 + 4*c^3 - 6*c^2 + 4*c - 1)/(a*b^2*c^2) (2*a^2*b + 4*a*b^2 + 2*a*b*c - 4*a*b + 2*b^3 + 2*b^2*c - 4*b^2 - 2*b*c + 2*b)*$.1^2 + (-2*a^4 + 4*a^3*b*c - 6*a^3*b - 4*a^3*c + 5*a^3 + 8*a^2*b^2*c - 6*a^2*b^2 + 4*a^2*b*c^2 - 16*a^2*b*c + 9*a^2*b - 2*a^2*c^2 + 6*a^2*c - 3*a^2 + 4*a*b^3*c - 2*a*b^3 + 4*a*b^2*c^2 - 12*a*b^2*c + 3*a*b^2 - 6*a*b*c^2 + 8*a*b*c + a*c^2 - a - b^3 - 2*b^2*c + 3*b^2 - b*c^2 + 4*b*c - 3*b + c^2 - 2*c + 1)/(a*c)*$.1 + (-2*a^4*b*c + a^4 + 2*a^3*b^2*c^2 - 6*a^3*b^2*c - 4*a^3*b*c^2 + 5*a^3*b*c + 4*a^3*b + 3*a^3*c - 4*a^3 + 4*a^2*b^3*c^2 - 6*a^2*b^3*c + 2*a^2*b^2*c^3 - 12*a^2*b^2*c^2 + 9*a^2*b^2*c + 6*a^2*b^2 - 2*a^2*b*c^3 + 6*a^2*b*c^2 + 6*a^2*b*c - 12*a^2*b + 3*a^2*c^2 - 9*a^2*c + 6*a^2 + 2*a*b^4*c^2 - 2*a*b^4*c + 2*a*b^3*c^3 - 8*a*b^3*c^2 + 3*a*b^3*c + 4*a*b^3 - 4*a*b^2*c^3 + 6*a*b^2*c^2 + 9*a*b^2*c - 12*a*b^2 + a*b*c^3 + 6*a*b*c^2 - 19*a*b*c + 12*a*b + a*c^3 - 6*a*c^2 + 9*a*c - 4*a - b^4*c + b^4 - 2*b^3*c^2 + 6*b^3*c - 4*b^3 - b^2*c^3 + 7*b^2*c^2 - 12*b^2*c + 6*b^2 + 2*b*c^3 - 8*b*c^2 + 10*b*c - 4*b - c^3 + 3*c^2 - 3*c + 1)/(a*b*c^2) (-a^3 - 3*a^2*b + 4*a^2 - 3*a*b^2 - 2*a*b*c + 8*a*b + a*c^2 + 2*a*c - 5*a - b^3 - 2*b^2*c + 4*b^2 - b*c^2 + 4*b*c - 5*b - 2*c + 2)*$.1^2 + (-2*a^4*b*c + a^4 - 6*a^3*b^2*c + 12*a^3*b*c + 5*a^3*b - 6*a^3 - 6*a^2*b^3*c - 4*a^2*b^2*c^2 + 24*a^2*b^2*c + 9*a^2*b^2 + 2*a^2*b*c^3 + 12*a^2*b*c^2 - 14*a^2*b*c - 21*a^2*b - 3*a^2*c^2 - 6*a^2*c + 12*a^2 - 2*a*b^4*c - 4*a*b^3*c^2 + 12*a*b^3*c + 7*a*b^3 - 2*a*b^2*c^3 + 16*a*b^2*c^2 - 10*a*b^2*c - 24*a*b^2 + 4*a*b*c^3 - 13*a*b*c^2 - 12*a*b*c + 27*a*b - 2*a*c^3 + 12*a*c - 10*a + 2*b^4 + 4*b^3*c - 9*b^3 + 2*b^2*c^2 - 14*b^2*c + 15*b^2 - 5*b*c^2 + 16*b*c - 11*b + 3*c^2 - 6*c + 3)/(a*b*c)*$.1 + (-a^5*b^2*c^2 - 3*a^4*b^3*c^2 + 8*a^4*b^2*c^2 + 3*a^4*b^2*c - 3*a^4*b*c^2 - 5*a^4*b*c - a^4*b + a^4*c + a^4 - 3*a^3*b^4*c^2 - 2*a^3*b^3*c^3 + 16*a^3*b^3*c^2 + 9*a^3*b^3*c + a^3*b^2*c^4 + 10*a^3*b^2*c^3 - 12*a^3*b^2*c^2 - 24*a^3*b^2*c - 4*a^3*b^2 - 6*a^3*b*c^3 - 6*a^3*b*c^2 + 15*a^3*b*c + 8*a^3*b + 3*a^3*c^2 - 4*a^3 - a^2*b^5*c^2 - 2*a^2*b^4*c^3 + 8*a^2*b^4*c^2 + 9*a^2*b^4*c - a^2*b^3*c^4 + 12*a^2*b^3*c^3 - 2*a^2*b^3*c^2 - 33*a^2*b^3*c - 6*a^2*b^3 + 4*a^2*b^2*c^4 - 11*a^2*b^2*c^3 - 26*a^2*b^2*c^2 + 33*a^2*b^2*c + 18*a^2*b^2 - 3*a^2*b*c^4 - 3*a^2*b*c^3 + 24*a^2*b*c^2 - 3*a^2*b*c - 18*a^2*b + 3*a^2*c^3 - 3*a^2*c^2 - 6*a^2*c + 6*a^2 + 3*a*b^5*c + 7*a*b^4*c^2 - 14*a*b^4*c - 4*a*b^4 + 5*a*b^3*c^3 - 26*a*b^3*c^2 + 16*a*b^3*c + 16*a*b^3 + a*b^2*c^4 - 14*a*b^2*c^3 + 28*a*b^2*c^2 + 6*a*b^2*c - 24*a*b^2 - 2*a*b*c^4 + 11*a*b*c^3 - 6*a*b*c^2 - 19*a*b*c + 16*a*b + a*c^4 - 2*a*c^3 - 3*a*c^2 + 8*a*c - 4*a - b^5 - 3*b^4*c + 5*b^4 - 3*b^3*c^2 + 12*b^3*c - 10*b^3 - b^2*c^3 + 9*b^2*c^2 - 18*b^2*c + 10*b^2 + 2*b*c^3 - 9*b*c^2 + 12*b*c - 5*b - c^3 + 3*c^2 - 3*c + 1)/(a^2*b^2*c^2) (-2*a^2*c - a^2 - 2*a*b*c - 2*a*b - 2*a*c^2 + 2*a*c + 2*a - b^2 + 2*b + c^2 - 1)*$.1^2 + (2*a^4*c + a^4 - 4*a^3*b*c^2 + 3*a^3*b + 4*a^3*c^2 - 2*a^3*c - 2*a^3 - 4*a^2*b^2*c^2 - 6*a^2*b^2*c + 3*a^2*b^2 - 4*a^2*b*c^3 + 4*a^2*b*c^2 + 8*a^2*b*c - 3*a^2*b + 2*a^2*c^3 - 3*a^2*c^2 - 4*a*b^3*c + a*b^3 - 4*a*b^2*c^2 + 10*a*b^2*c + 5*a*b*c^2 - 4*a*b*c - 3*a*b - 2*a*c + 2*a + b^3 + 2*b^2*c - 3*b^2 + b*c^2 - 4*b*c + 3*b - c^2 + 2*c - 1)/(a*b*c)*$.1 + (2*a^4*b*c^2 + 3*a^4*b*c - a^4*c - a^4 - 2*a^3*b^2*c^3 + a^3*b^2*c^2 + 9*a^3*b^2*c + 4*a^3*b*c^3 + 4*a^3*b*c^2 - 11*a^3*b*c - 4*a^3*b - 3*a^3*c^2 + 4*a^3 - 2*a^2*b^3*c^3 - 4*a^2*b^3*c^2 + 9*a^2*b^3*c - 2*a^2*b^2*c^4 + 2*a^2*b^2*c^3 + 18*a^2*b^2*c^2 - 18*a^2*b^2*c - 6*a^2*b^2 + 2*a^2*b*c^4 + 3*a^2*b*c^3 - 18*a^2*b*c^2 + 3*a^2*b*c + 12*a^2*b - 3*a^2*c^3 + 3*a^2*c^2 + 6*a^2*c - 6*a^2 - 3*a*b^4*c^2 + 3*a*b^4*c - 4*a*b^3*c^3 + 14*a*b^3*c^2 - 7*a*b^3*c - 4*a*b^3 - a*b^2*c^4 + 11*a*b^2*c^3 - 18*a*b^2*c^2 - 3*a*b^2*c + 12*a*b^2 + 2*a*b*c^4 - 9*a*b*c^3 + 4*a*b*c^2 + 15*a*b*c - 12*a*b - a*c^4 + 2*a*c^3 + 3*a*c^2 - 8*a*c + 4*a + b^4*c - b^4 + 2*b^3*c^2 - 6*b^3*c + 4*b^3 + b^2*c^3 - 7*b^2*c^2 + 12*b^2*c - 6*b^2 - 2*b*c^3 + 8*b*c^2 - 10*b*c + 4*b + c^3 - 3*c^2 + 3*c - 1)/(a*b^2*c^2) (-2*a^3 - 4*a^2*b - 2*a^2*c + 6*a^2 - 2*a*b^2 - 2*a*b*c + 8*a*b + 4*a*c - 6*a + 2*b^2 + 2*b*c - 4*b - 2*c + 2)*$.1^2 + (-4*a^4*b*c - 2*a^4*b + a^4 - 8*a^3*b^2*c - 6*a^3*b^2 - 4*a^3*b*c^2 + 8*a^3*b*c + 9*a^3*b + 2*a^3*c - 4*a^3 - 4*a^2*b^3*c - 6*a^2*b^3 - 4*a^2*b^2*c^2 + 8*a^2*b^2*c + 15*a^2*b^2 + 6*a^2*b*c^2 - 15*a^2*b + a^2*c^2 - 6*a^2*c + 6*a^2 - 2*a*b^4 + 7*a*b^3 + 2*a*b^2*c^2 + 2*a*b^2*c - 12*a*b^2 - a*b*c^2 - 8*a*b*c + 11*a*b - 2*a*c^2 + 6*a*c - 4*a - b^3 - 2*b^2*c + 3*b^2 - b*c^2 + 4*b*c - 3*b + c^2 - 2*c + 1)/(a*b*c)*$.1 + (-2*a^4*b*c^2 - 2*a^4*b*c + a^4*c + a^4 - 4*a^3*b^2*c^2 - 6*a^3*b^2*c - 2*a^3*b*c^3 + 2*a^3*b*c^2 + 9*a^3*b*c + 4*a^3*b + 2*a^3*c^2 - a^3*c - 4*a^3 - 2*a^2*b^3*c^2 - 6*a^2*b^3*c - 2*a^2*b^2*c^3 + 15*a^2*b^2*c + 6*a^2*b^2 + 2*a^2*b*c^3 + 6*a^2*b*c^2 - 6*a^2*b*c - 12*a^2*b + a^2*c^3 - 3*a^2*c^2 - 3*a^2*c + 6*a^2 - 2*a*b^4*c - 2*a*b^3*c^2 + 7*a*b^3*c + 4*a*b^3 + 6*a*b^2*c^2 - 3*a*b^2*c - 12*a*b^2 + a*b*c^3 - 4*a*b*c^2 - 7*a*b*c + 12*a*b - a*c^3 + 5*a*c - 4*a + b^4 + 2*b^3*c - 4*b^3 + b^2*c^2 - 6*b^2*c + 6*b^2 - 2*b*c^2 + 6*b*c - 4*b + c^2 - 2*c + 1)/(a*b*c^2) (2*a^2*c + 4*a*b*c + 2*a*c^2 - 4*a*c + 2*b^2*c + 2*b*c^2 - 4*b*c - 2*c^2 + 2*c)*$.1^2 + (4*a^3*b*c + 2*a^3*b - 3*a^3 + 8*a^2*b^2*c + 6*a^2*b^2 + 4*a^2*b*c^2 - 4*a^2*b*c - 15*a^2*b - 6*a^2*c + 9*a^2 + 4*a*b^3*c + 6*a*b^3 + 4*a*b^2*c^2 - 21*a*b^2 - 2*a*b*c^2 - 16*a*b*c + 24*a*b - 3*a*c^2 + 12*a*c - 9*a + 2*b^4 + 4*b^3*c - 9*b^3 + 2*b^2*c^2 - 14*b^2*c + 15*b^2 - 5*b*c^2 + 16*b*c - 11*b + 3*c^2 - 6*c + 3)/(a*b)*$.1 + (2*a^4*b^2*c^2 + 2*a^4*b^2*c - 3*a^4*b*c - a^4*b + a^4 + 4*a^3*b^3*c^2 + 6*a^3*b^3*c + 2*a^3*b^2*c^3 - 15*a^3*b^2*c - 4*a^3*b^2 - 6*a^3*b*c^2 + 6*a^3*b*c + 8*a^3*b + 3*a^3*c - 4*a^3 + 2*a^2*b^4*c^2 + 6*a^2*b^4*c + 2*a^2*b^3*c^3 + 4*a^2*b^3*c^2 - 21*a^2*b^3*c - 6*a^2*b^3 - 18*a^2*b^2*c^2 + 15*a^2*b^2*c + 18*a^2*b^2 - 3*a^2*b*c^3 + 9*a^2*b*c^2 + 9*a^2*b*c - 18*a^2*b + 3*a^2*c^2 - 9*a^2*c + 6*a^2 + 2*a*b^5*c + 4*a*b^4*c^2 - 9*a*b^4*c - 4*a*b^4 + 2*a*b^3*c^3 - 14*a*b^3*c^2 + 6*a*b^3*c + 16*a*b^3 - 5*a*b^2*c^3 + 10*a*b^2*c^2 + 16*a*b^2*c - 24*a*b^2 + 2*a*b*c^3 + 6*a*b*c^2 - 24*a*b*c + 16*a*b + a*c^3 - 6*a*c^2 + 9*a*c - 4*a - b^5 - 3*b^4*c + 5*b^4 - 3*b^3*c^2 + 12*b^3*c - 10*b^3 - b^2*c^3 + 9*b^2*c^2 - 18*b^2*c + 10*b^2 + 2*b*c^3 - 9*b*c^2 + 12*b*c - 5*b - c^3 + 3*c^2 - 3*c + 1)/(a^2*b^2*c) (-2*a^2*c - 4*a*b*c - 2*a*c^2 + 4*a*c - 2*b ** WARNING: Output too long, hence truncated. '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Thu Dec 8 16:46:13 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); A1:=Matrix(K,3,3,[[b,1,1],[0,a,1],[0,0,c]]); S:=Transpose(A)*A^(-1); F,T:=PrimaryRationalForm(S); B:=T*A*Transpose(T); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*B*Transpose(P)-B)); G:=GroebnerBasis(I); Q1:=Matrix(K,3,3,[(2*b-1)/(a+b-1),(a-b)/(a+b-1),0,(b-a)/(a+b-1),(2*a-1)/(a+b-1),0, 0,0,1]); Kt:=FunctionField(K); D:=(2*a*b*c-a-b-c+1)/(a*b*c); s:=t; px:=-(2*s+D); py:=1-s^2; q:=s^2+s*D+1; Q:=Matrix(Kt,3,3,[1,0,0,0,D*px/q+py/q,-px/q,0,px/q,py/q]); Q:=Q1*T^(-1)*Q*T; print (Q*A*Transpose(Q)-A1); for i in [1..3] do for j in [1..3] do print Numerator(Q[i,j])*(a+b+c-1)^2; end for; end for; Output: Magma V2.11-10 Thu Dec 8 2005 16:46:12 on modular [Seed = 2638640917] ------------------------------------- [0 0 0] [0 0 0] [0 0 0] (a^2 - 2*a*b*c + 2*a*b + 2*a*c - 2*a - 2*b^2*c + b^2 - 2*b*c^2 + 4*b*c - 2*b + c^2 - 2*c + 1)/(a + b - 1)*$.1^2 + (4*a^3*b*c + a^3*b - 2*a^3 - 4*a^2*b^2*c^2 + 6*a^2*b^2*c + 3*a^2*b^2 + 8*a^2*b*c^2 - 2*a^2*b*c - 9*a^2*b - 6*a^2*c + 6*a^2 - 4*a*b^3*c^2 + 3*a*b^3 - 4*a*b^2*c^3 + 8*a*b^2*c^2 + 8*a*b^2*c - 12*a*b^2 + 4*a*b*c^3 + a*b*c^2 - 20*a*b*c + 15*a*b - 6*a*c^2 + 12*a*c - 6*a - 2*b^4*c + b^4 - 4*b^3*c^2 + 10*b^3*c - 5*b^3 - 2*b^2*c^3 + 13*b^2*c^2 - 20*b^2*c + 9*b^2 + 4*b*c^3 - 15*b*c^2 + 18*b*c - 7*b - 2*c^3 + 6*c^2 - 6*c + 2)/(a^2*b*c + a*b^2*c - a*b*c)*$.1 + (3*a^4*b^2*c^2 + 3*a^4*b^2*c - 4*a^4*b*c - a^4*b + a^4 - 2*a^3*b^3*c^3 + 4*a^3*b^3*c^2 + 9*a^3*b^3*c + 6*a^3*b^2*c^3 + 6*a^3*b^2*c^2 - 20*a^3*b^2*c - 4*a^3*b^2 - 12*a^3*b*c^2 + 7*a^3*b*c + 8*a^3*b + 4*a^3*c - 4*a^3 - 2*a^2*b^4*c^3 - a^2*b^4*c^2 + 9*a^2*b^4*c - 2*a^2*b^3*c^4 + 4*a^2*b^3*c^3 + 22*a^2*b^3*c^2 - 27*a^2*b^3*c - 6*a^2*b^3 + 3*a^2*b^2*c^4 + 9*a^2*b^2*c^3 - 42*a^2*b^2*c^2 + 15*a^2*b^2*c + 18*a^2*b^2 - 12*a^2*b*c^3 + 15*a^2*b*c^2 + 15*a^2*b*c - 18*a^2*b + 6*a^2*c^2 - 12*a^2*c + 6*a^2 - 2*a*b^5*c^2 + 3*a*b^5*c - 4*a*b^4*c^3 + 16*a*b^4*c^2 - 10*a*b^4*c - 4*a*b^4 - 2*a*b^3*c^4 + 19*a*b^3*c^3 - 32*a*b^3*c^2 + 16*a*b^3 + 6*a*b^2*c^4 - 24*a*b^2*c^3 + 12*a*b^2*c^2 + 30*a*b^2*c - 24*a*b^2 - 4*a*b*c^4 + 5*a*b*c^3 + 18*a*b*c^2 - 35*a*b*c + 16*a*b + 4*a*c^3 - 12*a*c^2 + 12*a*c - 4*a + b^5*c - b^5 + 3*b^4*c^2 - 8*b^4*c + 5*b^4 + 3*b^3*c^3 - 15*b^3*c^2 + 22*b^3*c - 10*b^3 + b^2*c^4 - 10*b^2*c^3 + 27*b^2*c^2 - 28*b^2*c + 10*b^2 - 2*b*c^4 + 11*b*c^3 - 21*b*c^2 + 17*b*c - 5*b + c^4 - 4*c^3 + 6*c^2 - 4*c + 1)/(a^3*b^2*c^2 + a^2*b^3*c^2 - a^2*b^2*c^2) (-a^3 - 3*a^2*b - 2*a^2*c + 2*a^2 - 3*a*b^2 - 2*a*b*c + 4*a*b - a*c^2 + 2*a*c - a - b^3 + 2*b^2 + b*c^2 - b)/(a + b - 1)*$.1^2 + (-2*a^4*b*c + 2*a^4 - 6*a^3*b^2*c - 4*a^3*b*c^2 + 7*a^3*b + 6*a^3*c - 6*a^3 - 6*a^2*b^3*c - 4*a^2*b^2*c^2 + 9*a^2*b^2 - 2*a^2*b*c^3 - 4*a^2*b*c^2 + 20*a^2*b*c - 15*a^2*b + 6*a^2*c^2 - 12*a^2*c + 6*a^2 - 2*a*b^4*c + 5*a*b^3 + 2*a*b^2*c^3 - 8*a*b^2*c^2 + 16*a*b^2*c - 12*a*b^2 - 4*a*b*c^3 + 15*a*b*c^2 - 20*a*b*c + 9*a*b + 2*a*c^3 - 6*a*c^2 + 6*a*c - 2*a + b^4 + 2*b^3*c - 3*b^3 + b^2*c^2 - 4*b^2*c + 3*b^2 - b*c^2 + 2*b*c - b)/(a^2*b*c + a*b^2*c - a*b*c)*$.1 + (-a^4*b^2*c^2 + 3*a^4*b*c - a^4 - 3*a^3*b^3*c^2 - 2*a^3*b^2*c^3 - 2*a^3*b^2*c^2 + 9*a^3*b^2*c + 9*a^3*b*c^2 - 7*a^3*b*c - 4*a^3*b - 4*a^3*c + 4*a^3 - 3*a^2*b^4*c^2 - 2*a^2*b^3*c^3 - 4*a^2*b^3*c^2 + 9*a^2*b^3*c - a^2*b^2*c^4 - 6*a^2*b^2*c^3 + 24*a^2*b^2*c^2 - 12*a^2*b^2*c - 6*a^2*b^2 + 9*a^2*b*c^3 - 12*a^2*b*c^2 - 9*a^2*b*c + 12*a^2*b - 6*a^2*c^2 + 12*a^2*c - 6*a^2 - a*b^5*c^2 - 2*a*b^4*c^2 + 3*a*b^4*c + a*b^3*c^4 - 8*a*b^3*c^3 + 14*a*b^3*c^2 - 3*a*b^3*c - 4*a*b^3 - 4*a*b^2*c^4 + 15*a*b^2*c^3 - 8*a*b^2*c^2 - 15*a*b^2*c + 12*a*b^2 + 3*a*b*c^4 - 3*a*b*c^3 - 15*a*b*c^2 + 27*a*b*c - 12*a*b - 4*a*c^3 + 12*a*c^2 - 12*a*c + 4*a - b^4*c^2 + 2*b^4*c - b^4 - 2*b^3*c^3 + 8*b^3*c^2 - 10*b^3*c + 4*b^3 - b^2*c^4 + 8*b^2*c^3 - 19*b^2*c^2 + 18*b^2*c - 6*b^2 + 2*b*c^4 - 10*b*c^3 + 18*b*c^2 - 14*b*c + 4*b - c^4 + 4*c^3 - 6*c^2 + 4*c - 1)/(a^2*b^2*c^2 + a*b^3*c^2 - a*b^2*c^2) (2*a*b + 2*b^2 + 2*b*c - 2*b)*$.1^2 + (-2*a^3 + 4*a^2*b*c - 4*a^2*b - 4*a^2*c + 3*a^2 + 4*a*b^2*c - 2*a*b^2 + 4*a*b*c^2 - 8*a*b*c + 2*a*b - 2*a*c^2 + 2*a*c - b^2 - 2*b*c + 2*b - c^2 + 2*c - 1)/(a*c)*$.1 + (-2*a^3*b*c + a^3 + 2*a^2*b^2*c^2 - 4*a^2*b^2*c - 4*a^2*b*c^2 + 3*a^2*b*c + 3*a^2*b + 3*a^2*c - 3*a^2 + 2*a*b^3*c^2 - 2*a*b^3*c + 2*a*b^2*c^3 - 6*a*b^2*c^2 + 2*a*b^2*c + 3*a*b^2 - 2*a*b*c^3 + 2*a*b*c^2 + 6*a*b*c - 6*a*b + 3*a*c^2 - 6*a*c + 3*a - b^3*c + b^3 - 2*b^2*c^2 + 5*b^2*c - 3*b^2 - b*c^3 + 5*b*c^2 - 7*b*c + 3*b + c^3 - 3*c^2 + 3*c - 1)/(a*b*c^2) (-a^3 - 3*a^2*b + 4*a^2 - 3*a*b^2 - 2*a*b*c + 8*a*b + a*c^2 + 2*a*c - 5*a - b^3 - 2*b^2*c + 4*b^2 - b*c^2 + 4*b*c - 5*b - 2*c + 2)/(a + b - 1)*$.1^2 + (-2*a^4*b*c + a^4 - 6*a^3*b^2*c + 12*a^3*b*c + 5*a^3*b - 6*a^3 - 6*a^2*b^3*c - 4*a^2*b^2*c^2 + 24*a^2*b^2*c + 9*a^2*b^2 + 2*a^2*b*c^3 + 12*a^2*b*c^2 - 14*a^2*b*c - 21*a^2*b - 3*a^2*c^2 - 6*a^2*c + 12*a^2 - 2*a*b^4*c - 4*a*b^3*c^2 + 12*a*b^3*c + 7*a*b^3 - 2*a*b^2*c^3 + 16*a*b^2*c^2 - 10*a*b^2*c - 24*a*b^2 + 4*a*b*c^3 - 13*a*b*c^2 - 12*a*b*c + 27*a*b - 2*a*c^3 + 12*a*c - 10*a + 2*b^4 + 4*b^3*c - 9*b^3 + 2*b^2*c^2 - 14*b^2*c + 15*b^2 - 5*b*c^2 + 16*b*c - 11*b + 3*c^2 - 6*c + 3)/(a^2*b*c + a*b^2*c - a*b*c)*$.1 + (-a^5*b^2*c^2 - 3*a^4*b^3*c^2 + 8*a^4*b^2*c^2 + 3*a^4*b^2*c - 3*a^4*b*c^2 - 5*a^4*b*c - a^4*b + a^4*c + a^4 - 3*a^3*b^4*c^2 - 2*a^3*b^3*c^3 + 16*a^3*b^3*c^2 + 9*a^3*b^3*c + a^3*b^2*c^4 + 10*a^3*b^2*c^3 - 12*a^3*b^2*c^2 - 24*a^3*b^2*c - 4*a^3*b^2 - 6*a^3*b*c^3 - 6*a^3*b*c^2 + 15*a^3*b*c + 8*a^3*b + 3*a^3*c^2 - 4*a^3 - a^2*b^5*c^2 - 2*a^2*b^4*c^3 + 8*a^2*b^4*c^2 + 9*a^2*b^4*c - a^2*b^3*c^4 + 12*a^2*b^3*c^3 - 2*a^2*b^3*c^2 - 33*a^2*b^3*c - 6*a^2*b^3 + 4*a^2*b^2*c^4 - 11*a^2*b^2*c^3 - 26*a^2*b^2*c^2 + 33*a^2*b^2*c + 18*a^2*b^2 - 3*a^2*b*c^4 - 3*a^2*b*c^3 + 24*a^2*b*c^2 - 3*a^2*b*c - 18*a^2*b + 3*a^2*c^3 - 3*a^2*c^2 - 6*a^2*c + 6*a^2 + 3*a*b^5*c + 7*a*b^4*c^2 - 14*a*b^4*c - 4*a*b^4 + 5*a*b^3*c^3 - 26*a*b^3*c^2 + 16*a*b^3*c + 16*a*b^3 + a*b^2*c^4 - 14*a*b^2*c^3 + 28*a*b^2*c^2 + 6*a*b^2*c - 24*a*b^2 - 2*a*b*c^4 + 11*a*b*c^3 - 6*a*b*c^2 - 19*a*b*c + 16*a*b + a*c^4 - 2*a*c^3 - 3*a*c^2 + 8*a*c - 4*a - b^5 - 3*b^4*c + 5*b^4 - 3*b^3*c^2 + 12*b^3*c - 10*b^3 - b^2*c^3 + 9*b^2*c^2 - 18*b^2*c + 10*b^2 + 2*b*c^3 - 9*b*c^2 + 12*b*c - 5*b - c^3 + 3*c^2 - 3*c + 1)/(a^3*b^2*c^2 + a^2*b^3*c^2 - a^2*b^2*c^2) (-2*a^2*c - a^2 - 2*a*b*c - 2*a*b - 2*a*c^2 + 2*a*c + 2*a - b^2 + 2*b + c^2 - 1)/(a + b - 1)*$.1^2 + (2*a^4*c + a^4 - 4*a^3*b*c^2 + 3*a^3*b + 4*a^3*c^2 - 2*a^3*c - 2*a^3 - 4*a^2*b^2*c^2 - 6*a^2*b^2*c + 3*a^2*b^2 - 4*a^2*b*c^3 + 4*a^2*b*c^2 + 8*a^2*b*c - 3*a^2*b + 2*a^2*c^3 - 3*a^2*c^2 - 4*a*b^3*c + a*b^3 - 4*a*b^2*c^2 + 10*a*b^2*c + 5*a*b*c^2 - 4*a*b*c - 3*a*b - 2*a*c + 2*a + b^3 + 2*b^2*c - 3*b^2 + b*c^2 - 4*b*c + 3*b - c^2 + 2*c - 1)/(a^2*b*c + a*b^2*c - a*b*c)*$.1 + (2*a^4*b*c^2 + 3*a^4*b*c - a^4*c - a^4 - 2*a^3*b^2*c^3 + a^3*b^2*c^2 + 9*a^3*b^2*c + 4*a^3*b*c^3 + 4*a^3*b*c^2 - 11*a^3*b*c - 4*a^3*b - 3*a^3*c^2 + 4*a^3 - 2*a^2*b^3*c^3 - 4*a^2*b^3*c^2 + 9*a^2*b^3*c - 2*a^2*b^2*c^4 + 2*a^2*b^2*c^3 + 18*a^2*b^2*c^2 - 18*a^2*b^2*c - 6*a^2*b^2 + 2*a^2*b*c^4 + 3*a^2*b*c^3 - 18*a^2*b*c^2 + 3*a^2*b*c + 12*a^2*b - 3*a^2*c^3 + 3*a^2*c^2 + 6*a^2*c - 6*a^2 - 3*a*b^4*c^2 + 3*a*b^4*c - 4*a*b^3*c^3 + 14*a*b^3*c^2 - 7*a*b^3*c - 4*a*b^3 - a*b^2*c^4 + 11*a*b^2*c^3 - 18*a*b^2*c^2 - 3*a*b^2*c + 12*a*b^2 + 2*a*b*c^4 - 9*a*b*c^3 + 4*a*b*c^2 + 15*a*b*c - 12*a*b - a*c^4 + 2*a*c^3 + 3*a*c^2 - 8*a*c + 4*a + b^4*c - b^4 + 2*b^3*c^2 - 6*b^3*c + 4*b^3 + b^2*c^3 - 7*b^2*c^2 + 12*b^2*c - 6*b^2 - 2*b*c^3 + 8*b*c^2 - 10*b*c + 4*b + c^3 - 3*c^2 + 3*c - 1)/(a^2*b^2*c^2 + a*b^3*c^2 - a*b^2*c^2) (-2*a^2 - 2*a*b - 2*a*c + 4*a + 2*b + 2*c - 2)*$.1^2 + (-4*a^3*b*c - 2*a^3*b + a^3 - 4*a^2*b^2*c - 4*a^2*b^2 - 4*a^2*b*c^2 + 4*a^2*b*c + 6*a^2*b + 2*a^2*c - 3*a^2 - 2*a*b^3 + 5*a*b^2 + 2*a*b*c^2 + 2*a*b*c - 6*a*b + a*c^2 - 4*a*c + 3*a - b^2 - 2*b*c + 2*b - c^2 + 2*c - 1)/(a*b*c)*$.1 + (-2*a^3*b*c^2 - 2*a^3*b*c + a^3*c + a^3 - 2*a^2*b^2*c^2 - 4*a^2*b^2*c - 2*a^2*b*c^3 + 6*a^2*b*c + 3*a^2*b + 2*a^2*c^2 - 3*a^2 - 2*a*b^3*c - 2*a*b^2*c^2 + 5*a*b^2*c + 3*a*b^2 + 4*a*b*c^2 - 6*a*b + a*c^3 - a*c^2 - 3*a*c + 3*a + b^3 + 2*b^2*c - 3*b^2 + b*c^2 - 4*b*c + 3*b - c^2 + 2*c - 1)/(a*b*c^2) (2*a*c + 2*b*c + 2*c^2 - 2*c)*$.1^2 + (4*a^2*b*c + 2*a^2*b - 3*a^2 + 4*a*b^2*c + 4*a*b^2 + 4*a*b*c^2 - 10*a*b - 6*a*c + 6*a + 2*b^3 + 4*b^2*c - 7*b^2 + 2*b*c^2 - 10*b*c + 8*b - 3*c^2 + 6*c - 3)/(a*b)*$.1 + (2*a^3*b^2*c^2 + 2*a^3*b^2*c - 3*a^3*b*c - a^3*b + a^3 + 2*a^2*b^3*c^2 + 4*a^2*b^3*c + 2*a^2*b^2*c^3 + 2*a^2*b^2*c^2 - 10*a^2*b^2*c - 3*a^2*b^2 - 6*a^2*b*c^2 + 3*a^2*b*c + 6*a^2*b + 3*a^2*c - 3*a^2 + 2*a*b^4*c + 4*a*b^3*c^2 - 7*a*b^3*c - 3*a*b^3 + 2*a*b^2*c^3 - 10*a*b^2*c^2 + 2*a*b^2*c + 9*a*b^2 - 3*a*b*c^3 + 3*a*b*c^2 + 9*a*b*c - 9*a*b + 3*a*c^2 - 6*a*c + 3*a - b^4 - 3*b^3*c + 4*b^3 - 3*b^2*c^2 + 9*b^2*c - 6*b^2 - b*c^3 + 6*b*c^2 - 9*b*c + 4*b + c^3 - 3*c^2 + 3*c - 1)/(a^2*b^2*c) (-2*a*c - 2*b*c - 2*c^2 + 2*c)*$.1^2 + (2*a^3 - 4*a^2*b*c + 4*a^2*b + 4*a^2*c - 3*a^2 - 4*a*b^2*c + 2*a*b^2 - 4*a*b*c^2 + 8*a*b*c - 2*a*b + 2*a*c^2 - 2*a*c + b^2 + 2*b*c - 2*b + c^2 - 2*c + 1)/(a*b)*$.1 + (2*a^3*b*c - a^3 - 2*a^2*b^2*c^2 + 4*a^2*b^2*c + 4*a^2*b*c^2 - 3*a^2*b*c - 3*a^2*b - 3*a^2*c + 3*a^2 - 2*a*b^3*c^2 + 2*a*b^3*c - 2*a*b^2*c^3 + 6*a*b^2*c^2 - 2*a*b^2*c - 3*a*b^2 + 2*a*b*c^3 - 2*a*b*c^2 - 6*a*b*c + 6*a*b - 3*a*c^2 + 6*a*c - 3*a + b^3*c - b^3 + 2*b^2*c^2 - 5*b^2*c + 3*b^2 + b*c^3 - 5*b*c^2 + 7*b*c - 3*b - c^3 + 3*c^2 - 3*c + 1)/(a*b^2*c) (-a^2 - 2*a*b + 2*a - b^2 + 2*b + c^2 - 1)*$.1^2 + (-2*a^3*b - 4*a^2*b^2 + 4*a^2*b - a^2 - 2*a*b^3 + 4*a*b^2 + 2*a*b*c^2 - 4*a*b - 2*a*c + 2*a - b^2 - 2*b*c + 2*b - c^2 + 2*c - 1)/(a*b)*$.1 + (-a^3*b*c + a^3 - 2*a^2*b^2*c + 2*a^2*b*c + 3*a^2*b + 2*a^2*c - 3*a^2 - a*b^3*c + 2*a*b^2*c + 3*a*b^2 + a*b*c^3 + 3*a*b*c - 6*a*b + a*c^2 - 4*a*c + 3*a + b^3 + 2*b^2*c - 3*b^2 + b*c^2 - 4*b*c + 3*b - c^2 + 2*c - 1)/(a*b*c) Total time: 0.370 seconds, Total memory usage: 3.63MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Thu Dec 8 16:45:31 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); A1:=Matrix(K,3,3,[[b,1,1],[0,a,1],[0,0,c]]); S:=Transpose(A)*A^(-1); F,T:=PrimaryRationalForm(S); B:=T*A*Transpose(T); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*B*Transpose(P)-B)); G:=GroebnerBasis(I); Q1:=Matrix(K,3,3,[(2*b-1)/(a+b-1),(a-b)/(a+b-1),0,(b-a)/(a+b-1),(2*a-1)/(a+b-1),0, 0,0,1]); Kt:=FunctionField(K); D:=(2*a*b*c-a-b-c+1)/(a*b*c); s:=t; px:=-(2*s+D); py:=1-s^2; q:=s^2+s*D+1; Q:=Matrix(Kt,3,3,[1,0,0,0,D*px/q+py/q,-px/q,0,px/q,py/q]); Q:=Q1*T^(-1)*Q*T; print (Q*A*Transpose(Q)-A1); for i in [1..3] do for j in [1..3] do print Numerator(Q[i,j]); end for; end for; Output: Magma V2.11-10 Thu Dec 8 2005 16:45:30 on modular [Seed = 2153474774] ------------------------------------- [0 0 0] [0 0 0] [0 0 0] (a - 2*b*c + b + c - 1)/(a^2 + 2*a*b + a*c - 2*a + b^2 + b*c - 2*b - c + 1)*$.1^2 + (4*a^2*b*c + a^2*b - 2*a^2 - 4*a*b^2*c^2 + 2*a*b^2*c + 2*a*b^2 + 4*a*b*c^2 + a*b*c - 6*a*b - 4*a*c + 4*a - 2*b^3*c + b^3 - 2*b^2*c^2 + 7*b^2*c - 4*b^2 + 4*b*c^2 - 9*b*c + 5*b - 2*c^2 + 4*c - 2)/(a^3*b*c + 2*a^2*b^2*c + a^2*b*c^2 - 2*a^2*b*c + a*b^3*c + a*b^2*c^2 - 2*a*b^2*c - a*b*c^2 + a*b*c)*$.1 + (3*a^3*b^2*c^2 + 3*a^3*b^2*c - 4*a^3*b*c - a^3*b + a^3 - 2*a^2*b^3*c^3 + a^2*b^3*c^2 + 6*a^2*b^3*c + 3*a^2*b^2*c^3 + 6*a^2*b^2*c^2 - 13*a^2*b^2*c - 3*a^2*b^2 - 8*a^2*b*c^2 + 4*a^2*b*c + 6*a^2*b + 3*a^2*c - 3*a^2 - 2*a*b^4*c^2 + 3*a*b^4*c - 2*a*b^3*c^3 + 11*a*b^3*c^2 - 8*a*b^3*c - 3*a*b^3 + 6*a*b^2*c^3 - 15*a*b^2*c^2 + a*b^2*c + 9*a*b^2 - 4*a*b*c^3 + 3*a*b*c^2 + 10*a*b*c - 9*a*b + 3*a*c^2 - 6*a*c + 3*a + b^4*c - b^4 + 2*b^3*c^2 - 6*b^3*c + 4*b^3 + b^2*c^3 - 7*b^2*c^2 + 12*b^2*c - 6*b^2 - 2*b*c^3 + 8*b*c^2 - 10*b*c + 4*b + c^3 - 3*c^2 + 3*c - 1)/(a^4*b^2*c^2 + 2*a^3*b^3*c^2 + a^3*b^2*c^3 - 2*a^3*b^2*c^2 + a^2*b^4*c^2 + a^2*b^3*c^3 - 2*a^2*b^3*c^2 - a^2*b^2*c^3 + a^2*b^2*c^2) (-a^2 - 2*a*b - a*c + a - b^2 + b*c + b)/(a^2 + 2*a*b + a*c - 2*a + b^2 + b*c - 2*b - c + 1)*$.1^2 + (-2*a^3*b*c + 2*a^3 - 4*a^2*b^2*c - 2*a^2*b*c^2 - 2*a^2*b*c + 5*a^2*b + 4*a^2*c - 4*a^2 - 2*a*b^3*c + 2*a*b^2*c^2 - 2*a*b^2*c + 4*a*b^2 - 4*a*b*c^2 + 9*a*b*c - 6*a*b + 2*a*c^2 - 4*a*c + 2*a + b^3 + b^2*c - 2*b^2 - b*c + b)/(a^3*b*c + 2*a^2*b^2*c + a^2*b*c^2 - 2*a^2*b*c + a*b^3*c + a*b^2*c^2 - 2*a*b^2*c - a*b*c^2 + a*b*c)*$.1 + (-a^3*b^2*c^2 + 3*a^3*b*c - a^3 - 2*a^2*b^3*c^2 - a^2*b^2*c^3 - 3*a^2*b^2*c^2 + 6*a^2*b^2*c + 6*a^2*b*c^2 - 4*a^2*b*c - 3*a^2*b - 3*a^2*c + 3*a^2 - a*b^4*c^2 + a*b^3*c^3 - 3*a*b^3*c^2 + 3*a*b^3*c - 4*a*b^2*c^3 + 9*a*b^2*c^2 - 2*a*b^2*c - 3*a*b^2 + 3*a*b*c^3 - 2*a*b*c^2 - 7*a*b*c + 6*a*b - 3*a*c^2 + 6*a*c - 3*a - b^3*c^2 + 2*b^3*c - b^3 - b^2*c^3 + 5*b^2*c^2 - 7*b^2*c + 3*b^2 + 2*b*c^3 - 7*b*c^2 + 8*b*c - 3*b - c^3 + 3*c^2 - 3*c + 1)/(a^3*b^2*c^2 + 2*a^2*b^3*c^2 + a^2*b^2*c^3 - 2*a^2*b^2*c^2 + a*b^4*c^2 + a*b^3*c^3 - 2*a*b^3*c^2 - a*b^2*c^3 + a*b^2*c^2) 2*b/(a + b + c - 1)*$.1^2 + (-2*a^2 + 4*a*b*c - 2*a*b - 2*a*c + a - b - c + 1)/(a^2*c + a*b*c + a*c^2 - a*c)*$.1 + (-2*a^2*b*c + a^2 + 2*a*b^2*c^2 - 2*a*b^2*c - 2*a*b*c^2 + a*b*c + 2*a*b + 2*a*c - 2*a - b^2*c + b^2 - b*c^2 + 3*b*c - 2*b + c^2 - 2*c + 1)/(a^2*b*c^2 + a*b^2*c^2 + a*b*c^3 - a*b*c^2) (-a^2 - 2*a*b + a*c + 3*a - b^2 - b*c + 3*b - 2)/(a^2 + 2*a*b + a*c - 2*a + b^2 + b*c - 2*b - c + 1)*$.1^2 + (-2*a^3*b*c + a^3 - 4*a^2*b^2*c + 2*a^2*b*c^2 + 10*a^2*b*c + 4*a^2*b - a^2*c - 5*a^2 - 2*a*b^3*c - 2*a*b^2*c^2 + 10*a*b^2*c + 5*a*b^2 + 4*a*b*c^2 - 7*a*b*c - 12*a*b - 2*a*c^2 - 2*a*c + 7*a + 2*b^3 + 2*b^2*c - 7*b^2 - 5*b*c + 8*b + 3*c - 3)/(a^3*b*c + 2*a^2*b^2*c + a^2*b*c^2 - 2*a^2*b*c + a*b^3*c + a*b^2*c^2 - 2*a*b^2*c - a*b*c^2 + a*b*c)*$.1 + (-a^4*b^2*c^2 - 2*a^3*b^3*c^2 + a^3*b^2*c^3 + 7*a^3*b^2*c^2 + 3*a^3*b^2*c - 3*a^3*b*c^2 - 5*a^3*b*c - a^3*b + a^3*c + a^3 - a^2*b^4*c^2 - a^2*b^3*c^3 + 7*a^2*b^3*c^2 + 6*a^2*b^3*c + 4*a^2*b^2*c^3 - 5*a^2*b^2*c^2 - 16*a^2*b^2*c - 3*a^2*b^2 - 3*a^2*b*c^3 - 4*a^2*b*c^2 + 10*a^2*b*c + 6*a^2*b + 2*a^2*c^2 - 3*a^2 + 3*a*b^4*c + 4*a*b^3*c^2 - 11*a*b^3*c - 3*a*b^3 + a*b^2*c^3 - 11*a*b^2*c^2 + 10*a*b^2*c + 9*a*b^2 - 2*a*b*c^3 + 8*a*b*c^2 + a*b*c - 9*a*b + a*c^3 - a*c^2 - 3*a*c + 3*a - b^4 - 2*b^3*c + 4*b^3 - b^2*c^2 + 6*b^2*c - 6*b^2 + 2*b*c^2 - 6*b*c + 4*b - c^2 + 2*c - 1)/(a^4*b^2*c^2 + 2*a^3*b^3*c^2 + a^3*b^2*c^3 - 2*a^3*b^2*c^2 + a^2*b^4*c^2 + a^2*b^3*c^3 - 2*a^2*b^3*c^2 - a^2*b^2*c^3 + a^2*b^2*c^2) (-2*a*c - a - b + c + 1)/(a^2 + 2*a*b + a*c - 2*a + b^2 + b*c - 2*b - c + 1)*$.1^2 + (2*a^3*c + a^3 - 4*a^2*b*c^2 - 2*a^2*b*c + 2*a^2*b + 2*a^2*c^2 - a^2*c - a^2 - 4*a*b^2*c + a*b^2 + 5*a*b*c - a + b^2 + b*c - 2*b - c + 1)/(a^3*b*c + 2*a^2*b^2*c + a^2*b*c^2 - 2*a^2*b*c + a*b^3*c + a*b^2*c^2 - 2*a*b^2*c - a*b*c^2 + a*b*c)*$.1 + (2*a^3*b*c^2 + 3*a^3*b*c - a^3*c - a^3 - 2*a^2*b^2*c^3 - a^2*b^2*c^2 + 6*a^2*b^2*c + 2*a^2*b*c^3 + 3*a^2*b*c^2 - 7*a^2*b*c - 3*a^2*b - 2*a^2*c^2 + 3*a^2 - 3*a*b^3*c^2 + 3*a*b^3*c - a*b^2*c^3 + 8*a*b^2*c^2 - 5*a*b^2*c - 3*a*b^2 + 2*a*b*c^3 - 6*a*b*c^2 - a*b*c + 6*a*b - a*c^3 + a*c^2 + 3*a*c - 3*a + b^3*c - b^3 + b^2*c^2 - 4*b^2*c + 3*b^2 - 2*b*c^2 + 5*b*c - 3*b + c^2 - 2*c + 1)/(a^3*b^2*c^2 + 2*a^2*b^3*c^2 + a^2*b^2*c^3 - 2*a^2*b^2*c^2 + a*b^4*c^2 + a*b^3*c^3 - 2*a*b^3*c^2 - a*b^2*c^3 + a*b^2*c^2) (-2*a + 2)/(a + b + c - 1)*$.1^2 + (-4*a^2*b*c - 2*a^2*b + a^2 - 2*a*b^2 + 2*a*b*c + 3*a*b + a*c - 2*a - b - c + 1)/(a^2*b*c + a*b^2*c + a*b*c^2 - a*b*c)*$.1 + (-2*a^2*b*c^2 - 2*a^2*b*c + a^2*c + a^2 - 2*a*b^2*c + 3*a*b*c + 2*a*b + a*c^2 - 2*a + b^2 + b*c - 2*b - c + 1)/(a^2*b*c^2 + a*b^2*c^2 + a*b*c^3 - a*b*c^2) 2*c/(a + b + c - 1)*$.1^2 + (4*a*b*c + 2*a*b - 3*a + 2*b^2 + 2*b*c - 5*b - 3*c + 3)/(a^2*b + a*b^2 + a*b*c - a*b)*$.1 + (2*a^2*b^2*c^2 + 2*a^2*b^2*c - 3*a^2*b*c - a^2*b + a^2 + 2*a*b^3*c + 2*a*b^2*c^2 - 5*a*b^2*c - 2*a*b^2 - 3*a*b*c^2 + a*b*c + 4*a*b + 2*a*c - 2*a - b^3 - 2*b^2*c + 3*b^2 - b*c^2 + 4*b*c - 3*b + c^2 - 2*c + 1)/(a^3*b^2*c + a^2*b^3*c + a^2*b^2*c^2 - a^2*b^2*c) -2*c/(a + b + c - 1)*$.1^2 + (2*a^2 - 4*a*b*c + 2*a*b + 2*a*c - a + b + c - 1)/(a^2*b + a*b^2 + a*b*c - a*b)*$.1 + (2*a^2*b*c - a^2 - 2*a*b^2*c^2 + 2*a*b^2*c + 2*a*b*c^2 - a*b*c - 2*a*b - 2*a*c + 2*a + b^2*c - b^2 + b*c^2 - 3*b*c + 2*b - c^2 + 2*c - 1)/(a^2*b^2*c + a*b^3*c + a*b^2*c^2 - a*b^2*c) (-a - b + c + 1)/(a + b + c - 1)*$.1^2 + (-2*a^2*b - 2*a*b^2 + 2*a*b*c + 2*a*b - a - b - c + 1)/(a^2*b + a*b^2 + a*b*c - a*b)*$.1 + (-a^2*b*c + a^2 - a*b^2*c + a*b*c^2 + a*b*c + 2*a*b + a*c - 2*a + b^2 + b*c - 2*b - c + 1)/(a^2*b*c + a*b^2*c + a*b*c^2 - a*b*c) Total time: 0.360 seconds, Total memory usage: 3.63MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Thu Dec 8 16:45:05 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); A1:=Matrix(K,3,3,[[b,1,1],[0,a,1],[0,0,c]]); S:=Transpose(A)*A^(-1); F,T:=PrimaryRationalForm(S); B:=T*A*Transpose(T); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*B*Transpose(P)-B)); G:=GroebnerBasis(I); Q1:=Matrix(K,3,3,[(2*b-1)/(a+b-1),(a-b)/(a+b-1),0,(b-a)/(a+b-1),(2*a-1)/(a+b-1),0, 0,0,1]); Kt:=FunctionField(K); D:=(2*a*b*c-a-b-c+1)/(a*b*c); s:=t; px:=-(2*s+D); py:=1-s^2; q:=s^2+s*D+1; Q:=Matrix(Kt,3,3,[1,0,0,0,D*px/q+py/q,-px/q,0,px/q,py/q]); Q:=Q1*T^(-1)*Q*T; print (Q*A*Transpose(Q)-A1); for i in [1..3] do for j in [1..3] do print Evaluate(Numerator(Q[i,j]),1/(a+b+c-1)); end for; end for; Output: Magma V2.11-10 Thu Dec 8 2005 16:45:04 on modular [Seed = 2337300935] ------------------------------------- [0 0 0] [0 0 0] [0 0 0] (3*a^5*b^2*c^2 + 3*a^5*b^2*c - 4*a^5*b*c - a^5*b + a^5 - 2*a^4*b^3*c^3 + 7*a^4*b^3*c^2 + 12*a^4*b^3*c + 9*a^4*b^2*c^3 + 10*a^4*b^2*c^2 - 26*a^4*b^2*c - 5*a^4*b^2 - 16*a^4*b*c^2 + 8*a^4*b*c + 10*a^4*b + 5*a^4*c - 5*a^4 - 4*a^3*b^4*c^3 + 3*a^3*b^4*c^2 + 18*a^3*b^4*c - 4*a^3*b^3*c^4 + 12*a^3*b^3*c^3 + 39*a^3*b^3*c^2 - 53*a^3*b^3*c - 10*a^3*b^3 + 9*a^3*b^2*c^4 + 17*a^3*b^2*c^3 - 81*a^3*b^2*c^2 + 29*a^3*b^2*c + 30*a^3*b^2 - 24*a^3*b*c^3 + 28*a^3*b*c^2 + 26*a^3*b*c - 30*a^3*b + 10*a^3*c^2 - 20*a^3*c + 10*a^3 - 2*a^2*b^5*c^3 - 3*a^2*b^5*c^2 + 12*a^2*b^5*c - 4*a^2*b^4*c^4 - 3*a^2*b^4*c^3 + 48*a^2*b^4*c^2 - 43*a^2*b^4*c - 10*a^2*b^4 - 2*a^2*b^3*c^5 + 3*a^2*b^3*c^4 + 52*a^2*b^3*c^3 - 114*a^2*b^3*c^2 + 24*a^2*b^3*c + 40*a^2*b^3 + 3*a^2*b^2*c^5 + 16*a^2*b^2*c^4 - 85*a^2*b^2*c^3 + 63*a^2*b^2*c^2 + 63*a^2*b^2*c - 60*a^2*b^2 - 16*a^2*b*c^4 + 26*a^2*b*c^3 + 36*a^2*b*c^2 - 86*a^2*b*c + 40*a^2*b + 10*a^2*c^3 - 30*a^2*c^2 + 30*a^2*c - 10*a^2 - 2*a*b^6*c^2 + 3*a*b^6*c - 6*a*b^5*c^3 + 19*a*b^5*c^2 - 11*a*b^5*c - 5*a*b^5 - 6*a*b^4*c^4 + 35*a*b^4*c^3 - 45*a*b^4*c^2 - 7*a*b^4*c + 25*a*b^4 - 2*a*b^3*c^5 + 25*a*b^3*c^4 - 59*a*b^3*c^3 + 9*a*b^3*c^2 + 77*a*b^3*c - 50*a*b^3 + 6*a*b^2*c^5 - 29*a*b^2*c^4 + 16*a*b^2*c^3 + 81*a*b^2*c^2 - 124*a*b^2*c + 50*a*b^2 - 4*a*b*c^5 + 5*a*b*c^4 + 34*a*b*c^3 - 92*a*b*c^2 + 82*a*b*c - 25*a*b + 5*a*c^4 - 20*a*c^3 + 30*a*c^2 - 20*a*c + 5*a + b^6*c - b^6 + 4*b^5*c^2 - 10*b^5*c + 6*b^5 + 6*b^4*c^3 - 26*b^4*c^2 + 35*b^4*c - 15*b^4 + 4*b^3*c^4 - 28*b^3*c^3 + 64*b^3*c^2 - 60*b^3*c + 20*b^3 + b^2*c^5 - 13*b^2*c^4 + 48*b^2*c^3 - 76*b^2*c^2 + 55*b^2*c - 15*b^2 - 2*b*c^5 + 14*b*c^4 - 36*b*c^3 + 44*b*c^2 - 26*b*c + 6*b + c^5 - 5*c^4 + 10*c^3 - 10*c^2 + 5*c - 1)/(a^6*b^2*c^2 + 4*a^5*b^3*c^2 + 3*a^5*b^2*c^3 - 4*a^5*b^2*c^2 + 6*a^4*b^4*c^2 + 9*a^4*b^3*c^3 - 12*a^4*b^3*c^2 + 3*a^4*b^2*c^4 - 9*a^4*b^2*c^3 + 6*a^4*b^2*c^2 + 4*a^3*b^5*c^2 + 9*a^3*b^4*c^3 - 12*a^3*b^4*c^2 + 6*a^3*b^3*c^4 - 18*a^3*b^3*c^3 + 12*a^3*b^3*c^2 + a^3*b^2*c^5 - 6*a^3*b^2*c^4 + 9*a^3*b^2*c^3 - 4*a^3*b^2*c^2 + a^2*b^6*c^2 + 3*a^2*b^5*c^3 - 4*a^2*b^5*c^2 + 3*a^2*b^4*c^4 - 9*a^2*b^4*c^3 + 6*a^2*b^4*c^2 + a^2*b^3*c^5 - 6*a^2*b^3*c^4 + 9*a^2*b^3*c^3 - 4*a^2*b^3*c^2 - a^2*b^2*c^5 + 3*a^2*b^2*c^4 - 3*a^2*b^2*c^3 + a^2*b^2*c^2) (-a^5*b^2*c^2 + 3*a^5*b*c - a^5 - 4*a^4*b^3*c^2 - 3*a^4*b^2*c^3 - 3*a^4*b^2*c^2 + 12*a^4*b^2*c + 12*a^4*b*c^2 - 8*a^4*b*c - 5*a^4*b - 5*a^4*c + 5*a^4 - 6*a^3*b^4*c^2 - 7*a^3*b^3*c^3 - 9*a^3*b^3*c^2 + 18*a^3*b^3*c - 3*a^3*b^2*c^4 - 10*a^3*b^2*c^3 + 43*a^3*b^2*c^2 - 21*a^3*b^2*c - 10*a^3*b^2 + 18*a^3*b*c^3 - 22*a^3*b*c^2 - 16*a^3*b*c + 20*a^3*b - 10*a^3*c^2 + 20*a^3*c - 10*a^3 - 4*a^2*b^5*c^2 - 5*a^2*b^4*c^3 - 9*a^2*b^4*c^2 + 12*a^2*b^4*c - 2*a^2*b^3*c^4 - 20*a^2*b^3*c^3 + 49*a^2*b^3*c^2 - 15*a^2*b^3*c - 10*a^2*b^3 - a^2*b^2*c^5 - 11*a^2*b^2*c^4 + 49*a^2*b^2*c^3 - 35*a^2*b^2*c^2 - 33*a^2*b^2*c + 30*a^2*b^2 + 12*a^2*b*c^4 - 18*a^2*b*c^3 - 30*a^2*b*c^2 + 66*a^2*b*c - 30*a^2*b - 10*a^2*c^3 + 30*a^2*c^2 - 30*a^2*c + 10*a^2 - a*b^6*c^2 - a*b^5*c^3 - 3*a*b^5*c^2 + 3*a*b^5*c + a*b^4*c^4 - 10*a*b^4*c^3 + 17*a*b^4*c^2 + a*b^4*c - 5*a*b^4 + a*b^3*c^5 - 11*a*b^3*c^4 + 28*a*b^3*c^3 - 38*a*b^3*c + 20*a*b^3 - 4*a*b^2*c^5 + 17*a*b^2*c^4 - 3*a*b^2*c^3 - 61*a*b^2*c^2 + 81*a*b^2*c - 30*a*b^2 + 3*a*b*c^5 - 2*a*b*c^4 - 32*a*b*c^3 + 78*a*b*c^2 - 67*a*b*c + 20*a*b - 5*a*c^4 + 20*a*c^3 - 30*a*c^2 + 20*a*c - 5*a - b^5*c^2 + 3*b^5*c - b^5 - 3*b^4*c^3 + 13*b^4*c^2 - 16*b^4*c + 5*b^4 - 3*b^3*c^4 + 19*b^3*c^3 - 41*b^3*c^2 + 35*b^3*c - 10*b^3 - b^2*c^5 + 11*b^2*c^4 - 38*b^2*c^3 + 57*b^2*c^2 - 39*b^2*c + 10*b^2 + 2*b*c^5 - 13*b*c^4 + 32*b*c^3 - 38*b*c^2 + 22*b*c - 5*b - c^5 + 5*c^4 - 10*c^3 + 10*c^2 - 5*c + 1)/(a^5*b^2*c^2 + 4*a^4*b^3*c^2 + 3*a^4*b^2*c^3 - 4*a^4*b^2*c^2 + 6*a^3*b^4*c^2 + 9*a^3*b^3*c^3 - 12*a^3*b^3*c^2 + 3*a^3*b^2*c^4 - 9*a^3*b^2*c^3 + 6*a^3*b^2*c^2 + 4*a^2*b^5*c^2 + 9*a^2*b^4*c^3 - 12*a^2*b^4*c^2 + 6*a^2*b^3*c^4 - 18*a^2*b^3*c^3 + 12*a^2*b^3*c^2 + a^2*b^2*c^5 - 6*a^2*b^2*c^4 + 9*a^2*b^2*c^3 - 4*a^2*b^2*c^2 + a*b^6*c^2 + 3*a*b^5*c^3 - 4*a*b^5*c^2 + 3*a*b^4*c^4 - 9*a*b^4*c^3 + 6*a*b^4*c^2 + a*b^3*c^5 - 6*a*b^3*c^4 + 9*a*b^3*c^3 - 4*a*b^3*c^2 - a*b^2*c^5 + 3*a*b^2*c^4 - 3*a*b^2*c^3 + a*b^2*c^2) (-2*a^4*b*c + a^4 + 2*a^3*b^2*c^2 - 6*a^3*b^2*c - 6*a^3*b*c^2 + 3*a^3*b*c + 4*a^3*b + 4*a^3*c - 4*a^3 + 4*a^2*b^3*c^2 - 6*a^2*b^3*c + 4*a^2*b^2*c^3 - 12*a^2*b^2*c^2 + 5*a^2*b^2*c + 6*a^2*b^2 - 6*a^2*b*c^3 + 5*a^2*b*c^2 + 12*a^2*b*c - 12*a^2*b + 6*a^2*c^2 - 12*a^2*c + 6*a^2 + 2*a*b^4*c^2 - 2*a*b^4*c + 4*a*b^3*c^3 - 6*a*b^3*c^2 + a*b^3*c + 4*a*b^3 + 2*a*b^2*c^4 - 6*a*b^2*c^3 + 2*a*b^2*c^2 + 14*a*b^2*c - 12*a*b^2 - 2*a*b*c^4 + a*b*c^3 + 14*a*b*c^2 - 25*a*b*c + 12*a*b + 4*a*c^3 - 12*a*c^2 + 12*a*c - 4*a - b^4*c + b^4 - 3*b^3*c^2 + 6*b^3*c - 4*b^3 - 3*b^2*c^3 + 10*b^2*c^2 - 13*b^2*c + 6*b^2 - b*c^4 + 6*b*c^3 - 13*b*c^2 + 12*b*c - 4*b + c^4 - 4*c^3 + 6*c^2 - 4*c + 1)/(a^4*b*c^2 + 3*a^3*b^2*c^2 + 3*a^3*b*c^3 - 3*a^3*b*c^2 + 3*a^2*b^3*c^2 + 6*a^2*b^2*c^3 - 6*a^2*b^2*c^2 + 3*a^2*b*c^4 - 6*a^2*b*c^3 + 3*a^2*b*c^2 + a*b^4*c^2 + 3*a*b^3*c^3 - 3*a*b^3*c^2 + 3*a*b^2*c^4 - 6*a*b^2*c^3 + 3*a*b^2*c^2 + a*b*c^5 - 3*a*b*c^4 + 3*a*b*c^3 - a*b*c^2) (-a^6*b^2*c^2 - 4*a^5*b^3*c^2 - a^5*b^2*c^3 + 7*a^5*b^2*c^2 + 3*a^5*b^2*c - 3*a^5*b*c^2 - 4*a^5*b*c - a^5*b + a^5*c + a^5 - 6*a^4*b^4*c^2 - 5*a^4*b^3*c^3 + 21*a^4*b^3*c^2 + 12*a^4*b^3*c + a^4*b^2*c^4 + 18*a^4*b^2*c^3 - 9*a^4*b^2*c^2 - 27*a^4*b^2*c - 5*a^4*b^2 - 9*a^4*b*c^3 - 8*a^4*b*c^2 + 14*a^4*b*c + 10*a^4*b + 4*a^4*c^2 - 5*a^4 - 4*a^3*b^5*c^2 - 7*a^3*b^4*c^3 + 21*a^3*b^4*c^2 + 18*a^3*b^4*c - 2*a^3*b^3*c^4 + 36*a^3*b^3*c^3 + a^3*b^3*c^2 - 57*a^3*b^3*c - 10*a^3*b^3 + a^3*b^2*c^5 + 15*a^3*b^2*c^4 - 26*a^3*b^2*c^3 - 55*a^3*b^2*c^2 + 47*a^3*b^2*c + 30*a^3*b^2 - 9*a^3*b*c^4 - 6*a^3*b*c^3 + 42*a^3*b*c^2 + 2*a^3*b*c - 30*a^3*b + 6*a^3*c^3 - 6*a^3*c^2 - 10*a^3*c + 10*a^3 - a^2*b^6*c^2 - 3*a^2*b^5*c^3 + 7*a^2*b^5*c^2 + 12*a^2*b^5*c - 3*a^2*b^4*c^4 + 18*a^2*b^4*c^3 + 17*a^2*b^4*c^2 - 49*a^2*b^4*c - 10*a^2*b^4 - a^2*b^3*c^5 + 15*a^2*b^3*c^4 - 5*a^2*b^3*c^3 - 90*a^2*b^3*c^2 + 52*a^2*b^3*c + 40*a^2*b^3 + 4*a^2*b^2*c^5 - 13*a^2*b^2*c^4 - 45*a^2*b^2*c^3 + 97*a^2*b^2*c^2 + 15*a^2*b^2*c - 60*a^2*b^2 - 3*a^2*b*c^5 - 4*a^2*b*c^4 + 41*a^2*b*c^3 - 24*a^2*b*c^2 - 50*a^2*b*c + 40*a^2*b + 4*a^2*c^4 - 8*a^2*c^3 - 6*a^2*c^2 + 20*a^2*c - 10*a^2 + 3*a*b^6*c + 10*a*b^5*c^2 - 15*a*b^5*c - 5*a*b^5 + 12*a*b^4*c^3 - 43*a*b^4*c^2 + 14*a*b^4*c + 25*a*b^4 + 6*a*b^3*c^4 - 43*a*b^3*c^3 + 53*a*b^3*c^2 + 33*a*b^3*c - 50*a*b^3 + a*b^2*c^5 - 17*a*b^2*c^4 + 47*a*b^2*c^3 - 3*a*b^2*c^2 - 78*a*b^2*c + 50*a*b^2 - 2*a*b*c^5 + 14*a*b*c^4 - 14*a*b*c^3 - 31*a*b*c^2 + 58*a*b*c - 25*a*b + a*c^5 - 3*a*c^4 - 2*a*c^3 + 14*a*c^2 - 15*a*c + 5*a - b^6 - 4*b^5*c + 6*b^5 - 6*b^4*c^2 + 20*b^4*c - 15*b^4 - 4*b^3*c^3 + 24*b^3*c^2 - 40*b^3*c + 20*b^3 - b^2*c^4 + 12*b^2*c^3 - 36*b^2*c^2 + 40*b^2*c - 15*b^2 + 2*b*c^4 - 12*b*c^3 + 24*b*c^2 - 20*b*c + 6*b - c^4 + 4*c^3 - 6*c^2 + 4*c - 1)/(a^6*b^2*c^2 + 4*a^5*b^3*c^2 + 3*a^5*b^2*c^3 - 4*a^5*b^2*c^2 + 6*a^4*b^4*c^2 + 9*a^4*b^3*c^3 - 12*a^4*b^3*c^2 + 3*a^4*b^2*c^4 - 9*a^4*b^2*c^3 + 6*a^4*b^2*c^2 + 4*a^3*b^5*c^2 + 9*a^3*b^4*c^3 - 12*a^3*b^4*c^2 + 6*a^3*b^3*c^4 - 18*a^3*b^3*c^3 + 12*a^3*b^3*c^2 + a^3*b^2*c^5 - 6*a^3*b^2*c^4 + 9*a^3*b^2*c^3 - 4*a^3*b^2*c^2 + a^2*b^6*c^2 + 3*a^2*b^5*c^3 - 4*a^2*b^5*c^2 + 3*a^2*b^4*c^4 - 9*a^2*b^4*c^3 + 6*a^2*b^4*c^2 + a^2*b^3*c^5 - 6*a^2*b^3*c^4 + 9*a^2*b^3*c^3 - 4*a^2*b^3*c^2 - a^2*b^2*c^5 + 3*a^2*b^2*c^4 - 3*a^2*b^2*c^3 + a^2*b^2*c^2) (2*a^5*b*c^2 + 3*a^5*b*c - a^5*c - a^5 - 2*a^4*b^2*c^3 + 3*a^4*b^2*c^2 + 12*a^4*b^2*c + 6*a^4*b*c^3 + 7*a^4*b*c^2 - 14*a^4*b*c - 5*a^4*b - 4*a^4*c^2 + 5*a^4 - 4*a^3*b^3*c^3 - 3*a^3*b^3*c^2 + 18*a^3*b^3*c - 4*a^3*b^2*c^4 + 5*a^3*b^2*c^3 + 30*a^3*b^2*c^2 - 35*a^3*b^2*c - 10*a^3*b^2 + 6*a^3*b*c^4 + 7*a^3*b*c^3 - 38*a^3*b*c^2 + 8*a^3*b*c + 20*a^3*b - 6*a^3*c^3 + 6*a^3*c^2 + 10*a^3*c - 10*a^3 - 2*a^2*b^4*c^3 - 7*a^2*b^4*c^2 + 12*a^2*b^4*c - 4*a^2*b^3*c^4 - 8*a^2*b^3*c^3 + 39*a^2*b^3*c^2 - 31*a^2*b^3*c - 10*a^2*b^3 - 2*a^2*b^2*c^5 + a^2*b^2*c^4 + 32*a^2*b^2*c^3 - 65*a^2*b^2*c^2 + 9*a^2*b^2*c + 30*a^2*b^2 + 2*a^2*b*c^5 + 5*a^2*b*c^4 - 36*a^2*b*c^3 + 28*a^2*b*c^2 + 30*a^2*b*c - 30*a^2*b - 4*a^2*c^4 + 8*a^2*c^3 + 6*a^2*c^2 - 20*a^2*c + 10*a^2 - 3*a*b^5*c^2 + 3*a*b^5*c - 7*a*b^4*c^3 + 16*a*b^4*c^2 - 8*a*b^4*c - 5*a*b^4 - 5*a*b^3*c^4 + 25*a*b^3*c^3 - 28*a*b^3*c^2 - 6*a*b^3*c + 20*a*b^3 - a*b^2*c^5 + 14*a*b^2*c^4 - 31*a*b^2*c^3 + 9*a*b^2*c^2 + 39*a*b^2*c - 30*a*b^2 + 2*a*b*c^5 - 12*a*b*c^4 + 13*a*b*c^3 + 20*a*b*c^2 - 43*a*b*c + 20*a*b - a*c^5 + 3*a*c^4 + 2*a*c^3 - 14*a*c^2 + 15*a*c - 5*a + b^5*c - b^5 + 3*b^4*c^2 - 7*b^4*c + 5*b^4 + 3*b^3*c^3 - 13*b^3*c^2 + 19*b^3*c - 10*b^3 + b^2*c^4 - 9*b^2*c^3 + 23*b^2*c^2 - 25*b^2*c + 10*b^2 - 2*b*c^4 + 10*b*c^3 - 19*b*c^2 + 16*b*c - 5*b + c^4 - 4*c^3 + 6*c^2 - 4*c + 1)/(a^5*b^2*c^2 + 4*a^4*b^3*c^2 + 3*a^4*b^2*c^3 - 4*a^4*b^2*c^2 + 6*a^3*b^4*c^2 + 9*a^3*b^3*c^3 - 12*a^3*b^3*c^2 + 3*a^3*b^2*c^4 - 9*a^3*b^2*c^3 + 6*a^3*b^2*c^2 + 4*a^2*b^5*c^2 + 9*a^2*b^4*c^3 - 12*a^2*b^4*c^2 + 6*a^2*b^3*c^4 - 18*a^2*b^3*c^3 + 12*a^2*b^3*c^2 + a^2*b^2*c^5 - 6*a^2*b^2*c^4 + 9*a^2*b^2*c^3 - 4*a^2*b^2*c^2 + a*b^6*c^2 + 3*a*b^5*c^3 - 4*a*b^5*c^2 + 3*a*b^4*c^4 - 9*a*b^4*c^3 + 6*a*b^4*c^2 + a*b^3*c^5 - 6*a*b^3*c^4 + 9*a*b^3*c^3 - 4*a*b^3*c^2 - a*b^2*c^5 + 3*a*b^2*c^4 - 3*a*b^2*c^3 + a*b^2*c^2) (-2*a^4*b*c^2 - 2*a^4*b*c + a^4*c + a^4 - 4*a^3*b^2*c^2 - 6*a^3*b^2*c - ** WARNING: Output too long, hence truncated. '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Thu Dec 8 16:44:26 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); A1:=Matrix(K,3,3,[[b,1,1],[0,a,1],[0,0,c]]); S:=Transpose(A)*A^(-1); F,T:=PrimaryRationalForm(S); B:=T*A*Transpose(T); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*B*Transpose(P)-B)); G:=GroebnerBasis(I); Q1:=Matrix(K,3,3,[(2*b-1)/(a+b-1),(a-b)/(a+b-1),0,(b-a)/(a+b-1),(2*a-1)/(a+b-1),0, 0,0,1]); Kt:=FunctionField(K); D:=(2*a*b*c-a-b-c+1)/(a*b*c); s:=t; px:=-(2*s+D); py:=1-s^2; q:=s^2+s*D+1; Q:=Matrix(Kt,3,3,[1,0,0,0,D*px/q+py/q,-px/q,0,px/q,py/q]); Q:=Q1*T^(-1)*Q*T; print (Q*A*Transpose(Q)-A1); for i in [1..3] do for j in [1..3] do print Evaluate(Numerator(Q[i,j]),a+b+c-1); end for; end for; Output: Magma V2.11-10 Thu Dec 8 2005 16:44:25 on modular [Seed = 2387960987] ------------------------------------- [0 0 0] [0 0 0] [0 0 0] (a^5*b^2*c^2 - 2*a^4*b^3*c^3 + 3*a^4*b^3*c^2 + 3*a^4*b^2*c^3 + a^4*b^2*c^2 + a^4*b^2*c - 2*a^4*b*c - 4*a^3*b^4*c^3 + 3*a^3*b^4*c^2 - 4*a^3*b^3*c^4 + 6*a^3*b^3*c^3 + 3*a^3*b^3*c + 3*a^3*b^2*c^4 + 2*a^3*b^2*c^3 + 4*a^3*b^2*c^2 - 6*a^3*b^2*c - 6*a^3*b*c^2 + 2*a^3*b*c - a^3*b + a^3 - 2*a^2*b^5*c^3 + a^2*b^5*c^2 - 4*a^2*b^4*c^4 + 3*a^2*b^4*c^3 - 3*a^2*b^4*c^2 + 3*a^2*b^4*c - 2*a^2*b^3*c^5 + 3*a^2*b^3*c^4 - 2*a^2*b^3*c^3 + 12*a^2*b^3*c^2 - 6*a^2*b^3*c + a^2*b^2*c^5 + a^2*b^2*c^4 + 7*a^2*b^2*c^3 - 15*a^2*b^2*c^2 + 2*a^2*b^2*c - 3*a^2*b^2 - 6*a^2*b*c^3 + 4*a^2*b*c^2 - 2*a^2*b*c + 6*a^2*b + 3*a^2*c - 3*a^2 - 2*a*b^5*c^2 + a*b^5*c - 4*a*b^4*c^3 + 8*a*b^4*c^2 - 2*a*b^4*c - 2*a*b^3*c^4 + 11*a*b^3*c^3 - 9*a*b^3*c^2 + a*b^3*c - 3*a*b^3 + 4*a*b^2*c^4 - 9*a*b^2*c^3 + 3*a*b^2*c^2 - 6*a*b^2*c + 9*a*b^2 - 2*a*b*c^4 + 2*a*b*c^3 - 3*a*b*c^2 + 12*a*b*c - 9*a*b + 3*a*c^2 - 6*a*c + 3*a + b^4*c - b^4 + 2*b^3*c^2 - 6*b^3*c + 4*b^3 + b^2*c^3 - 7*b^2*c^2 + 12*b^2*c - 6*b^2 - 2*b*c^3 + 8*b*c^2 - 10*b*c + 4*b + c^3 - 3*c^2 + 3*c - 1)/(a^4*b^2*c^2 + 2*a^3*b^3*c^2 + a^3*b^2*c^3 - 2*a^3*b^2*c^2 + a^2*b^4*c^2 + a^2*b^3*c^3 - 2*a^2*b^3*c^2 - a^2*b^2*c^3 + a^2*b^2*c^2) (-a^5*b^2*c^2 - 4*a^4*b^3*c^2 - 3*a^4*b^2*c^3 + a^4*b^2*c^2 + 2*a^4*b*c - 6*a^3*b^4*c^2 - 7*a^3*b^3*c^3 + 3*a^3*b^3*c^2 - 3*a^3*b^2*c^4 + 2*a^3*b^2*c^3 - 4*a^3*b^2*c^2 + 7*a^3*b^2*c + 6*a^3*b*c^2 - 3*a^3*b*c - a^3 - 4*a^2*b^5*c^2 - 5*a^2*b^4*c^3 + 3*a^2*b^4*c^2 - 2*a^2*b^3*c^4 + 4*a^2*b^3*c^3 - 8*a^2*b^3*c^2 + 9*a^2*b^3*c - a^2*b^2*c^5 + a^2*b^2*c^4 - 8*a^2*b^2*c^3 + 18*a^2*b^2*c^2 - 9*a^2*b^2*c + 6*a^2*b*c^3 - 6*a^2*b*c^2 + 2*a^2*b*c - 3*a^2*b - 3*a^2*c + 3*a^2 - a*b^6*c^2 - a*b^5*c^3 + a*b^5*c^2 + a*b^4*c^4 + 2*a*b^4*c^3 - 4*a*b^4*c^2 + 5*a*b^4*c + a*b^3*c^5 + a*b^3*c^4 - 8*a*b^3*c^3 + 14*a*b^3*c^2 - 9*a*b^3*c - 4*a*b^2*c^4 + 11*a*b^2*c^3 - 11*a*b^2*c^2 + 7*a*b^2*c - 3*a*b^2 + 2*a*b*c^4 - 3*a*b*c^3 + 4*a*b*c^2 - 9*a*b*c + 6*a*b - 3*a*c^2 + 6*a*c - 3*a + b^5*c + 2*b^4*c^2 - 3*b^4*c + b^3*c^3 - 5*b^3*c^2 + 5*b^3*c - b^3 - 2*b^2*c^3 + 7*b^2*c^2 - 8*b^2*c + 3*b^2 + 2*b*c^3 - 7*b*c^2 + 8*b*c - 3*b - c^3 + 3*c^2 - 3*c + 1)/(a^3*b^2*c^2 + 2*a^2*b^3*c^2 + a^2*b^2*c^3 - 2*a^2*b^2*c^2 + a*b^4*c^2 + a*b^3*c^3 - 2*a*b^3*c^2 - a*b^2*c^3 + a*b^2*c^2) (2*a^3*b^2*c^2 - 2*a^3*b*c + 4*a^2*b^3*c^2 + 4*a^2*b^2*c^3 - 4*a^2*b^2*c - 4*a^2*b*c^2 + a^2*b*c + a^2 + 2*a*b^4*c^2 + 4*a*b^3*c^3 - 2*a*b^3*c + 2*a*b^2*c^4 - 4*a*b^2*c^2 - 2*a*b*c^3 + a*b*c + 2*a*b + 2*a*c - 2*a - b^3*c - 2*b^2*c^2 + b^2*c + b^2 - b*c^3 + b*c^2 + 2*b*c - 2*b + c^2 - 2*c + 1)/(a^2*b*c^2 + a*b^2*c^2 + a*b*c^3 - a*b*c^2) (-a^6*b^2*c^2 - 4*a^5*b^3*c^2 - a^5*b^2*c^3 + 3*a^5*b^2*c^2 + a^5*b*c - 6*a^4*b^4*c^2 - 5*a^4*b^3*c^3 + 9*a^4*b^3*c^2 + a^4*b^2*c^4 + 6*a^4*b^2*c^3 + 2*a^4*b^2*c^2 + 5*a^4*b^2*c - 6*a^4*b*c - 4*a^3*b^5*c^2 - 7*a^3*b^4*c^3 + 9*a^3*b^4*c^2 - 2*a^3*b^3*c^4 + 12*a^3*b^3*c^3 + 4*a^3*b^3*c^2 + 9*a^3*b^3*c + a^3*b^2*c^5 + 3*a^3*b^2*c^4 + 4*a^3*b^2*c^3 - 18*a^3*b^2*c - 3*a^3*b*c^3 - 9*a^3*b*c^2 + 7*a^3*b*c - a^3*b + a^3*c + a^3 - a^2*b^6*c^2 - 3*a^2*b^5*c^3 + 3*a^2*b^5*c^2 - 3*a^2*b^4*c^4 + 6*a^2*b^4*c^3 + 2*a^2*b^4*c^2 + 7*a^2*b^4*c - a^2*b^3*c^5 + 3*a^2*b^3*c^4 + 4*a^2*b^3*c^3 + 4*a^2*b^3*c^2 - 18*a^2*b^3*c + 2*a^2*b^2*c^4 - 5*a^2*b^2*c^3 - 19*a^2*b^2*c^2 + 11*a^2*b^2*c - 3*a^2*b^2 - 2*a^2*b*c^4 - 3*a^2*b*c^3 + 8*a^2*b*c^2 + 6*a^2*b + 2*a^2*c^2 - 3*a^2 + 2*a*b^5*c + 4*a*b^4*c^2 - 6*a*b^4*c + 2*a*b^3*c^3 - 10*a*b^3*c^2 + 4*a*b^3*c - 3*a*b^3 - 4*a*b^2*c^3 + 5*a*b^2*c^2 - a*b^2*c + 9*a*b^2 + a*b*c^3 + 2*a*b*c^2 + 4*a*b*c - 9*a*b + a*c^3 - a*c^2 - 3*a*c + 3*a - b^4 - 2*b^3*c + 4*b^3 - b^2*c^2 + 6*b^2*c - 6*b^2 + 2*b*c^2 - 6*b*c + 4*b - c^2 + 2*c - 1)/(a^4*b^2*c^2 + 2*a^3*b^3*c^2 + a^3*b^2*c^3 - 2*a^3*b^2*c^2 + a^2*b^4*c^2 + a^2*b^3*c^3 - 2*a^2*b^3*c^2 - a^2*b^2*c^3 + a^2*b^2*c^2) (-2*a^4*b^2*c^3 - a^4*b^2*c^2 + 2*a^4*b*c^2 + a^4*b*c - 4*a^3*b^3*c^3 - 3*a^3*b^3*c^2 - 4*a^3*b^2*c^4 - a^3*b^2*c^3 + 3*a^3*b^2*c^2 + 3*a^3*b^2*c + 4*a^3*b*c^3 + a^3*b*c - a^3*c - a^3 - 2*a^2*b^4*c^3 - 3*a^2*b^4*c^2 - 4*a^2*b^3*c^4 - 2*a^2*b^3*c^3 + 3*a^2*b^3*c - 2*a^2*b^2*c^5 + a^2*b^2*c^4 + 2*a^2*b^2*c^3 + 4*a^2*b^2*c^2 + 3*a^2*b^2*c + 2*a^2*b*c^4 - a^2*b*c^3 + 3*a^2*b*c^2 - 7*a^2*b*c - 3*a^2*b - 2*a^2*c^2 + 3*a^2 - a*b^5*c^2 - a*b^4*c^3 - a*b^4*c^2 + a*b^4*c + a*b^3*c^4 - 2*a*b^3*c^3 + 4*a*b^3*c^2 + 3*a*b^3*c + a*b^2*c^5 - a*b^2*c^4 + 3*a*b^2*c^3 + 5*a*b^2*c^2 - 8*a*b^2*c - 3*a*b^2 + 2*a*b*c^3 - 8*a*b*c^2 + a*b*c + 6*a*b - a*c^3 + a*c^2 + 3*a*c - 3*a + b^4*c + 2*b^3*c^2 - 2*b^3*c - b^3 + b^2*c^3 - 3*b^2*c^2 - b^2*c + 3*b^2 - b*c^3 + 4*b*c - 3*b + c^2 - 2*c + 1)/(a^3*b^2*c^2 + 2*a^2*b^3*c^2 + a^2*b^2*c^3 - 2*a^2*b^2*c^2 + a*b^4*c^2 + a*b^3*c^3 - 2*a*b^3*c^2 - a*b^2*c^3 + a*b^2*c^2) (-2*a^4*b*c^2 - 4*a^3*b^2*c^2 - 4*a^3*b*c^3 + 2*a^3*b*c^2 - 2*a^3*b*c + a^3*c - 2*a^2*b^3*c^2 - 4*a^2*b^2*c^3 + 4*a^2*b^2*c^2 - 4*a^2*b^2*c - 2*a^2*b*c^4 + 4*a^2*b*c^3 - 4*a^2*b*c^2 + 4*a^2*b*c + 2*a^2*c^2 - 2*a^2*c + a^2 + 2*a*b^3*c^2 - 2*a*b^3*c + 4*a*b^2*c^3 - 4*a*b^2*c^2 + 3*a*b^2*c + 2*a*b*c^4 - 2*a*b*c^3 + 4*a*b*c^2 - 3*a*b*c + 2*a*b + a*c^3 - 3*a*c^2 + 3*a*c - 2*a - b^2*c + b^2 - 2*b*c^2 + 3*b*c - 2*b - c^3 + 2*c^2 - 2*c + 1)/(a^2*b*c^2 + a*b^2*c^2 + a*b*c^3 - a*b*c^2) (2*a^4*b^2*c^2 + 4*a^3*b^3*c^2 + 4*a^3*b^2*c^3 + 2*a^3*b^2*c - 3*a^3*b*c + 2*a^2*b^4*c^2 + 4*a^2*b^3*c^3 + 4*a^2*b^3*c + 2*a^2*b^2*c^4 + 4*a^2*b^2*c^2 - 8*a^2*b^2*c - 6*a^2*b*c^2 + 3*a^2*b*c - a^2*b + a^2 + 2*a*b^4*c + 4*a*b^3*c^2 - 5*a*b^3*c + 2*a*b^2*c^3 - 8*a*b^2*c^2 + 3*a*b^2*c - 2*a*b^2 - 3*a*b*c^3 + 3*a*b*c^2 - 2*a*b*c + 4*a*b + 2*a*c - 2*a - b^3 - 2*b^2*c + 3*b^2 - b*c^2 + 4*b*c - 3*b + c^2 - 2*c + 1)/(a^3*b^2*c + a^2*b^3*c + a^2*b^2*c^2 - a^2*b^2*c) (-2*a^3*b^2*c^2 + 2*a^3*b*c - 4*a^2*b^3*c^2 - 4*a^2*b^2*c^3 + 4*a^2*b^2*c + 4*a^2*b*c^2 - a^2*b*c - a^2 - 2*a*b^4*c^2 - 4*a*b^3*c^3 + 2*a*b^3*c - 2*a*b^2*c^4 + 4*a*b^2*c^2 + 2*a*b*c^3 - a*b*c - 2*a*b - 2*a*c + 2*a + b^3*c + 2*b^2*c^2 - b^2*c - b^2 + b*c^3 - b*c^2 - 2*b*c + 2*b - c^2 + 2*c - 1)/(a^2*b^2*c + a*b^3*c + a*b^2*c^2 - a*b^2*c) (-a^4*b*c - 3*a^3*b^2*c - a^3*b*c^2 + a^3*b*c - 3*a^2*b^3*c - 2*a^2*b^2*c^2 + 2*a^2*b^2*c + a^2*b*c^3 + 2*a^2*b*c^2 - a^2*c + a^2 - a*b^4*c - a*b^3*c^2 + a*b^3*c + a*b^2*c^3 + 2*a*b^2*c^2 + a*b*c^4 + a*b*c^3 - 2*a*b*c + 2*a*b - 2*a*c^2 + 3*a*c - 2*a - b^2*c + b^2 - 2*b*c^2 + 3*b*c - 2*b - c^3 + 2*c^2 - 2*c + 1)/(a^2*b*c + a*b^2*c + a*b*c^2 - a*b*c) Total time: 0.370 seconds, Total memory usage: 3.63MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Thu Dec 8 16:00:31 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); A1:=Matrix(K,3,3,[[b,1,1],[0,a,1],[0,0,c]]); S:=Transpose(A)*A^(-1); F,T:=PrimaryRationalForm(S); B:=T*A*Transpose(T); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*B*Transpose(P)-B)); G:=GroebnerBasis(I); Q1:=Matrix(K,3,3,[(2*b-1)/(a+b-1),(a-b)/(a+b-1),0,(b-a)/(a+b-1),(2*a-1)/(a+b-1),0, 0,0,1]); Kt:=FunctionField(K); D:=(2*a*b*c-a-b-c+1)/(a*b*c); s:=t; px:=-(2*s+D); py:=1-s^2; q:=s^2+s*D+1; Q:=Matrix(Kt,3,3,[1,0,0,0,D*px/q+py/q,-px/q,0,px/q,py/q]); Q:=Q1*T^(-1)*Q*T; print (Q*A*Transpose(Q)-A1); for i in [1..3] do for j in [1..3] do print Numerator(Q[i,j]); end for; end for; Output: Magma V2.11-10 Thu Dec 8 2005 16:00:31 on modular [Seed = 1806743208] ------------------------------------- [0 0 0] [0 0 0] [0 0 0] (a - 2*b*c + b + c - 1)/(a^2 + 2*a*b + a*c - 2*a + b^2 + b*c - 2*b - c + 1)*$.1^2 + (4*a^2*b*c + a^2*b - 2*a^2 - 4*a*b^2*c^2 + 2*a*b^2*c + 2*a*b^2 + 4*a*b*c^2 + a*b*c - 6*a*b - 4*a*c + 4*a - 2*b^3*c + b^3 - 2*b^2*c^2 + 7*b^2*c - 4*b^2 + 4*b*c^2 - 9*b*c + 5*b - 2*c^2 + 4*c - 2)/(a^3*b*c + 2*a^2*b^2*c + a^2*b*c^2 - 2*a^2*b*c + a*b^3*c + a*b^2*c^2 - 2*a*b^2*c - a*b*c^2 + a*b*c)*$.1 + (3*a^3*b^2*c^2 + 3*a^3*b^2*c - 4*a^3*b*c - a^3*b + a^3 - 2*a^2*b^3*c^3 + a^2*b^3*c^2 + 6*a^2*b^3*c + 3*a^2*b^2*c^3 + 6*a^2*b^2*c^2 - 13*a^2*b^2*c - 3*a^2*b^2 - 8*a^2*b*c^2 + 4*a^2*b*c + 6*a^2*b + 3*a^2*c - 3*a^2 - 2*a*b^4*c^2 + 3*a*b^4*c - 2*a*b^3*c^3 + 11*a*b^3*c^2 - 8*a*b^3*c - 3*a*b^3 + 6*a*b^2*c^3 - 15*a*b^2*c^2 + a*b^2*c + 9*a*b^2 - 4*a*b*c^3 + 3*a*b*c^2 + 10*a*b*c - 9*a*b + 3*a*c^2 - 6*a*c + 3*a + b^4*c - b^4 + 2*b^3*c^2 - 6*b^3*c + 4*b^3 + b^2*c^3 - 7*b^2*c^2 + 12*b^2*c - 6*b^2 - 2*b*c^3 + 8*b*c^2 - 10*b*c + 4*b + c^3 - 3*c^2 + 3*c - 1)/(a^4*b^2*c^2 + 2*a^3*b^3*c^2 + a^3*b^2*c^3 - 2*a^3*b^2*c^2 + a^2*b^4*c^2 + a^2*b^3*c^3 - 2*a^2*b^3*c^2 - a^2*b^2*c^3 + a^2*b^2*c^2) (-a^2 - 2*a*b - a*c + a - b^2 + b*c + b)/(a^2 + 2*a*b + a*c - 2*a + b^2 + b*c - 2*b - c + 1)*$.1^2 + (-2*a^3*b*c + 2*a^3 - 4*a^2*b^2*c - 2*a^2*b*c^2 - 2*a^2*b*c + 5*a^2*b + 4*a^2*c - 4*a^2 - 2*a*b^3*c + 2*a*b^2*c^2 - 2*a*b^2*c + 4*a*b^2 - 4*a*b*c^2 + 9*a*b*c - 6*a*b + 2*a*c^2 - 4*a*c + 2*a + b^3 + b^2*c - 2*b^2 - b*c + b)/(a^3*b*c + 2*a^2*b^2*c + a^2*b*c^2 - 2*a^2*b*c + a*b^3*c + a*b^2*c^2 - 2*a*b^2*c - a*b*c^2 + a*b*c)*$.1 + (-a^3*b^2*c^2 + 3*a^3*b*c - a^3 - 2*a^2*b^3*c^2 - a^2*b^2*c^3 - 3*a^2*b^2*c^2 + 6*a^2*b^2*c + 6*a^2*b*c^2 - 4*a^2*b*c - 3*a^2*b - 3*a^2*c + 3*a^2 - a*b^4*c^2 + a*b^3*c^3 - 3*a*b^3*c^2 + 3*a*b^3*c - 4*a*b^2*c^3 + 9*a*b^2*c^2 - 2*a*b^2*c - 3*a*b^2 + 3*a*b*c^3 - 2*a*b*c^2 - 7*a*b*c + 6*a*b - 3*a*c^2 + 6*a*c - 3*a - b^3*c^2 + 2*b^3*c - b^3 - b^2*c^3 + 5*b^2*c^2 - 7*b^2*c + 3*b^2 + 2*b*c^3 - 7*b*c^2 + 8*b*c - 3*b - c^3 + 3*c^2 - 3*c + 1)/(a^3*b^2*c^2 + 2*a^2*b^3*c^2 + a^2*b^2*c^3 - 2*a^2*b^2*c^2 + a*b^4*c^2 + a*b^3*c^3 - 2*a*b^3*c^2 - a*b^2*c^3 + a*b^2*c^2) 2*b/(a + b + c - 1)*$.1^2 + (-2*a^2 + 4*a*b*c - 2*a*b - 2*a*c + a - b - c + 1)/(a^2*c + a*b*c + a*c^2 - a*c)*$.1 + (-2*a^2*b*c + a^2 + 2*a*b^2*c^2 - 2*a*b^2*c - 2*a*b*c^2 + a*b*c + 2*a*b + 2*a*c - 2*a - b^2*c + b^2 - b*c^2 + 3*b*c - 2*b + c^2 - 2*c + 1)/(a^2*b*c^2 + a*b^2*c^2 + a*b*c^3 - a*b*c^2) (-a^2 - 2*a*b + a*c + 3*a - b^2 - b*c + 3*b - 2)/(a^2 + 2*a*b + a*c - 2*a + b^2 + b*c - 2*b - c + 1)*$.1^2 + (-2*a^3*b*c + a^3 - 4*a^2*b^2*c + 2*a^2*b*c^2 + 10*a^2*b*c + 4*a^2*b - a^2*c - 5*a^2 - 2*a*b^3*c - 2*a*b^2*c^2 + 10*a*b^2*c + 5*a*b^2 + 4*a*b*c^2 - 7*a*b*c - 12*a*b - 2*a*c^2 - 2*a*c + 7*a + 2*b^3 + 2*b^2*c - 7*b^2 - 5*b*c + 8*b + 3*c - 3)/(a^3*b*c + 2*a^2*b^2*c + a^2*b*c^2 - 2*a^2*b*c + a*b^3*c + a*b^2*c^2 - 2*a*b^2*c - a*b*c^2 + a*b*c)*$.1 + (-a^4*b^2*c^2 - 2*a^3*b^3*c^2 + a^3*b^2*c^3 + 7*a^3*b^2*c^2 + 3*a^3*b^2*c - 3*a^3*b*c^2 - 5*a^3*b*c - a^3*b + a^3*c + a^3 - a^2*b^4*c^2 - a^2*b^3*c^3 + 7*a^2*b^3*c^2 + 6*a^2*b^3*c + 4*a^2*b^2*c^3 - 5*a^2*b^2*c^2 - 16*a^2*b^2*c - 3*a^2*b^2 - 3*a^2*b*c^3 - 4*a^2*b*c^2 + 10*a^2*b*c + 6*a^2*b + 2*a^2*c^2 - 3*a^2 + 3*a*b^4*c + 4*a*b^3*c^2 - 11*a*b^3*c - 3*a*b^3 + a*b^2*c^3 - 11*a*b^2*c^2 + 10*a*b^2*c + 9*a*b^2 - 2*a*b*c^3 + 8*a*b*c^2 + a*b*c - 9*a*b + a*c^3 - a*c^2 - 3*a*c + 3*a - b^4 - 2*b^3*c + 4*b^3 - b^2*c^2 + 6*b^2*c - 6*b^2 + 2*b*c^2 - 6*b*c + 4*b - c^2 + 2*c - 1)/(a^4*b^2*c^2 + 2*a^3*b^3*c^2 + a^3*b^2*c^3 - 2*a^3*b^2*c^2 + a^2*b^4*c^2 + a^2*b^3*c^3 - 2*a^2*b^3*c^2 - a^2*b^2*c^3 + a^2*b^2*c^2) (-2*a*c - a - b + c + 1)/(a^2 + 2*a*b + a*c - 2*a + b^2 + b*c - 2*b - c + 1)*$.1^2 + (2*a^3*c + a^3 - 4*a^2*b*c^2 - 2*a^2*b*c + 2*a^2*b + 2*a^2*c^2 - a^2*c - a^2 - 4*a*b^2*c + a*b^2 + 5*a*b*c - a + b^2 + b*c - 2*b - c + 1)/(a^3*b*c + 2*a^2*b^2*c + a^2*b*c^2 - 2*a^2*b*c + a*b^3*c + a*b^2*c^2 - 2*a*b^2*c - a*b*c^2 + a*b*c)*$.1 + (2*a^3*b*c^2 + 3*a^3*b*c - a^3*c - a^3 - 2*a^2*b^2*c^3 - a^2*b^2*c^2 + 6*a^2*b^2*c + 2*a^2*b*c^3 + 3*a^2*b*c^2 - 7*a^2*b*c - 3*a^2*b - 2*a^2*c^2 + 3*a^2 - 3*a*b^3*c^2 + 3*a*b^3*c - a*b^2*c^3 + 8*a*b^2*c^2 - 5*a*b^2*c - 3*a*b^2 + 2*a*b*c^3 - 6*a*b*c^2 - a*b*c + 6*a*b - a*c^3 + a*c^2 + 3*a*c - 3*a + b^3*c - b^3 + b^2*c^2 - 4*b^2*c + 3*b^2 - 2*b*c^2 + 5*b*c - 3*b + c^2 - 2*c + 1)/(a^3*b^2*c^2 + 2*a^2*b^3*c^2 + a^2*b^2*c^3 - 2*a^2*b^2*c^2 + a*b^4*c^2 + a*b^3*c^3 - 2*a*b^3*c^2 - a*b^2*c^3 + a*b^2*c^2) (-2*a + 2)/(a + b + c - 1)*$.1^2 + (-4*a^2*b*c - 2*a^2*b + a^2 - 2*a*b^2 + 2*a*b*c + 3*a*b + a*c - 2*a - b - c + 1)/(a^2*b*c + a*b^2*c + a*b*c^2 - a*b*c)*$.1 + (-2*a^2*b*c^2 - 2*a^2*b*c + a^2*c + a^2 - 2*a*b^2*c + 3*a*b*c + 2*a*b + a*c^2 - 2*a + b^2 + b*c - 2*b - c + 1)/(a^2*b*c^2 + a*b^2*c^2 + a*b*c^3 - a*b*c^2) 2*c/(a + b + c - 1)*$.1^2 + (4*a*b*c + 2*a*b - 3*a + 2*b^2 + 2*b*c - 5*b - 3*c + 3)/(a^2*b + a*b^2 + a*b*c - a*b)*$.1 + (2*a^2*b^2*c^2 + 2*a^2*b^2*c - 3*a^2*b*c - a^2*b + a^2 + 2*a*b^3*c + 2*a*b^2*c^2 - 5*a*b^2*c - 2*a*b^2 - 3*a*b*c^2 + a*b*c + 4*a*b + 2*a*c - 2*a - b^3 - 2*b^2*c + 3*b^2 - b*c^2 + 4*b*c - 3*b + c^2 - 2*c + 1)/(a^3*b^2*c + a^2*b^3*c + a^2*b^2*c^2 - a^2*b^2*c) -2*c/(a + b + c - 1)*$.1^2 + (2*a^2 - 4*a*b*c + 2*a*b + 2*a*c - a + b + c - 1)/(a^2*b + a*b^2 + a*b*c - a*b)*$.1 + (2*a^2*b*c - a^2 - 2*a*b^2*c^2 + 2*a*b^2*c + 2*a*b*c^2 - a*b*c - 2*a*b - 2*a*c + 2*a + b^2*c - b^2 + b*c^2 - 3*b*c + 2*b - c^2 + 2*c - 1)/(a^2*b^2*c + a*b^3*c + a*b^2*c^2 - a*b^2*c) (-a - b + c + 1)/(a + b + c - 1)*$.1^2 + (-2*a^2*b - 2*a*b^2 + 2*a*b*c + 2*a*b - a - b - c + 1)/(a^2*b + a*b^2 + a*b*c - a*b)*$.1 + (-a^2*b*c + a^2 - a*b^2*c + a*b*c^2 + a*b*c + 2*a*b + a*c - 2*a + b^2 + b*c - 2*b - c + 1)/(a^2*b*c + a*b^2*c + a*b*c^2 - a*b*c) Total time: 0.370 seconds, Total memory usage: 3.63MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Thu Dec 8 15:58:36 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); A1:=Matrix(K,3,3,[[b,1,1],[0,a,1],[0,0,c]]); S:=Transpose(A)*A^(-1); F,T:=PrimaryRationalForm(S); B:=T*A*Transpose(T); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*B*Transpose(P)-B)); G:=GroebnerBasis(I); Q1:=Matrix(K,3,3,[(2*b-1)/(a+b-1),(a-b)/(a+b-1),0,(b-a)/(a+b-1),(2*a-1)/(a+b-1),0, 0,0,1]); Kt:=FunctionField(K); D:=(2*a*b*c-a-b-c+1)/(a*b*c); s:=t; px:=-(2*s+D); py:=1-s^2; q:=s^2+s*D+1; Q:=Matrix(Kt,3,3,[1,0,0,0,D*px/q+py/q,-px/q,0,px/q,py/q]); Q:=Q1*T^(-1)*Q*T; print (Q*A*Transpose(Q)-A1); for i in [1..3] do for j in [1..3] do print Denominator(Q[i,j]); end for; end for; Output: Magma V2.11-10 Thu Dec 8 2005 15:58:35 on modular [Seed = 687076735] ------------------------------------- [0 0 0] [0 0 0] [0 0 0] $.1^2 + (2*a*b*c - a - b - c + 1)/(a*b*c)*$.1 + 1 $.1^2 + (2*a*b*c - a - b - c + 1)/(a*b*c)*$.1 + 1 $.1^2 + (2*a*b*c - a - b - c + 1)/(a*b*c)*$.1 + 1 $.1^2 + (2*a*b*c - a - b - c + 1)/(a*b*c)*$.1 + 1 $.1^2 + (2*a*b*c - a - b - c + 1)/(a*b*c)*$.1 + 1 $.1^2 + (2*a*b*c - a - b - c + 1)/(a*b*c)*$.1 + 1 $.1^2 + (2*a*b*c - a - b - c + 1)/(a*b*c)*$.1 + 1 $.1^2 + (2*a*b*c - a - b - c + 1)/(a*b*c)*$.1 + 1 $.1^2 + (2*a*b*c - a - b - c + 1)/(a*b*c)*$.1 + 1 Total time: 0.370 seconds, Total memory usage: 3.63MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Thu Dec 8 15:58:07 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); A1:=Matrix(K,3,3,[[b,1,1],[0,a,1],[0,0,c]]); S:=Transpose(A)*A^(-1); F,T:=PrimaryRationalForm(S); B:=T*A*Transpose(T); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*B*Transpose(P)-B)); G:=GroebnerBasis(I); Q1:=Matrix(K,3,3,[(2*b-1)/(a+b-1),(a-b)/(a+b-1),0,(b-a)/(a+b-1),(2*a-1)/(a+b-1),0, 0,0,1]); Kt:=FunctionField(K); D:=(2*a*b*c-a-b-c+1)/(a*b*c); s:=t; px:=-(2*s+D); py:=1-s^2; q:=s^2+s*D+1; Q:=Matrix(Kt,3,3,[1,0,0,0,D*px/q+py/q,-px/q,0,px/q,py/q]); Q:=Q1*T^(-1)*Q*T; print (Q*A*Transpose(Q)-A1); print Denominator(Q[3,3]); Output: Magma V2.11-10 Thu Dec 8 2005 15:58:06 on modular [Seed = 3540520608] ------------------------------------- [0 0 0] [0 0 0] [0 0 0] $.1^2 + (2*a*b*c - a - b - c + 1)/(a*b*c)*$.1 + 1 Total time: 0.360 seconds, Total memory usage: 3.63MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Thu Dec 8 15:57:42 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); A1:=Matrix(K,3,3,[[b,1,1],[0,a,1],[0,0,c]]); S:=Transpose(A)*A^(-1); F,T:=PrimaryRationalForm(S); B:=T*A*Transpose(T); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*B*Transpose(P)-B)); G:=GroebnerBasis(I); Q1:=Matrix(K,3,3,[(2*b-1)/(a+b-1),(a-b)/(a+b-1),0,(b-a)/(a+b-1),(2*a-1)/(a+b-1),0, 0,0,1]); Kt:=FunctionField(K); D:=(2*a*b*c-a-b-c+1)/(a*b*c); s:=t; px:=-(2*s+D); py:=1-s^2; q:=s^2+s*D+1; Q:=Matrix(Kt,3,3,[1,0,0,0,D*px/q+py/q,-px/q,0,px/q,py/q]); Q:=Q1*T^(-1)*Q*T; print (Q*A*Transpose(Q)-A1); print Numerator(Q[3,3]); Output: Magma V2.11-10 Thu Dec 8 2005 15:57:42 on modular [Seed = 3557755327] ------------------------------------- [0 0 0] [0 0 0] [0 0 0] (-a - b + c + 1)/(a + b + c - 1)*$.1^2 + (-2*a^2*b - 2*a*b^2 + 2*a*b*c + 2*a*b - a - b - c + 1)/(a^2*b + a*b^2 + a*b*c - a*b)*$.1 + (-a^2*b*c + a^2 - a*b^2*c + a*b*c^2 + a*b*c + 2*a*b + a*c - 2*a + b^2 + b*c - 2*b - c + 1)/(a^2*b*c + a*b^2*c + a*b*c^2 - a*b*c) Total time: 0.370 seconds, Total memory usage: 3.63MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Thu Dec 8 15:55:37 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); A1:=Matrix(K,3,3,[[b,1,1],[0,a,1],[0,0,c]]); S:=Transpose(A)*A^(-1); F,T:=PrimaryRationalForm(S); B:=T*A*Transpose(T); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*B*Transpose(P)-B)); G:=GroebnerBasis(I); Q1:=Matrix(K,3,3,[(2*b-1)/(a+b-1),(a-b)/(a+b-1),0,(b-a)/(a+b-1),(2*a-1)/(a+b-1),0, 0,0,1]); Kt:=FunctionField(K); D:=(2*a*b*c-a-b-c+1)/(a*b*c); s:=t; px:=-(2*s+D); py:=1-s^2; q:=s^2+s*D+1; Q:=Matrix(Kt,3,3,[1,0,0,0,D*px/q+py/q,-px/q,0,px/q,py/q]); Q:=Q1*T^(-1)*Q*T; print (Q*A*Transpose(Q)-A1); print Q; Output: Magma V2.11-10 Thu Dec 8 2005 15:55:37 on modular [Seed = 3640788797] ------------------------------------- [0 0 0] [0 0 0] [0 0 0] [((a - 2*b*c + b + c - 1)/(a^2 + 2*a*b + a*c - 2*a + b^2 + b*c - 2*b - c + 1)*t^2 + (4*a^2*b*c + a^2*b - 2*a^2 - 4*a*b^2*c^2 + 2*a*b^2*c + 2*a*b^2 + 4*a*b*c^2 + a*b*c - 6*a*b - 4*a*c + 4*a - 2*b^3*c + b^3 - 2*b^2*c^2 + 7*b^2*c - 4*b^2 + 4*b*c^2 - 9*b*c + 5*b - 2*c^2 + 4*c - 2)/(a^3*b*c + 2*a^2*b^2*c + a^2*b*c^2 - 2*a^2*b*c + a*b^3*c + a*b^2*c^2 - 2*a*b^2*c - a*b*c^2 + a*b*c)*t + (3*a^3*b^2*c^2 + 3*a^3*b^2*c - 4*a^3*b*c - a^3*b + a^3 - 2*a^2*b^3*c^3 + a^2*b^3*c^2 + 6*a^2*b^3*c + 3*a^2*b^2*c^3 + 6*a^2*b^2*c^2 - 13*a^2*b^2*c - 3*a^2*b^2 - 8*a^2*b*c^2 + 4*a^2*b*c + 6*a^2*b + 3*a^2*c - 3*a^2 - 2*a*b^4*c^2 + 3*a*b^4*c - 2*a*b^3*c^3 + 11*a*b^3*c^2 - 8*a*b^3*c - 3*a*b^3 + 6*a*b^2*c^3 - 15*a*b^2*c^2 + a*b^2*c + 9*a*b^2 - 4*a*b*c^3 + 3*a*b*c^2 + 10*a*b*c - 9*a*b + 3*a*c^2 - 6*a*c + 3*a + b^4*c - b^4 + 2*b^3*c^2 - 6*b^3*c + 4*b^3 + b^2*c^3 - 7*b^2*c^2 + 12*b^2*c - 6*b^2 - 2*b*c^3 + 8*b*c^2 - 10*b*c + 4*b + c^3 - 3*c^2 + 3*c - 1)/(a^4*b^2*c^2 + 2*a^3*b^3*c^2 + a^3*b^2*c^3 - 2*a^3*b^2*c^2 + a^2*b^4*c^2 + a^2*b^3*c^3 - 2*a^2*b^3*c^2 - a^2*b^2*c^3 + a^2*b^2*c^2))/(t^2 + (2*a*b*c - a - b - c + 1)/(a*b*c)*t + 1) ((-a^2 - 2*a*b - a*c + a - b^2 + b*c + b)/(a^2 + 2*a*b + a*c - 2*a + b^2 + b*c - 2*b - c + 1)*t^2 + (-2*a^3*b*c + 2*a^3 - 4*a^2*b^2*c - 2*a^2*b*c^2 - 2*a^2*b*c + 5*a^2*b + 4*a^2*c - 4*a^2 - 2*a*b^3*c + 2*a*b^2*c^2 - 2*a*b^2*c + 4*a*b^2 - 4*a*b*c^2 + 9*a*b*c - 6*a*b + 2*a*c^2 - 4*a*c + 2*a + b^3 + b^2*c - 2*b^2 - b*c + b)/(a^3*b*c + 2*a^2*b^2*c + a^2*b*c^2 - 2*a^2*b*c + a*b^3*c + a*b^2*c^2 - 2*a*b^2*c - a*b*c^2 + a*b*c)*t + (-a^3*b^2*c^2 + 3*a^3*b*c - a^3 - 2*a^2*b^3*c^2 - a^2*b^2*c^3 - 3*a^2*b^2*c^2 + 6*a^2*b^2*c + 6*a^2*b*c^2 - 4*a^2*b*c - 3*a^2*b - 3*a^2*c + 3*a^2 - a*b^4*c^2 + a*b^3*c^3 - 3*a*b^3*c^2 + 3*a*b^3*c - 4*a*b^2*c^3 + 9*a*b^2*c^2 - 2*a*b^2*c - 3*a*b^2 + 3*a*b*c^3 - 2*a*b*c^2 - 7*a*b*c + 6*a*b - 3*a*c^2 + 6*a*c - 3*a - b^3*c^2 + 2*b^3*c - b^3 - b^2*c^3 + 5*b^2*c^2 - 7*b^2*c + 3*b^2 + 2*b*c^3 - 7*b*c^2 + 8*b*c - 3*b - c^3 + 3*c^2 - 3*c + 1)/(a^3*b^2*c^2 + 2*a^2*b^3*c^2 + a^2*b^2*c^3 - 2*a^2*b^2*c^2 + a*b^4*c^2 + a*b^3*c^3 - 2*a*b^3*c^2 - a*b^2*c^3 + a*b^2*c^2))/(t^2 + (2*a*b*c - a - b - c + 1)/(a*b*c)*t + 1) (2*b/(a + b + c - 1)*t^2 + (-2*a^2 + 4*a*b*c - 2*a*b - 2*a*c + a - b - c + 1)/(a^2*c + a*b*c + a*c^2 - a*c)*t + (-2*a^2*b*c + a^2 + 2*a*b^2*c^2 - 2*a*b^2*c - 2*a*b*c^2 + a*b*c + 2*a*b + 2*a*c - 2*a - b^2*c + b^2 - b*c^2 + 3*b*c - 2*b + c^2 - 2*c + 1)/(a^2*b*c^2 + a*b^2*c^2 + a*b*c^3 - a*b*c^2))/(t^2 + (2*a*b*c - a - b - c + 1)/(a*b*c)*t + 1)] [((-a^2 - 2*a*b + a*c + 3*a - b^2 - b*c + 3*b - 2)/(a^2 + 2*a*b + a*c - 2*a + b^2 + b*c - 2*b - c + 1)*t^2 + (-2*a^3*b*c + a^3 - 4*a^2*b^2*c + 2*a^2*b*c^2 + 10*a^2*b*c + 4*a^2*b - a^2*c - 5*a^2 - 2*a*b^3*c - 2*a*b^2*c^2 + 10*a*b^2*c + 5*a*b^2 + 4*a*b*c^2 - 7*a*b*c - 12*a*b - 2*a*c^2 - 2*a*c + 7*a + 2*b^3 + 2*b^2*c - 7*b^2 - 5*b*c + 8*b + 3*c - 3)/(a^3*b*c + 2*a^2*b^2*c + a^2*b*c^2 - 2*a^2*b*c + a*b^3*c + a*b^2*c^2 - 2*a*b^2*c - a*b*c^2 + a*b*c)*t + (-a^4*b^2*c^2 - 2*a^3*b^3*c^2 + a^3*b^2*c^3 + 7*a^3*b^2*c^2 + 3*a^3*b^2*c - 3*a^3*b*c^2 - 5*a^3*b*c - a^3*b + a^3*c + a^3 - a^2*b^4*c^2 - a^2*b^3*c^3 + 7*a^2*b^3*c^2 + 6*a^2*b^3*c + 4*a^2*b^2*c^3 - 5*a^2*b^2*c^2 - 16*a^2*b^2*c - 3*a^2*b^2 - 3*a^2*b*c^3 - 4*a^2*b*c^2 + 10*a^2*b*c + 6*a^2*b + 2*a^2*c^2 - 3*a^2 + 3*a*b^4*c + 4*a*b^3*c^2 - 11*a*b^3*c - 3*a*b^3 + a*b^2*c^3 - 11*a*b^2*c^2 + 10*a*b^2*c + 9*a*b^2 - 2*a*b*c^3 + 8*a*b*c^2 + a*b*c - 9*a*b + a*c^3 - a*c^2 - 3*a*c + 3*a - b^4 - 2*b^3*c + 4*b^3 - b^2*c^2 + 6*b^2*c - 6*b^2 + 2*b*c^2 - 6*b*c + 4*b - c^2 + 2*c - 1)/(a^4*b^2*c^2 + 2*a^3*b^3*c^2 + a^3*b^2*c^3 - 2*a^3*b^2*c^2 + a^2*b^4*c^2 + a^2*b^3*c^3 - 2*a^2*b^3*c^2 - a^2*b^2*c^3 + a^2*b^2*c^2))/(t^2 + (2*a*b*c - a - b - c + 1)/(a*b*c)*t + 1) ((-2*a*c - a - b + c + 1)/(a^2 + 2*a*b + a*c - 2*a + b^2 + b*c - 2*b - c + 1)*t^2 + (2*a^3*c + a^3 - 4*a^2*b*c^2 - 2*a^2*b*c + 2*a^2*b + 2*a^2*c^2 - a^2*c - a^2 - 4*a*b^2*c + a*b^2 + 5*a*b*c - a + b^2 + b*c - 2*b - c + 1)/(a^3*b*c + 2*a^2*b^2*c + a^2*b*c^2 - 2*a^2*b*c + a*b^3*c + a*b^2*c^2 - 2*a*b^2*c - a*b*c^2 + a*b*c)*t + (2*a^3*b*c^2 + 3*a^3*b*c - a^3*c - a^3 - 2*a^2*b^2*c^3 - a^2*b^2*c^2 + 6*a^2*b^2*c + 2*a^2*b*c^3 + 3*a^2*b*c^2 - 7*a^2*b*c - 3*a^2*b - 2*a^2*c^2 + 3*a^2 - 3*a*b^3*c^2 + 3*a*b^3*c - a*b^2*c^3 + 8*a*b^2*c^2 - 5*a*b^2*c - 3*a*b^2 + 2*a*b*c^3 - 6*a*b*c^2 - a*b*c + 6*a*b - a*c^3 + a*c^2 + 3*a*c - 3*a + b^3*c - b^3 + b^2*c^2 - 4*b^2*c + 3*b^2 - 2*b*c^2 + 5*b*c - 3*b + c^2 - 2*c + 1)/(a^3*b^2*c^2 + 2*a^2*b^3*c^2 + a^2*b^2*c^3 - 2*a^2*b^2*c^2 + a*b^4*c^2 + a*b^3*c^3 - 2*a*b^3*c^2 - a*b^2*c^3 + a*b^2*c^2))/(t^2 + (2*a*b*c - a - b - c + 1)/(a*b*c)*t + 1) ((-2*a + 2)/(a + b + c - 1)*t^2 + (-4*a^2*b*c - 2*a^2*b + a^2 - 2*a*b^2 + 2*a*b*c + 3*a*b + a*c - 2*a - b - c + 1)/(a^2*b*c + a*b^2*c + a*b*c^2 - a*b*c)*t + (-2*a^2*b*c^2 - 2*a^2*b*c + a^2*c + a^2 - 2*a*b^2*c + 3*a*b*c + 2*a*b + a*c^2 - 2*a + b^2 + b*c - 2*b - c + 1)/(a^2*b*c^2 + a*b^2*c^2 + a*b*c^3 - a*b*c^2))/(t^2 + (2*a*b*c - a - b - c + 1)/(a*b*c)*t + 1)] [(2*c/(a + b + c - 1)*t^2 + (4*a*b*c + 2*a*b - 3*a + 2*b^2 + 2*b*c - 5*b - 3*c + 3)/(a^2*b + a*b^2 + a*b*c - a*b)*t + (2*a^2*b^2*c^2 + 2*a^2*b^2*c - 3*a^2*b*c - a^2*b + a^2 + 2*a*b^3*c + 2*a*b^2*c^2 - 5*a*b^2*c - 2*a*b^2 - 3*a*b*c^2 + a*b*c + 4*a*b + 2*a*c - 2*a - b^3 - 2*b^2*c + 3*b^2 - b*c^2 + 4*b*c - 3*b + c^2 - 2*c + 1)/(a^3*b^2*c + a^2*b^3*c + a^2*b^2*c^2 - a^2*b^2*c))/(t^2 + (2*a*b*c - a - b - c + 1)/(a*b*c)*t + 1) (-2*c/(a + b + c - 1)*t^2 + (2*a^2 - 4*a*b*c + 2*a*b + 2*a*c - a + b + c - 1)/(a^2*b + a*b^2 + a*b*c - a*b)*t + (2*a^2*b*c - a^2 - 2*a*b^2*c^2 + 2*a*b^2*c + 2*a*b*c^2 - a*b*c - 2*a*b - 2*a*c + 2*a + b^2*c - b^2 + b*c^2 - 3*b*c + 2*b - c^2 + 2*c - 1)/(a^2*b^2*c + a*b^3*c + a*b^2*c^2 - a*b^2*c))/(t^2 + (2*a*b*c - a - b - c + 1)/(a*b*c)*t + 1) ((-a - b + c + 1)/(a + b + c - 1)*t^2 + (-2*a^2*b - 2*a*b^2 + 2*a*b*c + 2*a*b - a - b - c + 1)/(a^2*b + a*b^2 + a*b*c - a*b)*t + (-a^2*b*c + a^2 - a*b^2*c + a*b*c^2 + a*b*c + 2*a*b + a*c - 2*a + b^2 + b*c - 2*b - c + 1)/(a^2*b*c + a*b^2*c + a*b*c^2 - a*b*c))/(t^2 + (2*a*b*c - a - b - c + 1)/(a*b*c)*t + 1)] Total time: 0.360 seconds, Total memory usage: 3.63MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Thu Dec 8 15:53:48 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); A1:=Matrix(K,3,3,[[b,1,1],[0,a,1],[0,0,c]]); S:=Transpose(A)*A^(-1); F,T:=PrimaryRationalForm(S); B:=T*A*Transpose(T); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*B*Transpose(P)-B)); G:=GroebnerBasis(I); Q1:=Matrix(K,3,3,[(2*b-1)/(a+b-1),(a-b)/(a+b-1),0,(b-a)/(a+b-1),(2*a-1)/(a+b-1),0, 0,0,1]); Kt:=FunctionField(K); D:=(2*a*b*c-a-b-c+1)/(a*b*c); s:=t-D/2; px:=-(2*s+D); py:=1-s^2; q:=s^2+s*D+1; Q:=Matrix(Kt,3,3,[1,0,0,0,D*px/q+py/q,-px/q,0,px/q,py/q]); Q:=Q1*T^(-1)*Q*T; print (Q*A*Transpose(Q)-A1); print Q; Output: Magma V2.11-10 Thu Dec 8 2005 15:53:47 on modular [Seed = 3256425652] ------------------------------------- [0 0 0] [0 0 0] [0 0 0] [((a - 2*b*c + b + c - 1)/(a^2 + 2*a*b + a*c - 2*a + b^2 + b*c - 2*b - c + 1)*t^2 + (2*a*b*c + a*b - a - 2*b^2*c + b^2 + 2*b*c - 2*b - c + 1)/(a^2*b*c + a*b^2*c - a*b*c)*t + (2*a^2*b^2*c - a^2*b*c - 1/2*a^2*b + 1/4*a^2 + 2*a*b^3*c + 2*a*b^2*c^2 - 3*a*b^2*c - a*b^2 - a*b*c^2 + 3/2*a*b + 1/2*a*c - 1/2*a - 1/2*b^3 - b^2*c + 5/4*b^2 - 1/2*b*c^2 + 3/2*b*c - b + 1/4*c^2 - 1/2*c + 1/4)/(a^3*b^2*c^2 + a^2*b^3*c^2 - a^2*b^2*c^2))/(t^2 + (a^2*b*c - 1/4*a^2 + a*b^2*c + a*b*c^2 - a*b*c - 1/2*a*b - 1/2*a*c + 1/2*a - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2)) ((-a^2 - 2*a*b - a*c + a - b^2 + b*c + b)/(a^2 + 2*a*b + a*c - 2*a + b^2 + b*c - 2*b - c + 1)*t^2 + (a^2 - 4*a*b*c + a*b + a*c - a + b*c)/(a^2*b*c + a*b^2*c - a*b*c)*t + (a^3*b*c - 1/4*a^3 + a^2*b*c^2 - a^2*b*c - 1/4*a^2*b - 1/2*a^2*c + 1/2*a^2 - a*b^3*c - a*b^2*c^2 + a*b^2*c + 1/4*a*b^2 - 1/4*a*c^2 + 1/2*a*c - 1/4*a + 1/4*b^3 + 1/2*b^2*c - 1/2*b^2 + 1/4*b*c^2 - 1/2*b*c + 1/4*b)/(a^3*b^2*c^2 + a^2*b^3*c^2 - a^2*b^2*c^2))/(t^2 + (a^2*b*c - 1/4*a^2 + a*b^2*c + a*b*c^2 - a*b*c - 1/2*a*b - 1/2*a*c + 1/2*a - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2)) (2*b/(a + b + c - 1)*t^2 + (-2*a + 1)/(a*c)*t)/(t^2 + (a^2*b*c - 1/4*a^2 + a*b^2*c + a*b*c^2 - a*b*c - 1/2*a*b - 1/2*a*c + 1/2*a - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2))] [((-a^2 - 2*a*b + a*c + 3*a - b^2 - b*c + 3*b - 2)/(a^2 + 2*a*b + a*c - 2*a + b^2 + b*c - 2*b - c + 1)*t^2 + (4*a*b*c + a*b - a*c - a + b^2 - b*c - 2*b + 1)/(a^2*b*c + a*b^2*c - a*b*c)*t + (-a^3*b*c + 1/4*a^3 - a^2*b*c^2 + a^2*b*c + 1/4*a^2*b + 1/2*a^2*c - 1/2*a^2 + a*b^3*c + a*b^2*c^2 - a*b^2*c - 1/4*a*b^2 + 1/4*a*c^2 - 1/2*a*c + 1/4*a - 1/4*b^3 - 1/2*b^2*c + 1/2*b^2 - 1/4*b*c^2 + 1/2*b*c - 1/4*b)/(a^3*b^2*c^2 + a^2*b^3*c^2 - a^2*b^2*c^2))/(t^2 + (a^2*b*c - 1/4*a^2 + a*b^2*c + a*b*c^2 - a*b*c - 1/2*a*b - 1/2*a*c + 1/2*a - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2)) ((-2*a*c - a - b + c + 1)/(a^2 + 2*a*b + a*c - 2*a + b^2 + b*c - 2*b - c + 1)*t^2 + (2*a^2*c + a^2 - 2*a*b*c + a*b - 2*a*c - a + c)/(a^2*b*c + a*b^2*c - a*b*c)*t + (2*a^3*b*c - 1/2*a^3 + 2*a^2*b^2*c + 2*a^2*b*c^2 - 3*a^2*b*c - a^2*b - a^2*c + 5/4*a^2 - a*b^2*c - 1/2*a*b^2 - a*b*c^2 + 3/2*a*b - 1/2*a*c^2 + 3/2*a*c - a + 1/4*b^2 + 1/2*b*c - 1/2*b + 1/4*c^2 - 1/2*c + 1/4)/(a^3*b^2*c^2 + a^2*b^3*c^2 - a^2*b^2*c^2))/(t^2 + (a^2*b*c - 1/4*a^2 + a*b^2*c + a*b*c^2 - a*b*c - 1/2*a*b - 1/2*a*c + 1/2*a - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2)) ((-2*a + 2)/(a + b + c - 1)*t^2 + (-2*a*b - a + 1)/(a*b*c)*t)/(t^2 + (a^2*b*c - 1/4*a^2 + a*b^2*c + a*b*c^2 - a*b*c - 1/2*a*b - 1/2*a*c + 1/2*a - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2))] [(2*c/(a + b + c - 1)*t^2 + (2*b - 1)/(a*b)*t)/(t^2 + (a^2*b*c - 1/4*a^2 + a*b^2*c + a*b*c^2 - a*b*c - 1/2*a*b - 1/2*a*c + 1/2*a - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2)) (-2*c/(a + b + c - 1)*t^2 + (2*a - 1)/(a*b)*t)/(t^2 + (a^2*b*c - 1/4*a^2 + a*b^2*c + a*b*c^2 - a*b*c - 1/2*a*b - 1/2*a*c + 1/2*a - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2)) ((-a - b + c + 1)/(a + b + c - 1)*t^2 + (-a - b + 1)/(a*b*c)*t + (a^2*b*c - 1/4*a^2 + a*b^2*c + a*b*c^2 - a*b*c - 1/2*a*b - 1/2*a*c + 1/2*a - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2))/(t^2 + (a^2*b*c - 1/4*a^2 + a*b^2*c + a*b*c^2 - a*b*c - 1/2*a*b - 1/2*a*c + 1/2*a - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2))] Total time: 0.310 seconds, Total memory usage: 3.53MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Thu Dec 8 15:52:11 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); A1:=Matrix(K,3,3,[[b,1,1],[0,a,1],[0,0,c]]); S:=Transpose(A)*A^(-1); F,T:=PrimaryRationalForm(S); B:=T*A*Transpose(T); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*B*Transpose(P)-B)); G:=GroebnerBasis(I); P1:=Matrix(R,3,3,[R.1,R.2,0,R.4,R.5,0,0,0,1]); I1:=Ideal(Eltseq(P1*A*Transpose(P1)-A1)); print GroebnerBasis(I1); Q1:=Matrix(K,3,3,[(2*b-1)/(a+b-1),(a-b)/(a+b-1),0,(b-a)/(a+b-1),(2*a-1)/(a+b-1),0, 0,0,1]); print (Q1*A*Transpose(Q1)-A1); Kt:=FunctionField(K); D:=(2*a*b*c-a-b-c+1)/(a*b*c); s:=t-D/2; px:=-(2*s+D); py:=1-s^2; q:=s^2+s*D+1; Q:=Matrix(Kt,3,3,[1,0,0,0,D*px/q+py/q,-px/q,0,px/q,py/q]); Q:=T^(-1)*Q*T; print (Q*A*Transpose(Q)-A); print Q; Output: Magma V2.11-10 Thu Dec 8 2005 15:52:10 on modular [Seed = 3390117149] ------------------------------------- [ $.1 + (-2*b + 1)/(a + b - 1), $.2 + (-a + b)/(a + b - 1), $.4 + (a - b)/(a + b - 1), $.5 + (-2*a + 1)/(a + b - 1) ] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [((a - b - c + 1)/(a + b + c - 1)*t^2 + (b + c - 1)/(a*b*c)*t + (a^2*b*c - 1/4*a^2 + a*b^2*c + a*b*c^2 - a*b*c - 1/2*a*b - 1/2*a*c + 1/2*a - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2))/(t^2 + (a^2*b*c - 1/4*a^2 + a*b^2*c + a*b*c^2 - a*b*c - 1/2*a*b - 1/2*a*c + 1/2*a - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2)) (-2*a/(a + b + c - 1)*t^2 + (-2*c + 1)/(b*c)*t)/(t^2 + (a^2*b*c - 1/4*a^2 + a*b^2*c + a*b*c^2 - a*b*c - 1/2*a*b - 1/2*a*c + 1/2*a - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2)) (2*a/(a + b + c - 1)*t^2 + (-2*b + 1)/(b*c)*t)/(t^2 + (a^2*b*c - 1/4*a^2 + a*b^2*c + a*b*c^2 - a*b*c - 1/2*a*b - 1/2*a*c + 1/2*a - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2))] [((-2*b + 2)/(a + b + c - 1)*t^2 + (2*b*c + b - 1)/(a*b*c)*t)/(t^2 + (a^2*b*c - 1/4*a^2 + a*b^2*c + a*b*c^2 - a*b*c - 1/2*a*b - 1/2*a*c + 1/2*a - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2)) ((-a + b - c - 1)/(a + b + c - 1)*t^2 + (a - c)/(a*b*c)*t + (a^2*b*c - 1/4*a^2 + a*b^2*c + a*b*c^2 - a*b*c - 1/2*a*b - 1/2*a*c + 1/2*a - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2))/(t^2 + (a^2*b*c - 1/4*a^2 + a*b^2*c + a*b*c^2 - a*b*c - 1/2*a*b - 1/2*a*c + 1/2*a - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2)) ((-2*b + 2)/(a + b + c - 1)*t^2 + (-2*a*b - b + 1)/(a*b*c)*t)/(t^2 + (a^2*b*c - 1/4*a^2 + a*b^2*c + a*b*c^2 - a*b*c - 1/2*a*b - 1/2*a*c + 1/2*a - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2))] [(2*c/(a + b + c - 1)*t^2 + (2*b - 1)/(a*b)*t)/(t^2 + (a^2*b*c - 1/4*a^2 + a*b^2*c + a*b*c^2 - a*b*c - 1/2*a*b - 1/2*a*c + 1/2*a - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2)) (-2*c/(a + b + c - 1)*t^2 + (2*a - 1)/(a*b)*t)/(t^2 + (a^2*b*c - 1/4*a^2 + a*b^2*c + a*b*c^2 - a*b*c - 1/2*a*b - 1/2*a*c + 1/2*a - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2)) ((-a - b + c + 1)/(a + b + c - 1)*t^2 + (-a - b + 1)/(a*b*c)*t + (a^2*b*c - 1/4*a^2 + a*b^2*c + a*b*c^2 - a*b*c - 1/2*a*b - 1/2*a*c + 1/2*a - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2))/(t^2 + (a^2*b*c - 1/4*a^2 + a*b^2*c + a*b*c^2 - a*b*c - 1/2*a*b - 1/2*a*c + 1/2*a - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2))] Total time: 0.270 seconds, Total memory usage: 3.43MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Thu Dec 8 15:51:48 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); A1:=Matrix(K,3,3,[[b,1,1],[0,a,1],[0,0,c]]); S:=Transpose(A)*A^(-1); F,T:=PrimaryRationalForm(S); B:=T*A*Transpose(T); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*B*Transpose(P)-B)); G:=GroebnerBasis(I); P1:=Matrix(R,3,3,[R.1,R.2,0,R.4,R.5,0,0,0,1]); I1:=Ideal(Eltseq(P1*A*Transpose(P1)-A1)); print GroebnerBasis(I1); Q1:=Matrix(K,3,3,[(2*b-1)/(a+b-1),(a-b)/(a+b-1),0,(b-a)/(a+b-1),(2*a-1)/(a+b-1),0, 0,0,0]); print (Q1*A*Transpose(Q1)-A1); Kt:=FunctionField(K); D:=(2*a*b*c-a-b-c+1)/(a*b*c); s:=t-D/2; px:=-(2*s+D); py:=1-s^2; q:=s^2+s*D+1; Q:=Matrix(Kt,3,3,[1,0,0,0,D*px/q+py/q,-px/q,0,px/q,py/q]); Q:=T^(-1)*Q*T; print (Q*A*Transpose(Q)-A); print Q; Output: Magma V2.11-10 Thu Dec 8 2005 15:51:48 on modular [Seed = 3474198534] ------------------------------------- [ $.1 + (-2*b + 1)/(a + b - 1), $.2 + (-a + b)/(a + b - 1), $.4 + (a - b)/(a + b - 1), $.5 + (-2*a + 1)/(a + b - 1) ] [ 0 0 -1] [ 0 0 -1] [ 0 0 -c] [0 0 0] [0 0 0] [0 0 0] [((a - b - c + 1)/(a + b + c - 1)*t^2 + (b + c - 1)/(a*b*c)*t + (a^2*b*c - 1/4*a^2 + a*b^2*c + a*b*c^2 - a*b*c - 1/2*a*b - 1/2*a*c + 1/2*a - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2))/(t^2 + (a^2*b*c - 1/4*a^2 + a*b^2*c + a*b*c^2 - a*b*c - 1/2*a*b - 1/2*a*c + 1/2*a - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2)) (-2*a/(a + b + c - 1)*t^2 + (-2*c + 1)/(b*c)*t)/(t^2 + (a^2*b*c - 1/4*a^2 + a*b^2*c + a*b*c^2 - a*b*c - 1/2*a*b - 1/2*a*c + 1/2*a - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2)) (2*a/(a + b + c - 1)*t^2 + (-2*b + 1)/(b*c)*t)/(t^2 + (a^2*b*c - 1/4*a^2 + a*b^2*c + a*b*c^2 - a*b*c - 1/2*a*b - 1/2*a*c + 1/2*a - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2))] [((-2*b + 2)/(a + b + c - 1)*t^2 + (2*b*c + b - 1)/(a*b*c)*t)/(t^2 + (a^2*b*c - 1/4*a^2 + a*b^2*c + a*b*c^2 - a*b*c - 1/2*a*b - 1/2*a*c + 1/2*a - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2)) ((-a + b - c - 1)/(a + b + c - 1)*t^2 + (a - c)/(a*b*c)*t + (a^2*b*c - 1/4*a^2 + a*b^2*c + a*b*c^2 - a*b*c - 1/2*a*b - 1/2*a*c + 1/2*a - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2))/(t^2 + (a^2*b*c - 1/4*a^2 + a*b^2*c + a*b*c^2 - a*b*c - 1/2*a*b - 1/2*a*c + 1/2*a - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2)) ((-2*b + 2)/(a + b + c - 1)*t^2 + (-2*a*b - b + 1)/(a*b*c)*t)/(t^2 + (a^2*b*c - 1/4*a^2 + a*b^2*c + a*b*c^2 - a*b*c - 1/2*a*b - 1/2*a*c + 1/2*a - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2))] [(2*c/(a + b + c - 1)*t^2 + (2*b - 1)/(a*b)*t)/(t^2 + (a^2*b*c - 1/4*a^2 + a*b^2*c + a*b*c^2 - a*b*c - 1/2*a*b - 1/2*a*c + 1/2*a - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2)) (-2*c/(a + b + c - 1)*t^2 + (2*a - 1)/(a*b)*t)/(t^2 + (a^2*b*c - 1/4*a^2 + a*b^2*c + a*b*c^2 - a*b*c - 1/2*a*b - 1/2*a*c + 1/2*a - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2)) ((-a - b + c + 1)/(a + b + c - 1)*t^2 + (-a - b + 1)/(a*b*c)*t + (a^2*b*c - 1/4*a^2 + a*b^2*c + a*b*c^2 - a*b*c - 1/2*a*b - 1/2*a*c + 1/2*a - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2))/(t^2 + (a^2*b*c - 1/4*a^2 + a*b^2*c + a*b*c^2 - a*b*c - 1/2*a*b - 1/2*a*c + 1/2*a - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2))] Total time: 0.280 seconds, Total memory usage: 3.43MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Thu Dec 8 15:50:03 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); A1:=Matrix(K,3,3,[[b,1,1],[0,a,1],[0,0,c]]); S:=Transpose(A)*A^(-1); F,T:=PrimaryRationalForm(S); B:=T*A*Transpose(T); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*B*Transpose(P)-B)); G:=GroebnerBasis(I); P1:=Matrix(R,3,3,[R.1,R.2,0,R.4,R.5,0,0,0,1]); I1:=Ideal(Eltseq(P1*A*Transpose(P1)-A1)); print GroebnerBasis(I1); Kt:=FunctionField(K); D:=(2*a*b*c-a-b-c+1)/(a*b*c); s:=t-D/2; px:=-(2*s+D); py:=1-s^2; q:=s^2+s*D+1; Q:=Matrix(Kt,3,3,[1,0,0,0,D*px/q+py/q,-px/q,0,px/q,py/q]); Q:=T^(-1)*Q*T; print (Q*A*Transpose(Q)-A); print Q; Output: Magma V2.11-10 Thu Dec 8 2005 15:50:03 on modular [Seed = 4029362086] ------------------------------------- [ $.1 + (-2*b + 1)/(a + b - 1), $.2 + (-a + b)/(a + b - 1), $.4 + (a - b)/(a + b - 1), $.5 + (-2*a + 1)/(a + b - 1) ] [0 0 0] [0 0 0] [0 0 0] [((a - b - c + 1)/(a + b + c - 1)*t^2 + (b + c - 1)/(a*b*c)*t + (a^2*b*c - 1/4*a^2 + a*b^2*c + a*b*c^2 - a*b*c - 1/2*a*b - 1/2*a*c + 1/2*a - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2))/(t^2 + (a^2*b*c - 1/4*a^2 + a*b^2*c + a*b*c^2 - a*b*c - 1/2*a*b - 1/2*a*c + 1/2*a - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2)) (-2*a/(a + b + c - 1)*t^2 + (-2*c + 1)/(b*c)*t)/(t^2 + (a^2*b*c - 1/4*a^2 + a*b^2*c + a*b*c^2 - a*b*c - 1/2*a*b - 1/2*a*c + 1/2*a - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2)) (2*a/(a + b + c - 1)*t^2 + (-2*b + 1)/(b*c)*t)/(t^2 + (a^2*b*c - 1/4*a^2 + a*b^2*c + a*b*c^2 - a*b*c - 1/2*a*b - 1/2*a*c + 1/2*a - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2))] [((-2*b + 2)/(a + b + c - 1)*t^2 + (2*b*c + b - 1)/(a*b*c)*t)/(t^2 + (a^2*b*c - 1/4*a^2 + a*b^2*c + a*b*c^2 - a*b*c - 1/2*a*b - 1/2*a*c + 1/2*a - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2)) ((-a + b - c - 1)/(a + b + c - 1)*t^2 + (a - c)/(a*b*c)*t + (a^2*b*c - 1/4*a^2 + a*b^2*c + a*b*c^2 - a*b*c - 1/2*a*b - 1/2*a*c + 1/2*a - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2))/(t^2 + (a^2*b*c - 1/4*a^2 + a*b^2*c + a*b*c^2 - a*b*c - 1/2*a*b - 1/2*a*c + 1/2*a - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2)) ((-2*b + 2)/(a + b + c - 1)*t^2 + (-2*a*b - b + 1)/(a*b*c)*t)/(t^2 + (a^2*b*c - 1/4*a^2 + a*b^2*c + a*b*c^2 - a*b*c - 1/2*a*b - 1/2*a*c + 1/2*a - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2))] [(2*c/(a + b + c - 1)*t^2 + (2*b - 1)/(a*b)*t)/(t^2 + (a^2*b*c - 1/4*a^2 + a*b^2*c + a*b*c^2 - a*b*c - 1/2*a*b - 1/2*a*c + 1/2*a - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2)) (-2*c/(a + b + c - 1)*t^2 + (2*a - 1)/(a*b)*t)/(t^2 + (a^2*b*c - 1/4*a^2 + a*b^2*c + a*b*c^2 - a*b*c - 1/2*a*b - 1/2*a*c + 1/2*a - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2)) ((-a - b + c + 1)/(a + b + c - 1)*t^2 + (-a - b + 1)/(a*b*c)*t + (a^2*b*c - 1/4*a^2 + a*b^2*c + a*b*c^2 - a*b*c - 1/2*a*b - 1/2*a*c + 1/2*a - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2))/(t^2 + (a^2*b*c - 1/4*a^2 + a*b^2*c + a*b*c^2 - a*b*c - 1/2*a*b - 1/2*a*c + 1/2*a - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2))] Total time: 0.270 seconds, Total memory usage: 3.43MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Thu Dec 8 15:49:12 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); A1:=Matrix(K,3,3,[[b,1,1],[0,a,1],[0,0,c]]); S:=Transpose(A)*A^(-1); F,T:=PrimaryRationalForm(S); B:=T*A*Transpose(T); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*B*Transpose(P)-B)); G:=GroebnerBasis(I); P1:=Matrix(R,3,3,[R.1,R.2,R.3,0,1,0,R.7,R.8,R.9]); I1:=Ideal(Eltseq(P*A*Transpose(P)-A1)); print GroebnerBasis(I1); Kt:=FunctionField(K); D:=(2*a*b*c-a-b-c+1)/(a*b*c); s:=t-D/2; px:=-(2*s+D); py:=1-s^2; q:=s^2+s*D+1; Q:=Matrix(Kt,3,3,[1,0,0,0,D*px/q+py/q,-px/q,0,px/q,py/q]); Q:=T^(-1)*Q*T; print (Q*A*Transpose(Q)-A); print Q; Output: Magma V2.11-10 Thu Dec 8 2005 15:49:10 on modular [Seed = 4113443486] ------------------------------------- [ $.1 + (-a*b + b*c - b - c + 1)/(a*c + b*c - c)*$.7 + (b^2 - b*c - 2*b + 1)/(a*c + b*c - c)*$.8 + (-2*b + 1)/(a + b - 1)*$.9, $.2 + (a*b + a*c - a)/(a*c + b*c - c)*$.7 + (-a - b^2 + b*c + b)/(a*c + b*c - c)*$.8 + (-a + b)/(a + b - 1)*$.9, $.3 + b/c*$.8, $.4 + (a^2 - a*c - a - b + 1)/(a*c + b*c - c)*$.7 + (-a*b + a - b*c)/(a*c + b*c - c)*$.8 + (a - b)/(a + b - 1)*$.9, $.5 + (-a^2 + a*c)/(a*c + b*c - c)*$.7 + (a*b - a*c - 2*a - b + c + 1)/(a*c + b*c - c)*$.8 + (-2*a + 1)/(a + b - 1)*$.9, $.6 + a/c*$.7 + 1/c*$.8, $.7^2 + (b + c - 1)/(a*b - 1/2*a)*$.7*$.9 + (a*b^2 - a*b + 1/2*b^2 - b + 1/2)/(a^2*b - 1/2*a^2)*$.8^2 + (a*b - a - b*c + c)/(a^2*b - 1/2*a^2)*$.8*$.9 + (a*b*c - a*c + 1/2*c^2)/(a^2*b - 1/2*a^2)*$.9^2 + (-a*b*c + a*c - 1/2*c^2)/(a^2*b - 1/2*a^2), $.7*$.8 + (-c + 1/2)/(b - 1/2)*$.7*$.9 + (1/2*a*b - 1/2*b^2 + b - 1/2)/(a*b - 1/2*a)*$.8^2 + (1/2*a + b*c - c)/(a*b - 1/2*a)*$.8*$.9 + (1/2*a*c - 1/2*c^2)/(a*b - 1/2*a)*$.9^2 + (-1/2*a*c + 1/2*c^2)/(a*b - 1/2*a), $.7*$.9^2 + (-a*b^2*c + a*b*c - 1/4*a*c + b^2*c^2 - b*c^2 + 1/4*c^2)/(a*b^2*c + a*b*c^2 - a*b*c - 1/4*a*b - 1/4*a*c + 1/4*a - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)*$.7 + (-1/2*a^2*b^3 + 1/4*a^2*b^2 - a*b^4 + 3/2*a*b^3 - 1/2*a*b^2 - 1/2*b^5 + 5/4*b^4 - b^3 + 1/4*b^2)/(a^2*b^2*c + a^2*b*c^2 - a^2*b*c - 1/4*a^2*b - 1/4*a^2*c + 1/4*a^2 - 1/4*a*b^2 - 1/2*a*b*c + 1/2*a*b - 1/4*a*c^2 + 1/2*a*c - 1/4*a)*$.8^3 + (-1/2*a^2*b^2*c - 3/4*a^2*b^2 + 1/2*a^2*b + a*b^3*c - 3/2*a*b^3 - a*b^2*c + 5/2*a*b^2 + 1/2*a*b*c - a*b + 3/2*b^4*c - 3/4*b^4 - 3*b^3*c + 2*b^3 + 2*b^2*c - 7/4*b^2 - 1/2*b*c + 1/2*b)/(a^2*b^2*c + a^2*b*c^2 - a^2*b*c - 1/4*a^2*b - 1/4*a^2*c + 1/4*a^2 - 1/4*a*b^2 - 1/2*a*b*c + 1/2*a*b - 1/4*a*c^2 + 1/2*a*c - 1/4*a)*$.8^2*$.9 + (-1/2*a^2*b^2*c - 1/4*a^2*b*c - 1/4*a^2*b + 1/4*a^2 - 3/2*a*b^3*c - 1/2*a*b^2*c^2 + 15/4*a*b^2*c - 1/4*a*b^2 + 3/4*a*b*c^2 - 2*a*b*c + 1/2*a*b + 1/4*a*c - 1/4*a - 3/2*b^3*c^2 + 3/2*b^3*c + 9/4*b^2*c^2 - 5/2*b^2*c - b*c^2 + b*c)/(a^2*b^2*c + a^2*b*c^2 - a^2*b*c - 1/4*a^2*b - 1/4*a^2*c + 1/4*a^2 - 1/4*a*b^2 - 1/2*a*b*c + 1/2*a*b - 1/4*a*c^2 + 1/2*a*c - 1/4*a)*$.8*$.9^2 + (1/2*a^2*b^2*c - 1/4*a^2*b*c + 3/2*a*b^3*c - 1/2*a*b^2*c^2 - 11/4*a*b^2*c + 1/4*a*b*c^2 + 3/2*a*b*c - 1/4*a*c + 1/2*b^3*c^2 - 1/4*b^2*c^2)/(a^2*b^2*c + a^2*b*c^2 - a^2*b*c - 1/4*a^2*b - 1/4*a^2*c + 1/4*a^2 - 1/4*a*b^2 - 1/2*a*b*c + 1/2*a*b - 1/4*a*c^2 + 1/2*a*c - 1/4*a)*$.8 + (-1/2*a^2*b*c^2 - 1/4*a^2*b*c + 1/4*a^2*c + 3/2*a*b^2*c^2 - 1/4*a*b^2*c + 1/2*a*b*c^3 - 5/4*a*b*c^2 + 1/2*a*b*c + 1/4*a*c^2 - 1/4*a*c + 1/2*b^2*c^3 - 3/4*b^2*c^2 - 1/2*b*c^3 + 1/2*b*c^2)/(a^2*b^2*c + a^2*b*c^2 - a^2*b*c - 1/4*a^2*b - 1/4*a^2*c + 1/4*a^2 - 1/4*a*b^2 - 1/2*a*b*c + 1/2*a*b - 1/4*a*c^2 + 1/2*a*c - 1/4*a)*$.9^3 + (1/2*a^2*b*c^2 + 1/4*a^2*b*c - 1/4*a^2*c - 3/2*a*b^2*c^2 + 1/4*a*b^2*c - 1/2*a*b*c^3 + 5/4*a*b*c^2 - 1/2*a*b*c - 1/4*a*c^2 + 1/4*a*c - 1/2*b^2*c^3 + 3/4*b^2*c^2 + 1/2*b*c^3 - 1/2*b*c^2)/(a^2*b^2*c + a^2*b*c^2 - a^2*b*c - 1/4*a^2*b - 1/4*a^2*c + 1/4*a^2 - 1/4*a*b^2 - 1/2*a*b*c + 1/2*a*b - 1/4*a*c^2 + 1/2*a*c - 1/4*a)*$.9, $.8^4 + (2*a - 4*b*c + 2*b + 2*c - 2)/(a*b + b^2 - b)*$.8^3*$.9 + (2*a^2*b*c + a^2 + 2*a*b^2*c + 2*a*b*c^2 - 8*a*b*c + 2*a*b + 2*a*c - 2*a + 6*b^2*c^2 - 6*b^2*c + b^2 - 6*b*c^2 + 8*b*c - 2*b + c^2 - 2*c + 1)/(a^2*b^2 + 2*a*b^3 - 2*a*b^2 + b^4 - 2*b^3 + b^2)*$.8^2*$.9^2 + (-2*a^2*b*c - 2*a*b^2*c + 2*a*b*c^2 + 2*a*b*c - 2*b^2*c^2 + 2*b*c^2 - c^2)/(a^2*b^2 + 2*a*b^3 - 2*a*b^2 + b^4 - 2*b^3 + b^2)*$.8^2 + (2*a^2*c - 4*a*b*c^2 + 2*a*b*c + 4*a*c^2 - 4*a*c - 4*b*c^3 + 6*b*c^2 - 2*b*c + 2*c^3 - 4*c^2 + 2*c)/(a^2*b^2 + 2*a*b^3 - 2*a*b^2 + b^4 - 2*b^3 + b^2)*$.8*$.9^3 + (-2*a^2*c + 4*a*b*c^2 - 2*a*b*c + 2*a*c + 4*b*c^3 - 2*b*c^2 - 2*c^3)/(a^2*b^2 + 2*a*b^3 - 2*a*b^2 + b^4 - 2*b^3 + b^2)*$.8*$.9 + (a^2*c^2 + 2*a*c^3 - 2*a*c^2 + c^4 - 2*c^3 + c^2)/(a^2*b^2 + 2*a*b^3 - 2*a*b^2 + b^4 - 2*b^3 + b^2)*$.9^4 + (-2*a^2*c^2 + 2*a*c^2 - 2*c^4 + 2*c^3 - c^2)/(a^2*b^2 + 2*a*b^3 - 2*a*b^2 + b^4 - 2*b^3 + b^2)*$.9^2 + (a^2*c^2 - 2*a*c^3 + c^4)/(a^2*b^2 + 2*a*b^3 - 2*a*b^2 + b^4 - 2*b^3 + b^2) ] [0 0 0] [0 0 0] [0 0 0] [((a - b - c + 1)/(a + b + c - 1)*t^2 + (b + c - 1)/(a*b*c)*t + (a^2*b*c - 1/4*a^2 + a*b^2*c + a*b*c^2 - a*b*c - 1/2*a*b - 1/2*a*c + 1/2*a - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2))/(t^2 + (a^2*b*c - 1/4*a^2 + a*b^2*c + a*b*c^2 - a*b*c - 1/2*a*b - 1/2*a*c + 1/2*a - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2)) (-2*a/(a + b + c - 1)*t^2 + (-2*c + 1)/(b*c)*t)/(t^2 + (a^2*b*c - 1/4*a^2 + a*b^2*c + a*b*c^2 - a*b*c - 1/2*a*b - 1/2*a*c + 1/2*a - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2)) (2*a/(a + b + c - 1)*t^2 + (-2*b + 1)/(b*c)*t)/(t^2 + (a^2*b*c - 1/4*a^2 + a*b^2*c + a*b*c^2 - a*b*c - 1/2*a*b - 1/2*a*c + 1/2*a - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2))] [((-2*b + 2)/(a + b + c - 1)*t^2 + (2*b*c + b - 1)/(a*b*c)*t)/(t^2 + (a^2*b*c - 1/4*a^2 + a*b^2*c + a*b*c^2 - a*b*c - 1/2*a*b - 1/2*a*c + 1/2*a - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2)) ((-a + b - c - 1)/(a + b + c - 1)*t^2 + (a - c)/(a*b*c)*t + (a^2*b*c - 1/4*a^2 + a*b^2*c + a*b*c^2 - a*b*c - 1/2*a*b - 1/2*a*c + 1/2*a - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2))/(t^2 + (a^2*b*c - 1/4*a^2 + a*b^2*c + a*b*c^2 - a*b*c - 1/2*a*b - 1/2*a*c + 1/2*a - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2)) ((-2*b + 2)/(a + b + c - 1)*t^2 + (-2*a*b - b + 1)/(a*b*c)*t)/(t^2 + (a^2*b*c - 1/4*a^2 + a*b^2*c + a*b*c^2 - a*b*c - 1/2*a*b - 1/2*a*c + 1/2*a - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2))] [(2*c/(a + b + c - 1)*t^2 + (2*b - 1)/(a*b)*t)/(t^2 + (a^2*b*c - 1/4*a^2 + a*b^2*c + a*b*c^2 - a*b*c - 1/2*a*b - 1/2*a*c + 1/2*a - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2)) (-2*c/(a + b + c - 1)*t^2 + (2*a - 1)/(a*b)*t)/(t^2 + (a^2*b*c - 1/4*a^2 + a*b^2*c + a*b*c^2 - a*b*c - 1/2*a*b - 1/2*a*c + 1/2*a - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2)) ((-a - b + c + 1)/(a + b + c - 1)*t^2 + (-a - b + 1)/(a*b*c)*t + (a^2*b*c - 1/4*a^2 + a*b^2*c + a*b*c^2 - a*b*c - 1/2*a*b - 1/2*a*c + 1/2*a - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2))/(t^2 + (a^2*b*c - 1/4*a^2 + a*b^2*c + a*b*c^2 - a*b*c - 1/2*a*b - 1/2*a*c + 1/2*a - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2))] Total time: 1.060 seconds, Total memory usage: 4.36MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Thu Dec 8 15:47:06 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); A1:=Matrix(K,3,3,[[c,1,1],[0,b,1],[0,0,a]]); S:=Transpose(A)*A^(-1); F,T:=PrimaryRationalForm(S); B:=T*A*Transpose(T); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*B*Transpose(P)-B)); G:=GroebnerBasis(I); P1:=Matrix(R,3,3,[R.1,R.2,R.3,0,1,0,R.7,R.8,R.9]); I1:=Ideal(Eltseq(P1*A*Transpose(P1)-A1)); print GroebnerBasis(I1); Kt:=FunctionField(K); D:=(2*a*b*c-a-b-c+1)/(a*b*c); s:=t-D/2; px:=-(2*s+D); py:=1-s^2; q:=s^2+s*D+1; Q:=Matrix(Kt,3,3,[1,0,0,0,D*px/q+py/q,-px/q,0,px/q,py/q]); Q:=T^(-1)*Q*T; print (Q*A*Transpose(Q)-A); print Q; Output: Magma V2.11-10 Thu Dec 8 2005 15:47:06 on modular [Seed = 4280559384] ------------------------------------- [ $.1 + (-2*b*c - b + 1)/(a*b + b*c + b - 1), $.2 + (-a + c)/(a*b + b*c + b - 1), $.3 + (a*b - b*c)/(a*b + b*c + b - 1), $.7 + (-a*b + b*c)/(a*b + b*c + b - 1), $.8 + (a - c)/(a*b + b*c + b - 1), $.9 + (-2*a*b - b + 1)/(a*b + b*c + b - 1) ] [0 0 0] [0 0 0] [0 0 0] [((a - b - c + 1)/(a + b + c - 1)*t^2 + (b + c - 1)/(a*b*c)*t + (a^2*b*c - 1/4*a^2 + a*b^2*c + a*b*c^2 - a*b*c - 1/2*a*b - 1/2*a*c + 1/2*a - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2))/(t^2 + (a^2*b*c - 1/4*a^2 + a*b^2*c + a*b*c^2 - a*b*c - 1/2*a*b - 1/2*a*c + 1/2*a - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2)) (-2*a/(a + b + c - 1)*t^2 + (-2*c + 1)/(b*c)*t)/(t^2 + (a^2*b*c - 1/4*a^2 + a*b^2*c + a*b*c^2 - a*b*c - 1/2*a*b - 1/2*a*c + 1/2*a - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2)) (2*a/(a + b + c - 1)*t^2 + (-2*b + 1)/(b*c)*t)/(t^2 + (a^2*b*c - 1/4*a^2 + a*b^2*c + a*b*c^2 - a*b*c - 1/2*a*b - 1/2*a*c + 1/2*a - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2))] [((-2*b + 2)/(a + b + c - 1)*t^2 + (2*b*c + b - 1)/(a*b*c)*t)/(t^2 + (a^2*b*c - 1/4*a^2 + a*b^2*c + a*b*c^2 - a*b*c - 1/2*a*b - 1/2*a*c + 1/2*a - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2)) ((-a + b - c - 1)/(a + b + c - 1)*t^2 + (a - c)/(a*b*c)*t + (a^2*b*c - 1/4*a^2 + a*b^2*c + a*b*c^2 - a*b*c - 1/2*a*b - 1/2*a*c + 1/2*a - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2))/(t^2 + (a^2*b*c - 1/4*a^2 + a*b^2*c + a*b*c^2 - a*b*c - 1/2*a*b - 1/2*a*c + 1/2*a - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2)) ((-2*b + 2)/(a + b + c - 1)*t^2 + (-2*a*b - b + 1)/(a*b*c)*t)/(t^2 + (a^2*b*c - 1/4*a^2 + a*b^2*c + a*b*c^2 - a*b*c - 1/2*a*b - 1/2*a*c + 1/2*a - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2))] [(2*c/(a + b + c - 1)*t^2 + (2*b - 1)/(a*b)*t)/(t^2 + (a^2*b*c - 1/4*a^2 + a*b^2*c + a*b*c^2 - a*b*c - 1/2*a*b - 1/2*a*c + 1/2*a - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2)) (-2*c/(a + b + c - 1)*t^2 + (2*a - 1)/(a*b)*t)/(t^2 + (a^2*b*c - 1/4*a^2 + a*b^2*c + a*b*c^2 - a*b*c - 1/2*a*b - 1/2*a*c + 1/2*a - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2)) ((-a - b + c + 1)/(a + b + c - 1)*t^2 + (-a - b + 1)/(a*b*c)*t + (a^2*b*c - 1/4*a^2 + a*b^2*c + a*b*c^2 - a*b*c - 1/2*a*b - 1/2*a*c + 1/2*a - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2))/(t^2 + (a^2*b*c - 1/4*a^2 + a*b^2*c + a*b*c^2 - a*b*c - 1/2*a*b - 1/2*a*c + 1/2*a - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2))] Total time: 0.280 seconds, Total memory usage: 3.43MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Thu Dec 8 15:40:45 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); S:=Transpose(A)*A^(-1); F,T:=PrimaryRationalForm(S); B:=T*A*Transpose(T); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*B*Transpose(P)-B)); G:=GroebnerBasis(I); Kt:=FunctionField(K); D:=(2*a*b*c-a-b-c+1)/(a*b*c); s:=t-D/2; px:=-(2*s+D); py:=1-s^2; q:=s^2+s*D+1; Q:=Matrix(Kt,3,3,[1,0,0,0,D*px/q+py/q,-px/q,0,px/q,py/q]); Q:=T^(-1)*Q*T; print (Q*A*Transpose(Q)-A); print Q; Output: Magma V2.11-10 Thu Dec 8 2005 15:40:45 on modular [Seed = 3778689153] ------------------------------------- [0 0 0] [0 0 0] [0 0 0] [((a - b - c + 1)/(a + b + c - 1)*t^2 + (b + c - 1)/(a*b*c)*t + (a^2*b*c - 1/4*a^2 + a*b^2*c + a*b*c^2 - a*b*c - 1/2*a*b - 1/2*a*c + 1/2*a - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2))/(t^2 + (a^2*b*c - 1/4*a^2 + a*b^2*c + a*b*c^2 - a*b*c - 1/2*a*b - 1/2*a*c + 1/2*a - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2)) (-2*a/(a + b + c - 1)*t^2 + (-2*c + 1)/(b*c)*t)/(t^2 + (a^2*b*c - 1/4*a^2 + a*b^2*c + a*b*c^2 - a*b*c - 1/2*a*b - 1/2*a*c + 1/2*a - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2)) (2*a/(a + b + c - 1)*t^2 + (-2*b + 1)/(b*c)*t)/(t^2 + (a^2*b*c - 1/4*a^2 + a*b^2*c + a*b*c^2 - a*b*c - 1/2*a*b - 1/2*a*c + 1/2*a - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2))] [((-2*b + 2)/(a + b + c - 1)*t^2 + (2*b*c + b - 1)/(a*b*c)*t)/(t^2 + (a^2*b*c - 1/4*a^2 + a*b^2*c + a*b*c^2 - a*b*c - 1/2*a*b - 1/2*a*c + 1/2*a - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2)) ((-a + b - c - 1)/(a + b + c - 1)*t^2 + (a - c)/(a*b*c)*t + (a^2*b*c - 1/4*a^2 + a*b^2*c + a*b*c^2 - a*b*c - 1/2*a*b - 1/2*a*c + 1/2*a - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2))/(t^2 + (a^2*b*c - 1/4*a^2 + a*b^2*c + a*b*c^2 - a*b*c - 1/2*a*b - 1/2*a*c + 1/2*a - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2)) ((-2*b + 2)/(a + b + c - 1)*t^2 + (-2*a*b - b + 1)/(a*b*c)*t)/(t^2 + (a^2*b*c - 1/4*a^2 + a*b^2*c + a*b*c^2 - a*b*c - 1/2*a*b - 1/2*a*c + 1/2*a - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2))] [(2*c/(a + b + c - 1)*t^2 + (2*b - 1)/(a*b)*t)/(t^2 + (a^2*b*c - 1/4*a^2 + a*b^2*c + a*b*c^2 - a*b*c - 1/2*a*b - 1/2*a*c + 1/2*a - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2)) (-2*c/(a + b + c - 1)*t^2 + (2*a - 1)/(a*b)*t)/(t^2 + (a^2*b*c - 1/4*a^2 + a*b^2*c + a*b*c^2 - a*b*c - 1/2*a*b - 1/2*a*c + 1/2*a - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2)) ((-a - b + c + 1)/(a + b + c - 1)*t^2 + (-a - b + 1)/(a*b*c)*t + (a^2*b*c - 1/4*a^2 + a*b^2*c + a*b*c^2 - a*b*c - 1/2*a*b - 1/2*a*c + 1/2*a - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2))/(t^2 + (a^2*b*c - 1/4*a^2 + a*b^2*c + a*b*c^2 - a*b*c - 1/2*a*b - 1/2*a*c + 1/2*a - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)/(a^2*b^2*c^2))] Total time: 0.270 seconds, Total memory usage: 3.43MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Thu Dec 8 15:36:44 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); S:=Transpose(A)*A^(-1); F,T:=PrimaryRationalForm(S); B:=T*A*Transpose(T); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*B*Transpose(P)-B)); G:=GroebnerBasis(I); Kt:=FunctionField(K); D:=(2*a*b*c-a-b-c+1)/(a*b*c); px:=-(2*t+D); py:=1-t^2; q:=t^2+t*D+1; Q:=Matrix(Kt,3,3,[1,0,0,0,D*px/q+py/q,-px/q,0,px/q,py/q]); Q:=T^(-1)*Q*T; print (Q*A*Transpose(Q)-A); print Q; Output: Magma V2.11-10 Thu Dec 8 2005 15:36:44 on modular [Seed = 2504937444] ------------------------------------- [0 0 0] [0 0 0] [0 0 0] [((a - b - c + 1)/(a + b + c - 1)*t^2 + (2*a^2*b*c - a^2 - 2*a*b^2*c - 2*a*b*c^2 + 2*a*b*c + a*b + a*c - a + 2*b^2 + 4*b*c - 4*b + 2*c^2 - 4*c + 2)/(a^2*b*c + a*b^2*c + a*b*c^2 - a*b*c)*t + (a^3*b^2*c^2 - a^2*b^3*c^2 - a^2*b^2*c^3 + a^2*b^2*c^2 + 3*a^2*b^2*c + 3*a^2*b*c^2 - 3*a^2*b*c - a^2*b - a^2*c + a^2 + 3*a*b^3*c + 6*a*b^2*c^2 - 6*a*b^2*c - 2*a*b^2 + 3*a*b*c^3 - 6*a*b*c^2 - a*b*c + 4*a*b - 2*a*c^2 + 4*a*c - 2*a - b^3 - 3*b^2*c + 3*b^2 - 3*b*c^2 + 6*b*c - 3*b - c^3 + 3*c^2 - 3*c + 1)/(a^3*b^2*c^2 + a^2*b^3*c^2 + a^2*b^2*c^3 - a^2*b^2*c^2))/(t^2 + (2*a*b*c - a - b - c + 1)/(a*b*c)*t + 1) (-2*a/(a + b + c - 1)*t^2 + (-4*a*b*c - 2*a*c + 3*a - 2*b*c + 3*b - 2*c^2 + 5*c - 3)/(a*b*c + b^2*c + b*c^2 - b*c)*t + (-2*a^2*b^2*c^2 - 2*a^2*b*c^2 + 3*a^2*b*c + a^2*c - a^2 - 2*a*b^2*c^2 + 3*a*b^2*c - 2*a*b*c^3 + 5*a*b*c^2 - a*b*c - 2*a*b + 2*a*c^2 - 4*a*c + 2*a + b^2*c - b^2 + 2*b*c^2 - 4*b*c + 2*b + c^3 - 3*c^2 + 3*c - 1)/(a^2*b^2*c^2 + a*b^3*c^2 + a*b^2*c^3 - a*b^2*c^2))/(t^2 + (2*a*b*c - a - b - c + 1)/(a*b*c)*t + 1) (2*a/(a + b + c - 1)*t^2 + (4*a*b*c - 2*a*b - a - 2*b^2 - 2*b*c + b - c + 1)/(a*b*c + b^2*c + b*c^2 - b*c)*t + (2*a^2*b*c^2 - 2*a^2*b*c - a^2*c + a^2 - 2*a*b^2*c - 2*a*b*c^2 + a*b*c + 2*a*b - a*c^2 + 3*a*c - 2*a + b^2 + 2*b*c - 2*b + c^2 - 2*c + 1)/(a^2*b*c^2 + a*b^2*c^2 + a*b*c^3 - a*b*c^2))/(t^2 + (2*a*b*c - a - b - c + 1)/(a*b*c)*t + 1)] [((-2*b + 2)/(a + b + c - 1)*t^2 + (-4*a*b^2*c + 6*a*b*c + 3*a*b - 3*a + 2*b^2*c + 3*b^2 + 2*b*c^2 + b*c - 6*b - 3*c + 3)/(a^2*b*c + a*b^2*c + a*b*c^2 - a*b*c)*t + (-2*a^2*b^3*c^2 + 4*a^2*b^2*c^2 + 3*a^2*b^2*c - 4*a^2*b*c - a^2*b + a^2 + 2*a*b^3*c^2 + 3*a*b^3*c + 2*a*b^2*c^3 + a*b^2*c^2 - 8*a*b^2*c - 2*a*b^2 - 5*a*b*c^2 + 3*a*b*c + 4*a*b + 2*a*c - 2*a - b^3*c - b^3 - 2*b^2*c^2 + 3*b^2 - b*c^3 + b*c^2 + 3*b*c - 3*b + c^2 - 2*c + 1)/(a^3*b^2*c^2 + a^2*b^3*c^2 + a^2*b^2*c^3 - a^2*b^2*c^2))/(t^2 + (2*a*b*c - a - b - c + 1)/(a*b*c)*t + 1) ((-a + b - c - 1)/(a + b + c - 1)*t^2 + (-2*a^2*b*c + 2*a^2 + 2*a*b^2*c - 2*a*b*c^2 - 2*a*b*c + a*b + 2*a*c - a - b^2 - b*c + 2*b + c - 1)/(a^2*b*c + a*b^2*c + a*b*c^2 - a*b*c)*t + (-a^2*b^2*c^2 + 3*a^2*b*c - a^2 + a*b^3*c^2 - a*b^2*c^3 - a*b^2*c^2 + 3*a*b^2*c + 4*a*b*c^2 - 3*a*b*c - 2*a*b - 2*a*c + 2*a + b^2*c^2 - b^2 + b*c^3 - b*c^2 - 2*b*c + 2*b - c^2 + 2*c - 1)/(a^2*b^2*c^2 + a*b^3*c^2 + a*b^2*c^3 - a*b^2*c^2))/(t^2 + (2*a*b*c - a - b - c + 1)/(a*b*c)*t + 1) ((-2*b + 2)/(a + b + c - 1)*t^2 + (-2*a^2*b - 4*a*b^2*c - 2*a*b^2 + 2*a*b*c + 3*a*b - a + b^2 + b*c - 2*b - c + 1)/(a^2*b*c + a*b^2*c + a*b*c^2 - a*b*c)*t + (-2*a^2*b*c + a^2 - 2*a*b^2*c^2 - 2*a*b^2*c + 3*a*b*c + 2*a*b + a*c - 2*a + b^2*c + b^2 + b*c^2 - 2*b - c + 1)/(a^2*b*c^2 + a*b^2*c^2 + a*b*c^3 - a*b*c^2))/(t^2 + (2*a*b*c - a - b - c + 1)/(a*b*c)*t + 1)] [(2*c/(a + b + c - 1)*t^2 + (4*a*b*c + 2*a*b - 3*a + 2*b^2 + 2*b*c - 5*b - 3*c + 3)/(a^2*b + a*b^2 + a*b*c - a*b)*t + (2*a^2*b^2*c^2 + 2*a^2*b^2*c - 3*a^2*b*c - a^2*b + a^2 + 2*a*b^3*c + 2*a*b^2*c^2 - 5*a*b^2*c - 2*a*b^2 - 3*a*b*c^2 + a*b*c + 4*a*b + 2*a*c - 2*a - b^3 - 2*b^2*c + 3*b^2 - b*c^2 + 4*b*c - 3*b + c^2 - 2*c + 1)/(a^3*b^2*c + a^2*b^3*c + a^2*b^2*c^2 - a^2*b^2*c))/(t^2 + (2*a*b*c - a - b - c + 1)/(a*b*c)*t + 1) (-2*c/(a + b + c - 1)*t^2 + (2*a^2 - 4*a*b*c + 2*a*b + 2*a*c - a + b + c - 1)/(a^2*b + a*b^2 + a*b*c - a*b)*t + (2*a^2*b*c - a^2 - 2*a*b^2*c^2 + 2*a*b^2*c + 2*a*b*c^2 - a*b*c - 2*a*b - 2*a*c + 2*a + b^2*c - b^2 + b*c^2 - 3*b*c + 2*b - c^2 + 2*c - 1)/(a^2*b^2*c + a*b^3*c + a*b^2*c^2 - a*b^2*c))/(t^2 + (2*a*b*c - a - b - c + 1)/(a*b*c)*t + 1) ((-a - b + c + 1)/(a + b + c - 1)*t^2 + (-2*a^2*b - 2*a*b^2 + 2*a*b*c + 2*a*b - a - b - c + 1)/(a^2*b + a*b^2 + a*b*c - a*b)*t + (-a^2*b*c + a^2 - a*b^2*c + a*b*c^2 + a*b*c + 2*a*b + a*c - 2*a + b^2 + b*c - 2*b - c + 1)/(a^2*b*c + a*b^2*c + a*b*c^2 - a*b*c))/(t^2 + (2*a*b*c - a - b - c + 1)/(a*b*c)*t + 1)] Total time: 0.310 seconds, Total memory usage: 3.53MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Thu Dec 8 15:36:36 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); S:=Transpose(A)*A^(-1); F,T:=PrimaryRationalForm(S); B:=T*A*Transpose(T); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*B*Transpose(P)-B)); G:=GroebnerBasis(I); Kt:=FunctionField(K); D:=(2*a*b*c-a-b-c+1)/(a*b*c); px:=-(2*t+D); py:=1-t^2; q:=t^2+t*D+1; Q:=Matrix(Kt,3,3,[1,0,0,0,D*px/q+py/q,-px/q,0,px/q,py/q]); Q:=T^(-1)*Q*T; print (Q*A*Transpose(Q)-A); Output: Magma V2.11-10 Thu Dec 8 2005 15:36:35 on modular [Seed = 2655339744] ------------------------------------- [0 0 0] [0 0 0] [0 0 0] Total time: 0.300 seconds, Total memory usage: 3.53MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Thu Dec 8 15:34:08 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); S:=Transpose(A)*A^(-1); F,T:=PrimaryRationalForm(S); B:=T*A*Transpose(T); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*B*Transpose(P)-B)); G:=GroebnerBasis(I); Kt:=FunctionField(K); D:=(2*a*b*c-a-b-c+1)/(a*b*c); px:=-(2*t+D); py:=1-t^2; q:=t^2+t*D+1; Q:=Matrix(Kt,3,3,[1,0,0,0,D*px/q+py/q,-px/q,0,px/q,py/q]); print Factorization(Numerator(D*px+py)); print (Q*B*Transpose(Q)-B); Output: Magma V2.11-10 Thu Dec 8 2005 15:34:07 on modular [Seed = 2203605009] ------------------------------------- [ <$.1 + (a*b*c - a - b - c + 1)/(a*b*c), 1>, <$.1 + (3*a*b*c - a - b - c + 1)/(a*b*c), 1> ] [0 0 0] [0 0 0] [0 0 0] Total time: 0.220 seconds, Total memory usage: 3.43MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Thu Dec 8 15:33:33 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); S:=Transpose(A)*A^(-1); F,T:=PrimaryRationalForm(S); B:=T*A*Transpose(T); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*B*Transpose(P)-B)); G:=GroebnerBasis(I); Kt:=FunctionField(K); D:=(2*a*b*c-a-b-c+1)/(a*b*c); px:=-(2*t+D); py:=1-t^2; q:=t^2+t*D+1; Q:=Matrix(Kt,3,3,[1,0,0,0,D*px/q+py/q,-px/q,0,px/q,py/q]); print D*px+py; print (Q*B*Transpose(Q)-B); Output: Magma V2.11-10 Thu Dec 8 2005 15:33:33 on modular [Seed = 2220842804] ------------------------------------- -t^2 + (-4*a*b*c + 2*a + 2*b + 2*c - 2)/(a*b*c)*t + (-3*a^2*b^2*c^2 + 4*a^2*b*c - a^2 + 4*a*b^2*c + 4*a*b*c^2 - 4*a*b*c - 2*a*b - 2*a*c + 2*a - b^2 - 2*b*c + 2*b - c^2 + 2*c - 1)/(a^2*b^2*c^2) [0 0 0] [0 0 0] [0 0 0] Total time: 0.210 seconds, Total memory usage: 3.43MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Thu Dec 8 15:31:53 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); S:=Transpose(A)*A^(-1); F,T:=PrimaryRationalForm(S); B:=T*A*Transpose(T); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*B*Transpose(P)-B)); G:=GroebnerBasis(I); Kt:=FunctionField(K); D:=(2*a*b*c-a-b-c+1)/(a*b*c); px:=-(2*t+D); py:=1-t^2; q:=t^2+t*D+1; Q:=Matrix(Kt,3,3,[1,0,0,0,D*px/q+py/q,-px/q,0,px/q,py/q]); print (Q*B*Transpose(Q)-B); Output: Magma V2.11-10 Thu Dec 8 2005 15:31:52 on modular [Seed = 2387957147] ------------------------------------- [0 0 0] [0 0 0] [0 0 0] Total time: 0.210 seconds, Total memory usage: 3.43MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Thu Dec 8 15:31:19 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); S:=Transpose(A)*A^(-1); F,T:=PrimaryRationalForm(S); B:=T*A*Transpose(T); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*B*Transpose(P)-B)); G:=GroebnerBasis(I); Kt:=FunctionField(K); D:=(2*a*b*c-a-b-c+1)/(a*b*c); px:=-(2*t+D); py:=1-t^2; q:=t^2+t*D+1; Q:=Matrix(Kt,3,3,[1,0,0,0,-D*px/q+py/q,-px/q,0,px/q,py/q]); print (Q*B*Transpose(Q)-B); Output: Magma V2.11-10 Thu Dec 8 2005 15:31:19 on modular [Seed = 3009964169] ------------------------------------- [0 0 0] [0 ((-16*a*b^2*c - 16*a*b*c^2 + 16*a*b*c + 8*a*b + 8*a*c - 8*a + 8*b^2 + 16*b*c - 16*b + 8*c^2 - 16*c + 8)/(a*b^2*c^2)*t^3 + (16*a^2*b^3*c^2 + 16*a^2*b^2*c^3 - 16*a^2*b^2*c^2 - 16*a^2*b^2*c - 16*a^2*b*c^2 + 16*a^2*b*c + 4*a^2*b + 4*a^2*c - 4*a^2 - 16*a*b^3*c - 32*a*b^2*c^2 + 32*a*b^2*c + 8*a*b^2 - 16*a*b*c^3 + 32*a*b*c^2 - 16*a*b + 8*a*c^2 - 16*a*c + 8*a + 4*b^3 + 12*b^2*c - 12*b^2 + 12*b*c^2 - 24*b*c + 12*b + 4*c^3 - 12*c^2 + 12*c - 4)/(a^2*b^3*c^3)*t^2 + (80*a^3*b^4*c^3 + 80*a^3*b^3*c^4 - 80*a^3*b^3*c^3 - 104*a^3*b^3*c^2 - 104*a^3*b^2*c^3 + 104*a^3*b^2*c^2 + 48*a^3*b^2*c + 48*a^3*b*c^2 - 48*a^3*b*c - 8*a^3*b - 8*a^3*c + 8*a^3 - 104*a^2*b^4*c^2 - 208*a^2*b^3*c^3 + 208*a^2*b^3*c^2 + 96*a^2*b^3*c - 104*a^2*b^2*c^4 + 208*a^2*b^2*c^3 + 88*a^2*b^2*c^2 - 192*a^2*b^2*c - 24*a^2*b^2 + 96*a^2*b*c^3 - 192*a^2*b*c^2 + 48*a^2*b*c + 48*a^2*b - 24*a^2*c^2 + 48*a^2*c - 24*a^2 + 48*a*b^4*c + 144*a*b^3*c^2 - 144*a*b^3*c - 24*a*b^3 + 144*a*b^2*c^3 - 288*a*b^2*c^2 + 72*a*b^2*c + 72*a*b^2 + 48*a*b*c^4 - 144*a*b*c^3 + 72*a*b*c^2 + 96*a*b*c - 72*a*b - 24*a*c^3 + 72*a*c^2 - 72*a*c + 24*a - 8*b^4 - 32*b^3*c + 32*b^3 - 48*b^2*c^2 + 96*b^2*c - 48*b^2 - 32*b*c^3 + 96*b*c^2 - 96*b*c + 32*b - 8*c^4 + 32*c^3 - 48*c^2 + 32*c - 8)/(a^3*b^4*c^4)*t + (48*a^4*b^5*c^4 + 48*a^4*b^4*c^5 - 48*a^4*b^4*c^4 - 80*a^4*b^4*c^3 - 80*a^4*b^3*c^4 + 80*a^4*b^3*c^3 + 52*a^4*b^3*c^2 + 52*a^4*b^2*c^3 - 52*a^4*b^2*c^2 - 16*a^4*b^2*c - 16*a^4*b*c^2 + 16*a^4*b*c + 2*a^4*b + 2*a^4*c - 2*a^4 - 80*a^3*b^5*c^3 - 160*a^3*b^4*c^4 + 160*a^3*b^4*c^3 + 104*a^3*b^4*c^2 - 80*a^3*b^3*c^5 + 160*a^3*b^3*c^4 + 128*a^3*b^3*c^3 - 208*a^3*b^3*c^2 - 48*a^3*b^3*c + 104*a^3*b^2*c^4 - 208*a^3*b^2*c^3 + 8*a^3*b^2*c^2 + 96*a^3*b^2*c + 8*a^3*b^2 - 48*a^3*b*c^3 + 96*a^3*b*c^2 - 32*a^3*b*c - 16*a^3*b + 8*a^3*c^2 - 16*a^3*c + 8*a^3 + 52*a^2*b^5*c^2 + 156*a^2*b^4*c^3 - 156*a^2*b^4*c^2 - 48*a^2*b^4*c + 156*a^2*b^3*c^4 - 312*a^2*b^3*c^3 + 12*a^2*b^3*c^2 + 144*a^2*b^3*c + 12*a^2*b^3 + 52*a^2*b^2*c^5 - 156*a^2*b^2*c^4 + 12*a^2*b^2*c^3 + 236*a^2*b^2*c^2 - 108*a^2*b^2*c - 36*a^2*b^2 - 48*a^2*b*c^4 + 144*a^2*b*c^3 - 108*a^2*b*c^2 - 24*a^2*b*c + 36*a^2*b + 12*a^2*c^3 - 36*a^2*c^2 + 36*a^2*c - 12*a^2 - 16*a*b^5*c - 64*a*b^4*c^2 + 64*a*b^4*c + 8*a*b^4 - 96*a*b^3*c^3 + 192*a*b^3*c^2 - 64*a*b^3*c - 32*a*b^3 - 64*a*b^2*c^4 + 192*a*b^2*c^3 - 144*a*b^2*c^2 - 32*a*b^2*c + 48*a*b^2 - 16*a*b*c^5 + 64*a*b*c^4 - 64*a*b*c^3 - 32*a*b*c^2 + 80*a*b*c - 32*a*b + 8*a*c^4 - 32*a*c^3 + 48*a*c^2 - 32*a*c + 8*a + 2*b^5 + 10*b^4*c - 10*b^4 + 20*b^3*c^2 - 40*b^3*c + 20*b^3 + 20*b^2*c^3 - 60*b^2*c^2 + 60*b^2*c - 20*b^2 + 10*b*c^4 - 40*b*c^3 + 60*b*c^2 - 40*b*c + 10*b + 2*c^5 - 10*c^4 + 20*c^3 - 20*c^2 + 10*c - 2)/(a^4*b^5*c^5))/(t^4 + (4*a*b*c - 2*a - 2*b - 2*c + 2)/(a*b*c)*t^3 + (6*a^2*b^2*c^2 - 4*a^2*b*c + a^2 - 4*a*b^2*c - 4*a*b*c^2 + 4*a*b*c + 2*a*b + 2*a*c - 2*a + b^2 + 2*b*c - 2*b + c^2 - 2*c + 1)/(a^2*b^2*c^2)*t^2 + (4*a*b*c - 2*a - 2*b - 2*c + 2)/(a*b*c)*t + 1) ((-8*a^2*b^3*c^2 - 8*a^2*b^2*c^3 + 8*a^2*b^2*c^2 + 12*a^2*b^2*c + 12*a^2*b*c^2 - 12*a^2*b*c - 4*a^2*b - 4*a^2*c + 4*a^2 + 12*a*b^3*c + 24*a*b^2*c^2 - 24*a*b^2*c - 8*a*b^2 + 12*a*b*c^3 - 24*a*b*c^2 - 4*a*b*c + 16*a*b - 8*a*c^2 + 16*a*c - 8*a - 4*b^3 - 12*b^2*c + 12*b^2 - 12*b*c^2 + 24*b*c - 12*b - 4*c^3 + 12*c^2 - 12*c + 4)/(a^2*b^3*c^3)*t^3 + (-24*a^3*b^4*c^3 - 24*a^3*b^3*c^4 + 24*a^3*b^3*c^3 + 24*a^3*b^3*c^2 + 24*a^3*b^2*c^3 - 24*a^3*b^2*c^2 - 10*a^3*b^2*c - 10*a^3*b*c^2 + 10*a^3*b*c + 2*a^3*b + 2*a^3*c - 2*a^3 + 24*a^2*b^4*c^2 + 48*a^2*b^3*c^3 - 48*a^2*b^3*c^2 - 20*a^2*b^3*c + 24*a^2*b^2*c^4 - 48*a^2*b^2*c^3 - 16*a^2*b^2*c^2 + 40*a^2*b^2*c + 6*a^2*b^2 - 20*a^2*b*c^3 + 40*a^2*b*c^2 - 8*a^2*b*c - 12*a^2*b + 6*a^2*c^2 - 12*a^2*c + 6*a^2 - 10*a*b^4*c - 30*a*b^3*c^2 + 30*a*b^3*c + 6*a*b^3 - 30*a*b^2*c^3 + 60*a*b^2*c^2 - 12*a*b^2*c - 18*a*b^2 - 10*a*b*c^4 + 30*a*b*c^3 - 12*a*b*c^2 - 26*a*b*c + 18*a*b + 6*a*c^3 - 18*a*c^2 + 18*a*c - 6*a + 2*b^4 + 8*b^3*c - 8*b^3 + 12*b^2*c^2 - 24*b^2*c + 12*b^2 + 8*b*c^3 - 24*b*c^2 + 24*b*c - 8*b + 2*c^4 - 8*c^3 + 12*c^2 - 8*c + 2)/(a^3*b^4*c^4)*t^2 + (-24*a^2*b^3*c^2 - 24*a^2*b^2*c^3 + 24*a^2*b^2*c^2 + 20*a^2*b^2*c + 20*a^2*b*c^2 - 20*a^2*b*c - 4*a^2*b - 4*a^2*c + 4*a^2 + 20*a*b^3*c + 40*a*b^2*c^2 - 40*a*b^2*c - 8*a*b^2 + 20*a*b*c^3 - 40*a*b*c^2 + 4*a*b*c + 16*a*b - 8*a*c^2 + 16*a*c - 8*a - 4*b^3 - 12*b^2*c + 12*b^2 - 12*b*c^2 + 24*b*c - 12*b - 4*c^3 + 12*c^2 - 12*c + 4)/(a^2*b^3*c^3)*t + (-8*a^2*b^3*c^2 - 8*a^2*b^2*c^3 + 8*a^2*b^2*c^2 + 8*a^2*b^2*c + 8*a^2*b*c^2 - 8*a^2*b*c - 2*a^2*b - 2*a^2*c + 2*a^2 + 8*a*b^3*c + 16*a*b^2*c^2 - 16*a*b^2*c - 4*a*b^2 + 8*a*b*c^3 - 16*a*b*c^2 + 8*a*b - 4*a*c^2 + 8*a*c - 4*a - 2*b^3 - 6*b^2*c + 6*b^2 - 6*b*c^2 + 12*b*c - 6*b - 2*c^3 + 6*c^2 - 6*c + 2)/(a^2*b^3*c^3))/(t^4 + (4*a*b*c - 2*a - 2*b - 2*c + 2)/(a*b*c)*t^3 + (6*a^2*b^2*c^2 - 4*a^2*b*c + a^2 - 4*a*b^2*c - 4*a*b*c^2 + 4*a*b*c + 2*a*b + 2*a*c - 2*a + b^2 + 2*b*c - 2*b + c^2 - 2*c + 1)/(a^2*b^2*c^2)*t^2 + (4*a*b*c - 2*a - 2*b - 2*c + 2)/(a*b*c)*t + 1)] [0 ((-8*a*b^2*c - 8*a*b*c^2 + 8*a*b*c + 4*a*b + 4*a*c - 4*a + 4*b^2 + 8*b*c - 8*b + 4*c^2 - 8*c + 4)/(a*b^2*c^2)*t^3 + (-24*a^2*b^3*c^2 - 24*a^2*b^2*c^3 + 24*a^2*b^2*c^2 + 16*a^2*b^2*c + 16*a^2*b*c^2 - 16*a^2*b*c - 2*a^2*b - 2*a^2*c + 2*a^2 + 16*a*b^3*c + 32*a*b^2*c^2 - 32*a*b^2*c - 4*a*b^2 + 16*a*b*c^3 - 32*a*b*c^2 + 8*a*b*c + 8*a*b - 4*a*c^2 + 8*a*c - 4*a - 2*b^3 - 6*b^2*c + 6*b^2 - 6*b*c^2 + 12*b*c - 6*b - 2*c^3 + 6*c^2 - 6*c + 2)/(a^2*b^3*c^3)*t^2 + (-24*a^2*b^3*c^2 - 24*a^2*b^2*c^3 + 24*a^2*b^2*c^2 + 28*a^2*b^2*c + 28*a^2*b*c^2 - 28*a^2*b*c - 8*a^2*b - 8*a^2*c + 8*a^2 + 28*a*b^3*c + 56*a*b^2*c^2 - 56*a*b^2*c - 16*a*b^2 + 28*a*b*c^3 - 56*a*b*c^2 - 4*a*b*c + 32*a*b - 16*a*c^2 + 32*a*c - 16*a - 8*b^3 - 24*b^2*c + 24*b^2 - 24*b*c^2 + 48*b*c - 24*b - 8*c^3 + 24*c^2 - 24*c + 8)/(a^2*b^3*c^3)*t + (-8*a^3*b^4*c^3 - 8*a^3*b^3*c^4 + 8*a^3*b^3*c^3 + 16*a^3*b^3*c^2 + 16*a^3*b^2*c^3 - 16*a^3*b^2*c^2 - 10*a^3*b^2*c - 10*a^3*b*c^2 + 10*a^3*b*c + 2*a^3*b + 2*a^3*c - 2*a^3 + 16*a^2*b^4*c^2 + 32*a^2*b^3*c^3 - 32*a^2*b^3*c^2 - 20*a^2*b^3*c + 16*a^2*b^2*c^4 - 32*a^2*b^2*c^3 - 24*a^2*b^2*c^2 + 40*a^2*b^2*c + 6*a^2*b^2 - 20*a^2*b*c^3 + 40*a^2*b*c^2 - 8*a^2*b*c - 12*a^2*b + 6*a^2*c^2 - 12*a^2*c + 6*a^2 - 10*a*b^4*c - 30*a*b^3*c^2 + 30*a*b^3*c + 6*a*b^3 - 30*a*b^2*c^3 + 60*a*b^2*c^2 - 12*a*b^2*c - 18*a*b^2 - 10*a*b*c^4 + 30*a*b*c^3 - 12*a*b*c^2 - 26*a*b*c + 18*a*b + 6*a*c^3 - 18*a*c^2 + 18*a*c - 6*a + 2*b^4 + 8*b^3*c - 8*b^3 + 12*b^2*c^2 - 24*b^2*c + 12*b^2 + 8*b*c^3 - 24*b*c^2 + 24*b*c - 8*b + 2*c^4 - 8*c^3 + 12*c^2 - 8*c + 2)/(a^3*b^4*c^4))/(t^4 + (4*a*b*c - 2*a - 2*b - 2*c + 2)/(a*b*c)*t^3 + (6*a^2*b^2*c^2 - 4*a^2*b*c + a^2 - 4*a*b^2*c - 4*a*b*c^2 + 4*a*b*c + 2*a*b + 2*a*c - 2*a + b^2 + 2*b*c - 2*b + c^2 - 2*c + 1)/(a^2*b^2*c^2)*t^2 + (4*a*b*c - 2*a - 2*b - 2*c + 2)/(a*b*c)*t + 1) 0] Total time: 0.220 seconds, Total memory usage: 3.43MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Thu Dec 8 15:29:55 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); S:=Transpose(A)*A^(-1); F,T:=PrimaryRationalForm(S); B:=T*A*Transpose(T); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*B*Transpose(P)-B)); G:=GroebnerBasis(I); Kt:=PolynomialRing(K); D:=(2*a*b*c-a-b-c+1)/(a*b*c); px:=-(2*t+D); py:=1-t^2; q:=t^2+t*D+1; print G[5]; print G[6]; print G[7]; print Evaluate(G[8],[0,0,0,0,0,0,0,px/q,py/q]); Output: Magma V2.11-10 Thu Dec 8 2005 15:29:55 on modular [Seed = 3027198042] ------------------------------------- $.5 + (-2*a*b*c + a + b + c - 1)/(a*b*c)*$.8 - $.9 $.6 + $.8 $.7 0 Total time: 0.200 seconds, Total memory usage: 3.34MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Thu Dec 8 15:28:46 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); S:=Transpose(A)*A^(-1); F,T:=PrimaryRationalForm(S); B:=T*A*Transpose(T); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*B*Transpose(P)-B)); G:=GroebnerBasis(I); Kt:=PolynomialRing(K); D:=(2*a*b*c-a-b-c+1)/(a*b*c); px:=-(2*t+D); q:=t^2+t*D+1; print G[5]; print G[6]; print G[7]; print Evaluate(G[8],[0,0,0,0,0,0,0,px/q,t*px/q+1]); Output: Magma V2.11-10 Thu Dec 8 2005 15:28:46 on modular [Seed = 3110235626] ------------------------------------- $.5 + (-2*a*b*c + a + b + c - 1)/(a*b*c)*$.8 - $.9 $.6 + $.8 $.7 0 Total time: 0.200 seconds, Total memory usage: 3.34MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Thu Dec 8 15:24:17 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); S:=Transpose(A)*A^(-1); F,T:=PrimaryRationalForm(S); B:=T*A*Transpose(T); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*B*Transpose(P)-B)); G:=GroebnerBasis(I); Kt:=PolynomialRing(K); D:=(2*a*b*c-a-b-c+1)/(a*b*c); px:=-(2*t+D); q:=t^2+t*D+1; print G[8]; print Evaluate(G[8],[0,0,0,0,0,0,0,px/q,t*px/q+1]); Output: Magma V2.11-10 Thu Dec 8 2005 15:24:17 on modular [Seed = 2108074017] ------------------------------------- $.8^2 + (2*a*b*c - a - b - c + 1)/(a*b*c)*$.8*$.9 + $.9^2 - 1 0 Total time: 0.210 seconds, Total memory usage: 3.34MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Thu Dec 8 15:20:46 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); S:=Transpose(A)*A^(-1); F,T:=PrimaryRationalForm(S); B:=T*A*Transpose(T); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*B*Transpose(P)-B)); G:=GroebnerBasis(I); Kt:=PolynomialRing(K); D:=(2*a*b*c-a-b-c+1)/(a*b*c); px:=2*t+D; q:=t^2+t*D+1; print G[8]; print Evaluate(G[8],[0,0,0,0,0,0,0,-px/q,t*px/q+1]); Output: Magma V2.11-10 Thu Dec 8 2005 15:20:46 on modular [Seed = 215472831] ------------------------------------- $.8^2 + (2*a*b*c - a - b - c + 1)/(a*b*c)*$.8*$.9 + $.9^2 - 1 (8*t^4 + (8*a*b*c - 4*a - 4*b - 4*c + 4)/(a*b*c)*t^3 + (-8*a^2*b^2*c^2 + 16*a^2*b*c - 4*a^2 + 16*a*b^2*c + 16*a*b*c^2 - 16*a*b*c - 8*a*b - 8*a*c + 8*a - 4*b^2 - 8*b*c + 8*b - 4*c^2 + 8*c - 4)/(a^2*b^2*c^2)*t^2 + (-8*a^3*b^3*c^3 + 20*a^3*b^2*c^2 - 12*a^3*b*c + 2*a^3 + 20*a^2*b^3*c^2 + 20*a^2*b^2*c^3 - 20*a^2*b^2*c^2 - 24*a^2*b^2*c - 24*a^2*b*c^2 + 24*a^2*b*c + 6*a^2*b + 6*a^2*c - 6*a^2 - 12*a*b^3*c - 24*a*b^2*c^2 + 24*a*b^2*c + 6*a*b^2 - 12*a*b*c^3 + 24*a*b*c^2 - 12*a*b + 6*a*c^2 - 12*a*c + 6*a + 2*b^3 + 6*b^2*c - 6*b^2 + 6*b*c^2 - 12*b*c + 6*b + 2*c^3 - 6*c^2 + 6*c - 2)/(a^3*b^3*c^3)*t)/(t^4 + (4*a*b*c - 2*a - 2*b - 2*c + 2)/(a*b*c)*t^3 + (6*a^2*b^2*c^2 - 4*a^2*b*c + a^2 - 4*a*b^2*c - 4*a*b*c^2 + 4*a*b*c + 2*a*b + 2*a*c - 2*a + b^2 + 2*b*c - 2*b + c^2 - 2*c + 1)/(a^2*b^2*c^2)*t^2 + (4*a*b*c - 2*a - 2*b - 2*c + 2)/(a*b*c)*t + 1) Total time: 0.200 seconds, Total memory usage: 3.34MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Thu Dec 8 15:20:29 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); S:=Transpose(A)*A^(-1); F,T:=PrimaryRationalForm(S); B:=T*A*Transpose(T); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*B*Transpose(P)-B)); G:=GroebnerBasis(I); Kt:=PolynomialRing(K); D:=2*a*b*c-a-b-c+1; px:=2*t+D; q:=t^2+t*D+1; print G[8]; print Evaluate(G[8],[0,0,0,0,0,0,0,-px/q,t*px/q+1]); Output: Magma V2.11-10 Thu Dec 8 2005 15:20:28 on modular [Seed = 837478288] ------------------------------------- $.8^2 + (2*a*b*c - a - b - c + 1)/(a*b*c)*$.8*$.9 + $.9^2 - 1 (8*t^4 + (20*a^2*b^2*c^2 - 10*a^2*b*c - 10*a*b^2*c - 10*a*b*c^2 - 2*a*b*c + 6*a + 6*b + 6*c - 6)/(a*b*c)*t^3 + (12*a^3*b^3*c^3 - 12*a^3*b^2*c^2 + 3*a^3*b*c - 12*a^2*b^3*c^2 - 12*a^2*b^2*c^3 - 16*a^2*b^2*c^2 + 6*a^2*b^2*c + 6*a^2*b*c^2 + 22*a^2*b*c - 7*a^2 + 3*a*b^3*c + 6*a*b^2*c^2 + 22*a*b^2*c + 3*a*b*c^3 + 22*a*b*c^2 - 17*a*b*c - 14*a*b - 14*a*c + 14*a - 7*b^2 - 14*b*c + 14*b - 7*c^2 + 14*c - 7)/(a*b*c)*t^2 + (-16*a^3*b^3*c^3 + 24*a^3*b^2*c^2 - 12*a^3*b*c + 2*a^3 + 24*a^2*b^3*c^2 + 24*a^2*b^2*c^3 - 12*a^2*b^2*c^2 - 24*a^2*b^2*c - 24*a^2*b*c^2 + 18*a^2*b*c + 6*a^2*b + 6*a^2*c - 6*a^2 - 12*a*b^3*c - 24*a*b^2*c^2 + 18*a*b^2*c + 6*a*b^2 - 12*a*b*c^3 + 18*a*b*c^2 + 2*a*b*c - 12*a*b + 6*a*c^2 - 12*a*c + 8*a + 2*b^3 + 6*b^2*c - 6*b^2 + 6*b*c^2 - 12*b*c + 8*b + 2*c^3 - 6*c^2 + 8*c - 4)/(a*b*c)*t + (4*a^3*b^3*c^3 - 4*a^3*b^2*c^2 + a^3*b*c - 4*a^2*b^3*c^2 - 4*a^2*b^2*c^3 + 2*a^2*b^2*c + 2*a^2*b*c^2 + 2*a^2*b*c - a^2 + a*b^3*c + 2*a*b^2*c^2 + 2*a*b^2*c + a*b*c^3 + 2*a*b*c^2 - 3*a*b*c - 2*a*b - 2*a*c + 2*a - b^2 - 2*b*c + 2*b - c^2 + 2*c - 1)/(a*b*c))/(t^4 + (4*a*b*c - 2*a - 2*b - 2*c + 2)*t^3 + (4*a^2*b^2*c^2 - 4*a^2*b*c + a^2 - 4*a*b^2*c - 4*a*b*c^2 + 4*a*b*c + 2*a*b + 2*a*c - 2*a + b^2 + 2*b*c - 2*b + c^2 - 2*c + 3)*t^2 + (4*a*b*c - 2*a - 2*b - 2*c + 2)*t + 1) Total time: 0.200 seconds, Total memory usage: 3.34MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Thu Dec 8 15:20:08 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); S:=Transpose(A)*A^(-1); F,T:=PrimaryRationalForm(S); B:=T*A*Transpose(T); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*B*Transpose(P)-B)); G:=GroebnerBasis(I); Kt:=FunctionField(K); D:=2*a*b*c-a-b-c+1; px:=2*t+D; q:=t^2+t*D+1; print Evaluate(G[8],[0,0,0,0,0,0,0,-px/q,t*px/q+1]); Output: Magma V2.11-10 Thu Dec 8 2005 15:20:08 on modular [Seed = 987884655] ------------------------------------- (8*t^4 + (20*a^2*b^2*c^2 - 10*a^2*b*c - 10*a*b^2*c - 10*a*b*c^2 - 2*a*b*c + 6*a + 6*b + 6*c - 6)/(a*b*c)*t^3 + (12*a^3*b^3*c^3 - 12*a^3*b^2*c^2 + 3*a^3*b*c - 12*a^2*b^3*c^2 - 12*a^2*b^2*c^3 - 16*a^2*b^2*c^2 + 6*a^2*b^2*c + 6*a^2*b*c^2 + 22*a^2*b*c - 7*a^2 + 3*a*b^3*c + 6*a*b^2*c^2 + 22*a*b^2*c + 3*a*b*c^3 + 22*a*b*c^2 - 17*a*b*c - 14*a*b - 14*a*c + 14*a - 7*b^2 - 14*b*c + 14*b - 7*c^2 + 14*c - 7)/(a*b*c)*t^2 + (-16*a^3*b^3*c^3 + 24*a^3*b^2*c^2 - 12*a^3*b*c + 2*a^3 + 24*a^2*b^3*c^2 + 24*a^2*b^2*c^3 - 12*a^2*b^2*c^2 - 24*a^2*b^2*c - 24*a^2*b*c^2 + 18*a^2*b*c + 6*a^2*b + 6*a^2*c - 6*a^2 - 12*a*b^3*c - 24*a*b^2*c^2 + 18*a*b^2*c + 6*a*b^2 - 12*a*b*c^3 + 18*a*b*c^2 + 2*a*b*c - 12*a*b + 6*a*c^2 - 12*a*c + 8*a + 2*b^3 + 6*b^2*c - 6*b^2 + 6*b*c^2 - 12*b*c + 8*b + 2*c^3 - 6*c^2 + 8*c - 4)/(a*b*c)*t + (4*a^3*b^3*c^3 - 4*a^3*b^2*c^2 + a^3*b*c - 4*a^2*b^3*c^2 - 4*a^2*b^2*c^3 + 2*a^2*b^2*c + 2*a^2*b*c^2 + 2*a^2*b*c - a^2 + a*b^3*c + 2*a*b^2*c^2 + 2*a*b^2*c + a*b*c^3 + 2*a*b*c^2 - 3*a*b*c - 2*a*b - 2*a*c + 2*a - b^2 - 2*b*c + 2*b - c^2 + 2*c - 1)/(a*b*c))/(t^4 + (4*a*b*c - 2*a - 2*b - 2*c + 2)*t^3 + (4*a^2*b^2*c^2 - 4*a^2*b*c + a^2 - 4*a*b^2*c - 4*a*b*c^2 + 4*a*b*c + 2*a*b + 2*a*c - 2*a + b^2 + 2*b*c - 2*b + c^2 - 2*c + 3)*t^2 + (4*a*b*c - 2*a - 2*b - 2*c + 2)*t + 1) Total time: 0.210 seconds, Total memory usage: 3.34MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Thu Dec 8 15:19:32 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); S:=Transpose(A)*A^(-1); F,T:=PrimaryRationalForm(S); B:=T*A*Transpose(T); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*B*Transpose(P)-B)); G:=GroebnerBasis(I); Kt:=PolynomialRing(K); D:=2*a*b*c-a-b-c+1; px:=2*t+D; q:=t^2+t*D+1; print Evaluate(G[8],[0,0,0,0,0,0,0,-px/q,t*px/q+1]); Output: Magma V2.11-10 Thu Dec 8 2005 15:19:32 on modular [Seed = 586806363] ------------------------------------- (8*t^4 + (20*a^2*b^2*c^2 - 10*a^2*b*c - 10*a*b^2*c - 10*a*b*c^2 - 2*a*b*c + 6*a + 6*b + 6*c - 6)/(a*b*c)*t^3 + (12*a^3*b^3*c^3 - 12*a^3*b^2*c^2 + 3*a^3*b*c - 12*a^2*b^3*c^2 - 12*a^2*b^2*c^3 - 16*a^2*b^2*c^2 + 6*a^2*b^2*c + 6*a^2*b*c^2 + 22*a^2*b*c - 7*a^2 + 3*a*b^3*c + 6*a*b^2*c^2 + 22*a*b^2*c + 3*a*b*c^3 + 22*a*b*c^2 - 17*a*b*c - 14*a*b - 14*a*c + 14*a - 7*b^2 - 14*b*c + 14*b - 7*c^2 + 14*c - 7)/(a*b*c)*t^2 + (-16*a^3*b^3*c^3 + 24*a^3*b^2*c^2 - 12*a^3*b*c + 2*a^3 + 24*a^2*b^3*c^2 + 24*a^2*b^2*c^3 - 12*a^2*b^2*c^2 - 24*a^2*b^2*c - 24*a^2*b*c^2 + 18*a^2*b*c + 6*a^2*b + 6*a^2*c - 6*a^2 - 12*a*b^3*c - 24*a*b^2*c^2 + 18*a*b^2*c + 6*a*b^2 - 12*a*b*c^3 + 18*a*b*c^2 + 2*a*b*c - 12*a*b + 6*a*c^2 - 12*a*c + 8*a + 2*b^3 + 6*b^2*c - 6*b^2 + 6*b*c^2 - 12*b*c + 8*b + 2*c^3 - 6*c^2 + 8*c - 4)/(a*b*c)*t + (4*a^3*b^3*c^3 - 4*a^3*b^2*c^2 + a^3*b*c - 4*a^2*b^3*c^2 - 4*a^2*b^2*c^3 + 2*a^2*b^2*c + 2*a^2*b*c^2 + 2*a^2*b*c - a^2 + a*b^3*c + 2*a*b^2*c^2 + 2*a*b^2*c + a*b*c^3 + 2*a*b*c^2 - 3*a*b*c - 2*a*b - 2*a*c + 2*a - b^2 - 2*b*c + 2*b - c^2 + 2*c - 1)/(a*b*c))/(t^4 + (4*a*b*c - 2*a - 2*b - 2*c + 2)*t^3 + (4*a^2*b^2*c^2 - 4*a^2*b*c + a^2 - 4*a*b^2*c - 4*a*b*c^2 + 4*a*b*c + 2*a*b + 2*a*c - 2*a + b^2 + 2*b*c - 2*b + c^2 - 2*c + 3)*t^2 + (4*a*b*c - 2*a - 2*b - 2*c + 2)*t + 1) Total time: 0.200 seconds, Total memory usage: 3.34MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Thu Dec 8 15:19:08 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); S:=Transpose(A)*A^(-1); F,T:=PrimaryRationalForm(S); B:=T*A*Transpose(T); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*B*Transpose(P)-B)); G:=GroebnerBasis(I); Kt:=PolynomialRing(K); D:=2*a*b*c-a-b-c+1; px:=2*t+D; q:=t^2+t*D+1; print Evaluate(G[8],[0,0,0,0,0,0,0,px/q,-t*px/q+1]); Output: Magma V2.11-10 Thu Dec 8 2005 15:19:07 on modular [Seed = 670888241] ------------------------------------- ((-4*a^2*b^2*c^2 + 2*a^2*b*c + 2*a*b^2*c + 2*a*b*c^2 - 6*a*b*c + 2*a + 2*b + 2*c - 2)/(a*b*c)*t^3 + (-4*a^3*b^3*c^3 + 4*a^3*b^2*c^2 - a^3*b*c + 4*a^2*b^3*c^2 + 4*a^2*b^2*c^3 - 8*a^2*b^2*c^2 - 2*a^2*b^2*c - 2*a^2*b*c^2 + 6*a^2*b*c - a^2 - a*b^3*c - 2*a*b^2*c^2 + 6*a*b^2*c - a*b*c^3 + 6*a*b*c^2 - 5*a*b*c - 2*a*b - 2*a*c + 2*a - b^2 - 2*b*c + 2*b - c^2 + 2*c - 1)/(a*b*c)*t^2 + (4*a^2*b^2*c^2 - 2*a^2*b*c - 2*a*b^2*c - 2*a*b*c^2 + 6*a*b*c - 2*a - 2*b - 2*c + 2)/(a*b*c)*t + (4*a^3*b^3*c^3 - 4*a^3*b^2*c^2 + a^3*b*c - 4*a^2*b^3*c^2 - 4*a^2*b^2*c^3 + 8*a^2*b^2*c^2 + 2*a^2*b^2*c + 2*a^2*b*c^2 - 6*a^2*b*c + a^2 + a*b^3*c + 2*a*b^2*c^2 - 6*a*b^2*c + a*b*c^3 - 6*a*b*c^2 + 5*a*b*c + 2*a*b + 2*a*c - 2*a + b^2 + 2*b*c - 2*b + c^2 - 2*c + 1)/(a*b*c))/(t^4 + (4*a*b*c - 2*a - 2*b - 2*c + 2)*t^3 + (4*a^2*b^2*c^2 - 4*a^2*b*c + a^2 - 4*a*b^2*c - 4*a*b*c^2 + 4*a*b*c + 2*a*b + 2*a*c - 2*a + b^2 + 2*b*c - 2*b + c^2 - 2*c + 3)*t^2 + (4*a*b*c - 2*a - 2*b - 2*c + 2)*t + 1) Total time: 0.200 seconds, Total memory usage: 3.34MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Thu Dec 8 15:18:04 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); S:=Transpose(A)*A^(-1); F,T:=PrimaryRationalForm(S); B:=T*A*Transpose(T); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*B*Transpose(P)-B)); G:=GroebnerBasis(I); Kt:=PolynomialRing(K); D:=2*a*b*c-a-b-c-1; px:=2*t+D; q:=t^2+t*D+1; print Evaluate(G[8],[0,0,0,0,0,0,0,R.8,t*R.8]); print Evaluate(G[8],[0,0,0,0,0,0,0,px/q,-t*px/q+1]); Output: Magma V2.11-10 Thu Dec 8 2005 15:18:03 on modular [Seed = 703790847] ------------------------------------- >> print Evaluate(G[8],[0,0,0,0,0,0,0,R.8,t*R.8]); ^ Runtime error in '*': Bad argument types Argument types given: RngUPolElt[FldFunRat], RngMPolElt ((-4*a^2*b^2*c^2 + 2*a^2*b*c + 2*a*b^2*c + 2*a*b*c^2 - 2*a*b*c + 2*a + 2*b + 2*c - 2)/(a*b*c)*t^3 + (-4*a^3*b^3*c^3 + 4*a^3*b^2*c^2 - a^3*b*c + 4*a^2*b^3*c^2 + 4*a^2*b^2*c^3 - 2*a^2*b^2*c - 2*a^2*b*c^2 + 2*a^2*b*c - a^2 - a*b^3*c - 2*a*b^2*c^2 + 2*a*b^2*c - a*b*c^3 + 2*a*b*c^2 - a*b*c - 2*a*b - 2*a*c - b^2 - 2*b*c - c^2 + 1)/(a*b*c)*t^2 + (4*a^2*b^2*c^2 - 2*a^2*b*c - 2*a*b^2*c - 2*a*b*c^2 + 2*a*b*c - 2*a - 2*b - 2*c + 2)/(a*b*c)*t + (4*a^3*b^3*c^3 - 4*a^3*b^2*c^2 + a^3*b*c - 4*a^2*b^3*c^2 - 4*a^2*b^2*c^3 + 2*a^2*b^2*c + 2*a^2*b*c^2 - 2*a^2*b*c + a^2 + a*b^3*c + 2*a*b^2*c^2 - 2*a*b^2*c + a*b*c^3 - 2*a*b*c^2 + a*b*c + 2*a*b + 2*a*c + b^2 + 2*b*c + c^2 - 1)/(a*b*c))/(t^4 + (4*a*b*c - 2*a - 2*b - 2*c - 2)*t^3 + (4*a^2*b^2*c^2 - 4*a^2*b*c + a^2 - 4*a*b^2*c - 4*a*b*c^2 - 4*a*b*c + 2*a*b + 2*a*c + 2*a + b^2 + 2*b*c + 2*b + c^2 + 2*c + 3)*t^2 + (4*a*b*c - 2*a - 2*b - 2*c - 2)*t + 1) Total time: 0.210 seconds, Total memory usage: 3.34MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Thu Dec 8 15:16:28 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); S:=Transpose(A)*A^(-1); F,T:=PrimaryRationalForm(S); B:=T*A*Transpose(T); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*B*Transpose(P)-B)); G:=GroebnerBasis(I); Kt:=PolynomialRing(K); D:=2*a*b*c-a-b-c-1; px:=2*t+D; q:=t^2+t*D+1; print Evaluate(G[8],[0,0,0,0,0,0,0,px/q,-t*px/q+1]); Output: Magma V2.11-10 Thu Dec 8 2005 15:16:28 on modular [Seed = 3523806560] ------------------------------------- ((-4*a^2*b^2*c^2 + 2*a^2*b*c + 2*a*b^2*c + 2*a*b*c^2 - 2*a*b*c + 2*a + 2*b + 2*c - 2)/(a*b*c)*t^3 + (-4*a^3*b^3*c^3 + 4*a^3*b^2*c^2 - a^3*b*c + 4*a^2*b^3*c^2 + 4*a^2*b^2*c^3 - 2*a^2*b^2*c - 2*a^2*b*c^2 + 2*a^2*b*c - a^2 - a*b^3*c - 2*a*b^2*c^2 + 2*a*b^2*c - a*b*c^3 + 2*a*b*c^2 - a*b*c - 2*a*b - 2*a*c - b^2 - 2*b*c - c^2 + 1)/(a*b*c)*t^2 + (4*a^2*b^2*c^2 - 2*a^2*b*c - 2*a*b^2*c - 2*a*b*c^2 + 2*a*b*c - 2*a - 2*b - 2*c + 2)/(a*b*c)*t + (4*a^3*b^3*c^3 - 4*a^3*b^2*c^2 + a^3*b*c - 4*a^2*b^3*c^2 - 4*a^2*b^2*c^3 + 2*a^2*b^2*c + 2*a^2*b*c^2 - 2*a^2*b*c + a^2 + a*b^3*c + 2*a*b^2*c^2 - 2*a*b^2*c + a*b*c^3 - 2*a*b*c^2 + a*b*c + 2*a*b + 2*a*c + b^2 + 2*b*c + c^2 - 1)/(a*b*c))/(t^4 + (4*a*b*c - 2*a - 2*b - 2*c - 2)*t^3 + (4*a^2*b^2*c^2 - 4*a^2*b*c + a^2 - 4*a*b^2*c - 4*a*b*c^2 - 4*a*b*c + 2*a*b + 2*a*c + 2*a + b^2 + 2*b*c + 2*b + c^2 + 2*c + 3)*t^2 + (4*a*b*c - 2*a - 2*b - 2*c - 2)*t + 1) Total time: 0.200 seconds, Total memory usage: 3.34MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Thu Dec 8 15:16:08 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); S:=Transpose(A)*A^(-1); F,T:=PrimaryRationalForm(S); B:=T*A*Transpose(T); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*B*Transpose(P)-B)); G:=GroebnerBasis(I); Kt:=PolynomialRing(K); D:=2*a*b*c-a-b-c-1; px:=2*t+D; q:=t^2+t*D+1; print Evaluate(G[8],[0,0,0,0,0,0,0,px/q,t*px/q+1]); Output: Magma V2.11-10 Thu Dec 8 2005 15:16:08 on modular [Seed = 3640790655] ------------------------------------- (8*t^4 + (20*a^2*b^2*c^2 - 10*a^2*b*c - 10*a*b^2*c - 10*a*b*c^2 + 2*a*b*c - 6*a - 6*b - 6*c + 6)/(a*b*c)*t^3 + (12*a^3*b^3*c^3 - 12*a^3*b^2*c^2 + 3*a^3*b*c - 12*a^2*b^3*c^2 - 12*a^2*b^2*c^3 + 16*a^2*b^2*c^2 + 6*a^2*b^2*c + 6*a^2*b*c^2 - 22*a^2*b*c + 7*a^2 + 3*a*b^3*c + 6*a*b^2*c^2 - 22*a*b^2*c + 3*a*b*c^3 - 22*a*b*c^2 + 11*a*b*c + 14*a*b + 14*a*c + 7*b^2 + 14*b*c + 7*c^2 - 7)/(a*b*c)*t^2 + (16*a^3*b^3*c^3 - 24*a^3*b^2*c^2 + 12*a^3*b*c - 2*a^3 - 24*a^2*b^3*c^2 - 24*a^2*b^2*c^3 + 4*a^2*b^2*c^2 + 24*a^2*b^2*c + 24*a^2*b*c^2 + 2*a^2*b*c - 6*a^2*b - 6*a^2*c - 2*a^2 + 12*a*b^3*c + 24*a*b^2*c^2 + 2*a*b^2*c - 6*a*b^2 + 12*a*b*c^3 + 2*a*b*c^2 - 18*a*b*c - 4*a*b - 6*a*c^2 - 4*a*c - 2*b^3 - 6*b^2*c - 2*b^2 - 6*b*c^2 - 4*b*c - 2*c^3 - 2*c^2 + 4)/(a*b*c)*t + (4*a^3*b^3*c^3 - 4*a^3*b^2*c^2 + a^3*b*c - 4*a^2*b^3*c^2 - 4*a^2*b^2*c^3 + 2*a^2*b^2*c + 2*a^2*b*c^2 - 2*a^2*b*c + a^2 + a*b^3*c + 2*a*b^2*c^2 - 2*a*b^2*c + a*b*c^3 - 2*a*b*c^2 + a*b*c + 2*a*b + 2*a*c + b^2 + 2*b*c + c^2 - 1)/(a*b*c))/(t^4 + (4*a*b*c - 2*a - 2*b - 2*c - 2)*t^3 + (4*a^2*b^2*c^2 - 4*a^2*b*c + a^2 - 4*a*b^2*c - 4*a*b*c^2 - 4*a*b*c + 2*a*b + 2*a*c + 2*a + b^2 + 2*b*c + 2*b + c^2 + 2*c + 3)*t^2 + (4*a*b*c - 2*a - 2*b - 2*c - 2)*t + 1) Total time: 0.200 seconds, Total memory usage: 3.34MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Thu Dec 8 15:10:16 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); S:=Transpose(A)*A^(-1); F,T:=PrimaryRationalForm(S); B:=T*A*Transpose(T); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*B*Transpose(P)-B)); print GroebnerBasis(I); Output: Magma V2.11-10 Thu Dec 8 2005 15:10:16 on modular [Seed = 3741585369] ------------------------------------- [ $.1^2 - 1, $.2, $.3, $.4, $.5 + (-2*a*b*c + a + b + c - 1)/(a*b*c)*$.8 - $.9, $.6 + $.8, $.7, $.8^2 + (2*a*b*c - a - b - c + 1)/(a*b*c)*$.8*$.9 + $.9^2 - 1 ] Total time: 0.200 seconds, Total memory usage: 3.34MB '207.191' ************** MAGMA ***************** Host 207.191.40.178 (207.191.40.178) Time: Thu Dec 8 14:35:51 2005 Input: 2003^Sin(2312312) Output: Magma V2.11-10 Thu Dec 8 2005 14:35:51 on modular [Seed = 2470992810] ------------------------------------- 0.00650119842810155572431836770133141839665 Total time: 0.190 seconds, Total memory usage: 3.24MB '207.191' ************** MAGMA ***************** Host 207.191.40.178 (207.191.40.178) Time: Thu Dec 8 14:35:42 2005 Input: 2003^sin(2312312) Output: Magma V2.11-10 Thu Dec 8 2005 14:35:41 on modular [Seed = 2521652917] ------------------------------------- >> 2003^sin(2312312); ^ User error: Identifier 'sin' has not been declared or assigned Total time: 0.190 seconds, Total memory usage: 3.24MB '207.191' ************** MAGMA ***************** Host 207.191.40.178 (207.191.40.178) Time: Thu Dec 8 14:35:23 2005 Input: 2+2 Output: Magma V2.11-10 Thu Dec 8 2005 14:35:23 on modular [Seed = 2638632008] ------------------------------------- 4 Total time: 0.190 seconds, Total memory usage: 3.24MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Thu Dec 8 12:48:27 2005 Input: K:=FunctionField(RationalField(),4); A:=Matrix(K,2,2,[a,b,c,d]); R:=PolynomialRing(K,4); P:=Matrix(R,2,2,[R.1,R.2,R.3,R.4]); I:=Ideal(Eltseq(P*A*Transpose(P)-Transpose(A))); print GroebnerBasis(I); Q:=Matrix(K,2,2,[-1,(b+c)/d,0,1]); print (Q*A*Transpose(Q)-Transpose(A)); Output: Magma V2.11-10 Thu Dec 8 2005 12:48:27 on modular [Seed = 637409248] ------------------------------------- [ $.1 + $.4, $.2 - a/d*$.3 + (-b - c)/d*$.4, $.3^2 + (b + c)/a*$.3*$.4 + d/a*$.4^2 - d/a ] [0 0] [0 0] Total time: 0.180 seconds, Total memory usage: 3.24MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Thu Dec 8 12:46:24 2005 Input: K:=FunctionField(RationalField(),4); A:=Matrix(K,2,2,[a,b,c,d]); R:=PolynomialRing(K,4); P:=Matrix(R,2,2,[R.1,R.2,R.3,R.4]); I:=Ideal(Eltseq(P*A*Transpose(P)-Transpose(A))); print GroebnerBasis(I); Output: Magma V2.11-10 Thu Dec 8 2005 12:46:23 on modular [Seed = 687020649] ------------------------------------- [ $.1 + $.4, $.2 - a/d*$.3 + (-b - c)/d*$.4, $.3^2 + (b + c)/a*$.3*$.4 + d/a*$.4^2 - d/a ] Total time: 0.190 seconds, Total memory usage: 3.24MB '217.136' ************** MAGMA ***************** Host 217.136.22.124 (217.136.22.124) Time: Thu Dec 8 10:53:55 2005 Input: factor(25478963) Output: Magma V2.11-10 Thu Dec 8 2005 10:53:54 on modular [Seed = 888086537] ------------------------------------- >> factor(25478963); ^ User error: Identifier 'factor' has not been declared or assigned Total time: 0.190 seconds, Total memory usage: 3.24MB '130.149' ************** MAGMA ***************** Host 130.149.168.47 (130.149.168.47) Time: Thu Dec 8 09:06:07 2005 Input: F := FreeAbelianGroup(4); A := quo< F | 2*a + 0*b + 0*c, b, 0, d >; print A; Output: Magma V2.11-10 Thu Dec 8 2005 09:06:07 on modular [Seed = 3406869805] ------------------------------------- Abelian Group isomorphic to Z/2 + Z Defined on 2 generators Relations: 2*l = 0 Total time: 0.190 seconds, Total memory usage: 3.24MB '130.149' ************** MAGMA ***************** Host 130.149.168.47 (130.149.168.47) Time: Thu Dec 8 09:06:02 2005 Input: F := FreeAbelianGroup(4); A := quo< F | 2*a + 0*b + 0*c, b, 0, d >; print A; Output: Magma V2.11-10 Thu Dec 8 2005 09:06:02 on modular [Seed = 4146891028] ------------------------------------- >> A := quo< F | 2*a + 0*b + 0*c, b, 0, d >; ^ Runtime error in 'AssignNames': Too many names given for this object Abelian Group isomorphic to Z/2 + Z Defined on 2 generators Relations: 2*A.1 = 0 Total time: 0.180 seconds, Total memory usage: 3.24MB '130.149' ************** MAGMA ***************** Host 130.149.168.47 (130.149.168.47) Time: Thu Dec 8 09:05:57 2005 Input: F := FreeAbelianGroup(4); A := quo< F | 2*a + 0*b + 0*c, b, 0, d >; print A; Output: Magma V2.11-10 Thu Dec 8 2005 09:05:57 on modular [Seed = 4079521055] ------------------------------------- >> A := quo< F | 2*a + 0*b + 0*c, b, 0, d >; ^ Runtime error in 'AssignNames': Too many names given for this object Abelian Group isomorphic to Z/2 + Z Defined on 2 generators Relations: 2*A.1 = 0 Total time: 0.190 seconds, Total memory usage: 3.24MB '130.149' ************** MAGMA ***************** Host 130.149.168.47 (130.149.168.47) Time: Thu Dec 8 09:05:44 2005 Input: F := FreeAbelianGroup(4); A := quo< F | 2*a + 0*b + 0*c, b, 0, d >; print A; Output: Magma V2.11-10 Thu Dec 8 2005 09:05:44 on modular [Seed = 4280582510] ------------------------------------- >> A := quo< F | 2*a + 0*b + 0*c, b, 0, d >; ^ Runtime error in 'AssignNames': Too many names given for this object Abelian Group isomorphic to Z/2 + Z Defined on 2 generators Relations: 2*A.1 = 0 Total time: 0.180 seconds, Total memory usage: 3.24MB '130.149' ************** MAGMA ***************** Host 130.149.168.47 (130.149.168.47) Time: Thu Dec 8 09:05:37 2005 Input: F := FreeAbelianGroup(4); A := quo< F | 2*a + 0*b + 0*c, b, 0, d >; print A; Output: Magma V2.11-10 Thu Dec 8 2005 09:05:36 on modular [Seed = 4213212514] ------------------------------------- >> A := quo< F | 2*a + 0*b + 0*c, b, 0, d >; ^ Runtime error in 'AssignNames': Too many names given for this object Abelian Group isomorphic to Z/2 + Z Defined on 2 generators Relations: 2*A.1 = 0 Total time: 0.190 seconds, Total memory usage: 3.24MB '130.149' ************** MAGMA ***************** Host 130.149.168.47 (130.149.168.47) Time: Thu Dec 8 09:04:39 2005 Input: F := FreeAbelianGroup(4); A := quo< F | 2*a + 0*b + 0*c, b, 0, d >; print A; Output: Magma V2.11-10 Thu Dec 8 2005 09:04:39 on modular [Seed = 3879508409] ------------------------------------- Abelian Group isomorphic to Z/2 + Z Defined on 2 generators Relations: 2*A.1 = 0 Total time: 0.180 seconds, Total memory usage: 3.24MB '130.149' ************** MAGMA ***************** Host 130.149.168.47 (130.149.168.47) Time: Thu Dec 8 09:04:23 2005 Input: F := FreeAbelianGroup(4); A := quo< F | 2*a + 0*b + 0*c, b, c, d >; print A; Output: Magma V2.11-10 Thu Dec 8 2005 09:04:23 on modular [Seed = 3812138413] ------------------------------------- Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*A.1 = 0 Total time: 0.180 seconds, Total memory usage: 3.24MB '130.149' ************** MAGMA ***************** Host 130.149.168.47 (130.149.168.47) Time: Thu Dec 8 09:04:17 2005 Input: F := FreeAbelianGroup(3); A := quo< F | 2*a + 0*b + 0*c, b, c, d >; print A; Output: Magma V2.11-10 Thu Dec 8 2005 09:04:17 on modular [Seed = 4013199783] ------------------------------------- >> A := quo< F | 2*a + 0*b + 0*c, b, c, d >; ^ User error: Identifier 'd' has not been declared or assigned >> print A; ^ User error: Identifier 'A' has not been declared or assigned Total time: 0.190 seconds, Total memory usage: 3.24MB '130.149' ************** MAGMA ***************** Host 130.149.168.47 (130.149.168.47) Time: Thu Dec 8 09:04:09 2005 Input: F := FreeAbelianGroup(3); A := quo< F | 2*a + 0*b + 0*c, b, c >; print A; Output: Magma V2.11-10 Thu Dec 8 2005 09:04:09 on modular [Seed = 3945829787] ------------------------------------- Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*A.1 = 0 Total time: 0.190 seconds, Total memory usage: 3.24MB '130.149' ************** MAGMA ***************** Host 130.149.168.47 (130.149.168.47) Time: Thu Dec 8 09:04:04 2005 Input: F := FreeAbelianGroup(3); A := quo< F | 2*a + 0*b + 0*c, b, 9*a + 6*b + 3*c >; print A; Output: Magma V2.11-10 Thu Dec 8 2005 09:04:04 on modular [Seed = 2538400234] ------------------------------------- Abelian Group isomorphic to Z/6 Defined on 1 generator Relations: 6*A.1 = 0 Total time: 0.190 seconds, Total memory usage: 3.24MB '130.149' ************** MAGMA ***************** Host 130.149.168.47 (130.149.168.47) Time: Thu Dec 8 09:03:57 2005 Input: F := FreeAbelianGroup(3); A := quo< F | 2*a + 0*b + 0*c, 8*a + 5*b + 2*c, 9*a + 6*b + 3*c >; print A; Output: Magma V2.11-10 Thu Dec 8 2005 09:03:57 on modular [Seed = 2471030263] ------------------------------------- Abelian Group isomorphic to Z/6 Defined on 1 generator Relations: 6*A.1 = 0 Total time: 0.180 seconds, Total memory usage: 3.24MB '130.149' ************** MAGMA ***************** Host 130.149.168.47 (130.149.168.47) Time: Thu Dec 8 09:03:38 2005 Input: F := FreeAbelianGroup(3); A := quo< F | 7*a + 4*b + c, 8*a + 5*b + 2*c, 9*a + 6*b + 3*c >; print A; Output: Magma V2.11-10 Thu Dec 8 2005 09:03:38 on modular [Seed = 2672091532] ------------------------------------- Abelian Group isomorphic to Z/3 + Z Defined on 2 generators Relations: 3*A.1 = 0 Total time: 0.180 seconds, Total memory usage: 3.24MB '130.149' ************** MAGMA ***************** Host 130.149.168.47 (130.149.168.47) Time: Thu Dec 8 09:03:16 2005 Input: F := FreeGroup(4); A := quo< F | 2*a, b, d >; Output: Magma V2.11-10 Thu Dec 8 2005 09:03:16 on modular [Seed = 2254305419] ------------------------------------- >> A := quo< F | 2*a, b, d >;; ^ Runtime error in '*': Bad argument types Argument types given: RngIntElt, GrpFPElt Total time: 0.190 seconds, Total memory usage: 3.24MB '130.149' ************** MAGMA ***************** Host 130.149.168.47 (130.149.168.47) Time: Thu Dec 8 09:02:05 2005 Input: F := FreeGroup(4); C:=sub< F| a,b,c,d >; f:=quo< F| C >; C Output: Magma V2.11-10 Thu Dec 8 2005 09:02:05 on modular [Seed = 2186935670] ------------------------------------- Finitely presented group C on 4 generators Generators as words in group F C.1 = a C.2 = b C.3 = c C.4 = d Total time: 0.190 seconds, Total memory usage: 3.24MB '130.149' ************** MAGMA ***************** Host 130.149.168.47 (130.149.168.47) Time: Thu Dec 8 09:01:55 2005 Input: F := FreeGroup(4); C:=sub< F| >; f:=quo< F| C >; C Output: Magma V2.11-10 Thu Dec 8 2005 09:01:55 on modular [Seed = 2387997044] ------------------------------------- Finitely presented group C on 1 generator Generators as words in group F C.1 = Id(F) Total time: 0.190 seconds, Total memory usage: 3.24MB '130.149' ************** MAGMA ***************** Host 130.149.168.47 (130.149.168.47) Time: Thu Dec 8 09:01:51 2005 Input: F := FreeGroup(4); C:=sub< F| >; f:=quo< F| C >; f Output: Magma V2.11-10 Thu Dec 8 2005 09:01:51 on modular [Seed = 2320627052] ------------------------------------- Finitely presented group f on 4 generators Relations Id(f) = Id(f) Total time: 0.190 seconds, Total memory usage: 3.24MB '130.149' ************** MAGMA ***************** Host 130.149.168.47 (130.149.168.47) Time: Thu Dec 8 09:01:35 2005 Input: F := FreeGroup(4); C:=sub< F| >; f:=quo< F| C >; C; F; Output: Magma V2.11-10 Thu Dec 8 2005 09:01:34 on modular [Seed = 3060648225] ------------------------------------- Finitely presented group C on 1 generator Generators as words in group F C.1 = Id(F) Finitely presented group F on 4 generators (free) Total time: 0.190 seconds, Total memory usage: 3.24MB '130.149' ************** MAGMA ***************** Host 130.149.168.47 (130.149.168.47) Time: Thu Dec 8 09:01:27 2005 Input: F := FreeGroup(4); D := FreeGroup(4); C:=sub< F| >; f:=quo< F| C >; C; F; Output: Magma V2.11-10 Thu Dec 8 2005 09:01:26 on modular [Seed = 2993278253] ------------------------------------- >> D := FreeGroup(4); ^ User error: bad syntax Finitely presented group C on 1 generator Generators as words in group F C.1 = Id(F) Finitely presented group F on 4 generators (free) Total time: 0.190 seconds, Total memory usage: 3.24MB '130.149' ************** MAGMA ***************** Host 130.149.168.47 (130.149.168.47) Time: Thu Dec 8 09:00:58 2005 Input: F := FreeGroup(4); D := FreeGroup(4); C:=sub< F| 2a,b,c >; f:=quo< F| C >; C; F; Output: Magma V2.11-10 Thu Dec 8 2005 09:00:57 on modular [Seed = 3194339598] ------------------------------------- >> D := FreeGroup(4); ^ User error: bad syntax >> C:=sub< F| 2a,b,c >; ^ User error: bad syntax >> f:=quo< F| C >; ^ User error: Identifier 'C' has not been declared or assigned >> C; ^ User error: Identifier 'C' has not been declared or assigned Finitely presented group F on 4 generators (free) Total time: 0.190 seconds, Total memory usage: 3.24MB '130.149' ************** MAGMA ***************** Host 130.149.168.47 (130.149.168.47) Time: Thu Dec 8 09:00:23 2005 Input: F := FreeGroup(4); D := FreeGroup(4); C:=sub< F| a^2,b,c >; f:=quo< F| C >; C; F; Output: Magma V2.11-10 Thu Dec 8 2005 09:00:23 on modular [Seed = 3126969828] ------------------------------------- >> D := FreeGroup(4); ^ User error: bad syntax Finitely presented group C on 3 generators Generators as words in group F C.1 = a^2 C.2 = b C.3 = c Finitely presented group F on 4 generators (free) Total time: 0.190 seconds, Total memory usage: 3.24MB '130.149' ************** MAGMA ***************** Host 130.149.168.47 (130.149.168.47) Time: Thu Dec 8 09:00:15 2005 Input: F := FreeGroup(4); D<2*a,b,c> := FreeGroup(4); C:=sub< F| a^2,b,c >; f:=quo< F| C >; C; F; Output: Magma V2.11-10 Thu Dec 8 2005 09:00:15 on modular [Seed = 2793265632] ------------------------------------- >> D<2*a,b,c> := FreeGroup(4); ^ User error: bad syntax Finitely presented group C on 3 generators Generators as words in group F C.1 = a^2 C.2 = b C.3 = c Finitely presented group F on 4 generators (free) Total time: 0.190 seconds, Total memory usage: 3.24MB '130.149' ************** MAGMA ***************** Host 130.149.168.47 (130.149.168.47) Time: Thu Dec 8 09:00:12 2005 Input: F := FreeGroup(4); D<2a,b,c> := FreeGroup(4); C:=sub< F| a^2,b,c >; f:=quo< F| C >; C; F; Output: Magma V2.11-10 Thu Dec 8 2005 09:00:11 on modular [Seed = 2725895648] ------------------------------------- >> D<2a,b,c> := FreeGroup(4); ^ User error: bad syntax Finitely presented group C on 3 generators Generators as words in group F C.1 = a^2 C.2 = b C.3 = c Finitely presented group F on 4 generators (free) Total time: 0.190 seconds, Total memory usage: 3.24MB '130.149' ************** MAGMA ***************** Host 130.149.168.47 (130.149.168.47) Time: Thu Dec 8 08:59:53 2005 Input: F := FreeGroup(4); C:=sub< F| a^2,b,c >; f:=quo< F| C >; C; F; Output: Magma V2.11-10 Thu Dec 8 2005 08:59:53 on modular [Seed = 2926957022] ------------------------------------- Finitely presented group C on 3 generators Generators as words in group F C.1 = a^2 C.2 = b C.3 = c Finitely presented group F on 4 generators (free) Total time: 0.180 seconds, Total memory usage: 3.24MB '130.149' ************** MAGMA ***************** Host 130.149.168.47 (130.149.168.47) Time: Thu Dec 8 08:59:44 2005 Input: F := FreeGroup(4); C:=sub< F| a^2,b,c >; f:=quo< F| C >; C F Output: Magma V2.11-10 Thu Dec 8 2005 08:59:44 on modular [Seed = 2859587027] ------------------------------------- >> F; ^ User error: bad syntax Total time: 0.180 seconds, Total memory usage: 3.24MB '130.149' ************** MAGMA ***************** Host 130.149.168.47 (130.149.168.47) Time: Thu Dec 8 08:59:39 2005 Input: F := FreeGroup(4); C:=sub< F| a^2,b,c >; f:=quo< F| C >; C Output: Magma V2.11-10 Thu Dec 8 2005 08:59:39 on modular [Seed = 1452091700] ------------------------------------- Finitely presented group C on 3 generators Generators as words in group F C.1 = a^2 C.2 = b C.3 = c Total time: 0.190 seconds, Total memory usage: 3.24MB '130.149' ************** MAGMA ***************** Host 130.149.168.47 (130.149.168.47) Time: Thu Dec 8 08:59:32 2005 Input: F := FreeGroup(4); C:=sub< F| a^2,b,c >; C f:=quo< F| C >; f Output: Magma V2.11-10 Thu Dec 8 2005 08:59:32 on modular [Seed = 1384721727] ------------------------------------- >> f:=quo< F| C >; ^ User error: bad syntax >> f; ^ User error: Identifier 'f' has not been declared or assigned Total time: 0.190 seconds, Total memory usage: 3.24MB '130.149' ************** MAGMA ***************** Host 130.149.168.47 (130.149.168.47) Time: Thu Dec 8 08:59:25 2005 Input: F := FreeGroup(4); C:=sub< F| a^2,b,c >; f:=quo< F| C >; f Output: Magma V2.11-10 Thu Dec 8 2005 08:59:25 on modular [Seed = 1585783114] ------------------------------------- Finitely presented group f on 4 generators Relations f.3 = Id(f) f.1^2 = Id(f) f.2 = Id(f) Total time: 0.190 seconds, Total memory usage: 3.24MB '130.149' ************** MAGMA ***************** Host 130.149.168.47 (130.149.168.47) Time: Thu Dec 8 08:59:20 2005 Input: F := FreeGroup(4); C:=sub< F| a^a,b,c >; f:=quo< F| C >; f Output: Magma V2.11-10 Thu Dec 8 2005 08:59:20 on modular [Seed = 1518413131] ------------------------------------- Finitely presented group f on 4 generators Relations f.3 = Id(f) f.2 = Id(f) f.1 = Id(f) Total time: 0.190 seconds, Total memory usage: 3.24MB '130.149' ************** MAGMA ***************** Host 130.149.168.47 (130.149.168.47) Time: Thu Dec 8 08:59:15 2005 Input: F := FreeGroup(4); C:=sub< F| aa,b,c >; f:=quo< F| C >; f Output: Magma V2.11-10 Thu Dec 8 2005 08:59:15 on modular [Seed = 1184708956] ------------------------------------- >> C:=sub< F| aa,b,c >; ^ User error: Identifier 'aa' has not been declared or assigned >> f:=quo< F| C >; ^ User error: Identifier 'C' has not been declared or assigned >> f; ^ User error: Identifier 'f' has not been declared or assigned Total time: 0.190 seconds, Total memory usage: 3.24MB '130.149' ************** MAGMA ***************** Host 130.149.168.47 (130.149.168.47) Time: Thu Dec 8 08:59:09 2005 Input: F := FreeGroup(4); C:=sub< F| a/2,b,c >; f:=quo< F| C >; f Output: Magma V2.11-10 Thu Dec 8 2005 08:59:08 on modular [Seed = 1117338951] ------------------------------------- >> C:=sub< F| a/2,b,c >; ^ Runtime error in '/': Bad argument types Argument types given: GrpFPElt, RngIntElt >> f:=quo< F| C >; ^ User error: Identifier 'C' has not been declared or assigned >> f; ^ User error: Identifier 'f' has not been declared or assigned Total time: 0.180 seconds, Total memory usage: 3.24MB '130.149' ************** MAGMA ***************** Host 130.149.168.47 (130.149.168.47) Time: Thu Dec 8 08:58:54 2005 Input: F := FreeGroup(4); C:=sub< F| 2*a,b,c >; f:=quo< F| C >; f Output: Magma V2.11-10 Thu Dec 8 2005 08:58:54 on modular [Seed = 1318400377] ------------------------------------- >> C:=sub< F| 2*a,b,c >; ^ Runtime error in '*': Bad argument types Argument types given: RngIntElt, GrpFPElt >> f:=quo< F| C >; ^ User error: Identifier 'C' has not been declared or assigned >> f; ^ User error: Identifier 'f' has not been declared or assigned Total time: 0.190 seconds, Total memory usage: 3.24MB '130.149' ************** MAGMA ***************** Host 130.149.168.47 (130.149.168.47) Time: Thu Dec 8 08:58:38 2005 Input: F := FreeGroup(4); C:=sub< F|a,b,c >; f:=quo< F| C >; f Output: Magma V2.11-10 Thu Dec 8 2005 08:58:37 on modular [Seed = 1251030382] ------------------------------------- Finitely presented group f on 4 generators Relations f.3 = Id(f) f.2 = Id(f) f.1 = Id(f) Total time: 0.190 seconds, Total memory usage: 3.24MB '130.149' ************** MAGMA ***************** Host 130.149.168.47 (130.149.168.47) Time: Thu Dec 8 08:58:33 2005 Input: F := FreeGroup(4); C:=sub< F|a,b,c >; f:=quo< F| a >; f Output: Magma V2.11-10 Thu Dec 8 2005 08:58:32 on modular [Seed = 1991051607] ------------------------------------- Finitely presented group f on 4 generators Relations f.1 = Id(f) Total time: 0.190 seconds, Total memory usage: 3.24MB '130.149' ************** MAGMA ***************** Host 130.149.168.47 (130.149.168.47) Time: Thu Dec 8 08:58:26 2005 Input: F := FreeGroup(4); C:=sub< F|a,b,c >; f:=quo< F| a >; Output: Magma V2.11-10 Thu Dec 8 2005 08:58:25 on modular [Seed = 1923681626] ------------------------------------- Total time: 0.190 seconds, Total memory usage: 3.24MB '130.149' ************** MAGMA ***************** Host 130.149.168.47 (130.149.168.47) Time: Thu Dec 8 08:58:13 2005 Input: F := FreeGroup(4); C:=sub< F|a,b,c >; f:=qou< F| a >; Output: Magma V2.11-10 Thu Dec 8 2005 08:58:12 on modular [Seed = 2141456290] ------------------------------------- >> f:=qou< F| a >; ^ User error: bad syntax Total time: 0.190 seconds, Total memory usage: 3.24MB '130.149' ************** MAGMA ***************** Host 130.149.168.47 (130.149.168.47) Time: Thu Dec 8 08:58:08 2005 Input: F := FreeGroup(4); C:=sub< F|a,b,c >; f:=qou< F| F.2 >; Output: Magma V2.11-10 Thu Dec 8 2005 08:58:08 on modular [Seed = 2074084258] ------------------------------------- >> f:=qou< F| F.2 >; ^ User error: bad syntax Total time: 0.180 seconds, Total memory usage: 3.24MB '130.149' ************** MAGMA ***************** Host 130.149.168.47 (130.149.168.47) Time: Thu Dec 8 08:57:51 2005 Input: F := FreeGroup(4); C:=sub< F|a,b,c >; f:=qou< F|C >; Output: Magma V2.11-10 Thu Dec 8 2005 08:57:50 on modular [Seed = 1740382096] ------------------------------------- >> f:=qou< F|C >; ^ User error: bad syntax Total time: 0.190 seconds, Total memory usage: 3.24MB '130.149' ************** MAGMA ***************** Host 130.149.168.47 (130.149.168.47) Time: Thu Dec 8 08:57:18 2005 Input: F := FreeGroup(4); C:=sub; f:=qou Output: Magma V2.11-10 Thu Dec 8 2005 08:57:18 on modular [Seed = 1673010164] ------------------------------------- >> f:=qou; ^ User error: bad syntax Total time: 0.180 seconds, Total memory usage: 3.24MB '130.149' ************** MAGMA ***************** Host 130.149.168.47 (130.149.168.47) Time: Thu Dec 8 08:56:59 2005 Input: F := FreeGroup(4); C:=sub; qou Output: Magma V2.11-10 Thu Dec 8 2005 08:56:59 on modular [Seed = 1874073581] ------------------------------------- >> qou; ^ User error: bad syntax Total time: 0.190 seconds, Total memory usage: 3.24MB '130.149' ************** MAGMA ***************** Host 130.149.168.47 (130.149.168.47) Time: Thu Dec 8 08:56:53 2005 Input: F := FreeGroup(4); C:=sub; qou Output: Magma V2.11-10 Thu Dec 8 2005 08:56:53 on modular [Seed = 1806701543] ------------------------------------- >> qou; ^ User error: Bad print expression Total time: 0.190 seconds, Total memory usage: 3.24MB '130.149' ************** MAGMA ***************** Host 130.149.168.47 (130.149.168.47) Time: Thu Dec 8 08:56:03 2005 Input: F := FreeGroup(4); C:=sub; C Output: Magma V2.11-10 Thu Dec 8 2005 08:56:03 on modular [Seed = 399273805] ------------------------------------- Finitely presented group C on 3 generators Generators as words in group F C.1 = a C.2 = b C.3 = c Total time: 0.190 seconds, Total memory usage: 3.24MB '130.149' ************** MAGMA ***************** Host 130.149.168.47 (130.149.168.47) Time: Thu Dec 8 08:55:58 2005 Input: F := FreeGroup(4); C:=sub; C Output: Magma V2.11-10 Thu Dec 8 2005 08:55:58 on modular [Seed = 331901780] ------------------------------------- >> C:=sub; ^ Runtime error in '+': Bad argument types Argument types given: GrpFPElt, GrpFPElt >> C; ^ User error: Identifier 'C' has not been declared or assigned Total time: 0.190 seconds, Total memory usage: 3.24MB '130.149' ************** MAGMA ***************** Host 130.149.168.47 (130.149.168.47) Time: Thu Dec 8 08:55:53 2005 Input: F := FreeGroup(4); C:=sub; C Output: Magma V2.11-10 Thu Dec 8 2005 08:55:53 on modular [Seed = 532965215] ------------------------------------- >> C:=sub; ^ Runtime error in '*': Bad argument types Argument types given: RngIntElt, GrpFPElt >> C; ^ User error: Identifier 'C' has not been declared or assigned Total time: 0.190 seconds, Total memory usage: 3.24MB '130.149' ************** MAGMA ***************** Host 130.149.168.47 (130.149.168.47) Time: Thu Dec 8 08:55:44 2005 Input: F := FreeGroup(4); C:=sub; C Output: Magma V2.11-10 Thu Dec 8 2005 08:55:44 on modular [Seed = 465593170] ------------------------------------- Finitely presented group C on 1 generator Generators as words in group F C.1 = a Total time: 0.190 seconds, Total memory usage: 3.24MB '130.149' ************** MAGMA ***************** Host 130.149.168.47 (130.149.168.47) Time: Thu Dec 8 08:55:25 2005 Input: F := FreeGroup(4); C:=sub; C Output: Magma V2.11-10 Thu Dec 8 2005 08:55:25 on modular [Seed = 131891043] ------------------------------------- >> C:=sub; ^ User error: bad syntax >> C; ^ User error: Identifier 'C' has not been declared or assigned Total time: 0.190 seconds, Total memory usage: 3.24MB '130.149' ************** MAGMA ***************** Host 130.149.168.47 (130.149.168.47) Time: Thu Dec 8 08:52:30 2005 Input: F := FreeGroup(4); C:=subGrpAb(F,a); C Output: Magma V2.11-10 Thu Dec 8 2005 08:52:30 on modular [Seed = 248870841] ------------------------------------- >> C:=subGrpAb(F,a); ^ User error: Identifier 'subGrpAb' has not been declared or assigned >> C; ^ User error: Identifier 'C' has not been declared or assigned Total time: 0.190 seconds, Total memory usage: 3.24MB '130.149' ************** MAGMA ***************** Host 130.149.168.47 (130.149.168.47) Time: Thu Dec 8 08:51:43 2005 Input: F := FreeGroup(4); C:=sub(F,a); C Output: Magma V2.11-10 Thu Dec 8 2005 08:51:42 on modular [Seed = 181498701] ------------------------------------- >> C:=sub(F,a); ^ User error: Identifier 'sub' has not been declared or assigned >> C; ^ User error: Identifier 'C' has not been declared or assigned Total time: 0.190 seconds, Total memory usage: 3.24MB '130.149' ************** MAGMA ***************** Host 130.149.168.47 (130.149.168.47) Time: Thu Dec 8 08:46:53 2005 Input: F := FreeGroup(4); rels := [ b^a = b*c ]; G := quo< GrpGPC : F | rels >; G; Output: Magma V2.11-10 Thu Dec 8 2005 08:46:52 on modular [Seed = 938233170] ------------------------------------- Consistency check (2,1,1) failed. >> G := quo< GrpGPC : F | rels >; ^ Runtime error in quo< ... >: Bad presentation for group >> G;; ^ User error: Identifier 'G' has not been declared or assigned Total time: 0.190 seconds, Total memory usage: 3.24MB '130.149' ************** MAGMA ***************** Host 130.149.168.47 (130.149.168.47) Time: Thu Dec 8 08:46:31 2005 Input: F := FreeGroup(4); rels := [ 2*a = 0 ]; G := quo< GrpGPC : F | rels >; G; Output: Magma V2.11-10 Thu Dec 8 2005 08:46:31 on modular [Seed = 870861101] ------------------------------------- >> rels := [ 2*a = 0 ]; ^ Runtime error in '*': Bad argument types Argument types given: RngIntElt, GrpFPElt >> G := quo< GrpGPC : F | rels >; ^ User error: Identifier 'rels' has not been declared or assigned >> G;; ^ User error: Identifier 'G' has not been declared or assigned Total time: 0.190 seconds, Total memory usage: 3.24MB '130.149' ************** MAGMA ***************** Host 130.149.168.47 (130.149.168.47) Time: Thu Dec 8 08:44:54 2005 Input: F := FreeGroup(4); rels := [ a = a+a ]; G := quo< GrpGPC : F | rels >; G; Output: Magma V2.11-10 Thu Dec 8 2005 08:44:54 on modular [Seed = 1071924416] ------------------------------------- >> rels := [ a = a+a ]; ^ Runtime error in '+': Bad argument types Argument types given: GrpFPElt, GrpFPElt >> G := quo< GrpGPC : F | rels >; ^ User error: Identifier 'rels' has not been declared or assigned >> G;; ^ User error: Identifier 'G' has not been declared or assigned Total time: 0.200 seconds, Total memory usage: 3.24MB '130.149' ************** MAGMA ***************** Host 130.149.168.47 (130.149.168.47) Time: Thu Dec 8 08:44:31 2005 Input: F := FreeGroup(4); rels := [ a = 2*a ]; G := quo< GrpGPC : F | rels >; G; Output: Magma V2.11-10 Thu Dec 8 2005 08:44:31 on modular [Seed = 1004552365] ------------------------------------- >> rels := [ a = 2*a ]; ^ Runtime error in '*': Bad argument types Argument types given: RngIntElt, GrpFPElt >> G := quo< GrpGPC : F | rels >; ^ User error: Identifier 'rels' has not been declared or assigned >> G;; ^ User error: Identifier 'G' has not been declared or assigned Total time: 0.190 seconds, Total memory usage: 3.24MB '130.149' ************** MAGMA ***************** Host 130.149.168.47 (130.149.168.47) Time: Thu Dec 8 08:43:27 2005 Input: F := FreeGroup(3); rels := [ b^a = b*c, b^(a^-1) = b*c^-1 ]; G := quo< GrpGPC : F | rels >; G; Output: Magma V2.11-10 Thu Dec 8 2005 08:43:20 on modular [Seed = 637427317] ------------------------------------- GrpGPC : G of infinite order on 3 PC-generators PC-Relations: b^a = b * c, b^(a^-1) = b * c^-1 Total time: 0.240 seconds, Total memory usage: 3.24MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Thu Dec 8 07:10:28 2005 Input: G :=DirichletGroup(3675); G; X :=Elements(G); X; Y :=X[8]; Conductor(Y); Order(Y); Output: Magma V2.11-10 Thu Dec 8 2005 07:10:26 on modular [Seed = 3256460123] ------------------------------------- Group of Dirichlet characters of modulus 3675 over Rational Field [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3 ] 105 2 Total time: 0.230 seconds, Total memory usage: 3.34MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Thu Dec 8 07:09:47 2005 Input: G :=DirichletGroup(3675); G; X :=Elements(G); X; Y :=X[7]; Conductor(Y); Order(Y); Output: Magma V2.11-10 Thu Dec 8 2005 07:09:16 on modular [Seed = 3474239112] ------------------------------------- Group of Dirichlet characters of modulus 3675 over Rational Field [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3 ] 35 2 Total time: 0.250 seconds, Total memory usage: 3.34MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Thu Dec 8 07:08:58 2005 Input: G :=DirichletGroup(3675); G; X :=Elements(G); X; Y :=X[8]; Conductor(Y); Order(Y); Output: Magma V2.11-10 Thu Dec 8 2005 07:08:54 on modular [Seed = 4163082629] ------------------------------------- Group of Dirichlet characters of modulus 3675 over Rational Field [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3 ] 105 2 Total time: 0.240 seconds, Total memory usage: 3.34MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Thu Dec 8 05:38:54 2005 Input: G :=DirichletGroup(105); G; X :=Elements(G); X; Y :=X[8]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; Coefficient(qEigenform(D[1],12),11);Parent($1); Output: Magma V2.11-10 Thu Dec 8 2005 05:38:54 on modular [Seed = 4130191596] ------------------------------------- Group of Dirichlet characters of modulus 105 over Rational Field [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3 ] 105 2 [] >> Coefficient(qEigenform(D[1],12),11);Parent($1);; ^ Runtime error in '[]': Sequence element 1 not defined Set of null sequences Total time: 0.210 seconds, Total memory usage: 3.91MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Thu Dec 8 05:37:09 2005 Input: G :=DirichletGroup(105); G; X :=Elements(G); X; Y :=X[7]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; Coefficient(qEigenform(D[1],12),11);Parent($1); Output: Magma V2.11-10 Thu Dec 8 2005 05:37:09 on modular [Seed = 3996492063] ------------------------------------- Group of Dirichlet characters of modulus 105 over Rational Field [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3 ] 35 2 [ Modular symbols space of level 105, weight 3, character $.2*$.3, and dimension 16 over Rational Field ] -4298657581773597647504145782596750932310358556832711002454219543201/1060354788\ 155084785331796352233842185911113329794742366431633553792856457447585916*a^\ 15 + 59792039146938966702085346471210765249524513188560315063883509432295/1\ 060354788155084785331796352233842185911113329794742366431633553792856457447\ 585916*a^14 + 2248116023755252986404452938356016622954425676202415019802671\ 354342037/30295851090145279580908467206681205311746095136992639040903815822\ 6530416413595976*a^13 - 106375893649896766138612912643006997350281429012339\ 2147140957671964814929/8482838305240678282654370817870737487288906638357938\ 931453068430342851659580687328*a^12 - 8025767737983380042130528078215951109\ 8785225632595715072720574844079253/1559345276698654096076171106226238508692\ 8137202863858329876964026365536138935087*a^11 + 454564293504204618256213652472228130198025352321178049906367490038175998773\ /42414191526203391413271854089353687436444533191789694657265342151714258297\ 90343664*a^10 + 46330602181772887430568752314793023560850992485147208453922\ 9748146003476351/3029585109014527958090846720668120531174609513699263904090\ 38158226530416413595976*a^9 - 372838139837664292581809634900339180485698524\ 348133251578366856627400443548187/84828383052406782826543708178707374872889\ 06638357938931453068430342851659580687328*a^8 - 586908927999659183743988816215739384065711763924034789343566565949655167205\ 7/6237381106794616384304684424904954034771254881145543331950785610546214455\ 5740348*a^7 + 1782427423253817741237532264633280299707060112858881754667069\ 0963198406953111017/2120709576310169570663592704467684371822226659589484732\ 863267107585712914895171832*a^6 - 88132348041601832812155114351918574435160\ 741509599262517612176007084653830584979/21207095763101695706635927044676843\ 71822226659589484732863267107585712914895171832*a^5 - 412607627058114982348201132587715925295396576903215390735685694561153455672\ 0390407/8482838305240678282654370817870737487288906638357938931453068430342\ 851659580687328*a^4 + 31511281196024029702444946733976629500285737560112208\ 52157554522243148399975157367/530177394077542392665898176116921092955556664\ 897371183215816776896428228723792958*a^3 - 117842641475610869683534570735827031431221433659772609683009641833477302642\ 068142067/42414191526203391413271854089353687436444533191789694657265342151\ 71425829790343664*a^2 + 206153836034717403743112319848430370292546919027209\ 544943233012443147466094824699047/21207095763101695706635927044676843718222\ 26659589484732863267107585712914895171832*a - 192001441789413192954514317740798175866107536955491792124958758103956350599\ 7580330285/8482838305240678282654370817870737487288906638357938931453068430\ 342851659580687328 Univariate Quotient Polynomial Algebra in a over Rational Field with modulus a^16 - 56*a^15 - 176*a^14 + 68728*a^13 - 1091948*a^12 - 22679096*a^11 + 785484976*a^10 - 2622829640*a^9 - 165895838642*a^8 + 2411431900824*a^7 - 2408011941072*a^6 - 240699299035992*a^5 + 2802555639497124*a^4 - 16324010107722792*a^3 + 61776032371713744*a^2 - 167208206400482520*a + 250585184476674681 Total time: 0.670 seconds, Total memory usage: 5.10MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Thu Dec 8 05:37:01 2005 Input: G :=DirichletGroup(105); G; X :=Elements(G); X; Y :=X[6]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; Coefficient(qEigenform(D[1],12),11);Parent($1); Output: Magma V2.11-10 Thu Dec 8 2005 05:37:00 on modular [Seed = 3778719241] ------------------------------------- Group of Dirichlet characters of modulus 105 over Rational Field [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3 ] 21 2 [] >> Coefficient(qEigenform(D[1],12),11);Parent($1);; ^ Runtime error in '[]': Sequence element 1 not defined Set of null sequences Total time: 0.210 seconds, Total memory usage: 4.38MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Thu Dec 8 05:35:26 2005 Input: G :=DirichletGroup(105); G; X :=Elements(G); X; Y :=X[5]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; Coefficient(qEigenform(D[1],12),11);Parent($1); Output: Magma V2.11-10 Thu Dec 8 2005 05:35:25 on modular [Seed = 3829377965] ------------------------------------- Group of Dirichlet characters of modulus 105 over Rational Field [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3 ] 7 2 [ Modular symbols space of level 105, weight 3, character $.3, and dimension 12 over Rational Field ] -9627753560054457599941856824348845429104226288761/2997533989399047188118278316\ 9642133506697637827789504145539724592*a^11 - 827853708124021379076764139093823015797391865310667/69942459752644434389426\ 494062498311515627821598175509672926024048*a^10 + 40644059523131479561661764583374110235005028651212103/699424597526444343894\ 26494062498311515627821598175509672926024048*a^9 + 989817221877944422541708558763314139114963392701732223/69942459752644434389\ 426494062498311515627821598175509672926024048*a^8 - 4261878610765066657170994510860016307411024111549809945/4995889982331745313\ 530463861607022251116272971298250690923287432*a^7 - 165719595913874323498843511999184560463227059528344202087/11657076625440739\ 064904415677083051919271303599695918278821004008*a^6 - 8550237862426105241240329493675908597396021803988777427/1665296660777248437\ 843487953869007417038757657099416896974429144*a^5 - 22112515395558585770232358054851227312561970994091911441063/388569220848024\ 6354968138559027683973090434533231972759607001336*a^4 + 750904150428656166015240182381075933603548512829283147506987/77713844169604\ 92709936277118055367946180869066463945519214002672*a^3 - 1556490164649547596071612342841201796183206846608735384947771/8634871574400\ 54745548475235339485327353429896273771724357111408*a^2 - 2844096403220102441323434035539965744585803674555527965456627/2878290524800\ 18248516158411779828442451143298757923908119037136*a - 15269237696891184072333869330633977403516889841562052248686571/587406229551\ 0576500329763505710784539819250995059671594266064 Univariate Quotient Polynomial Algebra in a over Rational Field with modulus a^12 + 36*a^11 - 2526*a^10 - 62724*a^9 + 4357263*a^8 + 70031160*a^7 - 2741991732*a^6 - 10848278808*a^5 + 934352883855*a^4 + 3118035044772*a^3 + 825882174951570*a^2 + 3724663386540060*a + 4212898119152289 Total time: 0.790 seconds, Total memory usage: 5.00MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Thu Dec 8 05:34:16 2005 Input: G :=DirichletGroup(105); G; X :=Elements(G); X; Y :=X[4]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; Coefficient(qEigenform(D[1],12),11);Parent($1); Output: Magma V2.11-10 Thu Dec 8 2005 05:34:15 on modular [Seed = 2672087196] ------------------------------------- Group of Dirichlet characters of modulus 105 over Rational Field [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3 ] 15 2 [ Modular symbols space of level 105, weight 3, character $.1*$.2, and dimension 24 over Rational Field ] 1204211855691391593183200709289952155056950680388873577693915493051674171073611\ 01921111/773542841130395463370175720228639654302353253795450016894911779720\ 33465700824042396292235962121107906560000*a^23 + 178413663326744086935855267582565035975997564952593413810130901598489068335\ 9326279354439/6446190342753295528084797668571997119186277114962083474124264\ 83100278880840200353302435299684342565888000*a^21 + 314163122012022777408157294806902082697836175760398973415297829811458435589\ 29161788688778211/154708568226079092674035144045727930860470650759090003378\ 98235594406693140164808479258447192424221581312000*a^19 + 303749024372985183227086387617877825746125847535800518380329791841520875051\ 37885941569536235047/386771420565197731685087860114319827151176626897725008\ 44745588986016732850412021198146117981060553953280000*a^17 + 724667128574550938868579220941148542781353982434056782045280667876824960971\ 064760280657235891487/42974602285021970187231984457146647461241847433080556\ 49416176554001859205601335688682901997895617105920000*a^15 + 184468297942586187153684218929043555888424249750515777012371263348874589630\ 922011126137916458194407/96692855141299432921271965028579956787794156724431\ 25211186397246504183212603005299536529495265138488320000*a^13 + 119723832376532744864758931160771929311384921478567895648026322866302922000\ 31120908049804612831370681/128923806855065910561695953371439942383725542299\ 24166948248529662005577616804007066048705993686851317760000*a^11 + 424031823655253626958763197922687939261200790731230192598618549025992165574\ 49705382056803442573887713/386771420565197731685087860114319827151176626897\ 7250084474558898601673285041202119814611798106055395328000*a^9 + 843999404082304051100875189403538073275387924102969608753468563437996319086\ 05326325926534126622691803/120677510316754362460245822188555328284298479531\ 271453493745987475871241342939223707164174667895644160000*a^7 + 272668624924044821167899473416090164973595095502511767773323075807539886151\ 8629513429037143619522487001/4834642757064971646063598251428997839389707836\ 221562605593198623252091606301502649768264747632569244160000*a^5 + 146381531928962722508261961449391819363941947189458556374555060833176093362\ 05324254941433278864341591873371/773542841130395463370175720228639654302353\ 25379545001689491177972033465700824042396292235962121107906560000*a^3 - 441582134098380488737104434011700011673746956771871179425637328829394160952\ 886464187339452627561547089377/15470856822607909267403514404572793086047065\ 07590900033789823559440669314016480847925844719242422158131200*a Univariate Quotient Polynomial Algebra in a over Rational Field with modulus a^24 + 1780*a^22 + 1308130*a^20 + 507142404*a^18 + 109322750831*a^16 + 12460050133096*a^14 + 617380119190876*a^12 + 7725436928267560*a^10 + 433854703302693743*a^8 + 769413012321755556*a^6 + 100884220801829193186*a^4 - 424438144655392806700*a^2 + 686517081253767450625 Total time: 0.720 seconds, Total memory usage: 5.38MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Thu Dec 8 05:33:42 2005 Input: G :=DirichletGroup(105); G; X :=Elements(G); X; Y :=X[4]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; qEigenform(D[1],12);Parent($1); Output: Magma V2.11-10 Thu Dec 8 2005 05:33:41 on modular [Seed = 2454314419] ------------------------------------- Group of Dirichlet characters of modulus 105 over Rational Field [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3 ] 15 2 [ Modular symbols space of level 105, weight 3, character $.1*$.2, and dimension 24 over Rational Field ] q + (-1204211855691391593183200709289952155056950680388873577693915493051674171\ 07361101921111/773542841130395463370175720228639654302353253795450016894911\ 77972033465700824042396292235962121107906560000*a^23 - 178413663326744086935855267582565035975997564952593413810130901598489068335\ 9326279354439/6446190342753295528084797668571997119186277114962083474124264\ 83100278880840200353302435299684342565888000*a^21 - 314163122012022777408157294806902082697836175760398973415297829811458435589\ 29161788688778211/154708568226079092674035144045727930860470650759090003378\ 98235594406693140164808479258447192424221581312000*a^19 - 303749024372985183227086387617877825746125847535800518380329791841520875051\ 37885941569536235047/386771420565197731685087860114319827151176626897725008\ 44745588986016732850412021198146117981060553953280000*a^17 - 724667128574550938868579220941148542781353982434056782045280667876824960971\ 064760280657235891487/42974602285021970187231984457146647461241847433080556\ 49416176554001859205601335688682901997895617105920000*a^15 - 184468297942586187153684218929043555888424249750515777012371263348874589630\ 922011126137916458194407/96692855141299432921271965028579956787794156724431\ 25211186397246504183212603005299536529495265138488320000*a^13 - 119723832376532744864758931160771929311384921478567895648026322866302922000\ 31120908049804612831370681/128923806855065910561695953371439942383725542299\ 24166948248529662005577616804007066048705993686851317760000*a^11 - 424031823655253626958763197922687939261200790731230192598618549025992165574\ 49705382056803442573887713/386771420565197731685087860114319827151176626897\ 7250084474558898601673285041202119814611798106055395328000*a^9 - 843999404082304051100875189403538073275387924102969608753468563437996319086\ 05326325926534126622691803/120677510316754362460245822188555328284298479531\ 271453493745987475871241342939223707164174667895644160000*a^7 - 272668624924044821167899473416090164973595095502511767773323075807539886151\ 8629513429037143619522487001/4834642757064971646063598251428997839389707836\ 221562605593198623252091606301502649768264747632569244160000*a^5 - 146381531928962722508261961449391819363941947189458556374555060833176093362\ 05324254941433278864341591873371/773542841130395463370175720228639654302353\ 25379545001689491177972033465700824042396292235962121107906560000*a^3 + 198866781635917141547745587446897932027845346436277121321546088827006347496\ 9367312113184171869983705220577/1547085682260790926740351440457279308604706\ 507590900033789823559440669314016480847925844719242422158131200*a)*q^2 + (12476961992900269994914694842672726036592375736303953796713579843574932262\ 621543224704133/47443960922664255086704110840689898797210999566120934369554\ 58915618052562983874600305923805676761284935680000*a^23 + 903595974794547123846916422064344864739315690818900717059310116401468361494\ 30943/814831443454823619459664702214699014956542904734928549800611533043183\ 9059534876248060213010825216000*a^22 + 133007050168195301953005387311837515633766020831799119301997881028159849995\ 84557381087875799/284663765535985530520224665044139392783265997396725606217\ 3275349370831537790324760183554283406056770961408000*a^21 + 357865506177126080130314451026606122453653922721176124837188427611487348007\ 8020927/1810736541010719154354810449365997811014539788299841221779136740095\ 96423545219472179115844685004800*a^20 + 974897171824789797692377662628740332528541515938800040501584199494134717216\ 3693943711385988549/2846637655359855305202246650441393927832659973967256062\ 173275349370831537790324760183554283406056770961408000*a^19 + 237122859406857588886674772392788810383911185280433205170360608553915486355\ 77963699093/162966288690964723891932940442939802991308580946985709960122306\ 6086367811906975249612042602165043200*a^18 + 188215803869345293851035002929306158336603517954081240851672910216800742344\ 82775471870754798609821/142331882767992765260112332522069696391632998698362\ 80310866376746854157688951623800917771417030283854807040000*a^17 + 460936402009265941459050048574968218631190986832770111355047874993530859849\ 68626858245597/814831443454823619459664702214699014956542904734928549800611\ 5330431839059534876248060213010825216000*a^16 + 671613875219589473470896334250658076060120771084512817606627993836243816340\ 401262078657390004269049/23721980461332127543352055420344949398605499783060\ 46718477729457809026281491937300152961902838380642467840000*a^15 + 166414687021068859430755703051148105662111510593905775943082868411673194568\ 0295629438861993/1358052405758039365766107837024498358260904841224880916334\ 352555071973176589146041343368835137536000*a^14 + 755015903911301101640126665642257862869747687738633291675159959248332901172\ 46029766302320265873167909/237219804613321275433520554203449493986054997830\ 6046718477729457809026281491937300152961902838380642467840000*a^13 + 574838437834069102056863264125090372795883678164347095185930084641058975423\ 874296643258353039/40741572172741180972983235110734950747827145236746427490\ 03057665215919529767438124030106505412608000*a^12 + 108013810666016199125403395251373191360736687993098688282623461087516170068\ 58620027283698568204882031337/711659413839963826300561662610348481958164993\ 4918140155433188373427078844475811900458885708515141927403520000*a^11 + 325951975220820429899352129081277103288253284678756994495452468357504059431\ 8863033269561505201/4526841352526797885887026123414994527536349470749603054\ 44784185023991058863048680447789611712512000*a^10 + 218130343648827737565803372280913862500672801900659825070260709851180442620\ 87242361986766483003219401209/142331882767992765260112332522069696391632998\ 6983628031086637674685415768895162380091777141703028385480704000*a^9 + 861904410485912602916314470610699386422004492306488529876067923010892809355\ 68512184146137205113/814831443454823619459664702214699014956542904734928549\ 800611533043183905953487624806021301082521600*a^8 + 245536630371036020664553019781851079231832069072760081613118395881714666314\ 76674851298429413095083696427/222046618982828026926852312826941804043109202\ 33753947442849261695560308407100817162118208138892798525440000*a^7 + 660104243408122437806728808695494411609075576792469569048106937921403333179\ 24306102007660562939/127118789930549706623972652451591109977619797930566076\ 41195187722982588236403863101497992216576000*a^6 - 784270088923878604488584132605570315486605782664341985567063583588721908770\ 0640491455239991661894817790081/1423318827679927652601123325220696963916329\ 9869836280310866376746854157688951623800917771417030283854807040000*a^5 + 158972368752598138639851865415021096570100607366720440374672617783107027228\ 592084988625677414079583/81483144345482361945966470221469901495654290473492\ 85498006115330431839059534876248060213010825216000*a^4 + 126177885362772698146987322954101513463415419823229999990285659191981727936\ 1294283120226611380015581913744663/4744396092266425508670411084068989879721\ 099956612093436955458915618052562983874600305923805676761284935680000*a^3 + 107548242445181351079847960998824707446537670615451239491034955486639086890\ 72587753799864188016071573/814831443454823619459664702214699014956542904734\ 9285498006115330431839059534876248060213010825216000*a^2 - 101578065370575548158380905662701243709132216675842064292224015995999904186\ 5443782424034611565293866032975871/5693275310719710610404493300882787855665\ 31994793451212434655069874166307558064952036710856681211354192281600*a - 876320634049716527955042884895808065835717919630735226929539927608466136519\ 633711982462702897508927/32593257738192944778386588088587960598261716189397\ 1419920244613217273562381395049922408520433008640)*q^3 + (30751561629235116440071308563747291375715714291036326509348297845284326516\ 529/73807195965110835095984121577418389035918741370917441105127856253911585\ 68419271963822656712704000*a^22 + 72634138526219137808122858123253617477379\ 34416031130241562849759391724397635561/984095946201477801279788287698911853\ 812249884945565881401704750052154475789236261843020895027200*a^20 + 397522895603599006568811105692766471482533174791540340530385729497940210610\ 2408427/7380719596511083509598412157741838903591874137091744110512785625391\ 15856841927196382265671270400*a^18 + 30475970835712802812665893951199093678\ 019436221306674628593504127798434523918249469807/14761439193022167019196824\ 315483677807183748274183488221025571250782317136838543927645313425408000*a^\ 16 + 1787863494995953277267595865643103885884753221899914866306019254841276\ 46655440712404143/410039977583949083866578453207879939088437452060652450584\ 043645855064364912181775767925372928000*a^14 + 350294452309956478262589394583688280814211180328226911724501572141809287269\ 613550399271559/73807195965110835095984121577418389035918741370917441105127\ 85625391158568419271963822656712704000*a^12 + 616448361674496555865156405854511542961024017207645493093892442093067447693\ 061064754152971/30752998318796181289993383990590995431632808904548933793803\ 2734391298273684136 ** WARNING: Output too long, hence truncated. '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Thu Dec 8 05:33:28 2005 Input: G :=DirichletGroup(105); G; X :=Elements(G); X; Y :=X[3]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; qEigenform(D[1],12);Parent($1); Output: Magma V2.11-10 Thu Dec 8 2005 05:33:27 on modular [Seed = 2504973641] ------------------------------------- Group of Dirichlet characters of modulus 105 over Rational Field [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3 ] 5 2 [] >> qEigenform(D[1],12);Parent($1);; ^ Runtime error in '[]': Sequence element 1 not defined Set of null sequences Total time: 0.220 seconds, Total memory usage: 4.50MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Thu Dec 8 05:32:37 2005 Input: G :=DirichletGroup(105); G; X :=Elements(G); X; Y :=X[2]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; qEigenform(D[1],12);Parent($1); Output: Magma V2.11-10 Thu Dec 8 2005 05:32:36 on modular [Seed = 2287190120] ------------------------------------- Group of Dirichlet characters of modulus 105 over Rational Field [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3 ] 3 2 [ Modular symbols space of level 105, weight 3, character $.1, and dimension 16 over Rational Field ] q + a*q^2 + (-355/468672*a^15 + 181/234336*a^14 - 3715/117168*a^13 + 2239/78112*a^12 - 78413/156224*a^11 + 2704/7323*a^10 - 27854/7323*a^9 + 73663/39056*a^8 - 6908345/468672*a^7 + 692717/234336*a^6 - 3457763/117168*a^5 - 52637/78112*a^4 - 4645727/156224*a^3 + 291167/117168*a^2 - 719393/58584*a + 4945/2441)*q^3 + (a^2 + 4)*q^4 + (-107/78112*a^15 - 3703/58584*a^13 - 88657/78112*a^11 - 588091/58584*a^9 - 3556185/78112*a^7 - 720082/7323*a^5 - 6160859/78112*a^3 - 408745/29292*a)*q^5 + (181/234336*a^15 + 245/78112*a^14 + 2239/78112*a^13 + 28463/234336*a^12 + 2704/7323*a^11 + 197951/117168*a^10 + 73663/39056*a^9 + 395555/39056*a^8 + 692717/234336*a^7 + 1924413/78112*a^6 - 52637/78112*a^5 + 3825007/234336*a^4 + 291167/117168*a^3 - 23593/58584*a^2 + 4945/2441*a + 1065/2441)*q^6 + (-187/78112*a^14 - 7587/78112*a^12 - 57065/39056*a^10 - 395867/39056*a^8 - 2620631/78112*a^6 - 3968375/78112*a^4 - 639803/19528*a^2 - 9730/2441)*q^7 + (a^3 + 8*a)*q^8 + (-1199/468672*a^15 + 37/4882*a^14 - 9041/78112*a^13 + 18001/58584*a^12 - 952757/468672*a^11 + 135191/29292*a^10 - 2062787/117168*a^9 + 311231/9764*a^8 - 37077241/468672*a^7 + 253210/2441*a^6 - 13759729/78112*a^5 + 8512241/58584*a^4 - 77690083/468672*a^3 + 1840435/29292*a^2 - 2881949/58584*a + 4001/4882)*q^9 + (-23/117168*a^14 - 149/19528*a^12 - 3059/29292*a^10 - 10817/19528*a^8 - 47809/117168*a^6 + 43213/9764*a^4 + 220415/29292*a^2 + 1926/2441)*q^10 + (2887/234336*a^15 + 5339/9764*a^13 + 2187631/234336*a^11 + 188875/2441*a^9 + 75768245/234336*a^7 + 1529956/2441*a^5 + 98626709/234336*a^3 + 597641/9764*a)*q^11 + O(q^12) Power series ring in q over Univariate Quotient Polynomial Algebra in a over Rational Field with modulus a^16 + 46*a^14 + 823*a^12 + 7252*a^10 + 32831*a^8 + 71486*a^6 + 60809*a^4 + 15680*a^2 + 576 Total time: 0.760 seconds, Total memory usage: 5.10MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Thu Dec 8 05:31:14 2005 Input: G :=DirichletGroup(105); G; X :=Elements(G); X; Y :=X[2]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; qEigenform(D[10],12);Parent($1); Output: Magma V2.11-10 Thu Dec 8 2005 05:31:11 on modular [Seed = 2186922697] ------------------------------------- Group of Dirichlet characters of modulus 105 over Rational Field [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3 ] 3 2 [ Modular symbols space of level 105, weight 3, character $.1, and dimension 16 over Rational Field ] >> qEigenform(D[10],12);Parent($1);; ^ Runtime error in '[]': Sequence element 10 not defined Set of sequences over Power Structure of ModSym Total time: 0.760 seconds, Total memory usage: 4.81MB '165.91.' ************** MAGMA ***************** Host 165.91.100.60 (165.91.100.60) Time: Thu Dec 8 00:05:45 2005 Input: S:=CuspForms(Gamma0(3),14); SetPrecision(S,15); Basis(S); for i:=1 to 3 do (Coefficient(S.i,14)-6*Coefficient(S.i,7))/7; end for; Output: Magma V2.11-10 Thu Dec 8 2005 00:05:45 on modular [Seed = 3489934558] ------------------------------------- [ q + 52*q^4 - 2430*q^5 + 8748*q^6 + 44900*q^7 - 340200*q^8 + 531441*q^9 + 1370160*q^10 - 6137208*q^11 + 5904900*q^12 + 10160150*q^13 - 30275856*q^14 + O(q^15), q^2 - 54*q^4 + 128*q^5 + 729*q^6 - 3456*q^7 + 3524*q^8 + 16902*q^10 - 36608*q^11 - 39366*q^12 + 89856*q^13 + 82808*q^14 + O(q^15), q^3 + 12*q^4 + 36*q^5 - 12*q^6 - 204*q^7 - 792*q^8 + 1104*q^10 + 7524*q^11 + 700*q^12 + 1416*q^13 - 39024*q^14 + O(q^15) ] -4363608 14792 -5400 Total time: 0.300 seconds, Total memory usage: 4.96MB '165.91.' ************** MAGMA ***************** Host 165.91.100.60 (165.91.100.60) Time: Thu Dec 8 00:00:23 2005 Input: S:=CuspForms(Gamma0(3),14); SetPrecision(S,15); Basis(S); for i:=1 to 3 do Coefficient(S.i,14)-6*Coefficient(S.i,7); end for; Output: Magma V2.11-10 Thu Dec 8 2005 00:00:23 on modular [Seed = 3390977034] ------------------------------------- [ q + 52*q^4 - 2430*q^5 + 8748*q^6 + 44900*q^7 - 340200*q^8 + 531441*q^9 + 1370160*q^10 - 6137208*q^11 + 5904900*q^12 + 10160150*q^13 - 30275856*q^14 + O(q^15), q^2 - 54*q^4 + 128*q^5 + 729*q^6 - 3456*q^7 + 3524*q^8 + 16902*q^10 - 36608*q^11 - 39366*q^12 + 89856*q^13 + 82808*q^14 + O(q^15), q^3 + 12*q^4 + 36*q^5 - 12*q^6 - 204*q^7 - 792*q^8 + 1104*q^10 + 7524*q^11 + 700*q^12 + 1416*q^13 - 39024*q^14 + O(q^15) ] -30545256 103544 -37800 Total time: 0.300 seconds, Total memory usage: 4.96MB '165.91.' ************** MAGMA ***************** Host 165.91.100.60 (165.91.100.60) Time: Wed Dec 7 23:58:52 2005 Input: S:=CuspForms(Gamma0(3),14); SetPrecision(S,15); Basis(S); Output: Magma V2.11-10 Wed Dec 7 2005 23:58:52 on modular [Seed = 2388291621] ------------------------------------- [ q + 52*q^4 - 2430*q^5 + 8748*q^6 + 44900*q^7 - 340200*q^8 + 531441*q^9 + 1370160*q^10 - 6137208*q^11 + 5904900*q^12 + 10160150*q^13 - 30275856*q^14 + O(q^15), q^2 - 54*q^4 + 128*q^5 + 729*q^6 - 3456*q^7 + 3524*q^8 + 16902*q^10 - 36608*q^11 - 39366*q^12 + 89856*q^13 + 82808*q^14 + O(q^15), q^3 + 12*q^4 + 36*q^5 - 12*q^6 - 204*q^7 - 792*q^8 + 1104*q^10 + 7524*q^11 + 700*q^12 + 1416*q^13 - 39024*q^14 + O(q^15) ] Total time: 0.310 seconds, Total memory usage: 4.96MB '165.91.' ************** MAGMA ***************** Host 165.91.100.60 (165.91.100.60) Time: Wed Dec 7 23:56:23 2005 Input: S:=CuspForms(Gamma0(3),14); Basis(S); Output: Magma V2.11-10 Wed Dec 7 2005 23:56:23 on modular [Seed = 2219866964] ------------------------------------- [ q + 52*q^4 - 2430*q^5 + 8748*q^6 + 44900*q^7 + O(q^8), q^2 - 54*q^4 + 128*q^5 + 729*q^6 - 3456*q^7 + O(q^8), q^3 + 12*q^4 + 36*q^5 - 12*q^6 - 204*q^7 + O(q^8) ] Total time: 0.310 seconds, Total memory usage: 4.96MB '200.177' ************** MAGMA ***************** Host 200.177.23.188 (200.177.23.188) Time: Wed Dec 7 23:02:24 2005 Input: K := FunctionField(Rationals(), 4); (a1*a2 + a1*a3 + a2*a3) eq a1*a2*a3 - (a1 - 1)*(a2 - 1)*(a3 - 1) + (a1 + a2 + a3) - 1; Output: Magma V2.11-10 Wed Dec 7 2005 23:02:23 on modular [Seed = 1824038660] ------------------------------------- true Total time: 0.190 seconds, Total memory usage: 3.24MB '200.177' ************** MAGMA ***************** Host 200.177.23.188 (200.177.23.188) Time: Wed Dec 7 23:00:46 2005 Input: K := FunctionField(Rationals(), 4); (a1 - 1)*(a2 - 1)*(a3 - 1); Output: Magma V2.11-10 Wed Dec 7 2005 23:00:46 on modular [Seed = 1656131243] ------------------------------------- a1*a2*a3 - a1*a2 - a1*a3 + a1 - a2*a3 + a2 + a3 - 1 Total time: 0.180 seconds, Total memory usage: 3.24MB '200.177' ************** MAGMA ***************** Host 200.177.23.188 (200.177.23.188) Time: Wed Dec 7 23:00:04 2005 Input: K := FunctionField(Rationals(), 4); (1-a1)*(1-a2)*(1-a3); Output: Magma V2.11-10 Wed Dec 7 2005 23:00:04 on modular [Seed = 1739819908] ------------------------------------- -a1*a2*a3 + a1*a2 + a1*a3 - a1 + a2*a3 - a2 - a3 + 1 Total time: 0.190 seconds, Total memory usage: 3.24MB '200.177' ************** MAGMA ***************** Host 200.177.23.188 (200.177.23.188) Time: Wed Dec 7 22:59:34 2005 Input: K := FunctionField(Rationals(), 4); (x-a1)*(x-a2)*(x-a3); Output: Magma V2.11-10 Wed Dec 7 2005 22:59:34 on modular [Seed = 482930845] ------------------------------------- -a1*a2*a3 + a1*a2*x + a1*a3*x - a1*x^2 + a2*a3*x - a2*x^2 - a3*x^2 + x^3 Total time: 0.180 seconds, Total memory usage: 3.24MB '200.177' ************** MAGMA ***************** Host 200.177.23.188 (200.177.23.188) Time: Wed Dec 7 22:59:23 2005 Input: K := FunctionField(Rationals(), 4); (x-a2)*(x-a3)*(x-a3); Output: Magma V2.11-10 Wed Dec 7 2005 22:59:22 on modular [Seed = 298188204] ------------------------------------- -a2*a3^2 + 2*a2*a3*x - a2*x^2 + a3^2*x - 2*a3*x^2 + x^3 Total time: 0.190 seconds, Total memory usage: 3.24MB '200.177' ************** MAGMA ***************** Host 200.177.23.188 (200.177.23.188) Time: Wed Dec 7 22:58:34 2005 Input: K := FunctionField(Rationals(), 3); (a1+a2)*(a1+a3)*(a2+a3); Output: Magma V2.11-10 Wed Dec 7 2005 22:58:34 on modular [Seed = 381876827] ------------------------------------- a1^2*a2 + a1^2*a3 + a1*a2^2 + 2*a1*a2*a3 + a1*a3^2 + a2^2*a3 + a2*a3^2 Total time: 0.190 seconds, Total memory usage: 3.24MB '200.177' ************** MAGMA ***************** Host 200.177.23.188 (200.177.23.188) Time: Wed Dec 7 22:58:24 2005 Input: K := FunctionField(Rationals(), 3); (a1-a2)*(a1-a3)*(a2-a3); Output: Magma V2.11-10 Wed Dec 7 2005 22:58:23 on modular [Seed = 199223155] ------------------------------------- a1^2*a2 - a1^2*a3 - a1*a2^2 + a1*a3^2 + a2^2*a3 - a2*a3^2 Total time: 0.190 seconds, Total memory usage: 3.24MB '200.177' ************** MAGMA ***************** Host 200.177.23.188 (200.177.23.188) Time: Wed Dec 7 22:45:09 2005 Input: K := FunctionField(Rationals(), 8); //(a1*x + b1 - y)*(a2*x + b2 - y)*(a3*x + b3 - y); (a1*x + b1 - y)*(a2*x + b2 - y)*(a3*x + b3 - y) eq a1*a2*a3*x^3 + b1*b2*b3 + (b1 + b2 + b3)*y^2 + (a1*a2*b3 + a1*a3*b2 + a2*a3*b1 - (a1*a2 + a1*a3 + a2*a3)*y)*x^2 + (a1*b2*b3 + a2*b1*b3 + a3*b1*b2 + (a1 + a2 + a3)*y^2 - (a1*b2 + a1*b3 + a2*b1 + a2*b3 + a3*b1 + a3*b2)*y)*x - (b1*b2 + b1*b3 + b2*b3 + y^2)*y ; Output: Magma V2.11-10 Wed Dec 7 2005 22:45:09 on modular [Seed = 1038733919] ------------------------------------- true Total time: 0.180 seconds, Total memory usage: 3.34MB '200.177' ************** MAGMA ***************** Host 200.177.23.188 (200.177.23.188) Time: Wed Dec 7 22:40:14 2005 Input: K := FunctionField(Rationals(), 8); //(a1*x + b1 - y)*(a2*x + b2 - y)*(a3*x + b3 - y); (a1*x + b1 - y)*(a2*x + b2 - y)*(a3*x + b3 - y) eq a1*a2*a3*x^3 + (a1*a2*b3 + a1*a3*b2 + a2*a3*b1 - (a1*a2 + a1*a3 + a2*a3)*y)*x^2 + (a1*b2*b3 + a2*b1*b3 + a3*b1*b2 + (a1 + a2 + a3)*y^2 - (a1*b2 + a1*b3 + a2*b1 + a2*b3 + a3*b1 + a3*b2)*y)*x + b1*b2*b3 + (b1 + b2 + b3)*y^2 - (b1*b2 + b1*b3 + b2*b3 + y^2)*y ; Output: Magma V2.11-10 Wed Dec 7 2005 22:40:11 on modular [Seed = 738181911] ------------------------------------- true Total time: 0.200 seconds, Total memory usage: 3.34MB '200.177' ************** MAGMA ***************** Host 200.177.23.188 (200.177.23.188) Time: Wed Dec 7 22:35:41 2005 Input: K := FunctionField(Rationals(), 8); //(a1*x + b1 - y)*(a2*x + b2 - y)*(a3*x + b3 - y); (a1*x + b1 - y)*(a2*x + b2 - y)*(a3*x + b3 - y) eq a1*a2*a3*x^3 + (a1*a2*b3 - a1*a2*y + a1*a3*b2 - a1*a3*y + a2*a3*b1 - a2*a3*y)*x^2 + (a1*b2*b3 - a1*b2*y - a1*b3*y + a1*y^2 + a2*b1*b3 - a2*b1*y - a2*b3*y + a2*y^2 + a3*b1*b2 - a3*b1*y - a3*b2*y + a3*y^2)*x + b1*b2*b3 - b1*b2*y - b1*b3*y + b1*y^2 - b2*b3*y + b2*y^2 + b3*y^2 - y^3 ; Output: Magma V2.11-10 Wed Dec 7 2005 22:35:41 on modular [Seed = 3624677877] ------------------------------------- true Total time: 0.190 seconds, Total memory usage: 3.34MB '200.177' ************** MAGMA ***************** Host 200.177.23.188 (200.177.23.188) Time: Wed Dec 7 22:31:08 2005 Input: K := FunctionField(Rationals(), 8); (a1*x + b1 - y)*(a2*x + b2 - y)*(a3*x + b3 - y); Output: Magma V2.11-10 Wed Dec 7 2005 22:31:08 on modular [Seed = 3523623182] ------------------------------------- a1*a2*a3*x^3 + a1*a2*b3*x^2 - a1*a2*x^2*y + a1*a3*b2*x^2 - a1*a3*x^2*y + a1*b2*b3*x - a1*b2*x*y - a1*b3*x*y + a1*x*y^2 + a2*a3*b1*x^2 - a2*a3*x^2*y + a2*b1*b3*x - a2*b1*x*y - a2*b3*x*y + a2*x*y^2 + a3*b1*b2*x - a3*b1*x*y - a3*b2*x*y + a3*x*y^2 + b1*b2*b3 - b1*b2*y - b1*b3*y + b1*y^2 - b2*b3*y + b2*y^2 + b3*y^2 - y^3 Total time: 0.200 seconds, Total memory usage: 3.24MB '200.177' ************** MAGMA ***************** Host 200.177.23.188 (200.177.23.188) Time: Wed Dec 7 22:30:51 2005 Input: K := FunctionField(Rationals(), (a1*x + b1 - y)*(a2*x + b2 - y)*(a3*x + b3 - y); Output: Magma V2.11-10 Wed Dec 7 2005 22:30:51 on modular [Seed = 3440977414] ------------------------------------- >> (a1*x + b1 - y)*(a2*x + b2 - y)*(a3*x + b3 - y);; ^ User error: bad syntax Total time: 0.190 seconds, Total memory usage: 3.24MB '132.239' ************** MAGMA ***************** Host 132.239.145.83 (132.239.145.83) Time: Wed Dec 7 22:11:42 2005 Input: G :=DirichletGroup(1575); G; X :=Elements(G); X; Y :=X[2]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; qEigenform(D[1],2);Parent($1); qEigenform(D[2],2);Parent($1); qEigenform(D[3],2);Parent($1); qEigenform(D[4],2);Parent($1); qEigenform(D[5],2);Parent($1); qEigenform(D[6],2);Parent($1); qEigenform(D[7],2);Parent($1); qEigenform(D[8],2);Parent($1); qEigenform(D[9],2);Parent($1); qEigenform(D[10],2);Parent($1); Output: ** WARNING: Computation used more memory than allowed. ** Magma V2.11-10 Wed Dec 7 2005 22:11:38 on modular [Seed = 4146270397] ------------------------------------- Group of Dirichlet characters of modulus 1575 over Rational Field [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3 ] 3 2 Current total memory usage: 71.1MB, failed memory request: 31.6MB System Error: User memory limit has been reached >> D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); ^ Runtime error: Variable 'M' has not been initialized >> D; ^ User error: Identifier 'D' has not been declared or assigned >> qEigenform(D[1],2);Parent($1); ^ User error: Identifier 'D' has not been declared or assigned Integer Ring >> qEigenform(D[2],2);Parent($1); ^ User error: Identifier 'D' has not been declared or assigned Power Structure of RngInt >> qEigenform(D[3],2);Parent($1); ^ User error: Identifier 'D' has not been declared or assigned Power Structure of PowStr >> qEigenform(D[4],2);Parent($1); ^ User error: Identifier 'D' has not been declared or assigned Power Structure of PowStr >> qEigenform(D[5],2);Parent($1); ^ User error: Identifier 'D' has not been declared or assigned Power Structure of PowStr >> qEigenform(D[6],2);Parent($1); ^ User error: Identifier 'D' has not been declared or assigned Power Structure of PowStr >> qEigenform(D[7],2);Parent($1); ^ User error: Identifier 'D' has not been declared or assigned Power Structure of PowStr >> qEigenform(D[8],2);Parent($1); ^ User error: Identifier 'D' has not been declared or assigned Power Structure of PowStr >> qEigenform(D[9],2);Parent($1); ^ User error: Identifier 'D' has not been declared or assigned Power Structure of PowStr >> qEigenform(D[10],2);Parent($1); ^ User error: Identifier 'D' has not been declared or assigned Power Structure of PowStr Total time: 3.430 seconds, Total memory usage: 71.10MB '132.239' ************** MAGMA ***************** Host 132.239.145.83 (132.239.145.83) Time: Wed Dec 7 21:52:29 2005 Input: G :=DirichletGroup(225); G; X :=Elements(G); X; Y :=X[2]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; qEigenform(D[1],2);Parent($1); qEigenform(D[2],2);Parent($1); qEigenform(D[3],2);Parent($1); qEigenform(D[4],2);Parent($1); qEigenform(D[5],2);Parent($1); qEigenform(D[6],2);Parent($1); qEigenform(D[7],2);Parent($1); qEigenform(D[8],2);Parent($1); qEigenform(D[9],2);Parent($1); qEigenform(D[10],2);Parent($1); Output: Magma V2.11-10 Wed Dec 7 2005 21:52:27 on modular [Seed = 2338281281] ------------------------------------- Group of Dirichlet characters of modulus 225 over Rational Field [ 1, $.1, $.2, $.1*$.2 ] 3 2 [ Modular symbols space of level 225, weight 3, character $.1, and dimension 2 over Rational Field, Modular symbols space of level 225, weight 3, character $.1, and dimension 2 over Rational Field, Modular symbols space of level 225, weight 3, character $.1, and dimension 4 over Rational Field, Modular symbols space of level 225, weight 3, character $.1, and dimension 4 over Rational Field ] q + O(q^2) Power series ring in q over Univariate Quotient Polynomial Algebra in a over Rational Field with modulus a^2 + 2 q + O(q^2) Power series ring in q over Univariate Quotient Polynomial Algebra in a over Rational Field with modulus a^2 + 2 q + O(q^2) Power series ring in q over Univariate Quotient Polynomial Algebra in a over Rational Field with modulus a^4 + 14*a^2 + 9 q + O(q^2) Power series ring in q over Univariate Quotient Polynomial Algebra in a over Rational Field with modulus a^4 - 238*a^2 + 17689 >> qEigenform(D[5],2);Parent($1); ^ Runtime error in '[]': Sequence element 5 not defined Power Structure of RngSerPow >> qEigenform(D[6],2);Parent($1); ^ Runtime error in '[]': Sequence element 6 not defined Power Structure of PowStr >> qEigenform(D[7],2);Parent($1); ^ Runtime error in '[]': Sequence element 7 not defined Power Structure of PowStr >> qEigenform(D[8],2);Parent($1); ^ Runtime error in '[]': Sequence element 8 not defined Power Structure of PowStr >> qEigenform(D[9],2);Parent($1); ^ Runtime error in '[]': Sequence element 9 not defined Power Structure of PowStr >> qEigenform(D[10],2);Parent($1); ^ Runtime error in '[]': Sequence element 10 not defined Power Structure of PowStr Total time: 1.389 seconds, Total memory usage: 6.42MB '132.239' ************** MAGMA ***************** Host 132.239.145.83 (132.239.145.83) Time: Wed Dec 7 21:45:50 2005 Input: G :=DirichletGroup(675); G; X :=Elements(G); X; Y :=X[3]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; qEigenform(D[1],2);Parent($1); qEigenform(D[2],2);Parent($1); qEigenform(D[3],2);Parent($1); qEigenform(D[4],2);Parent($1); qEigenform(D[5],2);Parent($1); qEigenform(D[6],2);Parent($1); qEigenform(D[7],2);Parent($1); qEigenform(D[8],2);Parent($1); qEigenform(D[9],2);Parent($1); qEigenform(D[10],2);Parent($1); Output: Magma V2.11-10 Wed Dec 7 2005 21:45:50 on modular [Seed = 2203541244] ------------------------------------- Group of Dirichlet characters of modulus 675 over Rational Field [ 1, $.1, $.2, $.1*$.2 ] 5 2 [] >> qEigenform(D[1],2);Parent($1); ^ Runtime error in '[]': Sequence element 1 not defined Set of null sequences >> qEigenform(D[2],2);Parent($1); ^ Runtime error in '[]': Sequence element 2 not defined Power Structure of PowSeqEnum >> qEigenform(D[3],2);Parent($1); ^ Runtime error in '[]': Sequence element 3 not defined Power Structure of PowStr >> qEigenform(D[4],2);Parent($1); ^ Runtime error in '[]': Sequence element 4 not defined Power Structure of PowStr >> qEigenform(D[5],2);Parent($1); ^ Runtime error in '[]': Sequence element 5 not defined Power Structure of PowStr >> qEigenform(D[6],2);Parent($1); ^ Runtime error in '[]': Sequence element 6 not defined Power Structure of PowStr >> qEigenform(D[7],2);Parent($1); ^ Runtime error in '[]': Sequence element 7 not defined Power Structure of PowStr >> qEigenform(D[8],2);Parent($1); ^ Runtime error in '[]': Sequence element 8 not defined Power Structure of PowStr >> qEigenform(D[9],2);Parent($1); ^ Runtime error in '[]': Sequence element 9 not defined Power Structure of PowStr >> qEigenform(D[10],2);Parent($1); ^ Runtime error in '[]': Sequence element 10 not defined Power Structure of PowStr Total time: 0.230 seconds, Total memory usage: 4.72MB '132.239' ************** MAGMA ***************** Host 132.239.145.83 (132.239.145.83) Time: Wed Dec 7 21:33:26 2005 Input: G :=DirichletGroup(1215); G; X :=Elements(G); X; Y :=X[3]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; qEigenform(D[1],2);Parent($1); qEigenform(D[2],2);Parent($1); qEigenform(D[3],2);Parent($1); qEigenform(D[4],2);Parent($1); qEigenform(D[5],2);Parent($1); qEigenform(D[6],2);Parent($1); qEigenform(D[7],2);Parent($1); qEigenform(D[8],2);Parent($1); qEigenform(D[9],2);Parent($1); qEigenform(D[10],2);Parent($1); Output: Magma V2.11-10 Wed Dec 7 2005 21:33:26 on modular [Seed = 2910418426] ------------------------------------- Group of Dirichlet characters of modulus 1215 over Rational Field [ 1, $.1, $.2, $.1*$.2 ] 5 2 [] >> qEigenform(D[1],2);Parent($1); ^ Runtime error in '[]': Sequence element 1 not defined Set of null sequences >> qEigenform(D[2],2);Parent($1); ^ Runtime error in '[]': Sequence element 2 not defined Power Structure of PowSeqEnum >> qEigenform(D[3],2);Parent($1); ^ Runtime error in '[]': Sequence element 3 not defined Power Structure of PowStr >> qEigenform(D[4],2);Parent($1); ^ Runtime error in '[]': Sequence element 4 not defined Power Structure of PowStr >> qEigenform(D[5],2);Parent($1); ^ Runtime error in '[]': Sequence element 5 not defined Power Structure of PowStr >> qEigenform(D[6],2);Parent($1); ^ Runtime error in '[]': Sequence element 6 not defined Power Structure of PowStr >> qEigenform(D[7],2);Parent($1); ^ Runtime error in '[]': Sequence element 7 not defined Power Structure of PowStr >> qEigenform(D[8],2);Parent($1); ^ Runtime error in '[]': Sequence element 8 not defined Power Structure of PowStr >> qEigenform(D[9],2);Parent($1); ^ Runtime error in '[]': Sequence element 9 not defined Power Structure of PowStr >> qEigenform(D[10],2);Parent($1); ^ Runtime error in '[]': Sequence element 10 not defined Power Structure of PowStr Total time: 0.220 seconds, Total memory usage: 4.45MB '132.239' ************** MAGMA ***************** Host 132.239.145.83 (132.239.145.83) Time: Wed Dec 7 21:28:05 2005 Input: G :=DirichletGroup(1215); G; X :=Elements(G); X; Y :=X[2]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; qEigenform(D[1],2);Parent($1); qEigenform(D[2],2);Parent($1); qEigenform(D[3],2);Parent($1); qEigenform(D[4],2);Parent($1); qEigenform(D[5],2);Parent($1); qEigenform(D[6],2);Parent($1); qEigenform(D[7],2);Parent($1); qEigenform(D[8],2);Parent($1); qEigenform(D[9],2);Parent($1); qEigenform(D[10],2);Parent($1); Output: ** WARNING: Computation time exceeded 20 seconds, so computation was terminated after 20 seconds. ** Magma V2.11-10 Wed Dec 7 2005 21:27:44 on modular [Seed = 1502445656] ------------------------------------- Group of Dirichlet characters of modulus 1215 over Rational Field [ 1, $.1, $.2, $.1*$.2 ] 3 2 Errors: /bin/sh: line 1: 25208 Alarm clock nice -n 19 /usr/local/bin/magma '132.239' ************** MAGMA ***************** Host 132.239.145.83 (132.239.145.83) Time: Wed Dec 7 21:21:37 2005 Input: G :=DirichletGroup(405); G; X :=Elements(G); X; Y :=X[2]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; qEigenform(D[1],2);Parent($1); qEigenform(D[2],2);Parent($1); qEigenform(D[3],2);Parent($1); qEigenform(D[4],2);Parent($1); qEigenform(D[5],2);Parent($1); qEigenform(D[6],2);Parent($1); qEigenform(D[7],2);Parent($1); qEigenform(D[8],2);Parent($1); qEigenform(D[9],2);Parent($1); qEigenform(D[10],2);Parent($1); Output: Magma V2.11-10 Wed Dec 7 2005 21:21:34 on modular [Seed = 1318753729] ------------------------------------- Group of Dirichlet characters of modulus 405 over Rational Field [ 1, $.1, $.2, $.1*$.2 ] 3 2 [ Modular symbols space of level 405, weight 3, character $.1, and dimension 16 over Rational Field, Modular symbols space of level 405, weight 3, character $.1, and dimension 16 over Rational Field ] q + O(q^2) Power series ring in q over Univariate Quotient Polynomial Algebra in a over Rational Field with modulus a^16 + 48*a^14 + 918*a^12 + 9004*a^10 + 48453*a^8 + 140868*a^6 + 198472*a^4 + 96984*a^2 + 2916 q + O(q^2) Power series ring in q over Univariate Quotient Polynomial Algebra in a over Rational Field with modulus a^16 + 48*a^14 + 912*a^12 + 8704*a^10 + 43602*a^8 + 109032*a^6 + 117844*a^4 + 36000*a^2 + 81 >> qEigenform(D[3],2);Parent($1); ^ Runtime error in '[]': Sequence element 3 not defined Power Structure of RngSerPow >> qEigenform(D[4],2);Parent($1); ^ Runtime error in '[]': Sequence element 4 not defined Power Structure of PowStr >> qEigenform(D[5],2);Parent($1); ^ Runtime error in '[]': Sequence element 5 not defined Power Structure of PowStr >> qEigenform(D[6],2);Parent($1); ^ Runtime error in '[]': Sequence element 6 not defined Power Structure of PowStr >> qEigenform(D[7],2);Parent($1); ^ Runtime error in '[]': Sequence element 7 not defined Power Structure of PowStr >> qEigenform(D[8],2);Parent($1); ^ Runtime error in '[]': Sequence element 8 not defined Power Structure of PowStr >> qEigenform(D[9],2);Parent($1); ^ Runtime error in '[]': Sequence element 9 not defined Power Structure of PowStr >> qEigenform(D[10],2);Parent($1); ^ Runtime error in '[]': Sequence element 10 not defined Power Structure of PowStr Total time: 3.049 seconds, Total memory usage: 11.31MB '132.239' ************** MAGMA ***************** Host 132.239.145.83 (132.239.145.83) Time: Wed Dec 7 21:21:10 2005 Input: G :=DirichletGroup(405); G; X :=Elements(G); X; Y :=X[3]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; qEigenform(D[1],2);Parent($1); qEigenform(D[2],2);Parent($1); qEigenform(D[3],2);Parent($1); qEigenform(D[4],2);Parent($1); qEigenform(D[5],2);Parent($1); qEigenform(D[6],2);Parent($1); qEigenform(D[7],2);Parent($1); qEigenform(D[8],2);Parent($1); qEigenform(D[9],2);Parent($1); qEigenform(D[10],2);Parent($1); Output: Magma V2.11-10 Wed Dec 7 2005 21:21:10 on modular [Seed = 1268748540] ------------------------------------- Group of Dirichlet characters of modulus 405 over Rational Field [ 1, $.1, $.2, $.1*$.2 ] 5 2 [] >> qEigenform(D[1],2);Parent($1); ^ Runtime error in '[]': Sequence element 1 not defined Set of null sequences >> qEigenform(D[2],2);Parent($1); ^ Runtime error in '[]': Sequence element 2 not defined Power Structure of PowSeqEnum >> qEigenform(D[3],2);Parent($1); ^ Runtime error in '[]': Sequence element 3 not defined Power Structure of PowStr >> qEigenform(D[4],2);Parent($1); ^ Runtime error in '[]': Sequence element 4 not defined Power Structure of PowStr >> qEigenform(D[5],2);Parent($1); ^ Runtime error in '[]': Sequence element 5 not defined Power Structure of PowStr >> qEigenform(D[6],2);Parent($1); ^ Runtime error in '[]': Sequence element 6 not defined Power Structure of PowStr >> qEigenform(D[7],2);Parent($1); ^ Runtime error in '[]': Sequence element 7 not defined Power Structure of PowStr >> qEigenform(D[8],2);Parent($1); ^ Runtime error in '[]': Sequence element 8 not defined Power Structure of PowStr >> qEigenform(D[9],2);Parent($1); ^ Runtime error in '[]': Sequence element 9 not defined Power Structure of PowStr >> qEigenform(D[10],2);Parent($1); ^ Runtime error in '[]': Sequence element 10 not defined Power Structure of PowStr Total time: 0.210 seconds, Total memory usage: 4.47MB '132.239' ************** MAGMA ***************** Host 132.239.145.83 (132.239.145.83) Time: Wed Dec 7 21:20:27 2005 Input: G :=DirichletGroup(405); G; X :=Elements(G); X; Y :=X[4]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; qEigenform(D[1],2);Parent($1); qEigenform(D[2],2);Parent($1); qEigenform(D[3],2);Parent($1); qEigenform(D[4],2);Parent($1); qEigenform(D[5],2);Parent($1); qEigenform(D[6],2);Parent($1); qEigenform(D[7],2);Parent($1); qEigenform(D[8],2);Parent($1); qEigenform(D[9],2);Parent($1); qEigenform(D[10],2);Parent($1); Output: Magma V2.11-10 Wed Dec 7 2005 21:20:25 on modular [Seed = 1167171085] ------------------------------------- Group of Dirichlet characters of modulus 405 over Rational Field [ 1, $.1, $.2, $.1*$.2 ] 15 2 [ Modular symbols space of level 405, weight 3, character $.1*$.2, and dimension 20 over Rational Field, Modular symbols space of level 405, weight 3, character $.1*$.2, and dimension 24 over Rational Field ] q + O(q^2) Power series ring in q over Univariate Quotient Polynomial Algebra in a over Rational Field with modulus a^20 + 1742*a^18 + 1290993*a^16 + 526424484*a^14 + 128074907970*a^12 + 18925916631936*a^10 + 1674883519400862*a^8 + 88428564569459820*a^6 + 3100955720461546389*a^4 + 68055510152463772226*a^2 + 672133169091045750841 q + O(q^2) Power series ring in q over Univariate Quotient Polynomial Algebra in a over Rational Field with modulus a^24 + 2856*a^22 + 3478968*a^20 + 2357272460*a^18 + 979084307250*a^16 + 260296493582268*a^14 + 44972144261339640*a^12 + 5000266398099461004*a^10 + 343570736862382838841*a^8 + 13461967850983877713964*a^6 + 267765601556510297904192*a^4 + 1663459504983123870604536*a^2 + 21207269458849925801875396 >> qEigenform(D[3],2);Parent($1); ^ Runtime error in '[]': Sequence element 3 not defined Power Structure of RngSerPow >> qEigenform(D[4],2);Parent($1); ^ Runtime error in '[]': Sequence element 4 not defined Power Structure of PowStr >> qEigenform(D[5],2);Parent($1); ^ Runtime error in '[]': Sequence element 5 not defined Power Structure of PowStr >> qEigenform(D[6],2);Parent($1); ^ Runtime error in '[]': Sequence element 6 not defined Power Structure of PowStr >> qEigenform(D[7],2);Parent($1); ^ Runtime error in '[]': Sequence element 7 not defined Power Structure of PowStr >> qEigenform(D[8],2);Parent($1); ^ Runtime error in '[]': Sequence element 8 not defined Power Structure of PowStr >> qEigenform(D[9],2);Parent($1); ^ Runtime error in '[]': Sequence element 9 not defined Power Structure of PowStr >> qEigenform(D[10],2);Parent($1); ^ Runtime error in '[]': Sequence element 10 not defined Power Structure of PowStr Total time: 2.370 seconds, Total memory usage: 11.23MB '132.239' ************** MAGMA ***************** Host 132.239.145.83 (132.239.145.83) Time: Wed Dec 7 21:19:40 2005 Input: G :=DirichletGroup(405); G; X :=Elements(G); X; Y :=X[4]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; qEigenform(D[1],12);Parent($1); qEigenform(D[2],12);Parent($1); qEigenform(D[3],12);Parent($1); qEigenform(D[4],12);Parent($1); qEigenform(D[5],12);Parent($1); qEigenform(D[6],12);Parent($1); qEigenform(D[7],12);Parent($1); qEigenform(D[8],12);Parent($1); qEigenform(D[9],12);Parent($1); qEigenform(D[10],12);Parent($1); Output: Magma V2.11-10 Wed Dec 7 2005 21:19:37 on modular [Seed = 2141955175] ------------------------------------- Group of Dirichlet characters of modulus 405 over Rational Field [ 1, $.1, $.2, $.1*$.2 ] 15 2 [ Modular symbols space of level 405, weight 3, character $.1*$.2, and dimension 20 over Rational Field, Modular symbols space of level 405, weight 3, character $.1*$.2, and dimension 24 over Rational Field ] q + (-75172966845177826177027655884569706026546198677196542977375869681023741/2\ 211769901855602660307810712383856679743406918230929364931136698836519859741\ 843244069859801216*a^19 - 7552738132943473814982161741830663703455891635585\ 87373766935004797958588/575981745274896526121825706349962677016512218289304\ 5221174835153220103801411050114765259899*a^17 - 784518838194958988522514326745813305163450401515685373272594473386546575947\ 55/368628316975933776717968452063976113290567819705154894155189449806086643\ 290307207344976633536*a^15 - 3470483133170729781768906603215984695878622618\ 6489578710261754713708779655010971/1843141584879668883589842260319880566452\ 83909852577447077594724903043321645153603672488316768*a^13 - 727209188475665381195775305894376304647298116717075796833910337588456913990\ 85355047/737256633951867553435936904127952226581135639410309788310378899612\ 173286580614414689953267072*a^11 - 5758224767195480410489139556162556127056\ 246361617702218194111572886929630276610416783/18431415848796688835898422603\ 1988056645283909852577447077594724903043321645153603672488316768*a^9 - 121587841934540189004247901390737109962254494949443910633582498140271249466\ 470176539259/20479350943107432039887136225776450738364878872508605230843858\ 322560369071683733741387590752*a^7 - 15554821111449934404193980110163480692\ 126214952085005255697031250980512379070897856148611/23039269810995861044873\ 028253998507080660488731572180884699340612880415205644200459061039596*a^5 - 271297661984924755572096649048038905508412214458276629785797113106076873853\ 772409787447305/57598174527489652612182570634996267701651221828930452211748\ 35153220103801411050114765259899*a^3 - 487987347838286670578425364656571359026184878602182967142523113821575514659\ 68699434736182407/345589047164937915673095423809977606209907330973582713270\ 49010919320622808466300688591559394*a)*q^2 + (-1332198671522142605793090023608424979360711597386148203331041/37787935916\ 59354592294855187844728140360814217624417056076049733242742643196983152*a^1\ 8 - 1224708941679031862907868784483477992397789344194941058564217379/839731\ 909257634353843301152854384031191292048360981568016899940720609476265996256\ *a^16 - 45359631817970327997932639559715715603133811207427255262289219703/1\ 749441477620071570506877401779966731648525100752044933368541543167936408887\ 4922*a^14 - 325666701914942179008129862029362211563559553828915412704661271\ 2828503/1259597863886451530764951729281576046786938072541472352025349911080\ 914214398994384*a^12 - 6642159545681721238362211969263563281525481738512494\ 23984234362895300179/419865954628817176921650576427192015595646024180490784\ 008449970360304738132998128*a^10 - 5674425264674729308176865280361167739563\ 8330270640884749759191261347472075/9330354547307048376036679476159822568792\ 1338706775729779655548968956608473999584*a^8 - 112941055719513271540920073814330092212006970945695247540413787922444267492\ 60/787248664929032206728094830800985029241836295338420220015843694425571383\ 99937149*a^6 - 201353155687143673387451032097467612777994659259052531369475\ 4564217925002161983/1049664886572042942304126441067980038989115060451226960\ 02112492590076184533249532*a^4 - 108570403027064976849333247960661247178548\ 77736526204936164823219149559446572627/874720738810035785253438700889983365\ 8242625503760224666842707715839682044437461*a^2 - 797865655748134371693973494965638974580588654167543528147085600260159316709\ 2491769/2361745994787096620184284492402955087725508886015260660047531083276\ 71415199811447)*q^4 + (6973157488129773926528696161305835386210716301759528\ 577632858889948842919/12533362777181748408410927370175187851879305869975266\ 401276441293406945871870445049729205540224*a^19 - 211271391000069870384217673493158214036267665140495925880318890957/48950492\ 186355279388587554103340608330233987375106698544409148244426488199973719751\ 008*a^18 + 1370109783685137453617201256655764546140003373641418717568634100\ 5841179114299/6266681388590874204205463685087593925939652934987633200638220\ 646703472935935222524864602770112*a^17 - 92543339883309267031774336293946139492704150154168724172982518741051/543894\ 357626169770984306156703784537002599859723407761604546091604738757777485775\ 0112*a^16 + 226370437755144704678186406900191038757049020030506484240592166\ 03344832238224903/626668138859087420420546368508759392593965293498763320063\ 8220646703472935935222524864602770112*a^15 - 51108352131546410076003974392086035621295860787552659090226758658611945/181\ 298119208723256994768718901261512334199953241135920534848697201579585925828\ 5916704*a^14 + 101627483925058269401926569690212709123067539440102725555221\ 72114953003538664981829/313334069429543710210273184254379696296982646749381\ 6600319110323351736467967611262432301385056*a^13 - 413940370195901077823070709302803583145609514522463276941157644082954586921\ /16316830728785093129529184701113536110077995791702232848136382748142162733\ 324573250336*a^12 + 2141244014708763041177392189988502679994879145072796532\ 4122109418879338479115920781923/1253336277718174840841092737017518785187930\ 5869975266401276441293406945871870445049729205540224*a^11 - 455025687380310041278871272507918130759957245672883958759932491940234503362\ 7/3399339735163561068651913479398653356266249123271298510028413072529617236\ 10928609382*a^10 + 33294832683069843523323249202430746261955452406660623065\ 16857230436055303940466371574223/626668138859087420420546368508759392593965\ 2934987633200638220646703472935935222524864602770112*a^9 - 377841494975630937752613046909187957587507598337539735002276242598529091235\ 4609/9064905960436162849738435945063075616709997662056796026742434860078979\ 29629142958352*a^8 + 732605484908948776668800806226144287843838404833098466\ 91122495172920519586662569545780919/783335173573859275525682960635949240742\ 456616873454150079777580837934116991902815608075346264*a^7 - 299600661901575804725954660218827716100516107723510384640958514432615241161\ 5352213/4079207682196273282382296175278384027519498947925558212034095687035\ 540683331143312584*a^6 + 68144447469254693087928413080376011710217803138442\ 55868836839501060944834850856745810699683/783335173573859275525682960635949\ 240742456616873454150079777580837934116991902815608075346264*a^5 - 115941367189763439641244619029546671930037424521829300123303948012122082498\ 12853882/169966986758178053432595673969932667813312456163564925501420653626\ 480861805464304691*a^4 + 17651435453646056674409171492265326216392185256389\ 3816886764579717041136469401171542633229841/3916675867869296377628414803179\ 74620371228308436727075039888790418967058495951407804037673132*a^3 - 131048466192529284289479144792918318288052904150155564233824776330432285168\ 621815407/37770441501817345207243483104429481736291656925236650111426811916\ 995747067880956598*a^2 + 10427042922351723684558441259500154954522029725985\ 21294418925609488911106865942991223605475010/979168966967324094407103700794\ 93655092807077109181768759972197604741764623987851951009418283*a - 121780864199294714824126779885933050749620577387413855130396299806425615803\ 040066877471/15297028808236024808933610657293940103198121054720843295127858\ 82638327756249178742219)*q^5 + (-751729668451778261770276558845697060265461\ 98677196542977375869681023741/663530970556680798092343213715157003923022075\ 4692788094793410096509559579225529732209579403648*a^19 - 755273813294347381498216174183066370345589163558587373766935004797958588/17\ 279452358246895783654771190498880310495366548679135663524505459660311404233\ 150344295779697*a^17 - 7845188381949589885225143267458133051634504015156853\ 7327259447338654657594755/1105884950927801330153905356191928339871703459115\ 464682465568349418259929870921622034929900608*a^15 - 347048313317072978176890660321598469587862261864895787102617547137087796550\ 10971/552942475463900665076952678095964169935851729557732341232784174709129\ 964935460811017464950304*a^13 - 7272091884756653811957753058943763046472981\ 1671707579683391033758845691399085355047/2211769901855602660307810712383856\ 679743406918230929364931136698836519859741843244069859801216*a^11 - 575822476719548041048913955616255612705624636161770221819411157288692963027\ 6610416783/5529424754639006650769526780959641699358517295577323412327841747\ 09129964935460811017464950304*a^9 - 121587841934540189004247901390737109962\ 254494949443910633582498140271249466470176539259/61438052829322296119661408\ 677329352215094636617525815692531574967681107215051201224162772256*a^7 - 155548211114499344041939801101634806921262149520850052556970312509805123790\ 70897856148611/691178094329875831346190847619955212419814661947165426540980\ 21838641245616932601377183118788*a^5 - 271297661984924755572096649048038905508412214458276629785797113106076873853\ 772409787447305/17279452358246895783654771190498880310495366548679135663524\ 505459660311404233150344295779697*a^3 - 833576395003224586251520788466548965236092209575765680413013223014781742744\ 35000123327741801/10367671414948137470192862714299328186297219929 ** WARNING: Output too long, hence truncated. '132.239' ************** MAGMA ***************** Host 132.239.145.83 (132.239.145.83) Time: Wed Dec 7 21:17:19 2005 Input: G :=DirichletGroup(135); G; X :=Elements(G); X; Y :=X[4]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; qEigenform(D[7],12);Parent($1); Output: Magma V2.11-10 Wed Dec 7 2005 21:17:19 on modular [Seed = 2024582390] ------------------------------------- Group of Dirichlet characters of modulus 135 over Rational Field [ 1, $.1, $.2, $.1*$.2 ] 15 2 [ Modular symbols space of level 135, weight 3, character $.1*$.2, and dimension 2 over Rational Field, Modular symbols space of level 135, weight 3, character $.1*$.2, and dimension 2 over Rational Field, Modular symbols space of level 135, weight 3, character $.1*$.2, and dimension 2 over Rational Field, Modular symbols space of level 135, weight 3, character $.1*$.2, and dimension 2 over Rational Field, Modular symbols space of level 135, weight 3, character $.1*$.2, and dimension 2 over Rational Field, Modular symbols space of level 135, weight 3, character $.1*$.2, and dimension 2 over Rational Field, Modular symbols space of level 135, weight 3, character $.1*$.2, and dimension 4 over Rational Field ] q + (-1/182*a^3 - 51/182*a)*q^2 + 6*q^4 + (-1/364*a^3 - 1/12*a^2 - 51/364*a - 71/12)*q^5 + (-1/546*a^3 - 233/546*a)*q^7 + (-1/91*a^3 - 51/91*a)*q^8 + (-5/546*a^3 - 1165/546*a + 5)*q^10 + (1/6*a^2 + 71/6)*q^11 + O(q^12) Power series ring in q over Univariate Quotient Polynomial Algebra in a over Rational Field with modulus a^4 + 142*a^2 + 8281 Total time: 0.700 seconds, Total memory usage: 4.91MB '132.239' ************** MAGMA ***************** Host 132.239.145.83 (132.239.145.83) Time: Wed Dec 7 21:16:53 2005 Input: G :=DirichletGroup(135); G; X :=Elements(G); X; Y :=X[4]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; qEigenform(D[3],12);Parent($1); Output: Magma V2.11-10 Wed Dec 7 2005 21:16:52 on modular [Seed = 1906681284] ------------------------------------- Group of Dirichlet characters of modulus 135 over Rational Field [ 1, $.1, $.2, $.1*$.2 ] 15 2 [ Modular symbols space of level 135, weight 3, character $.1*$.2, and dimension 2 over Rational Field, Modular symbols space of level 135, weight 3, character $.1*$.2, and dimension 2 over Rational Field, Modular symbols space of level 135, weight 3, character $.1*$.2, and dimension 2 over Rational Field, Modular symbols space of level 135, weight 3, character $.1*$.2, and dimension 2 over Rational Field, Modular symbols space of level 135, weight 3, character $.1*$.2, and dimension 2 over Rational Field, Modular symbols space of level 135, weight 3, character $.1*$.2, and dimension 2 over Rational Field, Modular symbols space of level 135, weight 3, character $.1*$.2, and dimension 4 over Rational Field ] q + q^2 - 3*q^4 + (-1/4*a + 3/4)*q^5 + (-1/2*a + 1/2)*q^7 - 7*q^8 + (-1/4*a + 3/4)*q^10 + (1/2*a - 1/2)*q^11 + O(q^12) Power series ring in q over Univariate Quotient Polynomial Algebra in a over Rational Field with modulus a^2 - 2*a + 397 Total time: 0.690 seconds, Total memory usage: 4.91MB '132.239' ************** MAGMA ***************** Host 132.239.145.83 (132.239.145.83) Time: Wed Dec 7 21:16:45 2005 Input: G :=DirichletGroup(135); G; X :=Elements(G); X; Y :=X[3]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; qEigenform(D[3],12);Parent($1); Output: Magma V2.11-10 Wed Dec 7 2005 21:16:45 on modular [Seed = 1689301192] ------------------------------------- Group of Dirichlet characters of modulus 135 over Rational Field [ 1, $.1, $.2, $.1*$.2 ] 5 2 [] >> qEigenform(D[3],12);Parent($1); ^ Runtime error in '[]': Sequence element 3 not defined Set of null sequences Total time: 0.220 seconds, Total memory usage: 4.45MB '132.239' ************** MAGMA ***************** Host 132.239.145.83 (132.239.145.83) Time: Wed Dec 7 21:16:12 2005 Input: G :=DirichletGroup(135); G; X :=Elements(G); X; Y :=X[2]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; qEigenform(D[3],12);Parent($1); Output: Magma V2.11-10 Wed Dec 7 2005 21:16:11 on modular [Seed = 1639298347] ------------------------------------- Group of Dirichlet characters of modulus 135 over Rational Field [ 1, $.1, $.2, $.1*$.2 ] 3 2 [ Modular symbols space of level 135, weight 3, character $.1, and dimension 2 over Rational Field, Modular symbols space of level 135, weight 3, character $.1, and dimension 4 over Rational Field, Modular symbols space of level 135, weight 3, character $.1, and dimension 4 over Rational Field ] q + a*q^2 + (a^2 + 4)*q^4 + (1/6*a^3 + 11/6*a)*q^5 + (a^2 + 5)*q^7 + (a^3 + 8*a)*q^8 + (-a^2 - 6)*q^10 + a*q^11 + O(q^12) Power series ring in q over Univariate Quotient Polynomial Algebra in a over Rational Field with modulus a^4 + 17*a^2 + 36 Total time: 0.840 seconds, Total memory usage: 4.91MB '132.239' ************** MAGMA ***************** Host 132.239.145.83 (132.239.145.83) Time: Wed Dec 7 21:15:26 2005 Input: G :=DirichletGroup(135); G; X :=Elements(G); X; Y :=X[2]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; qEigenform(D[2],12);Parent($1); Output: Magma V2.11-10 Wed Dec 7 2005 21:15:25 on modular [Seed = 449760774] ------------------------------------- Group of Dirichlet characters of modulus 135 over Rational Field [ 1, $.1, $.2, $.1*$.2 ] 3 2 [ Modular symbols space of level 135, weight 3, character $.1, and dimension 2 over Rational Field, Modular symbols space of level 135, weight 3, character $.1, and dimension 4 over Rational Field, Modular symbols space of level 135, weight 3, character $.1, and dimension 4 over Rational Field ] q + a*q^2 + (a^2 + 4)*q^4 + (1/6*a^3 + 11/6*a)*q^5 + (a^2 + 5)*q^7 + (a^3 + 8*a)*q^8 + (-a^2 - 6)*q^10 + a*q^11 + O(q^12) Power series ring in q over Univariate Quotient Polynomial Algebra in a over Rational Field with modulus a^4 + 17*a^2 + 36 Total time: 0.820 seconds, Total memory usage: 4.91MB '132.239' ************** MAGMA ***************** Host 132.239.145.83 (132.239.145.83) Time: Wed Dec 7 21:14:53 2005 Input: G :=DirichletGroup(135); G; X :=Elements(G); X; Y :=X[2]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; qEigenform(D[1],12);Parent($1); Output: Magma V2.11-10 Wed Dec 7 2005 21:14:52 on modular [Seed = 398705450] ------------------------------------- Group of Dirichlet characters of modulus 135 over Rational Field [ 1, $.1, $.2, $.1*$.2 ] 3 2 [ Modular symbols space of level 135, weight 3, character $.1, and dimension 2 over Rational Field, Modular symbols space of level 135, weight 3, character $.1, and dimension 4 over Rational Field, Modular symbols space of level 135, weight 3, character $.1, and dimension 4 over Rational Field ] q + a*q^2 - q^4 - a*q^5 + 12*q^7 + 3*a*q^8 + 5*q^10 + 7*a*q^11 + O(q^12) Power series ring in q over Univariate Quotient Polynomial Algebra in a over Rational Field with modulus a^2 + 5 Total time: 0.830 seconds, Total memory usage: 4.91MB '200.177' ************** MAGMA ***************** Host 200.177.23.188 (200.177.23.188) Time: Wed Dec 7 21:06:56 2005 Input: K := FunctionField(Rationals(), 6); //(u*x + v - y)*(zeta*u*x + v - y)*(zeta^2*u*x + v - y); g0 := u^3*x^3 + v^3 + 3*v*y^2 - 3*v^2*y - y^3; //(u*x + v + y)*(zeta*u*x + v + y)*(zeta^2*u*x + v + y); g1 := u^3*x^3 + v^3 + 3*v*y^2 + 3*v^2*y + y^3; //(u*x + v - y)*(zeta*u*x + v - y)*(zeta^2*u*x + v - y)*(u*x + v + y)*(zeta*u*x + v + y)*(zeta^2*u*x + v + y); g2 := (u^3*x^3 + v^3 + 3*v*y^2)^2 - (3*v^2*y + y^3)^2; g0*g1 eq g2; g3 := (u^3*x^3 + v^3 + 3*v*y^2)^2 - (3*v^2 + y^2)^2*y^2; g2 eq g3; (u*x + v - y)*(zeta*u*x + v + y)*(zeta^2*u*x + v - y); Output: Magma V2.11-10 Wed Dec 7 2005 21:06:55 on modular [Seed = 3742042562] ------------------------------------- true true zeta^3*u^3*x^3 + zeta^3*u^2*v*x^2 - zeta^3*u^2*x^2*y + zeta^2*u^2*v*x^2 + zeta^2*u^2*x^2*y + zeta^2*u*v^2*x - zeta^2*u*x*y^2 + zeta*u^2*v*x^2 - zeta*u^2*x^2*y + zeta*u*v^2*x - 2*zeta*u*v*x*y + zeta*u*x*y^2 + u*v^2*x - u*x*y^2 + v^3 - v^2*y - v*y^2 + y^3 Total time: 0.190 seconds, Total memory usage: 3.24MB '200.177' ************** MAGMA ***************** Host 200.177.23.188 (200.177.23.188) Time: Wed Dec 7 21:03:54 2005 Input: K := FunctionField(Rationals(), 6); //(u*x + v - y)*(zeta*u*x + v - y)*(zeta^2*u*x + v - y); g0 := u^3*x^3 + v^3 + 3*v*y^2 - 3*v^2*y - y^3; //(u*x + v + y)*(zeta*u*x + v + y)*(zeta^2*u*x + v + y); g1 := u^3*x^3 + v^3 + 3*v*y^2 + 3*v^2*y + y^3; //(u*x + v - y)*(zeta*u*x + v - y)*(zeta^2*u*x + v - y)*(u*x + v + y)*(zeta*u*x + v + y)*(zeta^2*u*x + v + y); g2 := (u^3*x^3 + v^3 + 3*v*y^2)^2 - (3*v^2*y + y^3)^2; g0*g1 eq g2; g3 := (u^3*x^3 + v^3 + 3*v*y^2)^2 - (3*v^2 + y^2)^2*y^2; g2 eq g3; (u*x + v - y)*(zeta*u*x + v + y); Output: Magma V2.11-10 Wed Dec 7 2005 21:03:53 on modular [Seed = 3658354033] ------------------------------------- true true zeta*u^2*x^2 + zeta*u*v*x - zeta*u*x*y + u*v*x + u*x*y + v^2 - y^2 Total time: 0.190 seconds, Total memory usage: 3.24MB '200.177' ************** MAGMA ***************** Host 200.177.23.188 (200.177.23.188) Time: Wed Dec 7 20:57:38 2005 Input: K := FunctionField(Rationals(), 6); //(u*x + v - y)*(zeta*u*x + v - y)*(zeta^2*u*x + v - y); g0 := u^3*x^3 + v^3 + 3*v*y^2 - 3*v^2*y - y^3; //(u*x + v + y)*(zeta*u*x + v + y)*(zeta^2*u*x + v + y); g1 := u^3*x^3 + v^3 + 3*v*y^2 + 3*v^2*y + y^3; //(u*x + v - y)*(zeta*u*x + v - y)*(zeta^2*u*x + v - y)*(u*x + v + y)*(zeta*u*x + v + y)*(zeta^2*u*x + v + y); g2 := (u^3*x^3 + v^3 + 3*v*y^2)^2 - (3*v^2*y + y^3)^2; g0*g1 eq g2; g3 := (u^3*x^3 + v^3 + 3*v*y^2)^2 - (3*v^2 + y^2)^2*y^2; g2 eq g3; Output: Magma V2.11-10 Wed Dec 7 2005 20:57:37 on modular [Seed = 3540452320] ------------------------------------- true true Total time: 0.190 seconds, Total memory usage: 3.24MB '200.177' ************** MAGMA ***************** Host 200.177.23.188 (200.177.23.188) Time: Wed Dec 7 20:50:22 2005 Input: K := FunctionField(Rationals(), 6); //(a*x + b - y)*(zeta*a*x + b - y)*(zeta^2*a*x + b - y); g0 := a^3*x^3 + b^3 + 3*b*y^2 - 3*b^2*y - y^3; //(a*x + b + y)*(zeta*a*x + b + y)*(zeta^2*a*x + b + y); g1 := a^3*x^3 + b^3 + 3*b*y^2 + 3*b^2*y + y^3; //(a*x + b - y)*(zeta*a*x + b - y)*(zeta^2*a*x + b - y)*(a*x + b + y)*(zeta*a*x + b + y)*(zeta^2*a*x + b + y); g2 := (a^3*x^3 + b^3 + 3*b*y^2)^2 - (3*b^2*y + y^3)^2; g0*g1 eq g2; g3 := (a^3*x^3 + b^3 + 3*b*y^2)^2 - (3*b^2 + y^2)^2*y^2; g2 eq g3; Output: Magma V2.11-10 Wed Dec 7 2005 20:50:22 on modular [Seed = 4163644036] ------------------------------------- true true Total time: 0.190 seconds, Total memory usage: 3.24MB '200.177' ************** MAGMA ***************** Host 200.177.23.188 (200.177.23.188) Time: Wed Dec 7 20:48:33 2005 Input: K := FunctionField(Rationals(), 6); //(a*x + b - y)*(zeta*a*x + b - y)*(zeta^2*a*x + b - y); g0 := a^3*x^3 + b^3 + 3*b*y^2 - 3*b^2*y - y^3; //(a*x + b + y)*(zeta*a*x + b + y)*(zeta^2*a*x + b + y); g1 := a^3*x^3 + b^3 + 3*b*y^2 + 3*b^2*y + y^3; //(a*x + b - y)*(zeta*a*x + b - y)*(zeta^2*a*x + b - y)*(a*x + b + y)*(zeta*a*x + b + y)*(zeta^2*a*x + b + y); g2 := (a^3*x^3 + b^3 + 3*b*y^2)^2 - (3*b^2*y + y^3)^2; g0*g1 eq g2; g2; Output: Magma V2.11-10 Wed Dec 7 2005 20:48:33 on modular [Seed = 4129432869] ------------------------------------- true a^6*x^6 + 2*a^3*b^3*x^3 + 6*a^3*b*x^3*y^2 + b^6 - 3*b^4*y^2 + 3*b^2*y^4 - y^6 Total time: 0.190 seconds, Total memory usage: 3.24MB '200.177' ************** MAGMA ***************** Host 200.177.23.188 (200.177.23.188) Time: Wed Dec 7 20:45:27 2005 Input: K := FunctionField(Rationals(), 6); a^3*x^3 + a^2*b*x^2 - a^2*x^2*y + a*b^2*x - 2*a*b*x*y + a*x*y^2 + b^3 - 3*b^2*y + 3*b*y^2 - y^3 + zeta^2*(a^2*b*x^2 - a^2*x^2*y + a*b^2*x - 2*a*b*x*y + a*x*y^2) + zeta *(a^2*b*x^2 - a^2*x^2*y + a*b^2*x - 2*a*b*x*y + a*x*y^2) eq a^3*x^3 + b^3 - 3*b^2*y + 3*b*y^2 - y^3 + (zeta^2 + zeta + 1)*(a^2*b*x^2 - a^2*x^2*y + a*b^2*x - 2*a*b*x*y + a*x*y^2) ; //(a*x + b - y)*(zeta*a*x + b - y)*(zeta^2*a*x + b - y)*(a*x + b + y)*(zeta*a*x + b + y)*(zeta^2*a*x + b + y); b^3 + 3*b^2*y + 3*b*y^2 + y^3 + a^3*x^3 + (a^2*b*x^2 + a^2*x^2*y + a*b^2*x + 2*a*b*x*y + a*x*y^2) + zeta^2*(a^2*b*x^2 + a^2*x^2*y + a*b^2*x + 2*a*b*x*y + a*x*y^2) + zeta* (a^2*b*x^2 + a^2*x^2*y + a*b^2*x + 2*a*b*x*y + a*x*y^2) eq b^3 + 3*b^2*y + 3*b*y^2 + y^3 + a^3*x^3 + (zeta^2 + zeta + 1)*(a^2*b*x^2 + a^2*x^2*y + a*b^2*x + 2*a*b*x*y + a*x*y^2) ; Output: Magma V2.11-10 Wed Dec 7 2005 20:45:27 on modular [Seed = 3963109369] ------------------------------------- true true Total time: 0.190 seconds, Total memory usage: 3.34MB '200.177' ************** MAGMA ***************** Host 200.177.23.188 (200.177.23.188) Time: Wed Dec 7 20:44:04 2005 Input: K := FunctionField(Rationals(), 6); a^3*x^3 + a^2*b*x^2 - a^2*x^2*y + a*b^2*x - 2*a*b*x*y + a*x*y^2 + b^3 - 3*b^2*y + 3*b*y^2 - y^3 + zeta^2*(a^2*b*x^2 - a^2*x^2*y + a*b^2*x - 2*a*b*x*y + a*x*y^2) + zeta *(a^2*b*x^2 - a^2*x^2*y + a*b^2*x - 2*a*b*x*y + a*x*y^2) eq a^3*x^3 + b^3 - 3*b^2*y + 3*b*y^2 - y^3 + (zeta^2 + zeta + 1)*(a^2*b*x^2 - a^2*x^2*y + a*b^2*x - 2*a*b*x*y + a*x*y^2) ; //(a*x + b - y)*(zeta*a*x + b - y)*(zeta^2*a*x + b - y)*(a*x + b + y)*(zeta*a*x + b + y)*(zeta^2*a*x + b + y); zeta^3*a^3*x^3 + zeta^3*a^2*b*x^2 + zeta^3*a^2*x^2*y + zeta^2*a^2*b*x^2 + zeta^2*a^2*x^2*y + zeta^2*a*b^2*x + 2*zeta^2*a*b*x*y + zeta^2*a*x*y^2 + zeta*a^2*b*x^2 + zeta*a^2*x^2*y + zeta*a*b^2*x + 2*zeta*a*b*x*y + zeta*a*x*y^2 + a*b^2*x + 2*a*b*x*y + a*x*y^2 + b^3 + 3*b^2*y + 3*b*y^2 + y^3 eq b^3 + 3*b^2*y + 3*b*y^2 + y^3 + zeta^3*(a^3*x^3 + a^2*b*x^2 + a^2*x^2*y) + a*b^2*x + 2*a*b*x*y + a*x*y^2 + zeta^2*(a^2*b*x^2 + a^2*x^2*y + a*b^2*x + 2*a*b*x*y + a*x*y^2) + zeta* (a^2*b*x^2 + a^2*x^2*y + a*b^2*x + 2*a*b*x*y + a*x*y^2) ; Output: Magma V2.11-10 Wed Dec 7 2005 20:44:04 on modular [Seed = 3946784557] ------------------------------------- true true Total time: 0.180 seconds, Total memory usage: 3.34MB '200.177' ************** MAGMA ***************** Host 200.177.23.188 (200.177.23.188) Time: Wed Dec 7 20:41:23 2005 Input: K := FunctionField(Rationals(), 6); a^3*x^3 + a^2*b*x^2 - a^2*x^2*y + a*b^2*x - 2*a*b*x*y + a*x*y^2 + b^3 - 3*b^2*y + 3*b*y^2 - y^3 + zeta^2*(a^2*b*x^2 - a^2*x^2*y + a*b^2*x - 2*a*b*x*y + a*x*y^2) + zeta *(a^2*b*x^2 - a^2*x^2*y + a*b^2*x - 2*a*b*x*y + a*x*y^2) eq a^3*x^3 + b^3 - 3*b^2*y + 3*b*y^2 - y^3 + (zeta^2 + zeta + 1)*(a^2*b*x^2 - a^2*x^2*y + a*b^2*x - 2*a*b*x*y + a*x*y^2) ; //(a*x + b - y)*(zeta*a*x + b - y)*(zeta^2*a*x + b - y)* (a*x + b + y)*(zeta*a*x + b + y)*(zeta^2*a*x + b + y); Output: Magma V2.11-10 Wed Dec 7 2005 20:41:23 on modular [Seed = 3778358656] ------------------------------------- true zeta^3*a^3*x^3 + zeta^3*a^2*b*x^2 + zeta^3*a^2*x^2*y + zeta^2*a^2*b*x^2 + zeta^2*a^2*x^2*y + zeta^2*a*b^2*x + 2*zeta^2*a*b*x*y + zeta^2*a*x*y^2 + zeta*a^2*b*x^2 + zeta*a^2*x^2*y + zeta*a*b^2*x + 2*zeta*a*b*x*y + zeta*a*x*y^2 + a*b^2*x + 2*a*b*x*y + a*x*y^2 + b^3 + 3*b^2*y + 3*b*y^2 + y^3 Total time: 0.190 seconds, Total memory usage: 3.34MB '200.177' ************** MAGMA ***************** Host 200.177.23.188 (200.177.23.188) Time: Wed Dec 7 20:40:57 2005 Input: K := FunctionField(Rationals(), 6); a^3*x^3 + a^2*b*x^2 - a^2*x^2*y + a*b^2*x - 2*a*b*x*y + a*x*y^2 + b^3 - 3*b^2*y + 3*b*y^2 - y^3 + zeta^2*(a^2*b*x^2 - a^2*x^2*y + a*b^2*x - 2*a*b*x*y + a*x*y^2) + zeta *(a^2*b*x^2 - a^2*x^2*y + a*b^2*x - 2*a*b*x*y + a*x*y^2) eq a^3*x^3 + b^3 - 3*b^2*y + 3*b*y^2 - y^3 + (zeta^2 + zeta + 1)*(a^2*b*x^2 - a^2*x^2*y + a*b^2*x - 2*a*b*x*y + a*x*y^2) ; (a*x + b - y)*(zeta*a*x + b - y)*(zeta^2*a*x + b - y)* (a*x + b + y)*(zeta*a*x + b + y)*(zeta^2*a*x + b + y); Output: Magma V2.11-10 Wed Dec 7 2005 20:40:56 on modular [Seed = 2621984918] ------------------------------------- true zeta^6*a^6*x^6 + 2*zeta^6*a^5*b*x^5 + zeta^6*a^4*b^2*x^4 - zeta^6*a^4*x^4*y^2 + 2*zeta^5*a^5*b*x^5 + 4*zeta^5*a^4*b^2*x^4 + 2*zeta^5*a^3*b^3*x^3 - 2*zeta^5*a^3*b*x^3*y^2 + 2*zeta^4*a^5*b*x^5 + 5*zeta^4*a^4*b^2*x^4 - zeta^4*a^4*x^4*y^2 + 4*zeta^4*a^3*b^3*x^3 - 4*zeta^4*a^3*b*x^3*y^2 + zeta^4*a^2*b^4*x^2 - 2*zeta^4*a^2*b^2*x^2*y^2 + zeta^4*a^2*x^2*y^4 + 4*zeta^3*a^4*b^2*x^4 + 8*zeta^3*a^3*b^3*x^3 + 4*zeta^3*a^2*b^4*x^2 - 4*zeta^3*a^2*b^2*x^2*y^2 + zeta^2*a^4*b^2*x^4 - zeta^2*a^4*x^4*y^2 + 4*zeta^2*a^3*b^3*x^3 - 4*zeta^2*a^3*b*x^3*y^2 + 5*zeta^2*a^2*b^4*x^2 - 6*zeta^2*a^2*b^2*x^2*y^2 + zeta^2*a^2*x^2*y^4 + 2*zeta^2*a*b^5*x - 4*zeta^2*a*b^3*x*y^2 + 2*zeta^2*a*b*x*y^4 + 2*zeta*a^3*b^3*x^3 - 2*zeta*a^3*b*x^3*y^2 + 4*zeta*a^2*b^4*x^2 - 4*zeta*a^2*b^2*x^2*y^2 + 2*zeta*a*b^5*x - 4*zeta*a*b^3*x*y^2 + 2*zeta*a*b*x*y^4 + a^2*b^4*x^2 - 2*a^2*b^2*x^2*y^2 + a^2*x^2*y^4 + 2*a*b^5*x - 4*a*b^3*x*y^2 + 2*a*b*x*y^4 + b^6 - 3*b^4*y^2 + 3*b^2*y^4 - y^6 Total time: 0.190 seconds, Total memory usage: 3.34MB '200.177' ************** MAGMA ***************** Host 200.177.23.188 (200.177.23.188) Time: Wed Dec 7 20:39:38 2005 Input: K := FunctionField(Rationals(), 6); //(a*x + b - y)*(zeta*a*x + b - y)*(zeta^2*a*x + b - y); a^3*x^3 + a^2*b*x^2 - a^2*x^2*y + a*b^2*x - 2*a*b*x*y + a*x*y^2 + b^3 - 3*b^2*y + 3*b*y^2 - y^3 + zeta^2*(a^2*b*x^2 - a^2*x^2*y + a*b^2*x - 2*a*b*x*y + a*x*y^2) + zeta *(a^2*b*x^2 - a^2*x^2*y + a*b^2*x - 2*a*b*x*y + a*x*y^2) eq a^3*x^3 + b^3 - 3*b^2*y + 3*b*y^2 - y^3 + (zeta^2 + zeta + 1)*(a^2*b*x^2 - a^2*x^2*y + a*b^2*x - 2*a*b*x*y + a*x*y^2) ; Output: Magma V2.11-10 Wed Dec 7 2005 20:39:38 on modular [Seed = 2605667939] ------------------------------------- true Total time: 0.190 seconds, Total memory usage: 3.34MB '200.177' ************** MAGMA ***************** Host 200.177.23.188 (200.177.23.188) Time: Wed Dec 7 20:38:33 2005 Input: K := FunctionField(Rationals(), 6); //(a*x + b - y)*(zeta*a*x + b - y)*(zeta^2*a*x + b - y); a^3*x^3 + a^2*b*x^2 - a^2*x^2*y + a*b^2*x - 2*a*b*x*y + a*x*y^2 + b^3 - 3*b^2*y + 3*b*y^2 - y^3 + zeta^2*(a^2*b*x^2 - a^2*x^2*y + a*b^2*x - 2*a*b*x*y + a*x*y^2) + zeta *(a^2*b*x^2 - a^2*x^2*y + a*b^2*x - 2*a*b*x*y + a*x*y^2) eq a^3*x^3 + a^2*b*x^2 - a^2*x^2*y + a*b^2*x - 2*a*b*x*y + a*x*y^2 + b^3 - 3*b^2*y + 3*b*y^2 - y^3 + (zeta^2 + zeta)*(a^2*b*x^2 - a^2*x^2*y + a*b^2*x - 2*a*b*x*y + a*x*y^2) ; Output: Magma V2.11-10 Wed Dec 7 2005 20:38:33 on modular [Seed = 2321434417] ------------------------------------- true Total time: 0.180 seconds, Total memory usage: 3.34MB '200.177' ************** MAGMA ***************** Host 200.177.23.188 (200.177.23.188) Time: Wed Dec 7 20:38:18 2005 Input: K := FunctionField(Rationals(), 5); //(a*x + b - y)*(zeta*a*x + b - y)*(zeta^2*a*x + b - y); a^3*(y^2 - mu) + a^2*b*x^2 - a^2*x^2*y + a*b^2*x - 2*a*b*x*y + a*x*y^2 + b^3 - 3*b^2*y + 3*b*y^2 - y^3 + zeta^2*(a^2*b*x^2 - a^2*x^2*y + a*b^2*x - 2*a*b*x*y + a*x*y^2) + zeta *(a^2*b*x^2 - a^2*x^2*y + a*b^2*x - 2*a*b*x*y + a*x*y^2) eq a^3*(y^2 - mu) + a^2*b*x^2 - a^2*x^2*y + a*b^2*x - 2*a*b*x*y + a*x*y^2 + b^3 - 3*b^2*y + 3*b*y^2 - y^3 + (zeta^2 + zeta)*(a^2*b*x^2 - a^2*x^2*y + a*b^2*x - 2*a*b*x*y + a*x*y^2) ; Output: Magma V2.11-10 Wed Dec 7 2005 20:38:18 on modular [Seed = 2169850921] ------------------------------------- >> K := FunctionField(Rationals(), 5); ^ Runtime error in 'AssignNames': Argument 2 should have length at most 5 >> a^3*(y^2 - mu) + a^2*b*x^2 - a^2*x^2*y + ^ User error: Identifier 'a' has not been declared or assigned Total time: 0.180 seconds, Total memory usage: 3.34MB '200.177' ************** MAGMA ***************** Host 200.177.23.188 (200.177.23.188) Time: Wed Dec 7 20:37:49 2005 Input: K := FunctionField(Rationals(), 5); //(a*x + b - y)*(zeta*a*x + b - y)*(zeta^2*a*x + b - y); a^3*(y^2 - mu) + a^2*b*x^2 - a^2*x^2*y + a*b^2*x - 2*a*b*x*y + a*x*y^2 + b^3 - 3*b^2*y + 3*b*y^2 - y^3 + zeta^2*(a^2*b*x^2 - a^2*x^2*y + a*b^2*x - 2*a*b*x*y + a*x*y^2) + zeta *(a^2*b*x^2 - a^2*x^2*y + a*b^2*x - 2*a*b*x*y + a*x*y^2); Output: Magma V2.11-10 Wed Dec 7 2005 20:37:49 on modular [Seed = 3194647537] ------------------------------------- >> K := FunctionField(Rationals(), 5); ^ Runtime error in 'AssignNames': Argument 2 should have length at most 5 >> a^3*(y^2 - mu) + a^2*b*x^2 - a^2*x^2*y + ^ User error: Identifier 'a' has not been declared or assigned Total time: 0.190 seconds, Total memory usage: 3.24MB '200.177' ************** MAGMA ***************** Host 200.177.23.188 (200.177.23.188) Time: Wed Dec 7 20:33:33 2005 Input: K := FunctionField(Rationals(), 5); (a*x + b - y)*(zeta*a*x + b - y)*(zeta^2*a*x + b - y); Output: Magma V2.11-10 Wed Dec 7 2005 20:33:33 on modular [Seed = 3026221819] ------------------------------------- zeta^3*a^3*x^3 + zeta^3*a^2*b*x^2 - zeta^3*a^2*x^2*y + zeta^2*a^2*b*x^2 - zeta^2*a^2*x^2*y + zeta^2*a*b^2*x - 2*zeta^2*a*b*x*y + zeta^2*a*x*y^2 + zeta*a^2*b*x^2 - zeta*a^2*x^2*y + zeta*a*b^2*x - 2*zeta*a*b*x*y + zeta*a*x*y^2 + a*b^2*x - 2*a*b*x*y + a*x*y^2 + b^3 - 3*b^2*y + 3*b*y^2 - y^3 Total time: 0.190 seconds, Total memory usage: 3.24MB '200.177' ************** MAGMA ***************** Host 200.177.23.188 (200.177.23.188) Time: Wed Dec 7 20:32:32 2005 Input: K := FunctionField(Rationals(), 5); (a*x + b - y)*(zeta*a*x + b - y)*(zeta^2*a*x + b - y); Output: Magma V2.11-10 Wed Dec 7 2005 20:32:32 on modular [Seed = 3009904033] ------------------------------------- a^3*x^3*zeta^3 + a^2*b*x^2*zeta^3 + a^2*b*x^2*zeta^2 + a^2*b*x^2*zeta - a^2*x^2*y*zeta^3 - a^2*x^2*y*zeta^2 - a^2*x^2*y*zeta + a*b^2*x*zeta^2 + a*b^2*x*zeta + a*b^2*x - 2*a*b*x*y*zeta^2 - 2*a*b*x*y*zeta - 2*a*b*x*y + a*x*y^2*zeta^2 + a*x*y^2*zeta + a*x*y^2 + b^3 - 3*b^2*y + 3*b*y^2 - y^3 Total time: 0.180 seconds, Total memory usage: 3.24MB '200.177' ************** MAGMA ***************** Host 200.177.23.188 (200.177.23.188) Time: Wed Dec 7 20:31:25 2005 Input: K := FunctionField(Rationals(), 4); (a*x + b - y)*(a*x + b + y); Output: Magma V2.11-10 Wed Dec 7 2005 20:31:24 on modular [Seed = 2843567719] ------------------------------------- a^2*x^2 + 2*a*b*x + b^2 - y^2 Total time: 0.180 seconds, Total memory usage: 3.24MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Wed Dec 7 18:28:16 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^9*3^6; C:=HyperellipticCurve(f); J:=Jacobian(C); //r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); //V:=RationalPoints(J:Bound:=1500); //B:=ReducedBasis(V); //B; //P:=J!B[1];//Q:=J![x+2,0]; //P1:=J![x+1,0];P2:=J![x,1]; //hP:=Height(P);hP; //HC:=HeightConstant(J:Effort:=20);HC; //Exp(hP/4+HC); //RationalPoints(J:Bound:=1000); //Chabauty(P,11); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Wed Dec 7 2005 18:28:15 on modular [Seed = 2708837005] ------------------------------------- Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] Total time: 0.230 seconds, Total memory usage: 4.51MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Wed Dec 7 18:28:10 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^9*3^5; C:=HyperellipticCurve(f); J:=Jacobian(C); //r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); //V:=RationalPoints(J:Bound:=1500); //B:=ReducedBasis(V); //B; //P:=J!B[1];//Q:=J![x+2,0]; //P1:=J![x+1,0];P2:=J![x,1]; //hP:=Height(P);hP; //HC:=HeightConstant(J:Effort:=20);HC; //Exp(hP/4+HC); //RationalPoints(J:Bound:=1000); //Chabauty(P,11); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Wed Dec 7 2005 18:28:10 on modular [Seed = 1586130295] ------------------------------------- Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] Total time: 0.230 seconds, Total memory usage: 4.51MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Wed Dec 7 18:28:02 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^9*3^3; C:=HyperellipticCurve(f); J:=Jacobian(C); //r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); //V:=RationalPoints(J:Bound:=1500); //B:=ReducedBasis(V); //B; //P:=J!B[1];//Q:=J![x+2,0]; //P1:=J![x+1,0];P2:=J![x,1]; //hP:=Height(P);hP; //HC:=HeightConstant(J:Effort:=20);HC; //Exp(hP/4+HC); //RationalPoints(J:Bound:=1000); //Chabauty(P,11); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Wed Dec 7 2005 18:28:02 on modular [Seed = 1536125050] ------------------------------------- Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] Total time: 0.230 seconds, Total memory usage: 4.51MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Wed Dec 7 18:27:53 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^8*3^2; C:=HyperellipticCurve(f); J:=Jacobian(C); //r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); //V:=RationalPoints(J:Bound:=1500); //B:=ReducedBasis(V); //B; //P:=J!B[1];//Q:=J![x+2,0]; //P1:=J![x+1,0];P2:=J![x,1]; //hP:=Height(P);hP; //HC:=HeightConstant(J:Effort:=20);HC; //Exp(hP/4+HC); //RationalPoints(J:Bound:=1000); //Chabauty(P,11); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Wed Dec 7 2005 18:27:53 on modular [Seed = 1350859914] ------------------------------------- Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] Total time: 0.220 seconds, Total memory usage: 4.13MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Wed Dec 7 18:27:45 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^7*3^9; C:=HyperellipticCurve(f); J:=Jacobian(C); //r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); //V:=RationalPoints(J:Bound:=1500); //B:=ReducedBasis(V); //B; //P:=J!B[1];//Q:=J![x+2,0]; //P1:=J![x+1,0];P2:=J![x,1]; //hP:=Height(P);hP; //HC:=HeightConstant(J:Effort:=20);HC; //Exp(hP/4+HC); //RationalPoints(J:Bound:=1000); //Chabauty(P,11); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Wed Dec 7 2005 18:27:44 on modular [Seed = 1301897630] ------------------------------------- Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] Total time: 0.230 seconds, Total memory usage: 4.51MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Wed Dec 7 18:27:40 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^7*3^7; C:=HyperellipticCurve(f); J:=Jacobian(C); //r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); //V:=RationalPoints(J:Bound:=1500); //B:=ReducedBasis(V); //B; //P:=J!B[1];//Q:=J![x+2,0]; //P1:=J![x+1,0];P2:=J![x,1]; //hP:=Height(P);hP; //HC:=HeightConstant(J:Effort:=20);HC; //Exp(hP/4+HC); //RationalPoints(J:Bound:=1000); //Chabauty(P,11); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Wed Dec 7 2005 18:27:39 on modular [Seed = 1251894914] ------------------------------------- Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] Total time: 0.220 seconds, Total memory usage: 4.51MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Wed Dec 7 18:27:32 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^6*3^9; C:=HyperellipticCurve(f); J:=Jacobian(C); //r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); //V:=RationalPoints(J:Bound:=1500); //B:=ReducedBasis(V); //B; //P:=J!B[1];//Q:=J![x+2,0]; //P1:=J![x+1,0];P2:=J![x,1]; //hP:=Height(P);hP; //HC:=HeightConstant(J:Effort:=20);HC; //Exp(hP/4+HC); //RationalPoints(J:Bound:=1000); //Chabauty(P,11); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Wed Dec 7 2005 18:27:32 on modular [Seed = 1200843648] ------------------------------------- Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] Total time: 0.220 seconds, Total memory usage: 4.13MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Wed Dec 7 18:27:27 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^6*3^8; C:=HyperellipticCurve(f); J:=Jacobian(C); //r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); //V:=RationalPoints(J:Bound:=1500); //B:=ReducedBasis(V); //B; //P:=J!B[1];//Q:=J![x+2,0]; //P1:=J![x+1,0];P2:=J![x,1]; //hP:=Height(P);hP; //HC:=HeightConstant(J:Effort:=20);HC; //Exp(hP/4+HC); //RationalPoints(J:Bound:=1000); //Chabauty(P,11); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Wed Dec 7 2005 18:27:26 on modular [Seed = 2091405441] ------------------------------------- Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] Total time: 0.220 seconds, Total memory usage: 4.51MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Wed Dec 7 18:27:23 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^6*3^6; C:=HyperellipticCurve(f); J:=Jacobian(C); //r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); //V:=RationalPoints(J:Bound:=1500); //B:=ReducedBasis(V); //B; //P:=J!B[1];//Q:=J![x+2,0]; //P1:=J![x+1,0];P2:=J![x,1]; //hP:=Height(P);hP; //HC:=HeightConstant(J:Effort:=20);HC; //Exp(hP/4+HC); //RationalPoints(J:Bound:=1000); //Chabauty(P,11); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Wed Dec 7 2005 18:27:22 on modular [Seed = 2041402752] ------------------------------------- Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] Total time: 0.220 seconds, Total memory usage: 4.13MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Wed Dec 7 18:27:17 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^3*3^9; C:=HyperellipticCurve(f); J:=Jacobian(C); //r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); //V:=RationalPoints(J:Bound:=1500); //B:=ReducedBasis(V); //B; //P:=J!B[1];//Q:=J![x+2,0]; //P1:=J![x+1,0];P2:=J![x,1]; //hP:=Height(P);hP; //HC:=HeightConstant(J:Effort:=20);HC; //Exp(hP/4+HC); //RationalPoints(J:Bound:=1000); //Chabauty(P,11); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Wed Dec 7 2005 18:27:17 on modular [Seed = 1990351488] ------------------------------------- Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] Total time: 0.230 seconds, Total memory usage: 4.51MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Wed Dec 7 18:27:13 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^1*3^9; C:=HyperellipticCurve(f); J:=Jacobian(C); //r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); //V:=RationalPoints(J:Bound:=1500); //B:=ReducedBasis(V); //B; //P:=J!B[1];//Q:=J![x+2,0]; //P1:=J![x+1,0];P2:=J![x,1]; //hP:=Height(P);hP; //HC:=HeightConstant(J:Effort:=20);HC; //Exp(hP/4+HC); //RationalPoints(J:Bound:=1000); //Chabauty(P,11); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Wed Dec 7 2005 18:27:12 on modular [Seed = 1824014495] ------------------------------------- Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] Total time: 0.230 seconds, Total memory usage: 4.51MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Wed Dec 7 18:27:08 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^0*3^9; C:=HyperellipticCurve(f); J:=Jacobian(C); //r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); //V:=RationalPoints(J:Bound:=1500); //B:=ReducedBasis(V); //B; //P:=J!B[1];//Q:=J![x+2,0]; //P1:=J![x+1,0];P2:=J![x,1]; //hP:=Height(P);hP; //HC:=HeightConstant(J:Effort:=20);HC; //Exp(hP/4+HC); //RationalPoints(J:Bound:=1000); //Chabauty(P,11); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Wed Dec 7 2005 18:27:07 on modular [Seed = 1774011782] ------------------------------------- Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] Total time: 0.220 seconds, Total memory usage: 4.13MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Wed Dec 7 18:26:54 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^9*3^6; C:=HyperellipticCurve(f); J:=Jacobian(C); //r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); //V:=RationalPoints(J:Bound:=1500); //B:=ReducedBasis(V); //B; //P:=J!B[1];//Q:=J![x+2,0]; //P1:=J![x+1,0];P2:=J![x,1]; //hP:=Height(P);hP; //HC:=HeightConstant(J:Effort:=20);HC; //Exp(hP/4+HC); //RationalPoints(J:Bound:=1000); //Chabauty(P,11); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Wed Dec 7 2005 18:26:54 on modular [Seed = 1722960635] ------------------------------------- Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] Total time: 0.220 seconds, Total memory usage: 4.13MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Wed Dec 7 18:26:47 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^7*3^7; C:=HyperellipticCurve(f); J:=Jacobian(C); //r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); //V:=RationalPoints(J:Bound:=1500); //B:=ReducedBasis(V); //B; //P:=J!B[1];//Q:=J![x+2,0]; //P1:=J![x+1,0];P2:=J![x,1]; //hP:=Height(P);hP; //HC:=HeightConstant(J:Effort:=20);HC; //Exp(hP/4+HC); //RationalPoints(J:Bound:=1000); //Chabauty(P,11); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Wed Dec 7 2005 18:26:47 on modular [Seed = 1672957943] ------------------------------------- Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] Total time: 0.220 seconds, Total memory usage: 4.13MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Wed Dec 7 18:26:39 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^5*3^9; C:=HyperellipticCurve(f); J:=Jacobian(C); //r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); //V:=RationalPoints(J:Bound:=1500); //B:=ReducedBasis(V); //B; //P:=J!B[1];//Q:=J![x+2,0]; //P1:=J![x+1,0];P2:=J![x,1]; //hP:=Height(P);hP; //HC:=HeightConstant(J:Effort:=20);HC; //Exp(hP/4+HC); //RationalPoints(J:Bound:=1000); //Chabauty(P,11); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Wed Dec 7 2005 18:26:39 on modular [Seed = 416035988] ------------------------------------- Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] Total time: 0.220 seconds, Total memory usage: 4.13MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Wed Dec 7 18:26:32 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^3*3^8; C:=HyperellipticCurve(f); J:=Jacobian(C); //r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); //V:=RationalPoints(J:Bound:=1500); //B:=ReducedBasis(V); //B; //P:=J!B[1];//Q:=J![x+2,0]; //P1:=J![x+1,0];P2:=J![x,1]; //hP:=Height(P);hP; //HC:=HeightConstant(J:Effort:=20);HC; //Exp(hP/4+HC); //RationalPoints(J:Bound:=1000); //Chabauty(P,11); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Wed Dec 7 2005 18:26:31 on modular [Seed = 364984721] ------------------------------------- Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] Total time: 0.220 seconds, Total memory usage: 4.13MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Wed Dec 7 18:26:26 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^3*3^3; C:=HyperellipticCurve(f); J:=Jacobian(C); //r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); //V:=RationalPoints(J:Bound:=1500); //B:=ReducedBasis(V); //B; //P:=J!B[1];//Q:=J![x+2,0]; //P1:=J![x+1,0];P2:=J![x,1]; //hP:=Height(P);hP; //HC:=HeightConstant(J:Effort:=20);HC; //Exp(hP/4+HC); //RationalPoints(J:Bound:=1000); //Chabauty(P,11); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Wed Dec 7 2005 18:26:26 on modular [Seed = 314982035] ------------------------------------- Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] Total time: 0.220 seconds, Total memory usage: 4.13MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Wed Dec 7 18:26:13 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^1*3^9; C:=HyperellipticCurve(f); J:=Jacobian(C); //r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); //V:=RationalPoints(J:Bound:=1500); //B:=ReducedBasis(V); //B; //P:=J!B[1];//Q:=J![x+2,0]; //P1:=J![x+1,0];P2:=J![x,1]; //hP:=Height(P);hP; //HC:=HeightConstant(J:Effort:=20);HC; //Exp(hP/4+HC); //RationalPoints(J:Bound:=1000); //Chabauty(P,11); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Wed Dec 7 2005 18:26:11 on modular [Seed = 266019775] ------------------------------------- Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] Total time: 0.220 seconds, Total memory usage: 4.13MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Wed Dec 7 18:22:24 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); S:=Transpose(A)*A^(-1); F,T:=PrimaryRationalForm(S); B:=T*A*Transpose(T); print T; print T^(-1); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*B*Transpose(P)-B)); print GroebnerBasis(I); Output: Magma V2.11-10 Wed Dec 7 2005 18:22:24 on modular [Seed = 64433808] ------------------------------------- [1/(b*c) -1/(b*c) 1/(b*c)] [0 -1/b (-b + 1)/(b*c)] [(-b - c + 1)/(a*b*c) (-a*b*c - a*b + a + b + c - 1)/(a*b^2*c) (-a*b^2*c + 2*a*b*c + a*b - a + b^2 + b*c - 2*b - c + 1)/(a*b^2*c^2)] [a*b*c/(a + b + c - 1) (a*b*c - a - b - c + 1)/(a + b + c - 1) -a*b*c/(a + b + c - 1)] [(-b^2*c + b*c)/(a + b + c - 1) (a*b^2*c - 2*a*b*c - a*b + a - b^2*c - b^2 - b*c^2 + 2*b + c - 1)/(a*b + a*c - a + b^2 + 2*b*c - 2*b + c^2 - 2*c + 1) (-a*b^2*c + a*b*c)/(a*b + a*c - a + b^2 + 2*b*c - 2*b + c^2 - 2*c + 1)] [b*c^2/(a + b + c - 1) (-a*b*c^2 - a*b*c + a*c - b^2*c - b*c^2 + 2*b*c + c^2 - c)/(a*b + a*c - a + b^2 + 2*b*c - 2*b + c^2 - 2*c + 1) a*b*c^2/(a*b + a*c - a + b^2 + 2*b*c - 2*b + c^2 - 2*c + 1)] [ $.1^2 - 1, $.2, $.3, $.4, $.5 + (-2*a*b*c + a + b + c - 1)/(a*b*c)*$.8 - $.9, $.6 + $.8, $.7, $.8^2 + (2*a*b*c - a - b - c + 1)/(a*b*c)*$.8*$.9 + $.9^2 - 1 ] Total time: 0.200 seconds, Total memory usage: 3.34MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Wed Dec 7 18:06:30 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); S:=Transpose(A)*A^(-1); F,T:=PrimaryRationalForm(S); B:=T*A*Transpose(T); print T; R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*B*Transpose(P)-B)); print GroebnerBasis(I); Output: Magma V2.11-10 Wed Dec 7 2005 18:06:30 on modular [Seed = 653398003] ------------------------------------- [1/(b*c) -1/(b*c) 1/(b*c)] [0 -1/b (-b + 1)/(b*c)] [(-b - c + 1)/(a*b*c) (-a*b*c - a*b + a + b + c - 1)/(a*b^2*c) (-a*b^2*c + 2*a*b*c + a*b - a + b^2 + b*c - 2*b - c + 1)/(a*b^2*c^2)] [ $.1^2 - 1, $.2, $.3, $.4, $.5 + (-2*a*b*c + a + b + c - 1)/(a*b*c)*$.8 - $.9, $.6 + $.8, $.7, $.8^2 + (2*a*b*c - a - b - c + 1)/(a*b*c)*$.8*$.9 + $.9^2 - 1 ] Total time: 0.200 seconds, Total memory usage: 3.34MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Wed Dec 7 18:03:50 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); S:=Transpose(A)*A^(-1); F,T:=PrimaryRationalForm(S); B:=T*A*Transpose(T); print B; R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*B*Transpose(P)-B)); print GroebnerBasis(I); Output: Magma V2.11-10 Wed Dec 7 2005 18:03:50 on modular [Seed = 603392598] ------------------------------------- [(a + b + c - 1)/(b^2*c^2) 0 0] [0 (b + c - 1)/(b*c) (a*b^2*c + a*b*c^2 - a*b*c - a*b - a*c + a - b^2 - 2*b*c + 2*b - c^2 + 2*c - 1)/(a*b^2*c^2)] [0 (b + c - 1)/(b*c) (b + c - 1)/(b*c)] [ $.1^2 - 1, $.2, $.3, $.4, $.5 + (-2*a*b*c + a + b + c - 1)/(a*b*c)*$.8 - $.9, $.6 + $.8, $.7, $.8^2 + (2*a*b*c - a - b - c + 1)/(a*b*c)*$.8*$.9 + $.9^2 - 1 ] Total time: 0.200 seconds, Total memory usage: 3.34MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Wed Dec 7 18:02:01 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); S:=Transpose(A)*A^(-1); F,T:=PrimaryRationalForm(S); print T*S*T^(-1)-F; B:=T*A*Transpose(T); print Transpose(B)*B^(-1)-F; R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*B*Transpose(P)-B)); print GroebnerBasis(I); Output: Magma V2.11-10 Wed Dec 7 2005 18:02:00 on modular [Seed = 3862088722] ------------------------------------- [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [ $.1^2 - 1, $.2, $.3, $.4, $.5 + (-2*a*b*c + a + b + c - 1)/(a*b*c)*$.8 - $.9, $.6 + $.8, $.7, $.8^2 + (2*a*b*c - a - b - c + 1)/(a*b*c)*$.8*$.9 + $.9^2 - 1 ] Total time: 0.200 seconds, Total memory usage: 3.34MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Wed Dec 7 18:01:34 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); S:=Transpose(A)*A^(-1); F,T:=PrimaryRationalForm(S); print T*S*T^(-1)-F; R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*B*Transpose(P)-B)); print GroebnerBasis(I); Output: Magma V2.11-10 Wed Dec 7 2005 18:01:33 on modular [Seed = 3913139000] ------------------------------------- [0 0 0] [0 0 0] [0 0 0] >> I:=Ideal(Eltseq(P*B*Transpose(P)-B)); ^ User error: Identifier 'B' has not been declared or assigned >> print GroebnerBasis(I); ^ User error: Identifier 'I' has not been declared or assigned Total time: 0.200 seconds, Total memory usage: 3.34MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Wed Dec 7 18:01:08 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); S:=Transpose(A)*A^(-1); F,T:=PrimaryRationalForm(S); print T*F*T^(-1)-S; R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*B*Transpose(P)-B)); print GroebnerBasis(I); Output: Magma V2.11-10 Wed Dec 7 2005 18:01:07 on modular [Seed = 3963143747] ------------------------------------- [-1/(a*b) (a*b^2*c + a*b*c^2 + a*b - a + b^2 + b*c - 2*b - c + 1)/(a*b^3*c + a*b^2*c^2 - a*b^2*c) (b^2 + b*c - 2*b - 2*c + 1)/(b^2*c + b*c^2 - b*c)] [(-a*b^3 - 2*a*b^2*c + 2*a*b^2 - a*b*c^2 + 2*a*b*c - a*b - a - b - c + 1)/(a^2*b + a*b^2 + a*b*c - a*b) (a^2*b^4*c + 2*a^2*b^3*c^2 - 2*a^2*b^3*c - a^2*b^3 + a^2*b^2*c^3 - 2*a^2*b^2*c^2 - 2*a^2*b^2*c + a^2*b^2 - a^2*b*c^2 + 3*a^2*b*c + a^2*b - a^2 - a*b^4 - 4*a*b^3*c + a*b^3 - 4*a*b^2*c^2 + 7*a*b^2*c + 3*a*b^2 - a*b*c^3 + 5*a*b*c^2 - a*b*c - 5*a*b - 2*a*c + 2*a - b^4 - b^3*c + 4*b^3 + b^2*c^2 + 4*b^2*c - 6*b^2 + b*c^3 - 5*b*c + 4*b - c^2 + 2*c - 1)/(a^2*b^3*c + a^2*b^2*c^2 - a^2*b^2*c + a*b^4*c + 2*a*b^3*c^2 - 2*a*b^3*c + a*b^2*c^3 - 2*a*b^2*c^2 + a*b^2*c) (-a^2*b^3*c - 2*a^2*b^2*c^2 + 2*a^2*b^2*c + a^2*b^2 - a^2*b*c^3 + 2*a^2*b*c^2 + a^2*b*c - a^2*b - a^2*c + a*b^3 + 3*a*b^2*c - a*b^2 + 2*a*b*c^2 - 3*a*b*c - a*b - a*c^2 + a + b^3 + 2*b^2*c - 3*b^2 + b*c^2 - 4*b*c + 3*b - c^2 + 2*c - 1)/(a^2*b^2*c + a^2*b*c^2 - a^2*b*c + a*b^3*c + 2*a*b^2*c^2 - 2*a*b^2*c + a*b*c^3 - 2*a*b*c^2 + a*b*c)] [(-a^2*b^4*c - 2*a^2*b^3*c^2 + 2*a^2*b^3*c + a^2*b^3 - a^2*b^2*c^3 + 2*a^2*b^2*c^2 + a^2*b^2*c - 2*a^2*b^2 + a^2*b*c^2 - 4*a^2*b*c + a^2 + a*b^4 + 3*a*b^3*c - 3*a*b^3 + 3*a*b^2*c^2 - 8*a*b^2*c + a*b^2 + a*b*c^3 - 5*a*b*c^2 + 3*a*b*c + 3*a*b + 2*a*c - 2*a - b^3 - 2*b^2*c + 3*b^2 - b*c^2 + 4*b*c - 3*b + c^2 - 2*c + 1)/(a^3*b^2*c + a^2*b^3*c + a^2*b^2*c^2 - a^2*b^2*c) (a^3*b^5*c^2 + 2*a^3*b^4*c^3 - 2*a^3*b^4*c^2 - 2*a^3*b^4*c + a^3*b^3*c^4 - 2*a^3*b^3*c^3 - 5*a^3*b^3*c^2 + 3*a^3*b^3*c + a^3*b^3 - 3*a^3*b^2*c^3 + 7*a^3*b^2*c^2 + 4*a^3*b^2*c - a^3*b^2 + a^3*b*c^2 - 5*a^3*b*c - a^3*b + a^3 - 2*a^2*b^5*c - 8*a^2*b^4*c^2 + 4*a^2*b^4*c + 2*a^2*b^4 - 9*a^2*b^3*c^3 + 16*a^2*b^3*c^2 + 10*a^2*b^3*c - 3*a^2*b^3 - 3*a^2*b^2*c^4 + 11*a^2*b^2*c^3 + 4*a^2*b^2*c^2 - 21*a^2*b^2*c - 3*a^2*b^2 + 2*a^2*b*c^3 - 12*a^2*b*c^2 + 6*a^2*b*c + 7*a^2*b + 3*a^2*c - 3*a^2 - a*b^5*c + a*b^5 - a*b^4*c^2 + 10*a*b^4*c - a*b^4 + a*b^3*c^3 + 14*a*b^3*c^2 - 18*a*b^3*c - 6*a*b^3 + a*b^2*c^4 + 6*a*b^2*c^3 - 22*a*b^2*c^2 + 4*a*b^2*c + 14*a*b^2 + a*b*c^4 - 7*a*b*c^3 + 6*a*b*c^2 + 11*a*b*c - 11*a*b + 3*a*c^2 - 6*a*c + 3*a + b^5 + 3*b^4*c - 5*b^4 + 3*b^3*c^2 - 12*b^3*c + 10*b^3 + b^2*c^3 - 9*b^2*c^2 + 18*b^2*c - 10*b^2 - 2*b*c^3 + 9*b*c^2 - 12*b*c + 5*b + c^3 - 3*c^2 + 3*c - 1)/(a^3*b^4*c^2 + a^3*b^3*c^3 - a^3*b^3*c^2 + a^2*b^5*c^2 + 2*a^2*b^4*c^3 - 2*a^2*b^4*c^2 + a^2*b^3*c^4 - 2*a^2*b^3*c^3 + a^2*b^3*c^2) (-a^2*b^4*c - 2*a^2*b^3*c^2 + 2*a^2*b^3*c + a^2*b^3 - a^2*b^2*c^3 + 2*a^2*b^2*c^2 + 2*a^2*b^2*c - a^2*b^2 + a^2*b*c^2 - 3*a^2*b*c - a^2*b + a^2 + a*b^4 + 4*a*b^3*c - a*b^3 + 4*a*b^2*c^2 - 6*a*b^2*c - 3*a*b^2 + a*b*c^3 - 4*a*b*c^2 + 5*a*b + 2*a*c - 2*a + b^4 + 2*b^3*c - 4*b^3 + b^2*c^2 - 6*b^2*c + 6*b^2 - 2*b*c^2 + 6*b*c - 4*b + c^2 - 2*c + 1)/(a^2*b^3*c + a^2*b^2*c^2 - a^2*b^2*c + a*b^4*c + 2*a*b^3*c^2 - 2*a*b^3*c + a*b^2*c^3 - 2*a*b^2*c^2 + a*b^2*c)] >> I:=Ideal(Eltseq(P*B*Transpose(P)-B)); ^ User error: Identifier 'B' has not been declared or assigned >> print GroebnerBasis(I); ^ User error: Identifier 'I' has not been declared or assigned Total time: 0.200 seconds, Total memory usage: 3.34MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Wed Dec 7 17:58:43 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); S:=Transpose(A)*A^(-1); F,T:=PrimaryRationalForm(S); print F; print T; Output: Magma V2.11-10 Wed Dec 7 2005 17:58:42 on modular [Seed = 2910422931] ------------------------------------- [1 0 0] [0 0 1] [0 -1 (2*a*b*c - a - b - c + 1)/(a*b*c)] [1/(b*c) -1/(b*c) 1/(b*c)] [0 -1/b (-b + 1)/(b*c)] [(-b - c + 1)/(a*b*c) (-a*b*c - a*b + a + b + c - 1)/(a*b^2*c) (-a*b^2*c + 2*a*b*c + a*b - a + b^2 + b*c - 2*b - c + 1)/(a*b^2*c^2)] Total time: 0.190 seconds, Total memory usage: 3.34MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Wed Dec 7 17:57:05 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); S:=Transpose(A)*A^(-1); print PrimaryRationalForm(S); Output: Magma V2.11-10 Wed Dec 7 2005 17:57:05 on modular [Seed = 3127808359] ------------------------------------- [1 0 0] [0 0 1] [0 -1 (2*a*b*c - a - b - c + 1)/(a*b*c)] Total time: 0.200 seconds, Total memory usage: 3.34MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Wed Dec 7 17:54:45 2005 Input: K:=RationalField(); A:=Matrix(K,4,4,[[1,1,1,1],[0,1,0,1],[0,0,1,1],[0,0,0,1]]); S:=Transpose(A)*(A^(-1)); print RationalForm(S); I4:=A*A^(-1); M:=KroneckerProduct(I4,S)-KroneckerProduct(Transpose(S),I4); L:=RowSequence(KernelMatrix(M)); R:=PolynomialRing(K,#L); P:=0; for i in [1..#L] do P:= P + R.(7-i)*Matrix(K,4,4,L[i]); end for; I:=Ideal(Eltseq(P*A*Transpose(P)-A)); G:=GroebnerBasis(I); Output: Magma V2.11-10 Wed Dec 7 2005 17:54:45 on modular [Seed = 3177813652] ------------------------------------- [1 0 0 0] [0 0 1 0] [0 0 0 1] [0 1 0 0] Total time: 0.190 seconds, Total memory usage: 3.43MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Wed Dec 7 17:51:33 2005 Input: K:=RationalField(); A:=Matrix(K,4,4,[[1,1,1,1],[0,1,0,1],[0,0,1,1],[0,0,0,1]]); S:=Transpose(A)*(A^(-1)); I4:=A*A^(-1); M:=KroneckerProduct(I4,S)-KroneckerProduct(Transpose(S),I4); L:=RowSequence(KernelMatrix(M)); R:=PolynomialRing(K,#L); P:=0; for i in [1..#L] do P:= P + R.(7-i)*Matrix(K,4,4,L[i]); end for; I:=Ideal(Eltseq(P*A*Transpose(P)-A)); G:=GroebnerBasis(I); print G; Output: Magma V2.11-10 Wed Dec 7 2005 17:51:33 on modular [Seed = 2236719596] ------------------------------------- [ $.1^2 + $.1*$.3 - $.1*$.5 - $.1*$.6 - $.3*$.5 - $.4*$.5, $.1*$.2 + $.1*$.5 + $.2*$.3 + $.3*$.5, $.1*$.3*$.5^2 + 2*$.1*$.5^3 + 3*$.1*$.5^2*$.6 + 2*$.1*$.5*$.6^2 + $.1*$.6^3 - $.1*$.6 + 2*$.2*$.3*$.5^2 + 3*$.2*$.4*$.5^2 + 2*$.2*$.4*$.5*$.6 + $.2*$.4*$.6^2 + $.2*$.5^3 + 3*$.2*$.5^2*$.6 + 2*$.2*$.5*$.6^2 + $.2*$.5 + $.2*$.6^3 + $.3*$.5^3 + 2*$.4*$.5^3 + 2*$.4*$.5^2*$.6 + $.4*$.5*$.6^2 - $.5^4 - 2*$.5^3*$.6 - 3*$.5^2*$.6^2 + $.5^2 - 2*$.5*$.6^3 + $.5*$.6 - $.6^4 + $.6^2, $.1*$.4 - $.1*$.5 - $.2*$.3 - $.2*$.5 - $.2*$.6 - $.3*$.5 - $.4*$.5 - $.4*$.6, $.1*$.5^4 + 2*$.1*$.5^3*$.6 + 3*$.1*$.5^2*$.6^2 + 2*$.1*$.5*$.6^3 + $.1*$.6^4 - $.1*$.6^2 + $.2*$.3*$.5^3 + 2*$.2*$.4*$.5^3 + 3*$.2*$.4*$.5^2*$.6 + 2*$.2*$.4*$.5*$.6^2 + $.2*$.4*$.6^3 + $.2*$.5^4 + 2*$.2*$.5^3*$.6 + 3*$.2*$.5^2*$.6^2 + 2*$.2*$.5*$.6^3 + $.2*$.5*$.6 + $.2*$.6^4 + $.3*$.5^4 + 2*$.4*$.5^4 + 3*$.4*$.5^3*$.6 + 2*$.4*$.5^2*$.6^2 + $.4*$.5*$.6^3 - $.5^4*$.6 - 2*$.5^3*$.6^2 - 3*$.5^2*$.6^3 + $.5^2*$.6 - 2*$.5*$.6^4 + $.5*$.6^2 - $.6^5 + $.6^3, $.2^2 + $.2*$.4 + $.2*$.5 + $.4*$.5, $.3^2 + $.3*$.4 + $.3*$.5 + $.4^2 + $.4*$.5 + $.4*$.6 + $.5^2 + $.5*$.6 + $.6^2 - 1, $.3*$.4*$.5 + $.4^2*$.5 + $.4^2*$.6 + $.4*$.5^2 + $.4*$.5*$.6 + $.4*$.6^2 + $.5^2*$.6 + $.5*$.6^2 + $.6^3 - $.6, $.3*$.6 - $.4*$.5, $.4^2*$.5^2 + $.4^2*$.5*$.6 + $.4^2*$.6^2 + $.4*$.5^2*$.6 + $.4*$.5*$.6^2 + $.4*$.6^3 + $.5^2*$.6^2 + $.5*$.6^3 + $.6^4 - $.6^2 ] Total time: 0.200 seconds, Total memory usage: 3.43MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Wed Dec 7 17:50:55 2005 Input: K:=RationalField(); A:=Matrix(K,4,4,[[1,1,1,1],[0,1,0,1],[0,0,1,1],[0,0,0,1]]); S:=Transpose(A)*(A^(-1)); I4:=A*A^(-1); M:=KroneckerProduct(I4,S)-KroneckerProduct(Transpose(S),I4); L:=RowSequence(KernelMatrix(M)); R:=PolynomialRing(K,#L); P:=0; for i in [1..#L] do P:= P + R.i*Matrix(K,4,4,L[i]); end for; I:=Ideal(Eltseq(P*A*Transpose(P)-A)); G:=GroebnerBasis(I); print G; Output: Magma V2.11-10 Wed Dec 7 2005 17:50:55 on modular [Seed = 2287774939] ------------------------------------- [ $.1^2 + $.1*$.2 - $.1*$.5 + $.2^2 - $.2*$.5 - $.2*$.6 + $.3^2 + $.3*$.4 + $.3*$.6 + $.4^2 - $.4*$.5 - 1, $.1*$.2*$.5 + $.2^2*$.5 + $.2^2*$.6 - $.2*$.5^2 - $.2*$.5*$.6 - $.2*$.6^2 + $.3^2*$.6 + $.3*$.4*$.6 - $.3*$.5^2 + $.3*$.5*$.6 + $.3*$.6^2 + $.4^2*$.6 - $.4*$.5^2 - $.4*$.5*$.6 + $.4*$.6^2 - $.5^3 + $.6^3 - $.6, $.1*$.3 + $.1*$.5 - $.3*$.5 - $.3*$.6 - $.5^2 - $.5*$.6, $.1*$.4 + $.2*$.5 + $.3*$.5 + $.5^2, $.1*$.6 - $.2*$.5 - $.3*$.5 - $.4*$.5 - $.4*$.6 - $.5^2 - $.5*$.6 - $.6^2, $.2^2*$.5^2 + $.2^2*$.5*$.6 + $.2^2*$.6^2 - $.2*$.5^2*$.6 - $.2*$.5*$.6^2 - $.2*$.6^3 + $.3*$.4*$.6^2 - 2*$.3*$.5^3 - 3*$.3*$.5^2*$.6 - 3*$.3*$.5*$.6^2 + $.3*$.5 - $.4^2*$.5^2 - 2*$.4^2*$.5*$.6 - $.4^2*$.6^2 - $.4*$.5^3 - 3*$.4*$.5^2*$.6 - 4*$.4*$.5*$.6^2 + $.4*$.5 - 2*$.4*$.6^3 + $.4*$.6 - 2*$.5^4 - 4*$.5^3*$.6 - 4*$.5^2*$.6^2 + $.5^2 - 2*$.5*$.6^3 + $.5*$.6, $.2*$.3 + $.2*$.5 + $.3*$.5 + $.5^2, $.2*$.4 + $.2*$.6 + $.4*$.5 + $.5*$.6, $.3^3 - 3*$.3*$.4*$.6 + 3*$.3*$.5*$.6 - $.3 - $.4^3 + 3*$.4^2*$.5 - 3*$.4*$.5^2 + $.4 + $.5^3 - $.5 - $.6^3 + $.6, $.3^2*$.4 + $.3^2*$.6 + $.3*$.4^2 + 2*$.3*$.4*$.6 + $.3*$.6^2 + $.4^3 - $.4^2*$.5 + $.4^2*$.6 + $.4*$.5^2 + $.4*$.6^2 - $.4 + $.5^2*$.6 + $.5*$.6^2 + $.6^3 - $.6, $.3^2*$.5 + $.3*$.4*$.6 + $.3*$.5^2 - $.3*$.5*$.6 - $.4^2*$.5 - $.4^2*$.6 + $.4*$.5^2 - $.4*$.6^2, $.3^2*$.6^2 + $.3*$.5^3 + 2*$.3*$.5^2*$.6 + 4*$.3*$.5*$.6^2 - $.3*$.5 + $.3*$.6^3 + $.4^2*$.5^2 + 2*$.4^2*$.5*$.6 + 2*$.4^2*$.6^2 + $.4*$.5^2*$.6 + 3*$.4*$.5*$.6^2 - $.4*$.5 + 3*$.4*$.6^3 - $.4*$.6 + $.5^4 + 2*$.5^3*$.6 + 3*$.5^2*$.6^2 - $.5^2 + 2*$.5*$.6^3 - $.5*$.6 + $.6^4 - $.6^2, $.3*$.4^2*$.6 + $.3*$.4*$.6^2 - $.4^3*$.5 + $.4^2*$.5^2 + $.4^2*$.6^2 - $.4*$.5^3 + $.4*$.5 + $.4*$.6^3 - $.5^3*$.6 - $.5^2*$.6^2 - $.5*$.6^3 + $.5*$.6, $.3*$.4*$.5 + $.3*$.5*$.6 + $.4^2*$.5 + $.4^2*$.6 + $.4*$.5*$.6 + $.4*$.6^2, $.3*$.4*$.6^3 - $.3*$.5^4 - 2*$.3*$.5^3*$.6 - 3*$.3*$.5^2*$.6^2 + $.3*$.5^2 - 2*$.3*$.5*$.6^3 - $.4^2*$.5^3 - 2*$.4^2*$.5^2*$.6 - 3*$.4^2*$.5*$.6^2 - $.4^2*$.6^3 - $.4*$.5^3*$.6 - 2*$.4*$.5^2*$.6^2 + $.4*$.5^2 - 3*$.4*$.5*$.6^3 + $.4*$.5*$.6 - $.4*$.6^4 - $.5^5 - 2*$.5^4*$.6 - 3*$.5^3*$.6^2 + $.5^3 - 2*$.5^2*$.6^3 + $.5^2*$.6 - $.5*$.6^4 + $.5*$.6^2, $.3*$.5^5 + 2*$.3*$.5^4*$.6 + 3*$.3*$.5^3*$.6^2 - $.3*$.5^3 + 2*$.3*$.5^2*$.6^3 + $.3*$.5*$.6^4 + $.4^2*$.5^4 + 2*$.4^2*$.5^3*$.6 + 3*$.4^2*$.5^2*$.6^2 + 2*$.4^2*$.5*$.6^3 + $.4^2*$.6^4 + $.4*$.5^4*$.6 + 2*$.4*$.5^3*$.6^2 - $.4*$.5^3 + 3*$.4*$.5^2*$.6^3 - $.4*$.5^2*$.6 + 2*$.4*$.5*$.6^4 + $.4*$.6^5 + $.5^6 + 2*$.5^5*$.6 + 3*$.5^4*$.6^2 - $.5^4 + 2*$.5^3*$.6^3 - $.5^3*$.6 + $.5^2*$.6^4 - $.5^2*$.6^2, $.4^3*$.5^2 + $.4^3*$.5*$.6 + $.4^3*$.6^2 - $.4^2*$.5^3 + $.4^2*$.6^3 + $.4*$.5^4 - $.4*$.5^2 - $.4*$.5*$.6^3 + $.5^4*$.6 + $.5^3*$.6^2 + $.5^2*$.6^3 - $.5^2*$.6 ] Total time: 0.190 seconds, Total memory usage: 3.43MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Wed Dec 7 17:49:34 2005 Input: K:=RationalField(); A:=Matrix(K,4,4,[[1,1,1,1],[0,1,0,1],[0,0,1,1],[0,0,0,1]]); S:=Transpose(A)*(A^(-1)); I4:=A*A^(-1); M:=KroneckerProduct(I4,S)-KroneckerProduct(Transpose(S),I4); L:=RowSequence(KernelMatrix(M)); R:=PolynomialRing(K,#L); P:=0; for i in [1..#L] do P:= P + R.i*Matrix(K,4,4,L[1]); end for; I:=Ideal(Eltseq(P*A*Transpose(P)-A)); G:=GroebnerBasis(I); print Factorization(G[1]); Output: Magma V2.11-10 Wed Dec 7 2005 17:49:33 on modular [Seed = 2470950514] ------------------------------------- [ <$.1 + $.2 + $.3 + $.4 + $.5 + $.6 - 1, 1>, <$.1 + $.2 + $.3 + $.4 + $.5 + $.6 + 1, 1> ] Total time: 0.200 seconds, Total memory usage: 3.34MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Wed Dec 7 17:47:31 2005 Input: K:=RationalField(); A:=Matrix(K,4,4,[[1,1,1,1],[0,1,0,1],[0,0,1,1],[0,0,0,1]]); S:=Transpose(A)*(A^(-1)); I4:=A*A^(-1); M:=KroneckerProduct(I4,S)-KroneckerProduct(Transpose(S),I4); L:=RowSequence(KernelMatrix(M)); R:=PolynomialRing(K,#L); P:=0; for i in [1..#L] do P:= P + R.i*Matrix(K,4,4,L[1]); end for; I:=Ideal(Eltseq(P*A*Transpose(P)-A)); print GroebnerBasis(I); Output: Magma V2.11-10 Wed Dec 7 2005 17:47:30 on modular [Seed = 2520953608] ------------------------------------- [ $.1^2 + 2*$.1*$.2 + 2*$.1*$.3 + 2*$.1*$.4 + 2*$.1*$.5 + 2*$.1*$.6 + $.2^2 + 2*$.2*$.3 + 2*$.2*$.4 + 2*$.2*$.5 + 2*$.2*$.6 + $.3^2 + 2*$.3*$.4 + 2*$.3*$.5 + 2*$.3*$.6 + $.4^2 + 2*$.4*$.5 + 2*$.4*$.6 + $.5^2 + 2*$.5*$.6 + $.6^2 - 1 ] Total time: 0.190 seconds, Total memory usage: 3.34MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Wed Dec 7 17:47:04 2005 Input: K:=RationalField(); A:=Matrix(K,4,4,[[1,1,1,1],[0,1,0,1],[0,0,1,1],[0,0,0,1]]); S:=Transpose(A)*(A^(-1)); I4:=A*A^(-1); M:=KroneckerProduct(I4,S)-KroneckerProduct(Transpose(S),I4); L:=RowSequence(KernelMatrix(M)); R:=PolynomialRing(K,#L); P:=0; for i in [1..#L] do P:= P + R.i*Matrix(K,4,4,L[1]); end for; Output: Magma V2.11-10 Wed Dec 7 2005 17:47:04 on modular [Seed = 2555166188] ------------------------------------- Total time: 0.190 seconds, Total memory usage: 3.34MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Wed Dec 7 17:46:24 2005 Input: K:=RationalField(); A:=Matrix(K,4,4,[[1,1,1,1],[0,1,0,1],[0,0,1,1],[0,0,0,1]]); S:=Transpose(A)*(A^(-1)); I4:=A*A^(-1); M:=KroneckerProduct(I4,S)-KroneckerProduct(Transpose(S),I4); L:=RowSequence(KernelMatrix(M)); print Matrix(K,4,4,L[1]); Output: Magma V2.11-10 Wed Dec 7 2005 17:46:24 on modular [Seed = 1672955455] ------------------------------------- [1 0 0 0] [0 0 1 0] [0 1 0 0] [0 0 0 1] Total time: 0.190 seconds, Total memory usage: 3.24MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Wed Dec 7 17:45:40 2005 Input: K:=RationalField(); A:=Matrix(K,4,4,[[1,1,1,1],[0,1,0,1],[0,0,1,1],[0,0,0,1]]); S:=Transpose(A)*(A^(-1)); I4:=A*A^(-1); M:=KroneckerProduct(I4,S)-KroneckerProduct(Transpose(S),I4); L:=RowSequence(KernelMatrix(M)); print #L; Output: Magma V2.11-10 Wed Dec 7 2005 17:45:40 on modular [Seed = 1722958702] ------------------------------------- 6 Total time: 0.190 seconds, Total memory usage: 3.24MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Wed Dec 7 17:43:53 2005 Input: K:=RationalField(); A:=Matrix(K,4,4,[[1,1,1,1],[0,1,0,1],[0,0,1,1],[0,0,0,1]]); S:=Transpose(A)*(A^(-1)); I4:=A*A^(-1); M:=KroneckerProduct(I4,S)-KroneckerProduct(Transpose(S),I4); print KernelMatrix(M); Output: Magma V2.11-10 Wed Dec 7 2005 17:43:53 on modular [Seed = 1774014428] ------------------------------------- [ 1 0 0 0 0 0 1 0 0 1 0 0 0 0 0 1] [ 0 1 0 0 0 0 0 1 -1 1 0 0 0 0 -1 1] [ 0 0 1 0 0 0 0 1 -1 0 1 0 0 -1 0 1] [ 0 0 0 1 0 0 -1 1 0 -1 0 1 1 -1 -1 1] [ 0 0 0 0 1 0 -1 1 -1 0 1 -1 0 0 0 0] [ 0 0 0 0 0 1 -1 0 0 -1 1 0 0 0 0 0] Total time: 0.190 seconds, Total memory usage: 3.24MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Wed Dec 7 17:42:44 2005 Input: K:=RationalField(); A:=Matrix(K,4,4,[[1,1,1,1],[0,1,0,1],[0,0,1,1],[0,0,0,1]]); S:=Transpose(A)*(A^(-1)); I4:=A*A^(-1); print KroneckerProduct(Transpose(S),I4); Output: Magma V2.11-10 Wed Dec 7 2005 17:42:44 on modular [Seed = 1824016916] ------------------------------------- [ 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0] [ 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0] [ 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0] [ 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1] [-1 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0] [ 0 -1 0 0 0 0 0 0 0 -1 0 0 0 0 0 0] [ 0 0 -1 0 0 0 0 0 0 0 -1 0 0 0 0 0] [ 0 0 0 -1 0 0 0 0 0 0 0 -1 0 0 0 0] [-1 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0] [ 0 -1 0 0 0 -1 0 0 0 0 0 0 0 0 0 0] [ 0 0 -1 0 0 0 -1 0 0 0 0 0 0 0 0 0] [ 0 0 0 -1 0 0 0 -1 0 0 0 0 0 0 0 0] [ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0] [ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0] [ 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0] [ 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0] Total time: 0.190 seconds, Total memory usage: 3.24MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Wed Dec 7 17:41:56 2005 Input: K:=RationalField(); A:=Matrix(K,4,4,[[1,1,1,1],[0,1,0,1],[0,0,1,1],[0,0,0,1]]); S:=Transpose(A)*(A^(-1)); I4:=A*A^(-1); print KroneckerProduct(I4,S); Output: Magma V2.11-10 Wed Dec 7 2005 17:41:56 on modular [Seed = 1990348894] ------------------------------------- [ 1 -1 -1 1 0 0 0 0 0 0 0 0 0 0 0 0] [ 1 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0] [ 1 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0] [ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0] [ 0 0 0 0 1 -1 -1 1 0 0 0 0 0 0 0 0] [ 0 0 0 0 1 0 -1 0 0 0 0 0 0 0 0 0] [ 0 0 0 0 1 -1 0 0 0 0 0 0 0 0 0 0] [ 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0] [ 0 0 0 0 0 0 0 0 1 -1 -1 1 0 0 0 0] [ 0 0 0 0 0 0 0 0 1 0 -1 0 0 0 0 0] [ 0 0 0 0 0 0 0 0 1 -1 0 0 0 0 0 0] [ 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0] [ 0 0 0 0 0 0 0 0 0 0 0 0 1 -1 -1 1] [ 0 0 0 0 0 0 0 0 0 0 0 0 1 0 -1 0] [ 0 0 0 0 0 0 0 0 0 0 0 0 1 -1 0 0] [ 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0] Total time: 0.190 seconds, Total memory usage: 3.24MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Wed Dec 7 17:36:45 2005 Input: K:=RationalField(); A:=Matrix(K,4,4,[[1,1,1,1],[0,1,0,1],[0,0,1,1],[0,0,0,1]]); B1:=Matrix(K,4,4,[[0,-1,1,0],[0,0,0,0],[0,0,0,0],[0,0,0,0]]); B2:=Matrix(K,4,4,[[0,0,0,1],[0,0,0,0],[0,0,0,0],[0,0,0,0]]); B3:=Matrix(K,4,4,[[0,1,0,0],[0,0,0,1],[0,0,0,0],[0,0,0,0]]); B4:=Matrix(K,4,4,[[0,0,0,0],[0,1,-1,0],[0,-1,1,0],[0,0,0,0]]); B5:=Matrix(K,4,4,[[0,1,0,0],[0,0,0,0],[0,0,0,1],[0,0,0,0]]); B6:=Matrix(K,4,4,[[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]); B:=[B1,B2,B3,B4,B5,B6]; R:=PolynomialRing(K,6); P:=0; for i in [1..6] do P:= P + R.i*B[i]; end for; print P; I:=Ideal(Eltseq(P*A*Transpose(P)-A)); print GroebnerBasis(I); Output: Magma V2.11-10 Wed Dec 7 2005 17:36:45 on modular [Seed = 1367699888] ------------------------------------- [ $.6 -$.1 + $.3 + $.5 $.1 $.2] [ 0 $.4 -$.4 + $.6 $.3] [ 0 -$.4 + $.6 $.4 $.5] [ 0 0 0 $.6] [ $.1, $.2, $.3, $.4^2 - $.4*$.6, $.5, $.6^2 - 1 ] Total time: 0.190 seconds, Total memory usage: 3.34MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Wed Dec 7 17:30:58 2005 Input: K:=RationalField(); A:=Matrix(K,4,4,[[1,1,1,1],[0,1,0,1],[0,0,1,1],[0,0,0,1]]); B1:=Matrix(K,4,4,[[0,-1,1,0],[0,0,0,0],[0,0,0,0],[0,0,0,0]]); B2:=Matrix(K,4,4,[[0,0,0,1],[0,0,0,0],[0,0,0,0],[0,0,0,0]]); B3:=Matrix(K,4,4,[[0,1,0,0],[0,0,0,1],[0,0,0,0],[0,0,0,0]]); B4:=Matrix(K,4,4,[[0,0,0,0],[0,1,-1,0],[0,-1,1,0],[0,0,0,0]]); B5:=Matrix(K,4,4,[[0,1,0,0],[0,0,0,0],[0,0,0,1],[0,0,0,0]]); B6:=Matrix(K,4,4,[[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]); B:=[B1,B2,B3,B4,B5,B6]; R:=PolynomialRing(K,6); P:=0; for i in [1..6] do P:= P + R.i*B[i]; end for; I:=Ideal(Eltseq(P*A*Transpose(P)-A)); print GroebnerBasis(I); Output: Magma V2.11-10 Wed Dec 7 2005 17:30:58 on modular [Seed = 148646326] ------------------------------------- [ $.1, $.2, $.3, $.4^2 - $.4*$.6, $.5, $.6^2 - 1 ] Total time: 0.190 seconds, Total memory usage: 3.34MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Wed Dec 7 17:30:25 2005 Input: K:=RationalField(); A:=Matrix(K,4,4,[[1,1,1,1],[0,1,0,1],[0,0,1,1],[0,0,0,1]]); B1:=Matrix(K,4,4,[[0,-1,1,0],[0,0,0,0],[0,0,0,0],[0,0,0,0]]); B2:=Matrix(K,4,4,[[0,0,0,1],[0,0,0,0],[0,0,0,0],[0,0,0,0]]); B3:=Matrix(K,4,4,[[0,1,0,0],[0,0,0,1],[0,0,0,0],[0,0,0,0]]); B4:=Matrix(K,4,4,[[0,0,0,0],[0,1,-1,0],[0,-1,1,0],[0,0,0,0]]); B5:=Matrix(K,4,4,[[0,1,0,0],[0,0,0,0],[0,0,0,1],[0,0,0,0]]); B6:=Matrix(K,4,4,[[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]); B:=[B1,B2,B3,B4,B5,B6]; R:=PolynomialRing(K,6); P:=0; for i in [1..6] do P:= P + R.i*B[i]; end for; Output: Magma V2.11-10 Wed Dec 7 2005 17:30:25 on modular [Seed = 331821710] ------------------------------------- Total time: 0.190 seconds, Total memory usage: 3.24MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Wed Dec 7 17:29:25 2005 Input: K:=RationalField(); A:=Matrix(K,4,4,[[1,1,1,1],[0,1,0,1],[0,0,1,1],[0,0,0,1]]); B1:=Matrix(K,4,4,[[0,-1,1,0],[0,0,0,0],[0,0,0,0],[0,0,0,0]]); B2:=Matrix(K,4,4,[[0,0,0,1],[0,0,0,0],[0,0,0,0],[0,0,0,0]]); B3:=Matrix(K,4,4,[[0,1,0,0],[0,0,0,1],[0,0,0,0],[0,0,0,0]]); B4:=Matrix(K,4,4,[[0,0,0,0],[0,1,-1,0],[0,-1,1,0],[0,0,0,0]]); B5:=Matrix(K,4,4,[[0,1,0,0],[0,0,0,0],[0,0,0,1],[0,0,0,0]]); B6:=Matrix(K,4,4,[[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]); B:=[B1,B2,B3,B4,B5,B6]; for i in [1..6] do print B[i]*A-A*B[i]; end for; Output: Magma V2.11-10 Wed Dec 7 2005 17:29:25 on modular [Seed = 3963145652] ------------------------------------- [0 0 0 0] [0 0 0 0] [0 0 0 0] [0 0 0 0] [0 0 0 0] [0 0 0 0] [0 0 0 0] [0 0 0 0] [0 0 0 0] [0 0 0 0] [0 0 0 0] [0 0 0 0] [0 0 0 0] [0 0 0 0] [0 0 0 0] [0 0 0 0] [0 0 0 0] [0 0 0 0] [0 0 0 0] [0 0 0 0] [0 0 0 0] [0 0 0 0] [0 0 0 0] [0 0 0 0] Total time: 0.180 seconds, Total memory usage: 3.24MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Wed Dec 7 17:23:13 2005 Input: K:=RationalField(); A:=Matrix(K,4,4,[[1,1,1,1],[0,1,0,1],[0,0,1,1],[0,0,0,1]]); R:=PolynomialRing(K,16); P:=Matrix(R,4,4,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9,R.10,R.11,R.12,R.13,R.14,R.15,R.16]); I:=Ideal(Eltseq(P*A-A*P)); print GroebnerBasis(I); Output: Magma V2.11-10 Wed Dec 7 2005 17:23:13 on modular [Seed = 2910424673] ------------------------------------- [ $.1 - $.16, $.2 + $.3 - $.8 - $.12, $.5, $.6 - $.11, $.7 + $.11 - $.16, $.9, $.10 + $.11 - $.16, $.13, $.14, $.15 ] Total time: 0.190 seconds, Total memory usage: 3.34MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Wed Dec 7 17:00:38 2005 Input: K:=RationalField(); A:=Matrix(K,4,4,[[1,0,1,1],[0,1,1,1],[0,0,1,1],[0,0,0,1]]); R:=PolynomialRing(K,16); P:=Matrix(R,4,4,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9,R.10,R.11,R.12,R.13,R.14,R.15,R.16]); I:=Ideal(Eltseq(P*A*Transpose(P)-A)); print GroebnerBasis(I); Output: Magma V2.11-10 Wed Dec 7 2005 17:00:29 on modular [Seed = 1502429497] ------------------------------------- [ $.1 - $.6 + $.8 + $.14 + $.15, $.2 + $.6 - $.13 - 2*$.14 - 2*$.15 - $.16, $.3 - $.8 - $.15, $.4 + $.8 + $.13 + $.14 + 2*$.15, $.5 + $.6 - $.8 - 2*$.13 - 2*$.14 - 3*$.15 - $.16, $.6^2 - 3*$.6*$.14 - 2*$.6*$.15 - $.6*$.16 - $.8*$.13 - $.8*$.16 + 3*$.14^2 + 4*$.14*$.15 + 2*$.14*$.16 + 2*$.15^2 + $.15*$.16 + $.16^2 - 1, $.6*$.8 + $.6*$.14 + $.6*$.15 - $.8*$.14 - $.8*$.15 - $.14^2 - 2*$.14*$.15 - $.15^2, $.6*$.13 - $.6*$.14 - $.8*$.13 - $.8*$.16 + 2*$.14^2 + 3*$.14*$.15 + $.14*$.16 + 2*$.15^2 + $.15*$.16 + $.16^2 - 1, $.6*$.14^2 + 2*$.6*$.14*$.16 + $.6*$.16^2 - $.6 - $.8*$.13*$.15 + $.8*$.13*$.16 - $.8*$.14^2 - $.8*$.14*$.15 - $.8*$.14*$.16 - $.8*$.15^2 - $.8*$.15*$.16 + $.8 - $.13*$.15^2 - $.13*$.16^2 + $.13 + 2*$.14^2*$.15 - 2*$.14^2*$.16 + 3*$.14*$.15^2 - $.14*$.16^2 + $.15^3 - $.16^3 + $.16, $.6*$.14*$.15 - $.6*$.14*$.16 + 1/2*$.6*$.15^2 + 1/2*$.6*$.15*$.16 - 1/2*$.6*$.16^2 + 1/2*$.6 + 1/2*$.8*$.13*$.15 - 1/2*$.8*$.13*$.16 + 1/2*$.8*$.14^2 + $.8*$.14*$.16 + 1/2*$.8*$.15*$.16 - 1/2*$.8 + 1/2*$.13*$.15^2 + 1/2*$.13*$.16^2 - 1/2*$.13 - $.14^3 - 7/2*$.14^2*$.15 + 1/2*$.14^2*$.16 - 4*$.14*$.15^2 - $.14*$.15*$.16 + 1/2*$.14 - 3/2*$.15^3 - 1/2*$.15^2*$.16 - 1/2*$.15*$.16^2 + 1/2*$.15 + 1/2*$.16^3 - 1/2*$.16, $.6*$.14*$.16^2 + $.6*$.14 - 1/2*$.6*$.15^3 + 1/2*$.6*$.15^2*$.16 - 1/2*$.6*$.15*$.16^2 + 3/2*$.6*$.15 + $.6*$.16^3 - $.6*$.16 + 1/2*$.8*$.13*$.15^2 + $.8*$.13*$.16^2 - 1/2*$.8*$.14^3 - 3/2*$.8*$.14^2*$.15 - 1/2*$.8*$.14^2*$.16 - 2*$.8*$.14*$.15^2 - 1/2*$.8*$.14*$.15*$.16 - 3/2*$.8*$.14*$.16^2 + 1/2*$.8*$.14 - 1/2*$.8*$.15^3 - 1/2*$.8*$.15^2*$.16 + $.8*$.16 + 1/2*$.13*$.15^3 - 1/2*$.13*$.15 - $.13*$.16^3 + $.13*$.16 - $.14^3*$.15 + 3/2*$.14^3*$.16 - 5/2*$.14^2*$.15^2 + 5/2*$.14^2*$.15*$.16 - 1/2*$.14^2*$.16^2 - $.14^2 - 2*$.14*$.15^3 + 2*$.14*$.15^2*$.16 - 1/2*$.14*$.15*$.16^2 - 3/2*$.14*$.15 - 1/2*$.14*$.16^3 - 1/2*$.14*$.16 - 1/2*$.15^4 + $.15^3*$.16 + 1/2*$.15^2*$.16^2 - 1/2*$.15^2 - 1/2*$.15*$.16^3 - $.16^4 + $.16^2, $.6*$.15^4 - 2*$.6*$.15^3*$.16 + 3*$.6*$.15^2*$.16^2 - 2*$.6*$.15^2 - 2*$.6*$.15*$.16^3 + 6*$.6*$.15*$.16 + $.6*$.16^4 - 2*$.6*$.16^2 + $.6 - $.8*$.13*$.15^3 + $.8*$.13*$.15 + $.8*$.13*$.16^3 - $.8*$.13*$.16 + 2*$.8*$.14^3*$.15 - 2*$.8*$.14^3*$.16 + 5*$.8*$.14^2*$.15^2 - 3*$.8*$.14^2*$.15*$.16 - $.8*$.14^2*$.16^2 + $.8*$.14^2 + 6*$.8*$.14*$.15^3 - 4*$.8*$.14*$.15^2*$.16 + 2*$.8*$.14*$.15*$.16^2 - 2*$.8*$.14*$.15 - 2*$.8*$.14*$.16^3 + 4*$.8*$.14*$.16 + 2*$.8*$.15^4 - $.8*$.15^3*$.16 - $.8*$.15^2*$.16^2 - $.8*$.15^2 + $.8*$.15*$.16^3 + $.8*$.16^2 - $.8 - $.13*$.15^4 + 2*$.13*$.15^2 - $.13*$.16^4 + 2*$.13*$.16^2 - $.13 + 3*$.14^3*$.15^2 - 4*$.14^3*$.15*$.16 + 3*$.14^3*$.16^2 - 5*$.14^3 + 7*$.14^2*$.15^3 - 7*$.14^2*$.15^2*$.16 + 4*$.14^2*$.15*$.16^2 - 11*$.14^2*$.15 + 2*$.14^2*$.16^3 - 4*$.14^2*$.16 + 6*$.14*$.15^4 - 5*$.14*$.15^3*$.16 + 3*$.14*$.15^2*$.16^2 - 12*$.14*$.15^2 + $.14*$.15*$.16^3 - 8*$.14*$.15*$.16 + $.14*$.16^4 - 5*$.14*$.16^2 + 4*$.14 + 2*$.15^5 - 2*$.15^4*$.16 - 6*$.15^3 + $.15^2*$.16^3 - 3*$.15^2*$.16 - 4*$.15*$.16^2 + 4*$.15 - $.16^5 + 2*$.16^3 - $.16, $.7 + $.8 + $.13 + $.14 + $.15, $.8^2 + $.8*$.13 + $.8*$.14 + 2*$.8*$.15 - $.15*$.16, $.9 - $.13 - $.14 - $.15, $.10 - $.13 - $.14 - $.15, $.11 - $.15 - $.16, $.12 + $.13 + $.14 + $.15, $.13^2 + $.13*$.15 + $.13*$.16 + $.14^2 + $.14*$.15 + $.14*$.16 + $.15^2 + $.15*$.16 + $.16^2 - 1, $.13*$.14 + $.13*$.15 + $.14*$.15 + $.15^2 + $.15*$.16, $.13*$.15*$.16 - $.14^3 - 2*$.14^2*$.15 - $.14^2*$.16 - 2*$.14*$.15^2 - $.14*$.15*$.16 - $.14*$.16^2 + $.14 - $.15^3 + $.15, $.14^4 + 3*$.14^3*$.15 + $.14^3*$.16 + 4*$.14^2*$.15^2 + 2*$.14^2*$.15*$.16 + $.14^2*$.16^2 - $.14^2 + 3*$.14*$.15^3 + 2*$.14*$.15^2*$.16 + $.14*$.15*$.16^2 - 2*$.14*$.15 + $.15^4 + $.15^3*$.16 + $.15^2*$.16^2 - $.15^2 ] Total time: 9.570 seconds, Total memory usage: 22.40MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Wed Dec 7 16:58:26 2005 Input: K:=RationalField(); A:=Matrix(K,4,4,[[1,1,1,1],[0,1,0,1],[0,0,1,1],[0,0,0,1]]); R:=PolynomialRing(K,16); P:=Matrix(R,4,4,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9,R.10,R.11,R.12,R.13,R.14,R.15,R.16]); I:=Ideal(Eltseq(P*A*Transpose(P)-A)); print GroebnerBasis(I); Output: Magma V2.11-10 Wed Dec 7 2005 16:58:22 on modular [Seed = 586562497] ------------------------------------- [ $.1 - $.13 - $.14 - $.15 - $.16, $.2 + $.13 + $.15, $.3 + $.13 + $.14, $.4 - $.13, $.5 + $.12 - $.13, $.6 - $.11 - $.12 - $.14, $.7 + $.11 + $.13 - $.15 - $.16, $.8 + $.12 + $.14 + $.15, $.9 - $.12 - $.13 - $.14 - $.15, $.10 + $.11 + $.12 + $.13 - $.16, $.11^2 + $.11*$.13 - $.11*$.15 - $.11*$.16 - $.13*$.15 - $.14*$.15, $.11*$.12 + $.11*$.15 + $.12*$.13 + $.13*$.15, $.11*$.13*$.15^2 + 2*$.11*$.15^3 + 3*$.11*$.15^2*$.16 + 2*$.11*$.15*$.16^2 + $.11*$.16^3 - $.11*$.16 + 2*$.12*$.13*$.15^2 + 3*$.12*$.14*$.15^2 + 2*$.12*$.14*$.15*$.16 + $.12*$.14*$.16^2 + $.12*$.15^3 + 3*$.12*$.15^2*$.16 + 2*$.12*$.15*$.16^2 + $.12*$.15 + $.12*$.16^3 + $.13*$.15^3 + 2*$.14*$.15^3 + 2*$.14*$.15^2*$.16 + $.14*$.15*$.16^2 - $.15^4 - 2*$.15^3*$.16 - 3*$.15^2*$.16^2 + $.15^2 - 2*$.15*$.16^3 + $.15*$.16 - $.16^4 + $.16^2, $.11*$.14 - $.11*$.15 - $.12*$.13 - $.12*$.15 - $.12*$.16 - $.13*$.15 - $.14*$.15 - $.14*$.16, $.11*$.15^4 + 2*$.11*$.15^3*$.16 + 3*$.11*$.15^2*$.16^2 + 2*$.11*$.15*$.16^3 + $.11*$.16^4 - $.11*$.16^2 + $.12*$.13*$.15^3 + 2*$.12*$.14*$.15^3 + 3*$.12*$.14*$.15^2*$.16 + 2*$.12*$.14*$.15*$.16^2 + $.12*$.14*$.16^3 + $.12*$.15^4 + 2*$.12*$.15^3*$.16 + 3*$.12*$.15^2*$.16^2 + 2*$.12*$.15*$.16^3 + $.12*$.15*$.16 + $.12*$.16^4 + $.13*$.15^4 + 2*$.14*$.15^4 + 3*$.14*$.15^3*$.16 + 2*$.14*$.15^2*$.16^2 + $.14*$.15*$.16^3 - $.15^4*$.16 - 2*$.15^3*$.16^2 - 3*$.15^2*$.16^3 + $.15^2*$.16 - 2*$.15*$.16^4 + $.15*$.16^2 - $.16^5 + $.16^3, $.12^2 + $.12*$.14 + $.12*$.15 + $.14*$.15, $.13^2 + $.13*$.14 + $.13*$.15 + $.14^2 + $.14*$.15 + $.14*$.16 + $.15^2 + $.15*$.16 + $.16^2 - 1, $.13*$.14*$.15 + $.14^2*$.15 + $.14^2*$.16 + $.14*$.15^2 + $.14*$.15*$.16 + $.14*$.16^2 + $.15^2*$.16 + $.15*$.16^2 + $.16^3 - $.16, $.13*$.16 - $.14*$.15, $.14^2*$.15^2 + $.14^2*$.15*$.16 + $.14^2*$.16^2 + $.14*$.15^2*$.16 + $.14*$.15*$.16^2 + $.14*$.16^3 + $.15^2*$.16^2 + $.15*$.16^3 + $.16^4 - $.16^2 ] Total time: 3.500 seconds, Total memory usage: 14.17MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Wed Dec 7 16:58:14 2005 Input: K:=RationalField(); A:=Matrix(K,4,4,[[1,1,1,1],[0,1,0,1],[0,0,1,1],[0,0,0,1]]); R:=PolynomialRing(K,16); P:=Matrix(R,4,4,R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9,R.10,R.11,R.12,R.13,R.14,R.15,R.16]); I:=Ideal(Eltseq(P*A*Transpose(P)-A)); print GroebnerBasis(I); Output: Magma V2.11-10 Wed Dec 7 2005 16:58:12 on modular [Seed = 670252750] ------------------------------------- >> 0,R.11,R.12,R.13,R.14,R.15,R.16]); ^ User error: bad syntax >> I:=Ideal(Eltseq(P*A*Transpose(P)-A)); ^ User error: Identifier 'P' has not been declared or assigned >> print GroebnerBasis(I); ^ User error: Identifier 'I' has not been declared or assigned Total time: 0.200 seconds, Total memory usage: 3.24MB '68.238.' ************** MAGMA ***************** Host 68.238.133.139 (68.238.133.139) Time: Wed Dec 7 15:10:38 2005 Input: [[1,0],[0,1]]*2 Output: Magma V2.11-10 Wed Dec 7 2005 15:10:38 on modular [Seed = 1218202344] ------------------------------------- >> [[1,0],[0,1]]*2 ^ Runtime error in '*': Bad argument types Argument types given: SeqEnum[SeqEnum[RngIntElt]], RngIntElt Total time: 0.190 seconds, Total memory usage: 3.24MB '68.238.' ************** MAGMA ***************** Host 68.238.133.139 (68.238.133.139) Time: Wed Dec 7 15:10:14 2005 Input: ((1,0),(0,1)) Output: Magma V2.11-10 Wed Dec 7 2005 15:10:13 on modular [Seed = 1335578100] ------------------------------------- >> ((1,0),(0,1)) ^ Runtime error in elt< ... >: No permutation group context in which to create cycle Total time: 0.190 seconds, Total memory usage: 3.24MB '68.238.' ************** MAGMA ***************** Host 68.238.133.139 (68.238.133.139) Time: Wed Dec 7 15:09:13 2005 Input: {{1,0},{0,1}}.{{2,0},{0,2}} Output: Magma V2.11-10 Wed Dec 7 2005 15:09:12 on modular [Seed = 47579252] ------------------------------------- >> {{1,0},{0,1}}.{{2,0},{0,2}}; ^ Runtime error in '.': Bad argument types Argument types given: SetEnum[SetEnum], SetEnum[SetEnum] Total time: 0.180 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Wed Dec 7 14:54:21 2005 Input: p := 1461501624496790265145448589920785493717258890819; q := p^2; allconjs := procedure(x) print "x = " * Sprint(x) * ", x^q = " * Sprint(x^q) * ", x^{q^2} = " * Sprint(x^(q^2)); end procedure; trace := function(x) return x + x^q + x^(q^2); end function; norm := function(x) return x^(1 + q + q^2); end function; Fp := GF(p); Fp2 := ExtensionField; xi := 1/(-8 + 8*i); "xi =", xi; Fp12 := ExtensionField; allconjs(1); allconjs(z^2); allconjs(z^4); print "trace(1) = " * Sprint(trace(1)) * ", norm(1) = " * Sprint(norm(1)); print "trace(z^2) = " * Sprint(trace(z^2)) * ", norm(z^2) = " * Sprint(norm(z^2)); print "trace(z^4) = " * Sprint(trace(z^4)) * ", norm(z^4) = " * Sprint(norm(z^4)); Output: Magma V2.11-10 Wed Dec 7 2005 14:54:20 on modular [Seed = 1301885886] ------------------------------------- xi = 1004782366841543307287495905570540026930615487438*i + 1004782366841543307287495905570540026930615487438 x = 1, x^q = 1, x^{q^2} = 1 x = z^2, x^q = 1461501624495162299985421915440573293973338433026*z^2, x^{q^2} = 1627965160026674480212199743920457792*z^2 x = z^4, x^q = 1627965160026674480212199743920457792*z^4, x^{q^2} = 1461501624495162299985421915440573293973338433026*z^4 trace(1) = 3, norm(1) = 1 trace(z^2) = 0, norm(z^2) = 1004782366841543307287495905570540026930615487438*i + 1004782366841543307287495905570540026930615487438 trace(z^4) = 0, norm(z^4) = 239777610269004652875425159283878870062987786775*i Total time: 0.240 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Wed Dec 7 14:54:01 2005 Input: p := 1461501624496790265145448589920785493717258890819; q := p^2; allconjs := procedure(x) print "x = " * Sprint(x); print "x^q = " * Sprint(x^q); print "x^{q^2} = " * Sprint(x^(q^2)); end procedure; trace := function(x) return x + x^q + x^(q^2); end function; norm := function(x) return x^(1 + q + q^2); end function; Fp := GF(p); Fp2 := ExtensionField; xi := 1/(-8 + 8*i); "xi =", xi; Fp12 := ExtensionField; allconjs(1); allconjs(z^2); allconjs(z^4); print "trace(1) = " * Sprint(trace(1)) * ", norm(1) = " * Sprint(norm(1)); print "trace(z^2) = " * Sprint(trace(z^2)) * ", norm(z^2) = " * Sprint(norm(z^2)); print "trace(z^4) = " * Sprint(trace(z^4)) * ", norm(z^4) = " * Sprint(norm(z^4)); Output: Magma V2.11-10 Wed Dec 7 2005 14:54:00 on modular [Seed = 249196745] ------------------------------------- xi = 1004782366841543307287495905570540026930615487438*i + 1004782366841543307287495905570540026930615487438 x = 1 x^q = 1 x^{q^2} = 1 x = z^2 x^q = 1461501624495162299985421915440573293973338433026*z^2 x^{q^2} = 1627965160026674480212199743920457792*z^2 x = z^4 x^q = 1627965160026674480212199743920457792*z^4 x^{q^2} = 1461501624495162299985421915440573293973338433026*z^4 trace(1) = 3, norm(1) = 1 trace(z^2) = 0, norm(z^2) = 1004782366841543307287495905570540026930615487438*i + 1004782366841543307287495905570540026930615487438 trace(z^4) = 0, norm(z^4) = 239777610269004652875425159283878870062987786775*i Total time: 0.240 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Wed Dec 7 14:53:44 2005 Input: p := 1461501624496790265145448589920785493717258890819; q := p^2; allconjs := procedure(x) print "x = " * Sprint(x); print "x^q = " * Sprint(x^q); print "x^{q^2} = " * Sprint(x^(q^2)); end function; trace := procedure(x) return x + x^q + x^(q^2); end function; norm := function(x) return x^(1 + q + q^2); end function; Fp := GF(p); Fp2 := ExtensionField; xi := 1/(-8 + 8*i); "xi =", xi; Fp12 := ExtensionField; allconjs(1); allconjs(z^2); allconjs(z^4); print "trace(1) = " * Sprint(trace(1)) * ", norm(1) = " * Sprint(norm(1)); print "trace(z^2) = " * Sprint(trace(z^2)) * ", norm(z^2) = " * Sprint(norm(z^2)); print "trace(z^4) = " * Sprint(trace(z^4)) * ", norm(z^4) = " * Sprint(norm(z^4)); Output: Magma V2.11-10 Wed Dec 7 2005 14:53:42 on modular [Seed = 3239427871] ------------------------------------- >> end function; ^ User error: bad syntax >> end function; ^ User error: bad syntax xi = 1004782366841543307287495905570540026930615487438*i + 1004782366841543307287495905570540026930615487438 >> allconjs(1); ^ User error: Identifier 'allconjs' has not been declared or assigned >> allconjs(z^2); ^ User error: Identifier 'allconjs' has not been declared or assigned >> allconjs(z^4); ^ User error: Identifier 'allconjs' has not been declared or assigned >> print "trace(1) = " * Sprint(trace(1)) * ", norm(1) = " * Sprint(norm(1)); ^ User error: Identifier 'trace' has not been declared or assigned >> print "trace(z^2) = " * Sprint(trace(z^2)) * ", norm(z^2) = " * Sprint(norm ^ User error: Identifier 'trace' has not been declared or assigned >> print "trace(z^4) = " * Sprint(trace(z^4)) * ", norm(z^4) = " * Sprint(norm ^ User error: Identifier 'trace' has not been declared or assigned Total time: 0.220 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Wed Dec 7 14:53:35 2005 Input: p := 1461501624496790265145448589920785493717258890819; q := p^2; allconjs := procedure(x) print "x = " * Sprint(x); print "x^q = " * Sprint(x^q); print "x^{q^2} = " * Sprint(x^(q^2); end function; trace := procedure(x) return x + x^q + x^(q^2); end function; norm := function(x) return x^(1 + q + q^2); end function; Fp := GF(p); Fp2 := ExtensionField; xi := 1/(-8 + 8*i); "xi =", xi; Fp12 := ExtensionField; allconjs(1); allconjs(z^2); allconjs(z^4); print "trace(1) = " * Sprint(trace(1)) * ", norm(1) = " * Sprint(norm(1)); print "trace(z^2) = " * Sprint(trace(z^2)) * ", norm(z^2) = " * Sprint(norm(z^2)); print "trace(z^4) = " * Sprint(trace(z^4)) * ", norm(z^4) = " * Sprint(norm(z^4)); Output: Magma V2.11-10 Wed Dec 7 2005 14:53:35 on modular [Seed = 3043085074] ------------------------------------- >> print "x^{q^2} = " * Sprint(x^(q^2); ^ User error: bad syntax >> end function; ^ User error: bad syntax >> end function; ^ User error: bad syntax xi = 1004782366841543307287495905570540026930615487438*i + 1004782366841543307287495905570540026930615487438 >> allconjs(1); ^ User error: Identifier 'allconjs' has not been declared or assigned >> allconjs(z^2); ^ User error: Identifier 'allconjs' has not been declared or assigned >> allconjs(z^4); ^ User error: Identifier 'allconjs' has not been declared or assigned >> print "trace(1) = " * Sprint(trace(1)) * ", norm(1) = " * Sprint(norm(1)); ^ User error: Identifier 'trace' has not been declared or assigned >> print "trace(z^2) = " * Sprint(trace(z^2)) * ", norm(z^2) = " * Sprint(norm ^ User error: Identifier 'trace' has not been declared or assigned >> print "trace(z^4) = " * Sprint(trace(z^4)) * ", norm(z^4) = " * Sprint(norm ^ User error: Identifier 'trace' has not been declared or assigned Total time: 0.210 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Wed Dec 7 14:38:51 2005 Input: p := 1461501624496790265145448589920785493717258890819; q := p^2; trace := function(x) return x + x^q + x^(q^2); end function; norm := function(x) return x^(1 + q + q^2); end function; Fp := GF(p); Fp2 := ExtensionField; xi := 1/(-8 + 8*i); "xi =", xi; Fp12 := ExtensionField; print "trace(1) = " * Sprint(trace(1)) * ", norm(1) = " * Sprint(norm(1)); print "trace(z^2) = " * Sprint(trace(z^2)) * ", norm(z^2) = " * Sprint(norm(z^2)); print "trace(z^4) = " * Sprint(trace(z^4)) * ", norm(z^4) = " * Sprint(norm(z^4)); Output: Magma V2.11-10 Wed Dec 7 2005 14:38:50 on modular [Seed = 1485578502] ------------------------------------- xi = 1004782366841543307287495905570540026930615487438*i + 1004782366841543307287495905570540026930615487438 trace(1) = 3, norm(1) = 1 trace(z^2) = 0, norm(z^2) = 1004782366841543307287495905570540026930615487438*i + 1004782366841543307287495905570540026930615487438 trace(z^4) = 0, norm(z^4) = 239777610269004652875425159283878870062987786775*i Total time: 0.220 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Wed Dec 7 14:38:13 2005 Input: p := 1461501624496790265145448589920785493717258890819; q := p^2; trace := function(x) return x + x^q + x^(q^2); end function; norm := function(x) return x^(1 + q + q^2); end function; Fp := GF(p); Fp2 := ExtensionField; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; print "trace(1) = " * Sprint(trace(1)) * ", norm(1) = " * Sprint(norm(1)); print "trace(z^2) = " * Sprint(trace(z^2)) * ", norm(z^2) = " * Sprint(norm(z^2)); print "trace(z^4) = " * Sprint(trace(z^4)) * ", norm(z^4) = " * Sprint(norm(z^4)); Output: Magma V2.11-10 Wed Dec 7 2005 14:38:12 on modular [Seed = 1401367139] ------------------------------------- trace(1) = 3, norm(1) = 1 trace(z^2) = 0, norm(z^2) = 1004782366841543307287495905570540026930615487438*i + 1004782366841543307287495905570540026930615487438 trace(z^4) = 0, norm(z^4) = 239777610269004652875425159283878870062987786775*i Total time: 0.220 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Wed Dec 7 14:34:37 2005 Input: p := 1461501624496790265145448589920785493717258890819; q := p^2; trace := function(x) "=========="; "*** x =", x; "*** x^q =", x^q; "*** x^(q^2) =", x^(q^2); "*** x^(-p) =", x^(-p); "*** x^(p^2) =", x^(p^2); "=========="; return x + x^q + x^(q^2); end function; norm := function(x) return x^(1 - p + p^2); end function; Fp := GF(p); Fp2 := ExtensionField; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; print "trace(1) = " * Sprint(trace(1)) * ", norm(1) = " * Sprint(norm(1)); print "trace(z^2) = " * Sprint(trace(z^2)) * ", norm(z^2) = " * Sprint(norm(z^2)); print "trace(z^4) = " * Sprint(trace(z^4)) * ", norm(z^4) = " * Sprint(norm(z^4)); Output: Magma V2.11-10 Wed Dec 7 2005 14:34:37 on modular [Seed = 738152818] ------------------------------------- ========== *** x = 1 *** x^q = 1 *** x^(q^2) = 1 *** x^(-p) = 1 *** x^(p^2) = 1 ========== trace(1) = 3, norm(1) = 1 ========== *** x = z^2 *** x^q = 1461501624495162299985421915440573293973338433026*z^2 *** x^(q^2) = 1627965160026674480212199743920457792*z^2 *** x^(-p) = (1461501624483766543865235194079087895765895228475*i + 1461501624483766543865235194079087895765895228475)*z^4 *** x^(p^2) = 1461501624495162299985421915440573293973338433026*z^2 ========== trace(z^2) = 0, norm(z^2) = 1461501624495162299985421915440573293973338433027*i\ *z^2 ========== *** x = z^4 *** x^q = 1627965160026674480212199743920457792*z^4 *** x^(q^2) = 1461501624495162299985421915440573293973338433026*z^4 *** x^(-p) = (1461501624483766543865235194079087895765895228483*i + 13023721280213395841697597951363662336)*z^2 *** x^(p^2) = 1627965160026674480212199743920457792*z^4 ========== trace(z^4) = 0, norm(z^4) = 1627965160026674480212199743920457793*z^4 Total time: 0.260 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Wed Dec 7 14:34:15 2005 Input: p := 1461501624496790265145448589920785493717258890819; q := p^2; trace := function(x) "=========="; "*** x =", x; "*** x^q =", x^q; "*** x^(q^2) =", x^(q^2); "*** x^(-p) =", x^(-p); "*** x^(p^2) =", x^(p^2); return x + x^q + x^(q^2); end function; norm := function(x) return x^(1 - p + p^2); end function; Fp := GF(p); Fp2 := ExtensionField; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; print "trace(1) = " * Sprint(trace(1)) * ", norm(1) = " * Sprint(norm(1)); print "trace(z^2) = " * Sprint(trace(z^2)) * ", norm(z^2) = " * Sprint(norm(z^2)); print "trace(z^4) = " * Sprint(trace(z^4)) * ", norm(z^4) = " * Sprint(norm(z^4)); Output: Magma V2.11-10 Wed Dec 7 2005 14:34:14 on modular [Seed = 788157452] ------------------------------------- ========== *** x = 1 *** x^q = 1 *** x^(q^2) = 1 *** x^(-p) = 1 *** x^(p^2) = 1 trace(1) = 3, norm(1) = 1 ========== *** x = z^2 *** x^q = 1461501624495162299985421915440573293973338433026*z^2 *** x^(q^2) = 1627965160026674480212199743920457792*z^2 *** x^(-p) = (1461501624483766543865235194079087895765895228475*i + 1461501624483766543865235194079087895765895228475)*z^4 *** x^(p^2) = 1461501624495162299985421915440573293973338433026*z^2 trace(z^2) = 0, norm(z^2) = 1461501624495162299985421915440573293973338433027*i\ *z^2 ========== *** x = z^4 *** x^q = 1627965160026674480212199743920457792*z^4 *** x^(q^2) = 1461501624495162299985421915440573293973338433026*z^4 *** x^(-p) = (1461501624483766543865235194079087895765895228483*i + 13023721280213395841697597951363662336)*z^2 *** x^(p^2) = 1627965160026674480212199743920457792*z^4 trace(z^4) = 0, norm(z^4) = 1627965160026674480212199743920457793*z^4 Total time: 0.260 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Wed Dec 7 14:32:25 2005 Input: p := 1461501624496790265145448589920785493717258890819; q := p^2; trace := function(x) return x + x^q + x^(q^2); end function; norm := function(x) return x^(1 - p + p^2); end function; Fp := GF(p); Fp2 := ExtensionField; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; print "trace(1) = " * Sprint(trace(1)) * ", norm(1) = " * Sprint(norm(1)); print "trace(z^2) = " * Sprint(trace(z^2)) * ", norm(z^2) = " * Sprint(norm(z^2)); print "trace(z^4) = " * Sprint(trace(z^4)) * ", norm(z^4) = " * Sprint(norm(z^4)); Output: Magma V2.11-10 Wed Dec 7 2005 14:32:25 on modular [Seed = 552885245] ------------------------------------- trace(1) = 3, norm(1) = 1 trace(z^2) = 0, norm(z^2) = 1461501624495162299985421915440573293973338433027*i\ *z^2 trace(z^4) = 0, norm(z^4) = 1627965160026674480212199743920457793*z^4 Total time: 0.210 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Wed Dec 7 14:32:06 2005 Input: p := 1461501624496790265145448589920785493717258890819; q := p^2; trace := function(x) return x + x^(-p) + x^(p^2); end function; norm := function(x) return x^(1 - p + p^2); end function; Fp := GF(p); Fp2 := ExtensionField; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; print "trace(1) = " * Sprint(trace(1)) * ", norm(1) = " * Sprint(norm(1)); print "trace(z^2) = " * Sprint(trace(z^2)) * ", norm(z^2) = " * Sprint(norm(z^2)); print "trace(z^4) = " * Sprint(trace(z^4)) * ", norm(z^4) = " * Sprint(norm(z^4)); Output: Magma V2.11-10 Wed Dec 7 2005 14:32:06 on modular [Seed = 988692888] ------------------------------------- trace(1) = 3, norm(1) = 1 trace(z^2) = (1461501624483766543865235194079087895765895228475*i + 1461501624483766543865235194079087895765895228475)*z^4 + 1461501624495162299985421915440573293973338433027*z^2, norm(z^2) = 1461501624495162299985421915440573293973338433027*i*z^2 trace(z^4) = 1627965160026674480212199743920457793*z^4 + (1461501624483766543865235194079087895765895228483*i + 13023721280213395841697597951363662336)*z^2, norm(z^4) = 1627965160026674480212199743920457793*z^4 Total time: 0.240 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Wed Dec 7 14:31:30 2005 Input: p := 1461501624496790265145448589920785493717258890819; q := p^2; trace := function(x) return x + x^q + x^(q^2); end function; norm := function(x) return x^(1 - p + p^2); end function; Fp := GF(p); Fp2 := ExtensionField; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; print "trace(1) = " * Sprint(trace(1)) * ", norm(1) = " * Sprint(norm(1)); print "trace(z^2) = " * Sprint(trace(z^2)) * ", norm(z^2) = " * Sprint(norm(z^2)); print "trace(z^4) = " * Sprint(trace(z^4)) * ", norm(z^4) = " * Sprint(norm(z^4)); Output: Magma V2.11-10 Wed Dec 7 2005 14:31:30 on modular [Seed = 1038698171] ------------------------------------- trace(1) = 3, norm(1) = 1 trace(z^2) = 0, norm(z^2) = 1461501624495162299985421915440573293973338433027*i\ *z^2 trace(z^4) = 0, norm(z^4) = 1627965160026674480212199743920457793*z^4 Total time: 0.220 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Wed Dec 7 14:30:52 2005 Input: p := 1461501624496790265145448589920785493717258890819; q := p^2; trace := function(x) return x + x^q + x^(q^2); end function; norm := function(x) return x^(1 - q + q^2); end function; Fp := GF(p); Fp2 := ExtensionField; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; print "trace(1) = " * Sprint(trace(1)) * ", norm(1) = " * Sprint(norm(1)); print "trace(z^2) = " * Sprint(trace(z^2)) * ", norm(z^2) = " * Sprint(norm(z^2)); print "trace(z^4) = " * Sprint(trace(z^4)) * ", norm(z^4) = " * Sprint(norm(z^4)); Output: Magma V2.11-10 Wed Dec 7 2005 14:30:51 on modular [Seed = 820267855] ------------------------------------- trace(1) = 3, norm(1) = 1 trace(z^2) = 0, norm(z^2) = 1461501624495162299985421915440573293973338433026*z\ ^2 trace(z^4) = 0, norm(z^4) = 1627965160026674480212199743920457792*z^4 Total time: 0.230 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Wed Dec 7 14:26:29 2005 Input: p := 1461501624496790265145448589920785493717258890819; trace := function(x) return x + x^(p^2) + x^(p^4); end function; norm := function(x) return x * x^(p^2) * x^(p^4); end function; Fp := GF(p); Fp2 := ExtensionField; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; print "trace(1) = " * Sprint(trace(1)) * ", norm(1) = " * Sprint(norm(1)); print "trace(z^2) = " * Sprint(trace(z^2)) * ", norm(z^2) = " * Sprint(norm(z^2)); print "trace(z^4) = " * Sprint(trace(z^4)) * ", norm(z^4) = " * Sprint(norm(z^4)); Output: Magma V2.11-10 Wed Dec 7 2005 14:26:29 on modular [Seed = 182353531] ------------------------------------- trace(1) = 3, norm(1) = 1 trace(z^2) = 0, norm(z^2) = 1004782366841543307287495905570540026930615487438*i + 1004782366841543307287495905570540026930615487438 trace(z^4) = 0, norm(z^4) = 239777610269004652875425159283878870062987786775*i Total time: 0.230 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Wed Dec 7 14:26:17 2005 Input: p := 1461501624496790265145448589920785493717258890819; trace := function(x) return x + x^(p^2) + x^(p^4); end function; norm := function(x) return x * x^(p^2) * x^(p^4); end function; Fp := GF(p); Fp2 := ExtensionField; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; print "trace(1) =" * Sprint(trace(1)) * ", norm(1) =" * Sprint(norm(1)); print "trace(z^2) =" * Sprint(trace(z^2)) * ", norm(z^2) =" * Sprint(norm(z^2)); print "trace(z^4) =" * Sprint(trace(z^4)) * ", norm(z^4) =" * Sprint(norm(z^4)); Output: Magma V2.11-10 Wed Dec 7 2005 14:26:17 on modular [Seed = 232356160] ------------------------------------- trace(1) =3, norm(1) =1 trace(z^2) =0, norm(z^2) =1004782366841543307287495905570540026930615487438*i + 1004782366841543307287495905570540026930615487438 trace(z^4) =0, norm(z^4) =239777610269004652875425159283878870062987786775*i Total time: 0.230 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Wed Dec 7 14:25:28 2005 Input: p := 1461501624496790265145448589920785493717258890819; trace := function(x) return x + x^(p^2) + x^(p^4); end function; norm := function(x) return x * x^(p^2) * x^(p^4); end function; Fp := GF(p); Fp2 := ExtensionField; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; "trace(1) =", trace(1), ", norm(1) =", norm(1); "trace(z^2) =", trace(z^2), ", norm(z^2) =", norm(z^2); "trace(z^4) =", trace(z^4), ", norm(z^4) =", norm(z^4); Output: Magma V2.11-10 Wed Dec 7 2005 14:25:27 on modular [Seed = 131298107] ------------------------------------- trace(1) = 3 , norm(1) = 1 trace(z^2) = 0 , norm(z^2) = 1004782366841543307287495905570540026930615487438*i + 1004782366841543307287495905570540026930615487438 trace(z^4) = 0 , norm(z^4) = 239777610269004652875425159283878870062987786775*i Total time: 0.230 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Wed Dec 7 14:16:46 2005 Input: p := 1461501624496790265145448589920785493717258890819; trace := function(x) return x + x^(-p^2) + x^(p^4); end function; norm := function(x) return x * x^(-p^2) * x^(p^4); end function; Fp := GF(p); Fp2 := ExtensionField; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; "trace(1) =", trace(1), ", norm(1) =", norm(1); "trace(z^2) =", trace(z^2), ", norm(z^2) =", norm(z^2); "trace(z^4) =", trace(z^4), ", norm(z^4) =", norm(z^4); Output: Magma V2.11-10 Wed Dec 7 2005 14:16:46 on modular [Seed = 331834649] ------------------------------------- trace(1) = 3 , norm(1) = 1 trace(z^2) = (13023721280213395841697597951363662336*i + 1461501624483766543865235194079087895765895228483)*z^4 + 1627965160026674480212199743920457793*z^2 , norm(z^2) = 1461501624495162299985421915440573293973338433026*z^2 trace(z^4) = 1461501624495162299985421915440573293973338433027*z^4 + (1461501624483766543865235194079087895765895228475*i + 13023721280213395841697597951363662344)*z^2 , norm(z^4) = 1627965160026674480212199743920457792*z^4 Total time: 0.270 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Wed Dec 7 14:16:22 2005 Input: p := 1461501624496790265145448589920785493717258890819; trace := function(x) return x + x^(p^2) + x^(p^4); end function; norm := function(x) return x * x^(-p^2) * x^(p^4); end function; Fp := GF(p); Fp2 := ExtensionField; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; "trace(1) =", trace(1), ", norm(1) =", norm(1); "trace(z^2) =", trace(z^2), ", norm(z^2) =", norm(z^2); "trace(z^4) =", trace(z^4), ", norm(z^4) =", norm(z^4); Output: Magma V2.11-10 Wed Dec 7 2005 14:16:22 on modular [Seed = 381837316] ------------------------------------- trace(1) = 3 , norm(1) = 1 trace(z^2) = 0 , norm(z^2) = 1461501624495162299985421915440573293973338433026*z^2 trace(z^4) = 0 , norm(z^4) = 1627965160026674480212199743920457792*z^4 Total time: 0.250 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Wed Dec 7 14:14:19 2005 Input: p := 1461501624496790265145448589920785493717258890819; trace := function(x) return x + x^(p^2) + x^(p^4); end function; norm := function(x) return x * x^(p^2) * x^(p^4); end function; Fp := GF(p); Fp2 := ExtensionField; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; "trace(1) =", trace(1), ", norm(1) =", norm(1); "trace(z^2) =", trace(z^2), ", norm(z^2) =", norm(z^2); "trace(z^4) =", trace(z^4), ", norm(z^4) =", norm(z^4); Output: Magma V2.11-10 Wed Dec 7 2005 14:14:18 on modular [Seed = 3896287094] ------------------------------------- trace(1) = 3 , norm(1) = 1 trace(z^2) = 0 , norm(z^2) = 1004782366841543307287495905570540026930615487438*i + 1004782366841543307287495905570540026930615487438 trace(z^4) = 0 , norm(z^4) = 239777610269004652875425159283878870062987786775*i Total time: 0.230 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Wed Dec 7 14:12:39 2005 Input: F := FunctionField(Rationals(), 4); (x - h0)*(x - h1)*(x - h2) eq x^3 - (h0 + h1 + h2)*x^2 + (h0*h1 + h0*h2 + h1*h2)*x - (h0*h1*h2); Output: Magma V2.11-10 Wed Dec 7 2005 14:12:39 on modular [Seed = 3979975391] ------------------------------------- true Total time: 0.190 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Wed Dec 7 14:12:00 2005 Input: F := FunctionField(Rationals(), 4); (x - h0)*(x - h1)*(x - h2); Output: Magma V2.11-10 Wed Dec 7 2005 14:11:59 on modular [Seed = 3878917901] ------------------------------------- x^3 - x^2*h0 - x^2*h1 - x^2*h2 + x*h0*h1 + x*h0*h2 + x*h1*h2 - h0*h1*h2 Total time: 0.190 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Wed Dec 7 14:11:50 2005 Input: F := FunctionField(Rationals(), 4); (x - h0)*(x - h1)*(x - h2); Output: Magma V2.11-10 Wed Dec 7 2005 14:11:49 on modular [Seed = 4163667992] ------------------------------------- -h0*h1*h2 + h0*h1*x + h0*h2*x - h0*x^2 + h1*h2*x - h1*x^2 - h2*x^2 + x^3 Total time: 0.190 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Wed Dec 7 13:55:46 2005 Input: p := 1461501624496790265145448589920785493717258890819; trace := function(x) return x + x^(p^2) + x^(p^4); end function; Fp := GF(p); Fp2 := ExtensionField; xi := 1/(-8 + 8*i); "3*xi =", 3*xi; Fp12 := ExtensionField; "trace(1) =", trace(1); "trace(z^2) =", trace(z^2); "trace(z^4) =", trace(z^4); "trace(z^6) =", trace(z^6); "trace(z^8) =", trace(z^8); Output: Magma V2.11-10 Wed Dec 7 2005 13:55:46 on modular [Seed = 2287795119] ------------------------------------- 3*xi = 91343851531049391571590536870049093357328680676*i + 91343851531049391571590536870049093357328680676 trace(1) = 3 trace(z^2) = 0 trace(z^4) = 0 trace(z^6) = 91343851531049391571590536870049093357328680676*i + 91343851531049391571590536870049093357328680676 trace(z^8) = 0 Total time: 0.230 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Wed Dec 7 13:55:27 2005 Input: p := 1461501624496790265145448589920785493717258890819; trace := function(x) return x + x^(p^2) + x^(p^4); end function; Fp := GF(p); Fp2 := ExtensionField; xi := 1/(-8 + 8*i); "xi =", xi; Fp12 := ExtensionField; "trace(1) =", trace(1); "trace(z^2) =", trace(z^2); "trace(z^4) =", trace(z^4); "trace(z^6) =", trace(z^6); "trace(z^8) =", trace(z^8); Output: Magma V2.11-10 Wed Dec 7 2005 13:55:27 on modular [Seed = 2371485319] ------------------------------------- xi = 1004782366841543307287495905570540026930615487438*i + 1004782366841543307287495905570540026930615487438 trace(1) = 3 trace(z^2) = 0 trace(z^4) = 0 trace(z^6) = 91343851531049391571590536870049093357328680676*i + 91343851531049391571590536870049093357328680676 trace(z^8) = 0 Total time: 0.240 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Wed Dec 7 13:53:46 2005 Input: p := 1461501624496790265145448589920785493717258890819; trace := function(x) return x + x^(p^2) + x^(p^4); end function; Fp := GF(p); Fp2 := ExtensionField; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; "trace(1) =", trace(1); "trace(z^2) =", trace(z^2); "trace(z^4) =", trace(z^4); "trace(z^6) =", trace(z^6); "trace(z^8) =", trace(z^8); Output: Magma V2.11-10 Wed Dec 7 2005 13:53:46 on modular [Seed = 2672554800] ------------------------------------- trace(1) = 3 trace(z^2) = 0 trace(z^4) = 0 trace(z^6) = 91343851531049391571590536870049093357328680676*i + 91343851531049391571590536870049093357328680676 trace(z^8) = 0 Total time: 0.240 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Wed Dec 7 13:53:20 2005 Input: p := 115792089237314936872688561244471742058375878355761205198700409522629664518163; trace := function(x) return x + x^(p^2) + x^(p^4); end function; Fp := GF(p); Fp2 := ExtensionField; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; "trace(1) =", trace(1); "trace(z^2) =", trace(z^2); "trace(z^4) =", trace(z^4); "trace(z^6) =", trace(z^6); "trace(z^8) =", trace(z^8); Output: Magma V2.11-10 Wed Dec 7 2005 13:53:19 on modular [Seed = 1790838496] ------------------------------------- trace(1) = 3 trace(z^2) = 0 trace(z^4) = 0 trace(z^6) = 723700557733218355454303507777948387864849239723507532491877559516\ 4354032385*i + 723700557733218355454303507777948387864849239723507532491877\ 5595164354032385 trace(z^8) = 0 Total time: 0.260 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Wed Dec 7 13:29:59 2005 Input: p := 115792089237314936872688561244471742058375878355761205198700409522629664518163; trace := function(x) return x + x^(p^2) + x^(p^4); end function; Fp := GF(p); Fp2 := ExtensionField; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; "trace(1) =", trace(1); "trace(i) =", trace(i); "trace(z^2) =", trace(z^2); "trace(z^4) =", trace(z^4); Output: Magma V2.11-10 Wed Dec 7 2005 13:29:58 on modular [Seed = 1167142006] ------------------------------------- trace(1) = 3 trace(i) = 3*i trace(z^2) = 0 trace(z^4) = 0 Total time: 0.230 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Wed Dec 7 13:28:52 2005 Input: p := 115792089237314936872688561244471742058375878355761205198700409522629664518163; trace := function(x) return x + x^(p^2) + x^(p^4); end function; Fp := GF(p); Fp2 := ExtensionField; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; "trace(1) =", trace(1); "trace(z^2) =", trace(z^2); "trace(z^4) =", trace(z^4); Output: Magma V2.11-10 Wed Dec 7 2005 13:28:51 on modular [Seed = 1569273771] ------------------------------------- trace(1) = 3 trace(z^2) = 0 trace(z^4) = 0 Total time: 0.240 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Wed Dec 7 13:28:29 2005 Input: p := 26959946667149205758383469736921695435015736735261155141423417423923; trace := function(x) return x + x^(p^2) + x^(p^4); end function; Fp := GF(p); Fp2 := ExtensionField; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; "trace(1) =", trace(1); "trace(z^2) =", trace(z^2); "trace(z^4) =", trace(z^4); Output: Magma V2.11-10 Wed Dec 7 2005 13:28:29 on modular [Seed = 1451375758] ------------------------------------- trace(1) = 3 trace(z^2) = 0 trace(z^4) = 0 Total time: 0.230 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Wed Dec 7 13:28:13 2005 Input: p := 6277101719531269400517043710060892862318604713139674509723; trace := function(x) return x + x^(p^2) + x^(p^4); end function; Fp := GF(p); Fp2 := ExtensionField; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; "trace(1) =", trace(1); "trace(z^2) =", trace(z^2); "trace(z^4) =", trace(z^4); Output: Magma V2.11-10 Wed Dec 7 2005 13:28:12 on modular [Seed = 738148738] ------------------------------------- trace(1) = 3 trace(z^2) = 0 trace(z^4) = 0 Total time: 0.220 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Wed Dec 7 13:27:53 2005 Input: p := 1461501624496790265145448589920785493717258890819; trace := function(x) return x + x^(p^2) + x^(p^4); end function; Fp := GF(p); Fp2 := ExtensionField; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; "trace(1) =", trace(1); "trace(z^2) =", trace(z^2); "trace(z^4) =", trace(z^4); Output: Magma V2.11-10 Wed Dec 7 2005 13:27:53 on modular [Seed = 754468496] ------------------------------------- trace(1) = 3 trace(z^2) = 0 trace(z^4) = 0 Total time: 0.210 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Wed Dec 7 13:27:25 2005 Input: trace := function(x, p) return x + x^(p^2) + x^(p^4); end function; p := 1461501624496790265145448589920785493717258890819; Fp := GF(p); Fp2 := ExtensionField; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; for j := 0 to 6 do j, ":", (z^4)^(p^j); end for; "trace(1) =", trace(1, p); "trace(z^2) =", trace(z^2, p); "trace(z^4) =", trace(z^4, p); Output: Magma V2.11-10 Wed Dec 7 2005 13:27:24 on modular [Seed = 569723832] ------------------------------------- 0 : z^4 1 : 1627965160026674480212199743920457793*z^4 2 : 1627965160026674480212199743920457792*z^4 3 : 1461501624496790265145448589920785493717258890818*z^4 4 : 1461501624495162299985421915440573293973338433026*z^4 5 : 1461501624495162299985421915440573293973338433027*z^4 6 : z^4 trace(1) = 3 trace(z^2) = 0 trace(z^4) = 0 Total time: 0.230 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Wed Dec 7 13:26:10 2005 Input: trace := function(x, p) return x + x^(p^2) + x^(p^4); end function; p := 1461501624496790265145448589920785493717258890819; t := 1208925814305217958863207; u := 448873741399; //a := 18*u^3 + 18*u^2 + 9*u + 1; a; Fp := GF(p); Fp2 := ExtensionField; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; for j := 0 to 6 do j, ":", (z^4)^(p^j); end for; "trace(1) =", trace(1, p); "trace(z^2) =", trace(z^2, p); "trace(z^4) =", trace(z^4, p); Output: Magma V2.11-10 Wed Dec 7 2005 13:26:10 on modular [Seed = 1005539692] ------------------------------------- 0 : z^4 1 : 1627965160026674480212199743920457793*z^4 2 : 1627965160026674480212199743920457792*z^4 3 : 1461501624496790265145448589920785493717258890818*z^4 4 : 1461501624495162299985421915440573293973338433026*z^4 5 : 1461501624495162299985421915440573293973338433027*z^4 6 : z^4 trace(1) = 3 trace(z^2) = 0 trace(z^4) = 0 Total time: 0.230 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Wed Dec 7 13:25:05 2005 Input: trace := function(x, p) return x + x^(p^2) + x^(p^4); end function; p := 1461501624496790265145448589920785493717258890819; t := 1208925814305217958863207; u := 448873741399; a := 18*u^3 + 18*u^2 + 9*u + 1; a; Fp := GF(p); Fp2 := ExtensionField; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; for j := 0 to 6 do j, ":", (z^4)^(p^j); end for; u0 := 1; u1 := u0^(p^2); u2 := u0^(p^4); "u0 =", u0; "u1 =", u1; "u2 =", u2; trace(u0, p); trace(u1, p); trace(u2, p); Output: Magma V2.11-10 Wed Dec 7 2005 13:25:05 on modular [Seed = 1021859368] ------------------------------------- 1627965160026674480212199743920457792 0 : z^4 1 : 1627965160026674480212199743920457793*z^4 2 : 1627965160026674480212199743920457792*z^4 3 : 1461501624496790265145448589920785493717258890818*z^4 4 : 1461501624495162299985421915440573293973338433026*z^4 5 : 1461501624495162299985421915440573293973338433027*z^4 6 : z^4 u0 = 1 u1 = 1 u2 = 1 3 3 3 Total time: 0.230 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Wed Dec 7 13:24:27 2005 Input: trace := function(x, p) return x + x^(p^2) + x^(p^4); end function; p := 1461501624496790265145448589920785493717258890819; t := 1208925814305217958863207; u := 448873741399; a := 18*u^3 + 18*u^2 + 9*u + 1; a; Fp := GF(p); Fp2 := ExtensionField; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; for j := 0 to 6 do j, ":", (z^4)^(p^j); end for; u0 := 1 + z^2 + z^4; u1 := u0^(p^2); u2 := u0^(p^4); "u0 =", u0; "u1 =", u1; "u2 =", u2; trace(u0, p); trace(u1, p); trace(u2, p); Output: Magma V2.11-10 Wed Dec 7 2005 13:24:27 on modular [Seed = 837114623] ------------------------------------- 1627965160026674480212199743920457792 0 : z^4 1 : 1627965160026674480212199743920457793*z^4 2 : 1627965160026674480212199743920457792*z^4 3 : 1461501624496790265145448589920785493717258890818*z^4 4 : 1461501624495162299985421915440573293973338433026*z^4 5 : 1461501624495162299985421915440573293973338433027*z^4 6 : z^4 u0 = z^4 + z^2 + 1 u1 = 1627965160026674480212199743920457792*z^4 + 1461501624495162299985421915440573293973338433026*z^2 + 1 u2 = 1461501624495162299985421915440573293973338433026*z^4 + 1627965160026674480212199743920457792*z^2 + 1 3 3 3 Total time: 0.430 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Wed Dec 7 13:23:32 2005 Input: trace := function(x, p) return x + x^(p^2) + x^(p^4); end function; p := 1461501624496790265145448589920785493717258890819; t := 1208925814305217958863207; u := 448873741399; a := 18*u^3 + 18*u^2 + 9*u + 1; a; Fp := GF(p); Fp2 := ExtensionField; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; for j := 0 to 6 do j, ":", (z^4)^(p^j); end for; u0 := (5 + 3*i) + (11 - 2*i)*z^2 + (47 + i)*z^4; u1 := u0^(p^2); u2 := u0^(p^4); "u0 =", u0; "u1 =", u1; "u2 =", u2; trace(u0, p); trace(u1, p); trace(u2, p); Output: Magma V2.11-10 Wed Dec 7 2005 13:23:31 on modular [Seed = 182346172] ------------------------------------- 1627965160026674480212199743920457792 0 : z^4 1 : 1627965160026674480212199743920457793*z^4 2 : 1627965160026674480212199743920457792*z^4 3 : 1461501624496790265145448589920785493717258890818*z^4 4 : 1461501624495162299985421915440573293973338433026*z^4 5 : 1461501624495162299985421915440573293973338433027*z^4 6 : z^4 u0 = (i + 47)*z^4 + (1461501624496790265145448589920785493717258890817*i + 11)*z^2 + 3*i + 5 u1 = (1627965160026674480212199743920457792*i + 76514362521253700569973387964261516224)*z^4 + (3255930320053348960424399487840915586*i + 1461501624478882648385155170638451296534133855096)*z^2 + 3*i + 5 u2 = (1461501624495162299985421915440573293973338433026*i + 1461501624420275902624194889350812105752997374548)*z^4 + (1461501624493534334825395240960361094229417975235*i + 17907616760293419282334197183125035712)*z^2 + 3*i + 5 9*i + 15 9*i + 15 9*i + 15 Total time: 0.460 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Wed Dec 7 13:22:04 2005 Input: trace := function(x, p) return x + x^(p^2) + x^(p^4); end function; p := 1461501624496790265145448589920785493717258890819; t := 1208925814305217958863207; u := 448873741399; a := 18*u^3 + 18*u^2 + 9*u + 1; a; Fp := GF(p); Fp2 := ExtensionField; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; for j := 0 to 6 do j, ":", (z^4)^(p^j); end for; u0 := (5 + 3*i) + (11 - 2*i)*z^2 + (47 + i)*z^4; u1 := u0^(p^2); u2 := u0^(p^4); "u0 =", u0; "u1 =", u1, ":", u1 eq -(a+1)*u0; "u2 =", u2, ":", u2 eq a*u0; p-a; u0 + u1 + u2; trace(u0, p); trace(u1, p); trace(u2, p); Output: Magma V2.11-10 Wed Dec 7 2005 13:22:04 on modular [Seed = 47606107] ------------------------------------- 1627965160026674480212199743920457792 0 : z^4 1 : 1627965160026674480212199743920457793*z^4 2 : 1627965160026674480212199743920457792*z^4 3 : 1461501624496790265145448589920785493717258890818*z^4 4 : 1461501624495162299985421915440573293973338433026*z^4 5 : 1461501624495162299985421915440573293973338433027*z^4 6 : z^4 u0 = (i + 47)*z^4 + (1461501624496790265145448589920785493717258890817*i + 11)*z^2 + 3*i + 5 u1 = (1627965160026674480212199743920457792*i + 76514362521253700569973387964261516224)*z^4 + (3255930320053348960424399487840915586*i + 1461501624478882648385155170638451296534133855096)*z^2 + 3*i + 5 : false u2 = (1461501624495162299985421915440573293973338433026*i + 1461501624420275902624194889350812105752997374548)*z^4 + (1461501624493534334825395240960361094229417975235*i + 17907616760293419282334197183125035712)*z^2 + 3*i + 5 : false 1461501624495162299985421915440573293973338433027 9*i + 15 9*i + 15 9*i + 15 9*i + 15 Total time: 0.460 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Wed Dec 7 13:20:47 2005 Input: trace := function(x, p) return x + x^(p^2) + x^(p^4); end function; p := 1461501624496790265145448589920785493717258890819; t := 1208925814305217958863207; u := 448873741399; a := 18*u^3 + 18*u^2 + 9*u + 1; a; Fp := GF(p); Fp2 := ExtensionField; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; for j := 0 to 6 do j, ":", (z^4)^(p^j); end for; u0 := z^2 + z^4; u1 := u0^(p^2); u2 := u0^(p^4); "u0 =", u0; "u1 =", u1, ":", u1 eq -(a+1)*u0; "u2 =", u2, ":", u2 eq a*u0; p-a; u0 + u1 + u2; trace(u0, p); trace(u1, p); trace(u2, p); Output: Magma V2.11-10 Wed Dec 7 2005 13:20:46 on modular [Seed = 131294188] ------------------------------------- 1627965160026674480212199743920457792 0 : z^4 1 : 1627965160026674480212199743920457793*z^4 2 : 1627965160026674480212199743920457792*z^4 3 : 1461501624496790265145448589920785493717258890818*z^4 4 : 1461501624495162299985421915440573293973338433026*z^4 5 : 1461501624495162299985421915440573293973338433027*z^4 6 : z^4 u0 = z^4 + z^2 u1 = 1627965160026674480212199743920457792*z^4 + 1461501624495162299985421915440573293973338433026*z^2 : false u2 = 1461501624495162299985421915440573293973338433026*z^4 + 1627965160026674480212199743920457792*z^2 : false 1461501624495162299985421915440573293973338433027 0 0 0 0 Total time: 0.420 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Wed Dec 7 13:19:47 2005 Input: trace := function(x, p) return x + x^(p^2) + x^(p^4); end function; p := 1461501624496790265145448589920785493717258890819; t := 1208925814305217958863207; u := 448873741399; a := 18*u^3 + 18*u^2 + 9*u + 1; a; Fp := GF(p); Fp2 := ExtensionField; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; for j := 0 to 6 do j, ":", (z^4)^(p^j); end for; u0 := z^2; u1 := u0^(p^2); u2 := u0^(p^4); "u0 =", u0; "u1 =", u1, ":", u1 eq -(a+1)*u0; "u2 =", u2, ":", u2 eq a*u0; p-a; u0 + u1 + u2; trace(u0, p); trace(u1, p); trace(u2, p); Output: Magma V2.11-10 Wed Dec 7 2005 13:19:47 on modular [Seed = 416054478] ------------------------------------- 1627965160026674480212199743920457792 0 : z^4 1 : 1627965160026674480212199743920457793*z^4 2 : 1627965160026674480212199743920457792*z^4 3 : 1461501624496790265145448589920785493717258890818*z^4 4 : 1461501624495162299985421915440573293973338433026*z^4 5 : 1461501624495162299985421915440573293973338433027*z^4 6 : z^4 u0 = z^2 u1 = 1461501624495162299985421915440573293973338433026*z^2 : true u2 = 1627965160026674480212199743920457792*z^2 : true 1461501624495162299985421915440573293973338433027 0 0 0 0 Total time: 0.250 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Wed Dec 7 13:19:37 2005 Input: trace := function(x, p) return x + x^(p^2) + x^(p^4); end function; p := 1461501624496790265145448589920785493717258890819; t := 1208925814305217958863207; u := 448873741399; a := 18*u^3 + 18*u^2 + 9*u + 1; a; Fp := GF(p); Fp2 := ExtensionField; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; for j := 0 to 6 do j, ":", (z^4)^(p^j); end for; u0 := z^2; u1 := u0^(p^2); u2 := u0^(p^4); "u0 =", u0; "u1 =", u1, ":", u1 eq (-a-1)*u0; "u2 =", u2, ":", u2 eq a*u0; p-a; u0 + u1 + u2; trace(u0, p); trace(u1, p); trace(u2, p); Output: Magma V2.11-10 Wed Dec 7 2005 13:19:36 on modular [Seed = 499743152] ------------------------------------- 1627965160026674480212199743920457792 0 : z^4 1 : 1627965160026674480212199743920457793*z^4 2 : 1627965160026674480212199743920457792*z^4 3 : 1461501624496790265145448589920785493717258890818*z^4 4 : 1461501624495162299985421915440573293973338433026*z^4 5 : 1461501624495162299985421915440573293973338433027*z^4 6 : z^4 u0 = z^2 u1 = 1461501624495162299985421915440573293973338433026*z^2 : true u2 = 1627965160026674480212199743920457792*z^2 : true 1461501624495162299985421915440573293973338433027 0 0 0 0 Total time: 0.250 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Wed Dec 7 13:19:30 2005 Input: trace := function(x, p) return x + x^(p^2) + x^(p^4); end function; p := 1461501624496790265145448589920785493717258890819; t := 1208925814305217958863207; u := 448873741399; a := 18*u^3 + 18*u^2 + 9*u + 1; a; Fp := GF(p); Fp2 := ExtensionField; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; for j := 0 to 6 do j, ":", (z^4)^(p^j); end for; u0 := z^2; u1 := u0^(p^2); u2 := u0^(p^4); "u0 =", u0; "u1 =", u1, ":", u1 eq (p-a-1)*u0; "u2 =", u2, ":", u2 eq a*u0; p-a; u0 + u1 + u2; trace(u0, p); trace(u1, p); trace(u2, p); Output: Magma V2.11-10 Wed Dec 7 2005 13:19:29 on modular [Seed = 381842098] ------------------------------------- 1627965160026674480212199743920457792 0 : z^4 1 : 1627965160026674480212199743920457793*z^4 2 : 1627965160026674480212199743920457792*z^4 3 : 1461501624496790265145448589920785493717258890818*z^4 4 : 1461501624495162299985421915440573293973338433026*z^4 5 : 1461501624495162299985421915440573293973338433027*z^4 6 : z^4 u0 = z^2 u1 = 1461501624495162299985421915440573293973338433026*z^2 : true u2 = 1627965160026674480212199743920457792*z^2 : true 1461501624495162299985421915440573293973338433027 0 0 0 0 Total time: 0.250 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Wed Dec 7 13:19:16 2005 Input: trace := function(x, p) return x + x^(p^2) + x^(p^4); end function; p := 1461501624496790265145448589920785493717258890819; t := 1208925814305217958863207; u := 448873741399; a := 18*u^3 + 18*u^2 + 9*u + 1; a; Fp := GF(p); Fp2 := ExtensionField; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; for j := 0 to 6 do j, ":", (z^4)^(p^j); end for; u0 := z^2; u1 := u0^(p^2); u2 := u0^(p^4); "u0 =", u0; "u1 =", u1, ":", u1 eq (p-a+1)*u0; "u2 =", u2, ":", u2 eq a*u0; p-a; u0 + u1 + u2; trace(u0, p); trace(u1, p); trace(u2, p); Output: Magma V2.11-10 Wed Dec 7 2005 13:19:16 on modular [Seed = 3896282202] ------------------------------------- 1627965160026674480212199743920457792 0 : z^4 1 : 1627965160026674480212199743920457793*z^4 2 : 1627965160026674480212199743920457792*z^4 3 : 1461501624496790265145448589920785493717258890818*z^4 4 : 1461501624495162299985421915440573293973338433026*z^4 5 : 1461501624495162299985421915440573293973338433027*z^4 6 : z^4 u0 = z^2 u1 = 1461501624495162299985421915440573293973338433026*z^2 : false u2 = 1627965160026674480212199743920457792*z^2 : true 1461501624495162299985421915440573293973338433027 0 0 0 0 Total time: 0.260 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Wed Dec 7 13:18:51 2005 Input: trace := function(x, p) return x + x^(p^2) + x^(p^4); end function; p := 1461501624496790265145448589920785493717258890819; t := 1208925814305217958863207; u := 448873741399; a := 18*u^3 + 18*u^2 + 9*u + 1; a; Fp := GF(p); Fp2 := ExtensionField; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; for j := 0 to 6 do j, ":", (z^4)^(p^j); end for; u0 := z^2; u1 := u0^(p^2); u2 := u0^(p^4); "u0 =", u0; "u1 =", u1, ":", u1 eq (p-a)*u0; "u2 =", u2, ":", u2 eq a*u0; p-a; u0 + u1 + u2; trace(u0, p); trace(u1, p); trace(u2, p); Output: Magma V2.11-10 Wed Dec 7 2005 13:18:30 on modular [Seed = 4045772176] ------------------------------------- 1627965160026674480212199743920457792 0 : z^4 1 : 1627965160026674480212199743920457793*z^4 2 : 1627965160026674480212199743920457792*z^4 3 : 1461501624496790265145448589920785493717258890818*z^4 4 : 1461501624495162299985421915440573293973338433026*z^4 5 : 1461501624495162299985421915440573293973338433027*z^4 6 : z^4 u0 = z^2 u1 = 1461501624495162299985421915440573293973338433026*z^2 : false u2 = 1627965160026674480212199743920457792*z^2 : true 1461501624495162299985421915440573293973338433027 0 0 0 0 Total time: 0.240 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Wed Dec 7 13:17:58 2005 Input: trace := function(x, p) return x + x^(p^2) + x^(p^4); end function; p := 1461501624496790265145448589920785493717258890819; t := 1208925814305217958863207; u := 448873741399; a := 18*u^3 + 18*u^2 + 9*u + 1; a; Fp := GF(p); Fp2 := ExtensionField; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; for j := 0 to 6 do j, ":", (z^4)^(p^j); end for; u0 := z^2; u1 := u0^(p^2); u2 := u0^(p^4); "u0 =", u0; "u1 =", u1, ":", u1 eq (p-a)*u0; "u2 =", u2, ":", u2 eq a*u0; u0 + u1 + u2; trace(u0, p); trace(u1, p); trace(u2, p); Output: Magma V2.11-10 Wed Dec 7 2005 13:17:58 on modular [Seed = 2470967250] ------------------------------------- 1627965160026674480212199743920457792 0 : z^4 1 : 1627965160026674480212199743920457793*z^4 2 : 1627965160026674480212199743920457792*z^4 3 : 1461501624496790265145448589920785493717258890818*z^4 4 : 1461501624495162299985421915440573293973338433026*z^4 5 : 1461501624495162299985421915440573293973338433027*z^4 6 : z^4 u0 = z^2 u1 = 1461501624495162299985421915440573293973338433026*z^2 : false u2 = 1627965160026674480212199743920457792*z^2 : true 0 0 0 0 Total time: 0.260 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Wed Dec 7 13:16:27 2005 Input: trace := function(x, p) return x + x^(p^2) + x^(p^4); end function; p := 1461501624496790265145448589920785493717258890819; t := 1208925814305217958863207; u := 448873741399; a := 18*u^3 + 18*u^2 + 9*u + 1; a; Fp := GF(p); Fp2 := ExtensionField; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; for j := 0 to 6 do j, ":", (z^4)^(p^j); end for; u0 := z^2; u1 := u0^(p^2); u2 := u0^(p^4); "u0 =", u0; "u1 =", u1, ":", u1 eq -a*u0; "u2 =", u2, ":", u2 eq a*u0; u0 + u1 + u2; trace(u0, p); trace(u1, p); trace(u2, p); Output: Magma V2.11-10 Wed Dec 7 2005 13:16:24 on modular [Seed = 636567756] ------------------------------------- 1627965160026674480212199743920457792 0 : z^4 1 : 1627965160026674480212199743920457793*z^4 2 : 1627965160026674480212199743920457792*z^4 3 : 1461501624496790265145448589920785493717258890818*z^4 4 : 1461501624495162299985421915440573293973338433026*z^4 5 : 1461501624495162299985421915440573293973338433027*z^4 6 : z^4 u0 = z^2 u1 = 1461501624495162299985421915440573293973338433026*z^2 : false u2 = 1627965160026674480212199743920457792*z^2 : true 0 0 0 0 Total time: 0.270 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Wed Dec 7 13:15:49 2005 Input: trace := function(x, p) return x + x^(p^2) + x^(p^4); end function; p := 1461501624496790265145448589920785493717258890819; t := 1208925814305217958863207; u := 448873741399; a := 18*u^3 + 18*u^2 + 9*u + 1; a; Fp := GF(p); Fp2 := ExtensionField; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; for j := 0 to 6 do j, ":", (z^4)^(p^j); end for; u0 := z^2; u1 := u0^(p^2); u2 := u0^(p^4); "u0 =", u0; "u1 =", u1, ":", u1 eq -a*u0; "u2 =", u2, ":", u1 eq a*u0; u0 + u1 + u2; trace(u0, p); trace(u1, p); trace(u2, p); Output: Magma V2.11-10 Wed Dec 7 2005 13:15:48 on modular [Seed = 3778380839] ------------------------------------- 1627965160026674480212199743920457792 0 : z^4 1 : 1627965160026674480212199743920457793*z^4 2 : 1627965160026674480212199743920457792*z^4 3 : 1461501624496790265145448589920785493717258890818*z^4 4 : 1461501624495162299985421915440573293973338433026*z^4 5 : 1461501624495162299985421915440573293973338433027*z^4 6 : z^4 u0 = z^2 u1 = 1461501624495162299985421915440573293973338433026*z^2 : false u2 = 1627965160026674480212199743920457792*z^2 : false 0 0 0 0 Total time: 0.250 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Wed Dec 7 13:15:32 2005 Input: trace := function(x, p) return x + x^(p^2) + x^(p^4); end function; p := 1461501624496790265145448589920785493717258890819; t := 1208925814305217958863207; u := 448873741399; a := 18*u^3 + 18*u^2 + 9*u + 1; Fp := GF(p); Fp2 := ExtensionField; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; for j := 0 to 6 do j, ":", (z^4)^(p^j); end for; u0 := z^2; u1 := u0^(p^2); u2 := u0^(p^4); "u0 =", u0; "u1 =", u1, ":", u1 eq -a*u0; "u2 =", u2, ":", u1 eq a*u0; u0 + u1 + u2; trace(u0, p); trace(u1, p); trace(u2, p); Output: Magma V2.11-10 Wed Dec 7 2005 13:15:32 on modular [Seed = 4197359411] ------------------------------------- 0 : z^4 1 : 1627965160026674480212199743920457793*z^4 2 : 1627965160026674480212199743920457792*z^4 3 : 1461501624496790265145448589920785493717258890818*z^4 4 : 1461501624495162299985421915440573293973338433026*z^4 5 : 1461501624495162299985421915440573293973338433027*z^4 6 : z^4 u0 = z^2 u1 = 1461501624495162299985421915440573293973338433026*z^2 : false u2 = 1627965160026674480212199743920457792*z^2 : false 0 0 0 0 Total time: 0.250 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Wed Dec 7 13:15:03 2005 Input: trace := function(x, p) return x + x^(p^2) + x^(p^4); end function; p := 1461501624496790265145448589920785493717258890819; t := 1208925814305217958863207; u := 448873741399; a := 18*u^3 + 18*u^2 + 9*u + 1; Fp := GF(p); Fp2 := ExtensionField; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; for j := 0 to 6 do j, ":", (z^4)^(p^j); end for; u0 := z^2; u1 := u0^(p^2); u2 := u0^(p^4); "u0 =", u0; "u1 =", u1, ":", u1 eq a*u0; "u2 =", u2, ":", u1 eq -a*u0; u0 + u1 + u2; trace(u0, p); trace(u1, p); trace(u2, p); Output: Magma V2.11-10 Wed Dec 7 2005 13:15:03 on modular [Seed = 4230519126] ------------------------------------- 0 : z^4 1 : 1627965160026674480212199743920457793*z^4 2 : 1627965160026674480212199743920457792*z^4 3 : 1461501624496790265145448589920785493717258890818*z^4 4 : 1461501624495162299985421915440573293973338433026*z^4 5 : 1461501624495162299985421915440573293973338433027*z^4 6 : z^4 u0 = z^2 u1 = 1461501624495162299985421915440573293973338433026*z^2 : false u2 = 1627965160026674480212199743920457792*z^2 : false 0 0 0 0 Total time: 0.250 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Wed Dec 7 13:13:50 2005 Input: trace := function(x, p) return x + x^(p^2) + x^(p^4); end function; p := 1461501624496790265145448589920785493717258890819; t := 1208925814305217958863207; u := 448873741399; t eq 6*u^2 + 1; (1627965160026674480212199743920457792 - 18*u^3 - 18*u^2 - 3*u - 1) mod u; 1627965160026674480212199743920457792 eq 18*u^3 + 18*u^2 + 9*u + 1; Fp := GF(p); Fp2 := ExtensionField; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; for j := 0 to 6 do j, ":", (z^4)^(p^j); end for; u0 := z^2; u1 := u0^(p^2); u2 := u0^(p^4); "u0 =", u0; "u1 =", u1; "u2 =", u2; u0 + u1 + u2; trace(u0, p); trace(u1, p); trace(u2, p); Output: Magma V2.11-10 Wed Dec 7 2005 13:13:50 on modular [Seed = 3778380978] ------------------------------------- true 0 true 0 : z^4 1 : 1627965160026674480212199743920457793*z^4 2 : 1627965160026674480212199743920457792*z^4 3 : 1461501624496790265145448589920785493717258890818*z^4 4 : 1461501624495162299985421915440573293973338433026*z^4 5 : 1461501624495162299985421915440573293973338433027*z^4 6 : z^4 u0 = z^2 u1 = 1461501624495162299985421915440573293973338433026*z^2 u2 = 1627965160026674480212199743920457792*z^2 0 0 0 0 Total time: 0.240 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Wed Dec 7 13:13:37 2005 Input: trace := function(x, p) return x + x^(p^2) + x^(p^4); end function; p := 1461501624496790265145448589920785493717258890819; t := 1208925814305217958863207; u := 448873741399; t eq 6*u^2 + 1; (1627965160026674480212199743920457792 - 18*u^3 - 18*u^2 - 3*u - 1) mod u; 1627965160026674480212199743920457792 - 18*u^3 - 18*u^2 - 9*u - 1 eq 0; Fp := GF(p); Fp2 := ExtensionField; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; for j := 0 to 6 do j, ":", (z^4)^(p^j); end for; u0 := z^2; u1 := u0^(p^2); u2 := u0^(p^4); "u0 =", u0; "u1 =", u1; "u2 =", u2; u0 + u1 + u2; trace(u0, p); trace(u1, p); trace(u2, p); Output: Magma V2.11-10 Wed Dec 7 2005 13:13:36 on modular [Seed = 3946811062] ------------------------------------- true 0 true 0 : z^4 1 : 1627965160026674480212199743920457793*z^4 2 : 1627965160026674480212199743920457792*z^4 3 : 1461501624496790265145448589920785493717258890818*z^4 4 : 1461501624495162299985421915440573293973338433026*z^4 5 : 1461501624495162299985421915440573293973338433027*z^4 6 : z^4 u0 = z^2 u1 = 1461501624495162299985421915440573293973338433026*z^2 u2 = 1627965160026674480212199743920457792*z^2 0 0 0 0 Total time: 0.250 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Wed Dec 7 13:13:25 2005 Input: trace := function(x, p) return x + x^(p^2) + x^(p^4); end function; p := 1461501624496790265145448589920785493717258890819; t := 1208925814305217958863207; u := 448873741399; t eq 6*u^2 + 1; (1627965160026674480212199743920457792 - 18*u^3 - 18*u^2 - 3*u - 1) mod u; 1627965160026674480212199743920457792 - 18*u^3 - 18*u^2 - 3*u - 1 eq 6*u; Fp := GF(p); Fp2 := ExtensionField; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; for j := 0 to 6 do j, ":", (z^4)^(p^j); end for; u0 := z^2; u1 := u0^(p^2); u2 := u0^(p^4); "u0 =", u0; "u1 =", u1; "u2 =", u2; u0 + u1 + u2; trace(u0, p); trace(u1, p); trace(u2, p); Output: Magma V2.11-10 Wed Dec 7 2005 13:13:25 on modular [Seed = 3590501278] ------------------------------------- true 0 true 0 : z^4 1 : 1627965160026674480212199743920457793*z^4 2 : 1627965160026674480212199743920457792*z^4 3 : 1461501624496790265145448589920785493717258890818*z^4 4 : 1461501624495162299985421915440573293973338433026*z^4 5 : 1461501624495162299985421915440573293973338433027*z^4 6 : z^4 u0 = z^2 u1 = 1461501624495162299985421915440573293973338433026*z^2 u2 = 1627965160026674480212199743920457792*z^2 0 0 0 0 Total time: 0.250 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Wed Dec 7 13:12:56 2005 Input: trace := function(x, p) return x + x^(p^2) + x^(p^4); end function; p := 1461501624496790265145448589920785493717258890819; t := 1208925814305217958863207; u := 448873741399; t eq 6*u^2 + 1; (1627965160026674480212199743920457792 - 18*u^3 - 18*u^2 - 3*u - 1) mod u; (1627965160026674480212199743920457792 - 18*u^3 - 18*u^2 - 3*u - 1) div u; Fp := GF(p); Fp2 := ExtensionField; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; for j := 0 to 6 do j, ":", (z^4)^(p^j); end for; u0 := z^2; u1 := u0^(p^2); u2 := u0^(p^4); "u0 =", u0; "u1 =", u1; "u2 =", u2; u0 + u1 + u2; trace(u0, p); trace(u1, p); trace(u2, p); Output: Magma V2.11-10 Wed Dec 7 2005 13:12:55 on modular [Seed = 2775695503] ------------------------------------- true 0 6 0 : z^4 1 : 1627965160026674480212199743920457793*z^4 2 : 1627965160026674480212199743920457792*z^4 3 : 1461501624496790265145448589920785493717258890818*z^4 4 : 1461501624495162299985421915440573293973338433026*z^4 5 : 1461501624495162299985421915440573293973338433027*z^4 6 : z^4 u0 = z^2 u1 = 1461501624495162299985421915440573293973338433026*z^2 u2 = 1627965160026674480212199743920457792*z^2 0 0 0 0 Total time: 0.260 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Wed Dec 7 13:12:13 2005 Input: trace := function(x, p) return x + x^(p^2) + x^(p^4); end function; p := 1461501624496790265145448589920785493717258890819; t := 1208925814305217958863207; u := 448873741399; t eq 6*u^2 + 1; (1627965160026674480212199743920457792 - 18*u^3 - 12*u^2 - 9*u - 1) mod u; (1627965160026674480212199743920457792 - 18*u^3 - 12*u^2 - 9*u - 1) div u; Fp := GF(p); Fp2 := ExtensionField; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; for j := 0 to 6 do j, ":", (z^4)^(p^j); end for; u0 := z^2; u1 := u0^(p^2); u2 := u0^(p^4); "u0 =", u0; "u1 =", u1; "u2 =", u2; u0 + u1 + u2; trace(u0, p); trace(u1, p); trace(u2, p); Output: Magma V2.11-10 Wed Dec 7 2005 13:12:09 on modular [Seed = 1468215712] ------------------------------------- true 0 2693242448394 0 : z^4 1 : 1627965160026674480212199743920457793*z^4 2 : 1627965160026674480212199743920457792*z^4 3 : 1461501624496790265145448589920785493717258890818*z^4 4 : 1461501624495162299985421915440573293973338433026*z^4 5 : 1461501624495162299985421915440573293973338433027*z^4 6 : z^4 u0 = z^2 u1 = 1461501624495162299985421915440573293973338433026*z^2 u2 = 1627965160026674480212199743920457792*z^2 0 0 0 0 Total time: 0.270 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Wed Dec 7 13:11:26 2005 Input: trace := function(x, p) return x + x^(p^2) + x^(p^4); end function; p := 1461501624496790265145448589920785493717258890819; t := 1208925814305217958863207; u := 448873741399; t eq 6*u^2 + 1; (1627965160026674480212199743920457792 - 18*u^3 - 12*u^2 - 3*u - 1) mod u; (1627965160026674480212199743920457792 - 18*u^3 - 12*u^2 - 3*u - 1) div u; Fp := GF(p); Fp2 := ExtensionField; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; for j := 0 to 6 do j, ":", (z^4)^(p^j); end for; u0 := z^2; u1 := u0^(p^2); u2 := u0^(p^4); "u0 =", u0; "u1 =", u1; "u2 =", u2; u0 + u1 + u2; trace(u0, p); trace(u1, p); trace(u2, p); Output: Magma V2.11-10 Wed Dec 7 2005 13:11:25 on modular [Seed = 1906644915] ------------------------------------- true 0 2693242448400 0 : z^4 1 : 1627965160026674480212199743920457793*z^4 2 : 1627965160026674480212199743920457792*z^4 3 : 1461501624496790265145448589920785493717258890818*z^4 4 : 1461501624495162299985421915440573293973338433026*z^4 5 : 1461501624495162299985421915440573293973338433027*z^4 6 : z^4 u0 = z^2 u1 = 1461501624495162299985421915440573293973338433026*z^2 u2 = 1627965160026674480212199743920457792*z^2 0 0 0 0 Total time: 0.260 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Wed Dec 7 13:11:05 2005 Input: trace := function(x, p) return x + x^(p^2) + x^(p^4); end function; p := 1461501624496790265145448589920785493717258890819; t := 1208925814305217958863207; u := 448873741399; t eq 6*u^2 + 1; (1627965160026674480212199743920457792 - 18*u^3 - 12*u^2 - 3*u - 1) mod u; Fp := GF(p); Fp2 := ExtensionField; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; for j := 0 to 6 do j, ":", (z^4)^(p^j); end for; u0 := z^2; u1 := u0^(p^2); u2 := u0^(p^4); "u0 =", u0; "u1 =", u1; "u2 =", u2; u0 + u1 + u2; trace(u0, p); trace(u1, p); trace(u2, p); Output: Magma V2.11-10 Wed Dec 7 2005 13:11:04 on modular [Seed = 1656095957] ------------------------------------- true 0 0 : z^4 1 : 1627965160026674480212199743920457793*z^4 2 : 1627965160026674480212199743920457792*z^4 3 : 1461501624496790265145448589920785493717258890818*z^4 4 : 1461501624495162299985421915440573293973338433026*z^4 5 : 1461501624495162299985421915440573293973338433027*z^4 6 : z^4 u0 = z^2 u1 = 1461501624495162299985421915440573293973338433026*z^2 u2 = 1627965160026674480212199743920457792*z^2 0 0 0 0 Total time: 0.260 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Wed Dec 7 13:10:45 2005 Input: trace := function(x, p) return x + x^(p^2) + x^(p^4); end function; p := 1461501624496790265145448589920785493717258890819; t := 1208925814305217958863207; u := 448873741399; t eq 6*u^2 + 1; (2*1627965160026674480212199743920457792 - 36*u^3 - 24*u^2 - 6*u - 1) mod u; Fp := GF(p); Fp2 := ExtensionField; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; for j := 0 to 6 do j, ":", (z^4)^(p^j); end for; u0 := z^2; u1 := u0^(p^2); u2 := u0^(p^4); "u0 =", u0; "u1 =", u1; "u2 =", u2; u0 + u1 + u2; trace(u0, p); trace(u1, p); trace(u2, p); Output: Magma V2.11-10 Wed Dec 7 2005 13:10:45 on modular [Seed = 1874528709] ------------------------------------- true 1 0 : z^4 1 : 1627965160026674480212199743920457793*z^4 2 : 1627965160026674480212199743920457792*z^4 3 : 1461501624496790265145448589920785493717258890818*z^4 4 : 1461501624495162299985421915440573293973338433026*z^4 5 : 1461501624495162299985421915440573293973338433027*z^4 6 : z^4 u0 = z^2 u1 = 1461501624495162299985421915440573293973338433026*z^2 u2 = 1627965160026674480212199743920457792*z^2 0 0 0 0 Total time: 0.250 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Wed Dec 7 13:09:45 2005 Input: trace := function(x, p) return x + x^(p^2) + x^(p^4); end function; p := 1461501624496790265145448589920785493717258890819; t := 1208925814305217958863207; u := 448873741399; t eq 6*u^2 + 1; (2*1627965160026674480212199743920457792 - (36*u^3 + 24*u^2 + 6*u + 1) - 1) mod u; Fp := GF(p); Fp2 := ExtensionField; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; for j := 0 to 6 do j, ":", (z^4)^(p^j); end for; u0 := z^2; u1 := u0^(p^2); u2 := u0^(p^4); "u0 =", u0; "u1 =", u1; "u2 =", u2; u0 + u1 + u2; trace(u0, p); trace(u1, p); trace(u2, p); Output: Magma V2.11-10 Wed Dec 7 2005 13:09:44 on modular [Seed = 603407684] ------------------------------------- true 0 0 : z^4 1 : 1627965160026674480212199743920457793*z^4 2 : 1627965160026674480212199743920457792*z^4 3 : 1461501624496790265145448589920785493717258890818*z^4 4 : 1461501624495162299985421915440573293973338433026*z^4 5 : 1461501624495162299985421915440573293973338433027*z^4 6 : z^4 u0 = z^2 u1 = 1461501624495162299985421915440573293973338433026*z^2 u2 = 1627965160026674480212199743920457792*z^2 0 0 0 0 Total time: 0.250 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Wed Dec 7 13:08:45 2005 Input: trace := function(x, p) return x + x^(p^2) + x^(p^4); end function; p := 1461501624496790265145448589920785493717258890819; t := 1208925814305217958863207; u := 448873741399; t eq 6*u^2 + 1; 2*1627965160026674480212199743920457792 - (36*u^3 + 24*u^2 + 6*u + 1); Fp := GF(p); Fp2 := ExtensionField; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; for j := 0 to 6 do j, ":", (z^4)^(p^j); end for; u0 := z^2; u1 := u0^(p^2); u2 := u0^(p^4); "u0 =", u0; "u1 =", u1; "u2 =", u2; u0 + u1 + u2; trace(u0, p); trace(u1, p); trace(u2, p); Output: Magma V2.11-10 Wed Dec 7 2005 13:08:45 on modular [Seed = 348156470] ------------------------------------- true 2417851628615822402623201 0 : z^4 1 : 1627965160026674480212199743920457793*z^4 2 : 1627965160026674480212199743920457792*z^4 3 : 1461501624496790265145448589920785493717258890818*z^4 4 : 1461501624495162299985421915440573293973338433026*z^4 5 : 1461501624495162299985421915440573293973338433027*z^4 6 : z^4 u0 = z^2 u1 = 1461501624495162299985421915440573293973338433026*z^2 u2 = 1627965160026674480212199743920457792*z^2 0 0 0 0 Total time: 0.250 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Wed Dec 7 13:08:40 2005 Input: trace := function(x, p) return x + x^(p^2) + x^(p^4); end function; p := 1461501624496790265145448589920785493717258890819; t := 1208925814305217958863207; u := 448873741399; t eq 6*u^2 + 1; 2*1627965160026674480212199743920457792 - (36*u^3 + 18*u^2 + 6*u + 1); Fp := GF(p); Fp2 := ExtensionField; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; for j := 0 to 6 do j, ":", (z^4)^(p^j); end for; u0 := z^2; u1 := u0^(p^2); u2 := u0^(p^4); "u0 =", u0; "u1 =", u1; "u2 =", u2; u0 + u1 + u2; trace(u0, p); trace(u1, p); trace(u2, p); Output: Magma V2.11-10 Wed Dec 7 2005 13:08:39 on modular [Seed = 432897841] ------------------------------------- true 3626777442921040361486407 0 : z^4 1 : 1627965160026674480212199743920457793*z^4 2 : 1627965160026674480212199743920457792*z^4 3 : 1461501624496790265145448589920785493717258890818*z^4 4 : 1461501624495162299985421915440573293973338433026*z^4 5 : 1461501624495162299985421915440573293973338433027*z^4 6 : z^4 u0 = z^2 u1 = 1461501624495162299985421915440573293973338433026*z^2 u2 = 1627965160026674480212199743920457792*z^2 0 0 0 0 Total time: 0.250 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Wed Dec 7 13:08:16 2005 Input: trace := function(x, p) return x + x^(p^2) + x^(p^4); end function; p := 1461501624496790265145448589920785493717258890819; t := 1208925814305217958863207; u := 448873741399; t eq 6*u^2 + 1; 2*1627965160026674480212199743920457792 - 36*u^3 + 24*u^2 + 6*u + 1; Fp := GF(p); Fp2 := ExtensionField; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; for j := 0 to 6 do j, ":", (z^4)^(p^j); end for; u0 := z^2; u1 := u0^(p^2); u2 := u0^(p^4); "u0 =", u0; "u1 =", u1; "u2 =", u2; u0 + u1 + u2; trace(u0, p); trace(u1, p); trace(u2, p); Output: Magma V2.11-10 Wed Dec 7 2005 13:08:16 on modular [Seed = 533428480] ------------------------------------- true 12089258143062952558425639 0 : z^4 1 : 1627965160026674480212199743920457793*z^4 2 : 1627965160026674480212199743920457792*z^4 3 : 1461501624496790265145448589920785493717258890818*z^4 4 : 1461501624495162299985421915440573293973338433026*z^4 5 : 1461501624495162299985421915440573293973338433027*z^4 6 : z^4 u0 = z^2 u1 = 1461501624495162299985421915440573293973338433026*z^2 u2 = 1627965160026674480212199743920457792*z^2 0 0 0 0 Total time: 0.250 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Wed Dec 7 13:08:10 2005 Input: trace := function(x, p) return x + x^(p^2) + x^(p^4); end function; p := 1461501624496790265145448589920785493717258890819; t := 1208925814305217958863207; u := 448873741399; t eq 6*u^2 + 1; 2*1627965160026674480212199743920457792 - 36*u^3 + 18*u^2 + 6*u + 1; Fp := GF(p); Fp2 := ExtensionField; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; for j := 0 to 6 do j, ":", (z^4)^(p^j); end for; u0 := z^2; u1 := u0^(p^2); u2 := u0^(p^4); "u0 =", u0; "u1 =", u1; "u2 =", u2; u0 + u1 + u2; trace(u0, p); trace(u1, p); trace(u2, p); Output: Magma V2.11-10 Wed Dec 7 2005 13:08:10 on modular [Seed = 80765441] ------------------------------------- true 10880332328757734599562433 0 : z^4 1 : 1627965160026674480212199743920457793*z^4 2 : 1627965160026674480212199743920457792*z^4 3 : 1461501624496790265145448589920785493717258890818*z^4 4 : 1461501624495162299985421915440573293973338433026*z^4 5 : 1461501624495162299985421915440573293973338433027*z^4 6 : z^4 u0 = z^2 u1 = 1461501624495162299985421915440573293973338433026*z^2 u2 = 1627965160026674480212199743920457792*z^2 0 0 0 0 Total time: 0.260 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Wed Dec 7 13:07:46 2005 Input: trace := function(x, p) return x + x^(p^2) + x^(p^4); end function; p := 1461501624496790265145448589920785493717258890819; t := 1208925814305217958863207; u := 448873741399; t eq 6*u^2 + 1; 36*u^3 + 18*u^2 + 6*u + 1; 2*1627965160026674480212199743920457792; Fp := GF(p); Fp2 := ExtensionField; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; for j := 0 to 6 do j, ":", (z^4)^(p^j); end for; u0 := z^2; u1 := u0^(p^2); u2 := u0^(p^4); "u0 =", u0; "u1 =", u1; "u2 =", u2; u0 + u1 + u2; trace(u0, p); trace(u1, p); trace(u2, p); Output: Magma V2.11-10 Wed Dec 7 2005 13:07:46 on modular [Seed = 215510001] ------------------------------------- true 3255930320049722182981478447479429177 3255930320053348960424399487840915584 0 : z^4 1 : 1627965160026674480212199743920457793*z^4 2 : 1627965160026674480212199743920457792*z^4 3 : 1461501624496790265145448589920785493717258890818*z^4 4 : 1461501624495162299985421915440573293973338433026*z^4 5 : 1461501624495162299985421915440573293973338433027*z^4 6 : z^4 u0 = z^2 u1 = 1461501624495162299985421915440573293973338433026*z^2 u2 = 1627965160026674480212199743920457792*z^2 0 0 0 0 Total time: 0.260 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Wed Dec 7 13:07:36 2005 Input: trace := function(x, p) return x + x^(p^2) + x^(p^4); end function; p := 1461501624496790265145448589920785493717258890819; t := 1208925814305217958863207; u := 448873741399; t eq 6*u^2 + 1; 36*u^3 + 18*u^2 + 6*u + 1; 1627965160026674480212199743920457792; Fp := GF(p); Fp2 := ExtensionField; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; for j := 0 to 6 do j, ":", (z^4)^(p^j); end for; u0 := z^2; u1 := u0^(p^2); u2 := u0^(p^4); "u0 =", u0; "u1 =", u1; "u2 =", u2; u0 + u1 + u2; trace(u0, p); trace(u1, p); trace(u2, p); Output: Magma V2.11-10 Wed Dec 7 2005 13:07:36 on modular [Seed = 4129460502] ------------------------------------- true 3255930320049722182981478447479429177 1627965160026674480212199743920457792 0 : z^4 1 : 1627965160026674480212199743920457793*z^4 2 : 1627965160026674480212199743920457792*z^4 3 : 1461501624496790265145448589920785493717258890818*z^4 4 : 1461501624495162299985421915440573293973338433026*z^4 5 : 1461501624495162299985421915440573293973338433027*z^4 6 : z^4 u0 = z^2 u1 = 1461501624495162299985421915440573293973338433026*z^2 u2 = 1627965160026674480212199743920457792*z^2 0 0 0 0 Total time: 0.250 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Wed Dec 7 13:07:31 2005 Input: trace := function(x, p) return x + x^(p^2) + x^(p^4); end function; p := 1461501624496790265145448589920785493717258890819; t := 1208925814305217958863207; u := 448873741399; t eq 6*u^2 + 1; 36*u^3 + 18*u^2 + 6*u + 1;] 1627965160026674480212199743920457792; Fp := GF(p); Fp2 := ExtensionField; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; for j := 0 to 6 do j, ":", (z^4)^(p^j); end for; u0 := z^2; u1 := u0^(p^2); u2 := u0^(p^4); "u0 =", u0; "u1 =", u1; "u2 =", u2; u0 + u1 + u2; trace(u0, p); trace(u1, p); trace(u2, p); Output: Magma V2.11-10 Wed Dec 7 2005 13:07:30 on modular [Seed = 4045772315] ------------------------------------- true 3255930320049722182981478447479429177 >> 36*u^3 + 18*u^2 + 6*u + 1;] ^ User error: bad syntax 1627965160026674480212199743920457792 0 : z^4 1 : 1627965160026674480212199743920457793*z^4 2 : 1627965160026674480212199743920457792*z^4 3 : 1461501624496790265145448589920785493717258890818*z^4 4 : 1461501624495162299985421915440573293973338433026*z^4 5 : 1461501624495162299985421915440573293973338433027*z^4 6 : z^4 u0 = z^2 u1 = 1461501624495162299985421915440573293973338433026*z^2 u2 = 1627965160026674480212199743920457792*z^2 0 0 0 0 Total time: 0.250 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Wed Dec 7 13:07:16 2005 Input: trace := function(x, p) return x + x^(p^2) + x^(p^4); end function; p := 1461501624496790265145448589920785493717258890819; t := 1208925814305217958863207; u := 448873741399; t eq 6*u^2 + 1; 36*u^3 + 18*u^2 + 6*u + 1 - 1627965160026674480212199743920457792; Fp := GF(p); Fp2 := ExtensionField; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; for j := 0 to 6 do j, ":", (z^4)^(p^j); end for; u0 := z^2; u1 := u0^(p^2); u2 := u0^(p^4); "u0 =", u0; "u1 =", u1; "u2 =", u2; u0 + u1 + u2; trace(u0, p); trace(u1, p); trace(u2, p); Output: Magma V2.11-10 Wed Dec 7 2005 13:07:16 on modular [Seed = 4230519584] ------------------------------------- true 1627965160023047702769278703558971385 0 : z^4 1 : 1627965160026674480212199743920457793*z^4 2 : 1627965160026674480212199743920457792*z^4 3 : 1461501624496790265145448589920785493717258890818*z^4 4 : 1461501624495162299985421915440573293973338433026*z^4 5 : 1461501624495162299985421915440573293973338433027*z^4 6 : z^4 u0 = z^2 u1 = 1461501624495162299985421915440573293973338433026*z^2 u2 = 1627965160026674480212199743920457792*z^2 0 0 0 0 Total time: 0.260 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Wed Dec 7 13:06:14 2005 Input: trace := function(x, p) return x + x^(p^2) + x^(p^4); end function; p := 1461501624496790265145448589920785493717258890819; t := 1208925814305217958863207; u := 448873741399; t eq 6*u^2 + 1; Fp := GF(p); Fp2 := ExtensionField; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; for j := 0 to 6 do j, ":", (z^4)^(p^j); end for; u0 := z^2; u1 := u0^(p^2); u2 := u0^(p^4); "u0 =", u0; "u1 =", u1; "u2 =", u2; u0 + u1 + u2; trace(u0, p); trace(u1, p); trace(u2, p); Output: Magma V2.11-10 Wed Dec 7 2005 13:06:14 on modular [Seed = 3862069624] ------------------------------------- true 0 : z^4 1 : 1627965160026674480212199743920457793*z^4 2 : 1627965160026674480212199743920457792*z^4 3 : 1461501624496790265145448589920785493717258890818*z^4 4 : 1461501624495162299985421915440573293973338433026*z^4 5 : 1461501624495162299985421915440573293973338433027*z^4 6 : z^4 u0 = z^2 u1 = 1461501624495162299985421915440573293973338433026*z^2 u2 = 1627965160026674480212199743920457792*z^2 0 0 0 0 Total time: 0.250 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Wed Dec 7 13:05:51 2005 Input: trace := function(x, p) return x + x^(p^2) + x^(p^4); end function; p := 1461501624496790265145448589920785493717258890819; t := 1208925814305217958863207; u := Sqrt((t - 1) div 6); "u =", u; Fp := GF(p); Fp2 := ExtensionField; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; for j := 0 to 6 do j, ":", (z^4)^(p^j); end for; u0 := z^2; u1 := u0^(p^2); u2 := u0^(p^4); "u0 =", u0; "u1 =", u1; "u2 =", u2; u0 + u1 + u2; trace(u0, p); trace(u1, p); trace(u2, p); Output: Magma V2.11-10 Wed Dec 7 2005 13:05:51 on modular [Seed = 3778381462] ------------------------------------- u = 448873741399.0000000000000000 0 : z^4 1 : 1627965160026674480212199743920457793*z^4 2 : 1627965160026674480212199743920457792*z^4 3 : 1461501624496790265145448589920785493717258890818*z^4 4 : 1461501624495162299985421915440573293973338433026*z^4 5 : 1461501624495162299985421915440573293973338433027*z^4 6 : z^4 u0 = z^2 u1 = 1461501624495162299985421915440573293973338433026*z^2 u2 = 1627965160026674480212199743920457792*z^2 0 0 0 0 Total time: 0.250 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Wed Dec 7 13:04:18 2005 Input: trace := function(x, p) return x + x^(p^2) + x^(p^4); end function; p := 1461501624496790265145448589920785493717258890819; Fp := GF(p); Fp2 := ExtensionField; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; for j := 0 to 6 do j, ":", (z^4)^(p^j); end for; u0 := z^2; u1 := u0^(p^2); u2 := u0^(p^4); "u0 =", u0; "u1 =", u1; "u2 =", u2; u0 + u1 + u2; trace(u0, p); trace(u1, p); trace(u2, p); F := FunctionField(Fp12, 3); (a + b*z^2 + c*z^4)^(p^2); Output: Magma V2.11-10 Wed Dec 7 2005 13:04:17 on modular [Seed = 3963128773] ------------------------------------- 0 : z^4 1 : 1627965160026674480212199743920457793*z^4 2 : 1627965160026674480212199743920457792*z^4 3 : 1461501624496790265145448589920785493717258890818*z^4 4 : 1461501624495162299985421915440573293973338433026*z^4 5 : 1461501624495162299985421915440573293973338433027*z^4 6 : z^4 u0 = z^2 u1 = 1461501624495162299985421915440573293973338433026*z^2 u2 = 1627965160026674480212199743920457792*z^2 0 0 0 0 >> (a + b*z^2 + c*z^4)^(p^2); ^ Runtime error in '^': Argument 1 (213598699840675693484172798829876117292923333\ 2694664962512687420744793421079272493992902162490761) is too large Total time: 0.250 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Wed Dec 7 13:04:00 2005 Input: trace := function(x, p) return x + x^(p^2) + x^(p^4); end function; p := 1461501624496790265145448589920785493717258890819; Fp := GF(p); Fp2 := ExtensionField; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; for j := 0 to 6 do j, ":", (z^4)^(p^j); end for; u0 := z^2; u1 := u0^(p^2); u2 := u0^(p^4); "u0 =", u0; "u1 =", u1; "u2 =", u2; u0 + u1 + u2; trace(u0, p); trace(u1, p); trace(u2, p); F := FunctionField(Fp12, 3); a + b*z^2 + c*z^4; Output: Magma V2.11-10 Wed Dec 7 2005 13:03:59 on modular [Seed = 3607343312] ------------------------------------- 0 : z^4 1 : 1627965160026674480212199743920457793*z^4 2 : 1627965160026674480212199743920457792*z^4 3 : 1461501624496790265145448589920785493717258890818*z^4 4 : 1461501624495162299985421915440573293973338433026*z^4 5 : 1461501624495162299985421915440573293973338433027*z^4 6 : z^4 u0 = z^2 u1 = 1461501624495162299985421915440573293973338433026*z^2 u2 = 1627965160026674480212199743920457792*z^2 0 0 0 0 a + z^2*b + z^4*c Total time: 0.260 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Wed Dec 7 13:03:42 2005 Input: trace := function(x, p) return x + x^(p^2) + x^(p^4); end function; p := 1461501624496790265145448589920785493717258890819; Fp := GF(p); Fp2 := ExtensionField; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; for j := 0 to 6 do j, ":", (z^4)^(p^j); end for; u0 := z^2; u1 := u0^(p^2); u2 := u0^(p^4); "u0 =", u0; "u1 =", u1; "u2 =", u2; u0 + u1 + u2; trace(u0, p); trace(u1, p); trace(u2, p); F := FunctionField(Fp12, 3); a + b*z^2 + c*z^4; Output: Magma V2.11-10 Wed Dec 7 2005 13:03:42 on modular [Seed = 3523654180] ------------------------------------- 0 : z^4 1 : 1627965160026674480212199743920457793*z^4 2 : 1627965160026674480212199743920457792*z^4 3 : 1461501624496790265145448589920785493717258890818*z^4 4 : 1461501624495162299985421915440573293973338433026*z^4 5 : 1461501624495162299985421915440573293973338433027*z^4 6 : z^4 u0 = z^2 u1 = 1461501624495162299985421915440573293973338433026*z^2 u2 = 1627965160026674480212199743920457792*z^2 0 0 0 0 a + z^2*b + z^4*c Total time: 0.250 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Wed Dec 7 13:03:16 2005 Input: trace := function(x, p) return x + x^(p^2) + x^(p^4); end function; p := 1461501624496790265145448589920785493717258890819; Fp := GF(p); Fp2 := ExtensionField; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; for j := 0 to 6 do j, ":", (z^4)^(p^j); end for; u0 := z^2; u1 := u0^(p^2); u2 := u0^(p^4); "u0 =", u0; "u1 =", u1; "u2 =", u2; u0 + u1 + u2; trace(u0, p); trace(u1, p); trace(u2, p); F := FunctionField(Fp2, 3); a + b*z^2 + c*z^4; Output: Magma V2.11-10 Wed Dec 7 2005 13:03:16 on modular [Seed = 3708402993] ------------------------------------- 0 : z^4 1 : 1627965160026674480212199743920457793*z^4 2 : 1627965160026674480212199743920457792*z^4 3 : 1461501624496790265145448589920785493717258890818*z^4 4 : 1461501624495162299985421915440573293973338433026*z^4 5 : 1461501624495162299985421915440573293973338433027*z^4 6 : z^4 u0 = z^2 u1 = 1461501624495162299985421915440573293973338433026*z^2 u2 = 1627965160026674480212199743920457792*z^2 0 0 0 0 >> a + b*z^2 + c*z^4; ^ Runtime error in '*': Bad argument types Argument types given: FldFunRatMElt, FldFinElt Total time: 0.260 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Wed Dec 7 13:03:02 2005 Input: trace := function(x, p) return x + x^(p^2) + x^(p^4); end function; p := 1461501624496790265145448589920785493717258890819; Fp := GF(p); Fp2 := ExtensionField; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; for j := 0 to 6 do j, ":", (z^4)^(p^j); end for; u0 := z^2; u1 := u0^(p^2); u2 := u0^(p^4); "u0 =", u0; "u1 =", u1; "u2 =", u2; u0 + u1 + u2; trace(u0, p); trace(u1, p); trace(u2, p); F := FunctionField(Fp2); a + b*z^2 + c*z^4; Output: Magma V2.11-10 Wed Dec 7 2005 13:03:02 on modular [Seed = 3624712198] ------------------------------------- 0 : z^4 1 : 1627965160026674480212199743920457793*z^4 2 : 1627965160026674480212199743920457792*z^4 3 : 1461501624496790265145448589920785493717258890818*z^4 4 : 1461501624495162299985421915440573293973338433026*z^4 5 : 1461501624495162299985421915440573293973338433027*z^4 6 : z^4 u0 = z^2 u1 = 1461501624495162299985421915440573293973338433026*z^2 u2 = 1627965160026674480212199743920457792*z^2 0 0 0 0 >> F := FunctionField(Fp2); ^ Runtime error in 'AssignNames': Argument 2 should have length at most 1 >> a + b*z^2 + c*z^4; ^ User error: Identifier 'a' has not been declared or assigned Total time: 0.250 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Wed Dec 7 13:01:05 2005 Input: trace := function(x, p) return x + x^(p^2) + x^(p^4); end function; p := 1461501624496790265145448589920785493717258890819; Fp := GF(p); Fp2 := ExtensionField; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; for j := 0 to 6 do j, ":", (z^4)^(p^j); end for; u0 := z^2; u1 := u0^(p^2); u2 := u0^(p^4); "u0 =", u0; "u1 =", u1; "u2 =", u2; u0 + u1 + u2; trace(u0, p); trace(u1, p); trace(u2, p); Output: Magma V2.11-10 Wed Dec 7 2005 13:01:04 on modular [Seed = 3256263338] ------------------------------------- 0 : z^4 1 : 1627965160026674480212199743920457793*z^4 2 : 1627965160026674480212199743920457792*z^4 3 : 1461501624496790265145448589920785493717258890818*z^4 4 : 1461501624495162299985421915440573293973338433026*z^4 5 : 1461501624495162299985421915440573293973338433027*z^4 6 : z^4 u0 = z^2 u1 = 1461501624495162299985421915440573293973338433026*z^2 u2 = 1627965160026674480212199743920457792*z^2 0 0 0 0 Total time: 0.250 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Wed Dec 7 12:59:55 2005 Input: trace := function(x, p) return x + x^(p^2) + x^(p^4); end function; p := 1461501624496790265145448589920785493717258890819; Fp := GF(p); Fp2 := ExtensionField; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; for j := 0 to 6 do j, ":", (z^4)^(p^j); end for; u0 := z; u1 := u0^(p^2); u2 := u0^(p^4); u0 + u1 + u2; trace(u0, p); trace(u1, p); trace(u2, p); Output: Magma V2.11-10 Wed Dec 7 2005 12:59:55 on modular [Seed = 3441012190] ------------------------------------- 0 : z^4 1 : 1627965160026674480212199743920457793*z^4 2 : 1627965160026674480212199743920457792*z^4 3 : 1461501624496790265145448589920785493717258890818*z^4 4 : 1461501624495162299985421915440573293973338433026*z^4 5 : 1461501624495162299985421915440573293973338433027*z^4 6 : z^4 1461501624493534334825395240960361094229417975235*z 1461501624493534334825395240960361094229417975235*z 1461501624493534334825395240960361094229417975233*z 1461501624496790265145448589920785493717258890817*z Total time: 0.250 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Wed Dec 7 12:57:43 2005 Input: trace := function(x, p) return x + x^(p^2) + x^(p^4); end function; p := 1461501624496790265145448589920785493717258890819; Fp := GF(p); Fp2 := ExtensionField; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; for j := 0 to 6 do j, ":", (z^2)^(p^j); end for; u0 := z; u1 := u0^(p^2); u2 := u0^(p^4); u0 + u1 + u2; trace(u0, p); trace(u1, p); trace(u2, p); Output: Magma V2.11-10 Wed Dec 7 2005 12:57:43 on modular [Seed = 3357321559] ------------------------------------- 0 : z^2 1 : 1627965160026674480212199743920457792*i*z^2 2 : 1461501624495162299985421915440573293973338433026*z^2 3 : i*z^2 4 : 1627965160026674480212199743920457792*z^2 5 : 1461501624495162299985421915440573293973338433026*i*z^2 6 : z^2 1461501624493534334825395240960361094229417975235*z 1461501624493534334825395240960361094229417975235*z 1461501624493534334825395240960361094229417975233*z 1461501624496790265145448589920785493717258890817*z Total time: 0.250 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Wed Dec 7 12:55:35 2005 Input: trace := function(x, p) return x + x^(p^2) + x^(p^4); end function; p := 1461501624496790265145448589920785493717258890819; Fp := GF(p); Fp2 := ExtensionField; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; (z^2)^(p^3); u0 := z; u1 := u0^(p^2); u2 := u0^(p^4); u0 + u1 + u2; trace(u0, p); trace(u1, p); trace(u2, p); Output: Magma V2.11-10 Wed Dec 7 2005 12:55:34 on modular [Seed = 2993083308] ------------------------------------- i*z^2 1461501624493534334825395240960361094229417975235*z 1461501624493534334825395240960361094229417975235*z 1461501624493534334825395240960361094229417975233*z 1461501624496790265145448589920785493717258890817*z Total time: 0.230 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Wed Dec 7 12:55:25 2005 Input: trace := function(x, p) return x + x^(p^2) + x^(p^4); end function; p := 1461501624496790265145448589920785493717258890819; Fp := GF(p); Fp2 := ExtensionField; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; (z^2)^(p^6); u0 := z; u1 := u0^(p^2); u2 := u0^(p^4); u0 + u1 + u2; trace(u0, p); trace(u1, p); trace(u2, p); Output: Magma V2.11-10 Wed Dec 7 2005 12:55:25 on modular [Seed = 3177831608] ------------------------------------- z^2 1461501624493534334825395240960361094229417975235*z 1461501624493534334825395240960361094229417975235*z 1461501624493534334825395240960361094229417975233*z 1461501624496790265145448589920785493717258890817*z Total time: 0.230 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Wed Dec 7 12:53:47 2005 Input: trace := function(x, p) return x + x^(p^2) + x^(p^4); end function; p := 1461501624496790265145448589920785493717258890819; Fp := GF(p); Fp2 := ExtensionField; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; u0 := z; u1 := u0^(p^2); u2 := u0^(p^4); u0*u1*u2 eq xi; u0 + u1 + u2; trace(u0, p); trace(u1, p); trace(u2, p); Output: Magma V2.11-10 Wed Dec 7 2005 12:53:47 on modular [Seed = 2809382148] ------------------------------------- false 1461501624493534334825395240960361094229417975235*z 1461501624493534334825395240960361094229417975235*z 1461501624493534334825395240960361094229417975233*z 1461501624496790265145448589920785493717258890817*z Total time: 0.230 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Wed Dec 7 12:53:34 2005 Input: trace := function(x, p) return x + x^(p^2) + x^(p^4); end function; p := 1461501624496790265145448589920785493717258890819; Fp := GF(p); Fp2 := ExtensionField; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; u0 := i*z^2; u1 := u0^(p^2); u2 := u0^(p^4); u0*u1*u2 eq xi; u0 + u1 + u2; trace(u0, p); trace(u1, p); trace(u2, p); Output: Magma V2.11-10 Wed Dec 7 2005 12:53:34 on modular [Seed = 2725691444] ------------------------------------- false 0 0 0 0 Total time: 0.230 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Wed Dec 7 12:53:23 2005 Input: trace := function(x, p) return x + x^(p^2) + x^(p^4); end function; p := 1461501624496790265145448589920785493717258890819; Fp := GF(p); Fp2 := ExtensionField; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; u0 := i*z^3; u1 := u0^(p^2); u2 := u0^(p^4); u0*u1*u2 eq xi; u0 + u1 + u2; trace(u0, p); trace(u1, p); trace(u2, p); Output: Magma V2.11-10 Wed Dec 7 2005 12:53:22 on modular [Seed = 2910441271] ------------------------------------- false i*z^3 i*z^3 1461501624496790265145448589920785493717258890818*i*z^3 i*z^3 Total time: 0.240 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Wed Dec 7 12:52:33 2005 Input: trace := function(x, p) return x + x^(p^2) + x^(p^4); end function; p := 1461501624496790265145448589920785493717258890819; Fp := GF(p); Fp2 := ExtensionField; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; u0 := z^3; u1 := u0^(p^2); u2 := u0^(p^4); u0*u1*u2 eq xi; u0 + u1 + u2; trace(u0, p); trace(u1, p); trace(u2, p); Output: Magma V2.11-10 Wed Dec 7 2005 12:52:33 on modular [Seed = 2520970495] ------------------------------------- false z^3 z^3 1461501624496790265145448589920785493717258890818*z^3 z^3 Total time: 0.240 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Wed Dec 7 12:51:18 2005 Input: trace := function(x, p) return x + x^(p^2) + x^(p^4); end function; p := 1461501624496790265145448589920785493717258890819; Fp := GF(p); Fp2 := ExtensionField; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; u0 := z^2; u1 := u0^(p^2); u2 := u0^(p^4); u0*u1*u2 eq xi; u0 + u1 + u2; trace(u0, p); trace(u1, p); trace(u2, p); Output: Magma V2.11-10 Wed Dec 7 2005 12:51:18 on modular [Seed = 2420437646] ------------------------------------- true 0 0 0 0 Total time: 0.230 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Wed Dec 7 12:51:00 2005 Input: trace := function(x, p) return x + x^(p^2) + x^(p^4); end function; p := 1461501624496790265145448589920785493717258890819; Fp := GF(p); Fp2 := ExtensionField; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; u0 := z^2; u1 := u0^(p^2); u2 := u0^(p^4); u0*u1*u2 eq xi; trace(u0, p); trace(u1, p); trace(u2, p); Output: Magma V2.11-10 Wed Dec 7 2005 12:51:00 on modular [Seed = 2605709719] ------------------------------------- true 0 0 0 Total time: 0.230 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Wed Dec 7 12:50:47 2005 Input: trace := function(x, p) return x + x^(p^2) + x^(p^4); end function; p := 1461501624496790265145448589920785493717258890819; Fp := GF(p); Fp2 := ExtensionField; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; u0 := z^2; u1 := u0^(p^2); u2 := u0^(p^4); u0*u1*u2; trace(u0, p); trace(u1, p); trace(u2, p); Output: Magma V2.11-10 Wed Dec 7 2005 12:50:46 on modular [Seed = 2253579640] ------------------------------------- 1004782366841543307287495905570540026930615487438*i + 1004782366841543307287495905570540026930615487438 0 0 0 Total time: 0.240 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Wed Dec 7 12:49:33 2005 Input: trace := function(x, p) return x + x^(p^2) + x^(p^4); end function; p := 1461501624496790265145448589920785493717258890819; Fp := GF(p); Fp2 := ExtensionField; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; u0 := z^2; u1 := u0^(p^2); u2 := u0^(p^4); trace(u0, p); trace(u1, p); trace(u2, p); Output: Magma V2.11-10 Wed Dec 7 2005 12:49:32 on modular [Seed = 2169889317] ------------------------------------- 0 0 0 Total time: 0.230 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Wed Dec 7 12:48:19 2005 Input: trace := function(x, p) return x + x^(p^2) + x^(p^4); end function; p := 1461501624496790265145448589920785493717258890819; Fp := GF(p); Fp2 := ExtensionField; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; u := z^2; trace(z^2, p); Output: Magma V2.11-10 Wed Dec 7 2005 12:48:19 on modular [Seed = 2354638823] ------------------------------------- 0 Total time: 0.220 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Wed Dec 7 12:47:02 2005 Input: p := 1461501624496790265145448589920785493717258890819; Fp := GF(p); Fp2 := ExtensionField; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; u := z^2; c0 := u; c1 := u^(p^4); c2 := u^(p^2); c0 + c1 + c2; Output: Magma V2.11-10 Wed Dec 7 2005 12:47:02 on modular [Seed = 1973491307] ------------------------------------- 0 Total time: 0.210 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Wed Dec 7 12:45:48 2005 Input: p := 1461501624496790265145448589920785493717258890819; Fp := GF(p); Fp2 := ExtensionField; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; u := z^2; u; -u^(p^2); u^(p^4); Output: Magma V2.11-10 Wed Dec 7 2005 12:45:48 on modular [Seed = 1889807540] ------------------------------------- z^2 1627965160026674480212199743920457793*z^2 1627965160026674480212199743920457792*z^2 Total time: 0.220 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Wed Dec 7 12:44:39 2005 Input: p := 1461501624496790265145448589920785493717258890819; Fp := GF(p); Fp2 := ExtensionField; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; u := z^2; u; u^(p^2); u^(p^4); Output: Magma V2.11-10 Wed Dec 7 2005 12:44:38 on modular [Seed = 2075076597] ------------------------------------- z^2 1461501624495162299985421915440573293973338433026*z^2 1627965160026674480212199743920457792*z^2 Total time: 0.220 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Wed Dec 7 12:43:56 2005 Input: p := 1461501624496790265145448589920785493717258890819; Fp := GF(p); Fp2 := ExtensionField; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; z^6 eq xi; Output: Magma V2.11-10 Wed Dec 7 2005 12:43:56 on modular [Seed = 1722956338] ------------------------------------- true Total time: 0.210 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Wed Dec 7 12:43:47 2005 Input: p := 1461501624496790265145448589920785493717258890819; Fp := GF(p); Fp2 := ExtensionField; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; z^6 eq xi; Output: Magma V2.11-10 Wed Dec 7 2005 12:43:46 on modular [Seed = 1639267643] ------------------------------------- true Total time: 0.210 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Wed Dec 7 12:43:35 2005 Input: p := 1461501624496790265145448589920785493717258890819; Fp := GF(p); Fp2 := ExtensionField; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; z^6; xi; Output: Magma V2.11-10 Wed Dec 7 2005 12:43:34 on modular [Seed = 1824010274] ------------------------------------- 1004782366841543307287495905570540026930615487438*i + 1004782366841543307287495905570540026930615487438 1004782366841543307287495905570540026930615487438*i + 1004782366841543307287495905570540026930615487438 Total time: 0.200 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Wed Dec 7 12:42:53 2005 Input: p := 1461501624496790265145448589920785493717258890819; Fp := GF(p); Fp2 := ExtensionField; xi := 1/(-8 + 8*i); Fp12 := ExtensionField; z^6; 1/xi; Output: Magma V2.11-10 Wed Dec 7 2005 12:42:53 on modular [Seed = 1468205890] ------------------------------------- 1004782366841543307287495905570540026930615487438*i + 1004782366841543307287495905570540026930615487438 8*i + 1461501624496790265145448589920785493717258890811 Total time: 0.200 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Wed Dec 7 12:40:33 2005 Input: p := 1461501624496790265145448589920785493717258890819; Fp := GF(p); Fp2 := ExtensionField; z := 1/(-8 + 8*i); z; Output: Magma V2.11-10 Wed Dec 7 2005 12:40:33 on modular [Seed = 1367674056] ------------------------------------- 1004782366841543307287495905570540026930615487438*i + 1004782366841543307287495905570540026930615487438 Total time: 0.180 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Wed Dec 7 12:40:19 2005 Input: p := 1461501624496790265145448589920785493717258890819; Fp := GF(p); Fp2 := ExtensionField; z := 1/(-8 + 8*i); z; Output: Magma V2.11-10 Wed Dec 7 2005 12:40:18 on modular [Seed = 1552417238] ------------------------------------- >> z := 1/(-8 + 8*i); ^ User error: Identifier 'i' has not been declared or assigned >> z; ^ User error: Identifier 'z' has not been declared or assigned Total time: 0.190 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Wed Dec 7 12:40:01 2005 Input: p := 1461501624496790265145448589920785493717258890819; Fp := GF(p); Fp2 := ext; z := 1/(-8 + 8*i); z; Output: Magma V2.11-10 Wed Dec 7 2005 12:40:01 on modular [Seed = 1200822846] ------------------------------------- >> Fp2 := ext; ^ User error: bad syntax >> z := 1/(-8 + 8*i); ^ User error: Identifier 'i' has not been declared or assigned >> z; ^ User error: Identifier 'z' has not been declared or assigned Total time: 0.190 seconds, Total memory usage: 3.24MB '143.107' ************** MAGMA ***************** Host 143.107.111.59 (143.107.111.59) Time: Wed Dec 7 12:38:14 2005 Input: p := 1461501624496790265145448589920785493717258890819; Fp := GF(p); Fp2 := ext; z := 1/(-8 + 8*i); z; Output: Magma V2.11-10 Wed Dec 7 2005 12:38:10 on modular [Seed = 920808252] ------------------------------------- >> Fp2 := ext; ^ User error: Identifier 'i' has not been declared or assigned >> z := 1/(-8 + 8*i); ^ User error: Identifier 'i' has not been declared or assigned >> z; ^ User error: Identifier 'z' has not been declared or assigned Total time: 0.230 seconds, Total memory usage: 3.24MB '88.106.' ************** MAGMA ***************** Host 88.106.224.134 (88.106.224.134) Time: Wed Dec 7 11:14:03 2005 Input: a*x^2+b*x+c=0, x Output: Magma V2.11-10 Wed Dec 7 2005 11:14:03 on modular [Seed = 2537807573] ------------------------------------- >> a*x^2+b*x+c=0, x; ^ User error: Identifier 'a' has not been declared or assigned Total time: 0.200 seconds, Total memory usage: 3.24MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Wed Dec 7 10:54:30 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*A*Transpose(P)-A)); G:=GroebnerBasis(I); print G; Output: Magma V2.11-10 Wed Dec 7 2005 10:54:28 on modular [Seed = 2405157170] ------------------------------------- [ $.1 + (-a^2 + a*c - b - c + 1)/(a*c + b*c - c)*$.7 + (a*b - a - b*c - b + 1)/(a*c + b*c - c)*$.8 - $.9, $.2 + (a^2 + a*c - a)/(a*c + b*c - c)*$.7 + (-a*b + a*c)/(a*c + b*c - c)*$.8, $.3 + (-a^2 + a*b)/(a*c + b*c - c)*$.7 + (2*a*b - a)/(a*c + b*c - c)*$.8, $.4 + (a*b - a - b*c - b + 1)/(a*c + b*c - c)*$.7 + (-b^2 - b*c + b)/(a*c + b*c - c)*$.8, $.5 + (-a*b + a*c)/(a*c + b*c - c)*$.7 + (-a + b^2 - b*c - 2*b + c + 1)/(a*c + b*c - c)*$.8 - $.9, $.6 + (2*a*b - a)/(a*c + b*c - c)*$.7 + (a*b - b^2 + 2*b - 1)/(a*c + b*c - c)*$.8, $.7^2 + (b + c - 1)/(a*b - 1/2*a)*$.7*$.9 + (a*b^2 - a*b + 1/2*b^2 - b + 1/2)/(a^2*b - 1/2*a^2)*$.8^2 + (a*b - a - b*c + c)/(a^2*b - 1/2*a^2)*$.8*$.9 + (a*b*c - a*c + 1/2*c^2)/(a^2*b - 1/2*a^2)*$.9^2 + (-a*b*c + a*c - 1/2*c^2)/(a^2*b - 1/2*a^2), $.7*$.8 + (-c + 1/2)/(b - 1/2)*$.7*$.9 + (1/2*a*b - 1/2*b^2 + b - 1/2)/(a*b - 1/2*a)*$.8^2 + (1/2*a + b*c - c)/(a*b - 1/2*a)*$.8*$.9 + (1/2*a*c - 1/2*c^2)/(a*b - 1/2*a)*$.9^2 + (-1/2*a*c + 1/2*c^2)/(a*b - 1/2*a), $.7*$.9^2 + (-a*b^2*c + a*b*c - 1/4*a*c + b^2*c^2 - b*c^2 + 1/4*c^2)/(a*b^2*c + a*b*c^2 - a*b*c - 1/4*a*b - 1/4*a*c + 1/4*a - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)*$.7 + (-1/2*a^2*b^3 + 1/4*a^2*b^2 - a*b^4 + 3/2*a*b^3 - 1/2*a*b^2 - 1/2*b^5 + 5/4*b^4 - b^3 + 1/4*b^2)/(a^2*b^2*c + a^2*b*c^2 - a^2*b*c - 1/4*a^2*b - 1/4*a^2*c + 1/4*a^2 - 1/4*a*b^2 - 1/2*a*b*c + 1/2*a*b - 1/4*a*c^2 + 1/2*a*c - 1/4*a)*$.8^3 + (-1/2*a^2*b^2*c - 3/4*a^2*b^2 + 1/2*a^2*b + a*b^3*c - 3/2*a*b^3 - a*b^2*c + 5/2*a*b^2 + 1/2*a*b*c - a*b + 3/2*b^4*c - 3/4*b^4 - 3*b^3*c + 2*b^3 + 2*b^2*c - 7/4*b^2 - 1/2*b*c + 1/2*b)/(a^2*b^2*c + a^2*b*c^2 - a^2*b*c - 1/4*a^2*b - 1/4*a^2*c + 1/4*a^2 - 1/4*a*b^2 - 1/2*a*b*c + 1/2*a*b - 1/4*a*c^2 + 1/2*a*c - 1/4*a)*$.8^2*$.9 + (-1/2*a^2*b^2*c - 1/4*a^2*b*c - 1/4*a^2*b + 1/4*a^2 - 3/2*a*b^3*c - 1/2*a*b^2*c^2 + 15/4*a*b^2*c - 1/4*a*b^2 + 3/4*a*b*c^2 - 2*a*b*c + 1/2*a*b + 1/4*a*c - 1/4*a - 3/2*b^3*c^2 + 3/2*b^3*c + 9/4*b^2*c^2 - 5/2*b^2*c - b*c^2 + b*c)/(a^2*b^2*c + a^2*b*c^2 - a^2*b*c - 1/4*a^2*b - 1/4*a^2*c + 1/4*a^2 - 1/4*a*b^2 - 1/2*a*b*c + 1/2*a*b - 1/4*a*c^2 + 1/2*a*c - 1/4*a)*$.8*$.9^2 + (1/2*a^2*b^2*c - 1/4*a^2*b*c + 3/2*a*b^3*c - 1/2*a*b^2*c^2 - 11/4*a*b^2*c + 1/4*a*b*c^2 + 3/2*a*b*c - 1/4*a*c + 1/2*b^3*c^2 - 1/4*b^2*c^2)/(a^2*b^2*c + a^2*b*c^2 - a^2*b*c - 1/4*a^2*b - 1/4*a^2*c + 1/4*a^2 - 1/4*a*b^2 - 1/2*a*b*c + 1/2*a*b - 1/4*a*c^2 + 1/2*a*c - 1/4*a)*$.8 + (-1/2*a^2*b*c^2 - 1/4*a^2*b*c + 1/4*a^2*c + 3/2*a*b^2*c^2 - 1/4*a*b^2*c + 1/2*a*b*c^3 - 5/4*a*b*c^2 + 1/2*a*b*c + 1/4*a*c^2 - 1/4*a*c + 1/2*b^2*c^3 - 3/4*b^2*c^2 - 1/2*b*c^3 + 1/2*b*c^2)/(a^2*b^2*c + a^2*b*c^2 - a^2*b*c - 1/4*a^2*b - 1/4*a^2*c + 1/4*a^2 - 1/4*a*b^2 - 1/2*a*b*c + 1/2*a*b - 1/4*a*c^2 + 1/2*a*c - 1/4*a)*$.9^3 + (1/2*a^2*b*c^2 + 1/4*a^2*b*c - 1/4*a^2*c - 3/2*a*b^2*c^2 + 1/4*a*b^2*c - 1/2*a*b*c^3 + 5/4*a*b*c^2 - 1/2*a*b*c - 1/4*a*c^2 + 1/4*a*c - 1/2*b^2*c^3 + 3/4*b^2*c^2 + 1/2*b*c^3 - 1/2*b*c^2)/(a^2*b^2*c + a^2*b*c^2 - a^2*b*c - 1/4*a^2*b - 1/4*a^2*c + 1/4*a^2 - 1/4*a*b^2 - 1/2*a*b*c + 1/2*a*b - 1/4*a*c^2 + 1/2*a*c - 1/4*a)*$.9, $.8^4 + (2*a - 4*b*c + 2*b + 2*c - 2)/(a*b + b^2 - b)*$.8^3*$.9 + (2*a^2*b*c + a^2 + 2*a*b^2*c + 2*a*b*c^2 - 8*a*b*c + 2*a*b + 2*a*c - 2*a + 6*b^2*c^2 - 6*b^2*c + b^2 - 6*b*c^2 + 8*b*c - 2*b + c^2 - 2*c + 1)/(a^2*b^2 + 2*a*b^3 - 2*a*b^2 + b^4 - 2*b^3 + b^2)*$.8^2*$.9^2 + (-2*a^2*b*c - 2*a*b^2*c + 2*a*b*c^2 + 2*a*b*c - 2*b^2*c^2 + 2*b*c^2 - c^2)/(a^2*b^2 + 2*a*b^3 - 2*a*b^2 + b^4 - 2*b^3 + b^2)*$.8^2 + (2*a^2*c - 4*a*b*c^2 + 2*a*b*c + 4*a*c^2 - 4*a*c - 4*b*c^3 + 6*b*c^2 - 2*b*c + 2*c^3 - 4*c^2 + 2*c)/(a^2*b^2 + 2*a*b^3 - 2*a*b^2 + b^4 - 2*b^3 + b^2)*$.8*$.9^3 + (-2*a^2*c + 4*a*b*c^2 - 2*a*b*c + 2*a*c + 4*b*c^3 - 2*b*c^2 - 2*c^3)/(a^2*b^2 + 2*a*b^3 - 2*a*b^2 + b^4 - 2*b^3 + b^2)*$.8*$.9 + (a^2*c^2 + 2*a*c^3 - 2*a*c^2 + c^4 - 2*c^3 + c^2)/(a^2*b^2 + 2*a*b^3 - 2*a*b^2 + b^4 - 2*b^3 + b^2)*$.9^4 + (-2*a^2*c^2 + 2*a*c^2 - 2*c^4 + 2*c^3 - c^2)/(a^2*b^2 + 2*a*b^3 - 2*a*b^2 + b^4 - 2*b^3 + b^2)*$.9^2 + (a^2*c^2 - 2*a*c^3 + c^4)/(a^2*b^2 + 2*a*b^3 - 2*a*b^2 + b^4 - 2*b^3 + b^2) ] Total time: 1.439 seconds, Total memory usage: 4.60MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Wed Dec 7 10:53:33 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.8,R.7,R.9]); I:=Ideal(Eltseq(P*A*Transpose(P)-A)); G:=GroebnerBasis(I); print G; Output: Magma V2.11-10 Wed Dec 7 2005 10:53:21 on modular [Seed = 2287783664] ------------------------------------- [ $.1 + (a*b - a - b*c - b + 1)/(a*c + b*c - c)*$.7 + (-a^2 + a*c - b - c + 1)/(a*c + b*c - c)*$.8 - $.9, $.2 + (-a*b + a*c)/(a*c + b*c - c)*$.7 + (a^2 + a*c - a)/(a*c + b*c - c)*$.8, $.3 + (2*a*b - a)/(a*c + b*c - c)*$.7 + (-a^2 + a*b)/(a*c + b*c - c)*$.8, $.4 + (-b^2 - b*c + b)/(a*c + b*c - c)*$.7 + (a*b - a - b*c - b + 1)/(a*c + b*c - c)*$.8, $.5 + (-a + b^2 - b*c - 2*b + c + 1)/(a*c + b*c - c)*$.7 + (-a*b + a*c)/(a*c + b*c - c)*$.8 - $.9, $.6 + (a*b - b^2 + 2*b - 1)/(a*c + b*c - c)*$.7 + (2*a*b - a)/(a*c + b*c - c)*$.8, $.7^2 + (a - c)/(a*b + 1/2*b - 1/2)*$.7*$.9 + (a^2*b - 1/2*a^2)/(a*b^2 - a*b + 1/2*b^2 - b + 1/2)*$.8^2 + (a*b + a*c - a)/(a*b^2 - a*b + 1/2*b^2 - b + 1/2)*$.8*$.9 + (a*b*c - a*c + 1/2*c^2)/(a*b^2 - a*b + 1/2*b^2 - b + 1/2)*$.9^2 + (-a*b*c + a*c - 1/2*c^2)/(a*b^2 - a*b + 1/2*b^2 - b + 1/2), $.7*$.8 + (b*c + 1/2*b - 1/2)/(a*b + 1/2*b - 1/2)*$.7*$.9 + (-1/2*a^2*b + 1/2*a*b^2 - a*b + 1/2*a)/(a*b^2 - a*b + 1/2*b^2 - b + 1/2)*$.8^2 + (-a*b*c + 1/2*b^2 - b + 1/2)/(a*b^2 - a*b + 1/2*b^2 - b + 1/2)*$.8*$.9 + (1/2*b^2*c - 1/2*b*c^2 - b*c + 1/2*c)/(a*b^2 - a*b + 1/2*b^2 - b + 1/2)*$.9^2 + (-1/2*b^2*c + 1/2*b*c^2 + b*c - 1/2*c)/(a*b^2 - a*b + 1/2*b^2 - b + 1/2), $.7*$.9^2 + (-a^2*b^4*c + a^2*b^3*c^2 + 2*a^2*b^3*c - a^2*b^2*c - a*b^4*c + a*b^3*c^2 + 3*a*b^3*c - a*b^2*c^2 - 3*a*b^2*c + a*b*c - 1/4*b^4*c + 1/4*b^3*c^2 + b^3*c - 1/2*b^2*c^2 - 3/2*b^2*c + 1/4*b*c^2 + b*c - 1/4*c)/(a^2*b^4*c - 2*a^2*b^3*c - 1/4*a^2*b^3 + a^2*b^2*c + 1/2*a^2*b^2 - 1/4*a^2*b + a*b^4*c^2 + a*b^4*c - 1/4*a*b^4 - 2*a*b^3*c^2 - 7/2*a*b^3*c + 1/2*a*b^3 + a*b^2*c^2 + 4*a*b^2*c - 3/2*a*b*c - 1/2*a*b + 1/4*a - 1/4*b^4*c - 1/4*b^4 - 1/4*b^3*c^2 + 1/2*b^3*c + b^3 + 1/2*b^2*c^2 - 3/2*b^2 - 1/4*b*c^2 - 1/2*b*c + b + 1/4*c - 1/4)*$.7 + (-1/2*a^5*b^3 - a^4*b^4 + 3/4*a^4*b^3 + 1/4*a^4*b^2 - 1/2*a^3*b^5 + 1/2*a^3*b^4 + 1/2*a^3*b^3 - 1/2*a^3*b^2 - 1/4*a^2*b^5 + 3/4*a^2*b^4 - 3/4*a^2*b^3 + 1/4*a^2*b^2)/(a^2*b^5*c - 3*a^2*b^4*c - 1/4*a^2*b^4 + 3*a^2*b^3*c + 3/4*a^2*b^3 - a^2*b^2*c - 3/4*a^2*b^2 + 1/4*a^2*b + a*b^5*c^2 + a*b^5*c - 1/4*a*b^5 - 3*a*b^4*c^2 - 9/2*a*b^4*c + 3/4*a*b^4 + 3*a*b^3*c^2 + 15/2*a*b^3*c - 1/2*a*b^3 - a*b^2*c^2 - 11/2*a*b^2*c - 1/2*a*b^2 + 3/2*a*b*c + 3/4*a*b - 1/4*a - 1/4*b^5*c - 1/4*b^5 - 1/4*b^4*c^2 + 3/4*b^4*c + 5/4*b^4 + 3/4*b^3*c^2 - 1/2*b^3*c - 5/2*b^3 - 3/4*b^2*c^2 - 1/2*b^2*c + 5/2*b^2 + 1/4*b*c^2 + 3/4*b*c - 5/4*b - 1/4*c + 1/4)*$.8^3 + (-3/2*a^4*b^3*c - 3/4*a^4*b^3 + 3/4*a^4*b^2 - a^3*b^4*c - 3/2*a^3*b^4 - a^3*b^3*c + 5/2*a^3*b^3 + 2*a^3*b^2*c - 1/2*a^3*b^2 - 1/2*a^3*b + 1/2*a^2*b^5*c - 3/4*a^2*b^5 - 3*a^2*b^4*c + 5/4*a^2*b^4 + 4*a^2*b^3*c + 3/4*a^2*b^3 - a^2*b^2*c - 9/4*a^2*b^2 - 1/2*a^2*b*c + a^2*b - 1/2*a*b^5 - 1/2*a*b^4*c + 2*a*b^4 + 3/2*a*b^3*c - 3*a*b^3 - 3/2*a*b^2*c + 2*a*b^2 + 1/2*a*b*c - 1/2*a*b)/(a^2*b^5*c - 3*a^2*b^4*c - 1/4*a^2*b^4 + 3*a^2*b^3*c + 3/4*a^2*b^3 - a^2*b^2*c - 3/4*a^2*b^2 + 1/4*a^2*b + a*b^5*c^2 + a*b^5*c - 1/4*a*b^5 - 3*a*b^4*c^2 - 9/2*a*b^4*c + 3/4*a*b^4 + 3*a*b^3*c^2 + 15/2*a*b^3*c - 1/2*a*b^3 - a*b^2*c^2 - 11/2*a*b^2*c - 1/2*a*b^2 + 3/2*a*b*c + 3/4*a*b - 1/4*a - 1/4*b^5*c - 1/4*b^5 - 1/4*b^4*c^2 + 3/4*b^4*c + 5/4*b^4 + 3/4*b^3*c^2 - 1/2*b^3*c - 5/2*b^3 - 3/4*b^2*c^2 - 1/2*b^2*c + 5/2*b^2 + 1/4*b*c^2 + 3/4*b*c - 5/4*b - 1/4*c + 1/4)*$.8^2*$.9 + (-3/2*a^3*b^4*c - 3/2*a^3*b^3*c^2 + 3/2*a^3*b^3*c - 1/2*a^2*b^5*c - 1/2*a^2*b^4*c^2 - 3/4*a^2*b^4*c - 1/4*a^2*b^4 - 5/4*a^2*b^3*c^2 + 11/4*a^2*b^3*c + 3/4*a^2*b^3 + 7/4*a^2*b^2*c^2 - 5/4*a^2*b^2*c - 3/4*a^2*b^2 - 1/4*a^2*b*c + 1/4*a^2*b + 1/4*a*b^5*c - 1/4*a*b^5 + 1/4*a*b^4*c^2 - 9/4*a*b^4*c + 3/4*a*b^4 - 7/4*a*b^3*c^2 + 5*a*b^3*c - 1/2*a*b^3 + 11/4*a*b^2*c^2 - 4*a*b^2*c - 1/2*a*b^2 - 5/4*a*b*c^2 + 3/4*a*b*c + 3/4*a*b + 1/4*a*c - 1/4*a - 1/4*b^5 - 1/2*b^4*c + 5/4*b^4 - 1/4*b^3*c^2 + 2*b^3*c - 5/2*b^3 + 3/4*b^2*c^2 - 3*b^2*c + 5/2*b^2 - 3/4*b*c^2 + 2*b*c - 5/4*b + 1/4*c^2 - 1/2*c + 1/4)/(a^2*b^5*c - 3*a^2*b^4*c - 1/4*a^2*b^4 + 3*a^2*b^3*c + 3/4*a^2*b^3 - a^2*b^2*c - 3/4*a^2*b^2 + 1/4*a^2*b + a*b^5*c^2 + a*b^5*c - 1/4*a*b^5 - 3*a*b^4*c^2 - 9/2*a*b^4*c + 3/4*a*b^4 + 3*a*b^3*c^2 + 15/2*a*b^3*c - 1/2*a*b^3 - a*b^2*c^2 - 11/2*a*b^2*c - 1/2*a*b^2 + 3/2*a*b*c + 3/4*a*b - 1/4*a - 1/4*b^5*c - 1/4*b^5 - 1/4*b^4*c^2 + 3/4*b^4*c + 5/4*b^4 + 3/4*b^3*c^2 - 1/2*b^3*c - 5/2*b^3 - 3/4*b^2*c^2 - 1/2*b^2*c + 5/2*b^2 + 1/4*b*c^2 + 3/4*b*c - 5/4*b - 1/4*c + 1/4)*$.8*$.9^2 + (3/2*a^3*b^4*c + 1/2*a^3*b^3*c^2 - 3*a^3*b^3*c + 3/2*a^3*b^2*c + 1/2*a^2*b^5*c - 1/2*a^2*b^4*c^2 - 1/4*a^2*b^4*c + 9/4*a^2*b^3*c^2 - 9/4*a^2*b^3*c - 7/4*a^2*b^2*c^2 + 13/4*a^2*b^2*c - 5/4*a^2*b*c + 1/4*a*b^5*c - 1/4*a*b^4*c^2 - 3/4*a*b^4*c + 7/4*a*b^3*c^2 + 1/2*a*b^3*c - 11/4*a*b^2*c^2 + 1/2*a*b^2*c + 5/4*a*b*c^2 - 3/4*a*b*c + 1/4*a*c + 1/4*b^3*c^2 - 3/4*b^2*c^2 + 3/4*b*c^2 - 1/4*c^2)/(a^2*b^5*c - 3*a^2*b^4*c - 1/4*a^2*b^4 + 3*a^2*b^3*c + 3/4*a^2*b^3 - a^2*b^2*c - 3/4*a^2*b^2 + 1/4*a^2*b + a*b^5*c^2 + a*b^5*c - 1/4*a*b^5 - 3*a*b^4*c^2 - 9/2*a*b^4*c + 3/4*a*b^4 + 3*a*b^3*c^2 + 15/2*a*b^3*c - 1/2*a*b^3 - a*b^2*c^2 - 11/2*a*b^2*c - 1/2*a*b^2 + 3/2*a*b*c + 3/4*a*b - 1/4*a - 1/4*b^5*c - 1/4*b^5 - 1/4*b^4*c^2 + 3/4*b^4*c + 5/4*b^4 + 3/4*b^3*c^2 - 1/2*b^3*c - 5/2*b^3 - 3/4*b^2*c^2 - 1/2*b^2*c + 5/2*b^2 + 1/4*b*c^2 + 3/4*b*c - 5/4*b - 1/4*c + 1/4)*$.8 + (-3/2*a^2*b^4*c^2 - 1/4*a^2*b^4*c - 1/2*a^2*b^3*c^3 + 9/4*a^2*b^3*c^2 + 3/4*a^2*b^3*c - 3/4*a^2*b^2*c^2 - 3/4*a^2*b^2*c + 1/4*a^2*b*c + 1/2*a*b^5*c^2 - 1/4*a*b^5*c - 1/2*a*b^4*c^3 - 13/4*a*b^4*c^2 + 3/4*a*b^4*c + 1/2*a*b^3*c^3 + 25/4*a*b^3*c^2 - 1/2*a*b^3*c - 19/4*a*b^2*c^2 - 1/2*a*b^2*c + 5/4*a*b*c^2 + 3/4*a*b*c - 1/4*a*c - 1/4*b^5*c - 1/4*b^4*c^2 + 5/4*b^4*c + b^3*c^2 - 5/2*b^3*c - 3/2*b^2*c^2 + 5/2*b^2*c + b*c^2 - 5/4*b*c - 1/4*c^2 + 1/4*c)/(a^2*b^5*c - 3*a^2*b^4*c - 1/4*a^2*b^4 + 3*a^2*b^3*c + 3/4*a^2*b^3 - a^2*b^2*c - 3/4*a^2*b^2 + 1/4*a^2*b + a*b^5*c^2 + a*b^5*c - 1/4*a*b^5 - 3*a*b^4*c^2 - 9/2*a*b^4*c + 3/4*a*b^4 + 3*a*b^3*c^2 + 15/2*a*b^3*c - 1/2*a*b^3 - a*b^2*c^2 - 11/2*a*b^2*c - 1/2*a*b^2 + 3/2*a*b*c + 3/4*a*b - 1/4*a - 1/4*b^5*c - 1/4*b^5 - 1/4*b^4*c^2 + 3/4*b^4*c + 5/4*b^4 + 3/4*b^3*c^2 - 1/2*b^3*c - 5/2*b^3 - 3/4*b^2*c^2 - 1/2*b^2*c + 5/2*b^2 + 1/4*b*c^2 + 3/4*b*c - 5/4*b - 1/4*c + 1/4)*$.9^3 + (3/2*a^2*b^4*c^2 + 1/4*a^2*b^4*c + 1/2*a^2*b^3*c^3 - 9/4*a^2*b^3*c^2 - 3/4*a^2*b^3*c + 3/4*a^2*b^2*c^2 + 3/4*a^2*b^2*c - 1/4*a^2*b*c - 1/2*a*b^5*c^2 + 1/4*a*b^5*c + 1/2*a*b^4*c^3 + 13/4*a*b^4*c^2 - 3/4*a*b^4*c - 1/2*a*b^3*c^3 - 25/4*a*b^3*c^2 + 1/2*a*b^3*c + 19/4*a*b^2*c^2 + 1/2*a*b^2*c - 5/4*a*b*c^2 - 3/4*a*b*c + 1/4*a*c + 1/4*b^5*c + 1/4*b^4*c^2 - 5/4*b^4*c - b^3*c^2 + 5/2*b^3*c + 3/2*b^2*c^2 - 5/2*b^2*c - b*c^2 + 5/4*b*c + 1/4*c^2 - 1/4*c)/(a^2*b^5*c - 3*a^2*b^4*c - 1/4*a^2*b^4 + 3*a^2*b^3*c + 3/4*a^2*b^3 - a^2*b^2*c - 3/4*a^2*b^2 + 1/4*a^2*b + a*b^5*c^2 + a*b^5*c - 1/4*a*b^5 - 3*a*b^4*c^2 - 9/2*a*b^4*c + 3/4*a*b^4 + 3*a*b^3*c^2 + 15/2*a*b^3*c - 1/2*a*b^3 - a*b^2*c^2 - 11/2*a*b^2*c - 1/2*a*b^2 + 3/2*a*b*c + 3/4*a*b - 1/4*a - 1/4*b^5*c - 1/4*b^5 - 1/4*b^4*c^2 + 3/4*b^4*c + 5/4*b^4 + 3/4*b^3*c^2 - 1/2*b^3*c - 5/2*b^3 - 3/4*b^2*c^2 - 1/2*b^2*c + 5/2*b^2 + 1/4*b*c^2 + 3/4*b*c - 5/4*b - 1/4*c + 1/4)*$.9, $.8^4 + (4*a*b*c + 2*a*b - 2*a + 2*b^2 + 2*b*c - 4*b - 2*c + 2)/(a^2*b + a*b^2 - a*b)*$.8^3*$.9 + (2*a^2*b^3*c + 6*a^2*b^2*c^2 + 2*a^2*b^2*c + a^2*b^2 - 4*a^2*b*c - 2*a^2*b + a^2 + 2*a*b^4*c + 2*a*b^3*c^2 + 2*a*b^3 + 2*a*b^2*c^2 - 4*a*b^2*c - 6*a*b^2 - 4*a*b*c^2 + 6*a*b + 2*a*c - 2*a + b^4 + 2*b^3*c - 4*b^3 + b^2*c^2 - 6*b^2*c + 6*b^2 - 2*b*c^2 + 6*b*c - 4*b + c^2 - 2*c + 1)/(a^4*b^2 + 2*a^3*b^3 - 2*a^3*b^2 + a^2*b^4 - 2*a^2*b^3 + a^2*b^2)*$.8^2*$.9^2 + (-2*a^2*b^3*c - 2*a^2*b^2*c^2 + 4*a^2*b^2*c - 2*a^2*b*c - 2*a*b^4*c + 2*a*b^3*c^2 + 6*a*b^3*c - 6*a*b^2*c^2 - 6*a*b^2*c + 4*a*b*c^2 + 2*a*b*c - b^2*c^2 + 2*b*c^2 - c^2)/(a^4*b^2 + 2*a^3*b^3 - 2*a^3*b^2 + a^2*b^4 - 2*a^2*b^3 + a^2*b^2)*$.8^2 + (4*a*b^2*c^2 + 2*a*b^2*c + 4*a*b*c^3 - 2*a*b*c^2 - 4*a*b*c - 2*a*c^2 + 2*a*c + 2*b^3*c + 4*b^2*c^2 - 6*b^2*c + 2*b*c^3 - 8*b*c^2 + 6*b*c - 2*c^3 + 4*c^2 - 2*c)/(a^4*b + 2*a^3*b^2 - 2*a^3*b + a^2*b^3 - 2*a^2*b^2 + a^2*b)*$.8*$.9^3 + (-4*a*b^3*c^2 - 2*a*b^3*c - 4*a*b^2*c^3 + 6*a*b^2*c^2 + 6*a*b^2*c - 2*a*b*c^2 - 6*a*b*c + 2*a*c - 2*b^4*c + 8*b^3*c - 2*b^2*c^3 - 12*b^2*c + 2*b*c^3 + 8*b*c - 2*c)/(a^4*b^2 + 2*a^3*b^3 - 2*a^3*b^2 + a^2*b^4 - 2*a^2*b^3 + a^2*b^2)*$.8*$.9 + (b^2*c^2 + 2*b*c^3 - 2*b*c^2 + c^4 - 2*c^3 + c^2)/(a^4 + 2*a^3*b - 2*a^3 + a^2*b^2 - 2*a^2*b + a^2)*$.9^4 + (-2*b^4*c^2 + 6*b^3*c^2 - 2*b^2*c^4 - 2*b^2*c^3 - 7*b^2*c^2 + 2*b*c^3 + 4*b*c^2 - c^2)/(a^4*b^2 + 2*a^3*b^3 - 2*a^3*b^2 + a^2*b^4 - 2*a^2*b^3 + a^2*b^2)*$.9^2 + (b^4*c^2 - 2*b^3*c^3 - 4*b^3*c^2 + b^2*c^4 + 4*b^2*c^3 + 6*b^2*c^2 - 2*b*c^3 - 4*b*c^2 + c^2)/(a^4*b^2 + 2*a^3*b^3 - 2*a^3*b^2 + a^2*b^4 - 2*a^2*b^3 + a^2*b^2) ] Total time: 12.500 seconds, Total memory usage: 10.35MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Wed Dec 7 10:53:14 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.8,R.7,R.9]); I:=Ideal(Eltseq(P*A*Transpose(P)-A)); G:=GroebnerBasis(I); Output: Magma V2.11-10 Wed Dec 7 2005 10:53:01 on modular [Seed = 1956655648] ------------------------------------- Total time: 12.470 seconds, Total memory usage: 10.35MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Wed Dec 7 10:51:18 2005 Input: K:=RationalField(); A:=Matrix(K,3,3,[[2,1,1],[1,2,1],[1,1,2]]); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*A*Transpose(P)-A)); print GroebnerBasis(I); Output: Magma V2.11-10 Wed Dec 7 2005 10:51:17 on modular [Seed = 2091400064] ------------------------------------- [ $.1^2 + 1/3*$.1*$.5 + 1/3*$.1*$.6 + 1/3*$.1*$.8 + 1/3*$.1*$.9 + 1/3*$.2*$.4 + 2/3*$.2*$.5 + 1/3*$.2*$.6 + 1/3*$.2*$.7 + 2/3*$.2*$.8 + 1/3*$.2*$.9 + 1/3*$.3*$.4 + 1/3*$.3*$.5 + 2/3*$.3*$.6 + 1/3*$.3*$.7 + 1/3*$.3*$.8 + 2/3*$.3*$.9 - $.4*$.5 - $.4*$.6 + 1/3*$.4*$.8 + 1/3*$.4*$.9 - $.5^2 - $.5*$.6 + 1/3*$.5*$.7 + 2/3*$.5*$.8 + 1/3*$.5*$.9 - $.6^2 + 1/3*$.6*$.7 + 1/3*$.6*$.8 + 2/3*$.6*$.9 - $.7*$.8 - $.7*$.9 - $.8^2 - $.8*$.9 - $.9^2, $.1*$.2 - 1/3*$.1*$.5 - 1/3*$.1*$.8 - 1/3*$.2*$.4 - 1/3*$.2*$.7 + $.4*$.5 - 1/3*$.4*$.8 - 1/3*$.5*$.7 + $.7*$.8 + 1/3, $.1*$.3 - 1/3*$.1*$.6 - 1/3*$.1*$.9 - 1/3*$.3*$.4 - 1/3*$.3*$.7 + $.4*$.6 - 1/3*$.4*$.9 - 1/3*$.6*$.7 + $.7*$.9 + 1/3, $.1*$.4 + 1/2*$.1*$.5 + 1/2*$.1*$.6 + 1/2*$.2*$.4 + $.2*$.5 + 1/2*$.2*$.6 + 1/2*$.3*$.4 + 1/2*$.3*$.5 + $.3*$.6 - 1/2, $.1*$.5^2 + 1/3*$.1*$.5*$.9 + 1/3*$.1*$.6*$.8 + $.1*$.6*$.9 - 2/3*$.1*$.8*$.9 - $.1*$.9^2 - 11/24*$.1 - $.2*$.4*$.5 - 1/6*$.2*$.4*$.9 - 1/6*$.2*$.5*$.9 - 1/6*$.2*$.6*$.7 - 1/6*$.2*$.6*$.8 + 1/3*$.2*$.6*$.9 + 1/3*$.2*$.7*$.9 + 1/3*$.2*$.8*$.9 - 1/3*$.2*$.9^2 - 1/24*$.2 - 1/6*$.3*$.4*$.8 - 1/2*$.3*$.4*$.9 - 1/6*$.3*$.5*$.7 + 1/3*$.3*$.5*$.8 - 1/6*$.3*$.5*$.9 - 1/2*$.3*$.6*$.7 - 1/6*$.3*$.6*$.8 + 1/3*$.3*$.7*$.8 + $.3*$.7*$.9 - 1/3*$.3*$.8^2 + 1/3*$.3*$.8*$.9 + 1/3*$.3 - 1/6*$.4*$.5*$.9 - 1/6*$.4*$.6*$.8 - 1/2*$.4*$.6*$.9 + 1/3*$.4*$.8*$.9 + 1/2*$.4*$.9^2 + 3/8*$.4 + 1/6*$.5^2*$.9 + 1/3*$.5*$.6*$.7 - 1/6*$.5*$.6*$.8 - 1/6*$.5*$.6*$.9 - 1/6*$.5*$.7*$.9 - 1/6*$.5*$.8*$.9 + 1/6*$.5*$.9^2 + 1/8*$.5 + 1/2*$.6^2*$.7 + 1/6*$.6^2*$.8 - 1/6*$.6*$.7*$.8 - 1/2*$.6*$.7*$.9 + 1/6*$.6*$.8^2 - 1/6*$.6*$.8*$.9 - 7/24*$.7 - 5/24*$.8 - 1/3*$.9, $.1*$.5*$.6 - 1/2*$.1*$.5*$.9 - 1/2*$.1*$.6*$.8 + $.1*$.8*$.9 + 3/8*$.1 - $.3*$.4*$.5 + 1/2*$.3*$.4*$.8 + 1/2*$.3*$.5*$.7 - $.3*$.7*$.8 - 3/8*$.3 + 1/2*$.4*$.5*$.9 - 1/2*$.4*$.8*$.9 - 1/8*$.4 - 1/2*$.5*$.6*$.7 + 1/2*$.6*$.7*$.8 + 1/8*$.6 - 1/8*$.7 + 1/8*$.9, $.1*$.5*$.8 + 1/3*$.1*$.5*$.9 + 1/3*$.1*$.6*$.8 + $.1*$.6*$.9 - 2/3*$.1 - $.2*$.5*$.7 - 1/3*$.2*$.5*$.9 - 1/3*$.2*$.6*$.7 + 1/3*$.2*$.6*$.9 - 1/3*$.2 - 1/3*$.3*$.5*$.7 + 1/3*$.3*$.5*$.8 - $.3*$.6*$.7 - 1/3*$.3*$.6*$.8 - 1/3*$.3 + 2/3*$.7 + 1/3*$.8 + 1/3*$.9, $.1*$.5*$.9^2 - 3/2*$.1*$.5 - $.1*$.6*$.8*$.9 - 1/2*$.1*$.6 + 3/4*$.1*$.8 + 1/4*$.1*$.9 - $.3*$.5*$.7*$.9 - 1/2*$.3*$.5 + $.3*$.6*$.7*$.8 + 1/2*$.3*$.6 + 1/4*$.3*$.8 - 1/4*$.3*$.9 + 3/4*$.5*$.7 + 1/4*$.5*$.9 + 1/4*$.6*$.7 - 1/4*$.6*$.9 - 3/4*$.7*$.8 - 1/4*$.7*$.9 - 1/4*$.8*$.9 + 1/4*$.9^2 - 1/4, $.1*$.6^2 - $.1*$.6*$.9 + $.1*$.9^2 - 9/8*$.1 - $.3*$.4*$.6 + 1/2*$.3*$.4*$.9 + 1/2*$.3*$.6*$.7 - $.3*$.7*$.9 - 3/8*$.3 + 1/2*$.4*$.6*$.9 - 1/2*$.4*$.9^2 + 3/8*$.4 - 1/2*$.6^2*$.7 + 1/2*$.6*$.7*$.9 + 1/8*$.6 + 3/8*$.7 + 1/8*$.9, $.1*$.7 + 1/2*$.1*$.8 + 1/2*$.1*$.9 + 1/2*$.2*$.7 + $.2*$.8 + 1/2*$.2*$.9 + 1/2*$.3*$.7 + 1/2*$.3*$.8 + $.3*$.9 - 1/2, $.1*$.8^2 + 2/3*$.1*$.8*$.9 + $.1*$.9^2 - 4/3*$.1 - $.2*$.7*$.8 - 1/3*$.2*$.7*$.9 - 1/3*$.2*$.8*$.9 + 1/3*$.2*$.9^2 - 2/3*$.2 - 1/3*$.3*$.7*$.8 - $.3*$.7*$.9 + 1/3*$.3*$.8^2 - 1/3*$.3*$.8*$.9 - 2/3*$.3 + 2/3*$.7 + 1/3*$.8 + 1/3*$.9, $.2^2 - 2/3*$.2*$.5 - 2/3*$.2*$.8 + $.5^2 - 2/3*$.5*$.8 + $.8^2 - 1, $.2*$.3 - 1/3*$.2*$.6 - 1/3*$.2*$.9 - 1/3*$.3*$.5 - 1/3*$.3*$.8 + $.5*$.6 - 1/3*$.5*$.9 - 1/3*$.6*$.8 + $.8*$.9 + 1/3, $.2*$.4*$.6 - 1/2*$.2*$.4*$.9 - 1/2*$.2*$.6*$.7 + $.2*$.7*$.9 + 3/8*$.2 - $.3*$.4*$.5 + 1/2*$.3*$.4*$.8 + 1/2*$.3*$.5*$.7 - $.3*$.7*$.8 - 3/8*$.3 + 1/2*$.4*$.5*$.9 - 1/2*$.4*$.6*$.8 - 1/2*$.5*$.7*$.9 - 1/8*$.5 + 1/2*$.6*$.7*$.8 + 1/8*$.6 - 1/8*$.8 + 1/8*$.9, $.2*$.4*$.8 + 1/3*$.2*$.4*$.9 - $.2*$.5*$.7 - 1/3*$.2*$.5*$.9 - 1/3*$.2*$.6*$.7 + 1/3*$.2*$.6*$.8 + 1/3*$.3*$.4*$.8 + $.3*$.4*$.9 - 1/3*$.3*$.5*$.7 + 1/3*$.3*$.5*$.9 - $.3*$.6*$.7 - 1/3*$.3*$.6*$.8 - 2/3*$.4 - 1/3*$.5 - 1/3*$.6 + 2/3*$.7 + 1/3*$.8 + 1/3*$.9, $.2*$.4*$.9^2 - 3/2*$.2*$.4 - $.2*$.6*$.7*$.9 - 1/2*$.2*$.6 + 3/4*$.2*$.7 + 1/4*$.2*$.9 - $.3*$.4*$.8*$.9 - 1/2*$.3*$.4 + $.3*$.6*$.7*$.8 + 1/2*$.3*$.6 + 1/4*$.3*$.7 - 1/4*$.3*$.9 + 3/4*$.4*$.8 + 1/4*$.4*$.9 + 1/4*$.6*$.8 - 1/4*$.6*$.9 - 3/4*$.7*$.8 - 1/4*$.7*$.9 - 1/4*$.8*$.9 + 1/4*$.9^2 - 1/4, $.2*$.5*$.6 - 1/2*$.2*$.5*$.9 - 1/2*$.2*$.6*$.8 + $.2*$.8*$.9 + 3/8*$.2 - $.3*$.5^2 + $.3*$.5*$.8 - $.3*$.8^2 + 9/8*$.3 + 1/2*$.5^2*$.9 - 1/2*$.5*$.6*$.8 - 1/2*$.5*$.8*$.9 - 1/8*$.5 + 1/2*$.6*$.8^2 - 3/8*$.6 - 1/8*$.8 - 3/8*$.9, $.2*$.5*$.9^2 - 3/2*$.2*$.5 - $.2*$.6*$.8*$.9 - 1/2*$.2*$.6 + 3/4*$.2*$.8 + 1/4*$.2*$.9 - $.3*$.5*$.8*$.9 - 1/2*$.3*$.5 + $.3*$.6*$.8^2 - 3/2*$.3*$.6 + 1/4*$.3*$.8 + 3/4*$.3*$.9 + 3/4*$.5*$.8 + 1/4*$.5*$.9 + 1/4*$.6*$.8 + 3/4*$.6*$.9 - 3/4*$.8^2 - 1/2*$.8*$.9 - 3/4*$.9^2 + 1/2, $.2*$.6^2 - $.2*$.6*$.9 + $.2*$.9^2 - 9/8*$.2 - $.3*$.5*$.6 + 1/2*$.3*$.5*$.9 + 1/2*$.3*$.6*$.8 - $.3*$.8*$.9 - 3/8*$.3 + 1/2*$.5*$.6*$.9 - 1/2*$.5*$.9^2 + 3/8*$.5 - 1/2*$.6^2*$.8 + 1/2*$.6*$.8*$.9 + 1/8*$.6 + 3/8*$.8 + 1/8*$.9, $.3^2 - 2/3*$.3*$.6 - 2/3*$.3*$.9 + $.6^2 - 2/3*$.6*$.9 + $.9^2 - 1, $.4^2 + $.4*$.5 + $.4*$.6 + $.5^2 + $.5*$.6 + $.6^2 - 1, $.4*$.5*$.8 + 1/3*$.4*$.5*$.9 + 1/3*$.4*$.6*$.8 + $.4*$.6*$.9 - 2/3*$.4 - $.5^2*$.7 - 1/3*$.5^2*$.9 - 2/3*$.5*$.6*$.7 + 1/3*$.5*$.6*$.8 + 1/3*$.5*$.6*$.9 - 1/3*$.5 - $.6^2*$.7 - 1/3*$.6^2*$.8 - 1/3*$.6 + 4/3*$.7 + 2/3*$.8 + 2/3*$.9, $.4*$.5*$.9^2 - 3/2*$.4*$.5 - $.4*$.6*$.8*$.9 - 1/2*$.4*$.6 + 3/4*$.4*$.8 + 1/4*$.4*$.9 - $.5*$.6*$.7*$.9 - 1/2*$.5*$.6 + 3/4*$.5*$.7 + 1/4*$.5*$.9 + $.6^2*$.7*$.8 + 1/2*$.6^2 + 1/4*$.6*$.7 + 1/4*$.6*$.8 - 1/2*$.6*$.9 - 3/2*$.7*$.8 - 1/2*$.7*$.9 - 1/2*$.8*$.9 + 1/2*$.9^2 - 3/4, $.4*$.7 + 1/2*$.4*$.8 + 1/2*$.4*$.9 + 1/2*$.5*$.7 + $.5*$.8 + 1/2*$.5*$.9 + 1/2*$.6*$.7 + 1/2*$.6*$.8 + $.6*$.9 - 1/2, $.4*$.8^2 + 2/3*$.4*$.8*$.9 + $.4*$.9^2 - 4/3*$.4 - $.5*$.7*$.8 - 1/3*$.5*$.7*$.9 - 1/3*$.5*$.8*$.9 + 1/3*$.5*$.9^2 - 2/3*$.5 - 1/3*$.6*$.7*$.8 - $.6*$.7*$.9 + 1/3*$.6*$.8^2 - 1/3*$.6*$.8*$.9 - 2/3*$.6 + 2/3*$.7 + 1/3*$.8 + 1/3*$.9, $.5^2*$.9^2 - 3/2*$.5^2 - 2*$.5*$.6*$.8*$.9 - $.5*$.6 + 3/2*$.5*$.8 + 1/2*$.5*$.9 + $.6^2*$.8^2 - 3/2*$.6^2 + 1/2*$.6*$.8 + 3/2*$.6*$.9 - 3/2*$.8^2 - $.8*$.9 - 3/2*$.9^2 + 3/2, $.7^2 + $.7*$.8 + $.7*$.9 + $.8^2 + $.8*$.9 + $.9^2 - 1 ] Total time: 0.210 seconds, Total memory usage: 3.86MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Wed Dec 7 10:50:11 2005 Input: K:=RationalField(); A:=Matrix(K,3,3,[[0,1,1],[-1,0,1],[-1,-1,0]]); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*A*Transpose(P)-A)); print GroebnerBasis(I); Output: Magma V2.11-10 Wed Dec 7 2005 10:50:11 on modular [Seed = 1739792605] ------------------------------------- [ $.1 - $.3 - $.4 + $.6 + $.7 - $.9, $.2 + $.3 - $.5 - $.6 + $.8 + $.9, $.4*$.8 + $.4*$.9 - $.5*$.7 + $.5*$.9 - $.6*$.7 - $.6*$.8 - 1 ] Total time: 0.200 seconds, Total memory usage: 3.34MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Wed Dec 7 10:33:41 2005 Input: K:=RationalField(); A:=Matrix(K,3,3,[[1,1,1],[0,1,1],[0,0,1]]); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.8,R.7,R.9]); I:=Ideal(Eltseq(P*A*Transpose(P)-A)); print GroebnerBasis(I); Output: Magma V2.11-10 Wed Dec 7 2005 10:33:40 on modular [Seed = 1183974811] ------------------------------------- [ $.1 - $.7 - $.8 - $.9, $.2 + $.8, $.3 + $.7, $.4 - $.7 - $.8, $.5 - $.7 - $.9, $.6 + $.7 + $.8, $.7^2 + $.7*$.8 + $.7*$.9 - $.8*$.9, $.8^2 + 2*$.8*$.9 + $.9^2 - 1 ] Total time: 0.200 seconds, Total memory usage: 3.53MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Wed Dec 7 10:31:07 2005 Input: K:=RationalField(); A:=Matrix(K,3,3,[[1,1,1],[0,1,1],[0,0,1]]); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*A*Transpose(P)-A)); print GroebnerBasis(I); Output: Magma V2.11-10 Wed Dec 7 2005 10:31:06 on modular [Seed = 1083442035] ------------------------------------- [ $.1 - $.7 - $.8 - $.9, $.2 + $.7, $.3 + $.8, $.4 - $.7 - $.8, $.5 - $.8 - $.9, $.6 + $.7 + $.8, $.7^2 + 2*$.7*$.9 + $.9^2 - 1, $.7*$.8 - $.7*$.9 + $.8^2 + $.8*$.9, $.7*$.9^2 - 1/2*$.8^3 - 1/2*$.8^2*$.9 - 1/2*$.8*$.9^2 + 1/2*$.8 + 1/2*$.9^3 - 1/2*$.9, $.8^4 + 2*$.8^2*$.9^2 - $.8^2 + 2*$.8*$.9 + $.9^4 - $.9^2 ] Total time: 0.220 seconds, Total memory usage: 3.53MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Wed Dec 7 10:25:30 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*A*Transpose(P)-A)); G:=GroebnerBasis(I); print G; Output: Magma V2.11-10 Wed Dec 7 2005 10:25:28 on modular [Seed = 920882135] ------------------------------------- [ $.1 + (-a^2 + a*c - b - c + 1)/(a*c + b*c - c)*$.7 + (a*b - a - b*c - b + 1)/(a*c + b*c - c)*$.8 - $.9, $.2 + (a^2 + a*c - a)/(a*c + b*c - c)*$.7 + (-a*b + a*c)/(a*c + b*c - c)*$.8, $.3 + (-a^2 + a*b)/(a*c + b*c - c)*$.7 + (2*a*b - a)/(a*c + b*c - c)*$.8, $.4 + (a*b - a - b*c - b + 1)/(a*c + b*c - c)*$.7 + (-b^2 - b*c + b)/(a*c + b*c - c)*$.8, $.5 + (-a*b + a*c)/(a*c + b*c - c)*$.7 + (-a + b^2 - b*c - 2*b + c + 1)/(a*c + b*c - c)*$.8 - $.9, $.6 + (2*a*b - a)/(a*c + b*c - c)*$.7 + (a*b - b^2 + 2*b - 1)/(a*c + b*c - c)*$.8, $.7^2 + (b + c - 1)/(a*b - 1/2*a)*$.7*$.9 + (a*b^2 - a*b + 1/2*b^2 - b + 1/2)/(a^2*b - 1/2*a^2)*$.8^2 + (a*b - a - b*c + c)/(a^2*b - 1/2*a^2)*$.8*$.9 + (a*b*c - a*c + 1/2*c^2)/(a^2*b - 1/2*a^2)*$.9^2 + (-a*b*c + a*c - 1/2*c^2)/(a^2*b - 1/2*a^2), $.7*$.8 + (-c + 1/2)/(b - 1/2)*$.7*$.9 + (1/2*a*b - 1/2*b^2 + b - 1/2)/(a*b - 1/2*a)*$.8^2 + (1/2*a + b*c - c)/(a*b - 1/2*a)*$.8*$.9 + (1/2*a*c - 1/2*c^2)/(a*b - 1/2*a)*$.9^2 + (-1/2*a*c + 1/2*c^2)/(a*b - 1/2*a), $.7*$.9^2 + (-a*b^2*c + a*b*c - 1/4*a*c + b^2*c^2 - b*c^2 + 1/4*c^2)/(a*b^2*c + a*b*c^2 - a*b*c - 1/4*a*b - 1/4*a*c + 1/4*a - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)*$.7 + (-1/2*a^2*b^3 + 1/4*a^2*b^2 - a*b^4 + 3/2*a*b^3 - 1/2*a*b^2 - 1/2*b^5 + 5/4*b^4 - b^3 + 1/4*b^2)/(a^2*b^2*c + a^2*b*c^2 - a^2*b*c - 1/4*a^2*b - 1/4*a^2*c + 1/4*a^2 - 1/4*a*b^2 - 1/2*a*b*c + 1/2*a*b - 1/4*a*c^2 + 1/2*a*c - 1/4*a)*$.8^3 + (-1/2*a^2*b^2*c - 3/4*a^2*b^2 + 1/2*a^2*b + a*b^3*c - 3/2*a*b^3 - a*b^2*c + 5/2*a*b^2 + 1/2*a*b*c - a*b + 3/2*b^4*c - 3/4*b^4 - 3*b^3*c + 2*b^3 + 2*b^2*c - 7/4*b^2 - 1/2*b*c + 1/2*b)/(a^2*b^2*c + a^2*b*c^2 - a^2*b*c - 1/4*a^2*b - 1/4*a^2*c + 1/4*a^2 - 1/4*a*b^2 - 1/2*a*b*c + 1/2*a*b - 1/4*a*c^2 + 1/2*a*c - 1/4*a)*$.8^2*$.9 + (-1/2*a^2*b^2*c - 1/4*a^2*b*c - 1/4*a^2*b + 1/4*a^2 - 3/2*a*b^3*c - 1/2*a*b^2*c^2 + 15/4*a*b^2*c - 1/4*a*b^2 + 3/4*a*b*c^2 - 2*a*b*c + 1/2*a*b + 1/4*a*c - 1/4*a - 3/2*b^3*c^2 + 3/2*b^3*c + 9/4*b^2*c^2 - 5/2*b^2*c - b*c^2 + b*c)/(a^2*b^2*c + a^2*b*c^2 - a^2*b*c - 1/4*a^2*b - 1/4*a^2*c + 1/4*a^2 - 1/4*a*b^2 - 1/2*a*b*c + 1/2*a*b - 1/4*a*c^2 + 1/2*a*c - 1/4*a)*$.8*$.9^2 + (1/2*a^2*b^2*c - 1/4*a^2*b*c + 3/2*a*b^3*c - 1/2*a*b^2*c^2 - 11/4*a*b^2*c + 1/4*a*b*c^2 + 3/2*a*b*c - 1/4*a*c + 1/2*b^3*c^2 - 1/4*b^2*c^2)/(a^2*b^2*c + a^2*b*c^2 - a^2*b*c - 1/4*a^2*b - 1/4*a^2*c + 1/4*a^2 - 1/4*a*b^2 - 1/2*a*b*c + 1/2*a*b - 1/4*a*c^2 + 1/2*a*c - 1/4*a)*$.8 + (-1/2*a^2*b*c^2 - 1/4*a^2*b*c + 1/4*a^2*c + 3/2*a*b^2*c^2 - 1/4*a*b^2*c + 1/2*a*b*c^3 - 5/4*a*b*c^2 + 1/2*a*b*c + 1/4*a*c^2 - 1/4*a*c + 1/2*b^2*c^3 - 3/4*b^2*c^2 - 1/2*b*c^3 + 1/2*b*c^2)/(a^2*b^2*c + a^2*b*c^2 - a^2*b*c - 1/4*a^2*b - 1/4*a^2*c + 1/4*a^2 - 1/4*a*b^2 - 1/2*a*b*c + 1/2*a*b - 1/4*a*c^2 + 1/2*a*c - 1/4*a)*$.9^3 + (1/2*a^2*b*c^2 + 1/4*a^2*b*c - 1/4*a^2*c - 3/2*a*b^2*c^2 + 1/4*a*b^2*c - 1/2*a*b*c^3 + 5/4*a*b*c^2 - 1/2*a*b*c - 1/4*a*c^2 + 1/4*a*c - 1/2*b^2*c^3 + 3/4*b^2*c^2 + 1/2*b*c^3 - 1/2*b*c^2)/(a^2*b^2*c + a^2*b*c^2 - a^2*b*c - 1/4*a^2*b - 1/4*a^2*c + 1/4*a^2 - 1/4*a*b^2 - 1/2*a*b*c + 1/2*a*b - 1/4*a*c^2 + 1/2*a*c - 1/4*a)*$.9, $.8^4 + (2*a - 4*b*c + 2*b + 2*c - 2)/(a*b + b^2 - b)*$.8^3*$.9 + (2*a^2*b*c + a^2 + 2*a*b^2*c + 2*a*b*c^2 - 8*a*b*c + 2*a*b + 2*a*c - 2*a + 6*b^2*c^2 - 6*b^2*c + b^2 - 6*b*c^2 + 8*b*c - 2*b + c^2 - 2*c + 1)/(a^2*b^2 + 2*a*b^3 - 2*a*b^2 + b^4 - 2*b^3 + b^2)*$.8^2*$.9^2 + (-2*a^2*b*c - 2*a*b^2*c + 2*a*b*c^2 + 2*a*b*c - 2*b^2*c^2 + 2*b*c^2 - c^2)/(a^2*b^2 + 2*a*b^3 - 2*a*b^2 + b^4 - 2*b^3 + b^2)*$.8^2 + (2*a^2*c - 4*a*b*c^2 + 2*a*b*c + 4*a*c^2 - 4*a*c - 4*b*c^3 + 6*b*c^2 - 2*b*c + 2*c^3 - 4*c^2 + 2*c)/(a^2*b^2 + 2*a*b^3 - 2*a*b^2 + b^4 - 2*b^3 + b^2)*$.8*$.9^3 + (-2*a^2*c + 4*a*b*c^2 - 2*a*b*c + 2*a*c + 4*b*c^3 - 2*b*c^2 - 2*c^3)/(a^2*b^2 + 2*a*b^3 - 2*a*b^2 + b^4 - 2*b^3 + b^2)*$.8*$.9 + (a^2*c^2 + 2*a*c^3 - 2*a*c^2 + c^4 - 2*c^3 + c^2)/(a^2*b^2 + 2*a*b^3 - 2*a*b^2 + b^4 - 2*b^3 + b^2)*$.9^4 + (-2*a^2*c^2 + 2*a*c^2 - 2*c^4 + 2*c^3 - c^2)/(a^2*b^2 + 2*a*b^3 - 2*a*b^2 + b^4 - 2*b^3 + b^2)*$.9^2 + (a^2*c^2 - 2*a*c^3 + c^4)/(a^2*b^2 + 2*a*b^3 - 2*a*b^2 + b^4 - 2*b^3 + b^2) ] Total time: 1.449 seconds, Total memory usage: 4.60MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Wed Dec 7 10:21:12 2005 Input: K:=FunctionField(RationalField(),2); A:=Matrix(K,2,2,[a,1,0,b]); R:=PolynomialRing(K,4); P:=Matrix(R,2,2,[R.1,R.2,R.3,R.4]); I:=Ideal(Eltseq(P*A*Transpose(P)-A)); print GroebnerBasis(I); print Transpose(A)*(A^(-1)); Output: Magma V2.11-10 Wed Dec 7 2005 10:21:11 on modular [Seed = 1055621328] ------------------------------------- [ $.1 - 1/b*$.3 - $.4, $.2 + a/b*$.3, $.3^2 + 1/a*$.3*$.4 + b/a*$.4^2 - b/a ] [ 1 -1/b] [ 1/a (a*b - 1)/(a*b)] Total time: 0.190 seconds, Total memory usage: 3.24MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Wed Dec 7 10:19:43 2005 Input: K:=FunctionField(RationalField(),2); A:=Matrix(K,2,2,[a,1,0,b]); R:=PolynomialRing(K,4); P:=Matrix(R,2,2,[R.1,R.2,R.3,R.4]); I:=Ideal(Eltseq(P*A*Transpose(P)-A)); print GroebnerBasis(I); Output: Magma V2.11-10 Wed Dec 7 2005 10:19:42 on modular [Seed = 1005487548] ------------------------------------- [ $.1 - 1/b*$.3 - $.4, $.2 + a/b*$.3, $.3^2 + 1/a*$.3*$.4 + b/a*$.4^2 - b/a ] Total time: 0.200 seconds, Total memory usage: 3.24MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Wed Dec 7 10:16:52 2005 Input: K:=RationalField(); A:=Matrix(K,2,2,[0,1,-1,0]); R:=PolynomialRing(K,4); P:=Matrix(R,2,2,[R.1,R.2,R.3,R.4]); I:=Ideal(Eltseq(P*A*Transpose(P)-A)); print GroebnerBasis(I); Output: Magma V2.11-10 Wed Dec 7 2005 10:16:52 on modular [Seed = 771265216] ------------------------------------- [ -$.1*$.4 + $.2*$.3 + 1 ] Total time: 0.190 seconds, Total memory usage: 3.24MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Wed Dec 7 10:16:06 2005 Input: K:=RationalField(); A:=Matrix(K,2,2,[1,0,0,1]); R:=PolynomialRing(K,4); P:=Matrix(R,2,2,[R.1,R.2,R.3,R.4]); I:=Ideal(Eltseq(P*A*Transpose(P)-A)); print GroebnerBasis(I); Output: Magma V2.11-10 Wed Dec 7 2005 10:16:06 on modular [Seed = 381921705] ------------------------------------- [ $.1^2 - $.4^2, $.1*$.2 + $.3*$.4, $.1*$.3 + $.2*$.4, $.1*$.4^2 - $.1 - $.2*$.3*$.4, $.2^2 + $.4^2 - 1, $.3^2 + $.4^2 - 1 ] Total time: 0.190 seconds, Total memory usage: 3.24MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Wed Dec 7 10:14:12 2005 Input: K:=RationalField(); A:=Matrix(K,2,2,[1,1,0,1]); R:=PolynomialRing(K,4); P:=Matrix(R,2,2,[R.1,R.2,R.3,R.4]); I:=Ideal(Eltseq(P*A*Transpose(P)-A)); print GroebnerBasis(I); B:=Matrix(K,2,2,[1,-1,1,0]); print B^3; Output: Magma V2.11-10 Wed Dec 7 2005 10:14:12 on modular [Seed = 331787358] ------------------------------------- [ $.1 - $.3 - $.4, $.2 + $.3, $.3^2 + $.3*$.4 + $.4^2 - 1 ] [-1 0] [ 0 -1] Total time: 0.190 seconds, Total memory usage: 3.24MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Wed Dec 7 10:14:05 2005 Input: K:=RationalField(); A:=Matrix(K,2,2,[1,1,0,1]); R:=PolynomialRing(K,4); P:=Matrix(R,2,2,[R.1,R.2,R.3,R.4]); I:=Ideal(Eltseq(P*A*Transpose(P)-A)); print GroebnerBasis(I); B:=Matrix(K,2,2,[1,-1,1,0]); print B^2; Output: Magma V2.11-10 Wed Dec 7 2005 10:14:04 on modular [Seed = 482976553] ------------------------------------- [ $.1 - $.3 - $.4, $.2 + $.3, $.3^2 + $.3*$.4 + $.4^2 - 1 ] [ 0 -1] [ 1 -1] Total time: 0.200 seconds, Total memory usage: 3.24MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Wed Dec 7 10:12:37 2005 Input: K:=RationalField(); A:=Matrix(K,2,2,[1,1,0,1]); R:=PolynomialRing(K,4); P:=Matrix(R,2,2,[R.1,R.2,R.3,R.4]); I:=Ideal(Eltseq(P*A*Transpose(P)-A)); print GroebnerBasis(I); Output: Magma V2.11-10 Wed Dec 7 2005 10:12:37 on modular [Seed = 97566185] ------------------------------------- [ $.1 - $.3 - $.4, $.2 + $.3, $.3^2 + $.3*$.4 + $.4^2 - 1 ] Total time: 0.190 seconds, Total memory usage: 3.24MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Wed Dec 7 10:12:27 2005 Input: K:=RationalField(); A:=Matrix(K,2,2,[1 1 0 1]); R:=PolynomialRing(K,4); P:=Matrix(R,2,2,[R.1,R.2,R.3,R.4]); I:=Ideal(Eltseq(P*A*Transpose(P)-A)); print GroebnerBasis(I); Output: Magma V2.11-10 Wed Dec 7 2005 10:12:27 on modular [Seed = 249280220] ------------------------------------- >> A:=Matrix(K,2,2,[1 1 0 1]); ^ User error: bad syntax >> I:=Ideal(Eltseq(P*A*Transpose(P)-A)); ^ User error: Identifier 'A' has not been declared or assigned >> print GroebnerBasis(I); ^ User error: Identifier 'I' has not been declared or assigned Total time: 0.190 seconds, Total memory usage: 3.24MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Wed Dec 7 09:00:35 2005 Input: ps Output: Magma V2.11-10 Wed Dec 7 2005 09:00:34 on modular [Seed = 1184099918] ------------------------------------- >> ps; ^ User error: Identifier 'ps' has not been declared or assigned Total time: 0.190 seconds, Total memory usage: 3.24MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Wed Dec 7 09:00:24 2005 Input: who Output: Magma V2.11-10 Wed Dec 7 2005 09:00:24 on modular [Seed = 1301865663] ------------------------------------- >> who; ^ User error: Identifier 'who' has not been declared or assigned Total time: 0.190 seconds, Total memory usage: 3.24MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Wed Dec 7 07:23:32 2005 Input: G :=DirichletGroup(3675); G; X :=Elements(G); X; Y Output: Magma V2.11-10 Wed Dec 7 2005 07:23:29 on modular [Seed = 2893626342] ------------------------------------- Group of Dirichlet characters of modulus 3675 over Rational Field [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3 ] >> Y; ^ User error: Identifier 'Y' has not been declared or assigned Total time: 0.220 seconds, Total memory usage: 3.34MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Wed Dec 7 07:01:29 2005 Input: G :=DirichletGroup(1715); G; X :=Elements(G); X; Output: Magma V2.11-10 Wed Dec 7 2005 07:01:26 on modular [Seed = 4146232293] ------------------------------------- Group of Dirichlet characters of modulus 1715 over Rational Field [ 1, $.1, $.2, $.1*$.2 ] Total time: 0.270 seconds, Total memory usage: 3.34MB '212.120' ************** MAGMA ***************** Host 212.120.88.54 (212.120.88.54) Time: Wed Dec 7 06:27:11 2005 Input: Q:=GaloisField(35098201); P:=PolynomialRing(Q,2); I:=ideal; Groebner(I); I; Output: Magma V2.11-10 Wed Dec 7 2005 06:27:10 on modular [Seed = 2588876432] ------------------------------------- Ideal of Polynomial ring of rank 2 over GF(35098201) Lexicographical Order Variables: x, y Dimension 0 Groebner basis: [ x + 33784728*y^33 + 15744019*y^32 + 14466235*y^31 + 14937582*y^30 + 9988153*y^29 + 24849537*y^28 + 13827463*y^27 + 10851940*y^26 + 25333828*y^25 + 29238403*y^24 + 35087366*y^23 + 3185785*y^22 + 12125255*y^21 + 11305600*y^20 + 713800*y^19 + 11882241*y^18 + 23388419*y^17 + 12677392*y^16 + 20159861*y^15 + 31143912*y^14 + 33062327*y^13 + 11580434*y^12 + 10629964*y^11 + 14094725*y^10 + 30606411*y^9 + 20913610*y^8 + 23355486*y^7 + 32139384*y^6 + 35026862*y^5 + 11038274*y^4 + 26690476*y^3 + 752845*y^2 + 9514380*y + 16409093, y^34 + 34*y^33 + 561*y^32 + 5984*y^31 + 46376*y^30 + 278256*y^29 + 1344904*y^28 + 5379616*y^27 + 18156204*y^26 + 17353055*y^25 + 25833537*y^24 + 5312152*y^23 + 21881025*y^22 + 15430534*y^21 + 23145801*y^20 + 30861068*y^19 + 27872968*y^18 + 17124952*y^17 + 27873223*y^16 + 30858484*y^15 + 23140973*y^14 + 15452362*y^13 + 21951303*y^12 + 5362540*y^11 + 25762851*y^10 + 17184126*y^9 + 18014271*y^8 + 5332492*y^7 + 1358334*y^6 + 297330*y^5 + 54824*y^4 + 7259*y^3 + 714*y^2 + 34*y + 1 ] Total time: 0.200 seconds, Total memory usage: 3.43MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Wed Dec 7 05:47:41 2005 Input: G :=DirichletGroup(2205); G; X :=Elements(G); X; YqEigenform(D[2],12);Parent($1); Output: Magma V2.11-10 Wed Dec 7 2005 05:47:40 on modular [Seed = 4214097202] ------------------------------------- Group of Dirichlet characters of modulus 2205 over Rational Field [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3 ] >> YqEigenform(D[2],12);Parent($1);; ^ User error: Identifier 'D' has not been declared or assigned Set of sequences over Group of Dirichlet characters of modulus 2205 over Rational Field Total time: 0.200 seconds, Total memory usage: 3.34MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Wed Dec 7 04:17:20 2005 Input: G :=DirichletGroup(735,CyclotomicField(3)); G; X :=Elements(G); X; Y :=X[4]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; qEigenform(D[1],12);Parent($1); Output: ** WARNING: Computation used more memory than allowed. ** Magma V2.11-10 Wed Dec 7 2005 04:17:15 on modular [Seed = 1840809715] ------------------------------------- Group of Dirichlet characters of modulus 735 over Cyclotomic Field of order 3 and degree 2 [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3, $.3^2, $.1*$.3^2, $.2*$.3^2, $.1*$.2*$.3^2, $.3^3, $.1*$.3^3, $.2*$.3^3, $.1*$.2*$.3^3, $.3^4, $.1*$.3^4, $.2*$.3^4, $.1*$.2*$.3^4, $.3^5, $.1*$.3^5, $.2*$.3^5, $.1*$.2*$.3^5 ] 15 2 Current total memory usage: 94.3MB, failed memory request: 6.9MB System Error: User memory limit has been reached >> D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); ^ Runtime error: Variable 'M' has not been initialized >> D; ^ User error: Identifier 'D' has not been declared or assigned >> qEigenform(D[1],12);Parent($1);; ^ User error: Identifier 'D' has not been declared or assigned Integer Ring Total time: 4.540 seconds, Total memory usage: 94.31MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Wed Dec 7 04:16:58 2005 Input: G :=DirichletGroup(735,CyclotomicField(2)); G; X :=Elements(G); X; Y :=X[4]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; qEigenform(D[1],12);Parent($1); Output: ** WARNING: Computation used more memory than allowed. ** Magma V2.11-10 Wed Dec 7 2005 04:16:53 on modular [Seed = 1519366618] ------------------------------------- Group of Dirichlet characters of modulus 735 over Cyclotomic Field of order 2 and degree 1 [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3 ] 15 2 Current total memory usage: 94.4MB, failed memory request: 6.9MB System Error: User memory limit has been reached >> D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); ^ Runtime error: Variable 'M' has not been initialized >> D; ^ User error: Identifier 'D' has not been declared or assigned >> qEigenform(D[1],12);Parent($1);; ^ User error: Identifier 'D' has not been declared or assigned Integer Ring Total time: 4.750 seconds, Total memory usage: 94.44MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Wed Dec 7 04:16:32 2005 Input: G :=DirichletGroup(735,CyclotomicField(3)); G; X :=Elements(G); X; Y :=X[4]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; qEigenform(D[1],12);Parent($1); Output: ** WARNING: Computation used more memory than allowed. ** Magma V2.11-10 Wed Dec 7 2005 04:16:28 on modular [Seed = 1602925622] ------------------------------------- Group of Dirichlet characters of modulus 735 over Cyclotomic Field of order 3 and degree 2 [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3, $.3^2, $.1*$.3^2, $.2*$.3^2, $.1*$.2*$.3^2, $.3^3, $.1*$.3^3, $.2*$.3^3, $.1*$.2*$.3^3, $.3^4, $.1*$.3^4, $.2*$.3^4, $.1*$.2*$.3^4, $.3^5, $.1*$.3^5, $.2*$.3^5, $.1*$.2*$.3^5 ] 15 2 Current total memory usage: 94.3MB, failed memory request: 6.9MB System Error: User memory limit has been reached >> D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); ^ Runtime error: Variable 'M' has not been initialized >> D; ^ User error: Identifier 'D' has not been declared or assigned >> qEigenform(D[1],12);Parent($1);; ^ User error: Identifier 'D' has not been declared or assigned Integer Ring Total time: 4.540 seconds, Total memory usage: 94.31MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Wed Dec 7 02:50:58 2005 Input: G :=DirichletGroup(945); G; X :=Elements(G); X; Y :=X[2]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; Output: ** WARNING: Computation time exceeded 20 seconds, so computation was terminated after 20 seconds. ** Magma V2.11-10 Wed Dec 7 2005 02:50:37 on modular [Seed = 2321505729] ------------------------------------- Group of Dirichlet characters of modulus 945 over Rational Field [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3 ] 3 2 Errors: /bin/sh: line 1: 17804 Alarm clock nice -n 19 /usr/local/bin/magma '71.107.' ************** MAGMA ***************** Host 71.107.113.52 (71.107.113.52) Time: Wed Dec 7 02:39:50 2005 Input: 2*5 Output: Magma V2.11-10 Wed Dec 7 2005 02:39:50 on modular [Seed = 2141892031] ------------------------------------- 10 Total time: 0.190 seconds, Total memory usage: 3.24MB '71.107.' ************** MAGMA ***************** Host 71.107.113.52 (71.107.113.52) Time: Wed Dec 7 02:39:45 2005 Input: 2*5 Output: Magma V2.11-10 Wed Dec 7 2005 02:39:45 on modular [Seed = 1757264559] ------------------------------------- 10 Total time: 0.200 seconds, Total memory usage: 3.24MB '132.239' ************** MAGMA ***************** Host 132.239.145.83 (132.239.145.83) Time: Tue Dec 6 22:51:38 2005 Input: G :=DirichletGroup(84); G; X :=Elements(G); X; Y :=X[8]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; Norm(Coefficient(qEigenform(D[5],12),11));Parent($1); Output: Magma V2.11-10 Tue Dec 6 2005 22:51:37 on modular [Seed = 670185583] ------------------------------------- Group of Dirichlet characters of modulus 84 over Rational Field [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3 ] 84 2 [ Modular symbols space of level 84, weight 3, character $.1*$.2*$.3, and dimension 1 over Rational Field, Modular symbols space of level 84, weight 3, character $.1*$.2*$.3, and dimension 1 over Rational Field, Modular symbols space of level 84, weight 3, character $.1*$.2*$.3, and dimension 1 over Rational Field, Modular symbols space of level 84, weight 3, character $.1*$.2*$.3, and dimension 1 over Rational Field, Modular symbols space of level 84, weight 3, character $.1*$.2*$.3, and dimension 24 over Rational Field ] 18446744073709551616 Rational Field Total time: 0.770 seconds, Total memory usage: 5.66MB '132.239' ************** MAGMA ***************** Host 132.239.145.83 (132.239.145.83) Time: Tue Dec 6 22:50:53 2005 Input: G :=DirichletGroup(84); G; X :=Elements(G); X; Y :=X[8]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; qEigenform(D[5],12);Parent($1); Output: Magma V2.11-10 Tue Dec 6 2005 22:50:53 on modular [Seed = 552943420] ------------------------------------- Group of Dirichlet characters of modulus 84 over Rational Field [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3 ] 84 2 [ Modular symbols space of level 84, weight 3, character $.1*$.2*$.3, and dimension 1 over Rational Field, Modular symbols space of level 84, weight 3, character $.1*$.2*$.3, and dimension 1 over Rational Field, Modular symbols space of level 84, weight 3, character $.1*$.2*$.3, and dimension 1 over Rational Field, Modular symbols space of level 84, weight 3, character $.1*$.2*$.3, and dimension 1 over Rational Field, Modular symbols space of level 84, weight 3, character $.1*$.2*$.3, and dimension 24 over Rational Field ] q + (-3154406192093409505440711462312941399852135706366107785045295182763186079\ 28428971344315339315045192045991571277525409859827663237/189995137843409027\ 617487587962791777330586787708604501827307026332673400350064776414910901612\ 630255870623517588050261987626733147108813190983697047190466795042504704*a^\ 23 + 2046713682813363551390719098081181580173919339183486331065409475571468\ 3675074421826010030637789674368851790787507432663/2808359505892003487982033\ 118596455881223287644722412087617624775237881768519330067813745105864690267\ 99198059093922373442235037760563158855880380306685952*a^22 + 806085296019685616144143947305046259006184635473299530724479575153186911398\ 1703355075673148713595769683513371437405201356793596757/1979116019202177371\ 015495707945747680526945705297963560701114857632014586979841420988655225131\ 565165318994974875523562371111803615716804072746844241567362448359424*a^21 - 594405873337565451457161013922161608814448343984476963661369465357742143789\ 108061944860489619859248937016028710929714103/29253744853041702999812844985\ 379748762742579632525125912683591408727935088743021539726511519423856958249\ 79782228358056689948310005866238082087294861312*a^20 - 127078750369517880433428667565193737420182227514108638612875763685033102954\ 84450452511051459387657914085644406883824703311612957974143/131941067946811\ 824734366380529716512035129713686530904046740990508800972465322761399243681\ 6754376776879329983250349041580741202410477869381831229494378241632239616*a\ ^19 + 115897702670860128055252011542100564997697371756860793042945494076141\ 7293168266372740607310102015816304808280296702173753557/1950249656869446866\ 654189665691983250849505308835008394178906093915195672582868102648434101294\ 923797216653188152238704459965540003910825388058196574208*a^18 - 801499971193552088123081136029973976441096960863138512484575007977890572381\ 15071165551896087726591097773005912938438963324363775295845/183251483259460\ 867686619972957939600048791269009070700064918042373334683979614946387838446\ 77144121901101805322921514466399183366812192630303211520755253356003328*a^1\ 7 - 93330274076902537849306929164657193536041977241136632494472590444585539\ 94399609467017138279145038758128068431535279159280137/270868007898534287035\ 304120234997673729097959560417832524848068599332732303176125367838069624294\ 97183564627613225537561943965833387650352611919396864*a^16 - 160919378681959651143345172158986210922269327431250461763250808566962073817\ 32467735601535570410881248587594453528434391417688187643051238151/593734805\ 760653211304648712383724304158083711589389068210334457289604376093952426296\ 596567539469549595698492462657068711333541084715041221824053272470208734507\ 8272*a^15 + 930823436402097914986428110859561869946661959994923991858824647\ 9834075435528806342021349063684299269281970919942420846948297780909/8776123\ 455912510899943853495613924628822773889757537773805077422618380526622906461\ 917953455827157087474939346685074170069844930017598714246261884583936*a^14 - 474400382573859373647744547665029784867702395709311220243662874342868577144\ 5285136571120498923793361328837341169135487875556688018469551624565/1236947\ 512001360856884684817466092300329341065811227225438196786020009116862400888\ 117909515707228228324371859297202226481944877259823002545466777650979601530\ 22464*a^13 + 17988148604197499466251189088797157487150628007331167213079988\ 6307322737443992720470915872459364480678344919118947877240625593563463/1828\ 359053315106437488302811586234297671411227032820369542724463045495943046438\ 84623290696996399105989061236389272378543121769375366639880130455928832*a^1\ 2 - 12077424639649578807401618997354726212754990341571861908738338954117449\ 72720643252948983280401574685353079380400382511684554800633233480797161501/\ 274877224889191301529929959436909400073186903513606050097377063560002025969\ 422419581757670157161828516527079843822716995987750502182889454548172811328\ 80034004992*a^11 + 70284478037285411782526662558559283016529395011013468253\ 869939853103932274198592143503509473048061476405720062849997393995361863252\ 831/40630201184780143055295618035249651059364693934062674878727210289899909\ 845476418805175710443644245775346941419838306342915948750081475528917879095\ 296*a^10 - 2275304208864881464093247236199218006147517499081666579033912627\ 397424583384308713264189478116631015764793505153774432667065177337510840579\ 90943591/103078959333446738073723734788841025027445088817602268786516398835\ 000759738533407343159126308935685693697654941433518873495406438318583545455\ 56480424830012751872*a^9 + 403624905434348741317428949808993986230233383536\ 112752648511952539299938277142816228039602617489234439902591785861675654751\ 84356950263245/152363254442925536457358567632186191472617602252735030795227\ 038587124661920536570519408914163665921657551030324393648785934807812805533\ 23344204660736*a^8 - 536863438013163848909686699813041801759222656207768838\ 267880930401537559768267360633888692139739679973718735231373760309391297820\ 07148112131983017571925/742168507200816514130810890479655380197604639486736\ 335262918071612005470117440532870745709424336936994623115578321335889166926\ 355893801527280066590587760918134784*a^7 + 242618769363091951561890266658236970786051566545089358542446419991784632475\ 9615479435587616221016583062314751340274420889162350986348673460167/1097015\ 431989063862492981686951740578602846736219692221725634677827297565827863307\ 739744181978394635934367418335634271258730616252199839280782735572992*a^6 - 815956106990515446621243672771862838230476458486154599474863160605389301281\ 596951288384298500076971815038264402537613999569586869147708263122822732015\ 3/2147478319446807043202577808100854688071772683700047266385758309062515827\ 886112652982481798102826785285367811279864976531154300798303823863657600088\ 50625265664*a^5 + 975751497144935641984257969495268233146482100189621161871\ 354807426121472387178178979731347495347495642696083323138270981563010814969\ 727776757/95227034026828460285849104770116369670386001407959394247016899116\ 952913700335356574630571352291201035969393952746030491209254883003458270901\ 2791296*a^4 - 4214282065319867674104287233039127493846095089120457888384077\ 344530577620215203451054009473787984621124170460749704387807920726244007592\ 9249811447833429/2236956582757090670002685216771723633408096545520882569151\ 831571940120654048034013523418539690444568005591470083192683886619063331566\ 483191310000092194013184*a^3 + 55633661622414685684625703326627082510787615\ 166410840928619784281858668663122204801269538594403523438448295127656780573\ 08196809025593323578383/330649423704265487103642724896237394688840282666525\ 674468808677489419839237275543661911706084344448041560395669257050316698801\ 6770953412184072192*a^2 - 9101952422759966972741079240584783508003137941846\ 924367312699275266145545290771848082338986500695633276654813330189767231611\ 35230292180453657443745361/606813309124644821506804800556565655763915078537\ 565801093704310964659465616328671203184282685124937067489005556421626488340\ 67473187024826153157554584256*a - 35192976472321009415300881048173611070273\ 254363617242903149999763674532848021383532918588950585547163810834389696410\ 705627955328530894755797/29898132207055255995338064678840910254705610050141\ 572127170923528774218681032583158086634303054872688943178138496188722213071\ 621554483255426)*q^2 + (627357858570778026301225016805693743264686948371932\ 321960948998522285788281795776772741615744531099314194630611362594028543/24\ 789857418426363455665403343370682139538163921238851899415376828487322017681\ 546396936563673651931777276544672985684441035157174854510970806654303638346\ 79296*a^22 - 15000640249084634727201454165311500627583747217900570190425267\ 315204987672052824682556923615126109025694218272440859805934647/25822768144\ 194128599651461816011127228685587417957137395224350863007627101751610830142\ 253826720762267996400701026754626078288723806782261256931566289944576*a^20 + 261243656123494266218453149588110177061989661234056624486624459104178885424\ 04887623205088104918756819355333376244859688263085101/172151787627960857331\ 009745440074181524570582786380915968162339086717514011677405534281692178138\ 41511997600467351169750718859149204521507504621044193296384*a^18 + 149350881659913989744502200475655919197936739705278835234382217831881625963\ 515709098691901658277964490975352277315632696968509719/23909970503883452407\ 084686866676969656190358720330682773355880428710765834955195213094679469185\ 8909888855562046544024315539710405618354270897514502684672*a^16 + 739444599610863832847240003865105196326492376110116060569125492543687683824\ 509741170623047203201900 ** WARNING: Output too long, hence truncated. '132.239' ************** MAGMA ***************** Host 132.239.145.83 (132.239.145.83) Time: Tue Dec 6 22:50:36 2005 Input: G :=DirichletGroup(84); G; X :=Elements(G); X; Y :=X[8]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; qEigenform(D[6],12);Parent($1); Output: Magma V2.11-10 Tue Dec 6 2005 22:50:35 on modular [Seed = 499646999] ------------------------------------- Group of Dirichlet characters of modulus 84 over Rational Field [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3 ] 84 2 [ Modular symbols space of level 84, weight 3, character $.1*$.2*$.3, and dimension 1 over Rational Field, Modular symbols space of level 84, weight 3, character $.1*$.2*$.3, and dimension 1 over Rational Field, Modular symbols space of level 84, weight 3, character $.1*$.2*$.3, and dimension 1 over Rational Field, Modular symbols space of level 84, weight 3, character $.1*$.2*$.3, and dimension 1 over Rational Field, Modular symbols space of level 84, weight 3, character $.1*$.2*$.3, and dimension 24 over Rational Field ] >> qEigenform(D[6],12);Parent($1); ^ Runtime error in '[]': Sequence element 6 not defined Set of sequences over Power Structure of ModSym Total time: 0.470 seconds, Total memory usage: 4.61MB '132.239' ************** MAGMA ***************** Host 132.239.145.83 (132.239.145.83) Time: Tue Dec 6 22:50:11 2005 Input: G :=DirichletGroup(84); G; X :=Elements(G); X; Y :=X[8]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; qEigenform(D[6],12);Parent)$1); Output: Magma V2.11-10 Tue Dec 6 2005 22:50:10 on modular [Seed = 381882085] ------------------------------------- Group of Dirichlet characters of modulus 84 over Rational Field [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3 ] 84 2 [ Modular symbols space of level 84, weight 3, character $.1*$.2*$.3, and dimension 1 over Rational Field, Modular symbols space of level 84, weight 3, character $.1*$.2*$.3, and dimension 1 over Rational Field, Modular symbols space of level 84, weight 3, character $.1*$.2*$.3, and dimension 1 over Rational Field, Modular symbols space of level 84, weight 3, character $.1*$.2*$.3, and dimension 1 over Rational Field, Modular symbols space of level 84, weight 3, character $.1*$.2*$.3, and dimension 24 over Rational Field ] >> qEigenform(D[6],12);Parent)$1); ^ Runtime error in '[]': Sequence element 6 not defined >> qEigenform(D[6],12);Parent)$1); ^ User error: bad syntax Total time: 0.470 seconds, Total memory usage: 4.61MB '132.239' ************** MAGMA ***************** Host 132.239.145.83 (132.239.145.83) Time: Tue Dec 6 22:48:09 2005 Input: G :=DirichletGroup(84); G; X :=Elements(G); X; Y :=X[8]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; Output: Magma V2.11-10 Tue Dec 6 2005 22:48:08 on modular [Seed = 232265076] ------------------------------------- Group of Dirichlet characters of modulus 84 over Rational Field [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3 ] 84 2 [ Modular symbols space of level 84, weight 3, character $.1*$.2*$.3, and dimension 1 over Rational Field, Modular symbols space of level 84, weight 3, character $.1*$.2*$.3, and dimension 1 over Rational Field, Modular symbols space of level 84, weight 3, character $.1*$.2*$.3, and dimension 1 over Rational Field, Modular symbols space of level 84, weight 3, character $.1*$.2*$.3, and dimension 1 over Rational Field, Modular symbols space of level 84, weight 3, character $.1*$.2*$.3, and dimension 24 over Rational Field ] Total time: 0.470 seconds, Total memory usage: 4.61MB '132.239' ************** MAGMA ***************** Host 132.239.145.83 (132.239.145.83) Time: Tue Dec 6 22:47:59 2005 Input: G :=DirichletGroup(84); G; X :=Elements(G); X; Y :=X[7]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; Output: Magma V2.11-10 Tue Dec 6 2005 22:47:58 on modular [Seed = 114499177] ------------------------------------- Group of Dirichlet characters of modulus 84 over Rational Field [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3 ] 21 2 [] Total time: 0.220 seconds, Total memory usage: 4.37MB '132.239' ************** MAGMA ***************** Host 132.239.145.83 (132.239.145.83) Time: Tue Dec 6 22:47:48 2005 Input: G :=DirichletGroup(84); G; X :=Elements(G); X; Y :=X[6]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; Output: Magma V2.11-10 Tue Dec 6 2005 22:47:48 on modular [Seed = 13965413] ------------------------------------- Group of Dirichlet characters of modulus 84 over Rational Field [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3 ] 28 2 [] Total time: 0.210 seconds, Total memory usage: 4.36MB '132.239' ************** MAGMA ***************** Host 132.239.145.83 (132.239.145.83) Time: Tue Dec 6 22:47:36 2005 Input: G :=DirichletGroup(84); G; X :=Elements(G); X; Y :=X[5]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; Output: Magma V2.11-10 Tue Dec 6 2005 22:47:35 on modular [Seed = 4264289963] ------------------------------------- Group of Dirichlet characters of modulus 84 over Rational Field [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3 ] 7 2 [ Modular symbols space of level 84, weight 3, character $.3, and dimension 2 over Rational Field ] Total time: 0.780 seconds, Total memory usage: 4.65MB '132.239' ************** MAGMA ***************** Host 132.239.145.83 (132.239.145.83) Time: Tue Dec 6 22:47:28 2005 Input: G :=DirichletGroup(84); G; X :=Elements(G); X; Y :=X[4]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; Output: Magma V2.11-10 Tue Dec 6 2005 22:47:27 on modular [Seed = 4112575928] ------------------------------------- Group of Dirichlet characters of modulus 84 over Rational Field [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3 ] 12 2 [] Total time: 0.230 seconds, Total memory usage: 4.43MB '132.239' ************** MAGMA ***************** Host 132.239.145.83 (132.239.145.83) Time: Tue Dec 6 22:43:42 2005 Input: G :=DirichletGroup(84); G; X :=Elements(G); X; Y :=X[3]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; Output: Magma V2.11-10 Tue Dec 6 2005 22:43:41 on modular [Seed = 3896373639] ------------------------------------- Group of Dirichlet characters of modulus 84 over Rational Field [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3 ] 3 2 [ Modular symbols space of level 84, weight 3, character $.2, and dimension 4 over Rational Field ] Total time: 0.780 seconds, Total memory usage: 4.70MB '132.239' ************** MAGMA ***************** Host 132.239.145.83 (132.239.145.83) Time: Tue Dec 6 22:43:34 2005 Input: G :=DirichletGroup(84); G; X :=Elements(G); X; Y :=X[3]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; Output: Magma V2.11-10 Tue Dec 6 2005 22:43:33 on modular [Seed = 3862167425] ------------------------------------- Group of Dirichlet characters of modulus 84 over Rational Field [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3 ] 3 2 [ Modular symbols space of level 84, weight 3, character $.2, and dimension 4 over Rational Field ] Total time: 0.790 seconds, Total memory usage: 4.70MB '132.239' ************** MAGMA ***************** Host 132.239.145.83 (132.239.145.83) Time: Tue Dec 6 22:43:23 2005 Input: G :=DirichletGroup(84); G; X :=Elements(G); X; Y :=X[2]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; Output: Magma V2.11-10 Tue Dec 6 2005 22:43:22 on modular [Seed = 3708339333] ------------------------------------- Group of Dirichlet characters of modulus 84 over Rational Field [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3 ] 4 2 [ Modular symbols space of level 84, weight 3, character $.1, and dimension 12 over Rational Field ] Total time: 0.750 seconds, Total memory usage: 4.72MB '213.78.' ************** MAGMA ***************** Host 213.78.42.15 (213.78.42.15) Time: Tue Dec 6 22:20:13 2005 Input: 12+2 Output: Magma V2.11-10 Tue Dec 6 2005 22:20:12 on modular [Seed = 2976163423] ------------------------------------- 14 Total time: 0.190 seconds, Total memory usage: 3.24MB '213.78.' ************** MAGMA ***************** Host 213.78.42.15 (213.78.42.15) Time: Tue Dec 6 22:08:27 2005 Input: 10^2 Output: Magma V2.11-10 Tue Dec 6 2005 22:08:26 on modular [Seed = 1824095556] ------------------------------------- 100 Total time: 0.190 seconds, Total memory usage: 3.24MB '213.78.' ************** MAGMA ***************** Host 213.78.42.15 (213.78.42.15) Time: Tue Dec 6 22:08:12 2005 Input: 10^2 Output: Magma V2.11-10 Tue Dec 6 2005 22:08:12 on modular [Seed = 1234982999] ------------------------------------- 100 Total time: 0.190 seconds, Total memory usage: 3.24MB '213.78.' ************** MAGMA ***************** Host 213.78.42.15 (213.78.42.15) Time: Tue Dec 6 22:08:06 2005 Input: 10^2 Output: Magma V2.11-10 Tue Dec 6 2005 22:08:05 on modular [Seed = 1150375251] ------------------------------------- 100 Total time: 0.190 seconds, Total memory usage: 3.24MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 21:40:57 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^7*3^4; C:=HyperellipticCurve(f); J:=Jacobian(C); //r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); //V:=RationalPoints(J:Bound:=1500); //B:=ReducedBasis(V); //B; //P:=J!B[1];//Q:=J![x+2,0]; //P1:=J![x+1,0];P2:=J![x,1]; //hP:=Height(P);hP; //HC:=HeightConstant(J:Effort:=20);HC; //Exp(hP/4+HC); //RationalPoints(J:Bound:=1000); //Chabauty(P,11); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 21:40:57 on modular [Seed = 3556891758] ------------------------------------- Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] Total time: 0.240 seconds, Total memory usage: 4.51MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 21:40:47 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^5*3^6; C:=HyperellipticCurve(f); J:=Jacobian(C); //r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); //V:=RationalPoints(J:Bound:=1500); //B:=ReducedBasis(V); //B; //P:=J!B[1];//Q:=J![x+2,0]; //P1:=J![x+1,0];P2:=J![x,1]; //hP:=Height(P);hP; //HC:=HeightConstant(J:Effort:=20);HC; //Exp(hP/4+HC); //RationalPoints(J:Bound:=1000); //Chabauty(P,11); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 21:40:46 on modular [Seed = 3506755964] ------------------------------------- Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] Total time: 0.230 seconds, Total memory usage: 4.51MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 21:40:38 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^5*3^4; C:=HyperellipticCurve(f); J:=Jacobian(C); //r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); //V:=RationalPoints(J:Bound:=1500); //B:=ReducedBasis(V); //B; //P:=J!B[1];//Q:=J![x+2,0]; //P1:=J![x+1,0];P2:=J![x,1]; //hP:=Height(P);hP; //HC:=HeightConstant(J:Effort:=20);HC; //Exp(hP/4+HC); //RationalPoints(J:Bound:=1000); //Chabauty(P,11); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 21:40:38 on modular [Seed = 3457926763] ------------------------------------- Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] Total time: 0.240 seconds, Total memory usage: 4.51MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 21:40:21 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^1*3^6; C:=HyperellipticCurve(f); J:=Jacobian(C); //r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); //V:=RationalPoints(J:Bound:=1500); //B:=ReducedBasis(V); //B; //P:=J!B[1];//Q:=J![x+2,0]; //P1:=J![x+1,0];P2:=J![x,1]; //hP:=Height(P);hP; //HC:=HeightConstant(J:Effort:=20);HC; //Exp(hP/4+HC); //RationalPoints(J:Bound:=1000); //Chabauty(P,11); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 21:40:21 on modular [Seed = 3407790941] ------------------------------------- Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] Total time: 0.230 seconds, Total memory usage: 4.51MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 21:40:11 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^1*3^2; C:=HyperellipticCurve(f); J:=Jacobian(C); //r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); //V:=RationalPoints(J:Bound:=1500); //B:=ReducedBasis(V); //B; //P:=J!B[1];//Q:=J![x+2,0]; //P1:=J![x+1,0];P2:=J![x,1]; //hP:=Height(P);hP; //HC:=HeightConstant(J:Effort:=20);HC; //Exp(hP/4+HC); //RationalPoints(J:Bound:=1000); //Chabauty(P,11); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 21:40:11 on modular [Seed = 3222653000] ------------------------------------- Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] Total time: 0.230 seconds, Total memory usage: 4.51MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 21:21:21 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^5*3^5; C:=HyperellipticCurve(f); J:=Jacobian(C); //r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); //V:=RationalPoints(J:Bound:=1500); //B:=ReducedBasis(V); //B; //P:=J!B[1];//Q:=J![x+2,0]; //P1:=J![x+1,0];P2:=J![x,1]; //hP:=Height(P);hP; //HC:=HeightConstant(J:Effort:=20);HC; //Exp(hP/4+HC); RationalPoints(J:Bound:=1000); //Chabauty(P,11); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 21:21:20 on modular [Seed = 2877206408] ------------------------------------- Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 to JacHyp: J given by a rule [no inverse] {@ (1, 0, 0), (x - 6, 0, 1) @} Total time: 0.470 seconds, Total memory usage: 38.54MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 21:21:07 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^5*3^5; C:=HyperellipticCurve(f); J:=Jacobian(C); //r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); //V:=RationalPoints(J:Bound:=1500); //B:=ReducedBasis(V); //B; //P:=J!B[1];//Q:=J![x+2,0]; //P1:=J![x+1,0];P2:=J![x,1]; //hP:=Height(P);hP; //HC:=HeightConstant(J:Effort:=20);HC; //Exp(hP/4+HC); RationalPoints(J:Bound:=12); //Chabauty(P,11); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 21:21:06 on modular [Seed = 2692064433] ------------------------------------- Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 to JacHyp: J given by a rule [no inverse] {@ (1, 0, 0) @} Total time: 0.310 seconds, Total memory usage: 38.54MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 21:20:25 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^5*3^5; C:=HyperellipticCurve(f); J:=Jacobian(C); //r:=TwoSelmerGroupData(J);r; //TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1];//Q:=J![x+2,0]; //P1:=J![x+1,0];P2:=J![x,1]; //hP:=Height(P);hP; //HC:=HeightConstant(J:Effort:=20);HC; //Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,11); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 21:20:25 on modular [Seed = 2638780535] ------------------------------------- [] >> P:=J!B[1];//Q:=J![x+2,0]; ^ Runtime error in '[]': Sequence element 1 not defined >> Chabauty(P,11); ^ User error: Identifier 'P' has not been declared or assigned Total time: 0.660 seconds, Total memory usage: 37.80MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 21:11:34 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^4*3^2; C:=HyperellipticCurve(f); J:=Jacobian(C); //r:=TwoSelmerGroupData(J);r; //TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1];//Q:=J![x+2,0]; //P1:=J![x+1,0];P2:=J![x,1]; //hP:=Height(P);hP; //HC:=HeightConstant(J:Effort:=20);HC; //Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,11); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 21:11:33 on modular [Seed = 2537721921] ------------------------------------- [ (x - 10, 316, 1) ] {@ <10, 1, 5, 1> @} Total time: 1.139 seconds, Total memory usage: 37.80MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 21:11:23 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^4*3^2; C:=HyperellipticCurve(f); J:=Jacobian(C); //r:=TwoSelmerGroupData(J);r; //TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1];//Q:=J![x+2,0]; //P1:=J![x+1,0];P2:=J![x,1]; //hP:=Height(P);hP; //HC:=HeightConstant(J:Effort:=20);HC; //Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,7); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 21:11:22 on modular [Seed = 2354678597] ------------------------------------- [ (x - 10, 316, 1) ] {@ <10, 1, 4, 1>, <381, 1, 4, 1> @} Total time: 1.179 seconds, Total memory usage: 37.80MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 21:10:57 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^4*3^2; C:=HyperellipticCurve(f); J:=Jacobian(C); //r:=TwoSelmerGroupData(J);r; //TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1];//Q:=J![x+2,0]; //P1:=J![x+1,0];P2:=J![x,1]; //hP:=Height(P);hP; //HC:=HeightConstant(J:Effort:=20);HC; //Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,7); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 21:10:56 on modular [Seed = 2304544314] ------------------------------------- [] >> P:=J!B[1];//Q:=J![x+2,0]; ^ Runtime error in '[]': Sequence element 1 not defined >> Chabauty(P,7); ^ User error: Identifier 'P' has not been declared or assigned Total time: 0.670 seconds, Total memory usage: 37.80MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 20:12:13 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^1*3^4; C:=HyperellipticCurve(f); J:=Jacobian(C); //r:=TwoSelmerGroupData(J);r; //TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1];//Q:=J![x+2,0]; //P1:=J![x+1,0];P2:=J![x,1]; //hP:=Height(P);hP; //HC:=HeightConstant(J:Effort:=20);HC; //Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,19); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 20:12:11 on modular [Seed = 670192335] ------------------------------------- [ (x^2 + 3/4*x - 9/4, -3/8*x - 99/8, 2) ] {@ @} Total time: 1.209 seconds, Total memory usage: 37.80MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 20:12:06 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^1*3^4; C:=HyperellipticCurve(f); J:=Jacobian(C); //r:=TwoSelmerGroupData(J);r; //TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1];//Q:=J![x+2,0]; //P1:=J![x+1,0];P2:=J![x,1]; //hP:=Height(P);hP; //HC:=HeightConstant(J:Effort:=20);HC; //Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,17); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 20:12:04 on modular [Seed = 499673338] ------------------------------------- [ (x^2 + 3/4*x - 9/4, -3/8*x - 99/8, 2) ] {@ <23738, 1, 4, 1>, <66775, 1, 4, 1>, <79574, 1, 4, 1>, <44999, 1, 4, 1>, <0, 1, 4, 1>, <49648, 1, 4, 1>, <7813, 1, 4, 1>, <44823, 1, 4, 1>, <25980, 1, 4, 1> @} Total time: 2.169 seconds, Total memory usage: 37.80MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 20:11:58 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^1*3^4; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; //TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1];//Q:=J![x+2,0]; //P1:=J![x+1,0];P2:=J![x,1]; //hP:=Height(P);hP; //HC:=HeightConstant(J:Effort:=20);HC; //Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,17); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 20:11:55 on modular [Seed = 449802178] ------------------------------------- TwoSelmerGroupData( J: Jacobian of Hyperelliptic Curve defined by y^2 = x^5 + 162 o... ) TwoSelmerWork( C: Hyperelliptic Curve defined by y^2 = x^5 + 162 over Rational..., BoundType: Default, Bound: 0, UseUnits: true, Points: {}, Fields: {} ) LocalImageOdd( f: x^5 + 162, p: 2, m1: Mapping from: Univariate Polynomial Ring in x over Rational ..., m2: Mapping from: Univariate Polynomial Ring in x over Rational ..., v: 6, Points: {} ) try( ) try( d: 2 ) SquarefreeRoots( F: $.1^4 + O(2^50)*$.1^3 - (661695299665*2^3 + O(2^53))*$.1^2 +... ) In file "/usr/local/magma/package/Geometry/Arith/sqfroots.m", line 84, column 15: >> return Roots(F:IsSquarefree); ^ Runtime error in 'Roots': Insufficient precision to find roots >> r:=TwoSelmerGroupData(J);r; ^ User error: Identifier 'r' has not been declared or assigned [ (x^2 + 3/4*x - 9/4, -3/8*x - 99/8, 2) ] {@ <23738, 1, 4, 1>, <66775, 1, 4, 1>, <79574, 1, 4, 1>, <44999, 1, 4, 1>, <0, 1, 4, 1>, <49648, 1, 4, 1>, <7813, 1, 4, 1>, <44823, 1, 4, 1>, <25980, 1, 4, 1> @} Total time: 2.569 seconds, Total memory usage: 39.03MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 20:10:42 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^0*3^5; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; //TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1];//Q:=J![x+2,0]; //P1:=J![x+1,0];P2:=J![x,1]; //hP:=Height(P);hP; //HC:=HeightConstant(J:Effort:=20);HC; //Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,29); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 20:10:40 on modular [Seed = 398615172] ------------------------------------- 1 [ (x^2 + 3*x + 9, -3*x + 9, 2) ] {@ @} Total time: 1.980 seconds, Total memory usage: 39.00MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 20:10:35 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^0*3^5; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; //TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1];//Q:=J![x+2,0]; //P1:=J![x+1,0];P2:=J![x,1]; //hP:=Height(P);hP; //HC:=HeightConstant(J:Effort:=20);HC; //Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,23); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 20:10:31 on modular [Seed = 215579016] ------------------------------------- 1 [ (x^2 + 3*x + 9, -3*x + 9, 2) ] {@ <274218, 1, 4, 1>, <271994, 1, 4, 1>, <34871, 1, 4, 1>, <0, 1, 4, 1>, <119449, 1, 4, 1>, <222506, 1, 4, 1>, <276781, 1, 4, 1>, <208779, 1, 4, 1>, <256392, 1, 4, 1>, <2850, 1, 4, 1> @} Total time: 3.290 seconds, Total memory usage: 38.97MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 20:10:26 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^0*3^5; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; //TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1];//Q:=J![x+2,0]; //P1:=J![x+1,0];P2:=J![x,1]; //hP:=Height(P);hP; //HC:=HeightConstant(J:Effort:=20);HC; //Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,19); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 20:10:24 on modular [Seed = 165444746] ------------------------------------- 1 [ (x^2 + 3*x + 9, -3*x + 9, 2) ] {@ <100876, 1, 4, 1>, <25523, 1, 4, 1> @} Total time: 2.060 seconds, Total memory usage: 38.96MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 20:10:18 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^0*3^5; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; //TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1];//Q:=J![x+2,0]; //P1:=J![x+1,0];P2:=J![x,1]; //hP:=Height(P);hP; //HC:=HeightConstant(J:Effort:=20);HC; //Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,17); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 20:10:16 on modular [Seed = 114520957] ------------------------------------- 1 [ (x^2 + 3*x + 9, -3*x + 9, 2) ] {@ <61454, 1, 4, 1>, <52643, 1, 4, 1>, <71747, 1, 4, 1>, <38993, 1, 4, 1>, <54970, 1, 4, 1>, <29585, 1, 4, 1>, <51208, 1, 4, 1>, <33843, 1, 4, 1> @} Total time: 2.790 seconds, Total memory usage: 38.97MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 20:10:10 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^0*3^5; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; //TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1];//Q:=J![x+2,0]; //P1:=J![x+1,0];P2:=J![x,1]; //hP:=Height(P);hP; //HC:=HeightConstant(J:Effort:=20);HC; //Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,13); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 20:10:08 on modular [Seed = 64386672] ------------------------------------- 1 [ (x^2 + 3*x + 9, -3*x + 9, 2) ] {@ <27266, 1, 4, 1>, <8341, 1, 4, 1>, <14210, 1, 4, 1>, <15232, 1, 4, 1>, <0, 1, 4, 1>, <7733, 1, 4, 1> @} Total time: 2.370 seconds, Total memory usage: 38.98MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 20:10:02 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^0*3^5; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; //TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1];//Q:=J![x+2,0]; //P1:=J![x+1,0];P2:=J![x,1]; //hP:=Height(P);hP; //HC:=HeightConstant(J:Effort:=20);HC; //Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,11); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 20:10:00 on modular [Seed = 2270360211] ------------------------------------- 1 [ (x^2 + 3*x + 9, -3*x + 9, 2) ] {@ <0, 1, 4, 1> @} Total time: 1.879 seconds, Total memory usage: 38.97MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 20:09:40 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^0*3^5; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1];//Q:=J![x+2,0]; //P1:=J![x+1,0];P2:=J![x,1]; //hP:=Height(P);hP; //HC:=HeightConstant(J:Effort:=20);HC; //Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,7); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 20:09:37 on modular [Seed = 2321547175] ------------------------------------- 1 Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 to JacHyp: J given by a rule [no inverse] [ (x^2 + 3*x + 9, -3*x + 9, 2) ] {@ <1458, 1, 4, 1>, <1350, 1, 4, 1>, <607, 1, 4, 1> @} Total time: 2.020 seconds, Total memory usage: 39.69MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 20:00:00 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^7*3^4; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); //V:=RationalPoints(J:Bound:=1500); //B:=ReducedBasis(V); //B; //P:=J!B[1];//Q:=J![x+2,0]; //P1:=J![x+1,0];P2:=J![x,1]; //hP:=Height(P);hP; //HC:=HeightConstant(J:Effort:=20);HC; //Exp(hP/4+HC); RationalPoints(J:Bound:=12); //Chabauty(P,29); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 19:59:58 on modular [Seed = 3812005009] ------------------------------------- 2 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] {@ (1, 0, 0) @} Total time: 2.069 seconds, Total memory usage: 40.90MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 19:59:57 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^7*3^4; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); //V:=RationalPoints(J:Bound:=1500); //B:=ReducedBasis(V); //B; //P:=J!B[1];//Q:=J![x+2,0]; //P1:=J![x+1,0];P2:=J![x,1]; //hP:=Height(P);hP; //HC:=HeightConstant(J:Effort:=20);HC; //Exp(hP/4+HC); RationalPoints(J:Bound:=12); //Chabauty(P,29); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 19:59:56 on modular [Seed = 3862139286] ------------------------------------- TwoSelmerGroupData( J: Jacobian of Hyperelliptic Curve defined by y^2 = x^5 + 10368... ) TwoSelmerWork( C: Hyperelliptic Curve defined by y^2 = x^5 + 10368 over Ration..., BoundType: Default, Bound: 0, UseUnits: true, Points: {}, Fields: {} ) LocalImageOdd( f: x^5 + 10368, p: 2, m1: Mapping from: Univariate Polynomial Ring in x over Rational ..., m2: Mapping from: Univariate Polynomial Ring in x over Rational ..., v: 30, Points: {} ) try( ) try( d: 2 ) SquarefreeRoots( F: (1 + O(2^44))*$.1^3 + O(2^50)*$.1^2 - (3053654638673*2^3 + O... ) In file "/usr/local/magma/package/Geometry/Arith/sqfroots.m", line 84, column 15: >> return Roots(F:IsSquarefree); ^ Runtime error in 'Roots': Insufficient precision to find roots >> r:=TwoSelmerGroupData(J);r; ^ User error: Identifier 'r' has not been declared or assigned Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] {@ (1, 0, 0) @} Total time: 1.149 seconds, Total memory usage: 39.75MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 19:59:54 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^7*3^4; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); //V:=RationalPoints(J:Bound:=1500); //B:=ReducedBasis(V); //B; //P:=J!B[1];//Q:=J![x+2,0]; //P1:=J![x+1,0];P2:=J![x,1]; //hP:=Height(P);hP; //HC:=HeightConstant(J:Effort:=20);HC; //Exp(hP/4+HC); RationalPoints(J:Bound:=12); //Chabauty(P,29); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 19:59:52 on modular [Seed = 3913064093] ------------------------------------- TwoSelmerGroupData( J: Jacobian of Hyperelliptic Curve defined by y^2 = x^5 + 10368... ) TwoSelmerWork( C: Hyperelliptic Curve defined by y^2 = x^5 + 10368 over Ration..., BoundType: Default, Bound: 0, UseUnits: true, Points: {}, Fields: {} ) LocalImageOdd( f: x^5 + 10368, p: 2, m1: Mapping from: Univariate Polynomial Ring in x over Rational ..., m2: Mapping from: Univariate Polynomial Ring in x over Rational ..., v: 30, Points: {} ) try( ) try( d: 2 ) SquarefreeRoots( F: $.1^4 + O(2^50)*$.1^3 - (511427965222993*2^3 + O(2^53))*$.1^... ) In file "/usr/local/magma/package/Geometry/Arith/sqfroots.m", line 84, column 15: >> return Roots(F:IsSquarefree); ^ Runtime error in 'Roots': Insufficient precision to find roots >> r:=TwoSelmerGroupData(J);r; ^ User error: Identifier 'r' has not been declared or assigned Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] {@ (1, 0, 0) @} Total time: 1.159 seconds, Total memory usage: 39.74MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 19:59:39 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^5*3^2; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); //V:=RationalPoints(J:Bound:=1500); //B:=ReducedBasis(V); //B; //P:=J!B[1];//Q:=J![x+2,0]; //P1:=J![x+1,0];P2:=J![x,1]; //hP:=Height(P);hP; //HC:=HeightConstant(J:Effort:=20);HC; //Exp(hP/4+HC); RationalPoints(J:Bound:=12); //Chabauty(P,29); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 19:59:38 on modular [Seed = 3963198570] ------------------------------------- 3 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] {@ (1, 0, 0), (x + 2, 16, 1), (x + 2, -16, 1), (x - 1, 17, 1), (x - 1, -17, 1), (x^2 - 8, 2*x + 16, 2), (x^2 - 8, -2*x - 16, 2) @} Total time: 0.760 seconds, Total memory usage: 39.02MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 19:59:34 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^5*3^1; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); //V:=RationalPoints(J:Bound:=1500); //B:=ReducedBasis(V); //B; //P:=J!B[1];//Q:=J![x+2,0]; //P1:=J![x+1,0];P2:=J![x,1]; //hP:=Height(P);hP; //HC:=HeightConstant(J:Effort:=20);HC; //Exp(hP/4+HC); RationalPoints(J:Bound:=12); //Chabauty(P,29); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 19:59:33 on modular [Seed = 4146235262] ------------------------------------- 2 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] {@ (1, 0, 0), (x + 2, 8, 1), (x + 2, -8, 1) @} Total time: 0.770 seconds, Total memory usage: 39.02MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 19:59:20 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^4*3^5; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); //V:=RationalPoints(J:Bound:=1500); //B:=ReducedBasis(V); //B; //P:=J!B[1];//Q:=J![x+2,0]; //P1:=J![x+1,0];P2:=J![x,1]; //hP:=Height(P);hP; //HC:=HeightConstant(J:Effort:=20);HC; //Exp(hP/4+HC); RationalPoints(J:Bound:=12); //Chabauty(P,29); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 19:59:20 on modular [Seed = 4197419630] ------------------------------------- 2 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] {@ (1, 0, 0) @} Total time: 0.700 seconds, Total memory usage: 39.41MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 19:59:09 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^4*3^1; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); //V:=RationalPoints(J:Bound:=1500); //B:=ReducedBasis(V); //B; //P:=J!B[1];//Q:=J![x+2,0]; //P1:=J![x+1,0];P2:=J![x,1]; //hP:=Height(P);hP; //HC:=HeightConstant(J:Effort:=20);HC; //Exp(hP/4+HC); RationalPoints(J:Bound:=12); //Chabauty(P,29); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 19:59:08 on modular [Seed = 4247293277] ------------------------------------- 2 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] {@ (1, 0, 0), (x + 2, 4, 1), (x + 2, -4, 1), (x - 1, 7, 1), (x - 1, -7, 1), (x^2 + x - 2, x + 6, 2), (x^2 + x - 2, -x - 6, 2) @} Total time: 0.810 seconds, Total memory usage: 39.30MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 19:58:55 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^1*3^8; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); //V:=RationalPoints(J:Bound:=1500); //B:=ReducedBasis(V); //B; //P:=J!B[1];//Q:=J![x+2,0]; //P1:=J![x+1,0];P2:=J![x,1]; //hP:=Height(P);hP; //HC:=HeightConstant(J:Effort:=20);HC; //Exp(hP/4+HC); RationalPoints(J:Bound:=12); //Chabauty(P,29); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 19:58:54 on modular [Seed = 13988018] ------------------------------------- 2 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] {@ (1, 0, 0) @} Total time: 0.790 seconds, Total memory usage: 39.44MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 19:58:46 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^1*3^1; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); //V:=RationalPoints(J:Bound:=1500); //B:=ReducedBasis(V); //B; //P:=J!B[1];//Q:=J![x+2,0]; //P1:=J![x+1,0];P2:=J![x,1]; //hP:=Height(P);hP; //HC:=HeightConstant(J:Effort:=20);HC; //Exp(hP/4+HC); RationalPoints(J:Bound:=12); //Chabauty(P,29); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 19:58:45 on modular [Seed = 199130034] ------------------------------------- 2 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] {@ (1, 0, 0), (x^2 - 2, x + 2, 2), (x^2 - 2, -x - 2, 2) @} Total time: 0.690 seconds, Total memory usage: 39.17MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 19:58:43 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^1*3^1; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); //V:=RationalPoints(J:Bound:=1500); //B:=ReducedBasis(V); //B; //P:=J!B[1];//Q:=J![x+2,0]; //P1:=J![x+1,0];P2:=J![x,1]; //hP:=Height(P);hP; //HC:=HeightConstant(J:Effort:=20);HC; //Exp(hP/4+HC); RationalPoints(J:Bound:=12); //Chabauty(P,29); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 19:58:42 on modular [Seed = 249265840] ------------------------------------- TwoSelmerGroupData( J: Jacobian of Hyperelliptic Curve defined by y^2 = x^5 + 6 ove... ) TwoSelmerWork( C: Hyperelliptic Curve defined by y^2 = x^5 + 6 over Rational F..., BoundType: Default, Bound: 0, UseUnits: true, Points: {}, Fields: {} ) LocalImageOdd( f: x^5 + 6, p: 2, m1: Mapping from: Univariate Polynomial Ring in x over Rational ..., m2: Mapping from: Univariate Polynomial Ring in x over Rational ..., v: 6, Points: {} ) try( ) try( d: 2 ) SquarefreeRoots( F: $.1^4 + O(2^50)*$.1^3 - (600047619*2^3 + O(2^53))*$.1^2 + O(... ) In file "/usr/local/magma/package/Geometry/Arith/sqfroots.m", line 84, column 15: >> return Roots(F:IsSquarefree); ^ Runtime error in 'Roots': Insufficient precision to find roots >> r:=TwoSelmerGroupData(J);r; ^ User error: Identifier 'r' has not been declared or assigned Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] {@ (1, 0, 0), (x^2 - 2, x + 2, 2), (x^2 - 2, -x - 2, 2) @} Total time: 0.630 seconds, Total memory usage: 39.40MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 19:58:32 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^1*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); //V:=RationalPoints(J:Bound:=1500); //B:=ReducedBasis(V); //B; //P:=J!B[1];//Q:=J![x+2,0]; //P1:=J![x+1,0];P2:=J![x,1]; //hP:=Height(P);hP; //HC:=HeightConstant(J:Effort:=20);HC; //Exp(hP/4+HC); RationalPoints(J:Bound:=12); //Chabauty(P,29); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 19:58:31 on modular [Seed = 298082778] ------------------------------------- 2 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] {@ (1, 0, 0), (x + 1, 1, 1), (x + 1, -1, 1), (x^2 + 2, x + 2, 2), (x^2 + 2, -x - 2, 2), (x^2 - x - 1, x + 2, 2), (x^2 - x - 1, -x - 2, 2) @} Total time: 0.610 seconds, Total memory usage: 39.19MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 19:58:29 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^1*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); //V:=RationalPoints(J:Bound:=1500); //B:=ReducedBasis(V); //B; //P:=J!B[1];//Q:=J![x+2,0]; //P1:=J![x+1,0];P2:=J![x,1]; //hP:=Height(P);hP; //HC:=HeightConstant(J:Effort:=20);HC; //Exp(hP/4+HC); RationalPoints(J:Bound:=12); //Chabauty(P,29); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 19:58:28 on modular [Seed = 348218588] ------------------------------------- TwoSelmerGroupData( J: Jacobian of Hyperelliptic Curve defined by y^2 = x^5 + 2 ove... ) TwoSelmerWork( C: Hyperelliptic Curve defined by y^2 = x^5 + 2 over Rational F..., BoundType: Default, Bound: 0, UseUnits: true, Points: {}, Fields: {} ) LocalImageOdd( f: x^5 + 2, p: 2, m1: Mapping from: Univariate Polynomial Ring in x over Rational ..., m2: Mapping from: Univariate Polynomial Ring in x over Rational ..., v: 6, Points: {} ) try( ) try( d: 2 ) SquarefreeRoots( F: $.1^4 + O(2^50)*$.1^3 + (8257011711*2^3 + O(2^53))*$.1^2 + O... ) In file "/usr/local/magma/package/Geometry/Arith/sqfroots.m", line 84, column 15: >> return Roots(F:IsSquarefree); ^ Runtime error in 'Roots': Insufficient precision to find roots >> r:=TwoSelmerGroupData(J);r; ^ User error: Identifier 'r' has not been declared or assigned Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] {@ (1, 0, 0), (x + 1, 1, 1), (x + 1, -1, 1), (x^2 + 2, x + 2, 2), (x^2 + 2, -x - 2, 2), (x^2 - x - 1, x + 2, 2), (x^2 - x - 1, -x - 2, 2) @} Total time: 0.640 seconds, Total memory usage: 39.56MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 19:58:24 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^1*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); //V:=RationalPoints(J:Bound:=1500); //B:=ReducedBasis(V); //B; //P:=J!B[1];//Q:=J![x+2,0]; //P1:=J![x+1,0];P2:=J![x,1]; //hP:=Height(P);hP; //HC:=HeightConstant(J:Effort:=20);HC; //Exp(hP/4+HC); RationalPoints(J:Bound:=12); //Chabauty(P,29); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 19:58:24 on modular [Seed = 533360595] ------------------------------------- TwoSelmerGroupData( J: Jacobian of Hyperelliptic Curve defined by y^2 = x^5 + 2 ove... ) TwoSelmerWork( C: Hyperelliptic Curve defined by y^2 = x^5 + 2 over Rational F..., BoundType: Default, Bound: 0, UseUnits: true, Points: {}, Fields: {} ) LocalImageOdd( f: x^5 + 2, p: 2, m1: Mapping from: Univariate Polynomial Ring in x over Rational ..., m2: Mapping from: Univariate Polynomial Ring in x over Rational ..., v: 6, Points: {} ) try( ) try( d: 2 ) SquarefreePart( F: $.1^4 + O(2^50)*$.1^3 + (4239300493311*2^3 + O(2^53))*$.1^2 ... ) In file "/usr/local/magma/package/Geometry/Arith/loclib.m", line 120, column 5: >> assert R eq 0; ^ Runtime error in assert: Assertion failed >> r:=TwoSelmerGroupData(J);r; ^ User error: Identifier 'r' has not been declared or assigned Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] {@ (1, 0, 0), (x + 1, 1, 1), (x + 1, -1, 1), (x^2 + 2, x + 2, 2), (x^2 + 2, -x - 2, 2), (x^2 - x - 1, x + 2, 2), (x^2 - x - 1, -x - 2, 2) @} Total time: 0.460 seconds, Total memory usage: 39.18MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 19:57:15 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^0*3^4; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); //V:=RationalPoints(J:Bound:=1500); //B:=ReducedBasis(V); //B; //P:=J!B[1];//Q:=J![x+2,0]; //P1:=J![x+1,0];P2:=J![x,1]; //hP:=Height(P);hP; //HC:=HeightConstant(J:Effort:=20);HC; //Exp(hP/4+HC); RationalPoints(J:Bound:=12); //Chabauty(P,29); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 19:57:14 on modular [Seed = 586633780] ------------------------------------- 2 Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 to JacHyp: J given by a rule [no inverse] {@ (1, 0, 0), (x^2, 9, 2), (x^2, -9, 2), (x + 2, 7, 1), (x + 2, -7, 1), (x, 9, 1), (x, -9, 1), (x - 3, 18, 1), (x - 3, -18, 1), (x^2 + 2*x, x + 9, 2), (x^2 + 2*x, -x - 9, 2), (x^2 + 2*x + 6, 2*x - 3, 2), (x^2 + 2*x + 6, -2*x + 3, 2) @} Total time: 0.680 seconds, Total memory usage: 39.48MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 19:39:48 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^7*3^6; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); //V:=RationalPoints(J:Bound:=1500); //B:=ReducedBasis(V); //B; //P:=J!B[1];//Q:=J![x+2,0]; //P1:=J![x+1,0];P2:=J![x,1]; //hP:=Height(P);hP; //HC:=HeightConstant(J:Effort:=20);HC; //Exp(hP/4+HC); RationalPoints(J:Bound:=12); //Chabauty(P,29); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 19:39:46 on modular [Seed = 804930305] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] {@ (1, 0, 0) @} Total time: 1.360 seconds, Total memory usage: 40.01MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 19:28:37 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^0*3^1; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); //V:=RationalPoints(J:Bound:=1500); //B:=ReducedBasis(V); //B; //P:=J!B[1];//Q:=J![x+2,0]; //P1:=J![x+1,0];P2:=J![x,1]; //hP:=Height(P);hP; //HC:=HeightConstant(J:Effort:=20);HC; //Exp(hP/4+HC); RationalPoints(J:Bound:=12); //Chabauty(P,29); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 19:28:36 on modular [Seed = 1100230281] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] {@ (1, 0, 0), (x - 1, 2, 1), (x - 1, -2, 1) @} Total time: 0.630 seconds, Total memory usage: 39.38MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 19:28:20 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^0*3^1; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); //V:=RationalPoints(J:Bound:=1500); //B:=ReducedBasis(V); //B; //P:=J!B[1];//Q:=J![x+2,0]; P1:=J![x+1,0];P2:=J![x,1]; //hP:=Height(P);hP; //HC:=HeightConstant(J:Effort:=20);HC; //Exp(hP/4+HC); RationalPoints(J:Bound:=12); //Chabauty(P,29); //Order(Q);Order(P); Order(P1);Order(P2); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 19:28:19 on modular [Seed = 1150364571] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] >> P1:=J![x+1,0];P2:=J![x,1]; ^ Runtime error in '!': Points specified by second polynomial are not on the curve >> P1:=J![x+1,0];P2:=J![x,1]; ^ Runtime error in '!': Points specified by second polynomial are not on the curve {@ (1, 0, 0), (x - 1, 2, 1), (x - 1, -2, 1) @} >> Order(P1);Order(P2); ^ Runtime error in 'Order': Bad argument types Argument types given: Intrinsic >> Order(P1);Order(P2); ^ Runtime error in 'Order': Bad argument types Argument types given: Intrinsic Total time: 0.620 seconds, Total memory usage: 39.39MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 19:15:43 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^0*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); //V:=RationalPoints(J:Bound:=1500); //B:=ReducedBasis(V); //B; //P:=J!B[1];//Q:=J![x+2,0]; P1:=J![x+1,0];P2:=J![x,1]; //hP:=Height(P);hP; //HC:=HeightConstant(J:Effort:=20);HC; //Exp(hP/4+HC); RationalPoints(J:Bound:=12); //Chabauty(P,29); //Order(Q);Order(P); Order(P1);Order(P2); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 19:15:42 on modular [Seed = 1468153221] ------------------------------------- 0 Abelian Group isomorphic to Z/10 Defined on 1 generator Relations: 10*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/10 Defined on 1 generator Relations: 10*P[1] = 0 to JacHyp: J given by a rule [no inverse] {@ (1, 0, 0), (x^2, 1, 2), (x^2, -1, 2), (x + 1, 0, 1), (x, 1, 1), (x, -1, 1), (x^2 + x, x + 1, 2), (x^2 + x, -x - 1, 2), (x^2 - 2*x + 2, 2*x - 3, 2), (x^2 - 2*x + 2, -2*x + 3, 2) @} 2 5 Total time: 0.610 seconds, Total memory usage: 39.65MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 19:14:30 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^0*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); //V:=RationalPoints(J:Bound:=1500); //B:=ReducedBasis(V); //B; //P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; //hP:=Height(P);hP; //HC:=HeightConstant(J:Effort:=20);HC; //Exp(hP/4+HC); RationalPoints(J:Bound:=12); //Chabauty(P,29); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 19:14:29 on modular [Seed = 1519333443] ------------------------------------- 0 Abelian Group isomorphic to Z/10 Defined on 1 generator Relations: 10*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/10 Defined on 1 generator Relations: 10*P[1] = 0 to JacHyp: J given by a rule [no inverse] {@ (1, 0, 0), (x^2, 1, 2), (x^2, -1, 2), (x + 1, 0, 1), (x, 1, 1), (x, -1, 1), (x^2 + x, x + 1, 2), (x^2 + x, -x - 1, 2), (x^2 - 2*x + 2, 2*x - 3, 2), (x^2 - 2*x + 2, -2*x + 3, 2) @} Total time: 0.610 seconds, Total memory usage: 39.64MB '134.219' ************** MAGMA ***************** Host 134.219.148.42 (134.219.148.42) Time: Tue Dec 6 16:45:14 2005 Input: 5*0.5; Output: Magma V2.11-10 Tue Dec 6 2005 16:45:14 on modular [Seed = 788228921] ------------------------------------- 2.500000000000000000000000000000000000000 Total time: 0.190 seconds, Total memory usage: 3.24MB '134.219' ************** MAGMA ***************** Host 134.219.148.42 (134.219.148.42) Time: Tue Dec 6 16:44:10 2005 Input: "Replace this by some code, then click [PARI] or [MAGMA]!" Output: Magma V2.11-10 Tue Dec 6 2005 16:44:10 on modular [Seed = 870741604] ------------------------------------- Replace this by some code, then click [PARI] or [MAGMA]! Total time: 0.180 seconds, Total memory usage: 3.24MB '134.219' ************** MAGMA ***************** Host 134.219.148.42 (134.219.148.42) Time: Tue Dec 6 16:43:21 2005 Input: 174801202447173743443223007264396063127947174680137917788974928763633474431825 923127812764804103556846639940324358013520109641947732704660271041589584499 909560268503525396803490530790881045657521939684848324648124706619281356934 214616009*0.5; Output: Magma V2.11-10 Tue Dec 6 2005 16:43:21 on modular [Seed = 887190192] ------------------------------------- >> 88974928763633474431825 923127812764804103556846639940324358013520109641 ^ User error: bad syntax Total time: 0.190 seconds, Total memory usage: 3.24MB '134.219' ************** MAGMA ***************** Host 134.219.148.42 (134.219.148.42) Time: Tue Dec 6 16:42:45 2005 Input: 174801202447173743443223007264396063127947174680137917788974928763633474431825 923127812764804103556846639940324358013520109641947732704660271041589584499 909560268503525396803490530790881045657521939684848324648124706619281356934 214616009/2; Output: Magma V2.11-10 Tue Dec 6 2005 16:42:45 on modular [Seed = 1133905387] ------------------------------------- >> 88974928763633474431825 923127812764804103556846639940324358013520109641 ^ User error: bad syntax Total time: 0.190 seconds, Total memory usage: 3.24MB '134.219' ************** MAGMA ***************** Host 134.219.148.42 (134.219.148.42) Time: Tue Dec 6 16:42:17 2005 Input: 174801202447173743443223007264396063127947174680137917788974928763633474431825 923127812764804103556846639940324358013520109641947732704660271041589584499 909560268503525396803490530790881045657521939684848324648124706619281356934 214616009*0.5; Output: Magma V2.11-10 Tue Dec 6 2005 16:42:17 on modular [Seed = 1150354048] ------------------------------------- >> 88974928763633474431825 923127812764804103556846639940324358013520109641 ^ User error: bad syntax Total time: 0.190 seconds, Total memory usage: 3.24MB '134.219' ************** MAGMA ***************** Host 134.219.148.42 (134.219.148.42) Time: Tue Dec 6 16:41:34 2005 Input: 2*3; Output: Magma V2.11-10 Tue Dec 6 2005 16:41:34 on modular [Seed = 1318783165] ------------------------------------- 6 Total time: 0.190 seconds, Total memory usage: 3.24MB '134.219' ************** MAGMA ***************** Host 134.219.148.42 (134.219.148.42) Time: Tue Dec 6 16:27:48 2005 Input: 0.5*174801202447173743443223007264396063127947174680137917788974928763633474431825 923127812764804103556846639940324358013520109641947732704660271041589584499 909560268503525396803490530790881045657521939684848324648124706619281356934 214616009 ; Output: Magma V2.11-10 Tue Dec 6 2005 16:27:48 on modular [Seed = 1824075077] ------------------------------------- >> 917788974928763633474431825 92312781276480410355684663994032435801352010 ^ User error: bad syntax Total time: 0.180 seconds, Total memory usage: 3.24MB '134.219' ************** MAGMA ***************** Host 134.219.148.42 (134.219.148.42) Time: Tue Dec 6 16:27:21 2005 Input: 0.5*174801202447173743443223007264396063127947174680137917788974928763633474431825 923127812764804103556846639940324358013520109641947732704660271041589584499 909560268503525396803490530790881045657521939684848324648124706619281356934 214616009; Output: Magma V2.11-10 Tue Dec 6 2005 16:27:21 on modular [Seed = 1906585653] ------------------------------------- >> 917788974928763633474431825 92312781276480410355684663994032435801352010 ^ User error: bad syntax Total time: 0.190 seconds, Total memory usage: 3.24MB '134.219' ************** MAGMA ***************** Host 134.219.148.42 (134.219.148.42) Time: Tue Dec 6 16:27:17 2005 Input: 0.5*174801202447173743443223007264396063127947174680137917788974928763633474431825 923127812764804103556846639940324358013520109641947732704660271041589584499 909560268503525396803490530790881045657521939684848324648124706619281356934 214616009; Output: Magma V2.11-10 Tue Dec 6 2005 16:27:17 on modular [Seed = 2058304306] ------------------------------------- >> 917788974928763633474431825 92312781276480410355684663994032435801352010 ^ User error: bad syntax Total time: 0.190 seconds, Total memory usage: 3.24MB '134.219' ************** MAGMA ***************** Host 134.219.148.42 (134.219.148.42) Time: Tue Dec 6 16:27:11 2005 Input: 0.5*174801202447173743443223007264396063127947174680137917788974928763633474431825 923127812764804103556846639940324358013520109641947732704660271041589584499 909560268503525396803490530790881045657521939684848324648124706619281356934 214616009 Output: Magma V2.11-10 Tue Dec 6 2005 16:27:11 on modular [Seed = 2141863469] ------------------------------------- >> 917788974928763633474431825 92312781276480410355684663994032435801352010 ^ User error: bad syntax Total time: 0.190 seconds, Total memory usage: 3.24MB '134.219' ************** MAGMA ***************** Host 134.219.148.42 (134.219.148.42) Time: Tue Dec 6 15:54:15 2005 Input: A := matrix[1,2,3,4]; Output: Magma V2.11-10 Tue Dec 6 2005 15:54:14 on modular [Seed = 3691668948] ------------------------------------- >> A := matrix[1,2,3,4]; ^ User error: Identifier 'matrix' has not been declared or assigned Total time: 0.190 seconds, Total memory usage: 3.24MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 15:52:50 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^9*3^5; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; hP:=Height(P);hP; HC:=HeightConstant(J:Effort:=20);HC; Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,29); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 15:52:47 on modular [Seed = 3845348484] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 + 8*x + 48, -4*x + 336, 2) ] 3.06508423551517870438515543448541065190957332594386690400587 10.615807194328795518188426595221995880700 87735.1813188090206788198021698402428989801619 {@ @} Total time: 2.319 seconds, Total memory usage: 39.30MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 15:52:42 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^9*3^5; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; hP:=Height(P);hP; HC:=HeightConstant(J:Effort:=20);HC; Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,23); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 15:52:38 on modular [Seed = 3896398208] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 + 8*x + 48, -4*x + 336, 2) ] 3.06508423551517870438515543448541065190957332594386690400587 10.615807194328795518188426595221995880700 87735.1813188090206788198021698402428989801619 {@ <74649, 1, 4, 1>, <249979, 1, 4, 1>, <36233, 1, 4, 1>, <33970, 1, 4, 1>, <158613, 1, 4, 1>, <0, 1, 4, 1>, <87568, 1, 4, 1>, <200366, 1, 4, 1>, <103472, 1, 4, 1>, <107491, 1, 4, 1>, <129032, 1, 4, 1> @} Total time: 3.740 seconds, Total memory usage: 39.46MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 15:52:31 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^9*3^5; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; hP:=Height(P);hP; HC:=HeightConstant(J:Effort:=20);HC; Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,19); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 15:52:29 on modular [Seed = 4013512348] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 + 8*x + 48, -4*x + 336, 2) ] 3.06508423551517870438515543448541065190957332594386690400587 10.615807194328795518188426595221995880700 87735.1813188090206788198021698402428989801619 {@ <22706, 1, 4, 1>, <26230, 1, 4, 1> @} Total time: 2.399 seconds, Total memory usage: 39.29MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 15:52:23 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^9*3^5; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; hP:=Height(P);hP; HC:=HeightConstant(J:Effort:=20);HC; Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,17); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 15:52:20 on modular [Seed = 4062464904] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 + 8*x + 48, -4*x + 336, 2) ] 3.06508423551517870438515543448541065190957332594386690400587 10.615807194328795518188426595221995880700 87735.1813188090206788198021698402428989801619 {@ <4933, 1, 4, 1>, <76884, 1, 4, 1>, <12222, 1, 4, 1>, <78542, 1, 4, 1>, <74569, 1, 4, 1>, <2337, 1, 4, 1>, <58863, 1, 4, 1> @} Total time: 3.140 seconds, Total memory usage: 39.31MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 15:52:13 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^9*3^5; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; hP:=Height(P);hP; HC:=HeightConstant(J:Effort:=20);HC; Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,13); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 15:52:10 on modular [Seed = 4180624253] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 + 8*x + 48, -4*x + 336, 2) ] 3.06508423551517870438515543448541065190957332594386690400587 10.615807194328795518188426595221995880700 87735.1813188090206788198021698402428989801619 {@ <11470, 1, 4, 1>, <20108, 1, 4, 1>, <17396, 1, 4, 1>, <24070, 1, 4, 1>, <18897, 1, 4, 1> @} Total time: 2.710 seconds, Total memory usage: 39.30MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 15:51:58 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^9*3^5; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; hP:=Height(P);hP; HC:=HeightConstant(J:Effort:=20);HC; Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,11); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 15:51:55 on modular [Seed = 4280894826] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 + 8*x + 48, -4*x + 336, 2) ] 3.06508423551517870438515543448541065190957332594386690400587 10.615807194328795518188426595221995880700 87735.1813188090206788198021698402428989801619 {@ <10632, 1, 4, 1>, <4110, 1, 4, 1>, <1421, 1, 4, 1>, <5279, 1, 4, 1> @} Total time: 2.600 seconds, Total memory usage: 39.28MB '146.6.1' ************** MAGMA ***************** Host 146.6.139.217 (146.6.139.217) Time: Tue Dec 6 15:51:33 2005 Input: count:=0; m:=6; q:=2^m; F1:=GF(q); F2:=GF(q^2); Trace0:={@0@}; for i:=1 to q-1 do dd:=F1.1^i; if Trace(dd) eq 0 then Trace0:=Trace0 join {@dd@}; end if; end for; d:=0; for i:=1 to 2 do a:=F1.1^i; for j:=1 to #Trace0 do b:=Trace0[j]; P:=PolynomialRing(F1); C:=HyperellipticCurve(a*x^5+b*x^3+x,x); N1:=#C; //over Fq R:=PolynomialRing(F2); CC:=HyperellipticCurve(a*z^5+b*z^3+z,z); N2:=#CC; //over Fq^2; a1:=N1-q-1; a2:=(N2-1-q^2+a1^2)/2; if not IsSquare(a1^2-4*a2+8*q) then if IsDivisibleBy(a2, 2^Ceiling(m/2)) then delta:=(a2+2*q)^2-4*q*a1^2; V:=0; if delta ne 0 then V:=Valuation(delta,2); end if; B:=delta/(2^V); if IsOdd(V) or (B-1 mod 8 ne 0) then // count:=count+1; // print C,"(a1,a2)= (", a1 ,",", a2,")"; //print "(",a1, a2,")"; end if; end if; // end if; J:=q^2+a1*q+a1+a2+1; Rat:=#RationalPoints(C); Jac:=Jacobian(C); R:=RingOfIntegers(); J:=q^2+a1*q+a1+a2+1; Rat:=#RationalPoints(C); Jac:=Jacobian(C); R:=RingOfIntegers(); if R!J mod 8 eq 0 then for ii:=1 to Floor(J) do DD:=Points(Jac)[ii]; if HasOrder(DD,8) then count:=count+1; //print DD, "has order 8, a=", a," b= ", b; //2*DD; //(F1.1^2*x^5+F1.1*x^3+x-x*(F1.1^2*x+F1.1^29)-(F1.1^2*x+F1.1^29)^2)/DD[1]^2; end if; end for; count; end if; end if; end for; end for; print "done"; Output: ** WARNING: Computation time exceeded 20 seconds, so computation was terminated after 20 seconds. ** Magma V2.11-10 Tue Dec 6 2005 15:51:12 on modular [Seed = 47495357] ------------------------------------- 4 Errors: /bin/sh: line 1: 15740 Alarm clock nice -n 19 /usr/local/bin/magma '146.6.1' ************** MAGMA ***************** Host 146.6.139.217 (146.6.139.217) Time: Tue Dec 6 15:50:34 2005 Input: count:=0; m:=6; q:=2^m; F1:=GF(q); F2:=GF(q^2); Trace0:={@0@}; for i:=1 to q-1 do dd:=F1.1^i; if Trace(dd) eq 0 then Trace0:=Trace0 join {@dd@}; end if; end for; d:=0; for i:=1 to 1 do a:=F1.1^i; for j:=1 to 2 do //#Trace0 do b:=Trace0[j]; P:=PolynomialRing(F1); C:=HyperellipticCurve(a*x^5+b*x^3+x,x); N1:=#C; //over Fq R:=PolynomialRing(F2); CC:=HyperellipticCurve(a*z^5+b*z^3+z,z); N2:=#CC; //over Fq^2; a1:=N1-q-1; a2:=(N2-1-q^2+a1^2)/2; if not IsSquare(a1^2-4*a2+8*q) then if IsDivisibleBy(a2, 2^Ceiling(m/2)) then delta:=(a2+2*q)^2-4*q*a1^2; V:=0; if delta ne 0 then V:=Valuation(delta,2); end if; B:=delta/(2^V); if IsOdd(V) or (B-1 mod 8 ne 0) then // count:=count+1; // print C,"(a1,a2)= (", a1 ,",", a2,")"; //print "(",a1, a2,")"; end if; end if; // end if; J:=q^2+a1*q+a1+a2+1; Rat:=#RationalPoints(C); Jac:=Jacobian(C); R:=RingOfIntegers(); J:=q^2+a1*q+a1+a2+1; Rat:=#RationalPoints(C); Jac:=Jacobian(C); R:=RingOfIntegers(); if R!J mod 8 eq 0 then for ii:=1 to Floor(J) do DD:=Points(Jac)[ii]; if HasOrder(DD,8) then count:=count+1; //print DD, "has order 8, a=", a," b= ", b; //2*DD; //(F1.1^2*x^5+F1.1*x^3+x-x*(F1.1^2*x+F1.1^29)-(F1.1^2*x+F1.1^29)^2)/DD[1]^2; end if; end for; count; end if; end if; end for; end for; print "done"; Output: Magma V2.11-10 Tue Dec 6 2005 15:50:34 on modular [Seed = 165655533] ------------------------------------- done Total time: 0.220 seconds, Total memory usage: 3.43MB '146.6.1' ************** MAGMA ***************** Host 146.6.139.217 (146.6.139.217) Time: Tue Dec 6 15:50:07 2005 Input: count:=0; m:=6; q:=2^m; F1:=GF(q); F2:=GF(q^2); Trace0:={@0@}; for i:=1 to q-1 do dd:=F1.1^i; if Trace(dd) eq 0 then Trace0:=Trace0 join {@dd@}; end if; end for; d:=0; for i:=1 to 1 do a:=F1.1^i; for j:=1 to #Trace0 do b:=Trace0[j]; P:=PolynomialRing(F1); C:=HyperellipticCurve(a*x^5+b*x^3+x,x); N1:=#C; //over Fq R:=PolynomialRing(F2); CC:=HyperellipticCurve(a*z^5+b*z^3+z,z); N2:=#CC; //over Fq^2; a1:=N1-q-1; a2:=(N2-1-q^2+a1^2)/2; if not IsSquare(a1^2-4*a2+8*q) then if IsDivisibleBy(a2, 2^Ceiling(m/2)) then delta:=(a2+2*q)^2-4*q*a1^2; V:=0; if delta ne 0 then V:=Valuation(delta,2); end if; B:=delta/(2^V); if IsOdd(V) or (B-1 mod 8 ne 0) then // count:=count+1; // print C,"(a1,a2)= (", a1 ,",", a2,")"; //print "(",a1, a2,")"; end if; end if; // end if; J:=q^2+a1*q+a1+a2+1; Rat:=#RationalPoints(C); Jac:=Jacobian(C); R:=RingOfIntegers(); J:=q^2+a1*q+a1+a2+1; Rat:=#RationalPoints(C); Jac:=Jacobian(C); R:=RingOfIntegers(); if R!J mod 8 eq 0 then for ii:=1 to Floor(J) do DD:=Points(Jac)[ii]; if HasOrder(DD,8) then count:=count+1; //print DD, "has order 8, a=", a," b= ", b; //2*DD; //(F1.1^2*x^5+F1.1*x^3+x-x*(F1.1^2*x+F1.1^29)-(F1.1^2*x+F1.1^29)^2)/DD[1]^2; end if; end for; count; end if; end if; end for; end for; print "done"; Output: ** WARNING: Computation time exceeded 20 seconds, so computation was terminated after 20 seconds. ** Magma V2.11-10 Tue Dec 6 2005 15:49:47 on modular [Seed = 215660057] ------------------------------------- 4 Errors: /bin/sh: line 1: 15730 Alarm clock nice -n 19 /usr/local/bin/magma '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 15:42:46 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^8*3^5; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; hP:=Height(P);hP; HC:=HeightConstant(J:Effort:=20);HC; Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,11); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 15:42:44 on modular [Seed = 314877608] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 + 8*x + 96, 20*x - 48, 2) ] 2.808141398815367927531209718024871984466 10.484172230478017379230331171078121089143 72128.720041723759240639205711309210490074464 {@ @} Total time: 2.049 seconds, Total memory usage: 39.63MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 15:42:37 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^8*3^5; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; hP:=Height(P);hP; HC:=HeightConstant(J:Effort:=20);HC; Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,7); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 15:42:35 on modular [Seed = 433040860] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 + 8*x + 96, 20*x - 48, 2) ] 2.808141398815367927531209718024871984466 10.484172230478017379230331171078121089143 72128.720041723759240639205711309210490074464 {@ <12, 1, 4, 1> @} Total time: 2.040 seconds, Total memory usage: 39.62MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 15:38:01 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^8*3^5; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; hP:=Height(P);hP; HC:=HeightConstant(J:Effort:=20);HC; Exp(hP/4+HC); //RationalPoints(J:Bound:=12); //Chabauty(P,29); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 15:37:59 on modular [Seed = 771463412] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 + 8*x + 96, 20*x - 48, 2) ] 2.808141398815367927531209718024871984466 10.484172230478017379230331171078121089143 72128.720041723759240639205711309210490074464 Total time: 1.700 seconds, Total memory usage: 39.58MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 15:37:20 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^8*3^2; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; hP:=Height(P);hP; HC:=HeightConstant(J:Effort:=20);HC; Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,29); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 15:37:16 on modular [Seed = 820413746] ------------------------------------- 1 Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 to JacHyp: J given by a rule [no inverse] [ (x^2 - 8*x - 32, -32*x - 48, 2) ] 2.1159528178465433219870294644907003355887 8.737886401785989643018017729228475458387 10581.628396140394363073703025963690658217679 {@ <0, 1, 4, 1> @} Total time: 3.930 seconds, Total memory usage: 39.47MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 15:37:10 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^8*3^2; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; hP:=Height(P);hP; HC:=HeightConstant(J:Effort:=20);HC; Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,23); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 15:37:05 on modular [Seed = 937525816] ------------------------------------- 1 Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 to JacHyp: J given by a rule [no inverse] [ (x^2 - 8*x - 32, -32*x - 48, 2) ] 2.1159528178465433219870294644907003355887 8.737886401785989643018017729228475458387 10581.628396140394363073703025963690658217679 {@ <261417, 1, 4, 1>, <44441, 1, 4, 1>, <0, 1, 4, 1>, <35100, 1, 4, 1>, <31761, 1, 4, 1>, <181934, 1, 4, 1>, <115431, 1, 4, 1>, <194062, 1, 4, 1>, <154635, 1, 4, 1>, <60434, 1, 4, 1> @} Total time: 5.070 seconds, Total memory usage: 39.47MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 15:36:56 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^8*3^2; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; hP:=Height(P);hP; HC:=HeightConstant(J:Effort:=20);HC; Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,19); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 15:36:52 on modular [Seed = 1038845991] ------------------------------------- 1 Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 to JacHyp: J given by a rule [no inverse] [ (x^2 - 8*x - 32, -32*x - 48, 2) ] 2.1159528178465433219870294644907003355887 8.737886401785989643018017729228475458387 10581.628396140394363073703025963690658217679 {@ <51030, 1, 4, 1>, <0, 1, 4, 1> @} Total time: 3.930 seconds, Total memory usage: 39.47MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 15:36:43 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^8*3^2; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; hP:=Height(P);hP; HC:=HeightConstant(J:Effort:=20);HC; Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,17); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 15:36:38 on modular [Seed = 1083635048] ------------------------------------- 1 Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 to JacHyp: J given by a rule [no inverse] [ (x^2 - 8*x - 32, -32*x - 48, 2) ] 2.1159528178465433219870294644907003355887 8.737886401785989643018017729228475458387 10581.628396140394363073703025963690658217679 {@ <7881, 1, 4, 1>, <0, 1, 4, 1>, <42540, 1, 4, 1>, <2153, 1, 4, 1>, <21258, 1, 4, 1>, <30750, 1, 4, 1>, <71880, 1, 4, 1> @} Total time: 4.589 seconds, Total memory usage: 39.47MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 15:36:32 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^8*3^2; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; hP:=Height(P);hP; HC:=HeightConstant(J:Effort:=20);HC; Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,13); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 15:36:27 on modular [Seed = 1200747122] ------------------------------------- 1 Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 to JacHyp: J given by a rule [no inverse] [ (x^2 - 8*x - 32, -32*x - 48, 2) ] 2.1159528178465433219870294644907003355887 8.737886401785989643018017729228475458387 10581.628396140394363073703025963690658217679 {@ <21308, 1, 4, 1>, <0, 1, 4, 1>, <24889, 1, 4, 1>, <9389, 1, 4, 1>, <3447, 1, 4, 1>, <2598, 1, 4, 1> @} Total time: 4.160 seconds, Total memory usage: 39.47MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 15:36:21 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^8*3^2; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; hP:=Height(P);hP; HC:=HeightConstant(J:Effort:=20);HC; Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,11); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 15:36:17 on modular [Seed = 1251802997] ------------------------------------- 1 Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 to JacHyp: J given by a rule [no inverse] [ (x^2 - 8*x - 32, -32*x - 48, 2) ] 2.1159528178465433219870294644907003355887 8.737886401785989643018017729228475458387 10581.628396140394363073703025963690658217679 {@ <5942, 1, 4, 1>, <5847, 1, 4, 1>, <0, 1, 4, 1>, <6498, 1, 4, 1>, <6885, 1, 4, 1>, <4110, 1, 4, 1> @} Total time: 4.160 seconds, Total memory usage: 39.47MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 15:36:06 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^8*3^2; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; hP:=Height(P);hP; HC:=HeightConstant(J:Effort:=20);HC; Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,7); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 15:36:00 on modular [Seed = 1367860255] ------------------------------------- 1 Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 to JacHyp: J given by a rule [no inverse] [ (x^2 - 8*x - 32, -32*x - 48, 2) ] 2.1159528178465433219870294644907003355887 8.737886401785989643018017729228475458387 10581.628396140394363073703025963690658217679 {@ <1912, 1, 4, 1>, <0, 1, 4, 1>, <943, 1, 4, 1>, <0, 1, 0, 6> @} Total time: 6.250 seconds, Total memory usage: 39.47MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 15:35:37 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^8*3^2; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=11000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; hP:=Height(P);hP; HC:=HeightConstant(J:Effort:=20);HC; Exp(hP/4+HC); //RationalPoints(J:Bound:=12); //Chabauty(P,29); //Order(Q);Order(P); //P+Q; Output: ** WARNING: Computation time exceeded 20 seconds, so computation was terminated after 20 seconds. ** Magma V2.11-10 Tue Dec 6 2005 15:35:16 on modular [Seed = 1502604323] ------------------------------------- 1 Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 to JacHyp: J given by a rule [no inverse] Errors: /bin/sh: line 1: 15655 Alarm clock nice -n 19 /usr/local/bin/magma '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 15:32:59 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^8*3^2; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; hP:=Height(P);hP; HC:=HeightConstant(J:Effort:=20);HC; Exp(hP/4+HC); //RationalPoints(J:Bound:=12); //Chabauty(P,29); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 15:32:56 on modular [Seed = 1552607834] ------------------------------------- 1 Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 to JacHyp: J given by a rule [no inverse] [ (x^2 - 8*x - 32, -32*x - 48, 2) ] 2.1159528178465433219870294644907003355887 8.737886401785989643018017729228475458387 10581.628396140394363073703025963690658217679 Total time: 3.350 seconds, Total memory usage: 39.47MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 15:28:16 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^8*3^2; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; hP:=Height(P);hP; HC:=HeightConstant(J:Effort:=20);HC; Exp(hP/4+HC); //RationalPoints(J:Bound:=12); //Chabauty(P,29); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 15:28:13 on modular [Seed = 1656018241] ------------------------------------- 1 Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 to JacHyp: J given by a rule [no inverse] [ (x^2 - 8*x - 32, -32*x - 48, 2) ] 2.1159528178465433219870294644907003355887 8.737886401785989643018017729228475458387 10581.628396140394363073703025963690658217679 Total time: 3.359 seconds, Total memory usage: 39.47MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 15:26:58 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^8*3^1; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; hP:=Height(P);hP; HC:=HeightConstant(J:Effort:=20);HC; Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,29); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 15:26:55 on modular [Seed = 1757339417] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 + 16, -4*x - 32, 2) ] 1.49967351695988847267651787151293279909398082122246414246079 6.386431030553623320704260894112207959006 863.807039365512812559094883980533584598159 {@ @} Total time: 2.220 seconds, Total memory usage: 39.60MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 15:26:47 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^8*3^1; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; hP:=Height(P);hP; HC:=HeightConstant(J:Effort:=20);HC; Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,23); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 15:26:44 on modular [Seed = 1874450453] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 + 16, -4*x - 32, 2) ] 1.49967351695988847267651787151293279909398082122246414246079 6.386431030553623320704260894112207959006 863.807039365512812559094883980533584598159 {@ <114991, 1, 4, 1>, <70244, 1, 4, 1>, <195507, 1, 4, 1>, <13960, 1, 4, 1>, <138074, 1, 4, 1>, <29547, 1, 4, 1>, <0, 1, 4, 1>, <82450, 1, 4, 1>, <202201, 1, 4, 1>, <148362, 1, 4, 1>, <232543, 1, 4, 1> @} Total time: 3.600 seconds, Total memory usage: 39.57MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 15:26:37 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^8*3^1; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; hP:=Height(P);hP; HC:=HeightConstant(J:Effort:=20);HC; Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,19); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 15:26:35 on modular [Seed = 1923400439] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 + 16, -4*x - 32, 2) ] 1.49967351695988847267651787151293279909398082122246414246079 6.386431030553623320704260894112207959006 863.807039365512812559094883980533584598159 {@ <128766, 1, 4, 1>, <2520, 1, 4, 1> @} Total time: 2.310 seconds, Total memory usage: 39.58MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 15:26:30 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^8*3^1; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; hP:=Height(P);hP; HC:=HeightConstant(J:Effort:=20);HC; Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,17); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 15:26:26 on modular [Seed = 2041563637] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 + 16, -4*x - 32, 2) ] 1.49967351695988847267651787151293279909398082122246414246079 6.386431030553623320704260894112207959006 863.807039365512812559094883980533584598159 {@ <33619, 1, 4, 1>, <15634, 1, 4, 1>, <79085, 1, 4, 1>, <41020, 1, 4, 1>, <22301, 1, 4, 1>, <71556, 1, 4, 1>, <57819, 1, 4, 1>, <73062, 1, 4, 1> @} Total time: 3.089 seconds, Total memory usage: 39.59MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 15:26:22 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^8*3^1; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; hP:=Height(P);hP; HC:=HeightConstant(J:Effort:=20);HC; Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,13); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 15:26:19 on modular [Seed = 2091568361] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 + 16, -4*x - 32, 2) ] 1.49967351695988847267651787151293279909398082122246414246079 6.386431030553623320704260894112207959006 863.807039365512812559094883980533584598159 {@ <5116, 1, 4, 1>, <3542, 1, 4, 1>, <20049, 1, 4, 1>, <0, 1, 4, 1>, <3726, 1, 4, 1>, <22, 1, 4, 1> @} Total time: 2.549 seconds, Total memory usage: 39.56MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 15:26:14 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^8*3^1; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; hP:=Height(P);hP; HC:=HeightConstant(J:Effort:=20);HC; Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,11); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 15:26:12 on modular [Seed = 2186555929] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 + 16, -4*x - 32, 2) ] 1.49967351695988847267651787151293279909398082122246414246079 6.386431030553623320704260894112207959006 863.807039365512812559094883980533584598159 {@ <0, 1, 4, 1> @} Total time: 1.940 seconds, Total memory usage: 39.53MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 15:26:06 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^8*3^1; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; hP:=Height(P);hP; HC:=HeightConstant(J:Effort:=20);HC; Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,7); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 15:26:04 on modular [Seed = 2270246692] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 + 16, -4*x - 32, 2) ] 1.49967351695988847267651787151293279909398082122246414246079 6.386431030553623320704260894112207959006 863.807039365512812559094883980533584598159 {@ <475, 1, 4, 1>, <2015, 1, 4, 1>, <488, 1, 4, 1>, <2354, 1, 4, 1> @} Total time: 2.200 seconds, Total memory usage: 39.57MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 15:24:40 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^8*3^1; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; hP:=Height(P);hP; HC:=HeightConstant(J:Effort:=20);HC; Exp(hP/4+HC); //RationalPoints(J:Bound:=12); //Chabauty(P,31); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 15:24:38 on modular [Seed = 2354723957] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 + 16, -4*x - 32, 2) ] 1.49967351695988847267651787151293279909398082122246414246079 6.386431030553623320704260894112207959006 863.807039365512812559094883980533584598159 Total time: 1.669 seconds, Total memory usage: 39.54MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 15:23:52 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^8*3^1; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; hP:=Height(P);hP; HC:=HeightConstant(J:Effort:=20);HC; Exp(hP/4+HC); //RationalPoints(J:Bound:=12); //Chabauty(P,31); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 15:23:50 on modular [Seed = 2437358008] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 + 16, -4*x - 32, 2) ] 1.49967351695988847267651787151293279909398082122246414246079 6.386431030553623320704260894112207959006 863.807039365512812559094883980533584598159 Total time: 1.659 seconds, Total memory usage: 39.52MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 15:22:40 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^7*3^2; C:=HyperellipticCurve(f); J:=Jacobian(C); //r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; hP:=Height(P);hP; HC:=HeightConstant(J:Effort:=20);HC; Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,31); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 15:22:39 on modular [Seed = 2555259112] ------------------------------------- Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 28/9*x - 8/9, 44/27*x + 928/27, 2) ] 4.6613668053397002241366926279695766743193744943846637819259 6.862593078943470323468275363544094477084 3065.404397974187850397290252109710706677000 {@ @} Total time: 1.540 seconds, Total memory usage: 38.22MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 15:22:29 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^7*3^2; C:=HyperellipticCurve(f); J:=Jacobian(C); //r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; hP:=Height(P);hP; HC:=HeightConstant(J:Effort:=20);HC; Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,29); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 15:22:27 on modular [Seed = 2638949857] ------------------------------------- Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 28/9*x - 8/9, 44/27*x + 928/27, 2) ] 4.6613668053397002241366926279695766743193744943846637819259 6.862593078943470323468275363544094477084 3065.404397974187850397290252109710706677000 {@ <392168, 1, 4, 1> @} Total time: 1.649 seconds, Total memory usage: 38.22MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 15:22:21 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^7*3^2; C:=HyperellipticCurve(f); J:=Jacobian(C); //r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; hP:=Height(P);hP; HC:=HeightConstant(J:Effort:=20);HC; Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,23); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 15:22:18 on modular [Seed = 2725516071] ------------------------------------- Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 28/9*x - 8/9, 44/27*x + 928/27, 2) ] 4.6613668053397002241366926279695766743193744943846637819259 6.862593078943470323468275363544094477084 3065.404397974187850397290252109710706677000 {@ <51383, 1, 4, 1>, <171963, 1, 4, 1>, <50185, 1, 4, 1>, <0, 1, 4, 1>, <96713, 1, 4, 1>, <155355, 1, 4, 1>, <180270, 1, 4, 1>, <128425, 1, 4, 1>, <31518, 1, 4, 1>, <106257, 1, 4, 1> @} Total time: 2.819 seconds, Total memory usage: 38.22MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 15:22:12 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^7*3^2; C:=HyperellipticCurve(f); J:=Jacobian(C); //r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; hP:=Height(P);hP; HC:=HeightConstant(J:Effort:=20);HC; Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,19); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 15:22:10 on modular [Seed = 2809206314] ------------------------------------- Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 28/9*x - 8/9, 44/27*x + 928/27, 2) ] 4.6613668053397002241366926279695766743193744943846637819259 6.862593078943470323468275363544094477084 3065.404397974187850397290252109710706677000 {@ <111296, 1, 4, 1> @} Total time: 1.540 seconds, Total memory usage: 38.22MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 15:22:05 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^7*3^2; C:=HyperellipticCurve(f); J:=Jacobian(C); //r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; hP:=Height(P);hP; HC:=HeightConstant(J:Effort:=20);HC; Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,17); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 15:22:03 on modular [Seed = 2893684024] ------------------------------------- Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 28/9*x - 8/9, 44/27*x + 928/27, 2) ] 4.6613668053397002241366926279695766743193744943846637819259 6.862593078943470323468275363544094477084 3065.404397974187850397290252109710706677000 {@ <17507, 1, 4, 1>, <76475, 1, 4, 1>, <28388, 1, 4, 1>, <30994, 1, 4, 1>, <76221, 1, 4, 1>, <35202, 1, 4, 1>, <73827, 1, 4, 1>, <0, 1, 4, 1> @} Total time: 2.450 seconds, Total memory usage: 38.22MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 15:21:58 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^7*3^2; C:=HyperellipticCurve(f); J:=Jacobian(C); //r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; hP:=Height(P);hP; HC:=HeightConstant(J:Effort:=20);HC; Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,13); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 15:21:57 on modular [Seed = 2992898856] ------------------------------------- Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 28/9*x - 8/9, 44/27*x + 928/27, 2) ] 4.6613668053397002241366926279695766743193744943846637819259 6.862593078943470323468275363544094477084 3065.404397974187850397290252109710706677000 {@ <16251, 1, 4, 1> @} Total time: 1.429 seconds, Total memory usage: 38.22MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 15:21:51 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^7*3^2; C:=HyperellipticCurve(f); J:=Jacobian(C); //r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; hP:=Height(P);hP; HC:=HeightConstant(J:Effort:=20);HC; Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,7); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 15:21:49 on modular [Seed = 3076589077] ------------------------------------- Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 28/9*x - 8/9, 44/27*x + 928/27, 2) ] 4.6613668053397002241366926279695766743193744943846637819259 6.862593078943470323468275363544094477084 3065.404397974187850397290252109710706677000 {@ <1850, 1, 4, 1>, <0, 1, 4, 1>, <801, 1, 4, 1> @} Total time: 1.520 seconds, Total memory usage: 38.22MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 15:21:06 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^7*3^2; C:=HyperellipticCurve(f); J:=Jacobian(C); //r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=4000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; hP:=Height(P);hP; HC:=HeightConstant(J:Effort:=20);HC; Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,11); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 15:21:02 on modular [Seed = 3161066861] ------------------------------------- Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 28/9*x - 8/9, 44/27*x + 928/27, 2) ] 4.6613668053397002241366926279695766743193744943846637819259 6.862593078943470323468275363544094477084 3065.404397974187850397290252109710706677000 {@ <9277, 1, 4, 1>, <6898, 1, 4, 1>, <11740, 1, 4, 1>, <2205, 1, 4, 1>, <13803, 1, 4, 1> @} Total time: 4.389 seconds, Total memory usage: 38.22MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 15:19:41 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^7*3^2; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; hP:=Height(P);hP; HC:=HeightConstant(J:Effort:=20);HC; Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,11); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 15:19:38 on modular [Seed = 3239538916] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 28/9*x - 8/9, 44/27*x + 928/27, 2) ] 4.6613668053397002241366926279695766743193744943846637819259 6.862593078943470323468275363544094477084 3065.404397974187850397290252109710706677000 {@ <9277, 1, 4, 1>, <6898, 1, 4, 1>, <11740, 1, 4, 1>, <2205, 1, 4, 1>, <13803, 1, 4, 1> @} Total time: 2.940 seconds, Total memory usage: 39.92MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 15:19:36 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^7*3^2; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; hP:=Height(P);hP; HC:=HeightConstant(J:Effort:=20);HC; Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,11); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 15:19:33 on modular [Seed = 3322968044] ------------------------------------- TwoSelmerGroupData( J: Jacobian of Hyperelliptic Curve defined by y^2 = x^5 + 1152 ... ) TwoSelmerWork( C: Hyperelliptic Curve defined by y^2 = x^5 + 1152 over Rationa..., BoundType: Default, Bound: 0, UseUnits: true, Points: {}, Fields: {} ) LocalImageOdd( f: x^5 + 1152, p: 2, m1: Mapping from: Univariate Polynomial Ring in x over Rational ..., m2: Mapping from: Univariate Polynomial Ring in x over Rational ..., v: 30, Points: {} ) try( ) try( d: 2 ) SquarefreeRoots( F: $.1^4 + O(2^50)*$.1^3 - (187785903341577*2^3 + O(2^53))*$.1^... ) In file "/usr/local/magma/package/Geometry/Arith/sqfroots.m", line 84, column 15: >> return Roots(F:IsSquarefree); ^ Runtime error in 'Roots': Insufficient precision to find roots >> r:=TwoSelmerGroupData(J);r; ^ User error: Identifier 'r' has not been declared or assigned Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 28/9*x - 8/9, 44/27*x + 928/27, 2) ] 4.6613668053397002241366926279695766743193744943846637819259 6.862593078943470323468275363544094477084 3065.404397974187850397290252109710706677000 {@ <9277, 1, 4, 1>, <6898, 1, 4, 1>, <11740, 1, 4, 1>, <2205, 1, 4, 1>, <13803, 1, 4, 1> @} Total time: 2.730 seconds, Total memory usage: 39.93MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 15:19:31 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^7*3^2; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; hP:=Height(P);hP; HC:=HeightConstant(J:Effort:=20);HC; Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,11); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 15:19:28 on modular [Seed = 3407706852] ------------------------------------- TwoSelmerGroupData( J: Jacobian of Hyperelliptic Curve defined by y^2 = x^5 + 1152 ... ) TwoSelmerWork( C: Hyperelliptic Curve defined by y^2 = x^5 + 1152 over Rationa..., BoundType: Default, Bound: 0, UseUnits: true, Points: {}, Fields: {} ) LocalImageOdd( f: x^5 + 1152, p: 2, m1: Mapping from: Univariate Polynomial Ring in x over Rational ..., m2: Mapping from: Univariate Polynomial Ring in x over Rational ..., v: 30, Points: {} ) try( ) try( d: 2 ) SquarefreeRoots( F: $.1^4 + O(2^50)*$.1^3 + (522236902506487*2^3 + O(2^53))*$.1^... ) In file "/usr/local/magma/package/Geometry/Arith/sqfroots.m", line 84, column 15: >> return Roots(F:IsSquarefree); ^ Runtime error in 'Roots': Insufficient precision to find roots >> r:=TwoSelmerGroupData(J);r; ^ User error: Identifier 'r' has not been declared or assigned Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 28/9*x - 8/9, 44/27*x + 928/27, 2) ] 4.6613668053397002241366926279695766743193744943846637819259 6.862593078943470323468275363544094477084 3065.404397974187850397290252109710706677000 {@ <9277, 1, 4, 1>, <6898, 1, 4, 1>, <11740, 1, 4, 1>, <2205, 1, 4, 1>, <13803, 1, 4, 1> @} Total time: 2.919 seconds, Total memory usage: 40.35MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 15:19:26 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^7*3^2; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; hP:=Height(P);hP; HC:=HeightConstant(J:Effort:=20);HC; Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,11); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 15:19:24 on modular [Seed = 3490079203] ------------------------------------- TwoSelmerGroupData( J: Jacobian of Hyperelliptic Curve defined by y^2 = x^5 + 1152 ... ) TwoSelmerWork( C: Hyperelliptic Curve defined by y^2 = x^5 + 1152 over Rationa..., BoundType: Default, Bound: 0, UseUnits: true, Points: {}, Fields: {} ) LocalImageOdd( f: x^5 + 1152, p: 2, m1: Mapping from: Univariate Polynomial Ring in x over Rational ..., m2: Mapping from: Univariate Polynomial Ring in x over Rational ..., v: 30, Points: {} ) try( ) try( d: 2 ) SquarefreeRoots( F: (1 + O(2^44))*$.1^3 + O(2^50)*$.1^2 + (4827400634359*2^3 + O... ) In file "/usr/local/magma/package/Geometry/Arith/sqfroots.m", line 84, column 15: >> return Roots(F:IsSquarefree); ^ Runtime error in 'Roots': Insufficient precision to find roots >> r:=TwoSelmerGroupData(J);r; ^ User error: Identifier 'r' has not been declared or assigned Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 28/9*x - 8/9, 44/27*x + 928/27, 2) ] 4.6613668053397002241366926279695766743193744943846637819259 6.862593078943470323468275363544094477084 3065.404397974187850397290252109710706677000 {@ <9277, 1, 4, 1>, <6898, 1, 4, 1>, <11740, 1, 4, 1>, <2205, 1, 4, 1>, <13803, 1, 4, 1> @} Total time: 2.279 seconds, Total memory usage: 39.34MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 15:19:22 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^7*3^2; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; hP:=Height(P);hP; HC:=HeightConstant(J:Effort:=20);HC; Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,11); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 15:19:19 on modular [Seed = 3573769453] ------------------------------------- TwoSelmerGroupData( J: Jacobian of Hyperelliptic Curve defined by y^2 = x^5 + 1152 ... ) TwoSelmerWork( C: Hyperelliptic Curve defined by y^2 = x^5 + 1152 over Rationa..., BoundType: Default, Bound: 0, UseUnits: true, Points: {}, Fields: {} ) LocalImageOdd( f: x^5 + 1152, p: 2, m1: Mapping from: Univariate Polynomial Ring in x over Rational ..., m2: Mapping from: Univariate Polynomial Ring in x over Rational ..., v: 30, Points: {} ) try( ) try( d: 2 ) SquarefreeRoots( F: $.1^4 + O(2^50)*$.1^3 - (414826611343369*2^3 + O(2^53))*$.1^... ) In file "/usr/local/magma/package/Geometry/Arith/sqfroots.m", line 84, column 15: >> return Roots(F:IsSquarefree); ^ Runtime error in 'Roots': Insufficient precision to find roots >> r:=TwoSelmerGroupData(J);r; ^ User error: Identifier 'r' has not been declared or assigned Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 28/9*x - 8/9, 44/27*x + 928/27, 2) ] 4.6613668053397002241366926279695766743193744943846637819259 6.862593078943470323468275363544094477084 3065.404397974187850397290252109710706677000 {@ <9277, 1, 4, 1>, <6898, 1, 4, 1>, <11740, 1, 4, 1>, <2205, 1, 4, 1>, <13803, 1, 4, 1> @} Total time: 2.660 seconds, Total memory usage: 39.85MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 15:19:17 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^7*3^2; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; hP:=Height(P);hP; HC:=HeightConstant(J:Effort:=20);HC; Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,11); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 15:19:15 on modular [Seed = 3658247142] ------------------------------------- TwoSelmerGroupData( J: Jacobian of Hyperelliptic Curve defined by y^2 = x^5 + 1152 ... ) TwoSelmerWork( C: Hyperelliptic Curve defined by y^2 = x^5 + 1152 over Rationa..., BoundType: Default, Bound: 0, UseUnits: true, Points: {}, Fields: {} ) LocalImageOdd( f: x^5 + 1152, p: 2, m1: Mapping from: Univariate Polynomial Ring in x over Rational ..., m2: Mapping from: Univariate Polynomial Ring in x over Rational ..., v: 30, Points: {} ) try( ) try( d: 2 ) SquarefreeRoots( F: $.1^4 + O(2^50)*$.1^3 - (139579446788105*2^3 + O(2^53))*$.1^... ) In file "/usr/local/magma/package/Geometry/Arith/sqfroots.m", line 84, column 15: >> return Roots(F:IsSquarefree); ^ Runtime error in 'Roots': Insufficient precision to find roots >> r:=TwoSelmerGroupData(J);r; ^ User error: Identifier 'r' has not been declared or assigned Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 28/9*x - 8/9, 44/27*x + 928/27, 2) ] 4.6613668053397002241366926279695766743193744943846637819259 6.862593078943470323468275363544094477084 3065.404397974187850397290252109710706677000 {@ <9277, 1, 4, 1>, <6898, 1, 4, 1>, <11740, 1, 4, 1>, <2205, 1, 4, 1>, <13803, 1, 4, 1> @} Total time: 2.279 seconds, Total memory usage: 39.36MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 15:19:07 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^7*3^2; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; hP:=Height(P);hP; HC:=HeightConstant(J:Effort:=20);HC; Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,11); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 15:19:04 on modular [Seed = 3741937304] ------------------------------------- TwoSelmerGroupData( J: Jacobian of Hyperelliptic Curve defined by y^2 = x^5 + 1152 ... ) TwoSelmerWork( C: Hyperelliptic Curve defined by y^2 = x^5 + 1152 over Rationa..., BoundType: Default, Bound: 0, UseUnits: true, Points: {}, Fields: {} ) LocalImageOdd( f: x^5 + 1152, p: 2, m1: Mapping from: Univariate Polynomial Ring in x over Rational ..., m2: Mapping from: Univariate Polynomial Ring in x over Rational ..., v: 30, Points: {} ) try( ) try( d: 2 ) SquarefreeRoots( F: $.1^4 + O(2^50)*$.1^3 + (121347758161911*2^3 + O(2^53))*$.1^... ) In file "/usr/local/magma/package/Geometry/Arith/sqfroots.m", line 84, column 15: >> return Roots(F:IsSquarefree); ^ Runtime error in 'Roots': Insufficient precision to find roots >> r:=TwoSelmerGroupData(J);r; ^ User error: Identifier 'r' has not been declared or assigned Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 28/9*x - 8/9, 44/27*x + 928/27, 2) ] 4.6613668053397002241366926279695766743193744943846637819259 6.862593078943470323468275363544094477084 3065.404397974187850397290252109710706677000 {@ <9277, 1, 4, 1>, <6898, 1, 4, 1>, <11740, 1, 4, 1>, <2205, 1, 4, 1>, <13803, 1, 4, 1> @} Total time: 2.660 seconds, Total memory usage: 39.84MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 15:19:01 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^7*3^2; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; hP:=Height(P);hP; HC:=HeightConstant(J:Effort:=20);HC; Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,11); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 15:18:59 on modular [Seed = 3828503976] ------------------------------------- TwoSelmerGroupData( J: Jacobian of Hyperelliptic Curve defined by y^2 = x^5 + 1152 ... ) TwoSelmerWork( C: Hyperelliptic Curve defined by y^2 = x^5 + 1152 over Rationa..., BoundType: Default, Bound: 0, UseUnits: true, Points: {}, Fields: {} ) LocalImageOdd( f: x^5 + 1152, p: 2, m1: Mapping from: Univariate Polynomial Ring in x over Rational ..., m2: Mapping from: Univariate Polynomial Ring in x over Rational ..., v: 30, Points: {} ) try( ) try( d: 2 ) SquarefreeRoots( F: $.1^4 + O(2^50)*$.1^3 + (96911345975287*2^3 + O(2^53))*$.1^2... ) In file "/usr/local/magma/package/Geometry/Arith/sqfroots.m", line 84, column 15: >> return Roots(F:IsSquarefree); ^ Runtime error in 'Roots': Insufficient precision to find roots >> r:=TwoSelmerGroupData(J);r; ^ User error: Identifier 'r' has not been declared or assigned Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 28/9*x - 8/9, 44/27*x + 928/27, 2) ] 4.6613668053397002241366926279695766743193744943846637819259 6.862593078943470323468275363544094477084 3065.404397974187850397290252109710706677000 {@ <9277, 1, 4, 1>, <6898, 1, 4, 1>, <11740, 1, 4, 1>, <2205, 1, 4, 1>, <13803, 1, 4, 1> @} Total time: 2.160 seconds, Total memory usage: 39.18MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 15:18:42 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^7*3^2; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; hP:=Height(P);hP; HC:=HeightConstant(J:Effort:=20);HC; Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,11); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 15:18:40 on modular [Seed = 3913242806] ------------------------------------- TwoSelmerGroupData( J: Jacobian of Hyperelliptic Curve defined by y^2 = x^5 + 1152 ... ) TwoSelmerWork( C: Hyperelliptic Curve defined by y^2 = x^5 + 1152 over Rationa..., BoundType: Default, Bound: 0, UseUnits: true, Points: {}, Fields: {} ) LocalImageOdd( f: x^5 + 1152, p: 2, m1: Mapping from: Univariate Polynomial Ring in x over Rational ..., m2: Mapping from: Univariate Polynomial Ring in x over Rational ..., v: 30, Points: {} ) try( ) try( d: 2 ) SquarefreeRoots( F: $.1^4 + O(2^50)*$.1^3 + (306779403452407*2^3 + O(2^53))*$.1^... ) In file "/usr/local/magma/package/Geometry/Arith/sqfroots.m", line 84, column 15: >> return Roots(F:IsSquarefree); ^ Runtime error in 'Roots': Insufficient precision to find roots >> r:=TwoSelmerGroupData(J);r; ^ User error: Identifier 'r' has not been declared or assigned Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 28/9*x - 8/9, 44/27*x + 928/27, 2) ] 4.6613668053397002241366926279695766743193744943846637819259 6.862593078943470323468275363544094477084 3065.404397974187850397290252109710706677000 {@ <9277, 1, 4, 1>, <6898, 1, 4, 1>, <11740, 1, 4, 1>, <2205, 1, 4, 1>, <13803, 1, 4, 1> @} Total time: 2.200 seconds, Total memory usage: 39.23MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 15:18:38 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^7*3^2; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; hP:=Height(P);hP; HC:=HeightConstant(J:Effort:=20);HC; Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,11); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 15:18:36 on modular [Seed = 3996671885] ------------------------------------- TwoSelmerGroupData( J: Jacobian of Hyperelliptic Curve defined by y^2 = x^5 + 1152 ... ) TwoSelmerWork( C: Hyperelliptic Curve defined by y^2 = x^5 + 1152 over Rationa..., BoundType: Default, Bound: 0, UseUnits: true, Points: {}, Fields: {} ) LocalImageOdd( f: x^5 + 1152, p: 2, m1: Mapping from: Univariate Polynomial Ring in x over Rational ..., m2: Mapping from: Univariate Polynomial Ring in x over Rational ..., v: 30, Points: {} ) try( ) try( d: 2 ) SquarefreeRoots( F: $.1^4 + O(2^50)*$.1^3 - (505458911543305*2^3 + O(2^53))*$.1^... ) In file "/usr/local/magma/package/Geometry/Arith/sqfroots.m", line 84, column 15: >> return Roots(F:IsSquarefree); ^ Runtime error in 'Roots': Insufficient precision to find roots >> r:=TwoSelmerGroupData(J);r; ^ User error: Identifier 'r' has not been declared or assigned Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 28/9*x - 8/9, 44/27*x + 928/27, 2) ] 4.6613668053397002241366926279695766743193744943846637819259 6.862593078943470323468275363544094477084 3065.404397974187850397290252109710706677000 {@ <9277, 1, 4, 1>, <6898, 1, 4, 1>, <11740, 1, 4, 1>, <2205, 1, 4, 1>, <13803, 1, 4, 1> @} Total time: 2.589 seconds, Total memory usage: 39.78MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 15:18:34 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^7*3^2; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; hP:=Height(P);hP; HC:=HeightConstant(J:Effort:=20);HC; Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,11); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 15:18:32 on modular [Seed = 4079305364] ------------------------------------- TwoSelmerGroupData( J: Jacobian of Hyperelliptic Curve defined by y^2 = x^5 + 1152 ... ) TwoSelmerWork( C: Hyperelliptic Curve defined by y^2 = x^5 + 1152 over Rationa..., BoundType: Default, Bound: 0, UseUnits: true, Points: {}, Fields: {} ) LocalImageOdd( f: x^5 + 1152, p: 2, m1: Mapping from: Univariate Polynomial Ring in x over Rational ..., m2: Mapping from: Univariate Polynomial Ring in x over Rational ..., v: 30, Points: {} ) try( ) try( d: 2 ) SquarefreeRoots( F: $.1^4 + O(2^50)*$.1^3 - (305167809028105*2^3 + O(2^53))*$.1^... ) In file "/usr/local/magma/package/Geometry/Arith/sqfroots.m", line 84, column 15: >> return Roots(F:IsSquarefree); ^ Runtime error in 'Roots': Insufficient precision to find roots >> r:=TwoSelmerGroupData(J);r; ^ User error: Identifier 'r' has not been declared or assigned Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 28/9*x - 8/9, 44/27*x + 928/27, 2) ] 4.6613668053397002241366926279695766743193744943846637819259 6.862593078943470323468275363544094477084 3065.404397974187850397290252109710706677000 {@ <9277, 1, 4, 1>, <6898, 1, 4, 1>, <11740, 1, 4, 1>, <2205, 1, 4, 1>, <13803, 1, 4, 1> @} Total time: 2.310 seconds, Total memory usage: 39.41MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 15:18:31 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^7*3^2; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; hP:=Height(P);hP; HC:=HeightConstant(J:Effort:=20);HC; Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,11); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 15:18:28 on modular [Seed = 4163783059] ------------------------------------- TwoSelmerGroupData( J: Jacobian of Hyperelliptic Curve defined by y^2 = x^5 + 1152 ... ) TwoSelmerWork( C: Hyperelliptic Curve defined by y^2 = x^5 + 1152 over Rationa..., BoundType: Default, Bound: 0, UseUnits: true, Points: {}, Fields: {} ) LocalImageOdd( f: x^5 + 1152, p: 2, m1: Mapping from: Univariate Polynomial Ring in x over Rational ..., m2: Mapping from: Univariate Polynomial Ring in x over Rational ..., v: 30, Points: {} ) try( ) try( d: 2 ) SquarefreeRoots( F: $.1^4 + O(2^50)*$.1^3 - (18461674176521*2^3 + O(2^53))*$.1^2... ) In file "/usr/local/magma/package/Geometry/Arith/sqfroots.m", line 84, column 15: >> return Roots(F:IsSquarefree); ^ Runtime error in 'Roots': Insufficient precision to find roots >> r:=TwoSelmerGroupData(J);r; ^ User error: Identifier 'r' has not been declared or assigned Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 28/9*x - 8/9, 44/27*x + 928/27, 2) ] 4.6613668053397002241366926279695766743193744943846637819259 6.862593078943470323468275363544094477084 3065.404397974187850397290252109710706677000 {@ <9277, 1, 4, 1>, <6898, 1, 4, 1>, <11740, 1, 4, 1>, <2205, 1, 4, 1>, <13803, 1, 4, 1> @} Total time: 3.080 seconds, Total memory usage: 40.54MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 15:18:26 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^7*3^2; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; hP:=Height(P);hP; HC:=HeightConstant(J:Effort:=20);HC; Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,11); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 15:18:24 on modular [Seed = 4247473298] ------------------------------------- TwoSelmerGroupData( J: Jacobian of Hyperelliptic Curve defined by y^2 = x^5 + 1152 ... ) TwoSelmerWork( C: Hyperelliptic Curve defined by y^2 = x^5 + 1152 over Rationa..., BoundType: Default, Bound: 0, UseUnits: true, Points: {}, Fields: {} ) LocalImageOdd( f: x^5 + 1152, p: 2, m1: Mapping from: Univariate Polynomial Ring in x over Rational ..., m2: Mapping from: Univariate Polynomial Ring in x over Rational ..., v: 30, Points: {} ) try( ) try( d: 2 ) SquarefreePart( F: $.1^4 + O(2^50)*$.1^3 + (382136867618807*2^3 + O(2^53))*$.1^... ) In file "/usr/local/magma/package/Geometry/Arith/loclib.m", line 120, column 5: >> assert R eq 0; ^ Runtime error in assert: Assertion failed >> r:=TwoSelmerGroupData(J);r; ^ User error: Identifier 'r' has not been declared or assigned Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 28/9*x - 8/9, 44/27*x + 928/27, 2) ] 4.6613668053397002241366926279695766743193744943846637819259 6.862593078943470323468275363544094477084 3065.404397974187850397290252109710706677000 {@ <9277, 1, 4, 1>, <6898, 1, 4, 1>, <11740, 1, 4, 1>, <2205, 1, 4, 1>, <13803, 1, 4, 1> @} Total time: 2.149 seconds, Total memory usage: 39.16MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 15:18:23 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^7*3^2; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; hP:=Height(P);hP; HC:=HeightConstant(J:Effort:=20);HC; Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,11); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 15:18:20 on modular [Seed = 47494001] ------------------------------------- TwoSelmerGroupData( J: Jacobian of Hyperelliptic Curve defined by y^2 = x^5 + 1152 ... ) TwoSelmerWork( C: Hyperelliptic Curve defined by y^2 = x^5 + 1152 over Rationa..., BoundType: Default, Bound: 0, UseUnits: true, Points: {}, Fields: {} ) LocalImageOdd( f: x^5 + 1152, p: 2, m1: Mapping from: Univariate Polynomial Ring in x over Rational ..., m2: Mapping from: Univariate Polynomial Ring in x over Rational ..., v: 30, Points: {} ) try( ) try( d: 2 ) SquarefreeRoots( F: $.1^4 + O(2^50)*$.1^3 + (220979510771703*2^3 + O(2^53))*$.1^... ) In file "/usr/local/magma/package/Geometry/Arith/sqfroots.m", line 84, column 15: >> return Roots(F:IsSquarefree); ^ Runtime error in 'Roots': Insufficient precision to find roots >> r:=TwoSelmerGroupData(J);r; ^ User error: Identifier 'r' has not been declared or assigned Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 28/9*x - 8/9, 44/27*x + 928/27, 2) ] 4.6613668053397002241366926279695766743193744943846637819259 6.862593078943470323468275363544094477084 3065.404397974187850397290252109710706677000 {@ <9277, 1, 4, 1>, <6898, 1, 4, 1>, <11740, 1, 4, 1>, <2205, 1, 4, 1>, <13803, 1, 4, 1> @} Total time: 2.930 seconds, Total memory usage: 40.35MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 15:18:18 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^7*3^2; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; hP:=Height(P);hP; HC:=HeightConstant(J:Effort:=20);HC; Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,11); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 15:18:15 on modular [Seed = 131184255] ------------------------------------- TwoSelmerGroupData( J: Jacobian of Hyperelliptic Curve defined by y^2 = x^5 + 1152 ... ) TwoSelmerWork( C: Hyperelliptic Curve defined by y^2 = x^5 + 1152 over Rationa..., BoundType: Default, Bound: 0, UseUnits: true, Points: {}, Fields: {} ) LocalImageOdd( f: x^5 + 1152, p: 2, m1: Mapping from: Univariate Polynomial Ring in x over Rational ..., m2: Mapping from: Univariate Polynomial Ring in x over Rational ..., v: 30, Points: {} ) try( ) try( d: 2 ) SquarefreeRoots( F: $.1^4 + O(2^50)*$.1^3 - (207781347065865*2^3 + O(2^53))*$.1^... ) In file "/usr/local/magma/package/Geometry/Arith/sqfroots.m", line 84, column 15: >> return Roots(F:IsSquarefree); ^ Runtime error in 'Roots': Insufficient precision to find roots >> r:=TwoSelmerGroupData(J);r; ^ User error: Identifier 'r' has not been declared or assigned Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 28/9*x - 8/9, 44/27*x + 928/27, 2) ] 4.6613668053397002241366926279695766743193744943846637819259 6.862593078943470323468275363544094477084 3065.404397974187850397290252109710706677000 {@ <9277, 1, 4, 1>, <6898, 1, 4, 1>, <11740, 1, 4, 1>, <2205, 1, 4, 1>, <13803, 1, 4, 1> @} Total time: 3.069 seconds, Total memory usage: 40.51MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 15:18:05 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^7*3^2; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; hP:=Height(P);hP; HC:=HeightConstant(J:Effort:=20);HC; Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,11); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 15:18:01 on modular [Seed = 215662218] ------------------------------------- TwoSelmerGroupData( J: Jacobian of Hyperelliptic Curve defined by y^2 = x^5 + 1152 ... ) TwoSelmerWork( C: Hyperelliptic Curve defined by y^2 = x^5 + 1152 over Rationa..., BoundType: Default, Bound: 0, UseUnits: true, Points: {}, Fields: {} ) LocalImageOdd( f: x^5 + 1152, p: 2, m1: Mapping from: Univariate Polynomial Ring in x over Rational ..., m2: Mapping from: Univariate Polynomial Ring in x over Rational ..., v: 30, Points: {} ) try( ) try( d: 2 ) SquarefreeRoots( F: $.1^4 + O(2^50)*$.1^3 + (40731028881399*2^3 + O(2^53))*$.1^2... ) In file "/usr/local/magma/package/Geometry/Arith/sqfroots.m", line 84, column 15: >> return Roots(F:IsSquarefree); ^ Runtime error in 'Roots': Insufficient precision to find roots >> r:=TwoSelmerGroupData(J);r; ^ User error: Identifier 'r' has not been declared or assigned Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 28/9*x - 8/9, 44/27*x + 928/27, 2) ] 4.6613668053397002241366926279695766743193744943846637819259 6.862593078943470323468275363544094477084 3065.404397974187850397290252109710706677000 {@ <9277, 1, 4, 1>, <6898, 1, 4, 1>, <11740, 1, 4, 1>, <2205, 1, 4, 1>, <13803, 1, 4, 1> @} Total time: 3.109 seconds, Total memory usage: 40.63MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 15:18:00 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^7*3^2; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; hP:=Height(P);hP; HC:=HeightConstant(J:Effort:=20);HC; Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,11); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 15:17:58 on modular [Seed = 298296216] ------------------------------------- TwoSelmerGroupData( J: Jacobian of Hyperelliptic Curve defined by y^2 = x^5 + 1152 ... ) TwoSelmerWork( C: Hyperelliptic Curve defined by y^2 = x^5 + 1152 over Rationa..., BoundType: Default, Bound: 0, UseUnits: true, Points: {}, Fields: {} ) LocalImageOdd( f: x^5 + 1152, p: 2, m1: Mapping from: Univariate Polynomial Ring in x over Rational ..., m2: Mapping from: Univariate Polynomial Ring in x over Rational ..., v: 30, Points: {} ) try( ) try( d: 2 ) SquarefreeRoots( F: $.1^4 + O(2^50)*$.1^3 + (396582569639927*2^3 + O(2^53))*$.1^... ) In file "/usr/local/magma/package/Geometry/Arith/sqfroots.m", line 84, column 15: >> return Roots(F:IsSquarefree); ^ Runtime error in 'Roots': Insufficient precision to find roots >> r:=TwoSelmerGroupData(J);r; ^ User error: Identifier 'r' has not been declared or assigned Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 28/9*x - 8/9, 44/27*x + 928/27, 2) ] 4.6613668053397002241366926279695766743193744943846637819259 6.862593078943470323468275363544094477084 3065.404397974187850397290252109710706677000 {@ <9277, 1, 4, 1>, <6898, 1, 4, 1>, <11740, 1, 4, 1>, <2205, 1, 4, 1>, <13803, 1, 4, 1> @} Total time: 2.180 seconds, Total memory usage: 39.21MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 15:17:56 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^7*3^2; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; hP:=Height(P);hP; HC:=HeightConstant(J:Effort:=20);HC; Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,11); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 15:17:54 on modular [Seed = 381723803] ------------------------------------- TwoSelmerGroupData( J: Jacobian of Hyperelliptic Curve defined by y^2 = x^5 + 1152 ... ) TwoSelmerWork( C: Hyperelliptic Curve defined by y^2 = x^5 + 1152 over Rationa..., BoundType: Default, Bound: 0, UseUnits: true, Points: {}, Fields: {} ) LocalImageOdd( f: x^5 + 1152, p: 2, m1: Mapping from: Univariate Polynomial Ring in x over Rational ..., m2: Mapping from: Univariate Polynomial Ring in x over Rational ..., v: 30, Points: {} ) try( ) try( d: 2 ) SquarefreeRoots( F: $.1^4 + O(2^50)*$.1^3 + (127222226747383*2^3 + O(2^53))*$.1^... ) In file "/usr/local/magma/package/Geometry/Arith/sqfroots.m", line 84, column 15: >> return Roots(F:IsSquarefree); ^ Runtime error in 'Roots': Insufficient precision to find roots >> r:=TwoSelmerGroupData(J);r; ^ User error: Identifier 'r' has not been declared or assigned Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 28/9*x - 8/9, 44/27*x + 928/27, 2) ] 4.6613668053397002241366926279695766743193744943846637819259 6.862593078943470323468275363544094477084 3065.404397974187850397290252109710706677000 {@ <9277, 1, 4, 1>, <6898, 1, 4, 1>, <11740, 1, 4, 1>, <2205, 1, 4, 1>, <13803, 1, 4, 1> @} Total time: 2.149 seconds, Total memory usage: 39.18MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 15:14:57 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^7*3^1; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; hP:=Height(P);hP; HC:=HeightConstant(J:Effort:=20);HC; Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,11); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 15:14:55 on modular [Seed = 533310815] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 12*x - 24, -44*x - 96, 2) ] 2.9055439367899368398441645250175263592357000094565176957970 5.574645113293738508088336630026289936746 545.1337064398618038830322605633276151897425 {@ @} Total time: 2.040 seconds, Total memory usage: 39.70MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 15:14:53 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^7*3^1; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; hP:=Height(P);hP; HC:=HeightConstant(J:Effort:=20);HC; Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,11); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 15:14:51 on modular [Seed = 619877984] ------------------------------------- TwoSelmerGroupData( J: Jacobian of Hyperelliptic Curve defined by y^2 = x^5 + 384 o... ) TwoSelmerWork( C: Hyperelliptic Curve defined by y^2 = x^5 + 384 over Rational..., BoundType: Default, Bound: 0, UseUnits: true, Points: {}, Fields: {} ) LocalImageOdd( f: x^5 + 384, p: 2, m1: Mapping from: Univariate Polynomial Ring in x over Rational ..., m2: Mapping from: Univariate Polynomial Ring in x over Rational ..., v: 30, Points: {} ) try( ) try( d: 2 ) SquarefreeRoots( F: $.1^4 + O(2^50)*$.1^3 + (201481855696893*2^3 + O(2^53))*$.1^... ) In file "/usr/local/magma/package/Geometry/Arith/sqfroots.m", line 84, column 15: >> return Roots(F:IsSquarefree); ^ Runtime error in 'Roots': Insufficient precision to find roots >> r:=TwoSelmerGroupData(J);r; ^ User error: Identifier 'r' has not been declared or assigned Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 12*x - 24, -44*x - 96, 2) ] 2.9055439367899368398441645250175263592357000094565176957970 5.574645113293738508088336630026289936746 545.1337064398618038830322605633276151897425 {@ @} Total time: 2.140 seconds, Total memory usage: 40.10MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 15:14:50 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^7*3^1; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; hP:=Height(P);hP; HC:=HeightConstant(J:Effort:=20);HC; Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,11); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 15:14:47 on modular [Seed = 704617313] ------------------------------------- TwoSelmerGroupData( J: Jacobian of Hyperelliptic Curve defined by y^2 = x^5 + 384 o... ) TwoSelmerWork( C: Hyperelliptic Curve defined by y^2 = x^5 + 384 over Rational..., BoundType: Default, Bound: 0, UseUnits: true, Points: {}, Fields: {} ) LocalImageOdd( f: x^5 + 384, p: 2, m1: Mapping from: Univariate Polynomial Ring in x over Rational ..., m2: Mapping from: Univariate Polynomial Ring in x over Rational ..., v: 30, Points: {} ) try( ) try( d: 2 ) SquarefreeRoots( F: $.1^4 + O(2^50)*$.1^3 - (489467683536899*2^3 + O(2^53))*$.1^... ) In file "/usr/local/magma/package/Geometry/Arith/sqfroots.m", line 84, column 15: >> return Roots(F:IsSquarefree); ^ Runtime error in 'Roots': Insufficient precision to find roots >> r:=TwoSelmerGroupData(J);r; ^ User error: Identifier 'r' has not been declared or assigned Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 12*x - 24, -44*x - 96, 2) ] 2.9055439367899368398441645250175263592357000094565176957970 5.574645113293738508088336630026289936746 545.1337064398618038830322605633276151897425 {@ @} Total time: 2.700 seconds, Total memory usage: 40.99MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 15:14:46 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^7*3^1; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; hP:=Height(P);hP; HC:=HeightConstant(J:Effort:=20);HC; Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,11); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 15:14:43 on modular [Seed = 788044901] ------------------------------------- TwoSelmerGroupData( J: Jacobian of Hyperelliptic Curve defined by y^2 = x^5 + 384 o... ) TwoSelmerWork( C: Hyperelliptic Curve defined by y^2 = x^5 + 384 over Rational..., BoundType: Default, Bound: 0, UseUnits: true, Points: {}, Fields: {} ) LocalImageOdd( f: x^5 + 384, p: 2, m1: Mapping from: Univariate Polynomial Ring in x over Rational ..., m2: Mapping from: Univariate Polynomial Ring in x over Rational ..., v: 30, Points: {} ) try( ) try( d: 2 ) SquarefreeRoots( F: $.1^4 + O(2^50)*$.1^3 + (102843116552189*2^3 + O(2^53))*$.1^... ) In file "/usr/local/magma/package/Geometry/Arith/sqfroots.m", line 84, column 15: >> return Roots(F:IsSquarefree); ^ Runtime error in 'Roots': Insufficient precision to find roots >> r:=TwoSelmerGroupData(J);r; ^ User error: Identifier 'r' has not been declared or assigned Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 12*x - 24, -44*x - 96, 2) ] 2.9055439367899368398441645250175263592357000094565176957970 5.574645113293738508088336630026289936746 545.1337064398618038830322605633276151897425 {@ @} Total time: 1.889 seconds, Total memory usage: 39.77MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 15:14:42 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^7*3^1; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; hP:=Height(P);hP; HC:=HeightConstant(J:Effort:=20);HC; Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,11); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 15:14:40 on modular [Seed = 870678908] ------------------------------------- TwoSelmerGroupData( J: Jacobian of Hyperelliptic Curve defined by y^2 = x^5 + 384 o... ) TwoSelmerWork( C: Hyperelliptic Curve defined by y^2 = x^5 + 384 over Rational..., BoundType: Default, Bound: 0, UseUnits: true, Points: {}, Fields: {} ) LocalImageOdd( f: x^5 + 384, p: 2, m1: Mapping from: Univariate Polynomial Ring in x over Rational ..., m2: Mapping from: Univariate Polynomial Ring in x over Rational ..., v: 30, Points: {} ) try( ) try( d: 2 ) SquarefreeRoots( F: (1 + O(2^44))*$.1^3 + O(2^50)*$.1^2 + (1735166787581*2^3 + O... ) In file "/usr/local/magma/package/Geometry/Arith/sqfroots.m", line 84, column 15: >> return Roots(F:IsSquarefree); ^ Runtime error in 'Roots': Insufficient precision to find roots >> r:=TwoSelmerGroupData(J);r; ^ User error: Identifier 'r' has not been declared or assigned Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 12*x - 24, -44*x - 96, 2) ] 2.9055439367899368398441645250175263592357000094565176957970 5.574645113293738508088336630026289936746 545.1337064398618038830322605633276151897425 {@ @} Total time: 2.180 seconds, Total memory usage: 40.13MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 15:14:39 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^7*3^1; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; hP:=Height(P);hP; HC:=HeightConstant(J:Effort:=20);HC; Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,11); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 15:14:37 on modular [Seed = 955157098] ------------------------------------- TwoSelmerGroupData( J: Jacobian of Hyperelliptic Curve defined by y^2 = x^5 + 384 o... ) TwoSelmerWork( C: Hyperelliptic Curve defined by y^2 = x^5 + 384 over Rational..., BoundType: Default, Bound: 0, UseUnits: true, Points: {}, Fields: {} ) LocalImageOdd( f: x^5 + 384, p: 2, m1: Mapping from: Univariate Polynomial Ring in x over Rational ..., m2: Mapping from: Univariate Polynomial Ring in x over Rational ..., v: 30, Points: {} ) try( ) try( d: 2 ) SquarefreeRoots( F: $.1^4 + O(2^50)*$.1^3 + (478662650494973*2^3 + O(2^53))*$.1^... ) In file "/usr/local/magma/package/Geometry/Arith/sqfroots.m", line 84, column 15: >> return Roots(F:IsSquarefree); ^ Runtime error in 'Roots': Insufficient precision to find roots >> r:=TwoSelmerGroupData(J);r; ^ User error: Identifier 'r' has not been declared or assigned Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 12*x - 24, -44*x - 96, 2) ] 2.9055439367899368398441645250175263592357000094565176957970 5.574645113293738508088336630026289936746 545.1337064398618038830322605633276151897425 {@ @} Total time: 1.870 seconds, Total memory usage: 39.75MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 15:14:36 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^7*3^1; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; hP:=Height(P);hP; HC:=HeightConstant(J:Effort:=20);HC; Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,11); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 15:14:34 on modular [Seed = 1038847848] ------------------------------------- TwoSelmerGroupData( J: Jacobian of Hyperelliptic Curve defined by y^2 = x^5 + 384 o... ) TwoSelmerWork( C: Hyperelliptic Curve defined by y^2 = x^5 + 384 over Rational..., BoundType: Default, Bound: 0, UseUnits: true, Points: {}, Fields: {} ) LocalImageOdd( f: x^5 + 384, p: 2, m1: Mapping from: Univariate Polynomial Ring in x over Rational ..., m2: Mapping from: Univariate Polynomial Ring in x over Rational ..., v: 30, Points: {} ) try( ) try( d: 2 ) SquarefreeRoots( F: $.1^4 + O(2^50)*$.1^3 - (95639888986115*2^3 + O(2^53))*$.1^2... ) In file "/usr/local/magma/package/Geometry/Arith/sqfroots.m", line 84, column 15: >> return Roots(F:IsSquarefree); ^ Runtime error in 'Roots': Insufficient precision to find roots >> r:=TwoSelmerGroupData(J);r; ^ User error: Identifier 'r' has not been declared or assigned Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 12*x - 24, -44*x - 96, 2) ] 2.9055439367899368398441645250175263592357000094565176957970 5.574645113293738508088336630026289936746 545.1337064398618038830322605633276151897425 {@ @} Total time: 2.020 seconds, Total memory usage: 39.92MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 15:14:32 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^7*3^1; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; hP:=Height(P);hP; HC:=HeightConstant(J:Effort:=20);HC; Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,11); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 15:14:30 on modular [Seed = 1117057051] ------------------------------------- TwoSelmerGroupData( J: Jacobian of Hyperelliptic Curve defined by y^2 = x^5 + 384 o... ) TwoSelmerWork( C: Hyperelliptic Curve defined by y^2 = x^5 + 384 over Rational..., BoundType: Default, Bound: 0, UseUnits: true, Points: {}, Fields: {} ) LocalImageOdd( f: x^5 + 384, p: 2, m1: Mapping from: Univariate Polynomial Ring in x over Rational ..., m2: Mapping from: Univariate Polynomial Ring in x over Rational ..., v: 30, Points: {} ) try( ) try( d: 2 ) SquarefreeRoots( F: $.1^4 + O(2^50)*$.1^3 - (403373980778499*2^3 + O(2^53))*$.1^... ) In file "/usr/local/magma/package/Geometry/Arith/sqfroots.m", line 84, column 15: >> return Roots(F:IsSquarefree); ^ Runtime error in 'Roots': Insufficient precision to find roots >> r:=TwoSelmerGroupData(J);r; ^ User error: Identifier 'r' has not been declared or assigned Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 12*x - 24, -44*x - 96, 2) ] 2.9055439367899368398441645250175263592357000094565176957970 5.574645113293738508088336630026289936746 545.1337064398618038830322605633276151897425 {@ @} Total time: 1.899 seconds, Total memory usage: 39.79MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 15:13:30 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^7*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); //r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; hP:=Height(P);hP; HC:=HeightConstant(J:Effort:=20);HC; Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,11); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 15:13:28 on modular [Seed = 1200747728] ------------------------------------- Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 4*x + 8, -4*x + 16, 2) ] 1.5030074136745814353823801003995556043806314222319625754371 5.393385283934348998903993780164142041461 320.261271901877284467762401417456847046362 {@ @} Total time: 1.449 seconds, Total memory usage: 38.33MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 15:13:12 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^7*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; hP:=Height(P);hP; HC:=HeightConstant(J:Effort:=20);HC; Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,7); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 15:13:09 on modular [Seed = 1285226470] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 4*x + 8, -4*x + 16, 2) ] 1.5030074136745814353823801003995556043806314222319625754371 5.393385283934348998903993780164142041461 320.261271901877284467762401417456847046362 {@ <2034, 1, 4, 1>, <0, 1, 4, 1>, <132, 1, 4, 1> @} Total time: 2.960 seconds, Total memory usage: 40.59MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 15:13:07 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^7*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; hP:=Height(P);hP; HC:=HeightConstant(J:Effort:=20);HC; Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,7); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 15:13:04 on modular [Seed = 1367859966] ------------------------------------- TwoSelmerGroupData( J: Jacobian of Hyperelliptic Curve defined by y^2 = x^5 + 128 o... ) TwoSelmerWork( C: Hyperelliptic Curve defined by y^2 = x^5 + 128 over Rational..., BoundType: Default, Bound: 0, UseUnits: true, Points: {}, Fields: {} ) LocalImageOdd( f: x^5 + 128, p: 2, m1: Mapping from: Univariate Polynomial Ring in x over Rational ..., m2: Mapping from: Univariate Polynomial Ring in x over Rational ..., v: 30, Points: {} ) try( ) try( d: 2 ) SquarefreeRoots( F: $.1^4 + O(2^50)*$.1^3 - (420621710327809*2^3 + O(2^53))*$.1^... ) In file "/usr/local/magma/package/Geometry/Arith/sqfroots.m", line 84, column 15: >> return Roots(F:IsSquarefree); ^ Runtime error in 'Roots': Insufficient precision to find roots >> r:=TwoSelmerGroupData(J);r; ^ User error: Identifier 'r' has not been declared or assigned Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 4*x + 8, -4*x + 16, 2) ] 1.5030074136745814353823801003995556043806314222319625754371 5.393385283934348998903993780164142041461 320.261271901877284467762401417456847046362 {@ <2034, 1, 4, 1>, <0, 1, 4, 1>, <132, 1, 4, 1> @} Total time: 2.540 seconds, Total memory usage: 40.09MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 15:13:01 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^7*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; hP:=Height(P);hP; HC:=HeightConstant(J:Effort:=20);HC; Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,7); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 15:12:59 on modular [Seed = 1451288034] ------------------------------------- TwoSelmerGroupData( J: Jacobian of Hyperelliptic Curve defined by y^2 = x^5 + 128 o... ) TwoSelmerWork( C: Hyperelliptic Curve defined by y^2 = x^5 + 128 over Rational..., BoundType: Default, Bound: 0, UseUnits: true, Points: {}, Fields: {} ) LocalImageOdd( f: x^5 + 128, p: 2, m1: Mapping from: Univariate Polynomial Ring in x over Rational ..., m2: Mapping from: Univariate Polynomial Ring in x over Rational ..., v: 30, Points: {} ) try( ) try( d: 2 ) SquarefreeRoots( F: $.1^4 + O(2^50)*$.1^3 - (369089060511745*2^3 + O(2^53))*$.1^... ) In file "/usr/local/magma/package/Geometry/Arith/sqfroots.m", line 84, column 15: >> return Roots(F:IsSquarefree); ^ Runtime error in 'Roots': Insufficient precision to find roots >> r:=TwoSelmerGroupData(J);r; ^ User error: Identifier 'r' has not been declared or assigned Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 4*x + 8, -4*x + 16, 2) ] 1.5030074136745814353823801003995556043806314222319625754371 5.393385283934348998903993780164142041461 320.261271901877284467762401417456847046362 {@ <2034, 1, 4, 1>, <0, 1, 4, 1>, <132, 1, 4, 1> @} Total time: 2.390 seconds, Total memory usage: 39.92MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 15:12:56 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^7*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; hP:=Height(P);hP; HC:=HeightConstant(J:Effort:=20);HC; Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,7); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 15:12:53 on modular [Seed = 1536026849] ------------------------------------- TwoSelmerGroupData( J: Jacobian of Hyperelliptic Curve defined by y^2 = x^5 + 128 o... ) TwoSelmerWork( C: Hyperelliptic Curve defined by y^2 = x^5 + 128 over Rational..., BoundType: Default, Bound: 0, UseUnits: true, Points: {}, Fields: {} ) LocalImageOdd( f: x^5 + 128, p: 2, m1: Mapping from: Univariate Polynomial Ring in x over Rational ..., m2: Mapping from: Univariate Polynomial Ring in x over Rational ..., v: 30, Points: {} ) try( ) try( d: 2 ) SquarefreeRoots( F: $.1^4 + O(2^50)*$.1^3 - (534561969995777*2^3 + O(2^53))*$.1^... ) In file "/usr/local/magma/package/Geometry/Arith/sqfroots.m", line 84, column 15: >> return Roots(F:IsSquarefree); ^ Runtime error in 'Roots': Insufficient precision to find roots >> r:=TwoSelmerGroupData(J);r; ^ User error: Identifier 'r' has not been declared or assigned Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 4*x + 8, -4*x + 16, 2) ] 1.5030074136745814353823801003995556043806314222319625754371 5.393385283934348998903993780164142041461 320.261271901877284467762401417456847046362 {@ <2034, 1, 4, 1>, <0, 1, 4, 1>, <132, 1, 4, 1> @} Total time: 2.740 seconds, Total memory usage: 40.55MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 15:10:19 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^7*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; hP:=Height(P);hP; HC:=HeightConstant(J:Effort:=20);HC; Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,19); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 15:10:15 on modular [Seed = 3094219751] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 4*x + 8, -4*x + 16, 2) ] 1.5030074136745814353823801003995556043806314222319625754371 5.393385283934348998903993780164142041461 320.261271901877284467762401417456847046362 {@ @} Total time: 2.950 seconds, Total memory usage: 40.67MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 15:10:14 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^7*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; hP:=Height(P);hP; HC:=HeightConstant(J:Effort:=20);HC; Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,19); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 15:10:12 on modular [Seed = 3691671424] ------------------------------------- TwoSelmerGroupData( J: Jacobian of Hyperelliptic Curve defined by y^2 = x^5 + 128 o... ) TwoSelmerWork( C: Hyperelliptic Curve defined by y^2 = x^5 + 128 over Rational..., BoundType: Default, Bound: 0, UseUnits: true, Points: {}, Fields: {} ) LocalImageOdd( f: x^5 + 128, p: 2, m1: Mapping from: Univariate Polynomial Ring in x over Rational ..., m2: Mapping from: Univariate Polynomial Ring in x over Rational ..., v: 30, Points: {} ) try( ) try( d: 2 ) SquarefreeRoots( F: $.1^4 + O(2^50)*$.1^3 - (487120654958593*2^3 + O(2^53))*$.1^... ) In file "/usr/local/magma/package/Geometry/Arith/sqfroots.m", line 84, column 15: >> return Roots(F:IsSquarefree); ^ Runtime error in 'Roots': Insufficient precision to find roots >> r:=TwoSelmerGroupData(J);r; ^ User error: Identifier 'r' has not been declared or assigned Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 4*x + 8, -4*x + 16, 2) ] 1.5030074136745814353823801003995556043806314222319625754371 5.393385283934348998903993780164142041461 320.261271901877284467762401417456847046362 {@ @} Total time: 1.840 seconds, Total memory usage: 39.29MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 15:10:10 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^7*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; hP:=Height(P);hP; HC:=HeightConstant(J:Effort:=20);HC; Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,19); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 15:10:08 on modular [Seed = 3861927358] ------------------------------------- TwoSelmerGroupData( J: Jacobian of Hyperelliptic Curve defined by y^2 = x^5 + 128 o... ) TwoSelmerWork( C: Hyperelliptic Curve defined by y^2 = x^5 + 128 over Rational..., BoundType: Default, Bound: 0, UseUnits: true, Points: {}, Fields: {} ) LocalImageOdd( f: x^5 + 128, p: 2, m1: Mapping from: Univariate Polynomial Ring in x over Rational ..., m2: Mapping from: Univariate Polynomial Ring in x over Rational ..., v: 30, Points: {} ) try( ) try( d: 2 ) SquarefreeRoots( F: $.1^4 + O(2^50)*$.1^3 - (108548325900289*2^3 + O(2^53))*$.1^... ) In file "/usr/local/magma/package/Geometry/Arith/sqfroots.m", line 84, column 15: >> return Roots(F:IsSquarefree); ^ Runtime error in 'Roots': Insufficient precision to find roots >> r:=TwoSelmerGroupData(J);r; ^ User error: Identifier 'r' has not been declared or assigned Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 4*x + 8, -4*x + 16, 2) ] 1.5030074136745814353823801003995556043806314222319625754371 5.393385283934348998903993780164142041461 320.261271901877284467762401417456847046362 {@ @} Total time: 2.049 seconds, Total memory usage: 39.55MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 15:10:03 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^7*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; hP:=Height(P);hP; HC:=HeightConstant(J:Effort:=20);HC; Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,19); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 15:10:01 on modular [Seed = 4197206427] ------------------------------------- TwoSelmerGroupData( J: Jacobian of Hyperelliptic Curve defined by y^2 = x^5 + 128 o... ) TwoSelmerWork( C: Hyperelliptic Curve defined by y^2 = x^5 + 128 over Rational..., BoundType: Default, Bound: 0, UseUnits: true, Points: {}, Fields: {} ) LocalImageOdd( f: x^5 + 128, p: 2, m1: Mapping from: Univariate Polynomial Ring in x over Rational ..., m2: Mapping from: Univariate Polynomial Ring in x over Rational ..., v: 30, Points: {} ) try( ) try( d: 2 ) SquarefreeRoots( F: $.1^4 + O(2^50)*$.1^3 - (70932133183489*2^3 + O(2^53))*$.1^2... ) In file "/usr/local/magma/package/Geometry/Arith/sqfroots.m", line 84, column 15: >> return Roots(F:IsSquarefree); ^ Runtime error in 'Roots': Insufficient precision to find roots >> r:=TwoSelmerGroupData(J);r; ^ User error: Identifier 'r' has not been declared or assigned Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 4*x + 8, -4*x + 16, 2) ] 1.5030074136745814353823801003995556043806314222319625754371 5.393385283934348998903993780164142041461 320.261271901877284467762401417456847046362 {@ @} Total time: 1.899 seconds, Total memory usage: 39.35MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 14:18:23 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^7*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); //r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; hP:=Height(P);hP; HC:=HeightConstant(J:Effort:=20);HC; Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,19); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 14:18:21 on modular [Seed = 2725511726] ------------------------------------- Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 4*x + 8, -4*x + 16, 2) ] 1.5030074136745814353823801003995556043806314222319625754371 5.393385283934348998903993780164142041461 320.261271901877284467762401417456847046362 {@ @} Total time: 1.520 seconds, Total memory usage: 38.33MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 14:18:17 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^7*3^2; C:=HyperellipticCurve(f); J:=Jacobian(C); //r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; hP:=Height(P);hP; HC:=HeightConstant(J:Effort:=20);HC; Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,19); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 14:18:15 on modular [Seed = 3211337529] ------------------------------------- Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 28/9*x - 8/9, 44/27*x + 928/27, 2) ] 4.6613668053397002241366926279695766743193744943846637819259 6.862593078943470323468275363544094477084 3065.404397974187850397290252109710706677000 {@ <111296, 1, 4, 1> @} Total time: 1.550 seconds, Total memory usage: 38.22MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 14:18:11 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^7*3^1; C:=HyperellipticCurve(f); J:=Jacobian(C); //r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; hP:=Height(P);hP; HC:=HeightConstant(J:Effort:=20);HC; Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,19); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 14:18:09 on modular [Seed = 3094222904] ------------------------------------- Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 12*x - 24, -44*x - 96, 2) ] 2.9055439367899368398441645250175263592357000094565176957970 5.574645113293738508088336630026289936746 545.1337064398618038830322605633276151897425 {@ <0, 1, 4, 1> @} Total time: 1.530 seconds, Total memory usage: 38.52MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 14:18:05 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^7*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); //r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; hP:=Height(P);hP; HC:=HeightConstant(J:Effort:=20);HC; Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,19); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 14:18:04 on modular [Seed = 3043169576] ------------------------------------- Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 4*x + 8, -4*x + 16, 2) ] 1.5030074136745814353823801003995556043806314222319625754371 5.393385283934348998903993780164142041461 320.261271901877284467762401417456847046362 {@ @} Total time: 1.530 seconds, Total memory usage: 38.33MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 14:16:42 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^6*3^7; C:=HyperellipticCurve(f); J:=Jacobian(C); //r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; hP:=Height(P);hP; HC:=HeightConstant(J:Effort:=20);HC; Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,19); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 14:16:41 on modular [Seed = 3339806511] ------------------------------------- Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 + 216, -36*x - 648, 2) ] 0.5805158111864935118169755456131714467017852891061103277760 11.0437339907318897295517289923878972803934 72320.502514550595206585416395842058447714122 {@ @} Total time: 1.620 seconds, Total memory usage: 38.52MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 14:16:30 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^6*3^7; C:=HyperellipticCurve(f); J:=Jacobian(C); //r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; hP:=Height(P);hP; HC:=HeightConstant(J:Effort:=20);HC; Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,17); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 14:16:28 on modular [Seed = 3222691901] ------------------------------------- Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 + 216, -36*x - 648, 2) ] 0.5805158111864935118169755456131714467017852891061103277760 11.0437339907318897295517289923878972803934 72320.502514550595206585416395842058447714122 {@ @} Total time: 1.649 seconds, Total memory usage: 38.52MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 14:16:23 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^6*3^7; C:=HyperellipticCurve(f); J:=Jacobian(C); //r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; hP:=Height(P);hP; HC:=HeightConstant(J:Effort:=20);HC; Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,13); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 14:16:21 on modular [Seed = 3590354466] ------------------------------------- Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 + 216, -36*x - 648, 2) ] 0.5805158111864935118169755456131714467017852891061103277760 11.0437339907318897295517289923878972803934 72320.502514550595206585416395842058447714122 {@ <15944, 1, 4, 1> @} Total time: 1.700 seconds, Total memory usage: 38.52MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 14:16:16 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^6*3^7; C:=HyperellipticCurve(f); J:=Jacobian(C); //r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; hP:=Height(P);hP; HC:=HeightConstant(J:Effort:=20);HC; Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,11); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 14:16:13 on modular [Seed = 3540349919] ------------------------------------- Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 + 216, -36*x - 648, 2) ] 0.5805158111864935118169755456131714467017852891061103277760 11.0437339907318897295517289923878972803934 72320.502514550595206585416395842058447714122 {@ <8251, 1, 4, 1>, <0, 1, 4, 1>, <11146, 1, 4, 1>, <448, 1, 4, 1>, <4373, 1, 4, 1>, <82, 1, 4, 1>, <11205, 1, 4, 1>, <14480, 1, 4, 1>, <7203, 1, 4, 1>, <6052, 1, 4, 1> @} Total time: 2.649 seconds, Total memory usage: 38.52MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 14:16:08 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^6*3^7; C:=HyperellipticCurve(f); J:=Jacobian(C); //r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; hP:=Height(P);hP; HC:=HeightConstant(J:Effort:=20);HC; Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,7); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 14:16:06 on modular [Seed = 3963243755] ------------------------------------- Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 + 216, -36*x - 648, 2) ] 0.5805158111864935118169755456131714467017852891061103277760 11.0437339907318897295517289923878972803934 72320.502514550595206585416395842058447714122 {@ <1865, 1, 4, 1>, <577, 1, 4, 1>, <440, 1, 4, 1>, <981, 1, 4, 1> @} Total time: 1.730 seconds, Total memory usage: 38.52MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 14:16:01 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^6*3^7; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; hP:=Height(P);hP; HC:=HeightConstant(J:Effort:=20);HC; Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,7); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 14:15:59 on modular [Seed = 3913239031] ------------------------------------- TwoSelmerGroupData( J: Jacobian of Hyperelliptic Curve defined by y^2 = x^5 + 13996... ) TwoSelmerWork( C: Hyperelliptic Curve defined by y^2 = x^5 + 139968 over Ratio..., BoundType: Default, Bound: 0, UseUnits: true, Points: {}, Fields: {} ) LocalImageOdd( f: x^5 + 139968, p: 2, m1: Mapping from: Univariate Polynomial Ring in x over Rational ..., m2: Mapping from: Univariate Polynomial Ring in x over Rational ..., v: 26, Points: {} ) try( ) try( d: 2 ) SquarefreeRoots( F: $.1^4 + O(2^50)*$.1^3 - (226206511859851*2^2 + O(2^52))*$.1^... ) In file "/usr/local/magma/package/Geometry/Arith/sqfroots.m", line 84, column 15: >> return Roots(F:IsSquarefree); ^ Runtime error in 'Roots': Insufficient precision to find roots >> r:=TwoSelmerGroupData(J);r; ^ User error: Identifier 'r' has not been declared or assigned Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 + 216, -36*x - 648, 2) ] 0.5805158111864935118169755456131714467017852891061103277760 11.0437339907318897295517289923878972803934 72320.502514550595206585416395842058447714122 {@ <1865, 1, 4, 1>, <577, 1, 4, 1>, <440, 1, 4, 1>, <981, 1, 4, 1> @} Total time: 2.080 seconds, Total memory usage: 39.47MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 14:14:06 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^6*3^7; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; hP:=Height(P);hP; HC:=HeightConstant(J:Effort:=20);HC; Exp(hP/4+HC); //RationalPoints(J:Bound:=12); //Chabauty(P,19); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 14:14:05 on modular [Seed = 3795075950] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 + 216, -36*x - 648, 2) ] 0.5805158111864935118169755456131714467017852891061103277760 11.0437339907318897295517289923878972803934 72320.502514550595206585416395842058447714122 Total time: 1.700 seconds, Total memory usage: 39.51MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 14:14:03 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^6*3^7; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; hP:=Height(P);hP; HC:=HeightConstant(J:Effort:=20);HC; Exp(hP/4+HC); //RationalPoints(J:Bound:=12); //Chabauty(P,19); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 14:14:02 on modular [Seed = 4280901236] ------------------------------------- TwoSelmerGroupData( J: Jacobian of Hyperelliptic Curve defined by y^2 = x^5 + 13996... ) TwoSelmerWork( C: Hyperelliptic Curve defined by y^2 = x^5 + 139968 over Ratio..., BoundType: Default, Bound: 0, UseUnits: true, Points: {}, Fields: {} ) LocalImageOdd( f: x^5 + 139968, p: 2, m1: Mapping from: Univariate Polynomial Ring in x over Rational ..., m2: Mapping from: Univariate Polynomial Ring in x over Rational ..., v: 26, Points: {} ) try( ) try( d: 2 ) SquarefreeRoots( F: $.1^4 + O(2^50)*$.1^3 - (68700777678987*2^2 + O(2^52))*$.1^2... ) In file "/usr/local/magma/package/Geometry/Arith/sqfroots.m", line 84, column 15: >> return Roots(F:IsSquarefree); ^ Runtime error in 'Roots': Insufficient precision to find roots >> r:=TwoSelmerGroupData(J);r; ^ User error: Identifier 'r' has not been declared or assigned Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 + 216, -36*x - 648, 2) ] 0.5805158111864935118169755456131714467017852891061103277760 11.0437339907318897295517289923878972803934 72320.502514550595206585416395842058447714122 Total time: 1.540 seconds, Total memory usage: 39.48MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 14:14:00 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^6*3^7; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; hP:=Height(P);hP; HC:=HeightConstant(J:Effort:=20);HC; Exp(hP/4+HC); //RationalPoints(J:Bound:=12); //Chabauty(P,19); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 14:13:57 on modular [Seed = 4163787132] ------------------------------------- TwoSelmerGroupData( J: Jacobian of Hyperelliptic Curve defined by y^2 = x^5 + 13996... ) TwoSelmerWork( C: Hyperelliptic Curve defined by y^2 = x^5 + 139968 over Ratio..., BoundType: Default, Bound: 0, UseUnits: true, Points: {}, Fields: {} ) LocalImageOdd( f: x^5 + 139968, p: 2, m1: Mapping from: Univariate Polynomial Ring in x over Rational ..., m2: Mapping from: Univariate Polynomial Ring in x over Rational ..., v: 26, Points: {} ) try( ) try( d: 2 ) SquarefreeRoots( F: $.1^4 + O(2^50)*$.1^3 + (113686586849141*2^2 + O(2^52))*$.1^... ) In file "/usr/local/magma/package/Geometry/Arith/sqfroots.m", line 84, column 15: >> return Roots(F:IsSquarefree); ^ Runtime error in 'Roots': Insufficient precision to find roots >> r:=TwoSelmerGroupData(J);r; ^ User error: Identifier 'r' has not been declared or assigned Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 + 216, -36*x - 648, 2) ] 0.5805158111864935118169755456131714467017852891061103277760 11.0437339907318897295517289923878972803934 72320.502514550595206585416395842058447714122 Total time: 2.069 seconds, Total memory usage: 40.18MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 14:13:55 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^6*3^7; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; hP:=Height(P);hP; HC:=HeightConstant(J:Effort:=20);HC; Exp(hP/4+HC); //RationalPoints(J:Bound:=12); //Chabauty(P,19); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 14:13:53 on modular [Seed = 265924239] ------------------------------------- TwoSelmerGroupData( J: Jacobian of Hyperelliptic Curve defined by y^2 = x^5 + 13996... ) TwoSelmerWork( C: Hyperelliptic Curve defined by y^2 = x^5 + 139968 over Ratio..., BoundType: Default, Bound: 0, UseUnits: true, Points: {}, Fields: {} ) LocalImageOdd( f: x^5 + 139968, p: 2, m1: Mapping from: Univariate Polynomial Ring in x over Rational ..., m2: Mapping from: Univariate Polynomial Ring in x over Rational ..., v: 26, Points: {} ) try( ) try( d: 2 ) SquarefreeRoots( F: $.1^4 + O(2^50)*$.1^3 + (137517039482741*2^2 + O(2^52))*$.1^... ) In file "/usr/local/magma/package/Geometry/Arith/sqfroots.m", line 84, column 15: >> return Roots(F:IsSquarefree); ^ Runtime error in 'Roots': Insufficient precision to find roots >> r:=TwoSelmerGroupData(J);r; ^ User error: Identifier 'r' has not been declared or assigned Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 + 216, -36*x - 648, 2) ] 0.5805158111864935118169755456131714467017852891061103277760 11.0437339907318897295517289923878972803934 72320.502514550595206585416395842058447714122 Total time: 1.830 seconds, Total memory usage: 39.85MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 14:03:08 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^6*3^4; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; hP:=Height(P);hP; HC:=HeightConstant(J:Effort:=20);HC; Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,19); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 14:03:04 on modular [Seed = 365147660] ------------------------------------- 1 Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 to JacHyp: J given by a rule [no inverse] [ (x - 12, 504, 1) ] 1.35594798448245585339093825048755382779205779027500869424170 9.424858926246048445585012146857651468011 17393.396859886899872080836208350879499314648 {@ <0, 1, 4, 1>, <30278, 1, 4, 1>, <123900, 1, 4, 1>, <12, 1, 4, 1>, <108970, 1, 4, 1> @} Total time: 4.129 seconds, Total memory usage: 39.47MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 14:02:58 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^6*3^4; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; hP:=Height(P);hP; HC:=HeightConstant(J:Effort:=20);HC; Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,17); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 14:02:53 on modular [Seed = 788040962] ------------------------------------- 1 Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 to JacHyp: J given by a rule [no inverse] [ (x - 12, 504, 1) ] 1.35594798448245585339093825048755382779205779027500869424170 9.424858926246048445585012146857651468011 17393.396859886899872080836208350879499314648 {@ <0, 1, 4, 1>, <18171, 1, 4, 1>, <12, 1, 4, 1>, <4493, 1, 4, 1>, <5609, 1, 4, 1>, <67229, 1, 4, 1>, <17513, 1, 4, 1>, <58828, 1, 4, 1> @} Total time: 4.429 seconds, Total memory usage: 39.47MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 14:02:42 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^6*3^4; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; hP:=Height(P);hP; HC:=HeightConstant(J:Effort:=20);HC; Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,13); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 14:02:38 on modular [Seed = 738035732] ------------------------------------- 1 Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 to JacHyp: J given by a rule [no inverse] [ (x - 12, 504, 1) ] 1.35594798448245585339093825048755382779205779027500869424170 9.424858926246048445585012146857651468011 17393.396859886899872080836208350879499314648 {@ <0, 1, 4, 1>, <16447, 1, 4, 1>, <12, 1, 4, 1>, <17053, 1, 4, 1>, <25317, 1, 4, 1>, <26687, 1, 4, 1> @} Total time: 4.000 seconds, Total memory usage: 39.47MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 14:02:30 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^6*3^4; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; hP:=Height(P);hP; HC:=HeightConstant(J:Effort:=20);HC; Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,11); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 14:02:26 on modular [Seed = 619874071] ------------------------------------- 1 Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 to JacHyp: J given by a rule [no inverse] [ (x - 12, 504, 1) ] 1.35594798448245585339093825048755382779205779027500869424170 9.424858926246048445585012146857651468011 17393.396859886899872080836208350879499314648 {@ <0, 1, 4, 1>, <12, 1, 4, 1>, <11287, 1, 4, 1> @} Total time: 3.649 seconds, Total memory usage: 39.47MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 14:01:58 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^6*3^4; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; hP:=Height(P);hP; HC:=HeightConstant(J:Effort:=20);HC; Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,7); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 14:01:55 on modular [Seed = 1055432044] ------------------------------------- 1 Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 to JacHyp: J given by a rule [no inverse] [ (x - 12, 504, 1) ] 1.35594798448245585339093825048755382779205779027500869424170 9.424858926246048445585012146857651468011 17393.396859886899872080836208350879499314648 {@ <0, 1, 4, 1>, <0, 1, 0, 1> @} Total time: 3.520 seconds, Total memory usage: 39.47MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 13:59:20 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^6*3^4; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; hP:=Height(P);hP; HC:=HeightConstant(J:Effort:=20);HC; Exp(hP/4+HC); //RationalPoints(J:Bound:=12); //Chabauty(P,13); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 13:59:17 on modular [Seed = 1639447957] ------------------------------------- 1 Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 to JacHyp: J given by a rule [no inverse] [ (x - 12, 504, 1) ] 1.35594798448245585339093825048755382779205779027500869424170 9.424858926246048445585012146857651468011 17393.396859886899872080836208350879499314648 Total time: 3.180 seconds, Total memory usage: 39.47MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 13:59:07 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^6*3^4; C:=HyperellipticCurve(f); J:=Jacobian(C); //r:=TwoSelmerGroupData(J);r; //TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; hP:=Height(P);hP; HC:=HeightConstant(J:Effort:=20);HC; Exp(hP/4+HC); //RationalPoints(J:Bound:=12); //Chabauty(P,13); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 13:59:05 on modular [Seed = 2058151538] ------------------------------------- [ (x - 12, 504, 1) ] 1.35594798448245585339093825048755382779205779027500869424170 9.424858926246048445585012146857651468011 17393.396859886899872080836208350879499314648 Total time: 2.229 seconds, Total memory usage: 37.80MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 13:58:34 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^6*3^2; C:=HyperellipticCurve(f); J:=Jacobian(C); //r:=TwoSelmerGroupData(J);r; //TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; hP:=Height(P);hP; HC:=HeightConstant(J:Effort:=20);HC; Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,13); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 13:58:29 on modular [Seed = 2007100243] ------------------------------------- [ (x^2 - 4*x, 4*x + 24, 2) ] 0.540534546721831473133330220087515914278 7.570601031141423279961310497319392860814 2221.0485158927615717179160532823465639394347 {@ <0, 1, 4, 1>, <27313, 1, 4, 1>, <14097, 1, 4, 1>, <4, 1, 4, 1>, <14724, 1, 4, 1>, <12362, 1, 4, 1>, <19554, 1, 4, 1> @} Total time: 4.809 seconds, Total memory usage: 37.80MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 13:58:09 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^6*3^2; C:=HyperellipticCurve(f); J:=Jacobian(C); //r:=TwoSelmerGroupData(J);r; //TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; hP:=Height(P);hP; HC:=HeightConstant(J:Effort:=20);HC; Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,11); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 13:58:03 on modular [Seed = 1889987642] ------------------------------------- [ (x^2 - 4*x, 4*x + 24, 2) ] 0.540534546721831473133330220087515914278 7.570601031141423279961310497319392860814 2221.0485158927615717179160532823465639394347 {@ <0, 1, 4, 1>, <12469, 1, 4, 1>, <2007, 1, 4, 1>, <11375, 1, 4, 1>, <4, 1, 4, 1>, <8660, 1, 4, 1>, <3067, 1, 4, 1>, <8037, 1, 4, 1>, <3970, 1, 4, 1>, <8725, 1, 4, 1> @} Total time: 5.280 seconds, Total memory usage: 37.80MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 13:57:54 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^6*3^2; C:=HyperellipticCurve(f); J:=Jacobian(C); //r:=TwoSelmerGroupData(J);r; //TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; hP:=Height(P);hP; HC:=HeightConstant(J:Effort:=20);HC; Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,7); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 13:57:47 on modular [Seed = 2388136663] ------------------------------------- [ (x^2 - 4*x, 4*x + 24, 2) ] 0.540534546721831473133330220087515914278 7.570601031141423279961310497319392860814 2221.0485158927615717179160532823465639394347 {@ <0, 1, 4, 1>, <125, 1, 4, 1>, <4, 1, 4, 1>, <0, 1, 0, 6> @} Total time: 6.790 seconds, Total memory usage: 37.80MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 13:49:55 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^6*3^2; C:=HyperellipticCurve(f); J:=Jacobian(C); //r:=TwoSelmerGroupData(J);r; //TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; hP:=Height(P);hP; HC:=HeightConstant(J:Effort:=20);HC; Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,43); //Order(Q);Order(P); //P+Q; Output: ** WARNING: Computation time exceeded 20 seconds, so computation was terminated after 20 seconds. ** Magma V2.11-10 Tue Dec 6 2005 13:49:35 on modular [Seed = 2169970636] ------------------------------------- [ (x^2 - 4*x, 4*x + 24, 2) ] 0.540534546721831473133330220087515914278 7.570601031141423279961310497319392860814 2221.0485158927615717179160532823465639394347 Errors: /bin/sh: line 1: 12787 Alarm clock nice -n 19 /usr/local/bin/magma '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 13:49:30 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^6*3^2; C:=HyperellipticCurve(f); J:=Jacobian(C); //r:=TwoSelmerGroupData(J);r; //TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; hP:=Height(P);hP; HC:=HeightConstant(J:Effort:=20);HC; Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,41); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 13:49:22 on modular [Seed = 2622096092] ------------------------------------- [ (x^2 - 4*x, 4*x + 24, 2) ] 0.540534546721831473133330220087515914278 7.570601031141423279961310497319392860814 2221.0485158927615717179160532823465639394347 {@ <0, 1, 4, 1>, <2597267, 1, 4, 1>, <1697915, 1, 4, 1>, <1774761, 1, 4, 1>, <2447684, 1, 4, 1>, <2657729, 1, 4, 1>, <1379556, 1, 4, 1>, <2659667, 1, 4, 1>, <1439973, 1, 4, 1>, <471083, 1, 4, 1>, <2507694, 1, 4, 1>, <1346880, 1, 4, 1>, <2522497, 1, 4, 1>, <661046, 1, 4, 1>, <1249728, 1, 4, 1>, <1545728, 1, 4, 1>, <4, 1, 4, 1>, <899637, 1, 4, 1>, <2090612, 1, 4, 1>, <1028885, 1, 4, 1> @} Total time: 7.490 seconds, Total memory usage: 37.80MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 13:49:14 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^6*3^2; C:=HyperellipticCurve(f); J:=Jacobian(C); //r:=TwoSelmerGroupData(J);r; //TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; hP:=Height(P);hP; HC:=HeightConstant(J:Effort:=20);HC; Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,37); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 13:49:08 on modular [Seed = 2605516237] ------------------------------------- [ (x^2 - 4*x, 4*x + 24, 2) ] 0.540534546721831473133330220087515914278 7.570601031141423279961310497319392860814 2221.0485158927615717179160532823465639394347 {@ <0, 1, 4, 1>, <60442, 1, 4, 1>, <990905, 1, 4, 1>, <777413, 1, 4, 1>, <784817, 1, 4, 1>, <1593893, 1, 4, 1>, <1634301, 1, 4, 1>, <4, 1, 4, 1>, <1813900, 1, 4, 1>, <905869, 1, 4, 1>, <666141, 1, 4, 1>, <1092020, 1, 4, 1>, <714265, 1, 4, 1>, <837924, 1, 4, 1>, <1520237, 1, 4, 1>, <146155, 1, 4, 1>, <777551, 1, 4, 1> @} Total time: 6.799 seconds, Total memory usage: 37.80MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 13:49:03 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^6*3^2; C:=HyperellipticCurve(f); J:=Jacobian(C); //r:=TwoSelmerGroupData(J);r; //TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; hP:=Height(P);hP; HC:=HeightConstant(J:Effort:=20);HC; Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,31); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 13:48:57 on modular [Seed = 2520779006] ------------------------------------- [ (x^2 - 4*x, 4*x + 24, 2) ] 0.540534546721831473133330220087515914278 7.570601031141423279961310497319392860814 2221.0485158927615717179160532823465639394347 {@ <0, 1, 4, 1>, <4, 1, 4, 1>, <474037, 1, 4, 1>, <784897, 1, 4, 1>, <267935, 1, 4, 1>, <386199, 1, 4, 1> @} Total time: 5.179 seconds, Total memory usage: 37.80MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 13:48:46 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^6*3^2; C:=HyperellipticCurve(f); J:=Jacobian(C); //r:=TwoSelmerGroupData(J);r; //TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; hP:=Height(P);hP; HC:=HeightConstant(J:Effort:=20);HC; Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,29); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 13:48:41 on modular [Seed = 2910532566] ------------------------------------- [ (x^2 - 4*x, 4*x + 24, 2) ] 0.540534546721831473133330220087515914278 7.570601031141423279961310497319392860814 2221.0485158927615717179160532823465639394347 {@ <0, 1, 4, 1>, <405390, 1, 4, 1>, <628452, 1, 4, 1>, <4, 1, 4, 1> @} Total time: 4.950 seconds, Total memory usage: 37.80MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 13:48:35 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^6*3^2; C:=HyperellipticCurve(f); J:=Jacobian(C); //r:=TwoSelmerGroupData(J);r; //TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; hP:=Height(P);hP; HC:=HeightConstant(J:Effort:=20);HC; Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,23); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 13:48:30 on modular [Seed = 2826843940] ------------------------------------- [ (x^2 - 4*x, 4*x + 24, 2) ] 0.540534546721831473133330220087515914278 7.570601031141423279961310497319392860814 2221.0485158927615717179160532823465639394347 {@ <0, 1, 4, 1>, <19419, 1, 4, 1>, <4, 1, 4, 1>, <51761, 1, 4, 1>, <109357, 1, 4, 1>, <148391, 1, 4, 1>, <98436, 1, 4, 1>, <39998, 1, 4, 1>, <272237, 1, 4, 1>, <129795, 1, 4, 1>, <135415, 1, 4, 1> @} Total time: 5.769 seconds, Total memory usage: 37.80MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 13:48:24 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^6*3^2; C:=HyperellipticCurve(f); J:=Jacobian(C); //r:=TwoSelmerGroupData(J);r; //TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; hP:=Height(P);hP; HC:=HeightConstant(J:Effort:=20);HC; Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,19); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 13:48:18 on modular [Seed = 2809215015] ------------------------------------- [ (x^2 - 4*x, 4*x + 24, 2) ] 0.540534546721831473133330220087515914278 7.570601031141423279961310497319392860814 2221.0485158927615717179160532823465639394347 {@ <0, 1, 4, 1>, <13972, 1, 4, 1>, <36661, 1, 4, 1>, <4, 1, 4, 1>, <45886, 1, 4, 1>, <26663, 1, 4, 1> @} Total time: 4.990 seconds, Total memory usage: 37.80MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 13:48:13 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^6*3^2; C:=HyperellipticCurve(f); J:=Jacobian(C); //r:=TwoSelmerGroupData(J);r; //TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; hP:=Height(P);hP; HC:=HeightConstant(J:Effort:=20);HC; Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,17); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 13:48:07 on modular [Seed = 2725526839] ------------------------------------- [ (x^2 - 4*x, 4*x + 24, 2) ] 0.540534546721831473133330220087515914278 7.570601031141423279961310497319392860814 2221.0485158927615717179160532823465639394347 {@ <0, 1, 4, 1>, <4, 1, 4, 1>, <13890, 1, 4, 1>, <4601, 1, 4, 1>, <5685, 1, 4, 1>, <1506, 1, 4, 1>, <62657, 1, 4, 1>, <25978, 1, 4, 1> @} Total time: 5.240 seconds, Total memory usage: 37.80MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 13:47:59 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^6*3^2; C:=HyperellipticCurve(f); J:=Jacobian(C); //r:=TwoSelmerGroupData(J);r; //TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; hP:=Height(P);hP; HC:=HeightConstant(J:Effort:=20);HC; Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,13); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 13:47:54 on modular [Seed = 3111069201] ------------------------------------- [ (x^2 - 4*x, 4*x + 24, 2) ] 0.540534546721831473133330220087515914278 7.570601031141423279961310497319392860814 2221.0485158927615717179160532823465639394347 {@ <0, 1, 4, 1>, <27313, 1, 4, 1>, <14097, 1, 4, 1>, <4, 1, 4, 1>, <14724, 1, 4, 1>, <12362, 1, 4, 1>, <19554, 1, 4, 1> @} Total time: 4.809 seconds, Total memory usage: 37.80MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 13:47:46 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^6*3^2; C:=HyperellipticCurve(f); J:=Jacobian(C); //r:=TwoSelmerGroupData(J);r; //TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; hP:=Height(P);hP; HC:=HeightConstant(J:Effort:=20);HC; Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,7); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 13:47:39 on modular [Seed = 3026332429] ------------------------------------- [ (x^2 - 4*x, 4*x + 24, 2) ] 0.540534546721831473133330220087515914278 7.570601031141423279961310497319392860814 2221.0485158927615717179160532823465639394347 {@ <0, 1, 4, 1>, <125, 1, 4, 1>, <4, 1, 4, 1>, <0, 1, 0, 6> @} Total time: 6.700 seconds, Total memory usage: 37.80MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 13:47:23 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^6*3^2; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; hP:=Height(P);hP; HC:=HeightConstant(J:Effort:=20);HC; Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,11); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 13:47:16 on modular [Seed = 2992909684] ------------------------------------- 1 Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 to JacHyp: J given by a rule [no inverse] [ (x^2 - 4*x, 4*x + 24, 2) ] 0.540534546721831473133330220087515914278 7.570601031141423279961310497319392860814 2221.0485158927615717179160532823465639394347 {@ <0, 1, 4, 1>, <12469, 1, 4, 1>, <2007, 1, 4, 1>, <11375, 1, 4, 1>, <4, 1, 4, 1>, <8660, 1, 4, 1>, <3067, 1, 4, 1>, <8037, 1, 4, 1>, <3970, 1, 4, 1>, <8725, 1, 4, 1> @} Total time: 6.320 seconds, Total memory usage: 39.52MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 13:47:03 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^6*3^2; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; hP:=Height(P);hP; HC:=HeightConstant(J:Effort:=20);HC; Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,19); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 13:46:57 on modular [Seed = 3357431579] ------------------------------------- 1 Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 to JacHyp: J given by a rule [no inverse] [ (x^2 - 4*x, 4*x + 24, 2) ] 0.540534546721831473133330220087515914278 7.570601031141423279961310497319392860814 2221.0485158927615717179160532823465639394347 {@ <0, 1, 4, 1>, <13972, 1, 4, 1>, <36661, 1, 4, 1>, <4, 1, 4, 1>, <45886, 1, 4, 1>, <26663, 1, 4, 1> @} Total time: 5.960 seconds, Total memory usage: 39.52MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 13:46:42 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^6*3^2; C:=HyperellipticCurve(f); J:=Jacobian(C); //r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=2500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; hP:=Height(P);hP; HC:=HeightConstant(J:Effort:=20);HC; Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,19); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 13:46:36 on modular [Seed = 3339802681] ------------------------------------- Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 to JacHyp: J given by a rule [no inverse] [ (x^2 - 4*x, 4*x + 24, 2) ] 0.540534546721831473133330220087515914278 7.570601031141423279961310497319392860814 2221.0485158927615717179160532823465639394347 {@ <0, 1, 4, 1>, <13972, 1, 4, 1>, <36661, 1, 4, 1>, <4, 1, 4, 1>, <45886, 1, 4, 1>, <26663, 1, 4, 1> @} Total time: 5.929 seconds, Total memory usage: 39.08MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 13:45:25 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^6*3^2; C:=HyperellipticCurve(f); J:=Jacobian(C); //r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; hP:=Height(P);hP; HC:=HeightConstant(J:Effort:=20);HC; Exp(hP/4+HC); //RationalPoints(J:Bound:=12); Chabauty(P,19); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 13:45:19 on modular [Seed = 3256114575] ------------------------------------- Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 to JacHyp: J given by a rule [no inverse] [ (x^2 - 4*x, 4*x + 24, 2) ] 0.540534546721831473133330220087515914278 7.570601031141423279961310497319392860814 2221.0485158927615717179160532823465639394347 {@ <0, 1, 4, 1>, <13972, 1, 4, 1>, <36661, 1, 4, 1>, <4, 1, 4, 1>, <45886, 1, 4, 1>, <26663, 1, 4, 1> @} Total time: 5.269 seconds, Total memory usage: 39.08MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 13:44:25 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^6*3^2; C:=HyperellipticCurve(f); J:=Jacobian(C); //r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; Height(P); HeightConstant(J:Effort:=20); Exp(5); //RationalPoints(J:Bound:=12); Chabauty(P,19); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 13:44:20 on modular [Seed = 3641657007] ------------------------------------- Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 to JacHyp: J given by a rule [no inverse] [ (x^2 - 4*x, 4*x + 24, 2) ] 0.540534546721831473133330220087515914278 7.570601031141423279961310497319392860814 5.392650408063836341891309004159701491924 148.413159102576603421115580040552279623351 {@ <0, 1, 4, 1>, <13972, 1, 4, 1>, <36661, 1, 4, 1>, <4, 1, 4, 1>, <45886, 1, 4, 1>, <26663, 1, 4, 1> @} Total time: 5.309 seconds, Total memory usage: 39.08MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 13:41:50 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^6*3^2; C:=HyperellipticCurve(f); J:=Jacobian(C); //r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; Height(P); HeightConstant(J:Effort:=20); //RationalPoints(J:Bound:=12); Chabauty(P,19); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 13:41:44 on modular [Seed = 3607187405] ------------------------------------- Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 to JacHyp: J given by a rule [no inverse] [ (x^2 - 4*x, 4*x + 24, 2) ] 0.540534546721831473133330220087515914278 7.570601031141423279961310497319392860814 5.392650408063836341891309004159701491924 {@ <0, 1, 4, 1>, <13972, 1, 4, 1>, <36661, 1, 4, 1>, <4, 1, 4, 1>, <45886, 1, 4, 1>, <26663, 1, 4, 1> @} Total time: 5.309 seconds, Total memory usage: 39.08MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 13:40:27 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^6*3^1; C:=HyperellipticCurve(f); J:=Jacobian(C); //r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; Height(P); HeightConstant(J:Effort:=20); //RationalPoints(J:Bound:=12); Chabauty(P,19); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 13:40:26 on modular [Seed = 3523496634] ------------------------------------- Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 8, -4*x - 8, 2) ] 1.42704817539261298800466592211697877707043854843712007257320 5.459212313203383107163086028098250589276 4.660242215618429880961145512878017688844 {@ @} Total time: 1.510 seconds, Total memory usage: 38.52MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 13:40:19 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^6*3^1; C:=HyperellipticCurve(f); J:=Jacobian(C); //r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; Height(P); HeightConstant(J:Effort:=20); //RationalPoints(J:Bound:=12); Chabauty(P,17); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 13:40:17 on modular [Seed = 3913249162] ------------------------------------- Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 8, -4*x - 8, 2) ] 1.42704817539261298800466592211697877707043854843712007257320 5.459212313203383107163086028098250589276 4.660242215618429880961145512878017688844 {@ <19799, 1, 4, 1>, <69794, 1, 4, 1>, <76881, 1, 4, 1>, <24561, 1, 4, 1>, <6404, 1, 4, 1>, <46483, 1, 4, 1>, <27624, 1, 4, 1>, <14777, 1, 4, 1> @} Total time: 2.419 seconds, Total memory usage: 38.52MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 13:40:05 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^6*3^1; C:=HyperellipticCurve(f); J:=Jacobian(C); //r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; Height(P); HeightConstant(J:Effort:=20); //RationalPoints(J:Bound:=12); Chabauty(P,13); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 13:40:02 on modular [Seed = 3828513940] ------------------------------------- Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 8, -4*x - 8, 2) ] 1.42704817539261298800466592211697877707043854843712007257320 5.459212313203383107163086028098250589276 4.660242215618429880961145512878017688844 {@ <22652, 1, 4, 1>, <6356, 1, 4, 1>, <13101, 1, 4, 1>, <0, 1, 4, 1>, <8760, 1, 4, 1>, <14809, 1, 4, 1> @} Total time: 2.009 seconds, Total memory usage: 38.52MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 13:39:56 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^6*3^1; C:=HyperellipticCurve(f); J:=Jacobian(C); //r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; Height(P); HeightConstant(J:Effort:=20); //RationalPoints(J:Bound:=12); Chabauty(P,11); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 13:39:54 on modular [Seed = 3811932779] ------------------------------------- Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 8, -4*x - 8, 2) ] 1.42704817539261298800466592211697877707043854843712007257320 5.459212313203383107163086028098250589276 4.660242215618429880961145512878017688844 {@ <7960, 1, 4, 1>, <0, 1, 4, 1>, <52, 1, 4, 1>, <4578, 1, 4, 1>, <2294, 1, 4, 1> @} Total time: 2.040 seconds, Total memory usage: 38.52MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 13:39:49 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^6*3^1; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; Height(P); HeightConstant(J:Effort:=20); //RationalPoints(J:Bound:=12); Chabauty(P,11); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 13:39:46 on modular [Seed = 4264060798] ------------------------------------- TwoSelmerGroupData( J: Jacobian of Hyperelliptic Curve defined by y^2 = x^5 + 192 o... ) TwoSelmerWork( C: Hyperelliptic Curve defined by y^2 = x^5 + 192 over Rational..., BoundType: Default, Bound: 0, UseUnits: true, Points: {}, Fields: {} ) LocalImageOdd( f: x^5 + 192, p: 2, m1: Mapping from: Univariate Polynomial Ring in x over Rational ..., m2: Mapping from: Univariate Polynomial Ring in x over Rational ..., v: 26, Points: {} ) try( ) try( d: 2 ) SquarefreeRoots( F: $.1^4 + O(2^50)*$.1^3 - (510006934241283*2^2 + O(2^52))*$.1^... ) In file "/usr/local/magma/package/Geometry/Arith/sqfroots.m", line 84, column 15: >> return Roots(F:IsSquarefree); ^ Runtime error in 'Roots': Insufficient precision to find roots >> r:=TwoSelmerGroupData(J);r; ^ User error: Identifier 'r' has not been declared or assigned Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 8, -4*x - 8, 2) ] 1.42704817539261298800466592211697877707043854843712007257320 5.459212313203383107163086028098250589276 4.660242215618429880961145512878017688844 {@ <7960, 1, 4, 1>, <0, 1, 4, 1>, <52, 1, 4, 1>, <4578, 1, 4, 1>, <2294, 1, 4, 1> @} Total time: 2.480 seconds, Total memory usage: 39.75MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 13:39:38 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^6*3^1; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; Height(P); HeightConstant(J:Effort:=20); //RationalPoints(J:Bound:=12); Chabauty(P,7); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 13:39:35 on modular [Seed = 4112739424] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 8, -4*x - 8, 2) ] 1.42704817539261298800466592211697877707043854843712007257320 5.459212313203383107163086028098250589276 4.660242215618429880961145512878017688844 {@ <876, 1, 4, 1>, <2267, 1, 4, 1>, <458, 1, 4, 1> @} Total time: 2.859 seconds, Total memory usage: 40.79MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 13:37:10 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^6*3^1; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; Height(P); HeightConstant(J:Effort:=20); //RationalPoints(J:Bound:=12); //Chabauty(P,7); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 13:37:09 on modular [Seed = 114335521] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 8, -4*x - 8, 2) ] 1.42704817539261298800466592211697877707043854843712007257320 5.459212313203383107163086028098250589276 4.660242215618429880961145512878017688844 Total time: 1.590 seconds, Total memory usage: 39.48MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 13:37:07 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^6*3^1; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; Height(P); HeightConstant(J:Effort:=20); //RationalPoints(J:Bound:=12); //Chabauty(P,7); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 13:37:05 on modular [Seed = 499876926] ------------------------------------- TwoSelmerGroupData( J: Jacobian of Hyperelliptic Curve defined by y^2 = x^5 + 192 o... ) TwoSelmerWork( C: Hyperelliptic Curve defined by y^2 = x^5 + 192 over Rational..., BoundType: Default, Bound: 0, UseUnits: true, Points: {}, Fields: {} ) LocalImageOdd( f: x^5 + 192, p: 2, m1: Mapping from: Univariate Polynomial Ring in x over Rational ..., m2: Mapping from: Univariate Polynomial Ring in x over Rational ..., v: 26, Points: {} ) try( ) try( d: 2 ) SquarefreeRoots( F: $.1^4 + O(2^50)*$.1^3 - (126189686554627*2^2 + O(2^52))*$.1^... ) In file "/usr/local/magma/package/Geometry/Arith/sqfroots.m", line 84, column 15: >> return Roots(F:IsSquarefree); ^ Runtime error in 'Roots': Insufficient precision to find roots >> r:=TwoSelmerGroupData(J);r; ^ User error: Identifier 'r' has not been declared or assigned Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 8, -4*x - 8, 2) ] 1.42704817539261298800466592211697877707043854843712007257320 5.459212313203383107163086028098250589276 4.660242215618429880961145512878017688844 Total time: 1.790 seconds, Total memory usage: 40.08MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 13:37:03 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^6*3^1; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; Height(P); HeightConstant(J:Effort:=20); //RationalPoints(J:Bound:=12); //Chabauty(P,7); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 13:37:01 on modular [Seed = 466453028] ------------------------------------- TwoSelmerGroupData( J: Jacobian of Hyperelliptic Curve defined by y^2 = x^5 + 192 o... ) TwoSelmerWork( C: Hyperelliptic Curve defined by y^2 = x^5 + 192 over Rational..., BoundType: Default, Bound: 0, UseUnits: true, Points: {}, Fields: {} ) LocalImageOdd( f: x^5 + 192, p: 2, m1: Mapping from: Univariate Polynomial Ring in x over Rational ..., m2: Mapping from: Univariate Polynomial Ring in x over Rational ..., v: 26, Points: {} ) try( ) try( d: 2 ) SquarefreeRoots( F: $.1^4 + O(2^50)*$.1^3 - (205148595421187*2^2 + O(2^52))*$.1^... ) In file "/usr/local/magma/package/Geometry/Arith/sqfroots.m", line 84, column 15: >> return Roots(F:IsSquarefree); ^ Runtime error in 'Roots': Insufficient precision to find roots >> r:=TwoSelmerGroupData(J);r; ^ User error: Identifier 'r' has not been declared or assigned Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 8, -4*x - 8, 2) ] 1.42704817539261298800466592211697877707043854843712007257320 5.459212313203383107163086028098250589276 4.660242215618429880961145512878017688844 Total time: 1.720 seconds, Total memory usage: 39.93MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 13:36:59 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^6*3^1; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; Height(P); HeightConstant(J:Effort:=20); //RationalPoints(J:Bound:=12); //Chabauty(P,7); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 13:36:58 on modular [Seed = 381718318] ------------------------------------- TwoSelmerGroupData( J: Jacobian of Hyperelliptic Curve defined by y^2 = x^5 + 192 o... ) TwoSelmerWork( C: Hyperelliptic Curve defined by y^2 = x^5 + 192 over Rational..., BoundType: Default, Bound: 0, UseUnits: true, Points: {}, Fields: {} ) LocalImageOdd( f: x^5 + 192, p: 2, m1: Mapping from: Univariate Polynomial Ring in x over Rational ..., m2: Mapping from: Univariate Polynomial Ring in x over Rational ..., v: 26, Points: {} ) try( ) try( d: 2 ) SquarefreeRoots( F: $.1^4 + O(2^50)*$.1^3 + (533042781290493*2^2 + O(2^52))*$.1^... ) In file "/usr/local/magma/package/Geometry/Arith/sqfroots.m", line 84, column 15: >> return Roots(F:IsSquarefree); ^ Runtime error in 'Roots': Insufficient precision to find roots >> r:=TwoSelmerGroupData(J);r; ^ User error: Identifier 'r' has not been declared or assigned Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 8, -4*x - 8, 2) ] 1.42704817539261298800466592211697877707043854843712007257320 5.459212313203383107163086028098250589276 4.660242215618429880961145512878017688844 Total time: 1.560 seconds, Total memory usage: 39.72MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 13:36:56 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^6*3^1; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; Height(P); HeightConstant(J:Effort:=20); //RationalPoints(J:Bound:=12); //Chabauty(P,7); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 13:36:54 on modular [Seed = 771470361] ------------------------------------- TwoSelmerGroupData( J: Jacobian of Hyperelliptic Curve defined by y^2 = x^5 + 192 o... ) TwoSelmerWork( C: Hyperelliptic Curve defined by y^2 = x^5 + 192 over Rational..., BoundType: Default, Bound: 0, UseUnits: true, Points: {}, Fields: {} ) LocalImageOdd( f: x^5 + 192, p: 2, m1: Mapping from: Univariate Polynomial Ring in x over Rational ..., m2: Mapping from: Univariate Polynomial Ring in x over Rational ..., v: 26, Points: {} ) try( ) try( d: 2 ) SquarefreeRoots( F: $.1^4 + O(2^50)*$.1^3 + (28027162198013*2^2 + O(2^52))*$.1^2... ) In file "/usr/local/magma/package/Geometry/Arith/sqfroots.m", line 84, column 15: >> return Roots(F:IsSquarefree); ^ Runtime error in 'Roots': Insufficient precision to find roots >> r:=TwoSelmerGroupData(J);r; ^ User error: Identifier 'r' has not been declared or assigned Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 8, -4*x - 8, 2) ] 1.42704817539261298800466592211697877707043854843712007257320 5.459212313203383107163086028098250589276 4.660242215618429880961145512878017688844 Total time: 1.429 seconds, Total memory usage: 39.54MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 13:36:52 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^6*3^1; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; Height(P); HeightConstant(J:Effort:=20); //RationalPoints(J:Bound:=12); //Chabauty(P,7); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 13:36:50 on modular [Seed = 687780120] ------------------------------------- TwoSelmerGroupData( J: Jacobian of Hyperelliptic Curve defined by y^2 = x^5 + 192 o... ) TwoSelmerWork( C: Hyperelliptic Curve defined by y^2 = x^5 + 192 over Rational..., BoundType: Default, Bound: 0, UseUnits: true, Points: {}, Fields: {} ) LocalImageOdd( f: x^5 + 192, p: 2, m1: Mapping from: Univariate Polynomial Ring in x over Rational ..., m2: Mapping from: Univariate Polynomial Ring in x over Rational ..., v: 26, Points: {} ) try( ) try( d: 2 ) SquarefreeRoots( F: (1 + O(2^46))*$.1^3 + O(2^50)*$.1^2 - (10673799036931*2^2 + ... ) In file "/usr/local/magma/package/Geometry/Arith/sqfroots.m", line 84, column 15: >> return Roots(F:IsSquarefree); ^ Runtime error in 'Roots': Insufficient precision to find roots >> r:=TwoSelmerGroupData(J);r; ^ User error: Identifier 'r' has not been declared or assigned Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 8, -4*x - 8, 2) ] 1.42704817539261298800466592211697877707043854843712007257320 5.459212313203383107163086028098250589276 4.660242215618429880961145512878017688844 Total time: 1.740 seconds, Total memory usage: 39.99MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 13:36:45 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^6*3^1; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; Height(P); HeightConstant(J:Effort:=20); //RationalPoints(J:Bound:=12); //Chabauty(P,7); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 13:36:43 on modular [Seed = 670154350] ------------------------------------- TwoSelmerGroupData( J: Jacobian of Hyperelliptic Curve defined by y^2 = x^5 + 192 o... ) TwoSelmerWork( C: Hyperelliptic Curve defined by y^2 = x^5 + 192 over Rational..., BoundType: Default, Bound: 0, UseUnits: true, Points: {}, Fields: {} ) LocalImageOdd( f: x^5 + 192, p: 2, m1: Mapping from: Univariate Polynomial Ring in x over Rational ..., m2: Mapping from: Univariate Polynomial Ring in x over Rational ..., v: 26, Points: {} ) try( ) try( d: 2 ) SquarefreeRoots( F: $.1^4 + O(2^50)*$.1^3 - (405529284706307*2^2 + O(2^52))*$.1^... ) In file "/usr/local/magma/package/Geometry/Arith/sqfroots.m", line 84, column 15: >> return Roots(F:IsSquarefree); ^ Runtime error in 'Roots': Insufficient precision to find roots >> r:=TwoSelmerGroupData(J);r; ^ User error: Identifier 'r' has not been declared or assigned Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 8, -4*x - 8, 2) ] 1.42704817539261298800466592211697877707043854843712007257320 5.459212313203383107163086028098250589276 4.660242215618429880961145512878017688844 Total time: 1.530 seconds, Total memory usage: 39.68MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 13:33:28 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^5*3^8; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; Height(P); HeightConstant(J:Effort:=20); //RationalPoints(J:Bound:=12); Chabauty(P,61); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 13:33:20 on modular [Seed = 586463264] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 9/4*x + 81/2, -9/8*x - 1863/4, 2) ] 3.8195095727722308631704825180304497768505522455180313861189 11.1605099928888306781579870179578106896334 9.325001442362978234527468537544353265163 {@ <1501825, 1, 4, 1>, <2735217, 1, 4, 1>, <1248183, 1, 4, 1>, <13168104, 1, 4, 1>, <6926292, 1, 4, 1>, <5473814, 1, 4, 1>, <11483073, 1, 4, 1>, <78936, 1, 4, 1>, <5601040, 1, 4, 1>, <9052154, 1, 4, 1>, <13602106, 1, 4, 1>, <11931435, 1, 4, 1>, <5826426, 1, 4, 1>, <10829460, 1, 4, 1>, <10412941, 1, 4, 1>, <6629779, 1, 4, 1>, <798658, 1, 4, 1>, <1503626, 1, 4, 1>, <1197885, 1, 4, 1>, <7465266, 1, 4, 1>, <11451148, 1, 4, 1>, <7160797, 1, 4, 1>, <10985804, 1, 4, 1>, <3096037, 1, 4, 1>, <10632461, 1, 4, 1>, <9349970, 1, 4, 1>, <7280407, 1, 4, 1>, <10751359, 1, 4, 1>, <9497510, 1, 4, 1>, <1050706, 1, 4, 1> @} Total time: 7.799 seconds, Total memory usage: 39.12MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 13:33:10 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^5*3^8; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; Height(P); HeightConstant(J:Effort:=20); //RationalPoints(J:Bound:=12); Chabauty(P,47); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 13:33:04 on modular [Seed = 972006187] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 9/4*x + 81/2, -9/8*x - 1863/4, 2) ] 3.8195095727722308631704825180304497768505522455180313861189 11.1605099928888306781579870179578106896334 9.325001442362978234527468537544353265163 {@ <18553, 1, 4, 1>, <2154130, 1, 4, 1>, <954111, 1, 4, 1>, <1916482, 1, 4, 1>, <2198198, 1, 4, 1>, <2186846, 1, 4, 1>, <3655441, 1, 4, 1>, <1347416, 1, 4, 1>, <1125178, 1, 4, 1>, <2414931, 1, 4, 1>, <3563582, 1, 4, 1>, <0, 1, 4, 1>, <3742107, 1, 4, 1>, <2076665, 1, 4, 1>, <2066139, 1, 4, 1>, <2001988, 1, 4, 1>, <3036467, 1, 4, 1>, <3616228, 1, 4, 1>, <1073009, 1, 4, 1>, <4230318, 1, 4, 1>, <758107, 1, 4, 1>, <1457617, 1, 4, 1> @} Total time: 6.280 seconds, Total memory usage: 39.12MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 13:32:59 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^5*3^8; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; Height(P); HeightConstant(J:Effort:=20); //RationalPoints(J:Bound:=12); Chabauty(P,43); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 13:32:55 on modular [Seed = 887270929] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 9/4*x + 81/2, -9/8*x - 1863/4, 2) ] 3.8195095727722308631704825180304497768505522455180313861189 11.1605099928888306781579870179578106896334 9.325001442362978234527468537544353265163 {@ <2769263, 1, 4, 1>, <1802899, 1, 4, 1>, <2073720, 1, 4, 1>, <436913, 1, 4, 1>, <1103776, 1, 4, 1>, <977490, 1, 4, 1>, <1077728, 1, 4, 1> @} Total time: 3.339 seconds, Total memory usage: 39.12MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 13:32:41 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^5*3^8; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; Height(P); HeightConstant(J:Effort:=20); //RationalPoints(J:Bound:=12); Chabauty(P,41); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 13:32:38 on modular [Seed = 1302058873] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 9/4*x + 81/2, -9/8*x - 1863/4, 2) ] 3.8195095727722308631704825180304497768505522455180313861189 11.1605099928888306781579870179578106896334 9.325001442362978234527468537544353265163 {@ <285375, 1, 4, 1>, <648401, 1, 4, 1>, <1191607, 1, 4, 1>, <2640996, 1, 4, 1>, <2667085, 1, 4, 1> @} Total time: 3.069 seconds, Total memory usage: 39.12MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 13:32:30 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^5*3^8; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; Height(P); HeightConstant(J:Effort:=20); //RationalPoints(J:Bound:=12); Chabauty(P,37); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 13:32:27 on modular [Seed = 1218368001] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 9/4*x + 81/2, -9/8*x - 1863/4, 2) ] 3.8195095727722308631704825180304497768505522455180313861189 11.1605099928888306781579870179578106896334 9.325001442362978234527468537544353265163 {@ <1705268, 1, 4, 1>, <97004, 1, 4, 1>, <109642, 1, 4, 1>, <21290, 1, 4, 1> @} Total time: 2.990 seconds, Total memory usage: 39.12MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 13:32:22 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^5*3^8; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; Height(P); HeightConstant(J:Effort:=20); //RationalPoints(J:Bound:=12); Chabauty(P,31); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 13:32:18 on modular [Seed = 1200741639] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 9/4*x + 81/2, -9/8*x - 1863/4, 2) ] 3.8195095727722308631704825180304497768505522455180313861189 11.1605099928888306781579870179578106896334 9.325001442362978234527468537544353265163 {@ <693692, 1, 4, 1>, <37303, 1, 4, 1>, <749589, 1, 4, 1>, <512357, 1, 4, 1>, <620638, 1, 4, 1> @} Total time: 2.910 seconds, Total memory usage: 39.12MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 13:32:07 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^5*3^8; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; Height(P); HeightConstant(J:Effort:=20); //RationalPoints(J:Bound:=12); Chabauty(P,29); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 13:32:04 on modular [Seed = 1117050900] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 9/4*x + 81/2, -9/8*x - 1863/4, 2) ] 3.8195095727722308631704825180304497768505522455180313861189 11.1605099928888306781579870179578106896334 9.325001442362978234527468537544353265163 {@ <336140, 1, 4, 1> @} Total time: 2.390 seconds, Total memory usage: 39.12MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 13:31:59 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^5*3^8; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; Height(P); HeightConstant(J:Effort:=20); //RationalPoints(J:Bound:=12); Chabauty(P,23); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 13:31:55 on modular [Seed = 1485750833] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 9/4*x + 81/2, -9/8*x - 1863/4, 2) ] 3.8195095727722308631704825180304497768505522455180313861189 11.1605099928888306781579870179578106896334 9.325001442362978234527468537544353265163 {@ <85580, 1, 4, 1>, <232404, 1, 4, 1>, <59226, 1, 4, 1>, <139180, 1, 4, 1>, <0, 1, 4, 1>, <11333, 1, 4, 1>, <20295, 1, 4, 1>, <254229, 1, 4, 1>, <255518, 1, 4, 1>, <251564, 1, 4, 1> @} Total time: 3.740 seconds, Total memory usage: 39.12MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 13:31:46 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^5*3^8; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; Height(P); HeightConstant(J:Effort:=20); //RationalPoints(J:Bound:=12); Chabauty(P,19); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 13:31:44 on modular [Seed = 1468124469] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 9/4*x + 81/2, -9/8*x - 1863/4, 2) ] 3.8195095727722308631704825180304497768505522455180313861189 11.1605099928888306781579870179578106896334 9.325001442362978234527468537544353265163 {@ <108284, 1, 4, 1> @} Total time: 2.359 seconds, Total memory usage: 39.12MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 13:31:38 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^5*3^8; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; Height(P); HeightConstant(J:Effort:=20); //RationalPoints(J:Bound:=12); Chabauty(P,17); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 13:31:35 on modular [Seed = 1384433703] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 9/4*x + 81/2, -9/8*x - 1863/4, 2) ] 3.8195095727722308631704825180304497768505522455180313861189 11.1605099928888306781579870179578106896334 9.325001442362978234527468537544353265163 {@ <52562, 1, 4, 1>, <80301, 1, 4, 1>, <72852, 1, 4, 1>, <16320, 1, 4, 1>, <0, 1, 4, 1>, <8676, 1, 4, 1>, <43621, 1, 4, 1>, <32832, 1, 4, 1>, <7697, 1, 4, 1> @} Total time: 3.220 seconds, Total memory usage: 39.12MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 13:31:29 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^5*3^8; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; Height(P); HeightConstant(J:Effort:=20); //RationalPoints(J:Bound:=12); Chabauty(P,13); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 13:31:26 on modular [Seed = 1774187237] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 9/4*x + 81/2, -9/8*x - 1863/4, 2) ] 3.8195095727722308631704825180304497768505522455180313861189 11.1605099928888306781579870179578106896334 9.325001442362978234527468537544353265163 {@ <27252, 1, 4, 1> @} Total time: 2.209 seconds, Total memory usage: 39.12MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 13:31:21 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^5*3^8; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; Height(P); HeightConstant(J:Effort:=20); //RationalPoints(J:Bound:=12); Chabauty(P,11); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 13:31:18 on modular [Seed = 1689452512] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 9/4*x + 81/2, -9/8*x - 1863/4, 2) ] 3.8195095727722308631704825180304497768505522455180313861189 11.1605099928888306781579870179578106896334 9.325001442362978234527468537544353265163 {@ <325, 1, 4, 1>, <9792, 1, 4, 1>, <3109, 1, 4, 1>, <12273, 1, 4, 1>, <11846, 1, 4, 1> @} Total time: 2.620 seconds, Total memory usage: 39.12MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 13:31:13 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^5*3^8; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; Height(P); HeightConstant(J:Effort:=20); //RationalPoints(J:Bound:=12); Chabauty(P,7); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 13:31:10 on modular [Seed = 1672870143] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 9/4*x + 81/2, -9/8*x - 1863/4, 2) ] 3.8195095727722308631704825180304497768505522455180313861189 11.1605099928888306781579870179578106896334 9.325001442362978234527468537544353265163 {@ <684, 1, 4, 1>, <0, 1, 4, 1>, <351, 1, 4, 1> @} Total time: 2.259 seconds, Total memory usage: 39.12MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 13:28:20 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^5*3^8; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; Height(P); HeightConstant(J:Effort:=20); //RationalPoints(J:Bound:=12); //Chabauty(P,7); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 13:28:18 on modular [Seed = 2074992715] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 9/4*x + 81/2, -9/8*x - 1863/4, 2) ] 3.8195095727722308631704825180304497768505522455180313861189 11.1605099928888306781579870179578106896334 9.325001442362978234527468537544353265163 Total time: 1.790 seconds, Total memory usage: 39.12MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 13:27:37 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^5*3^7; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; Height(P); HeightConstant(J:Effort:=20); //RationalPoints(J:Bound:=12); Chabauty(P,67); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 13:27:28 on modular [Seed = 1923410192] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 + 108, 18*x + 324, 2) ] 2.0602834384501336281543102556814931412814 10.2187885663225179085246161325220231662577 8.592593249917571773597305046262669462013 {@ <11197197, 1, 4, 1>, <3005276, 1, 4, 1>, <7014446, 1, 4, 1>, <15971564, 1, 4, 1>, <17044899, 1, 4, 1>, <1864961, 1, 4, 1>, <12575718, 1, 4, 1>, <7013876, 1, 4, 1>, <1506713, 1, 4, 1>, <269252, 1, 4, 1>, <14859615, 1, 4, 1>, <6565980, 1, 4, 1>, <7207186, 1, 4, 1>, <9024818, 1, 4, 1>, <0, 1, 4, 1>, <7340076, 1, 4, 1>, <5335980, 1, 4, 1>, <4780394, 1, 4, 1>, <15417502, 1, 4, 1>, <11231438, 1, 4, 1>, <9369949, 1, 4, 1>, <4531156, 1, 4, 1>, <11596801, 1, 4, 1>, <19892731, 1, 4, 1>, <194286, 1, 4, 1>, <13346200, 1, 4, 1>, <17158751, 1, 4, 1>, <18649052, 1, 4, 1>, <19865104, 1, 4, 1>, <2283723, 1, 4, 1>, <20021136, 1, 4, 1>, <12434465, 1, 4, 1>, <3079662, 1, 4, 1> @} Total time: 9.070 seconds, Total memory usage: 39.12MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 13:27:20 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^5*3^7; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; Height(P); HeightConstant(J:Effort:=20); //RationalPoints(J:Bound:=12); Chabauty(P,61); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 13:27:12 on modular [Seed = 2287866875] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 + 108, 18*x + 324, 2) ] 2.0602834384501336281543102556814931412814 10.2187885663225179085246161325220231662577 8.592593249917571773597305046262669462013 {@ <7404317, 1, 4, 1>, <11140375, 1, 4, 1>, <9649939, 1, 4, 1>, <7165503, 1, 4, 1>, <7103941, 1, 4, 1>, <1746578, 1, 4, 1>, <8125851, 1, 4, 1>, <8082873, 1, 4, 1>, <3826247, 1, 4, 1>, <511199, 1, 4, 1>, <6540532, 1, 4, 1>, <9013869, 1, 4, 1>, <11599251, 1, 4, 1>, <3294978, 1, 4, 1>, <58529, 1, 4, 1>, <2332353, 1, 4, 1>, <8852801, 1, 4, 1>, <3326549, 1, 4, 1>, <5261737, 1, 4, 1>, <9348995, 1, 4, 1>, <8457519, 1, 4, 1>, <245557, 1, 4, 1>, <6001473, 1, 4, 1>, <7770011, 1, 4, 1>, <7062636, 1, 4, 1>, <2777906, 1, 4, 1>, <1484683, 1, 4, 1>, <1439110, 1, 4, 1>, <2275404, 1, 4, 1>, <9047087, 1, 4, 1> @} Total time: 7.650 seconds, Total memory usage: 39.12MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 13:26:24 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^5*3^7; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; Height(P); HeightConstant(J:Effort:=20); //RationalPoints(J:Bound:=12); Chabauty(P,59); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 13:26:21 on modular [Seed = 2253394488] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 + 108, 18*x + 324, 2) ] 2.0602834384501336281543102556814931412814 10.2187885663225179085246161325220231662577 8.592593249917571773597305046262669462013 {@ <7214339, 1, 4, 1> @} Total time: 2.399 seconds, Total memory usage: 39.12MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 13:26:13 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^5*3^7; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; Height(P); HeightConstant(J:Effort:=20); //RationalPoints(J:Bound:=12); Chabauty(P,53); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 13:26:11 on modular [Seed = 2203391795] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 + 108, 18*x + 324, 2) ] 2.0602834384501336281543102556814931412814 10.2187885663225179085246161325220231662577 8.592593249917571773597305046262669462013 {@ <5277684, 1, 4, 1>, <4487334, 1, 4, 1>, <3155679, 1, 4, 1> @} Total time: 2.750 seconds, Total memory usage: 39.12MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 13:26:01 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^5*3^7; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; Height(P); HeightConstant(J:Effort:=20); //RationalPoints(J:Bound:=12); Chabauty(P,47); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 13:25:55 on modular [Seed = 2555251800] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 + 108, 18*x + 324, 2) ] 2.0602834384501336281543102556814931412814 10.2187885663225179085246161325220231662577 8.592593249917571773597305046262669462013 {@ <865641, 1, 4, 1>, <2039568, 1, 4, 1>, <335647, 1, 4, 1>, <4187170, 1, 4, 1>, <3965865, 1, 4, 1>, <3143254, 1, 4, 1>, <2527018, 1, 4, 1>, <0, 1, 4, 1>, <3041163, 1, 4, 1>, <2395431, 1, 4, 1>, <1358670, 1, 4, 1>, <349407, 1, 4, 1>, <3670730, 1, 4, 1>, <4339743, 1, 4, 1>, <1470315, 1, 4, 1>, <1904195, 1, 4, 1>, <3883581, 1, 4, 1>, <2555510, 1, 4, 1>, <3054728, 1, 4, 1>, <4230330, 1, 4, 1>, <1395649, 1, 4, 1>, <707559, 1, 4, 1> @} Total time: 6.089 seconds, Total memory usage: 39.12MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 13:25:46 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^5*3^7; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; Height(P); HeightConstant(J:Effort:=20); //RationalPoints(J:Bound:=12); Chabauty(P,43); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 13:25:41 on modular [Seed = 2504196419] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 + 108, 18*x + 324, 2) ] 2.0602834384501336281543102556814931412814 10.2187885663225179085246161325220231662577 8.592593249917571773597305046262669462013 {@ <84385, 1, 4, 1>, <1261711, 1, 4, 1>, <2763557, 1, 4, 1>, <351559, 1, 4, 1>, <1051948, 1, 4, 1>, <2904115, 1, 4, 1>, <956780, 1, 4, 1>, <3094272, 1, 4, 1>, <327840, 1, 4, 1>, <0, 1, 4, 1>, <1641952, 1, 4, 1>, <2424089, 1, 4, 1>, <152365, 1, 4, 1>, <1926920, 1, 4, 1>, <93440, 1, 4, 1>, <1368710, 1, 4, 1>, <1467527, 1, 4, 1>, <1661588, 1, 4, 1>, <1301014, 1, 4, 1>, <3117856, 1, 4, 1>, <2372065, 1, 4, 1> @} Total time: 5.389 seconds, Total memory usage: 39.12MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 13:25:34 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^5*3^7; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; Height(P); HeightConstant(J:Effort:=20); //RationalPoints(J:Bound:=12); Chabauty(P,41); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 13:25:31 on modular [Seed = 2453931642] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 + 108, 18*x + 324, 2) ] 2.0602834384501336281543102556814931412814 10.2187885663225179085246161325220231662577 8.592593249917571773597305046262669462013 {@ <1715400, 1, 4, 1>, <2614156, 1, 4, 1>, <2628725, 1, 4, 1>, <865815, 1, 4, 1>, <2608615, 1, 4, 1>, <2064431, 1, 4, 1>, <804, 1, 4, 1>, <1106307, 1, 4, 1>, <1734890, 1, 4, 1>, <1944940, 1, 4, 1> @} Total time: 3.490 seconds, Total memory usage: 39.12MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 13:25:23 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^5*3^7; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; Height(P); HeightConstant(J:Effort:=20); //RationalPoints(J:Bound:=12); Chabauty(P,5); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 13:25:21 on modular [Seed = 2943957837] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 + 108, 18*x + 324, 2) ] 2.0602834384501336281543102556814931412814 10.2187885663225179085246161325220231662577 8.592593249917571773597305046262669462013 >> Chabauty(P,5); ^ Runtime error in 'Chabauty': The curve must have good reduction at p. Total time: 1.629 seconds, Total memory usage: 39.12MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 13:25:14 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^5*3^7; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; Height(P); HeightConstant(J:Effort:=20); //RationalPoints(J:Bound:=12); Chabauty(P,37); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 13:25:09 on modular [Seed = 2758946898] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 + 108, 18*x + 324, 2) ] 2.0602834384501336281543102556814931412814 10.2187885663225179085246161325220231662577 8.592593249917571773597305046262669462013 {@ <273808, 1, 4, 1>, <615796, 1, 4, 1>, <1393398, 1, 4, 1>, <1321033, 1, 4, 1>, <1658043, 1, 4, 1>, <488625, 1, 4, 1>, <1611907, 1, 4, 1>, <174526, 1, 4, 1>, <869901, 1, 4, 1>, <910815, 1, 4, 1>, <1760878, 1, 4, 1>, <1240452, 1, 4, 1>, <585687, 1, 4, 1>, <892909, 1, 4, 1>, <1346326, 1, 4, 1>, <1673351, 1, 4, 1>, <1337456, 1, 4, 1> @} Total time: 5.019 seconds, Total memory usage: 39.12MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 13:25:03 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^5*3^7; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; Height(P); HeightConstant(J:Effort:=20); //RationalPoints(J:Bound:=12); Chabauty(P,31); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 13:25:00 on modular [Seed = 2708944302] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 + 108, 18*x + 324, 2) ] 2.0602834384501336281543102556814931412814 10.2187885663225179085246161325220231662577 8.592593249917571773597305046262669462013 {@ <190679, 1, 4, 1>, <383865, 1, 4, 1>, <39695, 1, 4, 1>, <556144, 1, 4, 1>, <217957, 1, 4, 1>, <920606, 1, 4, 1>, <350099, 1, 4, 1>, <436388, 1, 4, 1>, <650884, 1, 4, 1> @} Total time: 3.419 seconds, Total memory usage: 39.12MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 13:23:55 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^5*3^7; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; Height(P); HeightConstant(J:Effort:=20); //RationalPoints(J:Bound:=12); Chabauty(P,29); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 13:23:53 on modular [Seed = 3194497780] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 + 108, 18*x + 324, 2) ] 2.0602834384501336281543102556814931412814 10.2187885663225179085246161325220231662577 8.592593249917571773597305046262669462013 {@ <0, 1, 4, 1> @} Total time: 2.250 seconds, Total memory usage: 39.12MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 13:23:47 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^5*3^7; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; Height(P); HeightConstant(J:Effort:=20); //RationalPoints(J:Bound:=12); Chabauty(P,23); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 13:23:44 on modular [Seed = 3144495096] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 + 108, 18*x + 324, 2) ] 2.0602834384501336281543102556814931412814 10.2187885663225179085246161325220231662577 8.592593249917571773597305046262669462013 {@ <49292, 1, 4, 1>, <174996, 1, 4, 1>, <222417, 1, 4, 1>, <259938, 1, 4, 1>, <150597, 1, 4, 1>, <174407, 1, 4, 1>, <196353, 1, 4, 1>, <181489, 1, 4, 1>, <0, 1, 4, 1>, <113422, 1, 4, 1> @} Total time: 3.540 seconds, Total memory usage: 39.12MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 13:23:34 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^5*3^7; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; Height(P); HeightConstant(J:Effort:=20); //RationalPoints(J:Bound:=12); Chabauty(P,19); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 13:23:32 on modular [Seed = 2959484135] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 + 108, 18*x + 324, 2) ] 2.0602834384501336281543102556814931412814 10.2187885663225179085246161325220231662577 8.592593249917571773597305046262669462013 {@ <5335, 1, 4, 1> @} Total time: 2.120 seconds, Total memory usage: 39.12MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 13:23:25 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^5*3^7; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; Height(P); HeightConstant(J:Effort:=20); //RationalPoints(J:Bound:=12); Chabauty(P,17); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 13:23:22 on modular [Seed = 3441121700] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 + 108, 18*x + 324, 2) ] 2.0602834384501336281543102556814931412814 10.2187885663225179085246161325220231662577 8.592593249917571773597305046262669462013 {@ <45158, 1, 4, 1>, <18984, 1, 4, 1>, <16432, 1, 4, 1>, <71595, 1, 4, 1>, <37302, 1, 4, 1>, <9848, 1, 4, 1>, <20928, 1, 4, 1> @} Total time: 2.960 seconds, Total memory usage: 39.12MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 13:23:17 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^5*3^7; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; Height(P); HeightConstant(J:Effort:=20); //RationalPoints(J:Bound:=12); Chabauty(P,13); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 13:23:14 on modular [Seed = 3390856875] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 + 108, 18*x + 324, 2) ] 2.0602834384501336281543102556814931412814 10.2187885663225179085246161325220231662577 8.592593249917571773597305046262669462013 {@ <3374, 1, 4, 1>, <11705, 1, 4, 1>, <4444, 1, 4, 1>, <4094, 1, 4, 1>, <2759, 1, 4, 1>, <20875, 1, 4, 1> @} Total time: 2.520 seconds, Total memory usage: 39.12MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 13:23:09 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^5*3^7; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; Height(P); HeightConstant(J:Effort:=20); //RationalPoints(J:Bound:=12); Chabauty(P,11); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 13:23:06 on modular [Seed = 3339801534] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 + 108, 18*x + 324, 2) ] 2.0602834384501336281543102556814931412814 10.2187885663225179085246161325220231662577 8.592593249917571773597305046262669462013 {@ <8143, 1, 4, 1>, <12281, 1, 4, 1>, <8787, 1, 4, 1>, <1619, 1, 4, 1>, <11906, 1, 4, 1>, <2329, 1, 4, 1>, <340, 1, 4, 1>, <10996, 1, 4, 1>, <12684, 1, 4, 1> @} Total time: 2.970 seconds, Total memory usage: 39.12MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 13:23:01 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^5*3^7; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; Height(P); HeightConstant(J:Effort:=20); //RationalPoints(J:Bound:=12); Chabauty(P,7); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 13:22:58 on modular [Seed = 3725085359] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 + 108, 18*x + 324, 2) ] 2.0602834384501336281543102556814931412814 10.2187885663225179085246161325220231662577 8.592593249917571773597305046262669462013 {@ <1864, 1, 4, 1>, <138, 1, 4, 1>, <2204, 1, 4, 1> @} Total time: 2.120 seconds, Total memory usage: 39.12MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 13:20:25 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^5*3^7; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; Height(P); HeightConstant(J:Effort:=20); //RationalPoints(J:Bound:=12); //Chabauty(P,19); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 13:20:23 on modular [Seed = 3556917696] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 + 108, 18*x + 324, 2) ] 2.0602834384501336281543102556814931412814 10.2187885663225179085246161325220231662577 8.592593249917571773597305046262669462013 Total time: 1.620 seconds, Total memory usage: 39.12MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 13:09:56 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^5*3^3; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; Height(P); HeightConstant(J:Effort:=20); //RationalPoints(J:Bound:=12); Chabauty(P,19); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 13:09:54 on modular [Seed = 3996679791] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 + 7/4*x + 5/2, 1/8*x - 117/4, 2) ] 4.9453098449393330988475861193843410859111152078536662240177 6.851601072134541759765248377825164707390 5.662960480135945929876651081135934249649 {@ @} Total time: 1.929 seconds, Total memory usage: 39.12MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 13:09:49 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^5*3^3; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; Height(P); HeightConstant(J:Effort:=20); //RationalPoints(J:Bound:=12); Chabauty(P,17); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 13:09:45 on modular [Seed = 3946675057] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 + 7/4*x + 5/2, 1/8*x - 117/4, 2) ] 4.9453098449393330988475861193843410859111152078536662240177 6.851601072134541759765248377825164707390 5.662960480135945929876651081135934249649 {@ <24622, 1, 4, 1>, <66269, 1, 4, 1>, <8671, 1, 4, 1>, <51446, 1, 4, 1>, <48474, 1, 4, 1>, <76752, 1, 4, 1>, <14483, 1, 4, 1>, <81071, 1, 4, 1> @} Total time: 2.960 seconds, Total memory usage: 39.12MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 13:09:40 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^5*3^3; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; Height(P); HeightConstant(J:Effort:=20); //RationalPoints(J:Bound:=12); Chabauty(P,13); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 13:09:37 on modular [Seed = 3761664114] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 + 7/4*x + 5/2, 1/8*x - 117/4, 2) ] 4.9453098449393330988475861193843410859111152078536662240177 6.851601072134541759765248377825164707390 5.662960480135945929876651081135934249649 {@ <16291, 1, 4, 1>, <19040, 1, 4, 1>, <2542, 1, 4, 1>, <20245, 1, 4, 1>, <11372, 1, 4, 1> @} Total time: 2.379 seconds, Total memory usage: 39.12MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 13:09:32 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^5*3^3; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; Height(P); HeightConstant(J:Effort:=20); //RationalPoints(J:Bound:=12); Chabauty(P,11); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 13:09:30 on modular [Seed = 4247481684] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 + 7/4*x + 5/2, 1/8*x - 117/4, 2) ] 4.9453098449393330988475861193843410859111152078536662240177 6.851601072134541759765248377825164707390 5.662960480135945929876651081135934249649 {@ <10911, 1, 4, 1>, <7542, 1, 4, 1>, <11767, 1, 4, 1>, <2646, 1, 4, 1>, <2070, 1, 4, 1> @} Total time: 2.359 seconds, Total memory usage: 39.12MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 13:09:24 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^5*3^3; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1500); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; Height(P); HeightConstant(J:Effort:=20); //RationalPoints(J:Bound:=12); Chabauty(P,7); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 13:09:22 on modular [Seed = 4197214811] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 + 7/4*x + 5/2, 1/8*x - 117/4, 2) ] 4.9453098449393330988475861193843410859111152078536662240177 6.851601072134541759765248377825164707390 5.662960480135945929876651081135934249649 {@ <1935, 1, 4, 1>, <2274, 1, 4, 1>, <540, 1, 4, 1> @} Total time: 2.000 seconds, Total memory usage: 39.12MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 13:08:20 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^5*3^3; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=4000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; Height(P); HeightConstant(J:Effort:=20); //RationalPoints(J:Bound:=12); //Chabauty(P1,11); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 13:08:15 on modular [Seed = 199072759] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 + 7/4*x + 5/2, 1/8*x - 117/4, 2) ] 4.9453098449393330988475861193843410859111152078536662240177 6.851601072134541759765248377825164707390 5.662960480135945929876651081135934249649 Total time: 4.120 seconds, Total memory usage: 39.12MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 13:06:21 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^5*3^3; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1400); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; Height(P); HeightConstant(J:Effort:=20); //RationalPoints(J:Bound:=12); //Chabauty(P1,11); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 13:06:19 on modular [Seed = 64328555] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 + 7/4*x + 5/2, 1/8*x - 117/4, 2) ] 4.9453098449393330988475861193843410859111152078536662240177 6.851601072134541759765248377825164707390 5.662960480135945929876651081135934249649 Total time: 1.480 seconds, Total memory usage: 39.12MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 13:06:11 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^5*3^3; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1400); B:=ReducedBasis(V); B; //P:=J!B[1]; //Q:=J![x+2,0]; //P1:=J![x-2,8]; Height(P); HeightConstant(J:Effort:=20); //RationalPoints(J:Bound:=12); //Chabauty(P1,11); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 13:06:09 on modular [Seed = 416188538] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 + 7/4*x + 5/2, 1/8*x - 117/4, 2) ] >> Height(P); ^ User error: Identifier 'P' has not been declared or assigned 6.851601072134541759765248377825164707390 5.662960480135945929876651081135934249649 Total time: 1.449 seconds, Total memory usage: 39.12MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 13:02:25 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^5*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1400); B:=ReducedBasis(V); B; //P:=J!B[1]; //Q:=J![x+2,0]; P1:=J![x-2,8]; Height(P1); HeightConstant(J:Effort:=20); //RationalPoints(J:Bound:=12); Chabauty(P1,11); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 13:02:22 on modular [Seed = 365134920] ------------------------------------- 1 Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 to JacHyp: J given by a rule [no inverse] [ (x^2 - 4, 2*x + 4, 2) ] 0.9176949999123136287095316694682973525170029825697109902393 4.486499277983286154053028928413021210016 3.465735902799726547086160607290882840364 {@ <2, 1, 5, 1>, <32702, 1, 5, 1> @} Total time: 2.229 seconds, Total memory usage: 39.58MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 13:02:14 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^5*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1400); B:=ReducedBasis(V); B; //P:=J!B[1]; //Q:=J![x+2,0]; P1:=J![x-2,8]; Height(P1); HeightConstant(J:Effort:=20); //RationalPoints(J:Bound:=12); Chabauty(P1,101); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 13:02:10 on modular [Seed = 314868547] ------------------------------------- 1 Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 to JacHyp: J given by a rule [no inverse] [ (x^2 - 4, 2*x + 4, 2) ] 0.9176949999123136287095316694682973525170029825697109902393 4.486499277983286154053028928413021210016 3.465735902799726547086160607290882840364 {@ <27012122, 1, 4, 1>, <56559795, 1, 4, 1>, <99616790, 1, 4, 1>, <28246568, 1, 4, 1>, <30642484, 1, 4, 1>, <40333758, 1, 4, 1>, <9564867, 1, 4, 1>, <38013215, 1, 4, 1>, <2, 1, 4, 1>, <43796927, 1, 4, 1>, <9774641, 1, 4, 1>, <38047577, 1, 4, 1> @} Total time: 4.400 seconds, Total memory usage: 39.58MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 13:02:01 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^5*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1400); B:=ReducedBasis(V); B; //P:=J!B[1]; //Q:=J![x+2,0]; P1:=J![x-2,8]; Height(P1); HeightConstant(J:Effort:=20); //RationalPoints(J:Bound:=12); Chabauty(P1,97); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 13:01:47 on modular [Seed = 687782224] ------------------------------------- 1 Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 to JacHyp: J given by a rule [no inverse] [ (x^2 - 4, 2*x + 4, 2) ] 0.9176949999123136287095316694682973525170029825697109902393 4.486499277983286154053028928413021210016 3.465735902799726547086160607290882840364 {@ <2, 1, 4, 1>, <71795493, 1, 4, 1>, <36374146, 1, 4, 1>, <45174985, 1, 4, 1>, <36345198, 1, 4, 1>, <56254183, 1, 4, 1>, <2717480, 1, 4, 1>, <85025356, 1, 4, 1>, <78179222, 1, 4, 1>, <4109767, 1, 4, 1>, <46528428, 1, 4, 1>, <66044578, 1, 4, 1>, <17744430, 1, 4, 1>, <74555086, 1, 4, 1>, <82216558, 1, 4, 1>, <83103622, 1, 4, 1>, <39892898, 1, 4, 1>, <26067587, 1, 4, 1>, <66094937, 1, 4, 1>, <34142239, 1, 4, 1>, <68850693, 1, 4, 1>, <45456511, 1, 4, 1>, <74967757, 1, 4, 1>, <0, 1, 4, 1>, <65450611, 1, 4, 1>, <6021947, 1, 4, 1>, <64373942, 1, 4, 1>, <7256233, 1, 4, 1>, <50207547, 1, 4, 1>, <48021213, 1, 4, 1>, <17290430, 1, 4, 1>, <77710172, 1, 4, 1>, <41538690, 1, 4, 1>, <51844808, 1, 4, 1>, <11365604, 1, 4, 1>, <4724195, 1, 4, 1>, <62726853, 1, 4, 1>, <32037834, 1, 4, 1>, <10981780, 1, 4, 1>, <63646517, 1, 4, 1>, <37558129, 1, 4, 1>, <17899769, 1, 4, 1>, <4567421, 1, 4, 1>, <23122583, 1, 4, 1>, <26809605, 1, 4, 1>, <56574871, 1, 4, 1>, <14718989, 1, 4, 1> @} Total time: 13.300 seconds, Total memory usage: 39.58MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 13:01:28 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^5*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1400); B:=ReducedBasis(V); B; //P:=J!B[1]; //Q:=J![x+2,0]; P1:=J![x-2,8]; Height(P1); HeightConstant(J:Effort:=20); //RationalPoints(J:Bound:=12); Chabauty(P1,89); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 13:01:25 on modular [Seed = 636729251] ------------------------------------- 1 Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 to JacHyp: J given by a rule [no inverse] [ (x^2 - 4, 2*x + 4, 2) ] 0.9176949999123136287095316694682973525170029825697109902393 4.486499277983286154053028928413021210016 3.465735902799726547086160607290882840364 {@ <2, 1, 4, 1>, <24247596, 1, 4, 1> @} Total time: 2.839 seconds, Total memory usage: 39.58MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 13:01:12 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^5*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1400); B:=ReducedBasis(V); B; //P:=J!B[1]; //Q:=J![x+2,0]; P1:=J![x-2,8]; Height(P1); HeightConstant(J:Effort:=20); //RationalPoints(J:Bound:=12); Chabauty(P1,83); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 13:01:01 on modular [Seed = 586461324] ------------------------------------- 1 Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 to JacHyp: J given by a rule [no inverse] [ (x^2 - 4, 2*x + 4, 2) ] 0.9176949999123136287095316694682973525170029825697109902393 4.486499277983286154053028928413021210016 3.465735902799726547086160607290882840364 {@ <2, 1, 4, 1>, <2021557, 1, 4, 1>, <8664646, 1, 4, 1>, <11863020, 1, 4, 1>, <29433056, 1, 4, 1>, <13807776, 1, 4, 1>, <13089905, 1, 4, 1>, <38676359, 1, 4, 1>, <16496936, 1, 4, 1>, <7688636, 1, 4, 1>, <19040011, 1, 4, 1>, <20888026, 1, 4, 1>, <21039143, 1, 4, 1>, <6866840, 1, 4, 1>, <26238789, 1, 4, 1>, <7273001, 1, 4, 1>, <28512238, 1, 4, 1>, <30588134, 1, 4, 1>, <15681343, 1, 4, 1>, <34879780, 1, 4, 1>, <10256907, 1, 4, 1>, <22381330, 1, 4, 1>, <3506172, 1, 4, 1>, <17966338, 1, 4, 1>, <17338558, 1, 4, 1>, <21725622, 1, 4, 1>, <25899387, 1, 4, 1>, <330457, 1, 4, 1>, <46188223, 1, 4, 1>, <39542733, 1, 4, 1>, <37416054, 1, 4, 1>, <6578097, 1, 4, 1>, <7209682, 1, 4, 1>, <21209936, 1, 4, 1>, <22232558, 1, 4, 1>, <31086576, 1, 4, 1>, <12403862, 1, 4, 1>, <25319803, 1, 4, 1>, <19001234, 1, 4, 1>, <20288888, 1, 4, 1> @} Total time: 10.900 seconds, Total memory usage: 39.58MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 13:00:46 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^5*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1400); B:=ReducedBasis(V); B; //P:=J!B[1]; //Q:=J![x+2,0]; P1:=J![x-2,8]; Height(P1); HeightConstant(J:Effort:=20); //RationalPoints(J:Bound:=12); Chabauty(P1,73); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 13:00:36 on modular [Seed = 1072278920] ------------------------------------- 1 Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 to JacHyp: J given by a rule [no inverse] [ (x^2 - 4, 2*x + 4, 2) ] 0.9176949999123136287095316694682973525170029825697109902393 4.486499277983286154053028928413021210016 3.465735902799726547086160607290882840364 {@ <2, 1, 4, 1>, <10480108, 1, 4, 1>, <1286378, 1, 4, 1>, <10037775, 1, 4, 1>, <19888360, 1, 4, 1>, <13103320, 1, 4, 1>, <9262653, 1, 4, 1>, <4447586, 1, 4, 1>, <24792296, 1, 4, 1>, <21057293, 1, 4, 1>, <7770948, 1, 4, 1>, <28231748, 1, 4, 1>, <2452023, 1, 4, 1>, <25084182, 1, 4, 1>, <11151608, 1, 4, 1>, <10185541, 1, 4, 1>, <20878892, 1, 4, 1>, <22213178, 1, 4, 1>, <7446635, 1, 4, 1>, <26164840, 1, 4, 1>, <6523878, 1, 4, 1>, <7365467, 1, 4, 1>, <4805262, 1, 4, 1>, <17879037, 1, 4, 1>, <20681855, 1, 4, 1>, <9239797, 1, 4, 1>, <23016988, 1, 4, 1>, <11429561, 1, 4, 1>, <4417018, 1, 4, 1>, <21422353, 1, 4, 1>, <0, 1, 4, 1>, <27643269, 1, 4, 1>, <49607, 1, 4, 1>, <14177976, 1, 4, 1>, <14060588, 1, 4, 1>, <25094947, 1, 4, 1> @} Total time: 9.230 seconds, Total memory usage: 39.58MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 13:00:25 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^5*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1400); B:=ReducedBasis(V); B; //P:=J!B[1]; //Q:=J![x+2,0]; P1:=J![x-2,8]; Height(P1); HeightConstant(J:Effort:=20); //RationalPoints(J:Bound:=12); Chabauty(P1,79); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 13:00:22 on modular [Seed = 887269105] ------------------------------------- 1 Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 to JacHyp: J given by a rule [no inverse] [ (x^2 - 4, 2*x + 4, 2) ] 0.9176949999123136287095316694682973525170029825697109902393 4.486499277983286154053028928413021210016 3.465735902799726547086160607290882840364 {@ <2, 1, 4, 1>, <12329804, 1, 4, 1> @} Total time: 2.879 seconds, Total memory usage: 39.58MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 13:00:12 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^5*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1400); B:=ReducedBasis(V); B; //P:=J!B[1]; //Q:=J![x+2,0]; P1:=J![x-2,8]; Height(P1); HeightConstant(J:Effort:=20); //RationalPoints(J:Bound:=12); Chabauty(P1,73); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 13:00:03 on modular [Seed = 853844194] ------------------------------------- 1 Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 to JacHyp: J given by a rule [no inverse] [ (x^2 - 4, 2*x + 4, 2) ] 0.9176949999123136287095316694682973525170029825697109902393 4.486499277983286154053028928413021210016 3.465735902799726547086160607290882840364 {@ <2, 1, 4, 1>, <10480108, 1, 4, 1>, <1286378, 1, 4, 1>, <10037775, 1, 4, 1>, <19888360, 1, 4, 1>, <13103320, 1, 4, 1>, <9262653, 1, 4, 1>, <4447586, 1, 4, 1>, <24792296, 1, 4, 1>, <21057293, 1, 4, 1>, <7770948, 1, 4, 1>, <28231748, 1, 4, 1>, <2452023, 1, 4, 1>, <25084182, 1, 4, 1>, <11151608, 1, 4, 1>, <10185541, 1, 4, 1>, <20878892, 1, 4, 1>, <22213178, 1, 4, 1>, <7446635, 1, 4, 1>, <26164840, 1, 4, 1>, <6523878, 1, 4, 1>, <7365467, 1, 4, 1>, <4805262, 1, 4, 1>, <17879037, 1, 4, 1>, <20681855, 1, 4, 1>, <9239797, 1, 4, 1>, <23016988, 1, 4, 1>, <11429561, 1, 4, 1>, <4417018, 1, 4, 1>, <21422353, 1, 4, 1>, <0, 1, 4, 1>, <27643269, 1, 4, 1>, <49607, 1, 4, 1>, <14177976, 1, 4, 1>, <14060588, 1, 4, 1>, <25094947, 1, 4, 1> @} Total time: 9.230 seconds, Total memory usage: 39.58MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 12:59:49 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^5*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1400); B:=ReducedBasis(V); B; //P:=J!B[1]; //Q:=J![x+2,0]; P1:=J![x-2,8]; Height(P1); HeightConstant(J:Effort:=20); //RationalPoints(J:Bound:=12); Chabauty(P1,71); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 12:59:41 on modular [Seed = 1218370230] ------------------------------------- 1 Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 to JacHyp: J given by a rule [no inverse] [ (x^2 - 4, 2*x + 4, 2) ] 0.9176949999123136287095316694682973525170029825697109902393 4.486499277983286154053028928413021210016 3.465735902799726547086160607290882840364 {@ <2, 1, 4, 1>, <1413031, 1, 4, 1>, <5185722, 1, 4, 1>, <19457557, 1, 4, 1>, <7357455, 1, 4, 1>, <21138811, 1, 4, 1>, <21842205, 1, 4, 1>, <7523086, 1, 4, 1>, <16168209, 1, 4, 1>, <13862217, 1, 4, 1>, <14310106, 1, 4, 1>, <21052861, 1, 4, 1>, <7717318, 1, 4, 1>, <2309298, 1, 4, 1>, <10027565, 1, 4, 1>, <0, 1, 4, 1>, <4955708, 1, 4, 1>, <20492067, 1, 4, 1>, <3555447, 1, 4, 1>, <14780959, 1, 4, 1>, <18964423, 1, 4, 1>, <11674927, 1, 4, 1>, <14548571, 1, 4, 1>, <14945119, 1, 4, 1>, <11773912, 1, 4, 1>, <8971958, 1, 4, 1>, <12730745, 1, 4, 1>, <15502653, 1, 4, 1>, <836768, 1, 4, 1>, <15949237, 1, 4, 1>, <17632988, 1, 4, 1> @} Total time: 7.469 seconds, Total memory usage: 39.58MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 12:59:30 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^5*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1400); B:=ReducedBasis(V); B; //P:=J!B[1]; //Q:=J![x+2,0]; P1:=J![x-2,8]; Height(P1); HeightConstant(J:Effort:=20); //RationalPoints(J:Bound:=12); Chabauty(P1,67); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 12:59:21 on modular [Seed = 1167316815] ------------------------------------- 1 Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 to JacHyp: J given by a rule [no inverse] [ (x^2 - 4, 2*x + 4, 2) ] 0.9176949999123136287095316694682973525170029825697109902393 4.486499277983286154053028928413021210016 3.465735902799726547086160607290882840364 {@ <2, 1, 4, 1>, <7699685, 1, 4, 1>, <13602056, 1, 4, 1>, <4610343, 1, 4, 1>, <5459277, 1, 4, 1>, <9612913, 1, 4, 1>, <4606819, 1, 4, 1>, <8326665, 1, 4, 1>, <17870986, 1, 4, 1>, <792199, 1, 4, 1>, <13558230, 1, 4, 1>, <13335759, 1, 4, 1>, <17055999, 1, 4, 1>, <12810586, 1, 4, 1>, <2829083, 1, 4, 1>, <12618534, 1, 4, 1>, <15563003, 1, 4, 1>, <6806210, 1, 4, 1>, <2433608, 1, 4, 1>, <16723006, 1, 4, 1>, <14449298, 1, 4, 1>, <12219126, 1, 4, 1>, <18183838, 1, 4, 1>, <2081372, 1, 4, 1>, <5134728, 1, 4, 1>, <12447858, 1, 4, 1>, <5318474, 1, 4, 1>, <2860131, 1, 4, 1>, <11339111, 1, 4, 1>, <17572280, 1, 4, 1>, <3109435, 1, 4, 1>, <15440538, 1, 4, 1>, <7892848, 1, 4, 1> @} Total time: 8.980 seconds, Total memory usage: 39.58MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 12:59:15 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^5*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1400); B:=ReducedBasis(V); B; //P:=J!B[1]; //Q:=J![x+2,0]; P1:=J![x-2,8]; Height(P1); HeightConstant(J:Effort:=20); //RationalPoints(J:Bound:=12); Chabauty(P1,61); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 12:59:10 on modular [Seed = 1117048927] ------------------------------------- 1 Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 to JacHyp: J given by a rule [no inverse] [ (x^2 - 4, 2*x + 4, 2) ] 0.9176949999123136287095316694682973525170029825697109902393 4.486499277983286154053028928413021210016 3.465735902799726547086160607290882840364 {@ <2, 1, 4, 1>, <6115384, 1, 4, 1>, <4897665, 1, 4, 1>, <11935065, 1, 4, 1>, <13385356, 1, 4, 1>, <107975, 1, 4, 1>, <13692166, 1, 4, 1>, <7468882, 1, 4, 1>, <449555, 1, 4, 1>, <12105611, 1, 4, 1>, <682609, 1, 4, 1>, <4407031, 1, 4, 1>, <2694933, 1, 4, 1>, <2404111, 1, 4, 1>, <3452130, 1, 4, 1>, <12998416, 1, 4, 1>, <12047364, 1, 4, 1>, <12416374, 1, 4, 1>, <7869665, 1, 4, 1> @} Total time: 5.059 seconds, Total memory usage: 39.58MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 12:58:57 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^5*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1400); B:=ReducedBasis(V); B; //P:=J!B[1]; //Q:=J![x+2,0]; P1:=J![x-2,8]; Height(P1); HeightConstant(J:Effort:=20); //RationalPoints(J:Bound:=12); Chabauty(P1,59); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 12:58:54 on modular [Seed = 1602866547] ------------------------------------- 1 Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 to JacHyp: J given by a rule [no inverse] [ (x^2 - 4, 2*x + 4, 2) ] 0.9176949999123136287095316694682973525170029825697109902393 4.486499277983286154053028928413021210016 3.465735902799726547086160607290882840364 {@ <2, 1, 4, 1>, <4543845, 1, 4, 1> @} Total time: 2.740 seconds, Total memory usage: 39.58MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 12:58:40 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^5*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1400); B:=ReducedBasis(V); B; //P:=J!B[1]; //Q:=J![x+2,0]; P1:=J![x-2,8]; Height(P1); HeightConstant(J:Effort:=20); //RationalPoints(J:Bound:=12); Chabauty(P1,53); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 12:58:34 on modular [Seed = 1417856621] ------------------------------------- 1 Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 to JacHyp: J given by a rule [no inverse] [ (x^2 - 4, 2*x + 4, 2) ] 0.9176949999123136287095316694682973525170029825697109902393 4.486499277983286154053028928413021210016 3.465735902799726547086160607290882840364 {@ <2, 1, 4, 1>, <5120360, 1, 4, 1>, <4704672, 1, 4, 1>, <406634, 1, 4, 1>, <6031745, 1, 4, 1>, <1736850, 1, 4, 1>, <4614562, 1, 4, 1>, <3065420, 1, 4, 1>, <1669026, 1, 4, 1>, <5352906, 1, 4, 1>, <1574013, 1, 4, 1>, <440055, 1, 4, 1>, <5893703, 1, 4, 1>, <1807695, 1, 4, 1>, <3672395, 1, 4, 1>, <7409625, 1, 4, 1>, <1793342, 1, 4, 1>, <288960, 1, 4, 1>, <4301686, 1, 4, 1>, <2796213, 1, 4, 1>, <3524899, 1, 4, 1>, <4685631, 1, 4, 1>, <7093616, 1, 4, 1>, <476980, 1, 4, 1>, <5546670, 1, 4, 1>, <6073579, 1, 4, 1> @} Total time: 6.719 seconds, Total memory usage: 39.58MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 12:58:25 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^5*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1400); B:=ReducedBasis(V); B; //P:=J!B[1]; //Q:=J![x+2,0]; P1:=J![x-2,8]; Height(P1); HeightConstant(J:Effort:=20); //RationalPoints(J:Bound:=12); Chabauty(P1,47); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 12:58:19 on modular [Seed = 1367851801] ------------------------------------- 1 Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 to JacHyp: J given by a rule [no inverse] [ (x^2 - 4, 2*x + 4, 2) ] 0.9176949999123136287095316694682973525170029825697109902393 4.486499277983286154053028928413021210016 3.465735902799726547086160607290882840364 {@ <2, 1, 4, 1>, <17975, 1, 4, 1>, <782586, 1, 4, 1>, <1600382, 1, 4, 1>, <2115842, 1, 4, 1>, <2717370, 1, 4, 1>, <1231405, 1, 4, 1>, <1632457, 1, 4, 1>, <4793661, 1, 4, 1>, <100360, 1, 4, 1>, <3138765, 1, 4, 1>, <1590302, 1, 4, 1>, <3086344, 1, 4, 1>, <1220364, 1, 4, 1>, <3822439, 1, 4, 1>, <1218985, 1, 4, 1>, <2915915, 1, 4, 1>, <2441318, 1, 4, 1>, <3667531, 1, 4, 1>, <0, 1, 4, 1>, <2754918, 1, 4, 1>, <160337, 1, 4, 1>, <3166042, 1, 4, 1> @} Total time: 6.179 seconds, Total memory usage: 39.58MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 12:58:11 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^5*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1400); B:=ReducedBasis(V); B; //P:=J!B[1]; //Q:=J![x+2,0]; P1:=J![x-2,8]; Height(P1); HeightConstant(J:Effort:=20); //RationalPoints(J:Bound:=12); Chabauty(P1,43); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 12:58:07 on modular [Seed = 1857616946] ------------------------------------- 1 Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 to JacHyp: J given by a rule [no inverse] [ (x^2 - 4, 2*x + 4, 2) ] 0.9176949999123136287095316694682973525170029825697109902393 4.486499277983286154053028928413021210016 3.465735902799726547086160607290882840364 {@ <2, 1, 4, 1>, <2511769, 1, 4, 1>, <1801855, 1, 4, 1>, <3062749, 1, 4, 1>, <1636020, 1, 4, 1>, <1528725, 1, 4, 1>, <1530864, 1, 4, 1> @} Total time: 3.379 seconds, Total memory usage: 39.58MB '146.6.1' ************** MAGMA ***************** Host 146.6.139.217 (146.6.139.217) Time: Tue Dec 6 12:58:01 2005 Input: count:=0; m:=6; q:=2^m; F1:=GF(q); F2:=GF(q^2); a:=F1.1; b:=F1.1^2; a1:=F1.1^55; a2:=F1.1^59; P:=PolynomialRing(F1); u:=x^2+F1.1^27*x+F1.1^51; v:=F1.1^39+F1.1^59; f:=a*x^5+b*x^3+x; v^2+x*v-f; v^2+x*v-f mod u; Roots(x^2-t); P:=PolynomialRing(F1); C:=HyperellipticCurve(a*x^5+b*x^3+x,x); N1:=#C; //over Fq R:=PolynomialRing(F2); CC:=HyperellipticCurve(a*z^5+b*z^3+z,z); N2:=#CC; //over Fq^2; print "done"; Output: Magma V2.11-10 Tue Dec 6 2005 12:57:59 on modular [Seed = 1706293041] ------------------------------------- F1.1*x^5 + F1.1^2*x^3 + F1.1^38*x + F1.1^3 F1.1^5*x + F1.1^59 >> Roots(x^2-t); ^ User error: Identifier 't' has not been declared or assigned done Total time: 0.200 seconds, Total memory usage: 3.43MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 12:58:01 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^5*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1400); B:=ReducedBasis(V); B; //P:=J!B[1]; //Q:=J![x+2,0]; P1:=J![x-2,8]; Height(P1); HeightConstant(J:Effort:=20); //RationalPoints(J:Bound:=12); Chabauty(P1,41); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 12:57:55 on modular [Seed = 1622602288] ------------------------------------- 1 Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 to JacHyp: J given by a rule [no inverse] [ (x^2 - 4, 2*x + 4, 2) ] 0.9176949999123136287095316694682973525170029825697109902393 4.486499277983286154053028928413021210016 3.465735902799726547086160607290882840364 {@ <2, 1, 4, 1>, <1827852, 1, 4, 1>, <1930548, 1, 4, 1>, <1154948, 1, 4, 1>, <1100631, 1, 4, 1>, <1621336, 1, 4, 1>, <1974720, 1, 4, 1>, <2653637, 1, 4, 1>, <2812782, 1, 4, 1>, <2504748, 1, 4, 1>, <889326, 1, 4, 1>, <0, 1, 4, 1>, <231922, 1, 4, 1>, <2712831, 1, 4, 1>, <531580, 1, 4, 1>, <1132337, 1, 4, 1>, <2266332, 1, 4, 1>, <2492533, 1, 4, 1>, <2739753, 1, 4, 1>, <2680070, 1, 4, 1>, <2382434, 1, 4, 1>, <367077, 1, 4, 1>, <398225, 1, 4, 1>, <2304265, 1, 4, 1>, <1572938, 1, 4, 1> @} Total time: 5.839 seconds, Total memory usage: 39.58MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 12:57:45 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^5*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1400); B:=ReducedBasis(V); B; //P:=J!B[1]; //Q:=J![x+2,0]; P1:=J![x-2,8]; Height(P1); HeightConstant(J:Effort:=20); //RationalPoints(J:Bound:=12); Chabauty(P1,37); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 12:57:40 on modular [Seed = 2108419866] ------------------------------------- 1 Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 to JacHyp: J given by a rule [no inverse] [ (x^2 - 4, 2*x + 4, 2) ] 0.9176949999123136287095316694682973525170029825697109902393 4.486499277983286154053028928413021210016 3.465735902799726547086160607290882840364 {@ <2, 1, 4, 1>, <219670, 1, 4, 1>, <527290, 1, 4, 1>, <58100, 1, 4, 1>, <558655, 1, 4, 1>, <966095, 1, 4, 1>, <239803, 1, 4, 1>, <1271528, 1, 4, 1>, <768807, 1, 4, 1>, <1233186, 1, 4, 1>, <868458, 1, 4, 1>, <251903, 1, 4, 1>, <97833, 1, 4, 1>, <734224, 1, 4, 1>, <479277, 1, 4, 1>, <1958, 1, 4, 1>, <651779, 1, 4, 1>, <626724, 1, 4, 1> @} Total time: 5.139 seconds, Total memory usage: 39.58MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 12:57:33 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^5*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1400); B:=ReducedBasis(V); B; //P:=J!B[1]; //Q:=J![x+2,0]; P1:=J![x-2,8]; Height(P1); HeightConstant(J:Effort:=20); //RationalPoints(J:Bound:=12); Chabauty(P1,31); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 12:57:29 on modular [Seed = 2058151936] ------------------------------------- 1 Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 to JacHyp: J given by a rule [no inverse] [ (x^2 - 4, 2*x + 4, 2) ] 0.9176949999123136287095316694682973525170029825697109902393 4.486499277983286154053028928413021210016 3.465735902799726547086160607290882840364 {@ <2, 1, 4, 1>, <291824, 1, 4, 1>, <651910, 1, 4, 1>, <30464, 1, 4, 1>, <547473, 1, 4, 1>, <822159, 1, 4, 1>, <876805, 1, 4, 1>, <696689, 1, 4, 1>, <0, 1, 4, 1>, <715713, 1, 4, 1>, <48457, 1, 4, 1> @} Total time: 3.560 seconds, Total memory usage: 39.58MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 12:57:13 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^5*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1400); B:=ReducedBasis(V); B; //P:=J!B[1]; //Q:=J![x+2,0]; P1:=J![x-2,8]; Height(P1); HeightConstant(J:Effort:=20); //RationalPoints(J:Bound:=12); Chabauty(P1,29); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 12:57:11 on modular [Seed = 2487355151] ------------------------------------- 1 Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 to JacHyp: J given by a rule [no inverse] [ (x^2 - 4, 2*x + 4, 2) ] 0.9176949999123136287095316694682973525170029825697109902393 4.486499277983286154053028928413021210016 3.465735902799726547086160607290882840364 {@ <2, 1, 4, 1>, <468048, 1, 4, 1> @} Total time: 2.580 seconds, Total memory usage: 39.58MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 12:57:03 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^5*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1400); B:=ReducedBasis(V); B; //P:=J!B[1]; //Q:=J![x+2,0]; P1:=J![x-2,8]; Height(P1); HeightConstant(J:Effort:=20); //RationalPoints(J:Bound:=12); Chabauty(P1,23); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 12:56:59 on modular [Seed = 2236814850] ------------------------------------- 1 Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 to JacHyp: J given by a rule [no inverse] [ (x^2 - 4, 2*x + 4, 2) ] 0.9176949999123136287095316694682973525170029825697109902393 4.486499277983286154053028928413021210016 3.465735902799726547086160607290882840364 {@ <2, 1, 4, 1>, <254187, 1, 4, 1>, <246453, 1, 4, 1>, <5918, 1, 4, 1>, <69818, 1, 4, 1>, <5443, 1, 4, 1>, <141219, 1, 4, 1>, <225826, 1, 4, 1>, <0, 1, 4, 1>, <258262, 1, 4, 1>, <148240, 1, 4, 1> @} Total time: 3.740 seconds, Total memory usage: 39.58MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 12:56:51 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^5*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1400); B:=ReducedBasis(V); B; //P:=J!B[1]; //Q:=J![x+2,0]; P1:=J![x-2,8]; Height(P1); HeightConstant(J:Effort:=20); //RationalPoints(J:Bound:=12); Chabauty(P1,19); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 12:56:49 on modular [Seed = 2287868677] ------------------------------------- 1 Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 to JacHyp: J given by a rule [no inverse] [ (x^2 - 4, 2*x + 4, 2) ] 0.9176949999123136287095316694682973525170029825697109902393 4.486499277983286154053028928413021210016 3.465735902799726547086160607290882840364 {@ <104488, 1, 4, 1>, <58245, 1, 4, 1>, <2, 1, 4, 1> @} Total time: 2.569 seconds, Total memory usage: 39.58MB '146.6.1' ************** MAGMA ***************** Host 146.6.139.217 (146.6.139.217) Time: Tue Dec 6 12:56:45 2005 Input: count:=0; m:=6; q:=2^m; F1:=GF(q); F2:=GF(q^2); a:=F1.1; b:=F1.1^2; a1:=F1.1^55; a2:=F1.1^59; P:=PolynomialRing(F1); u:=x^2+F1.1^27*x+F1.1^51; v:=F1.1^39+F1.1^59; f:=a*x^5+b*x^3+x; v^2+x*v-f; v^2+x*v-f mod u; Roots(x^2-a); P:=PolynomialRing(F1); C:=HyperellipticCurve(a*x^5+b*x^3+x,x); N1:=#C; //over Fq R:=PolynomialRing(F2); CC:=HyperellipticCurve(a*z^5+b*z^3+z,z); N2:=#CC; //over Fq^2; print "done"; Output: Magma V2.11-10 Tue Dec 6 2005 12:56:44 on modular [Seed = 3009748995] ------------------------------------- F1.1*x^5 + F1.1^2*x^3 + F1.1^38*x + F1.1^3 F1.1^5*x + F1.1^59 [ ] done Total time: 0.200 seconds, Total memory usage: 3.43MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 12:56:42 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^5*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1400); B:=ReducedBasis(V); B; //P:=J!B[1]; //Q:=J![x+2,0]; P1:=J![x-2,8]; Height(P1); HeightConstant(J:Effort:=20); //RationalPoints(J:Bound:=12); Chabauty(P1,19); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 12:56:39 on modular [Seed = 3059754253] ------------------------------------- 1 Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 to JacHyp: J given by a rule [no inverse] [ (x^2 - 4, 2*x + 4, 2) ] 0.9176949999123136287095316694682973525170029825697109902393 4.486499277983286154053028928413021210016 3.465735902799726547086160607290882840364 {@ <104488, 1, 4, 1>, <58245, 1, 4, 1>, <2, 1, 4, 1> @} Total time: 2.580 seconds, Total memory usage: 39.58MB '146.6.1' ************** MAGMA ***************** Host 146.6.139.217 (146.6.139.217) Time: Tue Dec 6 12:56:27 2005 Input: count:=0; m:=6; q:=2^m; F1:=GF(q); F2:=GF(q^2); a:=F1.1; b:=F1.1^2; a1:=F1.1^55; a2:=F1.1^59; P:=PolynomialRing(F1); u:=x^2+F1.1^27*x+F1.1^51; v:=F1.1^39+F1.1^59; f:=a*x^5+b*x^3+x; v^2+x*v-f; v^2+x*v-f mod u; Roots(x^2-1); P:=PolynomialRing(F1); C:=HyperellipticCurve(a*x^5+b*x^3+x,x); N1:=#C; //over Fq R:=PolynomialRing(F2); CC:=HyperellipticCurve(a*z^5+b*z^3+z,z); N2:=#CC; //over Fq^2; Points(Jacobian(C)); print "done"; Output: Magma V2.11-10 Tue Dec 6 2005 12:56:25 on modular [Seed = 3194498331] ------------------------------------- F1.1*x^5 + F1.1^2*x^3 + F1.1^38*x + F1.1^3 F1.1^5*x + F1.1^59 [ <1, 2> ] {@ (1, 0, 0), (x^2 + F1.1^39*x + F1.1^26, F1.1^41*x + F1.1^45, 2), (x^2 + F1.1^51*x + F1.1^35, F1.1^22*x + F1.1^47, 2), (x^2 + F1.1^55*x + F1.1^5, F1.1^59*x + F1.1^22, 2), (x^2 + F1.1^16*x + F1.1^22, F1.1^7*x + F1.1^46, 2), (x^2 + F1.1^40*x + F1.1^48, F1.1^3*x + F1.1^26, 2), (x^2 + F1.1^59*x + F1.1^31, F1.1^45*x + F1.1^48, 2), (x^2 + F1.1^4*x + F1.1, F1.1^17*x + F1.1^18, 2), (x^2 + F1.1^31*x + F1.1^58, F1.1^55*x + F1.1^59, 2), (x^2 + F1.1*x + F1.1^5, F1.1^22*x + F1.1^14, 2), (x^2 + F1.1^26*x + F1.1^22, F1.1^41*x + F1.1^50, 2), (x^2 + F1.1^57*x + F1.1^51, F1.1^14*x + F1.1^33, 2), (x^2 + F1.1^59*x, F1.1^17*x, 2), (x^2 + F1.1^47*x + F1.1^26, F1.1^27*x + F1.1^58, 2), (x^2 + F1.1^59*x + F1.1^19, F1.1^30*x + F1.1^10, 2), (x^2 + F1.1^36*x + F1.1^52, F1.1^29*x + F1.1^33, 2), (x^2 + F1.1^60*x + F1.1^42, F1.1^21, 2), (x^2 + F1.1^11*x + F1.1^45, F1.1^43*x + F1.1^16, 2), (x^2 + F1.1^28*x + 1, F1.1^55*x + F1.1^12, 2), (x^2 + F1.1^21*x + F1.1^59, F1.1^21*x + F1.1^60, 2), (x^2 + F1.1^15*x + F1.1^42, F1.1^45*x + F1.1^17, 2), (x^2 + F1.1^60*x + F1.1^39, F1.1^56*x + F1.1^61, 2), (x^2 + F1.1^10*x + F1.1^9, F1.1^50*x + F1.1^53, 2), (x^2 + F1.1^42*x + F1.1^50, F1.1^61*x + F1.1^48, 2), (x^2 + F1.1^55, F1.1^34*x + F1.1^52, 2), (x^2 + F1.1^3*x + F1.1^18, F1.1^28*x + F1.1^24, 2), (x^2 + F1.1^62*x + F1.1^46, F1.1^12*x + F1.1^22, 2), (x^2 + F1.1^34*x + F1.1^51, F1.1^57*x + F1.1^11, 2), (x^2 + F1.1^37*x + F1.1^32, F1.1^11*x + F1.1^26, 2), (x^2 + F1.1^25*x + F1.1^13, F1.1^61*x + F1.1^28, 2), (x^2 + F1.1^14*x + F1.1^14, F1.1^44*x + F1.1^29, 2), (x^2 + F1.1^6*x + F1.1^32, F1.1^29*x + F1.1^42, 2), (x^2 + F1.1^22*x + F1.1^29, F1.1^17*x + F1.1^26, 2), (x^2 + F1.1^32*x + F1.1^53, F1.1^24*x + F1.1^59, 2), (x^2 + F1.1^54*x + F1.1^27, F1.1^15*x + F1.1^29, 2), (x^2 + F1.1^49*x + F1.1^30, F1.1^24*x + F1.1^44, 2), (x^2 + F1.1^16*x + F1.1, F1.1^9*x + F1.1^62, 2), (x^2 + F1.1^56*x + F1.1, F1.1^43*x + F1.1^14, 2), (x^2 + F1.1^60*x + F1.1^6, F1.1^34*x + F1.1^4, 2), (x^2 + F1.1^61*x + F1.1^9, F1.1*x + F1.1^31, 2), (x^2 + F1.1^38*x + F1.1^45, F1.1^28*x + F1.1^35, 2), (x^2 + F1.1^52*x + F1.1^19, F1.1^20*x + F1.1, 2), (x^2 + F1.1^39*x + F1.1^51, F1.1^37*x + F1.1^28, 2), (x^2 + F1.1^25*x + F1.1^55, F1.1^26*x + F1.1^17, 2), (x^2 + F1.1^14*x + F1.1^28, F1.1^33*x + F1.1^6, 2), (x^2 + x + F1.1^37, F1.1^18*x + F1.1, 2), (x^2 + F1.1^9*x + F1.1^37, F1.1^32*x + F1.1^42, 2), (x^2 + F1.1^30*x + F1.1^17, F1.1^26*x + F1.1^54, 2), (x^2 + F1.1^13*x + F1.1^3, F1.1^32*x + F1.1^33, 2), (x^2 + F1.1^62*x + F1.1^9, F1.1^12*x + F1.1^35, 2), (x^2 + F1.1^40*x + F1.1^26, F1.1^45*x + F1.1^7, 2), (x^2 + F1.1^62*x + F1.1^15, F1.1^31*x + F1.1^20, 2), (x^2 + F1.1^34*x + F1.1^14, F1.1^2*x + F1.1^61, 2), (x^2 + F1.1^6*x + F1.1^8, F1.1^9*x + F1.1^3, 2), (x^2 + F1.1^35*x + F1.1^34, F1.1^31*x + F1.1^53, 2), (x^2 + F1.1^34*x + F1.1^8, F1.1^28*x + F1.1^14, 2), (x^2 + F1.1^25*x + F1.1^58, F1.1^23*x + F1.1^27, 2), (x^2 + F1.1^42*x + F1.1^20, F1.1^6*x + F1.1^39, 2), (x^2 + F1.1^4*x + F1.1^58, F1.1^61*x + F1.1^34, 2), (x^2 + F1.1^55*x + F1.1^26, F1.1^3*x, 2), (x^2 + F1.1^4*x + 1, F1.1^61*x + F1.1^5, 2), (x^2 + F1.1^11*x + F1.1^52, F1.1^61*x + F1.1^47, 2), (x^2 + F1.1^24*x + F1.1^57, F1.1^3*x + F1.1^40, 2), (x^2 + F1.1^56*x + F1.1^58, F1.1^56*x + F1.1^23, 2), (x^2 + F1.1^13*x + F1.1^16, F1.1^2*x + F1.1^25, 2), (x^2 + F1.1^14*x + F1.1^30, F1.1^62*x + F1.1^18, 2), (x^2 + F1.1^40*x + F1.1^54, F1.1^14*x + F1.1^45, 2), (x^2 + F1.1^21*x + F1.1^23, F1.1^36*x + F1.1^11, 2), (x^2 + F1.1^56*x + F1.1^62, F1.1^25*x + F1.1^51, 2), (x^2 + F1.1^36*x + F1.1^55, F1.1^16*x + F1.1^28, 2), (x^2 + F1.1^8*x + F1.1^41, F1.1^49*x + F1.1^15, 2), (x^2 + F1.1^34*x + F1.1^12, F1.1^35*x + F1.1^8, 2), (x^2 + F1.1^9*x + F1.1^32, F1.1^32*x + F1.1^8, 2), (x^2 + F1.1^39*x + F1.1^8, F1.1^39*x + F1.1^60, 2), (x^2 + F1.1^12*x + F1.1^35, F1.1^43*x + F1.1^61, 2), (x + F1.1^8, F1.1^27, 1), (x^2 + F1.1^9*x + F1.1^2, F1.1^41*x + F1.1^43, 2), (x^2 + F1.1^29*x + F1.1^44, F1.1^23*x + F1.1^27, 2), (x^2 + F1.1^39*x + F1.1^53, F1.1^34*x + F1.1^23, 2), (x^2 + F1.1^50*x + F1.1^2, F1.1^46*x + F1.1^57, 2), (x^2 + F1.1^41*x + F1.1^14, F1.1^59*x + F1.1^56, 2), (x^2 + F1.1^23*x + F1.1^12, F1.1^33*x + F1.1^35, 2), (x^2 + F1.1^10*x + F1.1^5, F1.1^29*x + F1.1^34, 2), (x^2 + F1.1^44*x + F1.1^54, F1.1^22*x + F1.1^22, 2), (x^2 + F1.1^30*x + F1.1^29, F1.1^41*x + F1.1^39, 2), (x^2 + F1.1^50*x + F1.1^4, F1.1^19, 2), (x^2 + F1.1^27*x + F1.1^17, F1.1^40*x + F1.1^11, 2), (x^2 + F1.1^38*x + F1.1^58, F1.1^7*x + F1.1^13, 2), (x^2 + F1.1^26*x + F1.1^9, F1.1^52*x + F1.1^54, 2), (x^2 + F1.1^37*x + F1.1^6, F1.1^21*x + F1.1^57, 2), (x^2 + F1.1^19*x + F1.1^48, F1.1^17*x + F1.1^24, 2), (x^2 + F1.1^39*x + F1.1^10, F1.1^42*x + F1.1^15, 2), (x^2 + F1.1^44*x + F1.1^10, F1.1^47*x + F1.1^38, 2), (x^2 + x + F1.1^21, F1.1^24*x + F1.1^11, 2), (x^2 + F1.1^42*x + F1.1^15, F1.1^37*x + F1.1^37, 2), (x^2 + F1.1^50*x + F1.1^31, F1.1^28*x + F1.1^4, 2), (x^2 + F1.1^7*x + F1.1^16, F1.1^6*x + F1.1^11, 2), (x^2 + F1.1^18*x + F1.1^48, F1.1^26*x + F1.1^8, 2), (x^2 + F1.1^23*x + F1.1^37, F1.1^47*x + F1.1^48, 2), (x^2 + F1.1^2*x + F1.1^44, F1.1^53*x + F1.1^36, 2), (x^2 + F1.1^14*x + F1.1^61, F1.1^52*x + F1.1^20, 2), (x^2 + F1.1^37*x + F1.1^56, F1.1^19*x + F1.1^44, 2), (x^2 + F1.1^44*x + F1.1^5, F1.1^62*x + F1.1^17, 2), (x^2 + F1.1^52*x + F1.1^11, F1.1^23*x + F1.1^26, 2), (x^2 + F1.1^2*x + F1.1^61, F1.1^23*x + F1.1^27, 2), (x^2 + F1.1^2*x + F1.1^28, F1.1^56*x + F1.1^48, 2), (x^2 + F1.1^21*x + F1.1^19, F1.1^3*x + F1.1^47, 2), (x^2 + F1.1^53*x + F1.1^3, F1.1^12*x + F1.1^28, 2), (x^2 + F1.1^51*x + F1.1^34, F1.1^61*x + F1.1^20, 2), (x^2 + F1.1*x + F1.1^43, F1.1^22*x + F1.1^33, 2), (x^2 + F1.1^40*x + F1.1^38, x + F1.1^47, 2), (x^2 + x + F1.1^5, F1.1^9*x + F1.1^10, 2), (x^2 + F1.1^62*x + F1.1^44, F1.1^21*x + F1.1^2, 2), (x^2 + F1.1^22*x + F1.1^16, F1.1^4*x + F1.1^19, 2), (x^2 + F1.1^51*x + F1.1^58, F1.1^39*x + F1.1^28, 2), (x^2 + F1.1^45*x + F1.1^8, F1.1^36*x + F1.1^15, 2), (x^2 + F1.1^34*x + F1.1^16, F1.1^56*x + F1.1^13, 2), (x^2 + F1.1^9*x + F1.1^50, F1.1^37*x + F1.1^21, 2), (x + F1.1^47, F1.1^18, 1), (x^2 + F1.1^4*x + F1.1^56, F1.1^34*x + F1.1^37, 2), (x^2 + F1.1^44*x + F1.1^29, F1.1^19*x + F1.1^45, 2), (x^2 + F1.1^40*x + F1.1^2, F1.1^19*x + F1.1^9, 2), (x^2 + F1.1^21*x + F1.1^21, F1.1^33*x + F1.1^23, 2), (x^2 + F1.1^43*x + F1.1^18, F1.1^29*x + F1.1^41, 2), (x^2 + F1.1^6*x + F1.1^41, F1.1^42*x + F1.1^50, 2), (x^2 + F1.1^27*x + F1.1^47, F1.1^23*x + F1.1^27, 2), (x^2 + F1.1^15*x + F1.1^60, F1.1^26*x + F1.1^5, 2), (x^2 + F1.1^62*x + F1.1^49, F1.1*x + F1.1^41, 2), (x^2 + F1.1^28*x + F1.1^8, F1.1^56*x + F1.1^3, 2), (x^2 + F1.1^48*x + F1.1^17, F1.1^61*x + F1.1^50, 2), (x^2 + F1.1^49*x + F1.1^19, F1.1^48*x + F1.1^39, 2), (x^2 + F1.1^19*x + F1.1^27, F1.1^21*x + F1.1^10, 2), (x^2 + F1.1^37*x + F1.1^35, F1.1^10*x + F1.1^43, 2), (x^2 + F1.1^61*x + F1.1^8, F1.1^41*x + F1.1^42, 2), (x^2 + F1.1^49*x + F1.1^61, F1.1^55*x + F1.1^52, 2), (x^2 + F1.1^58*x + F1.1^35, F1.1^32*x + F1.1^42, 2), (x^2 + F1.1^45*x + F1.1^30, F1.1^5*x + F1.1^28, 2), (x^2 + F1.1^61*x + F1.1^37, F1.1^36*x + 1, 2), (x^2 + F1.1^46*x + F1.1^36, F1.1^40*x + F1.1^38, 2), (x^2 + F1.1^15*x + F1.1^52, F1.1^11*x + F1.1^16, 2), (x^2 + F1.1^42*x + F1.1^10, F1.1^44*x + F1.1^26, 2), (x^2 + F1.1^26*x + F1.1^48, F1.1^49*x + F1.1^6, 2), (x^2 + F1.1^17*x + F1.1^45, F1.1^51*x + F1.1^27, 2), (x^2 + F1.1^6*x + F1.1^38, F1.1^15*x + F1.1^30, 2), (x^2 + F1.1^34*x + F1.1^4, F1.1^44*x + F1.1^62, 2), (x^2 + F1.1^25*x + F1.1^14, F1.1^27*x + F1.1^57, 2), (x^2 + F1.1^45*x + F1.1^43, F1.1^13*x + F1.1^51, 2), (x^2 + F1.1^14*x + F1.1^25, F1.1*x + F1.1^39, 2), (x^2 + F1.1^23*x + F1.1^29, F1.1^51*x + F1.1^16, 2), (x^2 + F1.1^11*x + F1.1^55, F1.1^27*x + F1.1^38, 2), (x^2 + F1.1^16, F1.1^38*x + F1.1^12, 2), (x^2 + F1.1^24*x + F1.1^32, F1.1^44*x + F1.1^29, 2), (x^2 + F1.1^26*x + F1.1^18, F1.1^3*x + F1.1^23, 2), (x^2 + F1.1^8*x + F1.1^27, F1.1^42*x + F1.1^28, 2), (x^2 + F1.1^6*x + F1.1^55, F1.1^5*x + F1.1^29, 2), (x^2 + F1.1^55*x + F1.1^13, F1.1^46, 2), (x^2 + F1.1^58*x + F1.1^9, F1.1^40*x + F1.1^33, 2), (x^2 + F1.1^33*x + F1.1^30, F1.1^3*x + F1.1^38, 2), (x^2 + F1.1^33*x + F1.1^25, F1.1^24*x + F1.1^6, 2), (x^2 + F1.1^27*x + F1.1^16, F1.1^37*x + F1.1^44, 2), (x^2 + F1.1^22*x + F1.1^18, F1.1^33*x + F1.1^38, 2), (x^2 + F1.1^12*x + F1.1^14, F1.1^2*x + F1.1^54, 2), (x^2 + F1.1^35*x + 1, F1.1^16*x + F1.1^34, 2), (x^2 + F1.1^20*x + F1.1^61, F1.1^11*x + F1.1^31, 2), (x^2 + F1.1^11*x + F1.1^46, F1.1^43*x + F1.1^48, 2), (x^2 + F1.1^18*x + F1.1^62, F1.1^61*x + F1.1, 2), (x^2 + F1.1^3*x + F1.1, F1.1^41*x + F1.1^52, 2), (x^2 + F1.1^16*x + F1.1^29, F1.1^37*x + F1.1^28, 2), (x^2 + F1.1^24*x + F1.1^27, F1.1^40*x + F1.1^43, 2), (x^2 + F1.1^13*x + F1.1^55, F1.1^20*x + F1.1^50, 2), (x^2 + F1.1^4*x + F1.1^49, F1.1^52*x + F1.1^58, 2), (x^2 + F1.1^37*x + F1.1^15, F1.1^33*x + F1.1^59, 2), (x^2 + F1.1^25*x + F1.1^61, x + F1.1^45, 2), (x^2 + F1.1^27*x + F1.1^38, F1.1^30*x + F1.1^51, 2), (x^2 + F1.1^55*x + F1.1^37, F1.1^29*x + F1.1^5, 2), (x^2 + F1.1^45*x + F1.1^49, F1.1^21*x + F1.1^18, 2), (x^2 + F1.1^10*x + F1.1^24, F1.1^38*x + F1.1^22, 2), (x^2 + F1.1^62*x + F1.1^40, F1.1^4*x + F1.1^10, 2), (x^2 + F1.1^49*x + F1.1^42, F1.1^49*x + F1.1^32, 2), (x^2 + F1.1^43*x + F1.1^14, F1.1^45*x + F1.1^46, 2), (x^2 + F1.1^34*x + F1.1^30, F1.1^59*x + F1.1^43, 2), (x + F1.1^11, F1.1^52, 1), (x^2 + F1.1^44*x + F1.1^44, F1.1^52*x + F1.1^28, 2), (x^2 + F1.1^39*x + F1.1^56, x + F1.1^25, 2), (x^2 + F1.1^15*x + F1.1^15, F1.1^58*x + F1.1^ ** WARNING: Output too long, hence truncated. '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 12:56:26 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^5*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1400); B:=ReducedBasis(V); B; //P:=J!B[1]; //Q:=J![x+2,0]; P1:=J![x-2,8]; Height(P1); HeightConstant(J:Effort:=20); //RationalPoints(J:Bound:=12); Chabauty(P1,17); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 12:56:23 on modular [Seed = 2742366216] ------------------------------------- 1 Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 to JacHyp: J given by a rule [no inverse] [ (x^2 - 4, 2*x + 4, 2) ] 0.9176949999123136287095316694682973525170029825697109902393 4.486499277983286154053028928413021210016 3.465735902799726547086160607290882840364 {@ <2, 1, 4, 1>, <199, 1, 4, 1> @} Total time: 2.509 seconds, Total memory usage: 39.58MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 12:56:17 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^5*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1400); B:=ReducedBasis(V); B; //P:=J!B[1]; //Q:=J![x+2,0]; P1:=J![x-2,8]; Height(P1); HeightConstant(J:Effort:=20); //RationalPoints(J:Bound:=12); Chabauty(P1,13); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 12:56:14 on modular [Seed = 2775791223] ------------------------------------- 1 Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 to JacHyp: J given by a rule [no inverse] [ (x^2 - 4, 2*x + 4, 2) ] 0.9176949999123136287095316694682973525170029825697109902393 4.486499277983286154053028928413021210016 3.465735902799726547086160607290882840364 {@ <2, 1, 4, 1>, <27094, 1, 4, 1>, <13160, 1, 4, 1>, <19042, 1, 4, 1>, <19507, 1, 4, 1>, <9810, 1, 4, 1>, <25736, 1, 4, 1> @} Total time: 2.839 seconds, Total memory usage: 39.58MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 12:56:07 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^5*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1400); B:=ReducedBasis(V); B; //P:=J!B[1]; //Q:=J![x+2,0]; P1:=J![x-2,8]; Height(P1); HeightConstant(J:Effort:=20); //RationalPoints(J:Bound:=12); Chabauty(P1,11); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 12:56:05 on modular [Seed = 2826845041] ------------------------------------- 1 Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 to JacHyp: J given by a rule [no inverse] [ (x^2 - 4, 2*x + 4, 2) ] 0.9176949999123136287095316694682973525170029825697109902393 4.486499277983286154053028928413021210016 3.465735902799726547086160607290882840364 {@ <2, 1, 5, 1>, <32702, 1, 5, 1> @} Total time: 2.240 seconds, Total memory usage: 39.58MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 12:55:56 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^5*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1400); B:=ReducedBasis(V); B; //P:=J!B[1]; //Q:=J![x+2,0]; P1:=J![x-2,8]; Height(P1); HeightConstant(J:Effort:=20); //RationalPoints(J:Bound:=12); Chabauty(P1,7); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 12:55:54 on modular [Seed = 3573761555] ------------------------------------- 1 Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 to JacHyp: J given by a rule [no inverse] [ (x^2 - 4, 2*x + 4, 2) ] 0.9176949999123136287095316694682973525170029825697109902393 4.486499277983286154053028928413021210016 3.465735902799726547086160607290882840364 {@ <2, 1, 4, 1>, <1347, 1, 4, 1>, <0, 1, 4, 1> @} Total time: 2.350 seconds, Total memory usage: 39.58MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 12:55:39 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^5*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1400); B:=ReducedBasis(V); B; //P:=J!B[1]; //Q:=J![x+2,0]; P1:=J![x-2,8]; Height(P1); HeightConstant(J:Effort:=20); RationalPoints(J:Bound:=12); //Chabauty(P1,19); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 12:55:37 on modular [Seed = 3624815405] ------------------------------------- 1 Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 to JacHyp: J given by a rule [no inverse] [ (x^2 - 4, 2*x + 4, 2) ] 0.9176949999123136287095316694682973525170029825697109902393 4.486499277983286154053028928413021210016 3.465735902799726547086160607290882840364 {@ (1, 0, 0), (x + 2, 0, 1), (x - 2, 8, 1), (x - 2, -8, 1), (x^2 - 4, 2*x + 4, 2), (x^2 - 4, -2*x - 4, 2) @} Total time: 1.899 seconds, Total memory usage: 39.58MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 12:55:31 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^5*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1400); B:=ReducedBasis(V); B; //P:=J!B[1]; //Q:=J![x+2,0]; P1:=J![x-2,8]; Height(P1); HeightConstant(J:Effort:=20); RationalPoints(J:Bound:=12); //Chabauty(P1,17); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 12:55:29 on modular [Seed = 3272953918] ------------------------------------- 1 Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 to JacHyp: J given by a rule [no inverse] [ (x^2 - 4, 2*x + 4, 2) ] 0.9176949999123136287095316694682973525170029825697109902393 4.486499277983286154053028928413021210016 3.465735902799726547086160607290882840364 {@ (1, 0, 0), (x + 2, 0, 1), (x - 2, 8, 1), (x - 2, -8, 1), (x^2 - 4, 2*x + 4, 2), (x^2 - 4, -2*x - 4, 2) @} Total time: 1.899 seconds, Total memory usage: 39.58MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 12:55:23 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^5*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1400); B:=ReducedBasis(V); B; //P:=J!B[1]; //Q:=J![x+2,0]; P1:=J![x-2,8]; Height(P1); HeightConstant(J:Effort:=20); RationalPoints(J:Bound:=12); //Chabauty(P1,13); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 12:55:21 on modular [Seed = 3322959139] ------------------------------------- 1 Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 to JacHyp: J given by a rule [no inverse] [ (x^2 - 4, 2*x + 4, 2) ] 0.9176949999123136287095316694682973525170029825697109902393 4.486499277983286154053028928413021210016 3.465735902799726547086160607290882840364 {@ (1, 0, 0), (x + 2, 0, 1), (x - 2, 8, 1), (x - 2, -8, 1), (x^2 - 4, 2*x + 4, 2), (x^2 - 4, -2*x - 4, 2) @} Total time: 1.899 seconds, Total memory usage: 39.58MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 12:55:15 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^5*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1400); B:=ReducedBasis(V); B; //P:=J!B[1]; //Q:=J![x+2,0]; P1:=J![x-2,8]; Height(P1); HeightConstant(J:Effort:=20); RationalPoints(J:Bound:=12); //Chabauty(P1,11); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 12:55:13 on modular [Seed = 3374275108] ------------------------------------- 1 Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 to JacHyp: J given by a rule [no inverse] [ (x^2 - 4, 2*x + 4, 2) ] 0.9176949999123136287095316694682973525170029825697109902393 4.486499277983286154053028928413021210016 3.465735902799726547086160607290882840364 {@ (1, 0, 0), (x + 2, 0, 1), (x - 2, 8, 1), (x - 2, -8, 1), (x^2 - 4, 2*x + 4, 2), (x^2 - 4, -2*x - 4, 2) @} Total time: 1.919 seconds, Total memory usage: 39.58MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 12:52:42 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^5*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1400); B:=ReducedBasis(V); B; //P:=J!B[1]; //Q:=J![x+2,0]; P1:=J![x-2,8]; Height(P1); HeightConstant(J:Effort:=20); RationalPoints(J:Bound:=12); //Chabauty(P1,7); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Tue Dec 6 2005 12:52:40 on modular [Seed = 4062477942] ------------------------------------- 1 Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 to JacHyp: J given by a rule [no inverse] [ (x^2 - 4, 2*x + 4, 2) ] 0.9176949999123136287095316694682973525170029825697109902393 4.486499277983286154053028928413021210016 3.465735902799726547086160607290882840364 {@ (1, 0, 0), (x + 2, 0, 1), (x - 2, 8, 1), (x - 2, -8, 1), (x^2 - 4, 2*x + 4, 2), (x^2 - 4, -2*x - 4, 2) @} Total time: 1.909 seconds, Total memory usage: 39.58MB '146.6.1' ************** MAGMA ***************** Host 146.6.139.217 (146.6.139.217) Time: Tue Dec 6 12:52:33 2005 Input: count:=0; m:=6; q:=2^m; F1:=GF(q); F2:=GF(q^2); a:=F1.1; b:=F1.1^2; a1:=F1.1^55; a2:=F1.1^59; P:=PolynomialRing(F1); u:=x^2+F1.1^27*x+F1.1^51; v:=F1.1^39+F1.1^59; f:=a*x^5+b*x^3+x; v^2+x*v-f; v^2+x*v-f mod u; P:=PolynomialRing(F1); C:=HyperellipticCurve(a*x^5+b*x^3+x,x); N1:=#C; //over Fq R:=PolynomialRing(F2); CC:=HyperellipticCurve(a*z^5+b*z^3+z,z); N2:=#CC; //over Fq^2; Points(Jacobian(C)); print "done"; Output: Magma V2.11-10 Tue Dec 6 2005 12:52:31 on modular [Seed = 4112742726] ------------------------------------- F1.1*x^5 + F1.1^2*x^3 + F1.1^38*x + F1.1^3 F1.1^5*x + F1.1^59 {@ (1, 0, 0), (x^2 + F1.1^55*x + F1.1^5, F1.1^59*x + F1.1^22, 2), (x^2 + F1.1^59*x + F1.1^31, F1.1^45*x + F1.1^48, 2), (x^2 + F1.1*x + F1.1^5, F1.1^22*x + F1.1^14, 2), (x^2 + F1.1^59*x, F1.1^17*x, 2), (x^2 + F1.1^36*x + F1.1^52, F1.1^29*x + F1.1^33, 2), (x^2 + F1.1^28*x + 1, F1.1^55*x + F1.1^12, 2), (x^2 + F1.1^60*x + F1.1^39, F1.1^56*x + F1.1^61, 2), (x^2 + F1.1^55, F1.1^34*x + F1.1^52, 2), (x^2 + F1.1^34*x + F1.1^51, F1.1^57*x + F1.1^11, 2), (x^2 + F1.1^14*x + F1.1^14, F1.1^44*x + F1.1^29, 2), (x^2 + F1.1^32*x + F1.1^53, F1.1^24*x + F1.1^59, 2), (x^2 + F1.1^16*x + F1.1, F1.1^9*x + F1.1^62, 2), (x^2 + F1.1^61*x + F1.1^9, F1.1*x + F1.1^31, 2), (x^2 + F1.1^39*x + F1.1^51, F1.1^37*x + F1.1^28, 2), (x^2 + x + F1.1^37, F1.1^18*x + F1.1, 2), (x^2 + F1.1^13*x + F1.1^3, F1.1^32*x + F1.1^33, 2), (x^2 + F1.1^62*x + F1.1^15, F1.1^31*x + F1.1^20, 2), (x^2 + F1.1^35*x + F1.1^34, F1.1^31*x + F1.1^53, 2), (x^2 + F1.1^42*x + F1.1^20, F1.1^6*x + F1.1^39, 2), (x^2 + F1.1^4*x + 1, F1.1^61*x + F1.1^5, 2), (x^2 + F1.1^56*x + F1.1^58, F1.1^56*x + F1.1^23, 2), (x^2 + F1.1^40*x + F1.1^54, F1.1^14*x + F1.1^45, 2), (x^2 + F1.1^36*x + F1.1^55, F1.1^16*x + F1.1^28, 2), (x^2 + F1.1^9*x + F1.1^32, F1.1^32*x + F1.1^8, 2), (x + F1.1^8, F1.1^27, 1), (x^2 + F1.1^39*x + F1.1^53, F1.1^34*x + F1.1^23, 2), (x^2 + F1.1^23*x + F1.1^12, F1.1^33*x + F1.1^35, 2), (x^2 + F1.1^30*x + F1.1^29, F1.1^41*x + F1.1^39, 2), (x^2 + F1.1^38*x + F1.1^58, F1.1^7*x + F1.1^13, 2), (x^2 + F1.1^19*x + F1.1^48, F1.1^17*x + F1.1^24, 2), (x^2 + x + F1.1^21, F1.1^24*x + F1.1^11, 2), (x^2 + F1.1^7*x + F1.1^16, F1.1^6*x + F1.1^11, 2), (x^2 + F1.1^2*x + F1.1^44, F1.1^53*x + F1.1^36, 2), (x^2 + F1.1^44*x + F1.1^5, F1.1^62*x + F1.1^17, 2), (x^2 + F1.1^2*x + F1.1^28, F1.1^56*x + F1.1^48, 2), (x^2 + F1.1^51*x + F1.1^34, F1.1^61*x + F1.1^20, 2), (x^2 + x + F1.1^5, F1.1^9*x + F1.1^10, 2), (x^2 + F1.1^51*x + F1.1^58, F1.1^39*x + F1.1^28, 2), (x^2 + F1.1^9*x + F1.1^50, F1.1^37*x + F1.1^21, 2), (x^2 + F1.1^44*x + F1.1^29, F1.1^19*x + F1.1^45, 2), (x^2 + F1.1^43*x + F1.1^18, F1.1^29*x + F1.1^41, 2), (x^2 + F1.1^15*x + F1.1^60, F1.1^26*x + F1.1^5, 2), (x^2 + F1.1^48*x + F1.1^17, F1.1^61*x + F1.1^50, 2), (x^2 + F1.1^37*x + F1.1^35, F1.1^10*x + F1.1^43, 2), (x^2 + F1.1^58*x + F1.1^35, F1.1^32*x + F1.1^42, 2), (x^2 + F1.1^46*x + F1.1^36, F1.1^40*x + F1.1^38, 2), (x^2 + F1.1^26*x + F1.1^48, F1.1^49*x + F1.1^6, 2), (x^2 + F1.1^34*x + F1.1^4, F1.1^44*x + F1.1^62, 2), (x^2 + F1.1^14*x + F1.1^25, F1.1*x + F1.1^39, 2), (x^2 + F1.1^16, F1.1^38*x + F1.1^12, 2), (x^2 + F1.1^8*x + F1.1^27, F1.1^42*x + F1.1^28, 2), (x^2 + F1.1^58*x + F1.1^9, F1.1^40*x + F1.1^33, 2), (x^2 + F1.1^27*x + F1.1^16, F1.1^37*x + F1.1^44, 2), (x^2 + F1.1^35*x + 1, F1.1^16*x + F1.1^34, 2), (x^2 + F1.1^18*x + F1.1^62, F1.1^61*x + F1.1, 2), (x^2 + F1.1^24*x + F1.1^27, F1.1^40*x + F1.1^43, 2), (x^2 + F1.1^37*x + F1.1^15, F1.1^33*x + F1.1^59, 2), (x^2 + F1.1^55*x + F1.1^37, F1.1^29*x + F1.1^5, 2), (x^2 + F1.1^62*x + F1.1^40, F1.1^4*x + F1.1^10, 2), (x^2 + F1.1^34*x + F1.1^30, F1.1^59*x + F1.1^43, 2), (x^2 + F1.1^39*x + F1.1^56, x + F1.1^25, 2), (x^2 + F1.1^41*x + F1.1^9, F1.1^15*x + F1.1^7, 2), (x^2 + F1.1^49*x + F1.1^23, F1.1^26*x + F1.1^35, 2), (x^2 + F1.1^21*x + F1.1, F1.1^57*x + F1.1^58, 2), (x^2 + F1.1^9*x + F1.1^60, F1.1^62*x + F1.1^41, 2), (x^2 + F1.1^33*x + F1.1^19, F1.1^17*x + F1.1^22, 2), (x^2 + F1.1^34*x + F1.1^3, F1.1^53*x + F1.1^51, 2), (x^2 + F1.1^16*x + F1.1^51, F1.1^25*x + F1.1^14, 2), (x^2 + F1.1^59*x + F1.1^15, F1.1^43*x + F1.1^12, 2), (x^2 + F1.1^47*x + F1.1^30, F1.1^51*x + F1.1^32, 2), (x^2 + F1.1*x + F1.1^33, F1.1^17*x + F1.1^51, 2), (x^2 + F1.1^57*x + F1.1^40, F1.1^58*x + F1.1^52, 2), (x^2 + F1.1^48*x + F1.1^36, F1.1^11*x + F1.1^18, 2), (x^2 + F1.1^44*x + F1.1^16, F1.1^14*x + F1.1^33, 2), (x^2 + F1.1^14*x + F1.1^36, F1.1^38*x + F1.1^10, 2), (x^2 + F1.1^58*x + F1.1^40, F1.1^54*x, 2), (x^2 + F1.1^62*x + F1.1^56, F1.1^21*x + F1.1^8, 2), (x^2 + F1.1^54*x + F1.1^19, F1.1^16*x + F1.1^24, 2), (x^2 + F1.1^28*x + F1.1^41, F1.1^37*x + F1.1^55, 2), (x^2 + x + F1.1^3, F1.1^2*x + F1.1^40, 2), (x^2 + F1.1^47*x + F1.1^52, F1.1^52*x + F1.1^20, 2), (x^2 + F1.1^35*x + F1.1^21, F1.1^50*x + F1.1^36, 2), (x^2 + F1.1^8*x + F1.1^47, F1.1^31*x + F1.1^25, 2), (x^2 + F1.1^32*x + F1.1^24, F1.1^45*x + F1.1^21, 2), (x^2 + F1.1^56*x + F1.1^44, F1.1^3*x + F1.1^22, 2), (x^2 + F1.1^58*x + F1.1^31, F1.1^4*x + F1.1^54, 2), (x^2 + F1.1^40*x + F1.1^3, F1.1^27*x + F1.1^48, 2), (x^2 + F1.1^32*x + F1.1^38, F1.1^6*x + F1.1^35, 2), (x^2 + F1.1^55*x + F1.1^30, F1.1^50*x + F1.1^12, 2), (x^2 + F1.1^36*x + F1.1^42, F1.1^36*x + F1.1^3, 2), (x^2 + F1.1^35*x + F1.1^22, F1.1^7*x + F1.1^28, 2), (x^2 + F1.1^56*x + F1.1^40, F1.1^6*x + F1.1^54, 2), (x^2 + F1.1^7*x + F1.1^37, F1.1^29*x + F1.1^48, 2), (x^2 + F1.1^43*x + F1.1^39, F1.1^57*x + F1.1^3, 2), (x^2 + F1.1^5*x + F1.1^56, F1.1^42*x + F1.1^4, 2), (x^2 + F1.1^21*x + F1.1^3, F1.1^29*x + F1.1^28, 2), (x^2 + F1.1^16*x + F1.1^58, F1.1^11*x + F1.1^33, 2), (x^2 + F1.1^8*x + F1.1^37, F1.1^4*x + F1.1^53, 2), (x^2 + F1.1^25*x + F1.1^19, F1.1^27*x + F1.1^28, 2), (x^2 + F1.1^17*x + F1.1^60, F1.1^51*x + F1.1^3, 2), (x^2 + F1.1^40*x + F1.1^51, F1.1^29*x + F1.1^10, 2), (x^2 + F1.1^60*x + F1.1^3, F1.1^37*x + F1.1^21, 2), (x^2 + F1.1*x + F1.1^2, F1.1^24*x + F1.1^12, 2), (x^2 + F1.1^31*x + F1.1^45, F1.1^30*x + F1.1^58, 2), (x^2 + F1.1^21*x + F1.1^8, F1.1^16*x + F1.1^7, 2), (x^2 + F1.1^15*x + F1.1^58, F1.1^38*x + F1.1^58, 2), (x^2 + F1.1^36*x + F1.1^24, F1.1^60*x + F1.1^32, 2), (x^2 + F1.1^54*x + F1.1^39, F1.1*x + F1.1, 2), (x^2 + F1.1^53*x + F1.1^41, F1.1^38*x + F1.1^8, 2), (x^2 + F1.1^56*x + F1.1^41, F1.1^13*x + F1.1^52, 2), (x^2 + F1.1^19*x + F1.1^30, F1.1^22*x + F1.1^33, 2), (x^2 + x + F1.1^6, F1.1^58*x + F1.1^3, 2), (x^2 + F1.1^8*x + F1.1^40, F1.1^33*x + F1.1^15, 2), (x^2 + F1.1^14*x + F1.1^38, F1.1^52*x + F1.1^40, 2), (x^2 + F1.1^40*x + F1.1^18, F1.1^45*x + F1.1^3, 2), (x^2 + F1.1^3*x + F1.1^32, F1.1^46*x + F1.1^21, 2), (x^2 + F1.1^22*x + F1.1^54, F1.1^52*x + F1.1, 2), (x^2 + F1.1^58*x + F1.1^22, F1.1^58*x + F1.1^27, 2), (x^2 + F1.1^23*x + F1.1^57, F1.1^54*x + F1.1^39, 2), (x^2 + F1.1^12*x + F1.1^45, F1.1^58*x + F1.1^16, 2), (x^2 + F1.1^38*x + F1.1^59, F1.1^6*x + F1.1^19, 2), (x^2 + F1.1^62*x + F1.1^55, F1.1^6*x + F1.1^54, 2), (x^2 + F1.1^33*x + F1.1^46, F1.1^54*x + F1.1^22, 2), (x^2 + F1.1^9*x + F1.1^23, F1.1^44*x + F1.1^42, 2), (x^2 + F1.1^20*x + F1.1^37, F1.1^42*x + F1.1^27, 2), (x^2 + F1.1^16*x + F1.1^28, F1.1^34*x + F1.1^23, 2), (x^2 + F1.1^56*x + F1.1^35, F1.1^34*x + F1.1^58, 2), (x^2 + F1.1^25*x + F1.1^48, F1.1^44*x + F1.1^53, 2), (x^2 + F1.1^44*x + F1.1^11, F1.1^35*x + F1.1^3, 2), (x^2 + F1.1^14*x + F1.1^6, F1.1^26*x + F1.1^51, 2), (x^2 + F1.1^61*x + F1.1^36, F1.1^25*x + F1.1^43, 2), (x^2 + F1.1^52*x + F1.1^47, F1.1^9*x + F1.1^23, 2), (x^2 + F1.1^15*x, F1.1^49*x, 2), (x^2 + F1.1^61*x + F1.1^7, F1.1^59*x + F1.1^9, 2), (x^2 + F1.1^62*x + F1.1^51, F1.1^31*x + F1.1^38, 2), (x^2 + F1.1^23*x + F1.1^22, F1.1^45*x + F1.1^7, 2), (x^2 + F1.1^44*x + F1.1^11, F1.1^9*x + F1.1^12, 2), (x^2 + F1.1^13*x + F1.1^37, F1.1^46*x + F1.1^36, 2), (x^2 + F1.1^5*x + F1.1^55, F1.1^48*x + F1.1^32, 2), (x^2 + F1.1^56*x + F1.1^26, F1.1^22*x + F1.1^29, 2), (x^2 + F1.1^12*x + F1.1^17, F1.1^57*x + F1.1^27, 2), (x^2 + F1.1^55*x + F1.1^58, F1.1^21*x + F1.1^36, 2), (x^2 + F1.1^3*x + F1.1^8, F1.1^36*x + F1.1, 2), (x^2 + F1.1^16*x + F1.1^43, F1.1^20*x, 2), (x^2 + F1.1^25*x + F1.1^30, F1.1^30*x + F1.1^11, 2), (x^2 + F1.1^10*x + F1.1^57, F1.1^58*x + F1.1, 2), (x^2 + F1.1^61*x + F1.1^19, F1.1^15*x + F1.1^11, 2), (x^2 + F1.1^59*x + F1.1^39, F1.1^17*x + F1.1^53, 2), (x^2 + F1.1^38*x + F1.1^60, x + F1.1^13, 2), (x^2 + F1.1^42*x + F1.1^26, F1.1^13*x + F1.1^44, 2), (x^2 + F1.1^2*x + F1.1^23, F1.1^41*x + F1.1^61, 2), (x + F1.1^27, F1.1^8, 1), (x^2 + F1.1^33*x + F1.1^37, F1.1^61*x + F1.1^18, 2), (x^2 + F1.1^4*x + F1.1^60, F1.1^15*x + F1.1^39, 2), (x^2 + F1.1^12*x + F1.1^52, F1.1^56*x + F1.1^21, 2), (x^2 + F1.1^44*x + F1.1^39, F1.1^36*x + F1.1^10, 2), (x^2 + F1.1^10*x + F1.1^20, F1.1^43*x + F1.1^37, 2), (x^2 + F1.1^8*x + F1.1^39, F1.1^48*x + F1.1^47, 2), (x^2 + F1.1^42*x + F1.1^52, F1.1^32*x + F1.1^5, 2), (x^2 + F1.1^42*x + F1.1^51, F1.1^40*x + F1.1^38, 2), (x^2 + F1.1^35*x + F1.1^6, F1.1^20*x + F1.1^11, 2), (x^2 + F1.1^3*x + F1.1^3, F1.1^37*x + F1.1^37, 2), (x^2 + F1.1^38*x + F1.1^26, F1.1^22*x + F1.1^36, 2), (x^2 + F1.1^4*x + F1.1^35, F1.1^13*x + F1.1^49, 2), (x^2 + F1.1^2*x + F1.1^22, F1.1^28*x + F1.1^38, 2), (x^2 + F1.1^26*x + F1.1^50, F1.1^44*x + F1.1^11, 2), (x^2 + F1.1^60*x + F1.1^31, F1.1^20*x + F1.1^55, 2), (x^2 + F1.1^13*x + F1.1^31, F1.1^28*x + F1.1^47, 2), (x^2 + F1.1^2*x + F1.1^40, F1.1^3*x + F1.1^56, 2), (x^2 + F1.1^26*x + F1.1^12, F1.1^19*x + F1.1^36, 2), (x^2 + F1.1^8*x + F1.1^9, F1.1^51*x + F1.1^57, 2), (x^2 + F1.1^35*x, F1.1^26*x, 2), (x^2 + F1.1^4*x + F1.1^21, F1.1^5*x + F1.1^7, 2), (x^2 + F1.1^10*x + F1.1^19, F1.1^54*x + F1.1^34, 2), (x^2 + F1.1^47*x + F1.1^34, F1.1^11*x + F1.1^31, 2), (x^2 + F1.1^13*x + F1.1^31, F1.1^19*x + F1.1^62, 2), (x^2 + F1.1^56*x + F1.1^35, x + F1.1^56, 2), (x^2 + F1.1^10*x + F1.1^43, F1.1^21*x + F1.1^58, 2), (x^2 + F1.1^54*x + F1.1^59, F1.1^44*x + F1.1^29, 2), (x^2 + F1.1^31*x + F1.1^38, F1.1^34*x + F1.1^10, 2), (x^2 + F1.1^43*x + F1.1^17, F1.1^19*x + F1.1^31, 2), (x^2 + F1.1^40*x + F1.1^4, F1.1^28*x + F1.1^16, 2), (x^2 + F1.1^34*x + F1.1^31, F1.1^48*x + F1.1^35, 2), (x^2 + F1.1^8*x + F1.1^35, F1.1^47*x + F1.1^36, 2), (x^2 + F1.1^43*x + F1.1^45, F1.1^52*x + F1.1 ** WARNING: Output too long, hence truncated. '146.6.1' ************** MAGMA ***************** Host 146.6.139.217 (146.6.139.217) Time: Tue Dec 6 12:52:01 2005 Input: count:=0; m:=6; q:=2^m; F1:=GF(q); F2:=GF(q^2); a:=F1.1; b:=F1.1^2; a1:=F1.1^55; a2:=F1.1^59; P:=PolynomialRing(F1); u:=x^2+F1.1^27*x+F1.1^51; v:=F1.1^39+F1.1^59; f:=a*x^5+b*x^3+x; v^2+x*v-f; v^2+x*v-f mod u; P:=PolynomialRing(F1); C:=HyperellipticCurve(a*x^5+b*x^3+x,x); N1:=#C; //over Fq R:=PolynomialRing(F2); CC:=HyperellipticCurve(a*z^5+b*z^3+z,z); N2:=#CC; //over Fq^2; Points(Jac(C)); print "done"; Output: Magma V2.11-10 Tue Dec 6 2005 12:52:01 on modular [Seed = 3946669894] ------------------------------------- F1.1*x^5 + F1.1^2*x^3 + F1.1^38*x + F1.1^3 F1.1^5*x + F1.1^59 >> Points(Jac(C)); ^ User error: Identifier 'Jac' has not been declared or assigned done Total time: 0.200 seconds, Total memory usage: 3.43MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 12:51:11 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^5*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1400); B:=ReducedBasis(V); B; P:=J!B[1]; Q:=J![x+2,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=20); RationalPoints(J:Bound:=12); //Chabauty(P,7); //Order(Q);Order(P); P+Q; 2*J![x-2,8]; Output: Magma V2.11-10 Tue Dec 6 2005 12:51:09 on modular [Seed = 3980091660] ------------------------------------- 1 Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 to JacHyp: J given by a rule [no inverse] [ (x^2 - 4, 2*x + 4, 2) ] 0.917694999912313628709531669468297352510 4.486499277983286154053028928413021210016 3.465735902799726547086160607290882840364 {@ (1, 0, 0), (x + 2, 0, 1), (x - 2, 8, 1), (x - 2, -8, 1), (x^2 - 4, 2*x + 4, 2), (x^2 - 4, -2*x - 4, 2) @} (x - 2, 8, 1) (x^2 - 4*x + 4, 5*x - 2, 2) Total time: 1.899 seconds, Total memory usage: 39.58MB '146.6.1' ************** MAGMA ***************** Host 146.6.139.217 (146.6.139.217) Time: Tue Dec 6 12:51:08 2005 Input: count:=0; m:=6; q:=2^m; F1:=GF(q); F2:=GF(q^2); a:=F1.1; b:=F1.1^2; a1:=F1.1^55; a2:=F1.1^59; P:=PolynomialRing(F1); u:=x^2+F1.1^27*x+F1.1^51; v:=F1.1^39+F1.1^59; f:=a*x^5+b*x^3+x; v^2+x*v-f; v^2+x*v-f mod u; P:=PolynomialRing(F1); C:=HyperellipticCurve(a*x^5+b*x^3+x,x); N1:=#C; //over Fq R:=PolynomialRing(F2); CC:=HyperellipticCurve(a*z^5+b*z^3+z,z); N2:=#CC; //over Fq^2; Rationalpoints(C); print "done"; Output: Magma V2.11-10 Tue Dec 6 2005 12:51:08 on modular [Seed = 281451760] ------------------------------------- F1.1*x^5 + F1.1^2*x^3 + F1.1^38*x + F1.1^3 F1.1^5*x + F1.1^59 >> Rationalpoints(C); ^ User error: Identifier 'Rationalpoints' has not been declared or assigned done Total time: 0.210 seconds, Total memory usage: 3.43MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 12:50:34 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^5*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1400); B:=ReducedBasis(V); B; P:=J!B[1]; Q:=J![x+2,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=20); RationalPoints(J:Bound:=12); //Chabauty(P,7); //Order(Q);Order(P); 2*P; 2*J![x-2,8]; Output: Magma V2.11-10 Tue Dec 6 2005 12:50:32 on modular [Seed = 466458399] ------------------------------------- 1 Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 to JacHyp: J given by a rule [no inverse] [ (x^2 - 4, 2*x + 4, 2) ] 0.917694999912313628709531669468297352510 4.486499277983286154053028928413021210016 3.465735902799726547086160607290882840364 {@ (1, 0, 0), (x + 2, 0, 1), (x - 2, 8, 1), (x - 2, -8, 1), (x^2 - 4, 2*x + 4, 2), (x^2 - 4, -2*x - 4, 2) @} (x^2 - 4*x + 4, 5*x - 2, 2) (x^2 - 4*x + 4, 5*x - 2, 2) Total time: 1.929 seconds, Total memory usage: 39.58MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 12:50:11 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^5*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1400); B:=ReducedBasis(V); B; P:=J!B[1]; Q:=J![x+2,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=20); RationalPoints(J:Bound:=12); //Chabauty(P,7); //Order(Q);Order(P); //P+Q;P+2*Q; 2*J![x-2,8]; Output: Magma V2.11-10 Tue Dec 6 2005 12:50:09 on modular [Seed = 516461091] ------------------------------------- 1 Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 to JacHyp: J given by a rule [no inverse] [ (x^2 - 4, 2*x + 4, 2) ] 0.917694999912313628709531669468297352510 4.486499277983286154053028928413021210016 3.465735902799726547086160607290882840364 {@ (1, 0, 0), (x + 2, 0, 1), (x - 2, 8, 1), (x - 2, -8, 1), (x^2 - 4, 2*x + 4, 2), (x^2 - 4, -2*x - 4, 2) @} (x^2 - 4*x + 4, 5*x - 2, 2) Total time: 1.919 seconds, Total memory usage: 39.58MB '146.6.1' ************** MAGMA ***************** Host 146.6.139.217 (146.6.139.217) Time: Tue Dec 6 12:49:25 2005 Input: count:=0; m:=6; q:=2^m; F1:=GF(q); F2:=GF(q^2); a:=F1.1; b:=F1.1^2; a1:=F1.1^55; a2:=F1.1^59; P:=PolynomialRing(F1); u:=x^2+F1.1^27*x+F1.1^51; v:=F1.1^39+F1.1^59; f:=a*x^5+b*x^3+x; v^2+x*v-f; v^2+x*v-f mod u; P:=PolynomialRing(F1); C:=HyperellipticCurve(a*x^5+b*x^3+x,x); N1:=#C; //over Fq R:=PolynomialRing(F2); CC:=HyperellipticCurve(a*z^5+b*z^3+z,z); N2:=#CC; //over Fq^2; print "done"; Output: Magma V2.11-10 Tue Dec 6 2005 12:49:25 on modular [Seed = 14060617] ------------------------------------- F1.1*x^5 + F1.1^2*x^3 + F1.1^38*x + F1.1^3 F1.1^5*x + F1.1^59 done Total time: 0.210 seconds, Total memory usage: 3.43MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 12:49:06 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^5*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1400); B:=ReducedBasis(V); B; P:=J!B[1]; Q:=J![x+2,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=20); RationalPoints(J:Bound:=12); //Chabauty(P,7); //Order(Q);Order(P); //P+Q;P+2*Q; Order(Q); Output: Magma V2.11-10 Tue Dec 6 2005 12:49:03 on modular [Seed = 182224489] ------------------------------------- 1 Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 to JacHyp: J given by a rule [no inverse] [ (x^2 - 4, 2*x + 4, 2) ] 0.917694999912313628709531669468297352510 4.486499277983286154053028928413021210016 3.465735902799726547086160607290882840364 {@ (1, 0, 0), (x + 2, 0, 1), (x - 2, 8, 1), (x - 2, -8, 1), (x^2 - 4, 2*x + 4, 2), (x^2 - 4, -2*x - 4, 2) @} 2 Total time: 1.929 seconds, Total memory usage: 39.58MB '146.6.1' ************** MAGMA ***************** Host 146.6.139.217 (146.6.139.217) Time: Tue Dec 6 12:49:03 2005 Input: count:=0; m:=6; q:=2^m; F1:=GF(q); F2:=GF(q^2); a:=F1.1; b:=F1.1^2; a1:=F1.1^55; a2:=F1.1^59; P:=PolynomialRing(F1); u:=x^2+F1.1^27*x+F1.1^51; v:=F1.1^39+F1.1^59; f:=a*x^5+b*x^3+x; v^2+x*v-f; Mod(v^2+x*v-f,u); P:=PolynomialRing(F1); C:=HyperellipticCurve(a*x^5+b*x^3+x,x); N1:=#C; //over Fq R:=PolynomialRing(F2); CC:=HyperellipticCurve(a*z^5+b*z^3+z,z); N2:=#CC; //over Fq^2; print "done"; Output: Magma V2.11-10 Tue Dec 6 2005 12:49:03 on modular [Seed = 249070189] ------------------------------------- F1.1*x^5 + F1.1^2*x^3 + F1.1^38*x + F1.1^3 >> Mod(v^2+x*v-f,u); ^ Runtime error in procedure call: Attempting to call something that is not callable done Total time: 0.210 seconds, Total memory usage: 3.43MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 12:48:43 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^5*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1400); B:=ReducedBasis(V); B; P:=J!B[1]; Q:=J![x-2,8]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=20); RationalPoints(J:Bound:=12); //Chabauty(P,7); //Order(Q);Order(P); //P+Q;P+2*Q; Order(Q); Output: Magma V2.11-10 Tue Dec 6 2005 12:48:41 on modular [Seed = 904116384] ------------------------------------- 1 Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 to JacHyp: J given by a rule [no inverse] [ (x^2 - 4, 2*x + 4, 2) ] 0.917694999912313628709531669468297352510 4.486499277983286154053028928413021210016 3.465735902799726547086160607290882840364 {@ (1, 0, 0), (x + 2, 0, 1), (x - 2, 8, 1), (x - 2, -8, 1), (x^2 - 4, 2*x + 4, 2), (x^2 - 4, -2*x - 4, 2) @} 0 Total time: 1.919 seconds, Total memory usage: 39.58MB '146.6.1' ************** MAGMA ***************** Host 146.6.139.217 (146.6.139.217) Time: Tue Dec 6 12:48:36 2005 Input: count:=0; m:=6; q:=2^m; F1:=GF(q); F2:=GF(q^2); a:=F1.1; b:=F1.1^2; a1:=F1.1^55; a2:=F1.1^59; P:=PolynomialRing(F1); u:=x^2+F1.1^27*x+F1.1^51; v:=F1.1^39+F1.1^59; f:=a*x^5+b*x^3+x; v^2+x*v-f/u; //Mod(v^2+x*v-f P:=PolynomialRing(F1); C:=HyperellipticCurve(a*x^5+b*x^3+x,x); N1:=#C; //over Fq R:=PolynomialRing(F2); CC:=HyperellipticCurve(a*z^5+b*z^3+z,z); N2:=#CC; //over Fq^2; print "done"; Output: Magma V2.11-10 Tue Dec 6 2005 12:48:36 on modular [Seed = 955168160] ------------------------------------- (F1.1*x^5 + F1.1^61*x^3 + F1.1^23*x^2 + F1.1^19*x + F1.1^54)/(x^2 + F1.1^27*x + F1.1^51) done Total time: 0.200 seconds, Total memory usage: 3.43MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 12:48:19 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^5*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1400); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=20); RationalPoints(J:Bound:=12); //Chabauty(P,7); //Order(Q);Order(P); //P+Q;P+2*Q; Output: Magma V2.11-10 Tue Dec 6 2005 12:48:17 on modular [Seed = 603303598] ------------------------------------- 1 Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 to JacHyp: J given by a rule [no inverse] [ (x^2 - 4, 2*x + 4, 2) ] 0.917694999912313628709531669468297352510 4.486499277983286154053028928413021210016 3.465735902799726547086160607290882840364 {@ (1, 0, 0), (x + 2, 0, 1), (x - 2, 8, 1), (x - 2, -8, 1), (x^2 - 4, 2*x + 4, 2), (x^2 - 4, -2*x - 4, 2) @} Total time: 1.899 seconds, Total memory usage: 39.58MB '146.6.1' ************** MAGMA ***************** Host 146.6.139.217 (146.6.139.217) Time: Tue Dec 6 12:47:53 2005 Input: count:=0; m:=6; q:=2^m; F1:=GF(q); F2:=GF(q^2); a:=F1.1; b:=F1.1^2; a1:=F1.1^55; a2:=F1.1^59; P:=PolynomialRing(F1); u:=x^2+F1.1^27*x+F1.1^51; v:=F1.1^39+F1.1^59; f:=a*x^5+b*x^3+x; v^2-x*v-f/u; P:=PolynomialRing(F1); C:=HyperellipticCurve(a*x^5+b*x^3+x,x); N1:=#C; //over Fq R:=PolynomialRing(F2); CC:=HyperellipticCurve(a*z^5+b*z^3+z,z); N2:=#CC; //over Fq^2; print "done"; Output: Magma V2.11-10 Tue Dec 6 2005 12:47:52 on modular [Seed = 619882880] ------------------------------------- (F1.1*x^5 + F1.1^61*x^3 + F1.1^23*x^2 + F1.1^19*x + F1.1^54)/(x^2 + F1.1^27*x + F1.1^51) done Total time: 0.200 seconds, Total memory usage: 3.43MB '146.6.1' ************** MAGMA ***************** Host 146.6.139.217 (146.6.139.217) Time: Tue Dec 6 12:47:38 2005 Input: count:=0; m:=6; q:=2^m; F1:=GF(q); F2:=GF(q^2); a:=F1.1; b:=F1.1^2; a1:=F1.1^55; a2:=F1.1^59; P:=PolynomialRing(F1); u:=x^2+F1.1^27*x+F1.1^51; v:=F1.1^39+F1.1^59; f:=a*x^5+b*x^3+x; v^2-x*v-f; P:=PolynomialRing(F1); C:=HyperellipticCurve(a*x^5+b*x^3+x,x); N1:=#C; //over Fq R:=PolynomialRing(F2); CC:=HyperellipticCurve(a*z^5+b*z^3+z,z); N2:=#CC; //over Fq^2; print "done"; Output: Magma V2.11-10 Tue Dec 6 2005 12:47:38 on modular [Seed = 1519179660] ------------------------------------- F1.1*x^5 + F1.1^2*x^3 + F1.1^38*x + F1.1^3 done Total time: 0.200 seconds, Total memory usage: 3.43MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 12:46:39 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^5*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1400); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=20); //RationalPoints(J:Bound:=12); //Chabauty(P,7); //Order(Q);Order(P); //P+Q;P+2*Q; Output: Magma V2.11-10 Tue Dec 6 2005 12:46:37 on modular [Seed = 1552601421] ------------------------------------- 1 Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 to JacHyp: J given by a rule [no inverse] [ (x^2 - 4, 2*x + 4, 2) ] 0.917694999912313628709531669468297352510 4.486499277983286154053028928413021210016 3.465735902799726547086160607290882840364 Total time: 1.899 seconds, Total memory usage: 39.58MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Tue Dec 6 12:46:23 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^5*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1400); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=20); //RationalPoints(J:Bound:=12); //Chabauty(P,7); //Order(Q);Order(P); //P+Q;P+2*Q; Output: Magma V2.11-10 Tue Dec 6 2005 12:46:20 on modular [Seed = 1200736839] ------------------------------------- 1 Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 to JacHyp: J given by a rule [no inverse] [ (x^2 - 4, 2*x + 4, 2) ] 0.917694999912313628709531669468297352510 4.486499277983286154053028928413021210016 3.465735902799726547086160607290882840364 Total time: 1.919 seconds, Total memory usage: 39.58MB '146.6.1' ************** MAGMA ***************** Host 146.6.139.217 (146.6.139.217) Time: Tue Dec 6 12:44:13 2005 Input: count:=0; m:=6; q:=2^m; F1:=GF(q); F2:=GF(q^2); a:=F1.1; b:=F1.1^2; a1:=F1.1^55; a2:=F1.1^59; P:=PolynomialRing(F1); u:=x^2+F1.1^27*x+F1.1^51; v:=F1.1^39+F1.1^59; P:=PolynomialRing(F1); C:=HyperellipticCurve(a*x^5+b*x^3+x,x); N1:=#C; //over Fq R:=PolynomialRing(F2); CC:=HyperellipticCurve(a*z^5+b*z^3+z,z); N2:=#CC; //over Fq^2; print "done"; Output: Magma V2.11-10 Tue Dec 6 2005 12:44:13 on modular [Seed = 1251787979] ------------------------------------- done Total time: 0.210 seconds, Total memory usage: 3.43MB '146.6.1' ************** MAGMA ***************** Host 146.6.139.217 (146.6.139.217) Time: Tue Dec 6 12:42:05 2005 Input: count:=0; m:=6; q:=2^m; F1:=GF(q); F2:=GF(q^2); a:=F1.1; b:=F1.1^2; a1:=F1.1^55; a2:=F1.1^59; P:=PolynomialRing(F1); u(x):=x^2+F1.1^27*x+F1.1^51; v(x):=F1.1^39+F1.1^59; P:=PolynomialRing(F1); C:=HyperellipticCurve(a*x^5+b*x^3+x,x); N1:=#C; //over Fq R:=PolynomialRing(F2); CC:=HyperellipticCurve(a*z^5+b*z^3+z,z); N2:=#CC; //over Fq^2; print "done"; Output: Magma V2.11-10 Tue Dec 6 2005 12:42:04 on modular [Seed = 1302053709] ------------------------------------- >> u(x):=x^2+F1.1^27*x+F1.1^51; ^ User error: Illegal left hand side of an assignment statement >> v(x):=F1.1^39+F1.1^59; ^ User error: Illegal left hand side of an assignment statement done Total time: 0.210 seconds, Total memory usage: 3.43MB '146.6.1' ************** MAGMA ***************** Host 146.6.139.217 (146.6.139.217) Time: Tue Dec 6 12:41:19 2005 Input: count:=0; m:=6; q:=2^m; F1:=GF(q); F2:=GF(q^2); a:=F1.1; b:=F1.1^2; a1:=F1.1^55; a2:=F1.1^59; u(x):=x^2+F1.1^27*x+F1.1^51; v(x):=F1.1^39+F1.1^59; P:=PolynomialRing(F1); C:=HyperellipticCurve(a*x^5+b*x^3+x,x); N1:=#C; //over Fq R:=PolynomialRing(F2); CC:=HyperellipticCurve(a*z^5+b*z^3+z,z); N2:=#CC; //over Fq^2; print "done"; Output: Magma V2.11-10 Tue Dec 6 2005 12:41:19 on modular [Seed = 2007103780] ------------------------------------- >> u(x):=x^2+F1.1^27*x+F1.1^51; ^ User error: Identifier 'x' has not been declared or assigned >> v(x):=F1.1^39+F1.1^59; ^ User error: Illegal left hand side of an assignment statement done Total time: 0.210 seconds, Total memory usage: 3.43MB '146.6.1' ************** MAGMA ***************** Host 146.6.139.217 (146.6.139.217) Time: Tue Dec 6 12:41:03 2005 Input: count:=0; m:=6; q:=2^m; F1:=GF(q); F2:=GF(q^2); a:=F1.1^i; b:=F1.1^j; a1:=F1.1^55; a2:=F1.1^59; u(x):=x^2+F1.1^27*x+F1.1^51; v(x):=F1.1^39+F1.1^59; P:=PolynomialRing(F1); C:=HyperellipticCurve(a*x^5+b*x^3+x,x); N1:=#C; //over Fq R:=PolynomialRing(F2); CC:=HyperellipticCurve(a*z^5+b*z^3+z,z); N2:=#CC; //over Fq^2; print "done"; Output: Magma V2.11-10 Tue Dec 6 2005 12:41:03 on modular [Seed = 2058155065] ------------------------------------- >> a:=F1.1^i; ^ User error: Identifier 'i' has not been declared or assigned >> b:=F1.1^j; ^ User error: Identifier 'j' has not been declared or assigned >> u(x):=x^2+F1.1^27*x+F1.1^51; ^ User error: Identifier 'x' has not been declared or assigned >> v(x):=F1.1^39+F1.1^59; ^ User error: Illegal left hand side of an assignment statement >> C:=HyperellipticCurve(a*x^5+b*x^3+x,x); ^ User error: Identifier 'a' has not been declared or assigned >> N1:=#C; //over Fq ^ User error: Identifier 'C' has not been declared or assigned >> CC:=HyperellipticCurve(a*z^5+b*z^3+z,z); ^ User error: Identifier 'a' has not been declared or assigned >> N2:=#CC; //over Fq^2; ^ User error: Identifier 'CC' has not been declared or assigned done Total time: 0.200 seconds, Total memory usage: 3.34MB '218.44.' ************** MAGMA ***************** Host 218.44.255.46 (218.44.255.46) Time: Tue Dec 6 12:18:02 2005 Input: > k := Rationals(); > A := AffineSpace(k,2); > A; > f := x+y^2; > C := Curve(A,f); > C; > SingularPoints(C); > HasSingularPointsOverExtension(C); > Genus(C); > NumberOfPunctures(C); > P:=ProjectiveClosure(C); > P; > D:=AffinePatch(P,2); > D; > D2:=Curve(A,DefiningPolynomial(D)); > D2; > IsAnalyticallyIrreducible(D2,Origin(A)); > g := ResolutionGraph(D2,Origin(A)); > g; Output: Magma V2.11-10 Tue Dec 6 2005 12:18:02 on modular [Seed = 3272951652] ------------------------------------- Affine Space of dimension 2 Variables : x, y Curve over Rational Field defined by x + y^2 {@ @} false 0 1 Curve over Rational Field defined by $.1*$.3 + $.2^2 Curve over Rational Field defined by x*y + 1 Curve over Rational Field defined by x*y + 1 >> IsAnalyticallyIrreducible(D2,Origin(A)); ^ Runtime error in 'IsAnalyticallyIrreducible': Second argument must be a point of the first >> g := ResolutionGraph(D2,Origin(A)); ^ Runtime error in 'ResolutionGraph': Second argument must be a rational point on the first argument >> g; ^ User error: Identifier 'g' has not been declared or assigned Total time: 0.200 seconds, Total memory usage: 3.53MB '218.44.' ************** MAGMA ***************** Host 218.44.255.46 (218.44.255.46) Time: Tue Dec 6 12:17:45 2005 Input: > k := Rationals(); > A := AffineSpace(k,2); > A; > f := x+y^2; > C := Curve(A,f); > C; > SingularPoints(C); > HasSingularPointsOverExtension(C); > Genus(C); > NumberOfPunctures(C); > P:=ProjectiveClosure(C); > D:=AffinePatch(P,2); > D; > D2:=Curve(A,DefiningPolynomial(D)); > D2; > IsAnalyticallyIrreducible(D2,Origin(A)); > g := ResolutionGraph(D2,Origin(A)); > g; Output: Magma V2.11-10 Tue Dec 6 2005 12:17:45 on modular [Seed = 3289530384] ------------------------------------- Affine Space of dimension 2 Variables : x, y Curve over Rational Field defined by x + y^2 {@ @} false 0 1 Curve over Rational Field defined by x*y + 1 Curve over Rational Field defined by x*y + 1 >> IsAnalyticallyIrreducible(D2,Origin(A)); ^ Runtime error in 'IsAnalyticallyIrreducible': Second argument must be a point of the first >> g := ResolutionGraph(D2,Origin(A)); ^ Runtime error in 'ResolutionGraph': Second argument must be a rational point on the first argument >> g; ^ User error: Identifier 'g' has not been declared or assigned Total time: 0.210 seconds, Total memory usage: 3.53MB '218.44.' ************** MAGMA ***************** Host 218.44.255.46 (218.44.255.46) Time: Tue Dec 6 12:16:57 2005 Input: > k := Rationals(); > A := AffineSpace(k,2); > A; > f := x+y^2; > C := Curve(A,f); > C; > SingularPoints(C); > HasSingularPointsOverExtension(C); > Genus(C); > NumberOfPunctures(C); > P:=ProjectiveClosure(C); > P; > D:=AffinePatch(P,2); > D; > D2:=Curve(A,DefiningPolynomial(D)); > D2; > IsAnalyticallyIrreducible(D2,Origin(A)); > g := ResolutionGraph(D2,Origin(A)); > g; Output: Magma V2.11-10 Tue Dec 6 2005 12:16:56 on modular [Seed = 3441117480] ------------------------------------- Affine Space of dimension 2 Variables : x, y Curve over Rational Field defined by x + y^2 {@ @} false 0 1 Curve over Rational Field defined by $.1*$.3 + $.2^2 Curve over Rational Field defined by x*y + 1 Curve over Rational Field defined by x*y + 1 >> IsAnalyticallyIrreducible(D2,Origin(A)); ^ Runtime error in 'IsAnalyticallyIrreducible': Second argument must be a point of the first >> g := ResolutionGraph(D2,Origin(A)); ^ Runtime error in 'ResolutionGraph': Second argument must be a rational point on the first argument >> g; ^ User error: Identifier 'g' has not been declared or assigned Total time: 0.200 seconds, Total memory usage: 3.53MB '218.44.' ************** MAGMA ***************** Host 218.44.255.46 (218.44.255.46) Time: Tue Dec 6 12:16:10 2005 Input: > k := Rationals(); > A := AffineSpace(k,2); > A; > f := 1*x+0*y+0*(-x^3-y^2)+(0*x+1*y+1*(-x^3-y^2))^2; > C := Curve(A,f); > C; > SingularPoints(C); > HasSingularPointsOverExtension(C); > Genus(C); > NumberOfPunctures(C); > P:=ProjectiveClosure(C); > P; > D:=AffinePatch(P,2); > D; > D2:=Curve(A,DefiningPolynomial(D)); > D2; > IsAnalyticallyIrreducible(D2,Origin(A)); > g := ResolutionGraph(D2,Origin(A)); > g; Output: Magma V2.11-10 Tue Dec 6 2005 12:16:09 on modular [Seed = 4129321666] ------------------------------------- Affine Space of dimension 2 Variables : x, y Curve over Rational Field defined by x^6 + 2*x^3*y^2 - 2*x^3*y + x + y^4 - 2*y^3 + y^2 {@ @} false 2 1 Curve over Rational Field defined by $.1^6 + 2*$.1^3*$.2^2*$.3 - 2*$.1^3*$.2*$.3^2 + $.1*$.3^5 + $.2^4*$.3^2 - 2*$.2^3*$.3^3 + $.2^2*$.3^4 Curve over Rational Field defined by x^6 - 2*x^3*y^2 + 2*x^3*y + x*y^5 + y^4 - 2*y^3 + y^2 Curve over Rational Field defined by x^6 - 2*x^3*y^2 + 2*x^3*y + x*y^5 + y^4 - 2*y^3 + y^2 false The resolution graph on the Digraph Vertex Neighbours 1 ([ -2, 6, 3, 0 ]) 2 4 ; 2 ([ -2, 4, 2, 0 ]) 3 ; 3 ([ -2, 2, 1, 0 ]) ; 4 ([ -2, 8, 4, 0 ]) 5 ; 5 ([ -2, 10, 5, 0 ]) 6 ; 6 ([ -2, 12, 6, 0 ]) 7 ; 7 ([ -2, 14, 7, 0 ]) 8 ; 8 ([ -1, 16, 8, 2 ]) ; Total time: 0.320 seconds, Total memory usage: 4.26MB '218.44.' ************** MAGMA ***************** Host 218.44.255.46 (218.44.255.46) Time: Tue Dec 6 12:15:51 2005 Input: > k := Rationals(); > A := AffineSpace(k,2); > A; > f := 1*x+0*y+0*(-x^3-y^2)+(0*x+1*y+1*(-x^3-y^2))^2; > C := Curve(A,f); > C; > SingularPoints(C); > HasSingularPointsOverExtension(C); > Genus(C); > NumberOfPunctures(C); > P:=ProjectiveClosure(C); > D:=AffinePatch(P,2); > D; > D2:=Curve(A,DefiningPolynomial(D)); > D2; > IsAnalyticallyIrreducible(D2,Origin(A)); > g := ResolutionGraph(D2,Origin(A)); > g; Output: Magma V2.11-10 Tue Dec 6 2005 12:15:51 on modular [Seed = 4214063057] ------------------------------------- Affine Space of dimension 2 Variables : x, y Curve over Rational Field defined by x^6 + 2*x^3*y^2 - 2*x^3*y + x + y^4 - 2*y^3 + y^2 {@ @} false 2 1 Curve over Rational Field defined by x^6 - 2*x^3*y^2 + 2*x^3*y + x*y^5 + y^4 - 2*y^3 + y^2 Curve over Rational Field defined by x^6 - 2*x^3*y^2 + 2*x^3*y + x*y^5 + y^4 - 2*y^3 + y^2 false The resolution graph on the Digraph Vertex Neighbours 1 ([ -2, 6, 3, 0 ]) 2 4 ; 2 ([ -2, 4, 2, 0 ]) 3 ; 3 ([ -2, 2, 1, 0 ]) ; 4 ([ -2, 8, 4, 0 ]) 5 ; 5 ([ -2, 10, 5, 0 ]) 6 ; 6 ([ -2, 12, 6, 0 ]) 7 ; 7 ([ -2, 14, 7, 0 ]) 8 ; 8 ([ -1, 16, 8, 2 ]) ; Total time: 0.320 seconds, Total memory usage: 4.26MB '218.44.' ************** MAGMA ***************** Host 218.44.255.46 (218.44.255.46) Time: Tue Dec 6 12:04:53 2005 Input: > k := Rationals(); > A := AffineSpace(k,2); > A; > f := 1*x+0*y+0*(-x^3-y^2)+(0*x+1*y+1*(-x^3-y^2))^2; > C := Curve(A,f); > C; > SingularPoints(C); > HasSingularPointsOverExtension(C); > Genus(C); > NumberOfPunctures(C); > P:=ProjectiveClosure(C); > D:=AffinePatch(P,2); > D; > D2:=Curve(A,DefiningPolynomial(D)); > D2; > IsAnalyticallyIrreducible(D2,Origin(A)); > g := ResolutionGraph(D2,Origin(A)); > g; Output: Magma V2.11-10 Tue Dec 6 2005 12:04:53 on modular [Seed = 182226386] ------------------------------------- Affine Space of dimension 2 Variables : x, y Curve over Rational Field defined by x^6 + 2*x^3*y^2 - 2*x^3*y + x + y^4 - 2*y^3 + y^2 {@ @} false 2 1 Curve over Rational Field defined by x^6 - 2*x^3*y^2 + 2*x^3*y + x*y^5 + y^4 - 2*y^3 + y^2 Curve over Rational Field defined by x^6 - 2*x^3*y^2 + 2*x^3*y + x*y^5 + y^4 - 2*y^3 + y^2 false The resolution graph on the Digraph Vertex Neighbours 1 ([ -2, 6, 3, 0 ]) 2 4 ; 2 ([ -2, 4, 2, 0 ]) 3 ; 3 ([ -2, 2, 1, 0 ]) ; 4 ([ -2, 8, 4, 0 ]) 5 ; 5 ([ -2, 10, 5, 0 ]) 6 ; 6 ([ -2, 12, 6, 0 ]) 7 ; 7 ([ -2, 14, 7, 0 ]) 8 ; 8 ([ -1, 16, 8, 2 ]) ; Total time: 0.330 seconds, Total memory usage: 4.26MB '218.44.' ************** MAGMA ***************** Host 218.44.255.46 (218.44.255.46) Time: Tue Dec 6 11:58:04 2005 Input: > k := Rationals(); > A := AffineSpace(k,2); > A; > f := x; > C := Curve(A,f); > C; > SingularPoints(C); > HasSingularPointsOverExtension(C); > Genus(C); > NumberOfPunctures(C); > P:=ProjectiveClosure(C); > D:=AffinePatch(P,2); > D; > D2:=Curve(A,DefiningPolynomial(D)); > D2; > IsAnalyticallyIrreducible(D2,Origin(A)); > g := ResolutionGraph(D2,Origin(A)); > g; Output: Magma V2.11-10 Tue Dec 6 2005 11:58:03 on modular [Seed = 2487358689] ------------------------------------- Affine Space of dimension 2 Variables : x, y Curve over Rational Field defined by x {@ @} false 0 1 Curve over Rational Field defined by x Curve over Rational Field defined by x true >> g := ResolutionGraph(D2,Origin(A)); ^ Runtime error in 'ResolutionGraph': Point is not singular on curve >> g; ^ User error: Identifier 'g' has not been declared or assigned Total time: 0.220 seconds, Total memory usage: 3.53MB '129.162' ************** MAGMA ***************** Host 129.162.1.32 (129.162.1.32) Time: Tue Dec 6 10:49:56 2005 Input: 5+3 Output: Magma V2.11-10 Tue Dec 6 2005 10:49:56 on modular [Seed = 4197190320] ------------------------------------- 8 Total time: 0.200 seconds, Total memory usage: 3.24MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Tue Dec 6 10:18:40 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*A*Transpose(P)-A)); G:=GroebnerBasis(I); S:=FunctionField(K); p1 := (2*a-1)*c*t; p2 := c*t - a*(b-1/2)/(a-1/2); px := -a * p1; pz := p1 * p2; py := a*c*(c-a)*t^2 + a*(a^2-2*a*b*c+a*b-a+1/2*c)/(a-1/2)*t + a^2*(a+b-1)*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(c*(a-1/2)^2); q := a*c*(a+c-1)*t^2 - 2*a*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(a-1/2)*t + a^2*(a+b-1)*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(c*(a-1/2)^2); f7:=pz/q; f8:=px/q; f9:=py/q; f :=[-Evaluate(G[1],[0,0,0,0,0,0,f7,f8,f9]), -Evaluate(G[2],[0,0,0,0,0,0,f7,f8,f9]), -Evaluate(G[3],[0,0,0,0,0,0,f7,f8,f9]), -Evaluate(G[4],[0,0,0,0,0,0,f7,f8,f9]), -Evaluate(G[5],[0,0,0,0,0,0,f7,f8,f9]), -Evaluate(G[6],[0,0,0,0,0,0,f7,f8,f9]), f7, f8, f9]; Q:=Matrix(S,3,3,f); print (Q*A*Transpose(Q)-A); for i in [1..9] do d:=Evaluate(Derivative(f[i]),0); print i; print Factorization(Numerator(d)); print Factorization(Denominator(d)); end for; Output: Magma V2.11-10 Tue Dec 6 2005 10:18:38 on modular [Seed = 887282836] ------------------------------------- [0 0 0] [0 0 0] [0 0 0] 1 [ , , ] [ , , ] 2 [ , , ] [ , ] 3 [ , , ] [ , ] 4 [ , , ] [ , , ] 5 [ , , ] [ , , ] 6 [ , , ] [ , , ] 7 [ , , ] [ , , ] 8 [ , ] [ , , ] 9 [ , ] [ , ] Total time: 1.679 seconds, Total memory usage: 4.60MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Tue Dec 6 10:18:22 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*A*Transpose(P)-A)); G:=GroebnerBasis(I); S:=FunctionField(K); p1 := (2*a-1)*c*t; p2 := c*t - a*(b-1/2)/(a-1/2); px := -a * p1; pz := p1 * p2; py := a*c*(c-a)*t^2 + a*(a^2-2*a*b*c+a*b-a+1/2*c)/(a-1/2)*t + a^2*(a+b-1)*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(c*(a-1/2)^2); q := a*c*(a+c-1)*t^2 - 2*a*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(a-1/2)*t + a^2*(a+b-1)*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(c*(a-1/2)^2); f7:=pz/q; f8:=px/q; f9:=py/q; f :=[-Evaluate(G[1],[0,0,0,0,0,0,f7,f8,f9]), -Evaluate(G[2],[0,0,0,0,0,0,f7,f8,f9]), -Evaluate(G[3],[0,0,0,0,0,0,f7,f8,f9]), -Evaluate(G[4],[0,0,0,0,0,0,f7,f8,f9]), -Evaluate(G[5],[0,0,0,0,0,0,f7,f8,f9]), -Evaluate(G[6],[0,0,0,0,0,0,f7,f8,f9]), f7, f8, f9]; Q:=Matrix(S,3,3,f); print (Q*A*Transpose(Q)-A); for i in [1..9] do: d:=Evaluate(Derivative(f[1]),0); print i; print Factorization(Numerator(d)); print Factorization(Denominator(d)); end for; Output: Magma V2.11-10 Tue Dec 6 2005 10:18:20 on modular [Seed = 1005442025] ------------------------------------- [0 0 0] [0 0 0] [0 0 0] >> for i in [1..9] do: ^ User error: bad syntax >> print i; ^ User error: Identifier 'i' has not been declared or assigned [ , , ] [ , , ] >> end for; ^ User error: bad syntax Total time: 1.649 seconds, Total memory usage: 4.60MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Tue Dec 6 10:16:24 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*A*Transpose(P)-A)); G:=GroebnerBasis(I); S:=FunctionField(K); p1 := (2*a-1)*c*t; p2 := c*t - a*(b-1/2)/(a-1/2); px := -a * p1; pz := p1 * p2; py := a*c*(c-a)*t^2 + a*(a^2-2*a*b*c+a*b-a+1/2*c)/(a-1/2)*t + a^2*(a+b-1)*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(c*(a-1/2)^2); q := a*c*(a+c-1)*t^2 - 2*a*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(a-1/2)*t + a^2*(a+b-1)*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(c*(a-1/2)^2); f7:=pz/q; f8:=px/q; f9:=py/q; f :=[-Evaluate(G[1],[0,0,0,0,0,0,f7,f8,f9]), -Evaluate(G[2],[0,0,0,0,0,0,f7,f8,f9]), -Evaluate(G[3],[0,0,0,0,0,0,f7,f8,f9]), -Evaluate(G[4],[0,0,0,0,0,0,f7,f8,f9]), -Evaluate(G[5],[0,0,0,0,0,0,f7,f8,f9]), -Evaluate(G[6],[0,0,0,0,0,0,f7,f8,f9]), f7, f8, f9]; Q:=Matrix(S,3,3,f); print (Q*A*Transpose(Q)-A); d1:=Evaluate(Derivative(f[1]),0); print d1; print Factorization(Numerator(d1)); print Factorization(Denominator(d1)); Output: Magma V2.11-10 Tue Dec 6 2005 10:16:22 on modular [Seed = 586467850] ------------------------------------- [0 0 0] [0 0 0] [0 0 0] (-a^2*b*c - a^2*c^2 + a^2*c + a*b*c + a*c^2 - a*c - 1/4*b*c - 1/4*c^2 + 1/4*c)/(a^3*b*c - 1/4*a^3 + a^2*b^2*c - a^2*b*c - 1/2*a^2*b - 1/4*a^2*c + 1/2*a^2 - 1/4*a*b^2 - 1/4*a*b*c + 1/2*a*b + 1/4*a*c - 1/4*a) [ , , ] [ , , ] Total time: 1.649 seconds, Total memory usage: 4.60MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Tue Dec 6 10:13:57 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*A*Transpose(P)-A)); G:=GroebnerBasis(I); S:=FunctionField(K); p1 := (2*a-1)*c*t; p2 := c*t - a*(b-1/2)/(a-1/2); px := -a * p1; pz := p1 * p2; py := a*c*(c-a)*t^2 + a*(a^2-2*a*b*c+a*b-a+1/2*c)/(a-1/2)*t + a^2*(a+b-1)*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(c*(a-1/2)^2); q := a*c*(a+c-1)*t^2 - 2*a*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(a-1/2)*t + a^2*(a+b-1)*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(c*(a-1/2)^2); f7:=pz/q; f8:=px/q; f9:=py/q; f1:=-Evaluate(G[1],[0,0,0,0,0,0,f7,f8,f9]); f2:=-Evaluate(G[2],[0,0,0,0,0,0,f7,f8,f9]); f3:=-Evaluate(G[3],[0,0,0,0,0,0,f7,f8,f9]); f4:=-Evaluate(G[4],[0,0,0,0,0,0,f7,f8,f9]); f5:=-Evaluate(G[5],[0,0,0,0,0,0,f7,f8,f9]); f6:=-Evaluate(G[6],[0,0,0,0,0,0,f7,f8,f9]); Q:=Matrix(S,3,3,[f1,f2,f3,f4,f5,f6,f7,f8,f9]); print (Q*A*Transpose(Q)-A); d1:=Evaluate(Derivative(f1),0); print d1; print Factorization(Numerator(d1)); print Factorization(Denominator(d1)); Output: Magma V2.11-10 Tue Dec 6 2005 10:13:55 on modular [Seed = 704626932] ------------------------------------- [0 0 0] [0 0 0] [0 0 0] (-a^2*b*c - a^2*c^2 + a^2*c + a*b*c + a*c^2 - a*c - 1/4*b*c - 1/4*c^2 + 1/4*c)/(a^3*b*c - 1/4*a^3 + a^2*b^2*c - a^2*b*c - 1/2*a^2*b - 1/4*a^2*c + 1/2*a^2 - 1/4*a*b^2 - 1/4*a*b*c + 1/2*a*b + 1/4*a*c - 1/4*a) [ , , ] [ , , ] Total time: 1.649 seconds, Total memory usage: 4.60MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Tue Dec 6 10:13:04 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*A*Transpose(P)-A)); G:=GroebnerBasis(I); S:=FunctionField(K); p1 := (2*a-1)*c*t; p2 := c*t - a*(b-1/2)/(a-1/2); px := -a * p1; pz := p1 * p2; py := a*c*(c-a)*t^2 + a*(a^2-2*a*b*c+a*b-a+1/2*c)/(a-1/2)*t + a^2*(a+b-1)*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(c*(a-1/2)^2); q := a*c*(a+c-1)*t^2 - 2*a*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(a-1/2)*t + a^2*(a+b-1)*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(c*(a-1/2)^2); f7:=pz/q; f8:=px/q; f9:=py/q; f1:=-Evaluate(G[1],[0,0,0,0,0,0,f7,f8,f9]); f2:=-Evaluate(G[2],[0,0,0,0,0,0,f7,f8,f9]); f3:=-Evaluate(G[3],[0,0,0,0,0,0,f7,f8,f9]); f4:=-Evaluate(G[4],[0,0,0,0,0,0,f7,f8,f9]); f5:=-Evaluate(G[5],[0,0,0,0,0,0,f7,f8,f9]); f6:=-Evaluate(G[6],[0,0,0,0,0,0,f7,f8,f9]); Q:=Matrix(S,3,3,[f1,f2,f3,f4,f5,f6,f7,f8,f9]); print (Q*A*Transpose(Q)-A); d1:=Evaluate(Derivative(f1),0); print d1; print Factorization(Nominator(d1)); print Factorization(Denominator(d1)); Output: Magma V2.11-10 Tue Dec 6 2005 10:13:02 on modular [Seed = 754631996] ------------------------------------- [0 0 0] [0 0 0] [0 0 0] (-a^2*b*c - a^2*c^2 + a^2*c + a*b*c + a*c^2 - a*c - 1/4*b*c - 1/4*c^2 + 1/4*c)/(a^3*b*c - 1/4*a^3 + a^2*b^2*c - a^2*b*c - 1/2*a^2*b - 1/4*a^2*c + 1/2*a^2 - 1/4*a*b^2 - 1/4*a*b*c + 1/2*a*b + 1/4*a*c - 1/4*a) >> print Factorization(Nominator(d1)); ^ User error: Identifier 'Nominator' has not been declared or assigned [ , , ] Total time: 1.659 seconds, Total memory usage: 4.60MB '219.166' ************** MAGMA ***************** Host 219.166.29.134 (219.166.29.134) Time: Tue Dec 6 10:11:29 2005 Input: > k := Rationals(); > A := AffineSpace(k,2); > A; > f := 1+1*x+2*y+2*(-1*x*y-x^3)+(2+3*x+0*y+2*(-1*x*y-x^3))^2+(1+4*x+2*y)^3; > C := Curve(A,f); > C; > SingularPoints(C); > HasSingularPointsOverExtension(C); > Genus(C); > NumberOfPunctures(C); > P:=ProjectiveClosure(C); > D:=AffinePatch(P,2); > D; > D2:=Curve(A,DefiningPolynomial(D)); > D2; > IsAnalyticallyIrreducible(D2,Origin(A)); > g := ResolutionGraph(D2,Origin(A)); > g; Output: Magma V2.11-10 Tue Dec 6 2005 10:11:28 on modular [Seed = 1384430364] ------------------------------------- Affine Space of dimension 2 Variables : x, y Curve over Rational Field defined by 4*x^6 + 8*x^4*y - 12*x^4 + 54*x^3 + 4*x^2*y^2 + 84*x^2*y + 57*x^2 + 48*x*y^2 + 38*x*y + 25*x + 8*y^3 + 12*y^2 + 8*y + 6 {@ @} false 4 1 Curve over Rational Field defined by x^6 - 3*x^4*y^2 + 2*x^4*y + 27/2*x^3*y^3 + 57/4*x^2*y^4 + 21*x^2*y^3 + x^2*y^2 + 25/4*x*y^5 + 19/2*x*y^4 + 12*x*y^3 + 3/2*y^6 + 2*y^5 + 3*y^4 + 2*y^3 Curve over Rational Field defined by x^6 - 3*x^4*y^2 + 2*x^4*y + 27/2*x^3*y^3 + 57/4*x^2*y^4 + 21*x^2*y^3 + x^2*y^2 + 25/4*x*y^5 + 19/2*x*y^4 + 12*x*y^3 + 3/2*y^6 + 2*y^5 + 3*y^4 + 2*y^3 false The resolution graph on the Digraph Vertex Neighbours 1 ([ -1, 6, 2, 3 ]) 2 ; 2 ([ -2, 3, 1, 0 ]) ; Total time: 1.070 seconds, Total memory usage: 3.91MB '219.166' ************** MAGMA ***************** Host 219.166.29.134 (219.166.29.134) Time: Tue Dec 6 10:10:58 2005 Input: > k := Rationals(); > A := AffineSpace(k,2); > A; > f := 1+1*x+2*y+2*(-1*x*y-x^3)+(2+3*x+1*y+2*(-1*x*y-x^3))^2+(1+4*x+2*y)^3; > C := Curve(A,f); > C; > SingularPoints(C); > HasSingularPointsOverExtension(C); > Genus(C); > NumberOfPunctures(C); > P:=ProjectiveClosure(C); > D:=AffinePatch(P,2); > D; > D2:=Curve(A,DefiningPolynomial(D)); > D2; > IsAnalyticallyIrreducible(D2,Origin(A)); > g := ResolutionGraph(D2,Origin(A)); > g; Output: Magma V2.11-10 Tue Dec 6 2005 10:10:57 on modular [Seed = 1502588982] ------------------------------------- Affine Space of dimension 2 Variables : x, y Curve over Rational Field defined by 4*x^6 + 8*x^4*y - 12*x^4 - 4*x^3*y + 54*x^3 + 4*x^2*y^2 + 84*x^2*y + 57*x^2 + 44*x*y^2 + 44*x*y + 25*x + 8*y^3 + 13*y^2 + 12*y + 6 {@ @} false 4 1 Curve over Rational Field defined by x^6 - 3*x^4*y^2 + 2*x^4*y + 27/2*x^3*y^3 - x^3*y^2 + 57/4*x^2*y^4 + 21*x^2*y^3 + x^2*y^2 + 25/4*x*y^5 + 11*x*y^4 + 11*x*y^3 + 3/2*y^6 + 3*y^5 + 13/4*y^4 + 2*y^3 Curve over Rational Field defined by x^6 - 3*x^4*y^2 + 2*x^4*y + 27/2*x^3*y^3 - x^3*y^2 + 57/4*x^2*y^4 + 21*x^2*y^3 + x^2*y^2 + 25/4*x*y^5 + 11*x*y^4 + 11*x*y^3 + 3/2*y^6 + 3*y^5 + 13/4*y^4 + 2*y^3 false The resolution graph on the Digraph Vertex Neighbours 1 ([ -1, 6, 2, 3 ]) 2 ; 2 ([ -2, 3, 1, 0 ]) ; Total time: 1.350 seconds, Total memory usage: 4.01MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Tue Dec 6 10:10:56 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*A*Transpose(P)-A)); G:=GroebnerBasis(I); S:=FunctionField(K); p1 := (2*a-1)*c*t; p2 := c*t - a*(b-1/2)/(a-1/2); px := -a * p1; pz := p1 * p2; py := a*c*(c-a)*t^2 + a*(a^2-2*a*b*c+a*b-a+1/2*c)/(a-1/2)*t + a^2*(a+b-1)*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(c*(a-1/2)^2); q := a*c*(a+c-1)*t^2 - 2*a*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(a-1/2)*t + a^2*(a+b-1)*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(c*(a-1/2)^2); f7:=pz/q; f8:=px/q; f9:=py/q; f1:=-Evaluate(G[1],[0,0,0,0,0,0,f7,f8,f9]); f2:=-Evaluate(G[2],[0,0,0,0,0,0,f7,f8,f9]); f3:=-Evaluate(G[3],[0,0,0,0,0,0,f7,f8,f9]); f4:=-Evaluate(G[4],[0,0,0,0,0,0,f7,f8,f9]); f5:=-Evaluate(G[5],[0,0,0,0,0,0,f7,f8,f9]); f6:=-Evaluate(G[6],[0,0,0,0,0,0,f7,f8,f9]); Q:=Matrix(S,3,3,[f1,f2,f3,f4,f5,f6,f7,f8,f9]); print (Q*A*Transpose(Q)-A); L:=PolynomialRing(RationalField(),3); d1:=Evaluate(Derivative(f1),0); print d1; print Factorization(d1); Output: Magma V2.11-10 Tue Dec 6 2005 10:10:54 on modular [Seed = 1552594636] ------------------------------------- [0 0 0] [0 0 0] [0 0 0] (-a^2*b*c - a^2*c^2 + a^2*c + a*b*c + a*c^2 - a*c - 1/4*b*c - 1/4*c^2 + 1/4*c)/(a^3*b*c - 1/4*a^3 + a^2*b^2*c - a^2*b*c - 1/2*a^2*b - 1/4*a^2*c + 1/2*a^2 - 1/4*a*b^2 - 1/4*a*b*c + 1/2*a*b + 1/4*a*c - 1/4*a) >> print Factorization(d1); ^ Runtime error in 'Factorization': Bad argument types Argument types given: FldFunRatMElt Total time: 1.669 seconds, Total memory usage: 4.60MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Tue Dec 6 10:10:40 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*A*Transpose(P)-A)); G:=GroebnerBasis(I); S:=FunctionField(K); p1 := (2*a-1)*c*t; p2 := c*t - a*(b-1/2)/(a-1/2); px := -a * p1; pz := p1 * p2; py := a*c*(c-a)*t^2 + a*(a^2-2*a*b*c+a*b-a+1/2*c)/(a-1/2)*t + a^2*(a+b-1)*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(c*(a-1/2)^2); q := a*c*(a+c-1)*t^2 - 2*a*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(a-1/2)*t + a^2*(a+b-1)*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(c*(a-1/2)^2); f7:=pz/q; f8:=px/q; f9:=py/q; f1:=-Evaluate(G[1],[0,0,0,0,0,0,f7,f8,f9]); f2:=-Evaluate(G[2],[0,0,0,0,0,0,f7,f8,f9]); f3:=-Evaluate(G[3],[0,0,0,0,0,0,f7,f8,f9]); f4:=-Evaluate(G[4],[0,0,0,0,0,0,f7,f8,f9]); f5:=-Evaluate(G[5],[0,0,0,0,0,0,f7,f8,f9]); f6:=-Evaluate(G[6],[0,0,0,0,0,0,f7,f8,f9]); Q:=Matrix(S,3,3,[f1,f2,f3,f4,f5,f6,f7,f8,f9]); print (Q*A*Transpose(Q)-A); L:=PolynomialRing(RationalField(),3); d1:=Evaluate(Derivative(f1),0); print d1; print L!d1; Output: Magma V2.11-10 Tue Dec 6 2005 10:10:38 on modular [Seed = 1133881795] ------------------------------------- [0 0 0] [0 0 0] [0 0 0] (-a^2*b*c - a^2*c^2 + a^2*c + a*b*c + a*c^2 - a*c - 1/4*b*c - 1/4*c^2 + 1/4*c)/(a^3*b*c - 1/4*a^3 + a^2*b^2*c - a^2*b*c - 1/2*a^2*b - 1/4*a^2*c + 1/2*a^2 - 1/4*a*b^2 - 1/4*a*b*c + 1/2*a*b + 1/4*a*c - 1/4*a) >> print L!d1; ^ Runtime error in '!': Illegal coercion Total time: 1.649 seconds, Total memory usage: 4.60MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Tue Dec 6 10:09:17 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*A*Transpose(P)-A)); G:=GroebnerBasis(I); S:=FunctionField(K); p1 := (2*a-1)*c*t; p2 := c*t - a*(b-1/2)/(a-1/2); px := -a * p1; pz := p1 * p2; py := a*c*(c-a)*t^2 + a*(a^2-2*a*b*c+a*b-a+1/2*c)/(a-1/2)*t + a^2*(a+b-1)*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(c*(a-1/2)^2); q := a*c*(a+c-1)*t^2 - 2*a*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(a-1/2)*t + a^2*(a+b-1)*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(c*(a-1/2)^2); f7:=pz/q; f8:=px/q; f9:=py/q; f1:=-Evaluate(G[1],[0,0,0,0,0,0,f7,f8,f9]); f2:=-Evaluate(G[2],[0,0,0,0,0,0,f7,f8,f9]); f3:=-Evaluate(G[3],[0,0,0,0,0,0,f7,f8,f9]); f4:=-Evaluate(G[4],[0,0,0,0,0,0,f7,f8,f9]); f5:=-Evaluate(G[5],[0,0,0,0,0,0,f7,f8,f9]); f6:=-Evaluate(G[6],[0,0,0,0,0,0,f7,f8,f9]); Q:=Matrix(S,3,3,[f1,f2,f3,f4,f5,f6,f7,f8,f9]); print (Q*A*Transpose(Q)-A); L:=PolynomialRing(RationalField(),3); print (L!(Evaluate(Derivative(f1),0))); Output: Magma V2.11-10 Tue Dec 6 2005 10:09:15 on modular [Seed = 1218355901] ------------------------------------- [0 0 0] [0 0 0] [0 0 0] >> print (L!(Evaluate(Derivative(f1),0))); ^ Runtime error in '!': Illegal coercion Total time: 1.649 seconds, Total memory usage: 4.60MB '218.44.' ************** MAGMA ***************** Host 218.44.255.46 (218.44.255.46) Time: Tue Dec 6 10:09:01 2005 Input: > k := Rationals(); > A := AffineSpace(k,2); > A; > f := 1+1*x+1*y+2*(-1*x*y-x^3)+(2+3*x+1*y+2*(-1*x*y-x^3))^2+(1+4*x+2*y)^3; > C := Curve(A,f); > C; > SingularPoints(C); > HasSingularPointsOverExtension(C); > Genus(C); > NumberOfPunctures(C); > P:=ProjectiveClosure(C); > D:=AffinePatch(P,2); > D; > D2:=Curve(A,DefiningPolynomial(D)); > D2; > IsAnalyticallyIrreducible(D2,Origin(A)); > g := ResolutionGraph(D2,Origin(A)); > g; Output: Magma V2.11-10 Tue Dec 6 2005 10:09:00 on modular [Seed = 2074972569] ------------------------------------- Affine Space of dimension 2 Variables : x, y Curve over Rational Field defined by 4*x^6 + 8*x^4*y - 12*x^4 - 4*x^3*y + 54*x^3 + 4*x^2*y^2 + 84*x^2*y + 57*x^2 + 44*x*y^2 + 44*x*y + 25*x + 8*y^3 + 13*y^2 + 11*y + 6 {@ @} false 4 1 Curve over Rational Field defined by x^6 - 3*x^4*y^2 + 2*x^4*y + 27/2*x^3*y^3 - x^3*y^2 + 57/4*x^2*y^4 + 21*x^2*y^3 + x^2*y^2 + 25/4*x*y^5 + 11*x*y^4 + 11*x*y^3 + 3/2*y^6 + 11/4*y^5 + 13/4*y^4 + 2*y^3 Curve over Rational Field defined by x^6 - 3*x^4*y^2 + 2*x^4*y + 27/2*x^3*y^3 - x^3*y^2 + 57/4*x^2*y^4 + 21*x^2*y^3 + x^2*y^2 + 25/4*x*y^5 + 11*x*y^4 + 11*x*y^3 + 3/2*y^6 + 11/4*y^5 + 13/4*y^4 + 2*y^3 false The resolution graph on the Digraph Vertex Neighbours 1 ([ -1, 6, 2, 3 ]) 2 ; 2 ([ -2, 3, 1, 0 ]) ; Total time: 0.930 seconds, Total memory usage: 3.91MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Tue Dec 6 10:08:25 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*A*Transpose(P)-A)); G:=GroebnerBasis(I); S:=FunctionField(K); p1 := (2*a-1)*c*t; p2 := c*t - a*(b-1/2)/(a-1/2); px := -a * p1; pz := p1 * p2; py := a*c*(c-a)*t^2 + a*(a^2-2*a*b*c+a*b-a+1/2*c)/(a-1/2)*t + a^2*(a+b-1)*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(c*(a-1/2)^2); q := a*c*(a+c-1)*t^2 - 2*a*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(a-1/2)*t + a^2*(a+b-1)*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(c*(a-1/2)^2); f7:=pz/q; f8:=px/q; f9:=py/q; f1:=-Evaluate(G[1],[0,0,0,0,0,0,f7,f8,f9]); f2:=-Evaluate(G[2],[0,0,0,0,0,0,f7,f8,f9]); f3:=-Evaluate(G[3],[0,0,0,0,0,0,f7,f8,f9]); f4:=-Evaluate(G[4],[0,0,0,0,0,0,f7,f8,f9]); f5:=-Evaluate(G[5],[0,0,0,0,0,0,f7,f8,f9]); f6:=-Evaluate(G[6],[0,0,0,0,0,0,f7,f8,f9]); Q:=Matrix(S,3,3,[f1,f2,f3,f4,f5,f6,f7,f8,f9]); print (Q*A*Transpose(Q)-A); print Evaluate(Derivative(f1),0); Output: Magma V2.11-10 Tue Dec 6 2005 10:08:23 on modular [Seed = 2124977271] ------------------------------------- [0 0 0] [0 0 0] [0 0 0] (-a^2*b*c - a^2*c^2 + a^2*c + a*b*c + a*c^2 - a*c - 1/4*b*c - 1/4*c^2 + 1/4*c)/(a^3*b*c - 1/4*a^3 + a^2*b^2*c - a^2*b*c - 1/2*a^2*b - 1/4*a^2*c + 1/2*a^2 - 1/4*a*b^2 - 1/4*a*b*c + 1/2*a*b + 1/4*a*c - 1/4*a) Total time: 1.649 seconds, Total memory usage: 4.60MB '218.44.' ************** MAGMA ***************** Host 218.44.255.46 (218.44.255.46) Time: Tue Dec 6 10:08:03 2005 Input: > k := Rationals(); > A := AffineSpace(k,2); > A; > f := 1+1*x+1*y+2*(-0*x*y-x^3)+(2+3*x+1*y+2*(-0*x*y-x^3))^2+(1+4*x+2*y)^3; > C := Curve(A,f); > C; > SingularPoints(C); > HasSingularPointsOverExtension(C); > Genus(C); > NumberOfPunctures(C); > P:=ProjectiveClosure(C); > D:=AffinePatch(P,2); > D; > D2:=Curve(A,DefiningPolynomial(D)); > D2; > IsAnalyticallyIrreducible(D2,Origin(A)); > g := ResolutionGraph(D2,Origin(A)); > g; Output: Magma V2.11-10 Tue Dec 6 2005 10:08:02 on modular [Seed = 1706266457] ------------------------------------- Affine Space of dimension 2 Variables : x, y Curve over Rational Field defined by 4*x^6 - 12*x^4 - 4*x^3*y + 54*x^3 + 96*x^2*y + 57*x^2 + 48*x*y^2 + 54*x*y + 25*x + 8*y^3 + 13*y^2 + 11*y + 6 {@ @} false 4 1 Curve over Rational Field defined by x^6 - 3*x^4*y^2 + 27/2*x^3*y^3 - x^3*y^2 + 57/4*x^2*y^4 + 24*x^2*y^3 + 25/4*x*y^5 + 27/2*x*y^4 + 12*x*y^3 + 3/2*y^6 + 11/4*y^5 + 13/4*y^4 + 2*y^3 Curve over Rational Field defined by x^6 - 3*x^4*y^2 + 27/2*x^3*y^3 - x^3*y^2 + 57/4*x^2*y^4 + 24*x^2*y^3 + 25/4*x*y^5 + 27/2*x*y^4 + 12*x*y^3 + 3/2*y^6 + 11/4*y^5 + 13/4*y^4 + 2*y^3 false The resolution graph on the Digraph Vertex Neighbours 1 ([ -1, 6, 2, 3 ]) 2 ; 2 ([ -2, 3, 1, 0 ]) ; Total time: 0.870 seconds, Total memory usage: 4.26MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Tue Dec 6 10:07:21 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*A*Transpose(P)-A)); G:=GroebnerBasis(I); S:=FunctionField(K); p1 := (2*a-1)*c*t; p2 := c*t - a*(b-1/2)/(a-1/2); px := -a * p1; pz := p1 * p2; py := a*c*(c-a)*t^2 + a*(a^2-2*a*b*c+a*b-a+1/2*c)/(a-1/2)*t + a^2*(a+b-1)*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(c*(a-1/2)^2); q := a*c*(a+c-1)*t^2 - 2*a*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(a-1/2)*t + a^2*(a+b-1)*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(c*(a-1/2)^2); f7:=pz/q; f8:=px/q; f9:=py/q; f1:=-Evaluate(G[1],[0,0,0,0,0,0,f7,f8,f9]); f2:=-Evaluate(G[2],[0,0,0,0,0,0,f7,f8,f9]); f3:=-Evaluate(G[3],[0,0,0,0,0,0,f7,f8,f9]); f4:=-Evaluate(G[4],[0,0,0,0,0,0,f7,f8,f9]); f5:=-Evaluate(G[5],[0,0,0,0,0,0,f7,f8,f9]); f6:=-Evaluate(G[6],[0,0,0,0,0,0,f7,f8,f9]); Q:=Matrix(S,3,3,[f1,f2,f3,f4,f5,f6,f7,f8,f9]); print (Q*A*Transpose(Q)-A); Output: Magma V2.11-10 Tue Dec 6 2005 10:07:20 on modular [Seed = 1824163370] ------------------------------------- [0 0 0] [0 0 0] [0 0 0] Total time: 1.649 seconds, Total memory usage: 4.60MB '218.44.' ************** MAGMA ***************** Host 218.44.255.46 (218.44.255.46) Time: Tue Dec 6 10:07:21 2005 Input: > k := Rationals(); > A := AffineSpace(k,2); > A; > f := 1+1*x+0*y+2*(-0*x*y-x^3)+(2+3*x+1*y+2*(-0*x*y-x^3))^2+(2+4*x+3*y)^3; > C := Curve(A,f); > C; > SingularPoints(C); > HasSingularPointsOverExtension(C); > Genus(C); > NumberOfPunctures(C); > P:=ProjectiveClosure(C); > D:=AffinePatch(P,2); > D; > D2:=Curve(A,DefiningPolynomial(D)); > D2; > IsAnalyticallyIrreducible(D2,Origin(A)); > g := ResolutionGraph(D2,Origin(A)); > g; Output: Magma V2.11-10 Tue Dec 6 2005 10:07:19 on modular [Seed = 2420538072] ------------------------------------- Affine Space of dimension 2 Variables : x, y Curve over Rational Field defined by 4*x^6 - 12*x^4 - 4*x^3*y + 54*x^3 + 144*x^2*y + 105*x^2 + 108*x*y^2 + 150*x*y + 61*x + 27*y^3 + 55*y^2 + 40*y + 13 {@ @} false 4 1 Curve over Rational Field defined by x^6 - 3*x^4*y^2 + 27/2*x^3*y^3 - x^3*y^2 + 105/4*x^2*y^4 + 36*x^2*y^3 + 61/4*x*y^5 + 75/2*x*y^4 + 27*x*y^3 + 13/4*y^6 + 10*y^5 + 55/4*y^4 + 27/4*y^3 Curve over Rational Field defined by x^6 - 3*x^4*y^2 + 27/2*x^3*y^3 - x^3*y^2 + 105/4*x^2*y^4 + 36*x^2*y^3 + 61/4*x*y^5 + 75/2*x*y^4 + 27*x*y^3 + 13/4*y^6 + 10*y^5 + 55/4*y^4 + 27/4*y^3 false The resolution graph on the Digraph Vertex Neighbours 1 ([ -1, 6, 2, 3 ]) 2 ; 2 ([ -2, 3, 1, 0 ]) ; Total time: 2.189 seconds, Total memory usage: 4.24MB '218.44.' ************** MAGMA ***************** Host 218.44.255.46 (218.44.255.46) Time: Tue Dec 6 10:06:38 2005 Input: > k := Rationals(); > A := AffineSpace(k,2); > A; > f := 1+1*x+1*y+2*(-0*x*y-x^3)+(2+3*x+1*y+2*(-0*x*y-x^3))^2+(2+4*x+3*y)^3; > C := Curve(A,f); > C; > SingularPoints(C); > HasSingularPointsOverExtension(C); > Genus(C); > NumberOfPunctures(C); > P:=ProjectiveClosure(C); > D:=AffinePatch(P,2); > D; > D2:=Curve(A,DefiningPolynomial(D)); > D2; > IsAnalyticallyIrreducible(D2,Origin(A)); > g := ResolutionGraph(D2,Origin(A)); > g; Output: Magma V2.11-10 Tue Dec 6 2005 10:06:36 on modular [Seed = 2605543432] ------------------------------------- Affine Space of dimension 2 Variables : x, y Curve over Rational Field defined by 4*x^6 - 12*x^4 - 4*x^3*y + 54*x^3 + 144*x^2*y + 105*x^2 + 108*x*y^2 + 150*x*y + 61*x + 27*y^3 + 55*y^2 + 41*y + 13 {@ @} false 4 1 Curve over Rational Field defined by x^6 - 3*x^4*y^2 + 27/2*x^3*y^3 - x^3*y^2 + 105/4*x^2*y^4 + 36*x^2*y^3 + 61/4*x*y^5 + 75/2*x*y^4 + 27*x*y^3 + 13/4*y^6 + 41/4*y^5 + 55/4*y^4 + 27/4*y^3 Curve over Rational Field defined by x^6 - 3*x^4*y^2 + 27/2*x^3*y^3 - x^3*y^2 + 105/4*x^2*y^4 + 36*x^2*y^3 + 61/4*x*y^5 + 75/2*x*y^4 + 27*x*y^3 + 13/4*y^6 + 41/4*y^5 + 55/4*y^4 + 27/4*y^3 false The resolution graph on the Digraph Vertex Neighbours 1 ([ -1, 6, 2, 3 ]) 2 ; 2 ([ -2, 3, 1, 0 ]) ; Total time: 1.149 seconds, Total memory usage: 4.24MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Tue Dec 6 10:06:38 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*A*Transpose(P)-A)); G:=GroebnerBasis(I); S:=FunctionField(K); p1 := (2*a-1)*c*t; p2 := c*t - a*(b-1/2)/(a-1/2); px := -a * p1; pz := p1 * p2; py := a*c*(c-a)*t^2 + a*(a^2-2*a*b*c+a*b-a+1/2*c)/(a-1/2)*t + a^2*(a+b-1)*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(c*(a-1/2)^2); q := a*c*(a+c-1)*t^2 - 2*a*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(a-1/2)*t + a^2*(a+b-1)*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(c*(a-1/2)^2); f7:=pz/q; f8:=px/q; f9:=py/q; f1:=Evaluate(G[1],[0,0,0,0,0,0,f7,f8,f9]); f2:=Evaluate(G[2],[0,0,0,0,0,0,f7,f8,f9]); f3:=Evaluate(G[3],[0,0,0,0,0,0,f7,f8,f9]); f4:=Evaluate(G[4],[0,0,0,0,0,0,f7,f8,f9]); f5:=Evaluate(G[5],[0,0,0,0,0,0,f7,f8,f9]); f6:=Evaluate(G[6],[0,0,0,0,0,0,f7,f8,f9]); Q:=Matrix(S,3,3,[f1,f2,f3,f4,f5,f6,f7,f8,f9]); print (Q*A*Transpose(Q)-A); Output: Magma V2.11-10 Tue Dec 6 2005 10:06:36 on modular [Seed = 2555278091] ------------------------------------- [ 0 0 -2] [ 0 0 -2] [ 0 0 0] Total time: 1.659 seconds, Total memory usage: 4.60MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Tue Dec 6 10:06:21 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*A*Transpose(P)-A)); G:=GroebnerBasis(I); S:=FunctionField(K); p1 := (2*a-1)*c*t; p2 := c*t - a*(b-1/2)/(a-1/2); px := -a * p1; pz := p1 * p2; py := a*c*(c-a)*t^2 + a*(a^2-2*a*b*c+a*b-a+1/2*c)/(a-1/2)*t + a^2*(a+b-1)*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(c*(a-1/2)^2); q := a*c*(a+c-1)*t^2 - 2*a*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(a-1/2)*t + a^2*(a+b-1)*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(c*(a-1/2)^2); f7:=pz/q; f8:=px/q; f9:=py/q; f1:=-Evaluate(G[1],[0,0,0,0,0,0,f7,f8,f9]); f2:=-Evaluate(G[2],[0,0,0,0,0,0,f7,f8,f9]); f3:=-Evaluate(G[3],[0,0,0,0,0,0,f7,f8,f9]); f4:=-Evaluate(G[4],[0,0,0,0,0,0,f7,f8,f9]); f5:=-Evaluate(G[5],[0,0,0,0,0,0,f7,f8,f9]); f6:=-Evaluate(G[6],[0,0,0,0,0,0,f7,f8,f9]); Q:=Matrix(S,3,3,[f1,f2,f3,f4,f5,f6,f7,f8,f9]); print (Q*A*Transpose(Q)-A); Output: Magma V2.11-10 Tue Dec 6 2005 10:06:19 on modular [Seed = 2153147140] ------------------------------------- [0 0 0] [0 0 0] [0 0 0] Total time: 1.649 seconds, Total memory usage: 4.60MB '219.166' ************** MAGMA ***************** Host 219.166.29.134 (219.166.29.134) Time: Tue Dec 6 10:06:00 2005 Input: > k := Rationals(); > A := AffineSpace(k,2); > A; > f := 1+1*x+1*y+2*(-0*x*y-x^3)+(2+3*x+1*y+2*(-0*x*y-x^3))^2+(2+4*x+3*y)^3; > C := Curve(A,f > C; > SingularPoints(C); > HasSingularPointsOverExtension(C); > Genus(C); > NumberOfPunctures(C); > P:=ProjectiveClosure(C); > D:=AffinePatch(P,2); > D; > D2:=Curve(A,DefiningPolynomial(D)); > D2; > IsAnalyticallyIrreducible(D2,Origin(A)); > g := ResolutionGraph(D2,Origin(A)); > g; Output: Magma V2.11-10 Tue Dec 6 2005 10:05:59 on modular [Seed = 2270260287] ------------------------------------- Affine Space of dimension 2 Variables : x, y >> C; ^ User error: bad syntax >> SingularPoints(C); ^ User error: Identifier 'C' has not been declared or assigned >> HasSingularPointsOverExtension(C); ^ User error: Identifier 'C' has not been declared or assigned >> Genus(C); ^ User error: Identifier 'C' has not been declared or assigned >> NumberOfPunctures(C); ^ User error: Identifier 'C' has not been declared or assigned >> P:=ProjectiveClosure(C); ^ User error: Identifier 'C' has not been declared or assigned >> D:=AffinePatch(P,2); ^ User error: Identifier 'P' has not been declared or assigned >> D; ^ User error: Identifier 'D' has not been declared or assigned >> D2:=Curve(A,DefiningPolynomial(D)); ^ User error: Identifier 'D' has not been declared or assigned >> D2; ^ User error: Identifier 'D2' has not been declared or assigned >> IsAnalyticallyIrreducible(D2,Origin(A)); ^ User error: Identifier 'D2' has not been declared or assigned >> g := ResolutionGraph(D2,Origin(A)); ^ User error: Identifier 'D2' has not been declared or assigned >> g; ^ User error: Identifier 'g' has not been declared or assigned Total time: 0.190 seconds, Total memory usage: 3.24MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Tue Dec 6 10:06:00 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*A*Transpose(P)-A)); G:=GroebnerBasis(I); S:=PolynomialRing(K); p1 := (2*a-1)*c*t; p2 := c*t - a*(b-1/2)/(a-1/2); px := -a * p1; pz := p1 * p2; py := a*c*(c-a)*t^2 + a*(a^2-2*a*b*c+a*b-a+1/2*c)/(a-1/2)*t + a^2*(a+b-1)*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(c*(a-1/2)^2); q := a*c*(a+c-1)*t^2 - 2*a*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(a-1/2)*t + a^2*(a+b-1)*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(c*(a-1/2)^2); f7:=pz/q; f8:=px/q; f9:=py/q; f1:=-Evaluate(G[1],[0,0,0,0,0,0,f7,f8,f9]); f2:=-Evaluate(G[2],[0,0,0,0,0,0,f7,f8,f9]); f3:=-Evaluate(G[3],[0,0,0,0,0,0,f7,f8,f9]); f4:=-Evaluate(G[4],[0,0,0,0,0,0,f7,f8,f9]); f5:=-Evaluate(G[5],[0,0,0,0,0,0,f7,f8,f9]); f6:=-Evaluate(G[6],[0,0,0,0,0,0,f7,f8,f9]); Q:=Matrix(S,3,3,[f1,f2,f3,f4,f5,f6,f7,f8,f9]); print (Q*A*Transpose(Q)-A); Output: Magma V2.11-10 Tue Dec 6 2005 10:05:58 on modular [Seed = 2321310003] ------------------------------------- >> Q:=Matrix(S,3,3,[f1,f2,f3,f4,f5,f6,f7,f8,f9]); ^ Runtime error in 'Matrix': Can't coerce sequence element 1 into the coefficient ring >> print (Q*A*Transpose(Q)-A); ^ User error: Identifier 'Q' has not been declared or assigned Total time: 1.459 seconds, Total memory usage: 4.60MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Tue Dec 6 10:04:57 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*A*Transpose(P)-A)); G:=GroebnerBasis(I); S:=PolynomialRing(K); p1 := (2*a-1)*c*t; p2 := c*t - a*(b-1/2)/(a-1/2); px := -a * p1; pz := p1 * p2; py := a*c*(c-a)*t^2 + a*(a^2-2*a*b*c+a*b-a+1/2*c)/(a-1/2)*t + a^2*(a+b-1)*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(c*(a-1/2)^2); q := a*c*(a+c-1)*t^2 - 2*a*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(a-1/2)*t + a^2*(a+b-1)*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(c*(a-1/2)^2); f7:=pz/q; f8:=px/q; f9:=py/q; f1:=Evaluate(G[1],[0,0,0,0,0,0,f7,f8,f9]); Output: Magma V2.11-10 Tue Dec 6 2005 10:04:55 on modular [Seed = 2405000311] ------------------------------------- Total time: 1.449 seconds, Total memory usage: 4.60MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Tue Dec 6 10:03:58 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*A*Transpose(P)-A)); G:=GroebnerBasis(I); S:=PolynomialRing(K); p1 := (2*a-1)*c*t; p2 := c*t - a*(b-1/2)/(a-1/2); px := -a * p1; pz := p1 * p2; py := a*c*(c-a)*t^2 + a*(a^2-2*a*b*c+a*b-a+1/2*c)/(a-1/2)*t + a^2*(a+b-1)*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(c*(a-1/2)^2); q := a*c*(a+c-1)*t^2 - 2*a*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(a-1/2)*t + a^2*(a+b-1)*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(c*(a-1/2)^2); print Evaluate(G[1],[0,0,0,0,0,0,pz/q,px/q,py/q]); Output: Magma V2.11-10 Tue Dec 6 2005 10:03:57 on modular [Seed = 3026345862] ------------------------------------- ((-a^3 + a^2*b + a^2*c - a*b*c - 2*a*b - 2*a*c + 2*a + b + c - 1)/(a^3 + a^2*b + a^2*c - 2*a^2 + a*b*c - a*b - a*c + a)*t^2 + (2*a*b*c + a*b + a*c - 3/2*a - b - c + 1)/(a^2*c + a*c^2 - 3/2*a*c - 1/2*c^2 + 1/2*c)*t + (-a^3*b*c + 1/4*a^3 - a^2*b^2*c + a^2*b*c + 1/2*a^2*b + 1/4*a^2*c - 1/2*a^2 + 1/4*a*b^2 + 1/4*a*b*c - 1/2*a*b - 1/4*a*c + 1/4*a)/(a^3*c^2 + a^2*c^3 - 2*a^2*c^2 - a*c^3 + 5/4*a*c^2 + 1/4*c^3 - 1/4*c^2))/(t^2 + (-2*a*b*c + 1/2*a + 1/2*b + 1/2*c - 1/2)/(a^2*c + a*c^2 - 3/2*a*c - 1/2*c^2 + 1/2*c)*t + (a^3*b*c - 1/4*a^3 + a^2*b^2*c - a^2*b*c - 1/2*a^2*b - 1/4*a^2*c + 1/2*a^2 - 1/4*a*b^2 - 1/4*a*b*c + 1/2*a*b + 1/4*a*c - 1/4*a)/(a^3*c^2 + a^2*c^3 - 2*a^2*c^2 - a*c^3 + 5/4*a*c^2 + 1/4*c^3 - 1/4*c^2)) Total time: 1.439 seconds, Total memory usage: 4.60MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Tue Dec 6 10:02:46 2005 Input: K:=PolynomialRing(RationalField(),3); print Factorization((-a^3 + a^2*b + a^2*c - a*b*c - 2*a*b - 2*a*c + 2*a + b + c - 1)); print Factorization((a^3 + a^2*b + a^2*c - 2*a^2 + a*b*c - a*b - a*c + a)); print Factorization((2*a*b*c + a*b + a*c - 3/2*a - b - c + 1)); print Factorization((a^2*c + a*c^2 - 3/2*a*c - 1/2*c^2 + 1/2*c)); print Factorization((-a^3*b*c + 1/4*a^3 - a^2*b^2*c + a^2*b*c + 1/2*a^2*b + 1/4*a^2*c - 1/2*a^2 + 1/4*a*b^2 + 1/4*a*b*c - 1/2*a*b - 1/4*a*c + 1/4*a)); print Factorization((a^3*c^2 + a^2*c^3 - 2*a^2*c^2 - a*c^3 + 5/4*a*c^2 + 1/4*c^3 - 1/4*c^2)); Output: Magma V2.11-10 Tue Dec 6 2005 10:02:46 on modular [Seed = 3177928388] ------------------------------------- [ ] [ , , ] [ ] [ , , ] [ , , ] [ , , ] Total time: 0.190 seconds, Total memory usage: 3.24MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Tue Dec 6 10:00:13 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*A*Transpose(P)-A)); G:=GroebnerBasis(I); S:=PolynomialRing(K); p1 := (2*a-1)*c*t; p2 := c*t - a*(b-1/2)/(a-1/2); px := -a * p1; pz := p1 * p2; py := a*c*(c-a)*t^2 + a*(a^2-2*a*b*c+a*b-a+1/2*c)/(a-1/2)*t + a^2*(a+b-1)*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(c*(a-1/2)^2); q := a*c*(a+c-1)*t^2 - 2*a*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(a-1/2)*t + a^2*(a+b-1)*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(c*(a-1/2)^2); print Evaluate(G[1],[0,0,0,0,0,0,pz/q,px/q,py/q]); print Factorization((-a^3 + a^2*b + a^2*c - a*b*c - 2*a*b - 2*a*c + 2*a + b + c - 1)/(a^3 + a^2*b + a^2*c - 2*a^2 + a*b*c - a*b - a*c + a)*t^2 + (2*a*b*c + a*b + a*c - 3/2*a - b - c + 1)/(a^2*c + a*c^2 - 3/2*a*c - 1/2*c^2 + 1/2*c)*t + (-a^3*b*c + 1/4*a^3 - a^2*b^2*c + a^2*b*c + 1/2*a^2*b + 1/4*a^2*c - 1/2*a^2 + 1/4*a*b^2 + 1/4*a*b*c - 1/2*a*b - 1/4*a*c + 1/4*a)/(a^3*c^2 + a^2*c^3 - 2*a^2*c^2 - a*c^3 + 5/4*a*c^2 + 1/4*c^3 - 1/4*c^2)); print Factorization((t^2 + (-2*a*b*c + 1/2*a + 1/2*b + 1/2*c - 1/2)/(a^2*c + a*c^2 - 3/2*a*c - 1/2*c^2 + 1/2*c)*t + (a^3*b*c - 1/4*a^3 + a^2*b^2*c - a^2*b*c - 1/2*a^2*b - 1/4*a^2*c + 1/2*a^2 - 1/4*a*b^2 - 1/4*a*b*c + 1/2*a*b + 1/4*a*c - 1/4*a)/(a^3*c^2 + a^2*c^3 - 2*a^2*c^2 - a*c^3 + 5/4*a*c^2 + 1/4*c^3 - 1/4*c^2))); Output: Magma V2.11-10 Tue Dec 6 2005 10:00:12 on modular [Seed = 1874430844] ------------------------------------- ((-a^3 + a^2*b + a^2*c - a*b*c - 2*a*b - 2*a*c + 2*a + b + c - 1)/(a^3 + a^2*b + a^2*c - 2*a^2 + a*b*c - a*b - a*c + a)*t^2 + (2*a*b*c + a*b + a*c - 3/2*a - b - c + 1)/(a^2*c + a*c^2 - 3/2*a*c - 1/2*c^2 + 1/2*c)*t + (-a^3*b*c + 1/4*a^3 - a^2*b^2*c + a^2*b*c + 1/2*a^2*b + 1/4*a^2*c - 1/2*a^2 + 1/4*a*b^2 + 1/4*a*b*c - 1/2*a*b - 1/4*a*c + 1/4*a)/(a^3*c^2 + a^2*c^3 - 2*a^2*c^2 - a*c^3 + 5/4*a*c^2 + 1/4*c^3 - 1/4*c^2))/(t^2 + (-2*a*b*c + 1/2*a + 1/2*b + 1/2*c - 1/2)/(a^2*c + a*c^2 - 3/2*a*c - 1/2*c^2 + 1/2*c)*t + (a^3*b*c - 1/4*a^3 + a^2*b^2*c - a^2*b*c - 1/2*a^2*b - 1/4*a^2*c + 1/2*a^2 - 1/4*a*b^2 - 1/4*a*b*c + 1/2*a*b + 1/4*a*c - 1/4*a)/(a^3*c^2 + a^2*c^3 - 2*a^2*c^2 - a*c^3 + 5/4*a*c^2 + 1/4*c^3 - 1/4*c^2)) [ ] [ ] Total time: 1.449 seconds, Total memory usage: 4.60MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Tue Dec 6 09:59:46 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*A*Transpose(P)-A)); G:=GroebnerBasis(I); S:=PolynomialRing(K); p1 := (2*a-1)*c*t; p2 := c*t - a*(b-1/2)/(a-1/2); px := -a * p1; pz := p1 * p2; py := a*c*(c-a)*t^2 + a*(a^2-2*a*b*c+a*b-a+1/2*c)/(a-1/2)*t + a^2*(a+b-1)*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(c*(a-1/2)^2); q := a*c*(a+c-1)*t^2 - 2*a*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(a-1/2)*t + a^2*(a+b-1)*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(c*(a-1/2)^2); print Evaluate(G[1],[0,0,0,0,0,0,pz/q,px/q,py/q]); print Factorization((-a^3 + a^2*b + a^2*c - a*b*c - 2*a*b - 2*a*c + 2*a + b + c - 1)/(a^3 + a^2*b + a^2*c - 2*a^2 + a*b*c - a*b - a*c + a)*t^2 + (2*a*b*c + a*b + a*c - 3/2*a - b - c + 1)/(a^2*c + a*c^2 - 3/2*a*c - 1/2*c^2 + 1/2*c)*t + (-a^3*b*c + 1/4*a^3 - a^2*b^2*c + a^2*b*c + 1/2*a^2*b + 1/4*a^2*c - 1/2*a^2 + 1/4*a*b^2 + 1/4*a*b*c - 1/2*a*b - 1/4*a*c + 1/4*a)/(a^3*c^2 + a^2*c^3 - 2*a^2*c^2 - a*c^3 + 5/4*a*c^2 + 1/4*c^3 - 1/4*c^2)); print Factorization((t^2 + (-2*a*b*c + 1/2*a + 1/2*b + 1/2*c - 1/2)/(a^2*c + a*c^2 - 3/2*a*c - 1/2*c^2 + 1/2*c)*t + (a^3*b*c - 1/4*a^3 + a^2*b^2*c - a^2*b*c - 1/2*a^2*b - 1/4*a^2*c + 1/2*a^2 - 1/4*a*b^2 - 1/4*a*b*c + 1/2*a*b + 1/4*a*c - 1/4*a)/(a^3*c^2 + a^2*c^3 - 2*a^2*c^2 - a*c^3 + 5/4*a*c^2 + 1/4*c^3 - 1/4*c^2)); Output: Magma V2.11-10 Tue Dec 6 2005 09:59:44 on modular [Seed = 365162781] ------------------------------------- ((-a^3 + a^2*b + a^2*c - a*b*c - 2*a*b - 2*a*c + 2*a + b + c - 1)/(a^3 + a^2*b + a^2*c - 2*a^2 + a*b*c - a*b - a*c + a)*t^2 + (2*a*b*c + a*b + a*c - 3/2*a - b - c + 1)/(a^2*c + a*c^2 - 3/2*a*c - 1/2*c^2 + 1/2*c)*t + (-a^3*b*c + 1/4*a^3 - a^2*b^2*c + a^2*b*c + 1/2*a^2*b + 1/4*a^2*c - 1/2*a^2 + 1/4*a*b^2 + 1/4*a*b*c - 1/2*a*b - 1/4*a*c + 1/4*a)/(a^3*c^2 + a^2*c^3 - 2*a^2*c^2 - a*c^3 + 5/4*a*c^2 + 1/4*c^3 - 1/4*c^2))/(t^2 + (-2*a*b*c + 1/2*a + 1/2*b + 1/2*c - 1/2)/(a^2*c + a*c^2 - 3/2*a*c - 1/2*c^2 + 1/2*c)*t + (a^3*b*c - 1/4*a^3 + a^2*b^2*c - a^2*b*c - 1/2*a^2*b - 1/4*a^2*c + 1/2*a^2 - 1/4*a*b^2 - 1/4*a*b*c + 1/2*a*b + 1/4*a*c - 1/4*a)/(a^3*c^2 + a^2*c^3 - 2*a^2*c^2 - a*c^3 + 5/4*a*c^2 + 1/4*c^3 - 1/4*c^2)) [ ] >> a*c^3 + 5/4*a*c^2 + 1/4*c^3 - 1/4*c^2)); ^ User error: bad syntax Total time: 1.470 seconds, Total memory usage: 4.60MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Tue Dec 6 09:57:45 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*A*Transpose(P)-A)); G:=GroebnerBasis(I); S:=PolynomialRing(K); p1 := (2*a-1)*c*t; p2 := c*t - a*(b-1/2)/(a-1/2); px := -a * p1; pz := p1 * p2; py := a*c*(c-a)*t^2 + a*(a^2-2*a*b*c+a*b-a+1/2*c)/(a-1/2)*t + a^2*(a+b-1)*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(c*(a-1/2)^2); q := a*c*(a+c-1)*t^2 - 2*a*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(a-1/2)*t + a^2*(a+b-1)*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(c*(a-1/2)^2); print Evaluate(G[1],[0,0,0,0,0,0,pz/q,px/q,py/q]); Output: Magma V2.11-10 Tue Dec 6 2005 09:57:44 on modular [Seed = 670155430] ------------------------------------- ((-a^3 + a^2*b + a^2*c - a*b*c - 2*a*b - 2*a*c + 2*a + b + c - 1)/(a^3 + a^2*b + a^2*c - 2*a^2 + a*b*c - a*b - a*c + a)*t^2 + (2*a*b*c + a*b + a*c - 3/2*a - b - c + 1)/(a^2*c + a*c^2 - 3/2*a*c - 1/2*c^2 + 1/2*c)*t + (-a^3*b*c + 1/4*a^3 - a^2*b^2*c + a^2*b*c + 1/2*a^2*b + 1/4*a^2*c - 1/2*a^2 + 1/4*a*b^2 + 1/4*a*b*c - 1/2*a*b - 1/4*a*c + 1/4*a)/(a^3*c^2 + a^2*c^3 - 2*a^2*c^2 - a*c^3 + 5/4*a*c^2 + 1/4*c^3 - 1/4*c^2))/(t^2 + (-2*a*b*c + 1/2*a + 1/2*b + 1/2*c - 1/2)/(a^2*c + a*c^2 - 3/2*a*c - 1/2*c^2 + 1/2*c)*t + (a^3*b*c - 1/4*a^3 + a^2*b^2*c - a^2*b*c - 1/2*a^2*b - 1/4*a^2*c + 1/2*a^2 - 1/4*a*b^2 - 1/4*a*b*c + 1/2*a*b + 1/4*a*c - 1/4*a)/(a^3*c^2 + a^2*c^3 - 2*a^2*c^2 - a*c^3 + 5/4*a*c^2 + 1/4*c^3 - 1/4*c^2)) Total time: 1.449 seconds, Total memory usage: 4.60MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Tue Dec 6 09:56:34 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*A*Transpose(P)-A)); G:=GroebnerBasis(I); S:=PolynomialRing(K); p1 := (2*a-1)*c*t; p2 := c*t - a*(b-1/2)/(a-1/2); px := -a * p1; pz := p1 * p2; py := a*c*(c-a)*t^2 + a*(a^2-2*a*b*c+a*b-a+1/2*c)/(a-1/2)*t + a^2*(a+b-1)*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(c*(a-1/2)^2); q := a*c*(a+c-1)*t^2 - 2*a*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(a-1/2)*t + a^2*(a+b-1)*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(c*(a-1/2)^2); print Evaluate(G[7],[0,0,0,0,0,0,pz/q,px/q,py/q]); print Evaluate(G[8],[0,0,0,0,0,0,pz/q,px/q,py/q]); print Evaluate(G[9],[0,0,0,0,0,0,pz/q,px/q,py/q]); print Evaluate(G[10],[0,0,0,0,0,0,pz/q,px/q,py/q]); Output: Magma V2.11-10 Tue Dec 6 2005 09:56:31 on modular [Seed = 4264041227] ------------------------------------- 0 0 0 0 Total time: 2.919 seconds, Total memory usage: 5.67MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Tue Dec 6 09:51:02 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*A*Transpose(P)-A)); G:=GroebnerBasis(I); S:=FunctionField(K); p1 := (2*a-1)*c*t; p2 := c*t - a*(2*b-1)/(2*a-1); px := -a * p1; pz := p1 * p2; py := a*c*(c-a)*t^2 + a*(a^2-2*a*b*c+a*b-a+1/2*c)/(a-1/2)*t + a^2*(a+b-1)*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(c*(a-1/2)^2); q := a*c*(a+c-1)*t^2 - 2*a*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(a-1/2)*t + a^2*(a+b-1)*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(c*(a-1/2)^2); t0:=t*a*(a+b-1)*(a*b*c-1/4*a-1/4*b-1/4*c+1/4); print Evaluate(px/q,t0); print Evaluate(py/q,t0); print Evaluate(pz/q,t0); Output: Magma V2.11-10 Tue Dec 6 2005 09:51:00 on modular [Seed = 870700290] ------------------------------------- (-2*a + 1)/(a^4*b*c - 1/4*a^4 + a^3*b^2*c + a^3*b*c^2 - 2*a^3*b*c - 1/2*a^3*b - 1/2*a^3*c + 3/4*a^3 + a^2*b^2*c^2 - a^2*b^2*c - 1/4*a^2*b^2 - a^2*b*c^2 + 1/4*a^2*b*c + a^2*b - 1/4*a^2*c^2 + a^2*c - 3/4*a^2 - 1/4*a*b^2*c + 1/4*a*b^2 - 1/4*a*b*c^2 + 3/4*a*b*c - 1/2*a*b + 1/4*a*c^2 - 1/2*a*c + 1/4*a)*t/(t^2 - 2/(a^4*c + a^3*b*c + a^3*c^2 - 5/2*a^3*c + a^2*b*c^2 - 3/2*a^2*b*c - 3/2*a^2*c^2 + 2*a^2*c - 1/2*a*b*c^2 + 1/2*a*b*c + 1/2*a*c^2 - 1/2*a*c)*t + 1/(a^6*b*c^3 - 1/4*a^6*c^2 + a^5*b^2*c^3 + a^5*b*c^4 - 3*a^5*b*c^3 - 1/2*a^5*b*c^2 - 1/2*a^5*c^3 + a^5*c^2 + a^4*b^2*c^4 - 2*a^4*b^2*c^3 - 1/4*a^4*b^2*c^2 - 2*a^4*b*c^4 + 5/2*a^4*b*c^3 + 3/2*a^4*b*c^2 - 1/4*a^4*c^4 + 3/2*a^4*c^3 - 25/16*a^4*c^2 - a^3*b^2*c^4 + a^3*b^2*c^3 + 1/2*a^3*b^2*c^2 + a^3*b*c^4 - 13/8*a^3*b*c^2 + 1/2*a^3*c^4 - 13/8*a^3*c^3 + 19/16*a^3*c^2 + 1/4*a^2*b^2*c^4 - 5/16*a^2*b^2*c^2 - 11/16*a^2*b*c^3 + 3/4*a^2*b*c^2 - 5/16*a^2*c^4 + 3/4*a^2*c^3 - 7/16*a^2*c^2 - 1/16*a*b^2*c^3 + 1/16*a*b^2*c^2 - 1/16*a*b*c^4 + 3/16*a*b*c^3 - 1/8*a*b*c^2 + 1/16*a*c^4 - 1/8*a*c^3 + 1/16*a*c^2)) ((-a + c)/(a + c - 1)*t^2 + (a^2 - 2*a*b*c + a*b - a + 1/2*c)/(a^5*b*c^2 - 1/4*a^5*c + a^4*b^2*c^2 + a^4*b*c^3 - 5/2*a^4*b*c^2 - 1/2*a^4*b*c - 1/2*a^4*c^2 + 7/8*a^4*c + a^3*b^2*c^3 - 3/2*a^3*b^2*c^2 - 1/4*a^3*b^2*c - 3/2*a^3*b*c^3 + 5/4*a^3*b*c^2 + 5/4*a^3*b*c - 1/4*a^3*c^3 + 5/4*a^3*c^2 - 9/8*a^3*c - 1/2*a^2*b^2*c^3 + 1/4*a^2*b^2*c^2 + 3/8*a^2*b^2*c + 1/4*a^2*b*c^3 + 5/8*a^2*b*c^2 - a^2*b*c + 3/8*a^2*c^3 - a^2*c^2 + 5/8*a^2*c + 1/8*a*b^2*c^2 - 1/8*a*b^2*c + 1/8*a*b*c^3 - 3/8*a*b*c^2 + 1/4*a*b*c - 1/8*a*c^3 + 1/4*a*c^2 - 1/8*a*c)*t + 1/(a^6*b*c^3 - 1/4*a^6*c^2 + a^5*b^2*c^3 + a^5*b*c^4 - 3*a^5*b*c^3 - 1/2*a^5*b*c^2 - 1/2*a^5*c^3 + a^5*c^2 + a^4*b^2*c^4 - 2*a^4*b^2*c^3 - 1/4*a^4*b^2*c^2 - 2*a^4*b*c^4 + 5/2*a^4*b*c^3 + 3/2*a^4*b*c^2 - 1/4*a^4*c^4 + 3/2*a^4*c^3 - 25/16*a^4*c^2 - a^3*b^2*c^4 + a^3*b^2*c^3 + 1/2*a^3*b^2*c^2 + a^3*b*c^4 - 13/8*a^3*b*c^2 + 1/2*a^3*c^4 - 13/8*a^3*c^3 + 19/16*a^3*c^2 + 1/4*a^2*b^2*c^4 - 5/16*a^2*b^2*c^2 - 11/16*a^2*b*c^3 + 3/4*a^2*b*c^2 - 5/16*a^2*c^4 + 3/4*a^2*c^3 - 7/16*a^2*c^2 - 1/16*a*b^2*c^3 + 1/16*a*b^2*c^2 - 1/16*a*b*c^4 + 3/16*a*b*c^3 - 1/8*a*b*c^2 + 1/16*a*c^4 - 1/8*a*c^3 + 1/16*a*c^2))/(t^2 - 2/(a^4*c + a^3*b*c + a^3*c^2 - 5/2*a^3*c + a^2*b*c^2 - 3/2*a^2*b*c - 3/2*a^2*c^2 + 2*a^2*c - 1/2*a*b*c^2 + 1/2*a*b*c + 1/2*a*c^2 - 1/2*a*c)*t + 1/(a^6*b*c^3 - 1/4*a^6*c^2 + a^5*b^2*c^3 + a^5*b*c^4 - 3*a^5*b*c^3 - 1/2*a^5*b*c^2 - 1/2*a^5*c^3 + a^5*c^2 + a^4*b^2*c^4 - 2*a^4*b^2*c^3 - 1/4*a^4*b^2*c^2 - 2*a^4*b*c^4 + 5/2*a^4*b*c^3 + 3/2*a^4*b*c^2 - 1/4*a^4*c^4 + 3/2*a^4*c^3 - 25/16*a^4*c^2 - a^3*b^2*c^4 + a^3*b^2*c^3 + 1/2*a^3*b^2*c^2 + a^3*b*c^4 - 13/8*a^3*b*c^2 + 1/2*a^3*c^4 - 13/8*a^3*c^3 + 19/16*a^3*c^2 + 1/4*a^2*b^2*c^4 - 5/16*a^2*b^2*c^2 - 11/16*a^2*b*c^3 + 3/4*a^2*b*c^2 - 5/16*a^2*c^4 + 3/4*a^2*c^3 - 7/16*a^2*c^2 - 1/16*a*b^2*c^3 + 1/16*a*b^2*c^2 - 1/16*a*b*c^4 + 3/16*a*b*c^3 - 1/8*a*b*c^2 + 1/16*a*c^4 - 1/8*a*c^3 + 1/16*a*c^2)) ((2*a*c - c)/(a^2 + a*c - a)*t^2 + (-2*b + 1)/(a^4*b*c - 1/4*a^4 + a^3*b^2*c + a^3*b*c^2 - 2*a^3*b*c - 1/2*a^3*b - 1/2*a^3*c + 3/4*a^3 + a^2*b^2*c^2 - a^2*b^2*c - 1/4*a^2*b^2 - a^2*b*c^2 + 1/4*a^2*b*c + a^2*b - 1/4*a^2*c^2 + a^2*c - 3/4*a^2 - 1/4*a*b^2*c + 1/4*a*b^2 - 1/4*a*b*c^2 + 3/4*a*b*c - 1/2*a*b + 1/4*a*c^2 - 1/2*a*c + 1/4*a)*t)/(t^2 - 2/(a^4*c + a^3*b*c + a^3*c^2 - 5/2*a^3*c + a^2*b*c^2 - 3/2*a^2*b*c - 3/2*a^2*c^2 + 2*a^2*c - 1/2*a*b*c^2 + 1/2*a*b*c + 1/2*a*c^2 - 1/2*a*c)*t + 1/(a^6*b*c^3 - 1/4*a^6*c^2 + a^5*b^2*c^3 + a^5*b*c^4 - 3*a^5*b*c^3 - 1/2*a^5*b*c^2 - 1/2*a^5*c^3 + a^5*c^2 + a^4*b^2*c^4 - 2*a^4*b^2*c^3 - 1/4*a^4*b^2*c^2 - 2*a^4*b*c^4 + 5/2*a^4*b*c^3 + 3/2*a^4*b*c^2 - 1/4*a^4*c^4 + 3/2*a^4*c^3 - 25/16*a^4*c^2 - a^3*b^2*c^4 + a^3*b^2*c^3 + 1/2*a^3*b^2*c^2 + a^3*b*c^4 - 13/8*a^3*b*c^2 + 1/2*a^3*c^4 - 13/8*a^3*c^3 + 19/16*a^3*c^2 + 1/4*a^2*b^2*c^4 - 5/16*a^2*b^2*c^2 - 11/16*a^2*b*c^3 + 3/4*a^2*b*c^2 - 5/16*a^2*c^4 + 3/4*a^2*c^3 - 7/16*a^2*c^2 - 1/16*a*b^2*c^3 + 1/16*a*b^2*c^2 - 1/16*a*b*c^4 + 3/16*a*b*c^3 - 1/8*a*b*c^2 + 1/16*a*c^4 - 1/8*a*c^3 + 1/16*a*c^2)) Total time: 1.459 seconds, Total memory usage: 4.60MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Tue Dec 6 09:48:06 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*A*Transpose(P)-A)); G:=GroebnerBasis(I); S:=FunctionField(K); p1 := (2*a-1)*c*t; p2 := c*t - a*(2*b-1)/(2*a-1); px := -a * p1; pz := p1 * p2; py := a*c*(c-a)*t^2 + a*(a^2-2*a*b*c+a*b-a+1/2*c)/(a-1/2)*t + a^2*(a+b-1)*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(c*(a-1/2)^2); q := a*c*(a+c-1)*t^2 - 2*a*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(a-1/2)*t + a^2*(a+b-1)*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(c*(a-1/2)^2); t0:=t*a*(a+b-1)*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/((a-1/2)^2*c); print Evaluate(px/q,t0); print Evaluate(py/q,t0); print Evaluate(pz/q,t0); Output: Magma V2.11-10 Tue Dec 6 2005 09:48:05 on modular [Seed = 4163772218] ------------------------------------- (-2*a^3*c + 3*a^2*c - 3/2*a*c + 1/4*c)/(a^4*b*c - 1/4*a^4 + a^3*b^2*c + a^3*b*c^2 - 2*a^3*b*c - 1/2*a^3*b - 1/2*a^3*c + 3/4*a^3 + a^2*b^2*c^2 - a^2*b^2*c - 1/4*a^2*b^2 - a^2*b*c^2 + 1/4*a^2*b*c + a^2*b - 1/4*a^2*c^2 + a^2*c - 3/4*a^2 - 1/4*a*b^2*c + 1/4*a*b^2 - 1/4*a*b*c^2 + 3/4*a*b*c - 1/2*a*b + 1/4*a*c^2 - 1/2*a*c + 1/4*a)*t/(t^2 + (-2*a + 1)/(a^3 + a^2*b + a^2*c - 2*a^2 + a*b*c - a*b - a*c + a)*t + (a^2 - a + 1/4)/(a^4*b*c - 1/4*a^4 + a^3*b^2*c + a^3*b*c^2 - 2*a^3*b*c - 1/2*a^3*b - 1/2*a^3*c + 3/4*a^3 + a^2*b^2*c^2 - a^2*b^2*c - 1/4*a^2*b^2 - a^2*b*c^2 + 1/4*a^2*b*c + a^2*b - 1/4*a^2*c^2 + a^2*c - 3/4*a^2 - 1/4*a*b^2*c + 1/4*a*b^2 - 1/4*a*b*c^2 + 3/4*a*b*c - 1/2*a*b + 1/4*a*c^2 - 1/2*a*c + 1/4*a)) ((-a + c)/(a + c - 1)*t^2 + (a^3 - 2*a^2*b*c + a^2*b - 3/2*a^2 + a*b*c - 1/2*a*b + 1/2*a*c + 1/2*a - 1/4*c)/(a^4*b*c - 1/4*a^4 + a^3*b^2*c + a^3*b*c^2 - 2*a^3*b*c - 1/2*a^3*b - 1/2*a^3*c + 3/4*a^3 + a^2*b^2*c^2 - a^2*b^2*c - 1/4*a^2*b^2 - a^2*b*c^2 + 1/4*a^2*b*c + a^2*b - 1/4*a^2*c^2 + a^2*c - 3/4*a^2 - 1/4*a*b^2*c + 1/4*a*b^2 - 1/4*a*b*c^2 + 3/4*a*b*c - 1/2*a*b + 1/4*a*c^2 - 1/2*a*c + 1/4*a)*t + (a^2 - a + 1/4)/(a^4*b*c - 1/4*a^4 + a^3*b^2*c + a^3*b*c^2 - 2*a^3*b*c - 1/2*a^3*b - 1/2*a^3*c + 3/4*a^3 + a^2*b^2*c^2 - a^2*b^2*c - 1/4*a^2*b^2 - a^2*b*c^2 + 1/4*a^2*b*c + a^2*b - 1/4*a^2*c^2 + a^2*c - 3/4*a^2 - 1/4*a*b^2*c + 1/4*a*b^2 - 1/4*a*b*c^2 + 3/4*a*b*c - 1/2*a*b + 1/4*a*c^2 - 1/2*a*c + 1/4*a))/(t^2 + (-2*a + 1)/(a^3 + a^2*b + a^2*c - 2*a^2 + a*b*c - a*b - a*c + a)*t + (a^2 - a + 1/4)/(a^4*b*c - 1/4*a^4 + a^3*b^2*c + a^3*b*c^2 - 2*a^3*b*c - 1/2*a^3*b - 1/2*a^3*c + 3/4*a^3 + a^2*b^2*c^2 - a^2*b^2*c - 1/4*a^2*b^2 - a^2*b*c^2 + 1/4*a^2*b*c + a^2*b - 1/4*a^2*c^2 + a^2*c - 3/4*a^2 - 1/4*a*b^2*c + 1/4*a*b^2 - 1/4*a*b*c^2 + 3/4*a*b*c - 1/2*a*b + 1/4*a*c^2 - 1/2*a*c + 1/4*a)) ((2*a*c - c)/(a^2 + a*c - a)*t^2 + (-2*a^2*b*c + a^2*c + 2*a*b*c - a*c - 1/2*b*c + 1/4*c)/(a^4*b*c - 1/4*a^4 + a^3*b^2*c + a^3*b*c^2 - 2*a^3*b*c - 1/2*a^3*b - 1/2*a^3*c + 3/4*a^3 + a^2*b^2*c^2 - a^2*b^2*c - 1/4*a^2*b^2 - a^2*b*c^2 + 1/4*a^2*b*c + a^2*b - 1/4*a^2*c^2 + a^2*c - 3/4*a^2 - 1/4*a*b^2*c + 1/4*a*b^2 - 1/4*a*b*c^2 + 3/4*a*b*c - 1/2*a*b + 1/4*a*c^2 - 1/2*a*c + 1/4*a)*t)/(t^2 + (-2*a + 1)/(a^3 + a^2*b + a^2*c - 2*a^2 + a*b*c - a*b - a*c + a)*t + (a^2 - a + 1/4)/(a^4*b*c - 1/4*a^4 + a^3*b^2*c + a^3*b*c^2 - 2*a^3*b*c - 1/2*a^3*b - 1/2*a^3*c + 3/4*a^3 + a^2*b^2*c^2 - a^2*b^2*c - 1/4*a^2*b^2 - a^2*b*c^2 + 1/4*a^2*b*c + a^2*b - 1/4*a^2*c^2 + a^2*c - 3/4*a^2 - 1/4*a*b^2*c + 1/4*a*b^2 - 1/4*a*b*c^2 + 3/4*a*b*c - 1/2*a*b + 1/4*a*c^2 - 1/2*a*c + 1/4*a)) Total time: 1.459 seconds, Total memory usage: 4.60MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Tue Dec 6 09:47:27 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*A*Transpose(P)-A)); G:=GroebnerBasis(I); S:=FunctionField(K); p1 := (2*a-1)*c*t; p2 := c*t - a*(2*b-1)/(2*a-1); px := -a * p1; pz := p1 * p2; py := a*c*(c-a)*t^2 + a*(a^2-2*a*b*c+a*b-a+1/2*c)/(a-1/2)*t + a^2*(a+b-1)*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(c*(a-1/2)^2); q := a*c*(a+c-1)*t^2 - 2*a*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(a-1/2)*t + a^2*(a+b-1)*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(c*(a-1/2)^2); t0:=t*a*(a+b-1)*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/((a-1/2)^2*c); print Evaluate(p1,t0); print Evaluate(p2,t0); print Evaluate(py,t0); print Evaluate(q,t0); Output: Magma V2.11-10 Tue Dec 6 2005 09:47:26 on modular [Seed = 1351006025] ------------------------------------- (2*a^3*b*c - 1/2*a^3 + 2*a^2*b^2*c - 2*a^2*b*c - a^2*b - 1/2*a^2*c + a^2 - 1/2*a*b^2 - 1/2*a*b*c + a*b + 1/2*a*c - 1/2*a)/(a - 1/2)*t (a^3*b*c - 1/4*a^3 + a^2*b^2*c - a^2*b*c - 1/2*a^2*b - 1/4*a^2*c + 1/2*a^2 - 1/4*a*b^2 - 1/4*a*b*c + 1/2*a*b + 1/4*a*c - 1/4*a)/(a^2 - a + 1/4)*t + (-a*b + 1/2*a)/(a - 1/2) (-a^8*b^2*c^2 + 1/2*a^8*b*c - 1/16*a^8 - 2*a^7*b^3*c^2 + a^7*b^2*c^3 + 2*a^7*b^2*c^2 + 3/2*a^7*b^2*c - 3/2*a^7*b*c - 1/4*a^7*b - 1/16*a^7*c + 1/4*a^7 - a^6*b^4*c^2 + 2*a^6*b^3*c^3 + 2*a^6*b^3*c^2 + 3/2*a^6*b^3*c - 2*a^6*b^2*c^3 - 3/2*a^6*b^2*c^2 - 3*a^6*b^2*c - 3/8*a^6*b^2 - 1/2*a^6*b*c^3 + 1/2*a^6*b*c^2 + 11/8*a^6*b*c + 3/4*a^6*b + 1/16*a^6*c^2 + 1/8*a^6*c - 3/8*a^6 + a^5*b^4*c^3 + 1/2*a^5*b^4*c - 2*a^5*b^3*c^3 - a^5*b^3*c^2 - 3/2*a^5*b^3*c - 1/4*a^5*b^3 + 2*a^5*b^2*c^2 + 3/2*a^5*b^2*c + 3/4*a^5*b^2 + a^5*b*c^3 - 3/4*a^5*b*c^2 - 1/2*a^5*b*c - 3/4*a^5*b + 1/16*a^5*c^3 - 1/4*a^5*c^2 + 1/4*a^5 - 1/2*a^4*b^4*c^2 - 1/16*a^4*b^4 - 1/2*a^4*b^3*c^3 + 3/2*a^4*b^3*c^2 + 1/8*a^4*b^3*c + 1/4*a^4*b^3 + a^4*b^2*c^3 - 19/16*a^4*b^2*c^2 - 3/8*a^4*b^2*c - 3/8*a^4*b^2 - 3/8*a^4*b*c^3 - 1/8*a^4*b*c^2 + 3/8*a^4*b*c + 1/4*a^4*b - 1/8*a^4*c^3 + 5/16*a^4*c^2 - 1/8*a^4*c - 1/16*a^4 + 1/16*a^3*b^4*c + 1/8*a^3*b^3*c^2 - 1/4*a^3*b^3*c + 1/16*a^3*b^2*c^3 - 3/8*a^3*b^2*c^2 + 3/8*a^3*b^2*c - 1/8*a^3*b*c^3 + 3/8*a^3*b*c^2 - 1/4*a^3*b*c + 1/16*a^3*c^3 - 1/8*a^3*c^2 + 1/16*a^3*c)/(a^4*c - 2*a^3*c + 3/2*a^2*c - 1/2*a*c + 1/16*c)*t^2 + (a^6*b*c - 1/4*a^6 - 2*a^5*b^2*c^2 + 2*a^5*b^2*c - 3/2*a^5*b*c - 3/4*a^5*b - 1/4*a^5*c + 3/4*a^5 - 2*a^4*b^3*c^2 + a^4*b^3*c + 2*a^4*b^2*c^2 - a^4*b^2*c - 3/4*a^4*b^2 + a^4*b*c^2 - 1/2*a^4*b*c + 3/2*a^4*b + 3/8*a^4*c - 3/4*a^4 + 1/2*a^3*b^3*c - 1/4*a^3*b^3 + a^3*b^2*c^2 - 5/4*a^3*b^2*c + 3/4*a^3*b^2 - a^3*b*c^2 + 3/4*a^3*b*c - 3/4*a^3*b - 1/8*a^3*c^2 + 1/4*a^3 - 1/8*a^2*b^2*c - 1/8*a^2*b*c^2 + 1/4*a^2*b*c + 1/8*a^2*c^2 - 1/8*a^2*c)/(a^3*c - 3/2*a^2*c + 3/4*a*c - 1/8*c)*t + (a^4*b*c - 1/4*a^4 + a^3*b^2*c - a^3*b*c - 1/2*a^3*b - 1/4*a^3*c + 1/2*a^3 - 1/4*a^2*b^2 - 1/4*a^2*b*c + 1/2*a^2*b + 1/4*a^2*c - 1/4*a^2)/(a^2*c - a*c + 1/4*c) (a^8*b^2*c^2 - 1/2*a^8*b*c + 1/16*a^8 + 2*a^7*b^3*c^2 + a^7*b^2*c^3 - 3*a^7*b^2*c^2 - 3/2*a^7*b^2*c - a^7*b*c^2 + 2*a^7*b*c + 1/4*a^7*b + 3/16*a^7*c - 5/16*a^7 + a^6*b^4*c^2 + 2*a^6*b^3*c^3 - 4*a^6*b^3*c^2 - 3/2*a^6*b^3*c - 2*a^6*b^2*c^3 + 1/2*a^6*b^2*c^2 + 9/2*a^6*b^2*c + 3/8*a^6*b^2 - 1/2*a^6*b*c^3 + 3*a^6*b*c^2 - 19/8*a^6*b*c - a^6*b + 3/16*a^6*c^2 - 3/4*a^6*c + 5/8*a^6 + a^5*b^4*c^3 - a^5*b^4*c^2 - 1/2*a^5*b^4*c - 2*a^5*b^3*c^3 + 3*a^5*b^3*c + 1/4*a^5*b^3 + 4*a^5*b^2*c^2 - 15/4*a^5*b^2*c - 9/8*a^5*b^2 + a^5*b*c^3 - 5/2*a^5*b*c^2 + 1/8*a^5*b*c + 3/2*a^5*b + 1/16*a^5*c^3 - 9/16*a^5*c^2 + 9/8*a^5*c - 5/8*a^5 - 1/2*a^4*b^4*c^2 + 1/2*a^4*b^4*c + 1/16*a^4*b^4 - 1/2*a^4*b^3*c^3 + 2*a^4*b^3*c^2 - 9/8*a^4*b^3*c - 1/2*a^4*b^3 + a^4*b^2*c^3 - 33/16*a^4*b^2*c^2 + 9/8*a^4*b^2 - 3/8*a^4*b*c^3 + 11/8*a^4*b*c - a^4*b - 1/8*a^4*c^3 + 9/16*a^4*c^2 - 3/4*a^4*c + 5/16*a^4 + 1/16*a^3*b^4*c - 1/16*a^3*b^4 + 1/8*a^3*b^3*c^2 - 3/8*a^3*b^3*c + 1/4*a^3*b^3 + 1/16*a^3*b^2*c^3 - 7/16*a^3*b^2*c^2 + 3/4*a^3*b^2*c - 3/8*a^3*b^2 - 1/8*a^3*b*c^3 + 1/2*a^3*b*c^2 - 5/8*a^3*b*c + 1/4*a^3*b + 1/16*a^3*c^3 - 3/16*a^3*c^2 + 3/16*a^3*c - 1/16*a^3)/(a^4*c - 2*a^3*c + 3/2*a^2*c - 1/2*a*c + 1/16*c)*t^2 + (-2*a^5*b^2*c^2 + a^5*b*c - 1/8*a^5 - 2*a^4*b^3*c^2 + 2*a^4*b^2*c^2 + 2*a^4*b^2*c + a^4*b*c^2 - 2*a^4*b*c - 3/8*a^4*b - 1/4*a^4*c + 3/8*a^4 + a^3*b^3*c + a^3*b^2*c^2 - 2*a^3*b^2*c - 3/8*a^3*b^2 - a^3*b*c^2 + 1/2*a^3*b*c + 3/4*a^3*b - 1/8*a^3*c^2 + 1/2*a^3*c - 3/8*a^3 - 1/8*a^2*b^3 - 1/4*a^2*b^2*c + 3/8*a^2*b^2 - 1/8*a^2*b*c^2 + 1/2*a^2*b*c - 3/8*a^2*b + 1/8*a^2*c^2 - 1/4*a^2*c + 1/8*a^2)/(a^3*c - 3/2*a^2*c + 3/4*a*c - 1/8*c)*t + (a^4*b*c - 1/4*a^4 + a^3*b^2*c - a^3*b*c - 1/2*a^3*b - 1/4*a^3*c + 1/2*a^3 - 1/4*a^2*b^2 - 1/4*a^2*b*c + 1/2*a^2*b + 1/4*a^2*c - 1/4*a^2)/(a^2*c - a*c + 1/4*c) Total time: 1.449 seconds, Total memory usage: 4.60MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Tue Dec 6 09:02:41 2005 Input: K:=PolynomialRing(RationalField(),3); print Factorization((-2*a^2*b*c^2 + a^2*c^2 + 2*a*b*c^2 - a*c^2 - 1/2*b*c^2 + 1/4*c^2)); print Factorization((a^3*b*c - 1/4*a^3 + a^2*b^2*c - a^2*b*c - 1/2*a^2*b - 1/4*a^2*c + 1/2*a^2 - 1/4*a*b^2 - 1/4*a*b*c + 1/2*a*b + 1/4*a*c - 1/4*a)); print Factorization((-2*a^3*c^2 + 3*a^2*c^2 - 3/2*a*c^2 + 1/4*c^2)); print Factorization((a^3*b*c - 1/4*a^3 + a^2*b^2*c - a^2*b*c - 1/2*a^2*b - 1/4*a^2*c + 1/2*a^2 - 1/4*a*b^2 - 1/4*a*b*c + 1/2*a*b + 1/4*a*c - 1/4*a)); print Factorization((a^2*c - a*c + 1/4*c)); print Factorization((a^2*b*c - 1/4*a^2 - 1/4*a*b - 1/4*a*c + 1/4*a)); Output: Magma V2.11-10 Tue Dec 6 2005 09:02:40 on modular [Seed = 2809233133] ------------------------------------- [ , , ] [ , , ] [ , ] [ , , ] [ , ] [ , ] Total time: 0.200 seconds, Total memory usage: 3.24MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Tue Dec 6 09:00:32 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*A*Transpose(P)-A)); G:=GroebnerBasis(I); S:=FunctionField(K); print S; p1 := (2*a-1)*c*t; p2 := c*t - a*(2*b-1)/(2*a-1); px := -a * p1; pz := p1 * p2; py := a*c*(c-a)*t^2 + a*(a^2-2*a*b*c+a*b-a+1/2*c)/(a-1/2)*t + a^2*(a+b-1)*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(c*(a-1/2)^2); q := a*c*(a+c-1)*t^2 - 2*a*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(a-1/2)*t + a^2*(a+b-1)*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(c*(a-1/2)^2); print Evaluate(Derivative(pz/q),0); print Evaluate(Derivative(px/q),0); print Evaluate(Derivative(py/q),0); Output: Magma V2.11-10 Tue Dec 6 2005 09:00:30 on modular [Seed = 1906813765] ------------------------------------- Univariate rational function field over Multivariate Rational function field of rank 3 over Rational Field Variables: t (-2*a^2*b*c^2 + a^2*c^2 + 2*a*b*c^2 - a*c^2 - 1/2*b*c^2 + 1/4*c^2)/(a^3*b*c - 1/4*a^3 + a^2*b^2*c - a^2*b*c - 1/2*a^2*b - 1/4*a^2*c + 1/2*a^2 - 1/4*a*b^2 - 1/4*a*b*c + 1/2*a*b + 1/4*a*c - 1/4*a) (-2*a^3*c^2 + 3*a^2*c^2 - 3/2*a*c^2 + 1/4*c^2)/(a^3*b*c - 1/4*a^3 + a^2*b^2*c - a^2*b*c - 1/2*a^2*b - 1/4*a^2*c + 1/2*a^2 - 1/4*a*b^2 - 1/4*a*b*c + 1/2*a*b + 1/4*a*c - 1/4*a) (a^2*c - a*c + 1/4*c)/(a^2*b*c - 1/4*a^2 - 1/4*a*b - 1/4*a*c + 1/4*a) Total time: 1.459 seconds, Total memory usage: 4.60MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Tue Dec 6 08:59:59 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*A*Transpose(P)-A)); G:=GroebnerBasis(I); S:=FunctionField(K); print S; p1 := (2*a-1)*c*t; p2 := c*t - a*(2*b-1)/(2*a-1); px := -a * p1; pz := p1 * p2; py := a*c*(c-a)*t^2 + a*(a^2-2*a*b*c+a*b-a+1/2*c)/(a-1/2)*t + a^2*(a+b-1)*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(c*(a-1/2)^2); q := a*c*(a+c-1)*t^2 - 2*a*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(a-1/2)*t + a^2*(a+b-1)*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(c*(a-1/2)^2); print Derivative(p1); Output: Magma V2.11-10 Tue Dec 6 2005 08:59:58 on modular [Seed = 2091558504] ------------------------------------- Univariate rational function field over Multivariate Rational function field of rank 3 over Rational Field Variables: t 2*a*c - c Total time: 1.439 seconds, Total memory usage: 4.60MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Tue Dec 6 08:59:36 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*A*Transpose(P)-A)); G:=GroebnerBasis(I); S:=FunctionField(K); print S; p1 := (2*a-1)*c*t; p2 := c*t - a*(2*b-1)/(2*a-1); px := -a * p1; pz := p1 * p2; py := a*c*(c-a)*t^2 + a*(a^2-2*a*b*c+a*b-a+1/2*c)/(a-1/2)*t + a^2*(a+b-1)*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(c*(a-1/2)^2); q := a*c*(a+c-1)*t^2 - 2*a*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(a-1/2)*t + a^2*(a+b-1)*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(c*(a-1/2)^2); print Derivative(S!(px/q)); Output: Magma V2.11-10 Tue Dec 6 2005 08:59:35 on modular [Seed = 2074976630] ------------------------------------- Univariate rational function field over Multivariate Rational function field of rank 3 over Rational Field Variables: t ((2*a - 1)/(a + c - 1)*t^2 + (-2*a^3*b*c + 1/2*a^3 - 2*a^2*b^2*c + 2*a^2*b*c + a^2*b + 1/2*a^2*c - a^2 + 1/2*a*b^2 + 1/2*a*b*c - a*b - 1/2*a*c + 1/2*a)/(a^3*c^2 + 2*a^2*c^3 - 5/2*a^2*c^2 + a*c^4 - 3*a*c^3 + 2*a*c^2 - 1/2*c^4 + c^3 - 1/2*c^2))/(t^4 + (-4*a*b*c + a + b + c - 1)/(a^2*c + a*c^2 - 3/2*a*c - 1/2*c^2 + 1/2*c)*t^3 + (2*a^4*b*c - 1/2*a^4 + 2*a^3*b^2*c + 2*a^3*b*c^2 - 4*a^3*b*c - a^3*b - a^3*c + 3/2*a^3 + 6*a^2*b^2*c^2 - 2*a^2*b^2*c - 1/2*a^2*b^2 - 2*a^2*b*c^2 - 3/2*a^2*b*c + 2*a^2*b - 1/2*a^2*c^2 + 2*a^2*c - 5/4*a^2 - 5/2*a*b^2*c + 1/2*a*b^2 - 5/2*a*b*c^2 + 7/2*a*b*c - 1/2*a*b + 1/2*a*c^2 - 1/2*a*c + 1/4*b^2 + 1/2*b*c - 1/2*b + 1/4*c^2 - 1/2*c + 1/4)/(a^4*c^2 + 2*a^3*c^3 - 3*a^3*c^2 + a^2*c^4 - 4*a^2*c^3 + 13/4*a^2*c^2 - a*c^4 + 5/2*a*c^3 - 3/2*a*c^2 + 1/4*c^4 - 1/2*c^3 + 1/4*c^2)*t^2 + (-4*a^4*b^2*c^2 + 2*a^4*b*c - 1/4*a^4 - 4*a^3*b^3*c^2 + 4*a^3*b^2*c^2 + 4*a^3*b^2*c + 2*a^3*b*c^2 - 4*a^3*b*c - 3/4*a^3*b - 1/2*a^3*c + 3/4*a^3 + 2*a^2*b^3*c + 2*a^2*b^2*c^2 - 4*a^2*b^2*c - 3/4*a^2*b^2 - 2*a^2*b*c^2 + a^2*b*c + 3/2*a^2*b - 1/4*a^2*c^2 + a^2*c - 3/4*a^2 - 1/4*a*b^3 - 1/2*a*b^2*c + 3/4*a*b^2 - 1/4*a*b*c^2 + a*b*c - 3/4*a*b + 1/4*a*c^2 - 1/2*a*c + 1/4*a)/(a^5*c^3 + 2*a^4*c^4 - 7/2*a^4*c^3 + a^3*c^5 - 5*a^3*c^4 + 19/4*a^3*c^3 - 3/2*a^2*c^5 + 9/2*a^2*c^4 - 25/8*a^2*c^3 + 3/4*a*c^5 - 7/4*a*c^4 + a*c^3 - 1/8*c^5 + 1/4*c^4 - 1/8*c^3)*t + (a^6*b^2*c^2 - 1/2*a^6*b*c + 1/16*a^6 + 2*a^5*b^3*c^2 - 2*a^5*b^2*c^2 - 3/2*a^5*b^2*c - 1/2*a^5*b*c^2 + 3/2*a^5*b*c + 1/4*a^5*b + 1/8*a^5*c - 1/4*a^5 + a^4*b^4*c^2 - 2*a^4*b^3*c^2 - 3/2*a^4*b^3*c + 3*a^4*b^2*c + 3/8*a^4*b^2 + a^4*b*c^2 - 9/8*a^4*b*c - 3/4*a^4*b + 1/16*a^4*c^2 - 3/8*a^4*c + 3/8*a^4 - 1/2*a^3*b^4*c - 1/2*a^3*b^3*c^2 + 3/2*a^3*b^3*c + 1/4*a^3*b^3 + a^3*b^2*c^2 - 9/8*a^3*b^2*c - 3/4*a^3*b^2 - 3/8*a^3*b*c^2 - 1/4*a^3*b*c + 3/4*a^3*b - 1/8*a^3*c^2 + 3/8*a^3*c - 1/4*a^3 + 1/16*a^2*b^4 + 1/8*a^2*b^3*c - 1/4*a^2*b^3 + 1/16*a^2*b^2*c^2 - 3/8*a^2*b^2*c + 3/8*a^2*b^2 - 1/8*a^2*b*c^2 + 3/8*a^2*b*c - 1/4*a^2*b + 1/16*a^2*c^2 - 1/8*a^2*c + 1/16*a^2)/(a^6*c^4 + 2*a^5*c^5 - 4*a^5*c^4 + a^4*c^6 - 6*a^4*c^5 + 13/2*a^4*c^4 - 2*a^3*c^6 + 7*a^3*c^5 - 11/2*a^3*c^4 + 3/2*a^2*c^6 - 4*a^2*c^5 + 41/16*a^2*c^4 - 1/2*a*c^6 + 9/8*a*c^5 - 5/8*a*c^4 + 1/16*c^6 - 1/8*c^5 + 1/16*c^4)) Total time: 1.459 seconds, Total memory usage: 4.60MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Tue Dec 6 08:58:51 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*A*Transpose(P)-A)); G:=GroebnerBasis(I); S:=FunctionField(K,1); print S; p1 := (2*a-1)*c*t; p2 := c*t - a*(2*b-1)/(2*a-1); px := -a * p1; pz := p1 * p2; py := a*c*(c-a)*t^2 + a*(a^2-2*a*b*c+a*b-a+1/2*c)/(a-1/2)*t + a^2*(a+b-1)*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(c*(a-1/2)^2); q := a*c*(a+c-1)*t^2 - 2*a*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(a-1/2)*t + a^2*(a+b-1)*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(c*(a-1/2)^2); print Derivative(S!(px/q)); Output: Magma V2.11-10 Tue Dec 6 2005 08:58:50 on modular [Seed = 1722858550] ------------------------------------- Multivariate Rational function field of rank 1 over Multivariate Rational function field of rank 3 over Rational Field Variables: t >> print Derivative(S!(px/q)); ^ Runtime error in 'Derivative': Bad argument types Argument types given: FldFunRatMElt Total time: 1.449 seconds, Total memory usage: 4.60MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Tue Dec 6 08:58:39 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*A*Transpose(P)-A)); G:=GroebnerBasis(I); S:=FunctionField(K,1); p1 := (2*a-1)*c*t; p2 := c*t - a*(2*b-1)/(2*a-1); px := -a * p1; pz := p1 * p2; py := a*c*(c-a)*t^2 + a*(a^2-2*a*b*c+a*b-a+1/2*c)/(a-1/2)*t + a^2*(a+b-1)*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(c*(a-1/2)^2); q := a*c*(a+c-1)*t^2 - 2*a*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(a-1/2)*t + a^2*(a+b-1)*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(c*(a-1/2)^2); print Derivative(S!(px/q)); Output: Magma V2.11-10 Tue Dec 6 2005 08:58:38 on modular [Seed = 1841018665] ------------------------------------- >> print Derivative(S!(px/q)); ^ Runtime error in 'Derivative': Bad argument types Argument types given: FldFunRatMElt Total time: 1.459 seconds, Total memory usage: 4.60MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Tue Dec 6 08:55:26 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*A*Transpose(P)-A)); G:=GroebnerBasis(I); S:=FunctionField(K,1); p1 := (2*a-1)*c*t; p2 := c*t - a*(2*b-1)/(2*a-1); px := -a * p1; pz := p1 * p2; py := a*c*(c-a)*t^2 + a*(a^2-2*a*b*c+a*b-a+1/2*c)/(a-1/2)*t + a^2*(a+b-1)*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(c*(a-1/2)^2); q := a*c*(a+c-1)*t^2 - 2*a*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(a-1/2)*t + a^2*(a+b-1)*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(c*(a-1/2)^2); print Derivative(px/q); Output: Magma V2.11-10 Tue Dec 6 2005 08:55:24 on modular [Seed = 1807593581] ------------------------------------- >> print Derivative(px/q); ^ Runtime error in 'Derivative': Bad argument types Argument types given: FldFunRatMElt Total time: 1.439 seconds, Total memory usage: 4.60MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Tue Dec 6 08:52:47 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*A*Transpose(P)-A)); G:=GroebnerBasis(I); S:=FunctionField(K,1); p1 := (2*a-1)*c*t; p2 := c*t - a*(2*b-1)/(2*a-1); px := -a * p1; pz := p1 * p2; py := a*c*(c-a)*t^2 + a*(a^2-2*a*b*c+a*b-a+1/2*c)/(a-1/2)*t + a^2*(a+b-1)*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(c*(a-1/2)^2); q := a*c*(a+c-1)*t^2 - 2*a*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(a-1/2)*t + a^2*(a+b-1)*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(c*(a-1/2)^2); print Derivative(p1); print Evaluate(G[7],[0,0,0,0,0,0,pz/q,px/q,py/q]); print Evaluate(G[8],[0,0,0,0,0,0,pz/q,px/q,py/q]); print Evaluate(G[9],[0,0,0,0,0,0,pz/q,px/q,py/q]); print Evaluate(G[10],[0,0,0,0,0,0,pz/q,px/q,py/q]); Output: Magma V2.11-10 Tue Dec 6 2005 08:52:44 on modular [Seed = 1350993078] ------------------------------------- >> print Derivative(p1); ^ Runtime error in 'Derivative': Bad argument types Argument types given: FldFunRatMElt 0 0 0 0 Total time: 2.690 seconds, Total memory usage: 5.54MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Tue Dec 6 08:52:18 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*A*Transpose(P)-A)); G:=GroebnerBasis(I); S:=PolynomialRing(K,1); p1 := (2*a-1)*c*t; p2 := c*t - a*(2*b-1)/(2*a-1); px := -a * p1; pz := p1 * p2; py := a*c*(c-a)*t^2 + a*(a^2-2*a*b*c+a*b-a+1/2*c)/(a-1/2)*t + a^2*(a+b-1)*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(c*(a-1/2)^2); q := a*c*(a+c-1)*t^2 - 2*a*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(a-1/2)*t + a^2*(a+b-1)*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(c*(a-1/2)^2); print Derivative(p1); print Evaluate(G[7],[0,0,0,0,0,0,pz/q,px/q,py/q]); print Evaluate(G[8],[0,0,0,0,0,0,pz/q,px/q,py/q]); print Evaluate(G[9],[0,0,0,0,0,0,pz/q,px/q,py/q]); print Evaluate(G[10],[0,0,0,0,0,0,pz/q,px/q,py/q]); Output: Magma V2.11-10 Tue Dec 6 2005 08:52:15 on modular [Seed = 1602849690] ------------------------------------- >> print Derivative(p1); ^ Runtime error in 'Derivative': Bad argument types Argument types given: RngMPolElt 0 0 0 0 Total time: 2.690 seconds, Total memory usage: 5.50MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Tue Dec 6 08:52:01 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*A*Transpose(P)-A)); G:=GroebnerBasis(I); S:=PolynomialRing(K,1); p1 := (2*a-1)*c*t; p2 := c*t - a*(2*b-1)/(2*a-1); px := -a * p1; pz := p1 * p2; py := a*c*(c-a)*t^2 + a*(a^2-2*a*b*c+a*b-a+1/2*c)/(a-1/2)*t + a^2*(a+b-1)*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(c*(a-1/2)^2); q := a*c*(a+c-1)*t^2 - 2*a*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(a-1/2)*t + a^2*(a+b-1)*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(c*(a-1/2)^2); print Evaluate(G[7],[0,0,0,0,0,0,pz/q,px/q,py/q]); print Evaluate(G[8],[0,0,0,0,0,0,pz/q,px/q,py/q]); print Evaluate(G[9],[0,0,0,0,0,0,pz/q,px/q,py/q]); print Evaluate(G[10],[0,0,0,0,0,0,pz/q,px/q,py/q]); Output: Magma V2.11-10 Tue Dec 6 2005 08:51:58 on modular [Seed = 1519160942] ------------------------------------- 0 0 0 0 Total time: 2.680 seconds, Total memory usage: 5.76MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Tue Dec 6 08:45:47 2005 Input: K:=FunctionField(RationalField(),4); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*A*Transpose(P)-A)); G:=GroebnerBasis(I); p1 := (2*a-1)*c*t; p2 := c*t - a*(2*b-1)/(2*a-1); px := -a * p1; pz := p1 * p2; py := a*c*(c-a)*t^2 + a*(a^2-2*a*b*c+a*b-a+1/2*c)/(a-1/2)*t + a^2*(a+b-1)*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(c*(a-1/2)^2); q := a*c*(a+c-1)*t^2 - 2*a*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(a-1/2)*t + a^2*(a+b-1)*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(c*(a-1/2)^2); print G[1]; print Evaluate(G[1],[0,0,0,0,0,0,pz/q,px/q,py/q]); print Evaluate(G[7],[0,0,0,0,0,0,pz/q,px/q,py/q]); print Evaluate(G[8],[0,0,0,0,0,0,pz/q,px/q,py/q]); print Evaluate(G[9],[0,0,0,0,0,0,pz/q,px/q,py/q]); print Evaluate(G[10],[0,0,0,0,0,0,pz/q,px/q,py/q]); Output: Magma V2.11-10 Tue Dec 6 2005 08:45:45 on modular [Seed = 1133876736] ------------------------------------- $.1 + (-a^2 + a*c - b - c + 1)/(a*c + b*c - c)*$.7 + (a*b - a - b*c - b + 1)/(a*c + b*c - c)*$.8 - $.9 (-a^5*b*c - a^5*c^2*t^2 + 1/4*a^5 - 2*a^4*b^2*c + a^4*b*c^2*t^2 + 2*a^4*b*c^2*t + a^4*b*c*t + 2*a^4*b*c + 3/4*a^4*b + a^4*c^3*t^2 + a^4*c^2*t^2 + a^4*c^2*t - 3/2*a^4*c*t + 1/4*a^4*c - 3/4*a^4 - a^3*b^3*c + 2*a^3*b^2*c^2*t + a^3*b^2*c*t + 2*a^3*b^2*c + 3/4*a^3*b^2 - a^3*b*c^3*t^2 - 3*a^3*b*c^2*t^2 - 2*a^3*b*c^2*t - 4*a^3*b*c*t - 1/2*a^3*b*c - 3/2*a^3*b - 3*a^3*c^3*t^2 + 7/4*a^3*c^2*t^2 - 5/2*a^3*c^2*t + 13/4*a^3*c*t - 1/2*a^3*c + 3/4*a^3 + 1/4*a^2*b^3 - a^2*b^2*c^2*t - 3/2*a^2*b^2*c*t + 1/4*a^2*b^2*c - 3/4*a^2*b^2 + a^2*b*c^3*t^2 + 13/4*a^2*b*c^2*t^2 - 1/2*a^2*b*c^2*t + 15/4*a^2*b*c*t - 1/2*a^2*b*c + 3/4*a^2*b + 13/4*a^2*c^3*t^2 - 3*a^2*c^2*t^2 + 2*a^2*c^2*t - 9/4*a^2*c*t + 1/4*a^2*c - 1/4*a^2 + 1/2*a*b^2*c*t - 1/4*a*b*c^3*t^2 - 3/2*a*b*c^2*t^2 + 1/2*a*b*c^2*t - a*b*c*t - 3/2*a*c^3*t^2 + 3/2*a*c^2*t^2 - 1/2*a*c^2*t + 1/2*a*c*t + 1/4*b*c^2*t^2 + 1/4*c^3*t^2 - 1/4*c^2*t^2)/(a^5*b*c + a^5*c^2*t^2 - 1/4*a^5 + 2*a^4*b^2*c + a^4*b*c^2*t^2 - 2*a^4*b*c^2*t - 2*a^4*b*c - 3/4*a^4*b + a^4*c^3*t^2 - 3*a^4*c^2*t^2 + 1/2*a^4*c*t - 1/4*a^4*c + 3/4*a^4 + a^3*b^3*c - 2*a^3*b^2*c^2*t - 2*a^3*b^2*c - 3/4*a^3*b^2 + a^3*b*c^3*t^2 - 2*a^3*b*c^2*t^2 + 3*a^3*b*c^2*t + a^3*b*c*t + 1/2*a^3*b*c + 3/2*a^3*b - 2*a^3*c^3*t^2 + 13/4*a^3*c^2*t^2 + 1/2*a^3*c^2*t - 5/4*a^3*c*t + 1/2*a^3*c - 3/4*a^3 - 1/4*a^2*b^3 + a^2*b^2*c^2*t + 1/2*a^2*b^2*c*t - 1/4*a^2*b^2*c + 3/4*a^2*b^2 - a^2*b*c^3*t^2 + 5/4*a^2*b*c^2*t^2 - 1/2*a^2*b*c^2*t - 3/2*a^2*b*c*t + 1/2*a^2*b*c - 3/4*a^2*b + 5/4*a^2*c^3*t^2 - 3/2*a^2*c^2*t^2 - 3/4*a^2*c^2*t + a^2*c*t - 1/4*a^2*c + 1/4*a^2 - 1/4*a*b^2*c*t + 1/4*a*b*c^3*t^2 - 1/4*a*b*c^2*t^2 - 1/4*a*b*c^2*t + 1/2*a*b*c*t - 1/4*a*c^3*t^2 + 1/4*a*c^2*t^2 + 1/4*a*c^2*t - 1/4*a*c*t) 0 0 0 0 Total time: 1.909 seconds, Total memory usage: 7.44MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Tue Dec 6 08:44:17 2005 Input: K:=FunctionField(RationalField(),4); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*A*Transpose(P)-A)); G:=GroebnerBasis(I); S:=PolynomialRing(K,1); p1 := (2*a-1)*c*t; p2 := c*t - a*(2*b-1)/(2*a-1); px := -a * p1; pz := p1 * p2; py := a*c*(c-a)*t^2 + a*(a^2-2*a*b*c+a*b-a+1/2*c)/(a-1/2)*t + a^2*(a+b-1)*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(c*(a-1/2)^2); q := a*c*(a+c-1)*t^2 - 2*a*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(a-1/2)*t + a^2*(a+b-1)*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(c*(a-1/2)^2); print G[1]; print Evaluate(G[1],[0,0,0,0,0,0,pz/q,px/q,py/q]); print Evaluate(G[7],[0,0,0,0,0,0,pz/q,px/q,py/q]); print Evaluate(G[8],[0,0,0,0,0,0,pz/q,px/q,py/q]); print Evaluate(G[9],[0,0,0,0,0,0,pz/q,px/q,py/q]); print Evaluate(G[10],[0,0,0,0,0,0,pz/q,px/q,py/q]); Output: Magma V2.11-10 Tue Dec 6 2005 08:44:15 on modular [Seed = 1318623403] ------------------------------------- $.1 + (-a^2 + a*c - b - c + 1)/(a*c + b*c - c)*$.7 + (a*b - a - b*c - b + 1)/(a*c + b*c - c)*$.8 - $.9 ((-a^3 + a^2*b + a^2*c - a*b*c - 2*a*b - 2*a*c + 2*a + b + c - 1)/(a^3 + a^2*b + a^2*c - 2*a^2 + a*b*c - a*b - a*c + a)*$.1^2 + (2*a*b*c + a*b + a*c - 3/2*a - b - c + 1)/(a^2*c + a*c^2 - 3/2*a*c - 1/2*c^2 + 1/2*c)*$.1 + (-a^3*b*c + 1/4*a^3 - a^2*b^2*c + a^2*b*c + 1/2*a^2*b + 1/4*a^2*c - 1/2*a^2 + 1/4*a*b^2 + 1/4*a*b*c - 1/2*a*b - 1/4*a*c + 1/4*a)/(a^3*c^2 + a^2*c^3 - 2*a^2*c^2 - a*c^3 + 5/4*a*c^2 + 1/4*c^3 - 1/4*c^2))/($.1^2 + (-2*a*b*c + 1/2*a + 1/2*b + 1/2*c - 1/2)/(a^2*c + a*c^2 - 3/2*a*c - 1/2*c^2 + 1/2*c)*$.1 + (a^3*b*c - 1/4*a^3 + a^2*b^2*c - a^2*b*c - 1/2*a^2*b - 1/4*a^2*c + 1/2*a^2 - 1/4*a*b^2 - 1/4*a*b*c + 1/2*a*b + 1/4*a*c - 1/4*a)/(a^3*c^2 + a^2*c^3 - 2*a^2*c^2 - a*c^3 + 5/4*a*c^2 + 1/4*c^3 - 1/4*c^2)) 0 0 0 0 Total time: 2.509 seconds, Total memory usage: 5.67MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Tue Dec 6 08:43:06 2005 Input: K:=FunctionField(RationalField(),4); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*A*Transpose(P)-A)); G:=GroebnerBasis(I); S:=PolynomialRing(R,1); p1 := (2*a-1)*c*t; p2 := c*t - a*(2*b-1)/(2*a-1); px := -a * p1; pz := p1 * p2; py := a*c*(c-a)*t^2 + a*(a^2-2*a*b*c+a*b-a+1/2*c)/(a-1/2)*t + a^2*(a+b-1)*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(c*(a-1/2)^2); q := a*c*(a+c-1)*t^2 - 2*a*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(a-1/2)*t + a^2*(a+b-1)*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(c*(a-1/2)^2); print Evaluate(G[1],[R.1,0,0,0,0,0,pz/q,px/q,py/q]); print Evaluate(G[7],[0,0,0,0,0,0,pz/q,px/q,py/q]); print Evaluate(G[8],[0,0,0,0,0,0,pz/q,px/q,py/q]); print Evaluate(G[9],[0,0,0,0,0,0,pz/q,px/q,py/q]); print Evaluate(G[10],[0,0,0,0,0,0,pz/q,px/q,py/q]); Output: Magma V2.11-10 Tue Dec 6 2005 08:43:04 on modular [Seed = 3026346647] ------------------------------------- (($.1 + (-a^3 + a^2*b + a^2*c - a*b*c - 2*a*b - 2*a*c + 2*a + b + c - 1)/(a^3 + a^2*b + a^2*c - 2*a^2 + a*b*c - a*b - a*c + a))*$.1^2 + ((-2*a*b*c + 1/2*a + 1/2*b + 1/2*c - 1/2)/(a^2*c + a*c^2 - 3/2*a*c - 1/2*c^2 + 1/2*c)*$.1 + (2*a*b*c + a*b + a*c - 3/2*a - b - c + 1)/(a^2*c + a*c^2 - 3/2*a*c - 1/2*c^2 + 1/2*c))*$.1 + ((a^3*b*c - 1/4*a^3 + a^2*b^2*c - a^2*b*c - 1/2*a^2*b - 1/4*a^2*c + 1/2*a^2 - 1/4*a*b^2 - 1/4*a*b*c + 1/2*a*b + 1/4*a*c - 1/4*a)/(a^3*c^2 + a^2*c^3 - 2*a^2*c^2 - a*c^3 + 5/4*a*c^2 + 1/4*c^3 - 1/4*c^2)*$.1 + (-a^3*b*c + 1/4*a^3 - a^2*b^2*c + a^2*b*c + 1/2*a^2*b + 1/4*a^2*c - 1/2*a^2 + 1/4*a*b^2 + 1/4*a*b*c - 1/2*a*b - 1/4*a*c + 1/4*a)/(a^3*c^2 + a^2*c^3 - 2*a^2*c^2 - a*c^3 + 5/4*a*c^2 + 1/4*c^3 - 1/4*c^2)))/($.1^2 + (-2*a*b*c + 1/2*a + 1/2*b + 1/2*c - 1/2)/(a^2*c + a*c^2 - 3/2*a*c - 1/2*c^2 + 1/2*c)*$.1 + (a^3*b*c - 1/4*a^3 + a^2*b^2*c - a^2*b*c - 1/2*a^2*b - 1/4*a^2*c + 1/2*a^2 - 1/4*a*b^2 - 1/4*a*b*c + 1/2*a*b + 1/4*a*c - 1/4*a)/(a^3*c^2 + a^2*c^3 - 2*a^2*c^2 - a*c^3 + 5/4*a*c^2 + 1/4*c^3 - 1/4*c^2)) 0 0 0 0 Total time: 2.540 seconds, Total memory usage: 5.14MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Tue Dec 6 08:42:22 2005 Input: K:=FunctionField(RationalField(),4); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*A*Transpose(P)-A)); G:=GroebnerBasis(I); S:=PolynomialRing(K,1); p1 := (2*a-1)*c*t; p2 := c*t - a*(2*b-1)/(2*a-1); px := -a * p1; pz := p1 * p2; py := a*c*(c-a)*t^2 + a*(a^2-2*a*b*c+a*b-a+1/2*c)/(a-1/2)*t + a^2*(a+b-1)*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(c*(a-1/2)^2); q := a*c*(a+c-1)*t^2 - 2*a*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(a-1/2)*t + a^2*(a+b-1)*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(c*(a-1/2)^2); print Evaluate(G[1],[R.1,0,0,0,0,0,pz/q,px/q,py/q]); print Evaluate(G[7],[0,0,0,0,0,0,pz/q,px/q,py/q]); print Evaluate(G[8],[0,0,0,0,0,0,pz/q,px/q,py/q]); print Evaluate(G[9],[0,0,0,0,0,0,pz/q,px/q,py/q]); print Evaluate(G[10],[0,0,0,0,0,0,pz/q,px/q,py/q]); Output: Magma V2.11-10 Tue Dec 6 2005 08:42:20 on modular [Seed = 2976344006] ------------------------------------- >> print Evaluate(G[1],[R.1,0,0,0,0,0,pz/q,px/q,py/q]); ^ Runtime error in [ ... ]: Could not find a valid universe 0 0 0 0 Total time: 2.490 seconds, Total memory usage: 5.79MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Tue Dec 6 08:40:46 2005 Input: K:=FunctionField(RationalField(),4); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*A*Transpose(P)-A)); G:=GroebnerBasis(I); S:=PolynomialRing(K,1); p1 := (2*a-1)*c*t; p2 := c*t - a*(2*b-1)/(2*a-1); px := -a * p1; pz := p1 * p2; py := a*c*(c-a)*t^2 + a*(a^2-2*a*b*c+a*b-a+1/2*c)/(a-1/2)*t + a^2*(a+b-1)*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(c*(a-1/2)^2); q := a*c*(a+c-1)*t^2 - 2*a*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(a-1/2)*t + a^2*(a+b-1)*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(c*(a-1/2)^2); print Evaluate(G[7],[0,0,0,0,0,0,pz/q,px/q,py/q]); print Evaluate(G[8],[0,0,0,0,0,0,pz/q,px/q,py/q]); print Evaluate(G[9],[0,0,0,0,0,0,pz/q,px/q,py/q]); print Evaluate(G[10],[0,0,0,0,0,0,pz/q,px/q,py/q]); Output: Magma V2.11-10 Tue Dec 6 2005 08:40:43 on modular [Seed = 3211356207] ------------------------------------- 0 0 0 0 Total time: 2.500 seconds, Total memory usage: 5.68MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Tue Dec 6 08:40:33 2005 Input: G :=DirichletGroup(1680); G; X :=Elements(G); X; Y :=X[2]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; qEigenform(D[1],12);Parent($1); Output: ** WARNING: Computation used more memory than allowed. ** Magma V2.11-10 Tue Dec 6 2005 08:40:27 on modular [Seed = 2758964004] ------------------------------------- Group of Dirichlet characters of modulus 1680 over Rational Field [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3, $.4, $.1*$.4, $.2*$.4, $.1*$.2*$.4, $.3*$.4, $.1*$.3*$.4, $.2*$.3*$.4, $.1*$.2*$.3*$.4, $.5, $.1*$.5, $.2*$.5, $.1*$.2*$.5, $.3*$.5, $.1*$.3*$.5, $.2*$.3*$.5, $.1*$.2*$.3*$.5, $.4*$.5, $.1*$.4*$.5, $.2*$.4*$.5, $.1*$.2*$.4*$.5, $.3*$.4*$.5, $.1*$.3*$.4*$.5, $.2*$.3*$.4*$.5, $.1*$.2*$.3*$.4*$.5 ] 4 2 Current total memory usage: 92.0MB, failed memory request: 80.9MB System Error: User memory limit has been reached >> D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); ^ Runtime error: Variable 'M' has not been initialized >> D; ^ User error: Identifier 'D' has not been declared or assigned >> qEigenform(D[1],12);Parent($1); ^ User error: Identifier 'D' has not been declared or assigned Integer Ring Total time: 5.690 seconds, Total memory usage: 92.02MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Tue Dec 6 08:37:51 2005 Input: K:=FunctionField(RationalField(),4); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*A*Transpose(P)-A)); G:=GroebnerBasis(I); S:=PolynomialRing(K,1); p1 := (2*a-1)*c*t; p2 := c*t - a*(2*b-1)/(2*a-1); px := -a * p1; pz := p1 * p2; py := a*c*(c-a)*t^2 + a*(a^2-2*a*b*c+a*b-a+1/2*c)/(a-1/2)*t + a^2*(a+b-1)*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(a-1/2); q := a*c*(a+c-1)*t^2 - 2*a*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(a-1/2)*t + a^2*(a+b-1)*(a*b*c-1/4*a-1/4*b-1/4*c+1/4)/(a-1/2); print Evaluate(G[7],[0,0,0,0,0,0,pz/q,px/q,py/q]); print Evaluate(G[8],[0,0,0,0,0,0,pz/q,px/q,py/q]); print Evaluate(G[9],[0,0,0,0,0,0,pz/q,px/q,py/q]); print Evaluate(G[10],[0,0,0,0,0,0,pz/q,px/q,py/q]); Output: Magma V2.11-10 Tue Dec 6 2005 08:37:48 on modular [Seed = 2537636080] ------------------------------------- (-4*a^3*b^2*c^2 + 4*a^3*b*c^2 + a^3*b*c - a^3*c - 4*a^2*b^3*c^2 + 10*a^2*b^2*c^2 + 6*a^2*b^2*c - 5*a^2*b*c^2 - 17/2*a^2*b*c - a^2*b - a^2*c^2 + 5/2*a^2*c + a^2 + 2*a*b^3*c^2 + 5*a*b^3*c - 3*a*b^2*c^2 - 12*a*b^2*c - 2*a*b^2 - 1/2*a*b*c^2 + 8*a*b*c + 4*a*b + 3/2*a*c^2 - a*c - 2*a - 1/2*b^3*c - b^3 - 1/2*b^2*c^2 + 1/2*b^2*c + 3*b^2 + b*c^2 + 1/2*b*c - 3*b - 1/2*c^2 - 1/2*c + 1)/(a^3*b*c - 1/2*a^3*c + 2*a^2*b*c^2 - 2*a^2*b*c - a^2*c^2 + a^2*c + a*b*c^3 - 2*a*b*c^2 + a*b*c - 1/2*a*c^3 + a*c^2 - 1/2*a*c)*$.1^2/($.1^4 + (-4*a*b*c + a + b + c - 1)/(a^2*c + a*c^2 - 3/2*a*c - 1/2*c^2 + 1/2*c)*$.1^3 + (2*a^5*b*c^2 - 1/2*a^5*c + 2*a^4*b^2*c^2 + 2*a^4*b*c^3 - 5*a^4*b*c^2 - a^4*b*c - a^4*c^2 + 7/4*a^4*c + 2*a^3*b^2*c^3 - 3*a^3*b^2*c^2 - 1/2*a^3*b^2*c - 3*a^3*b*c^3 + 5/2*a^3*b*c^2 + 5/2*a^3*b*c - 1/2*a^3*c^3 + 5/2*a^3*c^2 - 9/4*a^3*c - a^2*b^2*c^3 + 9/2*a^2*b^2*c^2 + 3/4*a^2*b^2*c + 1/2*a^2*b*c^3 + 5/4*a^2*b*c^2 - 4*a^2*b*c + 3/4*a^2*c^3 - 2*a^2*c^2 + 5/4*a^2*c + 1/4*a^2 + 1/4*a*b^2*c^2 - 9/4*a*b^2*c + 1/4*a*b*c^3 - 11/4*a*b*c^2 + 5/2*a*b*c + 1/2*a*b - 1/4*a*c^3 + 1/2*a*c^2 + 1/4*a*c - 1/2*a + 1/4*b^2 + 1/2*b*c - 1/2*b + 1/4*c^2 - 1/2*c + 1/4)/(a^4*c^2 + 2*a^3*c^3 - 3*a^3*c^2 + a^2*c^4 - 4*a^2*c^3 + 13/4*a^2*c^2 - a*c^4 + 5/2*a*c^3 - 3/2*a*c^2 + 1/4*c^4 - 1/2*c^3 + 1/4*c^2)*$.1^2 + (-4*a^4*b^2*c^2 + 2*a^4*b*c - 1/4*a^4 - 4*a^3*b^3*c^2 + 4*a^3*b^2*c^2 + 4*a^3*b^2*c + 2*a^3*b*c^2 - 4*a^3*b*c - 3/4*a^3*b - 1/2*a^3*c + 3/4*a^3 + 2*a^2*b^3*c + 2*a^2*b^2*c^2 - 4*a^2*b^2*c - 3/4*a^2*b^2 - 2*a^2*b*c^2 + a^2*b*c + 3/2*a^2*b - 1/4*a^2*c^2 + a^2*c - 3/4*a^2 - 1/4*a*b^3 - 1/2*a*b^2*c + 3/4*a*b^2 - 1/4*a*b*c^2 + a*b*c - 3/4*a*b + 1/4*a*c^2 - 1/2*a*c + 1/4*a)/(a^4*c^2 + 2*a^3*c^3 - 3*a^3*c^2 + a^2*c^4 - 4*a^2*c^3 + 13/4*a^2*c^2 - a*c^4 + 5/2*a*c^3 - 3/2*a*c^2 + 1/4*c^4 - 1/2*c^3 + 1/4*c^2)*$.1 + (a^6*b^2*c^2 - 1/2*a^6*b*c + 1/16*a^6 + 2*a^5*b^3*c^2 - 2*a^5*b^2*c^2 - 3/2*a^5*b^2*c - 1/2*a^5*b*c^2 + 3/2*a^5*b*c + 1/4*a^5*b + 1/8*a^5*c - 1/4*a^5 + a^4*b^4*c^2 - 2*a^4*b^3*c^2 - 3/2*a^4*b^3*c + 3*a^4*b^2*c + 3/8*a^4*b^2 + a^4*b*c^2 - 9/8*a^4*b*c - 3/4*a^4*b + 1/16*a^4*c^2 - 3/8*a^4*c + 3/8*a^4 - 1/2*a^3*b^4*c - 1/2*a^3*b^3*c^2 + 3/2*a^3*b^3*c + 1/4*a^3*b^3 + a^3*b^2*c^2 - 9/8*a^3*b^2*c - 3/4*a^3*b^2 - 3/8*a^3*b*c^2 - 1/4*a^3*b*c + 3/4*a^3*b - 1/8*a^3*c^2 + 3/8*a^3*c - 1/4*a^3 + 1/16*a^2*b^4 + 1/8*a^2*b^3*c - 1/4*a^2*b^3 + 1/16*a^2*b^2*c^2 - 3/8*a^2*b^2*c + 3/8*a^2*b^2 - 1/8*a^2*b*c^2 + 3/8*a^2*b*c - 1/4*a^2*b + 1/16*a^2*c^2 - 1/8*a^2*c + 1/16*a^2)/(a^4*c^2 + 2*a^3*c^3 - 3*a^3*c^2 + a^2*c^4 - 4*a^2*c^3 + 13/4*a^2*c^2 - a*c^4 + 5/2*a*c^3 - 3/2*a*c^2 + 1/4*c^4 - 1/2*c^3 + 1/4*c^2)) (-2*a^3*b*c^2 + 1/2*a^3*c - 2*a^2*b^2*c^2 + 3*a^2*b*c^2 + 3*a^2*b*c + 1/2*a^2*c^2 - 5/4*a^2*c - 1/2*a^2 + a*b^2*c^2 + 5/2*a*b^2*c - 1/2*a*b*c^2 - 7/2*a*b*c - a*b - 3/4*a*c^2 + 1/2*a*c + a - 1/4*b^2*c - 1/2*b^2 - 1/4*b*c^2 + b + 1/4*c^2 + 1/4*c - 1/2)/(a^2*b*c - 1/2*a^2*c + 2*a*b*c^2 - 2*a*b*c - a*c^2 + a*c + b*c^3 - 2*b*c^2 + b*c - 1/2*c^3 + c^2 - 1/2*c)*$.1^2/($.1^4 + (-4*a*b*c + a + b + c - 1)/(a^2*c + a*c^2 - 3/2*a*c - 1/2*c^2 + 1/2*c)*$.1^3 + (2*a^5*b*c^2 - 1/2*a^5*c + 2*a^4*b^2*c^2 + 2*a^4*b*c^3 - 5*a^4*b*c^2 - a^4*b*c - a^4*c^2 + 7/4*a^4*c + 2*a^3*b^2*c^3 - 3*a^3*b^2*c^2 - 1/2*a^3*b^2*c - 3*a^3*b*c^3 + 5/2*a^3*b*c^2 + 5/2*a^3*b*c - 1/2*a^3*c^3 + 5/2*a^3*c^2 - 9/4*a^3*c - a^2*b^2*c^3 + 9/2*a^2*b^2*c^2 + 3/4*a^2*b^2*c + 1/2*a^2*b*c^3 + 5/4*a^2*b*c^2 - 4*a^2*b*c + 3/4*a^2*c^3 - 2*a^2*c^2 + 5/4*a^2*c + 1/4*a^2 + 1/4*a*b^2*c^2 - 9/4*a*b^2*c + 1/4*a*b*c^3 - 11/4*a*b*c^2 + 5/2*a*b*c + 1/2*a*b - 1/4*a*c^3 + 1/2*a*c^2 + 1/4*a*c - 1/2*a + 1/4*b^2 + 1/2*b*c - 1/2*b + 1/4*c^2 - 1/2*c + 1/4)/(a^4*c^2 + 2*a^3*c^3 - 3*a^3*c^2 + a^2*c^4 - 4*a^2*c^3 + 13/4*a^2*c^2 - a*c^4 + 5/2*a*c^3 - 3/2*a*c^2 + 1/4*c^4 - 1/2*c^3 + 1/4*c^2)*$.1^2 + (-4*a^4*b^2*c^2 + 2*a^4*b*c - 1/4*a^4 - 4*a^3*b^3*c^2 + 4*a^3*b^2*c^2 + 4*a^3*b^2*c + 2*a^3*b*c^2 - 4*a^3*b*c - 3/4*a^3*b - 1/2*a^3*c + 3/4*a^3 + 2*a^2*b^3*c + 2*a^2*b^2*c^2 - 4*a^2*b^2*c - 3/4*a^2*b^2 - 2*a^2*b*c^2 + a^2*b*c + 3/2*a^2*b - 1/4*a^2*c^2 + a^2*c - 3/4*a^2 - 1/4*a*b^3 - 1/2*a*b^2*c + 3/4*a*b^2 - 1/4*a*b*c^2 + a*b*c - 3/4*a*b + 1/4*a*c^2 - 1/2*a*c + 1/4*a)/(a^4*c^2 + 2*a^3*c^3 - 3*a^3*c^2 + a^2*c^4 - 4*a^2*c^3 + 13/4*a^2*c^2 - a*c^4 + 5/2*a*c^3 - 3/2*a*c^2 + 1/4*c^4 - 1/2*c^3 + 1/4*c^2)*$.1 + (a^6*b^2*c^2 - 1/2*a^6*b*c + 1/16*a^6 + 2*a^5*b^3*c^2 - 2*a^5*b^2*c^2 - 3/2*a^5*b^2*c - 1/2*a^5*b*c^2 + 3/2*a^5*b*c + 1/4*a^5*b + 1/8*a^5*c - 1/4*a^5 + a^4*b^4*c^2 - 2*a^4*b^3*c^2 - 3/2*a^4*b^3*c + 3*a^4*b^2*c + 3/8*a^4*b^2 + a^4*b*c^2 - 9/8*a^4*b*c - 3/4*a^4*b + 1/16*a^4*c^2 - 3/8*a^4*c + 3/8*a^4 - 1/2*a^3*b^4*c - 1/2*a^3*b^3*c^2 + 3/2*a^3*b^3*c + 1/4*a^3*b^3 + a^3*b^2*c^2 - 9/8*a^3*b^2*c - 3/4*a^3*b^2 - 3/8*a^3*b*c^2 - 1/4*a^3*b*c + 3/4*a^3*b - 1/8*a^3*c^2 + 3/8*a^3*c - 1/4*a^3 + 1/16*a^2*b^4 + 1/8*a^2*b^3*c - 1/4*a^2*b^3 + 1/16*a^2*b^2*c^2 - 3/8*a^2*b^2*c + 3/8*a^2*b^2 - 1/8*a^2*b*c^2 + 3/8*a^2*b*c - 1/4*a^2*b + 1/16*a^2*c^2 - 1/8*a^2*c + 1/16*a^2)/(a^4*c^2 + 2*a^3*c^3 - 3*a^3*c^2 + a^2*c^4 - 4*a^2*c^3 + 13/4*a^2*c^2 - a*c^4 + 5/2*a*c^3 - 3/2*a*c^2 + 1/4*c^4 - 1/2*c^3 + 1/4*c^2)) ((-2*a^4*b*c^2 - a^4*b*c + a^4*c - 4*a^3*b^2*c^2 - 2*a^3*b^2*c + 2*a^3*b*c^3 + 6*a^3*b*c^2 + 13/2*a^3*b*c + a^3*b - a^3*c^2 - 5/2*a^3*c - a^3 - 2*a^2*b^3*c^2 - a^2*b^3*c - 4*a^2*b^2*c^3 + 12*a^2*b^2*c^2 + 8*a^2*b^2*c + 2*a^2*b^2 + 3*a^2*b*c^3 - 29/2*a^2*b*c^2 - 10*a^2*b*c - 4*a^2*b - 2*a^2*c^3 + 7/2*a^2*c^2 + 3*a^2*c + 2*a^2 - 6*a*b^3*c^3 + 6*a*b^3*c^2 + 5/2*a*b^3*c + a*b^3 + 14*a*b^2*c^3 - 12*a*b^2*c^2 - 23/2*a*b^2*c - 3*a*b^2 - 10*a*b*c^3 + 9*a*b*c^2 + 23/2*a*b*c + 3*a*b + 3*a*c^3 - 3/2*a*c^2 - 7/2*a*c - a + 3*b^3*c^3 + 7/2*b^3*c^2 - 5*b^3*c - 6*b^2*c^3 - 13/2*b^2*c^2 + 11*b^2*c + 4*b*c^3 + 4*b*c^2 - 8*b*c - c^3 - c^2 + 2*c)/(a^3*b*c + a^3*c^2 - a^3*c + 3*a^2*b*c^2 - 3*a^2*b*c + 3*a^2*c^3 - 6*a^2*c^2 + 3*a^2*c + 3*a*b*c^3 - 6*a*b*c^2 + 3*a*b*c + 3*a*c^4 - 9*a*c^3 + 9*a*c^2 - 3*a*c + b*c^4 - 3*b*c^3 + 3*b*c^2 - b*c + c^5 - 4*c^4 + 6*c^3 - 4*c^2 + c)*$.1^4 + (-4*a^5*b^2*c^2 + 4*a^5*b*c^2 + a^5*b*c - a^5*c - 8*a^4*b^3*c^2 - 4*a^4*b^2*c^3 + 18*a^4*b^2*c^2 + 7*a^4*b^2*c - 8*a^4*b*c^2 - 21/2*a^4*b*c - a^4*b - a^4*c^2 + 7/2*a^4*c + a^4 - 4*a^3*b^4*c^2 + 8*a^3*b^3*c^3 + 16*a^3*b^3*c^2 + 11*a^3*b^3*c - 6*a^3*b^2*c^3 - 15*a^3*b^2*c^2 - 53/2*a^3*b^2*c - 3*a^3*b^2 + 5*a^3*b*c^3 + 4*a^3*b*c^2 + 18*a^3*b*c + 6*a^3*b + 3/2*a^3*c^2 - 7/2*a^3*c - 3*a^3 + 12*a^2*b^4*c^3 + 2*a^2*b^4*c^2 + 5*a^2*b^4*c - 28*a^2*b^3*c^3 - 18*a^2*b^3*c^2 - 35/2*a^2*b^3*c - 3*a^2*b^3 + 18*a^2*b^2*c^3 + 41/2*a^2*b^2*c^2 + 43/2*a^2*b^2*c + 9*a^2*b^2 - 9/2*a^2*b*c^3 - 19/2*a^2*b*c^2 - 19/2*a^2*b*c - 9*a^2*b - a^2*c^3 + 1/2*a^2*c^2 + 3/2*a^2*c + 3*a^2 - 6*a*b^4*c^3 - 15*a*b^4*c^2 - 1/2*a*b^4*c - a*b^4 + 9*a*b^3*c^3 + 35*a*b^3*c^2 + 6*a*b^3*c + 4*a*b^3 - a*b^2*c^3 - 55/2*a*b^2*c^2 - 10*a*b^2*c - 6*a*b^2 - 3*a*b*c^3 + 9*a*b*c^2 + 6*a*b*c + 4*a*b + 3/2*a*c^3 - 1/2*a*c^2 - 3/2*a*c - a + 3/2*b^4*c^2 + 3*b^4*c + 3/2*b^3*c^3 - 3/2*b^3*c^2 - 9*b^3*c - 3*b^2*c^3 - b^2*c^2 + 10*b^2*c + 2*b*c^3 + 3/2*b*c^2 - 5*b*c - 1/2*c^3 - 1/2*c^2 + c)/(a^4*b*c^2 + a^4*c^3 - a^4*c^2 + 3*a^3*b*c^3 - 7/2*a^3*b*c^2 + 3*a^3*c^4 - 13/2*a^3*c^3 + 7/2*a^3*c^2 + 3*a^2*b*c^4 - 15/2*a^2*b*c^3 + 9/2*a^2*b*c^2 + 3*a^2*c^5 - 21/2*a^2*c^4 + 12*a^2*c^3 - 9/2*a^2*c^2 + a*b*c^5 - 9/2*a*b*c^4 + 6*a*b*c^3 - 5/2*a*b*c^2 + a*c^6 - 11/2*a*c^5 + 21/2*a*c^4 - 17/2*a*c^3 + 5/2*a*c^2 - 1/2*b*c^5 + 3/2*b*c^4 - 3/2*b*c^3 + 1/2*b*c^2 - 1/2*c^6 + 2*c^5 - 3*c^4 + 2*c^3 - 1/2*c^2)*$.1^3 + (2*a^6*b^2*c^3 + a^6*b^2*c^2 - 3/2*a^6*b*c^2 - 1/4*a^6*b*c + 1/4*a^6*c - 2*a^5*b^3*c^3 + 3*a^5*b^3*c^2 + a^5*b^2*c^3 - 17/2*a^5*b^2*c^2 - 2*a^5*b^2*c - 5/2*a^5*b*c^3 + 7/2*a^5*b*c^2 + 29/8*a^5*b*c + 1/4*a^5*b + 3/4*a^5*c^2 - 9/8*a^5*c - 1/4*a^5 - 10*a^4*b^4*c^3 + 3*a^4*b^4*c^2 + 21*a^4*b^3*c^3 - 11/2*a^4*b^3*c^2 - 9/2*a^4*b^3*c - 29/2*a^4*b^2*c^3 + 13/4*a^4*b^2*c^2 + 11*a^4*b^2*c + a^4*b^2 + 19/4*a^4*b*c^3 + 33/8*a^4*b*c^2 - 33/4*a^4*b*c - 2*a^4*b + 1/2*a^4*c^3 - 21/8*a^4*c^2 + 5/4*a^4*c + a^4 - 6*a^3*b^5*c^3 + a^3*b^5*c^2 + 23*a^3*b^4*c^3 + 17/2*a^3*b^4*c^2 - 4*a^3*b^4*c - 55/2*a^3*b^3*c^3 - 95/4*a^3*b^3*c^2 + 43/4*a^3*b^3*c + 3/2*a^3*b^3 + 47/4*a^3*b^2*c^3 + 193/8*a^3*b^2*c^2 - 21/2*a^3*b^2*c - 9/2*a^3*b^2 - 1/4*a^3*b*c^3 - 53/4*a^3*b*c^2 + 13/4*a^3*b*c + 9/2*a^3*b - 5/4*a^3*c^3 + 23/8*a^3*c^2 + 1/2*a^3*c - 3/2*a^3 + 3*a^2*b^5*c^3 + 7*a^2*b^5*c^2 - 5/4*a^2*b^5*c - 15/2*a^2*b^4*c^3 - 97/4*a^2*b^4*c^2 + 7/4*a^2*b^4*c + a^2*b^4 + 17/4*a^2*b^3*c^3 + 251/8*a^2*b^3*c^2 + 9/4*a^2*b^3*c - 4*a^2*b^3 + 5/2*a^2*b^2*c^3 - 159/8*a^2*b^2*c^2 - 25/4*a^2*b^2*c + 6*a^2*b^2 - 13/4*a^2*b*c^3 + 53/8*a^2*b*c^2 + 5*a^2*b*c - 4*a^2*b + a^2*c^3 - 7/8*a^2*c^2 - 3/2*a^2*c + a^2 - 3/4*a*b^5*c^2 - 11/8*a*b^5*c + 1/4*a*b^5 - 3/4*a*b^4*c^3 + 13/8*a*b^4*c^2 + 45/8*a*b^4*c - 5/4*a*b^4 + 9/4*a*b^3*c^3 - 3/4*a*b^3*c^2 - 37/4*a*b^3*c + 5/2*a*b^3 - 5/2*a*b^2*c^3 - 1/2*a*b^2*c^2 + 31/4*a*b^2*c - 5/2*a*b^2 + 5/4*a*b*c^3 + 1/2*a*b*c^2 - 27/8*a*b*c + 5/4*a*b - 1/4*a*c^3 - 1/8*a*c^2 + 5/8*a*c - 1/4*a)/(a^4*b*c^2 + a^4*c^3 - a^4*c^2 + 3*a^3*b*c^3 - 7/2*a^3*b*c^2 + 3*a^3*c^4 - 13/2*a^3*c^3 + 7/2*a^3*c^2 + 3*a^2*b*c^4 - 15/2*a^2*b*c^3 + 9/2*a^2*b*c^2 + 3*a^2*c^5 - 21/2*a^2*c^4 + 12*a^2*c^3 - 9/2*a^2*c^2 + a*b*c^5 - 9/2*a*b*c^4 + 6*a*b*c^3 - 5/2*a*b*c^2 + a*c^6 - 11/2*a*c^5 + 21/2*a*c^4 - 17/2*a*c^3 + 5/2*a*c^2 - 1/2*b*c^5 + 3/2*b*c^4 - 3/2*b*c^3 + 1/2*b*c^2 - 1/2*c^6 + 2*c^5 - 3*c^4 + 2*c^3 - 1/2*c^2)*$.1^2)/ ** WARNING: Output too long, hence truncated. '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Tue Dec 6 08:31:46 2005 Input: R:=PolynomialRing(RationalField(),3); print Factorization((a^2*c + a*c^2 - a*c)); print Factorization((-2*a^2*b*c + 1/2*a^2 + 1/2*a*b + 1/2*a*c - 1/2*a)); print Factorization((a^4*b*c - 1/4*a^4 + a^3*b^2*c - a^3*b*c - 1/2*a^3*b - 1/4*a^3*c + 1/2*a^3 - 1/4*a^2*b^2 - 1/4*a^2*b*c + 1/2*a^2*b + 1/4*a^2*c - 1/4*a^2)); print Factorization((a^2*c - a*c + 1/4*c)); Output: Magma V2.11-10 Tue Dec 6 2005 08:31:46 on modular [Seed = 2655538967] ------------------------------------- [ , , ] [ , ] [ , , ] [ , ] Total time: 0.200 seconds, Total memory usage: 3.24MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Tue Dec 6 08:27:45 2005 Input: R:=PolynomialRing(RationalField(),3); print Factorization((-a^2*c + a*c^2)); print Factorization((a^3 - 2*a^2*b*c + a^2*b - a^2 + 1/2*a*c)); print Factorization((a^4*b*c - 1/4*a^4 + a^3*b^2*c - a^3*b*c - 1/2*a^3*b - 1/4*a^3*c + 1/2*a^3 - 1/4*a^2*b^2 - 1/4*a^2*b*c + 1/2*a^2*b + 1/4*a^2*c - 1/4*a^2)); print Factorization((a^2*c - a*c + 1/4*c)); Output: Magma V2.11-10 Tue Dec 6 2005 08:27:44 on modular [Seed = 2605533440] ------------------------------------- [ , , ] [ , ] [ , , ] [ , ] Total time: 0.190 seconds, Total memory usage: 3.24MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Tue Dec 6 08:26:25 2005 Input: K:=FunctionField(RationalField(),3); S:=PolynomialRing(K,1); p1:= (2*a*c - c)*t; p2:= c*t + (-a*b + 1/2*a)/(a - 1/2); py:= (-a^2*c + a*c^2)*t^2 + (a^3 - 2*a^2*b*c + a^2*b - a^2 + 1/2*a*c)/(a - 1/2)*t + (a^4*b*c - 1/4*a^4 + a^3*b^2*c - a^3*b*c - 1/2*a^3*b - 1/4*a^3*c + 1/2*a^3 - 1/4*a^2*b^2 - 1/4*a^2*b*c + 1/2*a^2*b + 1/4*a^2*c - 1/4*a^2)/(a^2*c - a*c + 1/4*c); q := (a^2*c + a*c^2 - a*c)*t^2 + (-2*a^2*b*c + 1/2*a^2 + 1/2*a*b + 1/2*a*c - 1/2*a)/(a - 1/2)*t + (a^4*b*c - 1/4*a^4 + a^3*b^2*c - a^3*b*c - 1/2*a^3*b - 1/4*a^3*c + 1/2*a^3 - 1/4*a^2*b^2 - 1/4*a^2*b*c + 1/2*a^2*b + 1/4*a^2*c - 1/4*a^2)/(a^2*c - a*c + 1/4*c); print Factorization(p1); print Factorization(p2); print Factorization(py); print Factorization(q); Output: Magma V2.11-10 Tue Dec 6 2005 08:26:24 on modular [Seed = 2169983818] ------------------------------------- [ ] [ ] [ ] [ ] Total time: 0.210 seconds, Total memory usage: 3.34MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Tue Dec 6 08:24:56 2005 Input: K:=FunctionField(RationalField(),3); S:=PolynomialRing(K,1); print Factorization((-a^2*c + a*c^2)); Output: Magma V2.11-10 Tue Dec 6 2005 08:24:56 on modular [Seed = 2287885035] ------------------------------------- >> print Factorization((-a^2*c + a*c^2)); ^ Runtime error in 'Factorization': Bad argument types Argument types given: FldFunRatMElt Total time: 0.190 seconds, Total memory usage: 3.24MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Tue Dec 6 08:24:03 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*A*Transpose(P)-A)); G:=GroebnerBasis(I); S:=PolynomialRing(K,1); p1 := t * c*(2*a-1) + (a+b-1); p2 := t * c + (1-b); px := -a * p1; pz := p1 * p2; py := t^2 * a*c*(c-a) + t *a*c*(1-2*b) + a*b*(a+b-1); q := t^2 * a*c*(a+c-1) + t *a*(a-2*b*c+b+c-1) + a*b*(a+b-1); t0 := (a+b-1)/(c*(1-2*a)); print Evaluate(p1,[t+t0]); print Evaluate(p2,[t+t0]); print Evaluate(py,[t+t0]); print Evaluate(q,[t+t0]); print Evaluate(G[7],[0,0,0,0,0,0,pz/q,px/q,py/q]); print Evaluate(G[8],[0,0,0,0,0,0,pz/q,px/q,py/q]); print Evaluate(G[9],[0,0,0,0,0,0,pz/q,px/q,py/q]); print Evaluate(G[10],[0,0,0,0,0,0,pz/q,px/q,py/q]); Output: Magma V2.11-10 Tue Dec 6 2005 08:24:02 on modular [Seed = 4045873736] ------------------------------------- (2*a*c - c)*t c*t + (-a*b + 1/2*a)/(a - 1/2) (-a^2*c + a*c^2)*t^2 + (a^3 - 2*a^2*b*c + a^2*b - a^2 + 1/2*a*c)/(a - 1/2)*t + (a^4*b*c - 1/4*a^4 + a^3*b^2*c - a^3*b*c - 1/2*a^3*b - 1/4*a^3*c + 1/2*a^3 - 1/4*a^2*b^2 - 1/4*a^2*b*c + 1/2*a^2*b + 1/4*a^2*c - 1/4*a^2)/(a^2*c - a*c + 1/4*c) (a^2*c + a*c^2 - a*c)*t^2 + (-2*a^2*b*c + 1/2*a^2 + 1/2*a*b + 1/2*a*c - 1/2*a)/(a - 1/2)*t + (a^4*b*c - 1/4*a^4 + a^3*b^2*c - a^3*b*c - 1/2*a^3*b - 1/4*a^3*c + 1/2*a^3 - 1/4*a^2*b^2 - 1/4*a^2*b*c + 1/2*a^2*b + 1/4*a^2*c - 1/4*a^2)/(a^2*c - a*c + 1/4*c) 0 0 0 0 Total time: 1.730 seconds, Total memory usage: 4.60MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Tue Dec 6 08:20:28 2005 Input: K:=FunctionField(RationalField(),4); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*A*Transpose(P)-A)); G:=GroebnerBasis(I); S:=PolynomialRing(K,1); p1 := t * c*(2*a-1) + (a+b-1); p2 := t * c + (1-b); px := -a * p1; pz := p1 * p2; py := t^2 * a*c*(c-a) + t *a*c*(1-2*b) + a*b*(a+b-1); q := t^2 * a*c*(a+c-1) + t *a*(a-2*b*c+b+c-1) + a*b*(a+b-1); t0 := (a+b-1)/(c*(1-2*a)); print Evaluate(p1,[t+t0]); print Evaluate(p2,[t+t0]); print Evaluate(py,[t+t0]); print Evaluate(q,[t+t0]); print Evaluate(G[7],[0,0,0,0,0,0,pz/q,px/q,py/q]); print Evaluate(G[8],[0,0,0,0,0,0,pz/q,px/q,py/q]); print Evaluate(G[9],[0,0,0,0,0,0,pz/q,px/q,py/q]); print Evaluate(G[10],[0,0,0,0,0,0,pz/q,px/q,py/q]); Output: Magma V2.11-10 Tue Dec 6 2005 08:20:26 on modular [Seed = 3590325639] ------------------------------------- (2*a*c - c)*t c*t + (-a*b + 1/2*a)/(a - 1/2) (-a^2*c + a*c^2)*t^2 + (a^3 - 2*a^2*b*c + a^2*b - a^2 + 1/2*a*c)/(a - 1/2)*t + (a^4*b*c - 1/4*a^4 + a^3*b^2*c - a^3*b*c - 1/2*a^3*b - 1/4*a^3*c + 1/2*a^3 - 1/4*a^2*b^2 - 1/4*a^2*b*c + 1/2*a^2*b + 1/4*a^2*c - 1/4*a^2)/(a^2*c - a*c + 1/4*c) (a^2*c + a*c^2 - a*c)*t^2 + (-2*a^2*b*c + 1/2*a^2 + 1/2*a*b + 1/2*a*c - 1/2*a)/(a - 1/2)*t + (a^4*b*c - 1/4*a^4 + a^3*b^2*c - a^3*b*c - 1/2*a^3*b - 1/4*a^3*c + 1/2*a^3 - 1/4*a^2*b^2 - 1/4*a^2*b*c + 1/2*a^2*b + 1/4*a^2*c - 1/4*a^2)/(a^2*c - a*c + 1/4*c) 0 0 0 0 Total time: 1.610 seconds, Total memory usage: 4.37MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Tue Dec 6 08:19:52 2005 Input: K:=FunctionField(RationalField(),4); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*A*Transpose(P)-A)); G:=GroebnerBasis(I); S:=PolynomialRing(K,1); p1 := t * c*(2*a-1) + (a+b-1); p2 := t * c + (1-b); px := -a * p1; pz := p1 * p2; py := t^2 * a*c*(c-a) + t *a*c*(1-2*b) + a*b*(a+b-1); q := t^2 * a*c*(a+c-1) + t *a*(a-2*b*c+b+c-1) + a*b*(a+b-1); t0 := (a+b-1)/(c*(1-2*a)); print Evaluate(p1,[t+t0]); print Evaluate(G[7],[0,0,0,0,0,0,pz/q,px/q,py/q]); print Evaluate(G[8],[0,0,0,0,0,0,pz/q,px/q,py/q]); print Evaluate(G[9],[0,0,0,0,0,0,pz/q,px/q,py/q]); print Evaluate(G[10],[0,0,0,0,0,0,pz/q,px/q,py/q]); Output: Magma V2.11-10 Tue Dec 6 2005 08:19:50 on modular [Seed = 3658225255] ------------------------------------- (2*a*c - c)*t 0 0 0 0 Total time: 1.610 seconds, Total memory usage: 4.37MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Tue Dec 6 08:18:46 2005 Input: K:=FunctionField(RationalField(),4); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*A*Transpose(P)-A)); G:=GroebnerBasis(I); S:=PolynomialRing(K,1); p1 := t * c*(2*a-1) + (a+b-1); p2 := t * c + (1-b); px := -a * p1; pz := p1 * p2; py := t^2 * a*c*(c-a) + t *a*c*(1-2*b) + a*b*(a+b-1); q := t^2 * a*c*(a+c-1) + t *a*(a-2*b*c+b+c-1) + a*b*(a+b-1); print Evaluate(px,[(a+b-1)/(c*(1-2*a))]); print Evaluate(py,[(a+b-1)/(c*(1-2*a))]); print Evaluate(pz,[(a+b-1)/(c*(1-2*a))]); print Evaluate(q,[(a+b-1)/(c*(1-2*a))]); print Evaluate(G[7],[0,0,0,0,0,0,pz/q,px/q,py/q]); print Evaluate(G[8],[0,0,0,0,0,0,pz/q,px/q,py/q]); print Evaluate(G[9],[0,0,0,0,0,0,pz/q,px/q,py/q]); print Evaluate(G[10],[0,0,0,0,0,0,pz/q,px/q,py/q]); Output: Magma V2.11-10 Tue Dec 6 2005 08:18:44 on modular [Seed = 3239513406] ------------------------------------- 0 (a^4*b*c - 1/4*a^4 + a^3*b^2*c - a^3*b*c - 1/2*a^3*b - 1/4*a^3*c + 1/2*a^3 - 1/4*a^2*b^2 - 1/4*a^2*b*c + 1/2*a^2*b + 1/4*a^2*c - 1/4*a^2)/(a^2*c - a*c + 1/4*c) 0 (a^4*b*c - 1/4*a^4 + a^3*b^2*c - a^3*b*c - 1/2*a^3*b - 1/4*a^3*c + 1/2*a^3 - 1/4*a^2*b^2 - 1/4*a^2*b*c + 1/2*a^2*b + 1/4*a^2*c - 1/4*a^2)/(a^2*c - a*c + 1/4*c) 0 0 0 0 Total time: 1.600 seconds, Total memory usage: 4.37MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Tue Dec 6 08:18:18 2005 Input: K:=FunctionField(RationalField(),4); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*A*Transpose(P)-A)); G:=GroebnerBasis(I); S:=PolynomialRing(K,1); p1 := t * c*(2*a-1) + (a+b-1); p2 := t * c + (1-b); px := -a * p1; pz := p1 * p2; py := t^2 * a*c*(c-a) + t *a*c*(1-2*b) + a*b*(a+b-1); q := t^2 * a*c*(a+c-1) + t *a*(a-2*b*c+b+c-1) + a*b*(a+b-1); print Evaluate(px,[(a+b-1)/(c*(1-2*a))]); print Evaluate(G[7],[0,0,0,0,0,0,pz/q,px/q,py/q]); print Evaluate(G[8],[0,0,0,0,0,0,pz/q,px/q,py/q]); print Evaluate(G[9],[0,0,0,0,0,0,pz/q,px/q,py/q]); print Evaluate(G[10],[0,0,0,0,0,0,pz/q,px/q,py/q]); Output: Magma V2.11-10 Tue Dec 6 2005 08:18:16 on modular [Seed = 3457679901] ------------------------------------- 0 0 0 0 0 Total time: 1.629 seconds, Total memory usage: 4.37MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Tue Dec 6 08:17:34 2005 Input: K:=FunctionField(RationalField(),4); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*A*Transpose(P)-A)); G:=GroebnerBasis(I); p1 := t * c*(2*a-1) + (a+b-1); p2 := t * c + (1-b); px := -a * p1; pz := p1 * p2; py := t^2 * a*c*(c-a) + t *a*c*(1-2*b) + a*b*(a+b-1); q := t^2 * a*c*(a+c-1) + t *a*(a-2*b*c+b+c-1) + a*b*(a+b-1); print Evaluate(px,[(a+b-1)/(c*(1-2*a))]); print Evaluate(G[7],[0,0,0,0,0,0,pz/q,px/q,py/q]); print Evaluate(G[8],[0,0,0,0,0,0,pz/q,px/q,py/q]); print Evaluate(G[9],[0,0,0,0,0,0,pz/q,px/q,py/q]); print Evaluate(G[10],[0,0,0,0,0,0,pz/q,px/q,py/q]); Output: Magma V2.11-10 Tue Dec 6 2005 08:17:33 on modular [Seed = 820444182] ------------------------------------- >> print Evaluate(px,[(a+b-1)/(c*(1-2*a))]); ^ Runtime error in 'Evaluate': Bad argument types Argument types given: FldFunRatMElt, SeqEnum[FldFunRatMElt] 0 0 0 0 Total time: 1.409 seconds, Total memory usage: 4.46MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Tue Dec 6 08:17:18 2005 Input: K:=FunctionField(RationalField(),4); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*A*Transpose(P)-A)); G:=GroebnerBasis(I); p1 := t * c*(2*a-1) + (a+b-1); p2 := t * c + (1-b); px := -a * p1; pz := p1 * p2; py := t^2 * a*c*(c-a) + t *a*c*(1-2*b) + a*b*(a+b-1); q := t^2 * a*c*(a+c-1) + t *a*(a-2*b*c+b+c-1) + a*b*(a+b-1); print Evaluate(px,(a+b-1)/(c*(1-2*a))); print Evaluate(G[7],[0,0,0,0,0,0,pz/q,px/q,py/q]); print Evaluate(G[8],[0,0,0,0,0,0,pz/q,px/q,py/q]); print Evaluate(G[9],[0,0,0,0,0,0,pz/q,px/q,py/q]); print Evaluate(G[10],[0,0,0,0,0,0,pz/q,px/q,py/q]); Output: Magma V2.11-10 Tue Dec 6 2005 08:17:16 on modular [Seed = 1055453737] ------------------------------------- >> print Evaluate(px,(a+b-1)/(c*(1-2*a))); ^ Runtime error in 'Evaluate': Bad argument types Argument types given: FldFunRatMElt, FldFunRatMElt 0 0 0 0 Total time: 1.419 seconds, Total memory usage: 4.46MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Tue Dec 6 08:16:01 2005 Input: K:=FunctionField(RationalField(),4); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*A*Transpose(P)-A)); G:=GroebnerBasis(I); p1 := t * c*(2*a-1) + (a+b-1); p2 := t * c + (1-b); px := -a * p1; pz := p1 * p2; py := t^2 * a*c*(c-a) + t *a*c*(1-2*b) + a*b*(a+b-1); q := t^2 * a*c*(a+c-1) + t *a*(a-2*b*c+b+c-1) + a*b*(a+b-1); print Evaluate(G[7],[0,0,0,0,0,0,pz/q,px/q,py/q]); print Evaluate(G[8],[0,0,0,0,0,0,pz/q,px/q,py/q]); print Evaluate(G[9],[0,0,0,0,0,0,pz/q,px/q,py/q]); print Evaluate(G[10],[0,0,0,0,0,0,pz/q,px/q,py/q]); Output: Magma V2.11-10 Tue Dec 6 2005 08:16:00 on modular [Seed = 569896314] ------------------------------------- 0 0 0 0 Total time: 1.409 seconds, Total memory usage: 4.46MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Tue Dec 6 08:15:39 2005 Input: K:=FunctionField(RationalField(),4); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*A*Transpose(P)-A)); G:=GroebnerBasis(I); p1 := t * c*(2*a-1) + (a+b-1); p2 := t * c + (1-b); px := -a * p1; pz := p1 * p2; py := t^2 * a*c*(c-a) + t *a*c*(1-2*b) + a*b*(a+b-1); q := t^2 * a*c*(a+c-1) + t *a*(a-2*b*c+b+c-1) + a*b*(a+b-1); print Evaluate(G[7],[0,0,0,0,0,0,pz/q,px/q,py/q]); Output: Magma V2.11-10 Tue Dec 6 2005 08:15:37 on modular [Seed = 788063132] ------------------------------------- 0 Total time: 1.340 seconds, Total memory usage: 4.33MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Tue Dec 6 08:14:35 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*A*Transpose(P)-A)); G:=GroebnerBasis(I); p1 := t * c*(2*a-1) + (a+b-1); p2 := t * c + (1-b); px := -a * p1; pz := p1 * p2; py := t^2 * a*c*(c-a) + t *a*c*(1-2*b) + a*b*(a+b-1); q := t^2 * a*c*(a+c-1) + t *a*(a-2*b*c+b+c-1) + a*b*(a+b-1); print Evaluate(G[7],[0,0,0,0,0,0,pz/q,px/q,py/q]); Output: Magma V2.11-10 Tue Dec 6 2005 08:14:33 on modular [Seed = 348314086] ------------------------------------- >> p1 := t * c*(2*a-1) + (a+b-1); ^ User error: Identifier 't' has not been declared or assigned >> p2 := t * c + (1-b); ^ User error: Identifier 't' has not been declared or assigned >> px := -a * p1; ^ User error: Identifier 'p1' has not been declared or assigned >> pz := p1 * p2; ^ User error: Identifier 'p1' has not been declared or assigned >> py := t^2 * a*c*(c-a) + t *a*c*(1-2*b) + a*b*(a+b-1); ^ User error: Identifier 't' has not been declared or assigned >> q := t^2 * a*c*(a+c-1) + t *a*(a-2*b*c+b+c-1) + a*b*(a+b-1); ^ User error: Identifier 't' has not been declared or assigned >> print Evaluate(G[7],[0,0,0,0,0,0,pz/q,px/q,py/q]); ^ User error: Identifier 'pz' has not been declared or assigned Total time: 1.459 seconds, Total memory usage: 4.60MB '66.69.1' ************** MAGMA ***************** Host 66.69.190.149 (66.69.190.149) Time: Tue Dec 6 08:03:45 2005 Input: "Replace this by some code, then click [PARI] or [MAGMA]!" Output: Magma V2.11-10 Tue Dec 6 2005 08:03:45 on modular [Seed = 2692116437] ------------------------------------- Replace this by some code, then click [PARI] or [MAGMA]! Total time: 0.210 seconds, Total memory usage: 3.24MB '129.67.' ************** MAGMA ***************** Host 129.67.62.181 (129.67.62.181) Time: Tue Dec 6 07:09:58 2005 Input: G:=GroupOfLieType("A2", 9) Output: Magma V2.11-10 Tue Dec 6 2005 07:09:57 on modular [Seed = 2091532460] ------------------------------------- Total time: 0.240 seconds, Total memory usage: 4.95MB '129.67.' ************** MAGMA ***************** Host 129.67.62.181 (129.67.62.181) Time: Tue Dec 6 07:06:40 2005 Input: ?GroupOfLieType ?2 Output: Magma V2.11-10 Tue Dec 6 2005 07:06:39 on modular [Seed = 904071356] ------------------------------------- 12 matches: 1 I /magma/group/fiurg/related/GroupOfLieType 2 I /magma/group/groups-lie-type/construction/GroupOfLieType 3 I /magma/group/groups-lie-type/construction/GroupOfLieType 4 I /magma/group/groups-lie-type/construction/GroupOfLieType 5 I /magma/group/groups-lie-type/construction/GroupOfLieType 6 I /magma/group/groups-lie-type/construction/GroupOfLieType 7 I /magma/group/groups-lie-type/construction/GroupOfLieType 8 I /magma/group/groups-lie-type/construction/GroupOfLieType 9 I /magma/group/groups-lie-type/construction/GroupOfLieType 10 I /magma/group/root-data/related-structures/GroupOfLieType 11 I /magma/lie-theory/coxeter-description/related/GroupOfLieType 12 I /magma/lie-theory/finite-coxeter-groups/related/GroupOfLieType To view an entry, type ? followed by the number next to it =============================================================================== PATH: /magma/group/groups-lie-type/construction/GroupOfLieType KIND: Intrinsic =============================================================================== GroupOfLieType(C, k) : Mtrx, Rng -> GrpLie GroupOfLieType(D, k) : Mtrx, Rng -> GrpLie Isogeny: BoolElt Default: "Ad" Normalising: BoolElt Default: true Construct the group of Lie type with Cartan matrix C or Dynkin digraph D, over the ring k. The optional parameter Isogeny can take the values described in Section H84E17. =============================================================================== Total time: 0.210 seconds, Total memory usage: 3.24MB '129.67.' ************** MAGMA ***************** Host 129.67.62.181 (129.67.62.181) Time: Tue Dec 6 07:06:25 2005 Input: ?GroupOfLieType Output: Magma V2.11-10 Tue Dec 6 2005 07:06:25 on modular [Seed = 687740335] ------------------------------------- 12 matches: 1 I /magma/group/fiurg/related/GroupOfLieType 2 I /magma/group/groups-lie-type/construction/GroupOfLieType 3 I /magma/group/groups-lie-type/construction/GroupOfLieType 4 I /magma/group/groups-lie-type/construction/GroupOfLieType 5 I /magma/group/groups-lie-type/construction/GroupOfLieType 6 I /magma/group/groups-lie-type/construction/GroupOfLieType 7 I /magma/group/groups-lie-type/construction/GroupOfLieType 8 I /magma/group/groups-lie-type/construction/GroupOfLieType 9 I /magma/group/groups-lie-type/construction/GroupOfLieType 10 I /magma/group/root-data/related-structures/GroupOfLieType 11 I /magma/lie-theory/coxeter-description/related/GroupOfLieType 12 I /magma/lie-theory/finite-coxeter-groups/related/GroupOfLieType To view an entry, type ? followed by the number next to it Total time: 0.200 seconds, Total memory usage: 3.24MB '129.67.' ************** MAGMA ***************** Host 129.67.62.181 (129.67.62.181) Time: Tue Dec 6 07:06:16 2005 Input: GroupOfLieType? Output: Magma V2.11-10 Tue Dec 6 2005 07:06:16 on modular [Seed = 804852433] ------------------------------------- >> GroupOfLieType?; ^ User error: bad syntax Total time: 0.200 seconds, Total memory usage: 3.24MB '129.67.' ************** MAGMA ***************** Host 129.67.62.181 (129.67.62.181) Time: Tue Dec 6 07:05:57 2005 Input: ? Output: Magma V2.11-10 Tue Dec 6 2005 07:05:56 on modular [Seed = 586424256] ------------------------------------- =============================================================================== PATH: /language/statement KIND: Overview =============================================================================== A statement is a complete command to Magma. Every statement must end with a semicolon ( ; ). If you press the key before typing ; then Magma may give you a special prompt symbol such as print> or if> to remind you that the statement is not yet finished. EXAMPLE: > print 5+8; > DivBy3 := func< n | IsZero(n mod 3) >; =============================================================================== Total time: 0.200 seconds, Total memory usage: 3.24MB '129.67.' ************** MAGMA ***************** Host 129.67.62.181 (129.67.62.181) Time: Tue Dec 6 07:05:52 2005 Input: "Replace this by some code, then click [PARI] or [MAGMA]!" Output: Magma V2.11-10 Tue Dec 6 2005 07:05:48 on modular [Seed = 653270012] ------------------------------------- Replace this by some code, then click [PARI] or [MAGMA]! Total time: 0.230 seconds, Total memory usage: 3.24MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Tue Dec 6 04:29:16 2005 Input: G :=DirichletGroup(784,CyclotomicField(168)); G; X :=Elements(G); X; Output: Magma V2.11-10 Tue Dec 6 2005 04:29:16 on modular [Seed = 1757311683] ------------------------------------- Group of Dirichlet characters of modulus 784 over Cyclotomic Field of order 168 and degree 48 [ 1, $.1, $.2, $.1*$.2, $.2^2, $.1*$.2^2, $.2^3, $.1*$.2^3, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3, $.2^2*$.3, $.1*$.2^2*$.3, $.2^3*$.3, $.1*$.2^3*$.3, $.3^2, $.1*$.3^2, $.2*$.3^2, $.1*$.2*$.3^2, $.2^2*$.3^2, $.1*$.2^2*$.3^2, $.2^3*$.3^2, $.1*$.2^3*$.3^2, $.3^3, $.1*$.3^3, $.2*$.3^3, $.1*$.2*$.3^3, $.2^2*$.3^3, $.1*$.2^2*$.3^3, $.2^3*$.3^3, $.1*$.2^3*$.3^3, $.3^4, $.1*$.3^4, $.2*$.3^4, $.1*$.2*$.3^4, $.2^2*$.3^4, $.1*$.2^2*$.3^4, $.2^3*$.3^4, $.1*$.2^3*$.3^4, $.3^5, $.1*$.3^5, $.2*$.3^5, $.1*$.2*$.3^5, $.2^2*$.3^5, $.1*$.2^2*$.3^5, $.2^3*$.3^5, $.1*$.2^3*$.3^5, $.3^6, $.1*$.3^6, $.2*$.3^6, $.1*$.2*$.3^6, $.2^2*$.3^6, $.1*$.2^2*$.3^6, $.2^3*$.3^6, $.1*$.2^3*$.3^6, $.3^7, $.1*$.3^7, $.2*$.3^7, $.1*$.2*$.3^7, $.2^2*$.3^7, $.1*$.2^2*$.3^7, $.2^3*$.3^7, $.1*$.2^3*$.3^7, $.3^8, $.1*$.3^8, $.2*$.3^8, $.1*$.2*$.3^8, $.2^2*$.3^8, $.1*$.2^2*$.3^8, $.2^3*$.3^8, $.1*$.2^3*$.3^8, $.3^9, $.1*$.3^9, $.2*$.3^9, $.1*$.2*$.3^9, $.2^2*$.3^9, $.1*$.2^2*$.3^9, $.2^3*$.3^9, $.1*$.2^3*$.3^9, $.3^10, $.1*$.3^10, $.2*$.3^10, $.1*$.2*$.3^10, $.2^2*$.3^10, $.1*$.2^2*$.3^10, $.2^3*$.3^10, $.1*$.2^3*$.3^10, $.3^11, $.1*$.3^11, $.2*$.3^11, $.1*$.2*$.3^11, $.2^2*$.3^11, $.1*$.2^2*$.3^11, $.2^3*$.3^11, $.1*$.2^3*$.3^11, $.3^12, $.1*$.3^12, $.2*$.3^12, $.1*$.2*$.3^12, $.2^2*$.3^12, $.1*$.2^2*$.3^12, $.2^3*$.3^12, $.1*$.2^3*$.3^12, $.3^13, $.1*$.3^13, $.2*$.3^13, $.1*$.2*$.3^13, $.2^2*$.3^13, $.1*$.2^2*$.3^13, $.2^3*$.3^13, $.1*$.2^3*$.3^13, $.3^14, $.1*$.3^14, $.2*$.3^14, $.1*$.2*$.3^14, $.2^2*$.3^14, $.1*$.2^2*$.3^14, $.2^3*$.3^14, $.1*$.2^3*$.3^14, $.3^15, $.1*$.3^15, $.2*$.3^15, $.1*$.2*$.3^15, $.2^2*$.3^15, $.1*$.2^2*$.3^15, $.2^3*$.3^15, $.1*$.2^3*$.3^15, $.3^16, $.1*$.3^16, $.2*$.3^16, $.1*$.2*$.3^16, $.2^2*$.3^16, $.1*$.2^2*$.3^16, $.2^3*$.3^16, $.1*$.2^3*$.3^16, $.3^17, $.1*$.3^17, $.2*$.3^17, $.1*$.2*$.3^17, $.2^2*$.3^17, $.1*$.2^2*$.3^17, $.2^3*$.3^17, $.1*$.2^3*$.3^17, $.3^18, $.1*$.3^18, $.2*$.3^18, $.1*$.2*$.3^18, $.2^2*$.3^18, $.1*$.2^2*$.3^18, $.2^3*$.3^18, $.1*$.2^3*$.3^18, $.3^19, $.1*$.3^19, $.2*$.3^19, $.1*$.2*$.3^19, $.2^2*$.3^19, $.1*$.2^2*$.3^19, $.2^3*$.3^19, $.1*$.2^3*$.3^19, $.3^20, $.1*$.3^20, $.2*$.3^20, $.1*$.2*$.3^20, $.2^2*$.3^20, $.1*$.2^2*$.3^20, $.2^3*$.3^20, $.1*$.2^3*$.3^20, $.3^21, $.1*$.3^21, $.2*$.3^21, $.1*$.2*$.3^21, $.2^2*$.3^21, $.1*$.2^2*$.3^21, $.2^3*$.3^21, $.1*$.2^3*$.3^21, $.3^22, $.1*$.3^22, $.2*$.3^22, $.1*$.2*$.3^22, $.2^2*$.3^22, $.1*$.2^2*$.3^22, $.2^3*$.3^22, $.1*$.2^3*$.3^22, $.3^23, $.1*$.3^23, $.2*$.3^23, $.1*$.2*$.3^23, $.2^2*$.3^23, $.1*$.2^2*$.3^23, $.2^3*$.3^23, $.1*$.2^3*$.3^23, $.3^24, $.1*$.3^24, $.2*$.3^24, $.1*$.2*$.3^24, $.2^2*$.3^24, $.1*$.2^2*$.3^24, $.2^3*$.3^24, $.1*$.2^3*$.3^24, $.3^25, $.1*$.3^25, $.2*$.3^25, $.1*$.2*$.3^25, $.2^2*$.3^25, $.1*$.2^2*$.3^25, $.2^3*$.3^25, $.1*$.2^3*$.3^25, $.3^26, $.1*$.3^26, $.2*$.3^26, $.1*$.2*$.3^26, $.2^2*$.3^26, $.1*$.2^2*$.3^26, $.2^3*$.3^26, $.1*$.2^3*$.3^26, $.3^27, $.1*$.3^27, $.2*$.3^27, $.1*$.2*$.3^27, $.2^2*$.3^27, $.1*$.2^2*$.3^27, $.2^3*$.3^27, $.1*$.2^3*$.3^27, $.3^28, $.1*$.3^28, $.2*$.3^28, $.1*$.2*$.3^28, $.2^2*$.3^28, $.1*$.2^2*$.3^28, $.2^3*$.3^28, $.1*$.2^3*$.3^28, $.3^29, $.1*$.3^29, $.2*$.3^29, $.1*$.2*$.3^29, $.2^2*$.3^29, $.1*$.2^2*$.3^29, $.2^3*$.3^29, $.1*$.2^3*$.3^29, $.3^30, $.1*$.3^30, $.2*$.3^30, $.1*$.2*$.3^30, $.2^2*$.3^30, $.1*$.2^2*$.3^30, $.2^3*$.3^30, $.1*$.2^3*$.3^30, $.3^31, $.1*$.3^31, $.2*$.3^31, $.1*$.2*$.3^31, $.2^2*$.3^31, $.1*$.2^2*$.3^31, $.2^3*$.3^31, $.1*$.2^3*$.3^31, $.3^32, $.1*$.3^32, $.2*$.3^32, $.1*$.2*$.3^32, $.2^2*$.3^32, $.1*$.2^2*$.3^32, $.2^3*$.3^32, $.1*$.2^3*$.3^32, $.3^33, $.1*$.3^33, $.2*$.3^33, $.1*$.2*$.3^33, $.2^2*$.3^33, $.1*$.2^2*$.3^33, $.2^3*$.3^33, $.1*$.2^3*$.3^33, $.3^34, $.1*$.3^34, $.2*$.3^34, $.1*$.2*$.3^34, $.2^2*$.3^34, $.1*$.2^2*$.3^34, $.2^3*$.3^34, $.1*$.2^3*$.3^34, $.3^35, $.1*$.3^35, $.2*$.3^35, $.1*$.2*$.3^35, $.2^2*$.3^35, $.1*$.2^2*$.3^35, $.2^3*$.3^35, $.1*$.2^3*$.3^35, $.3^36, $.1*$.3^36, $.2*$.3^36, $.1*$.2*$.3^36, $.2^2*$.3^36, $.1*$.2^2*$.3^36, $.2^3*$.3^36, $.1*$.2^3*$.3^36, $.3^37, $.1*$.3^37, $.2*$.3^37, $.1*$.2*$.3^37, $.2^2*$.3^37, $.1*$.2^2*$.3^37, $.2^3*$.3^37, $.1*$.2^3*$.3^37, $.3^38, $.1*$.3^38, $.2*$.3^38, $.1*$.2*$.3^38, $.2^2*$.3^38, $.1*$.2^2*$.3^38, $.2^3*$.3^38, $.1*$.2^3*$.3^38, $.3^39, $.1*$.3^39, $.2*$.3^39, $.1*$.2*$.3^39, $.2^2*$.3^39, $.1*$.2^2*$.3^39, $.2^3*$.3^39, $.1*$.2^3*$.3^39, $.3^40, $.1*$.3^40, $.2*$.3^40, $.1*$.2*$.3^40, $.2^2*$.3^40, $.1*$.2^2*$.3^40, $.2^3*$.3^40, $.1*$.2^3*$.3^40, $.3^41, $.1*$.3^41, $.2*$.3^41, $.1*$.2*$.3^41, $.2^2*$.3^41, $.1*$.2^2*$.3^41, $.2^3*$.3^41, $.1*$.2^3*$.3^41 ] Total time: 0.210 seconds, Total memory usage: 3.53MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Tue Dec 6 04:21:54 2005 Input: G :=DirichletGroup(784,CyclotomicField(168)); G; X :=Elements(G); X; Y :=X[2]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; qEigenform(D[1],12);Parent($1); Output: ** WARNING: Computation used more memory than allowed. ** Magma V2.11-10 Tue Dec 6 2005 04:21:50 on modular [Seed = 3161106386] ------------------------------------- Group of Dirichlet characters of modulus 784 over Cyclotomic Field of order 168 and degree 48 [ 1, $.1, $.2, $.1*$.2, $.2^2, $.1*$.2^2, $.2^3, $.1*$.2^3, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3, $.2^2*$.3, $.1*$.2^2*$.3, $.2^3*$.3, $.1*$.2^3*$.3, $.3^2, $.1*$.3^2, $.2*$.3^2, $.1*$.2*$.3^2, $.2^2*$.3^2, $.1*$.2^2*$.3^2, $.2^3*$.3^2, $.1*$.2^3*$.3^2, $.3^3, $.1*$.3^3, $.2*$.3^3, $.1*$.2*$.3^3, $.2^2*$.3^3, $.1*$.2^2*$.3^3, $.2^3*$.3^3, $.1*$.2^3*$.3^3, $.3^4, $.1*$.3^4, $.2*$.3^4, $.1*$.2*$.3^4, $.2^2*$.3^4, $.1*$.2^2*$.3^4, $.2^3*$.3^4, $.1*$.2^3*$.3^4, $.3^5, $.1*$.3^5, $.2*$.3^5, $.1*$.2*$.3^5, $.2^2*$.3^5, $.1*$.2^2*$.3^5, $.2^3*$.3^5, $.1*$.2^3*$.3^5, $.3^6, $.1*$.3^6, $.2*$.3^6, $.1*$.2*$.3^6, $.2^2*$.3^6, $.1*$.2^2*$.3^6, $.2^3*$.3^6, $.1*$.2^3*$.3^6, $.3^7, $.1*$.3^7, $.2*$.3^7, $.1*$.2*$.3^7, $.2^2*$.3^7, $.1*$.2^2*$.3^7, $.2^3*$.3^7, $.1*$.2^3*$.3^7, $.3^8, $.1*$.3^8, $.2*$.3^8, $.1*$.2*$.3^8, $.2^2*$.3^8, $.1*$.2^2*$.3^8, $.2^3*$.3^8, $.1*$.2^3*$.3^8, $.3^9, $.1*$.3^9, $.2*$.3^9, $.1*$.2*$.3^9, $.2^2*$.3^9, $.1*$.2^2*$.3^9, $.2^3*$.3^9, $.1*$.2^3*$.3^9, $.3^10, $.1*$.3^10, $.2*$.3^10, $.1*$.2*$.3^10, $.2^2*$.3^10, $.1*$.2^2*$.3^10, $.2^3*$.3^10, $.1*$.2^3*$.3^10, $.3^11, $.1*$.3^11, $.2*$.3^11, $.1*$.2*$.3^11, $.2^2*$.3^11, $.1*$.2^2*$.3^11, $.2^3*$.3^11, $.1*$.2^3*$.3^11, $.3^12, $.1*$.3^12, $.2*$.3^12, $.1*$.2*$.3^12, $.2^2*$.3^12, $.1*$.2^2*$.3^12, $.2^3*$.3^12, $.1*$.2^3*$.3^12, $.3^13, $.1*$.3^13, $.2*$.3^13, $.1*$.2*$.3^13, $.2^2*$.3^13, $.1*$.2^2*$.3^13, $.2^3*$.3^13, $.1*$.2^3*$.3^13, $.3^14, $.1*$.3^14, $.2*$.3^14, $.1*$.2*$.3^14, $.2^2*$.3^14, $.1*$.2^2*$.3^14, $.2^3*$.3^14, $.1*$.2^3*$.3^14, $.3^15, $.1*$.3^15, $.2*$.3^15, $.1*$.2*$.3^15, $.2^2*$.3^15, $.1*$.2^2*$.3^15, $.2^3*$.3^15, $.1*$.2^3*$.3^15, $.3^16, $.1*$.3^16, $.2*$.3^16, $.1*$.2*$.3^16, $.2^2*$.3^16, $.1*$.2^2*$.3^16, $.2^3*$.3^16, $.1*$.2^3*$.3^16, $.3^17, $.1*$.3^17, $.2*$.3^17, $.1*$.2*$.3^17, $.2^2*$.3^17, $.1*$.2^2*$.3^17, $.2^3*$.3^17, $.1*$.2^3*$.3^17, $.3^18, $.1*$.3^18, $.2*$.3^18, $.1*$.2*$.3^18, $.2^2*$.3^18, $.1*$.2^2*$.3^18, $.2^3*$.3^18, $.1*$.2^3*$.3^18, $.3^19, $.1*$.3^19, $.2*$.3^19, $.1*$.2*$.3^19, $.2^2*$.3^19, $.1*$.2^2*$.3^19, $.2^3*$.3^19, $.1*$.2^3*$.3^19, $.3^20, $.1*$.3^20, $.2*$.3^20, $.1*$.2*$.3^20, $.2^2*$.3^20, $.1*$.2^2*$.3^20, $.2^3*$.3^20, $.1*$.2^3*$.3^20, $.3^21, $.1*$.3^21, $.2*$.3^21, $.1*$.2*$.3^21, $.2^2*$.3^21, $.1*$.2^2*$.3^21, $.2^3*$.3^21, $.1*$.2^3*$.3^21, $.3^22, $.1*$.3^22, $.2*$.3^22, $.1*$.2*$.3^22, $.2^2*$.3^22, $.1*$.2^2*$.3^22, $.2^3*$.3^22, $.1*$.2^3*$.3^22, $.3^23, $.1*$.3^23, $.2*$.3^23, $.1*$.2*$.3^23, $.2^2*$.3^23, $.1*$.2^2*$.3^23, $.2^3*$.3^23, $.1*$.2^3*$.3^23, $.3^24, $.1*$.3^24, $.2*$.3^24, $.1*$.2*$.3^24, $.2^2*$.3^24, $.1*$.2^2*$.3^24, $.2^3*$.3^24, $.1*$.2^3*$.3^24, $.3^25, $.1*$.3^25, $.2*$.3^25, $.1*$.2*$.3^25, $.2^2*$.3^25, $.1*$.2^2*$.3^25, $.2^3*$.3^25, $.1*$.2^3*$.3^25, $.3^26, $.1*$.3^26, $.2*$.3^26, $.1*$.2*$.3^26, $.2^2*$.3^26, $.1*$.2^2*$.3^26, $.2^3*$.3^26, $.1*$.2^3*$.3^26, $.3^27, $.1*$.3^27, $.2*$.3^27, $.1*$.2*$.3^27, $.2^2*$.3^27, $.1*$.2^2*$.3^27, $.2^3*$.3^27, $.1*$.2^3*$.3^27, $.3^28, $.1*$.3^28, $.2*$.3^28, $.1*$.2*$.3^28, $.2^2*$.3^28, $.1*$.2^2*$.3^28, $.2^3*$.3^28, $.1*$.2^3*$.3^28, $.3^29, $.1*$.3^29, $.2*$.3^29, $.1*$.2*$.3^29, $.2^2*$.3^29, $.1*$.2^2*$.3^29, $.2^3*$.3^29, $.1*$.2^3*$.3^29, $.3^30, $.1*$.3^30, $.2*$.3^30, $.1*$.2*$.3^30, $.2^2*$.3^30, $.1*$.2^2*$.3^30, $.2^3*$.3^30, $.1*$.2^3*$.3^30, $.3^31, $.1*$.3^31, $.2*$.3^31, $.1*$.2*$.3^31, $.2^2*$.3^31, $.1*$.2^2*$.3^31, $.2^3*$.3^31, $.1*$.2^3*$.3^31, $.3^32, $.1*$.3^32, $.2*$.3^32, $.1*$.2*$.3^32, $.2^2*$.3^32, $.1*$.2^2*$.3^32, $.2^3*$.3^32, $.1*$.2^3*$.3^32, $.3^33, $.1*$.3^33, $.2*$.3^33, $.1*$.2*$.3^33, $.2^2*$.3^33, $.1*$.2^2*$.3^33, $.2^3*$.3^33, $.1*$.2^3*$.3^33, $.3^34, $.1*$.3^34, $.2*$.3^34, $.1*$.2*$.3^34, $.2^2*$.3^34, $.1*$.2^2*$.3^34, $.2^3*$.3^34, $.1*$.2^3*$.3^34, $.3^35, $.1*$.3^35, $.2*$.3^35, $.1*$.2*$.3^35, $.2^2*$.3^35, $.1*$.2^2*$.3^35, $.2^3*$.3^35, $.1*$.2^3*$.3^35, $.3^36, $.1*$.3^36, $.2*$.3^36, $.1*$.2*$.3^36, $.2^2*$.3^36, $.1*$.2^2*$.3^36, $.2^3*$.3^36, $.1*$.2^3*$.3^36, $.3^37, $.1*$.3^37, $.2*$.3^37, $.1*$.2*$.3^37, $.2^2*$.3^37, $.1*$.2^2*$.3^37, $.2^3*$.3^37, $.1*$.2^3*$.3^37, $.3^38, $.1*$.3^38, $.2*$.3^38, $.1*$.2*$.3^38, $.2^2*$.3^38, $.1*$.2^2*$.3^38, $.2^3*$.3^38, $.1*$.2^3*$.3^38, $.3^39, $.1*$.3^39, $.2*$.3^39, $.1*$.2*$.3^39, $.2^2*$.3^39, $.1*$.2^2*$.3^39, $.2^3*$.3^39, $.1*$.2^3*$.3^39, $.3^40, $.1*$.3^40, $.2*$.3^40, $.1*$.2*$.3^40, $.2^2*$.3^40, $.1*$.2^2*$.3^40, $.2^3*$.3^40, $.1*$.2^3*$.3^40, $.3^41, $.1*$.3^41, $.2*$.3^41, $.1*$.2*$.3^41, $.2^2*$.3^41, $.1*$.2^2*$.3^41, $.2^3*$.3^41, $.1*$.2^3*$.3^41 ] 4 2 Current total memory usage: 95.4MB, failed memory request: 0.0MB System Error: User memory limit has been reached >> D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); ^ Runtime error: Variable 'M' has not been initialized >> D; ^ User error: Identifier 'D' has not been declared or assigned >> qEigenform(D[1],12);Parent($1); ^ User error: Identifier 'D' has not been declared or assigned Integer Ring Total time: 3.520 seconds, Total memory usage: 95.37MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Tue Dec 6 04:16:34 2005 Input: G :=DirichletGroup(56,CyclotomicField(12)); G; X :=Elements(G); X; Y :=X[12]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; qEigenform(D[1],12);Parent($1); Output: Magma V2.11-10 Tue Dec 6 2005 04:16:33 on modular [Seed = 2758979861] ------------------------------------- Group of Dirichlet characters of modulus 56 over Cyclotomic Field of order 12 and degree 4 [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3, $.3^2, $.1*$.3^2, $.2*$.3^2, $.1*$.2*$.3^2, $.3^3, $.1*$.3^3, $.2*$.3^3, $.1*$.2*$.3^3, $.3^4, $.1*$.3^4, $.2*$.3^4, $.1*$.2*$.3^4, $.3^5, $.1*$.3^5, $.2*$.3^5, $.1*$.2*$.3^5 ] 56 6 [ Modular symbols space of level 56, weight 3, character $.1*$.2*$.3^2, and dimension 1 over Cyclotomic Field of order 12 and degree 4, Modular symbols space of level 56, weight 3, character $.1*$.2*$.3^2, and dimension 1 over Cyclotomic Field of order 12 and degree 4, Modular symbols space of level 56, weight 3, character $.1*$.2*$.3^2, and dimension 6 over Cyclotomic Field of order 12 and degree 4, Modular symbols space of level 56, weight 3, character $.1*$.2*$.3^2, and dimension 6 over Cyclotomic Field of order 12 and degree 4 ] q + 2*zeta_12^2*q^2 - zeta_12^2*q^3 + (4*zeta_12^2 - 4)*q^4 + (3*zeta_12^2 + 3)*q^5 + (-2*zeta_12^2 + 2)*q^6 + (8*zeta_12^2 - 3)*q^7 - 8*q^8 + (-8*zeta_12^2 + 8)*q^9 + (12*zeta_12^2 - 6)*q^10 - 17*zeta_12^2*q^11 + O(q^12) Power series ring in q over Cyclotomic Field of order 12 and degree 4 Total time: 0.560 seconds, Total memory usage: 4.94MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Tue Dec 6 04:16:25 2005 Input: G :=DirichletGroup(56,CyclotomicField(12)); G; X :=Elements(G); X; Y :=X[11]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; qEigenform(D[1],12);Parent($1); Output: Magma V2.11-10 Tue Dec 6 2005 04:16:24 on modular [Seed = 2672396855] ------------------------------------- Group of Dirichlet characters of modulus 56 over Cyclotomic Field of order 12 and degree 4 [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3, $.3^2, $.1*$.3^2, $.2*$.3^2, $.1*$.2*$.3^2, $.3^3, $.1*$.3^3, $.2*$.3^3, $.1*$.2*$.3^3, $.3^4, $.1*$.3^4, $.2*$.3^4, $.1*$.2*$.3^4, $.3^5, $.1*$.3^5, $.2*$.3^5, $.1*$.2*$.3^5 ] 56 6 [] >> qEigenform(D[1],12);Parent($1); ^ Runtime error in '[]': Sequence element 1 not defined Set of null sequences Total time: 0.210 seconds, Total memory usage: 4.32MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Tue Dec 6 04:16:16 2005 Input: G :=DirichletGroup(56,CyclotomicField(12)); G; X :=Elements(G); X; Y :=X[10]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; qEigenform(D[1],12);Parent($1); Output: Magma V2.11-10 Tue Dec 6 2005 04:16:15 on modular [Seed = 2588708613] ------------------------------------- Group of Dirichlet characters of modulus 56 over Cyclotomic Field of order 12 and degree 4 [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3, $.3^2, $.1*$.3^2, $.2*$.3^2, $.1*$.2*$.3^2, $.3^3, $.1*$.3^3, $.2*$.3^3, $.1*$.2*$.3^3, $.3^4, $.1*$.3^4, $.2*$.3^4, $.1*$.2*$.3^4, $.3^5, $.1*$.3^5, $.2*$.3^5, $.1*$.2*$.3^5 ] 28 6 [] >> qEigenform(D[1],12);Parent($1); ^ Runtime error in '[]': Sequence element 1 not defined Set of null sequences Total time: 0.700 seconds, Total memory usage: 4.94MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Tue Dec 6 04:15:26 2005 Input: G :=DirichletGroup(56,CyclotomicField(12)); G; X :=Elements(G); X; Y :=X[9]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; qEigenform(D[1],12);Parent($1); Output: Magma V2.11-10 Tue Dec 6 2005 04:15:25 on modular [Seed = 2504228984] ------------------------------------- Group of Dirichlet characters of modulus 56 over Cyclotomic Field of order 12 and degree 4 [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3, $.3^2, $.1*$.3^2, $.2*$.3^2, $.1*$.2*$.3^2, $.3^3, $.1*$.3^3, $.2*$.3^3, $.1*$.2*$.3^3, $.3^4, $.1*$.3^4, $.2*$.3^4, $.1*$.2*$.3^4, $.3^5, $.1*$.3^5, $.2*$.3^5, $.1*$.2*$.3^5 ] 7 3 [] >> qEigenform(D[1],12);Parent($1); ^ Runtime error in '[]': Sequence element 1 not defined Set of null sequences Total time: 0.210 seconds, Total memory usage: 4.33MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Tue Dec 6 04:15:15 2005 Input: G :=DirichletGroup(56,CyclotomicField(12)); G; X :=Elements(G); X; Y :=X[8]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; qEigenform(D[1],12);Parent($1); Output: Magma V2.11-10 Tue Dec 6 2005 04:15:14 on modular [Seed = 2420540786] ------------------------------------- Group of Dirichlet characters of modulus 56 over Cyclotomic Field of order 12 and degree 4 [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3, $.3^2, $.1*$.3^2, $.2*$.3^2, $.1*$.2*$.3^2, $.3^3, $.1*$.3^3, $.2*$.3^3, $.1*$.2*$.3^3, $.3^4, $.1*$.3^4, $.2*$.3^4, $.1*$.2*$.3^4, $.3^5, $.1*$.3^5, $.2*$.3^5, $.1*$.2*$.3^5 ] 56 6 [] >> qEigenform(D[1],12);Parent($1); ^ Runtime error in '[]': Sequence element 1 not defined Set of null sequences Total time: 0.210 seconds, Total memory usage: 4.32MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Tue Dec 6 04:14:55 2005 Input: G :=DirichletGroup(56,CyclotomicField(12)); G; X :=Elements(G); X; Y :=X[7]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; qEigenform(D[1],12);Parent($1); Output: Magma V2.11-10 Tue Dec 6 2005 04:14:54 on modular [Seed = 2338168389] ------------------------------------- Group of Dirichlet characters of modulus 56 over Cyclotomic Field of order 12 and degree 4 [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3, $.3^2, $.1*$.3^2, $.2*$.3^2, $.1*$.2*$.3^2, $.3^3, $.1*$.3^3, $.2*$.3^3, $.1*$.2*$.3^3, $.3^4, $.1*$.3^4, $.2*$.3^4, $.1*$.2*$.3^4, $.3^5, $.1*$.3^5, $.2*$.3^5, $.1*$.2*$.3^5 ] 56 6 [ Modular symbols space of level 56, weight 3, character $.2*$.3, and dimension 14 over Cyclotomic Field of order 12 and degree 4 ] q + (1/2348959680*(4579*zeta_12^2 + 4579)*a^13 + 15397/104398208*zeta_12^2*a^12 + 1/2348959680*(1467542*zeta_12^2 - 733771)*a^11 + 1/26099552*(249503*zeta_12^2 - 249503)*a^10 + 1/2348959680*(34198157*zeta_12^2 - 68396314)*a^9 - 21938035/104398208*a^8 + 1/391493280*(-107476613*zeta_12^2 - 107476613)*a^7 - 50610471/26099552*zeta_12^2*a^6 + 1/2348959680*(-10540068074*zeta_12^2 + 5270034037)*a^5 + 1/104398208*(-738593311*zeta_12^2 + 738593311)*a^4 + 1/2348959680*(-16484360123*zeta_12^2 + 32968720246)*a^3 + 11367885/1631222*a^2 + 1/2348959680*(11603758463*zeta_12^2 + 11603758463)*a + 9801249/104398208*zeta_12^2)*q^2 + a*q^3 + (1/1565973120*(52502*zeta_12^2 - 26251)*a^13 + 1/104398208*(19851*zeta_12^2 - 19851)*a^12 + 1/130497760*(141337*zeta_12^2 - 282674)*a^11 - 622539/52199104*a^10 + 1/521991040*(-12229781*zeta_12^2 - 12229781)*a^9 - 25556357/104398208*zeta_12^2*a^8 + 1/130497760*(-52658222*zeta_12^2 + 26329111)*a^7 + 1/6524888*(-12557077*zeta_12^2 + 12557077)*a^6 + 1/521991040*(-262718751*zeta_12^2 + 525437502)*a^5 + 421293737/104398208*a^4 + 1/32624440*(-34379021*zeta_12^2 - 34379021)*a^3 - 321222443/52199104*zeta_12^2*a^2 + 1/1565973120*(-9362800046*zeta_12^2 + 4681400023)*a + 1/104398208*(-438492713*zeta_12^2 + 438492713))*q^4 + (1/195746640*(-9343*zeta_12^2 + 9343)*a^13 + 1276409/391493280*a^11 + 31072933/391493280*zeta_12^2*a^9 + 1/65248880*(58403837*zeta_12^2 - 58403837)*a^7 - 60668884/12234165*a^5 - 5142634777/391493280*zeta_12^2*a^3 + 1/391493280*(-5843345917*zeta_12^2 + 5843345917)*a)*q^5 + (15397/104398208*zeta_12^2*a^13 + 1/78298656*(24802*zeta_12^2 - 12401)*a^12 + 1/26099552*(249503*zeta_12^2 - 249503)*a^11 + 1/78298656*(783845*zeta_12^2 - 1567690)*a^10 - 21938035/104398208*a^9 + 1/156597312*(-33018499*zeta_12^2 - 33018499)*a^8 - 50610471/26099552*zeta_12^2*a^7 + 1/39149328*(-142014746*zeta_12^2 + 71007373)*a^6 + 1/104398208*(-738593311*zeta_12^2 + 738593311)*a^5 + 1/39149328*(-227569759*zeta_12^2 + 455139518)*a^4 + 11367885/1631222*a^3 + 1/78298656*(324656903*zeta_12^2 + 324656903)*a^2 + 9801249/104398208*zeta_12^2*a + 1/52199104*(-4807950*zeta_12^2 + 2403975))*q^6 + (1/78298656*(8393*zeta_12^2 + 40739)*a^12 + 1/78298656*(3157175*zeta_12^2 - 609496)*a^10 + 1/26099552*(17440828*zeta_12^2 - 22769715)*a^8 + 1/39149328*(-95698738*zeta_12^2 - 210647131)*a^6 + 1/78298656*(-2129129966*zeta_12^2 + 1052988271)*a^4 + 1/26099552*(45576087*zeta_12^2 + 706336949)*a^2 + 1/26099552*(238877441*zeta_12^2 - 24851810))*q^7 + (1/2348959680*(-291967*zeta_12^2 + 583934)*a^13 - 183/3262444*a^12 + 1/2348959680*(20993293*zeta_12^2 + 20993293)*a^11 - 210395/52199104*zeta_12^2*a^10 + 1/1174479840*(548226986*zeta_12^2 - 274113493)*a^9 + 1/52199104*(-5382883*zeta_12^2 + 5382883)*a^8 + 1/130497760*(370662333*zeta_12^2 - 741324666)*a^7 + 30698417/26099552*a^6 + 1/2348959680*(-38895998071*zeta_12^2 - 38895998071)*a^5 + 158397319/26099552*zeta_12^2*a^4 + 1/2348959680*(-188548838578*zeta_12^2 + 94274419289)*a^3 + 1/52199104*(670342267*zeta_12^2 - 670342267)*a^2 + 1/146809980*(-2731818449*zeta_12^2 + 5463636898)*a - 237835863/52199104)*q^8 + (a^2 + 9*zeta_12^2)*q^9 + (-47433/260995520*a^13 + 1/104398208*(-41981*zeta_12^2 - 41981)*a^12 - 1435511/130497760*zeta_12^2*a^11 + 1/156597312*(-8089606*zeta_12^2 + 4044803)*a^10 + 1/65248880*(-13715531*zeta_12^2 + 13715531)*a^9 + 1/313194624*(-174679391*zeta_12^2 + 349358782)*a^8 + 45371317/32624440*a^7 + 1/3262444*(16183267*zeta_12^2 + 16183267)*a^6 + 196747729/260995520*zeta_12^2*a^5 + 1/313194624*(10380831650*zeta_12^2 - 5190415825)*a^4 + 1/130497760*(-1423689617*zeta_12^2 + 1423689617)*a^3 + 1/156597312*(2019684241*zeta_12^2 - 4039368482)*a^2 - 505558583/130497760*a + 1/104398208*(-504246155*zeta_12^2 - 504246155))*q^10 + (1/293619960*(-28826*zeta_12^2 + 14413)*a^13 + 1/73404990*(-139903*zeta_12^2 + 279806)*a^11 + 1/293619960*(-3530371*zeta_12^2 - 3530371)*a^9 + 1/8156110*(-18231238*zeta_12^2 + 9115619)*a^7 + 1/293619960*(-3824890421*zeta_12^2 + 7649780842)*a^5 + 1/36702495*(1660220933*zeta_12^2 + 1660220933)*a^3 + 1/293619960*(12225510218*zeta_12^2 - 6112755109)*a)*q^11 + O(q^12) Power series ring in q over Univariate Quotient Polynomial Algebra in a over Cyclotomic Field of order 12 and degree 4 with modulus a^14 + 79*zeta_12^2*a^12 + (2333*zeta_12^2 - 2333)*a^10 - 32667*a^8 - 220483*zeta_12^2*a^6 + (-618077*zeta_12^2 + 618077)*a^4 + 407087*a^2 + 23625*zeta_12^2 Total time: 0.680 seconds, Total memory usage: 4.94MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Tue Dec 6 04:14:43 2005 Input: G :=DirichletGroup(56,CyclotomicField(12)); G; X :=Elements(G); X; Y :=X[6]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; qEigenform(D[1],12);Parent($1); Output: Magma V2.11-10 Tue Dec 6 2005 04:14:43 on modular [Seed = 2253427525] ------------------------------------- Group of Dirichlet characters of modulus 56 over Cyclotomic Field of order 12 and degree 4 [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3, $.3^2, $.1*$.3^2, $.2*$.3^2, $.1*$.2*$.3^2, $.3^3, $.1*$.3^3, $.2*$.3^3, $.1*$.2*$.3^3, $.3^4, $.1*$.3^4, $.2*$.3^4, $.1*$.2*$.3^4, $.3^5, $.1*$.3^5, $.2*$.3^5, $.1*$.2*$.3^5 ] 28 6 [] >> qEigenform(D[1],12);Parent($1); ^ Runtime error in '[]': Sequence element 1 not defined Set of null sequences Total time: 0.220 seconds, Total memory usage: 4.33MB '60.225.' ************** MAGMA ***************** Host 60.225.131.213 (60.225.131.213) Time: Tue Dec 6 03:59:20 2005 Input: LogIntegral(100); Output: Magma V2.11-10 Tue Dec 6 2005 03:59:20 on modular [Seed = 820388248] ------------------------------------- 30.12614158407962992590172 Total time: 0.180 seconds, Total memory usage: 3.24MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Tue Dec 6 03:58:40 2005 Input: G :=DirichletGroup(56,CyclotomicField(12)); G; X :=Elements(G); X; Y :=X[5]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; qEigenform(D[1],12);Parent($1); Output: Magma V2.11-10 Tue Dec 6 2005 03:58:39 on modular [Seed = 738015956] ------------------------------------- Group of Dirichlet characters of modulus 56 over Cyclotomic Field of order 12 and degree 4 [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3, $.3^2, $.1*$.3^2, $.2*$.3^2, $.1*$.2*$.3^2, $.3^3, $.1*$.3^3, $.2*$.3^3, $.1*$.2*$.3^3, $.3^4, $.1*$.3^4, $.2*$.3^4, $.1*$.2*$.3^4, $.3^5, $.1*$.3^5, $.2*$.3^5, $.1*$.2*$.3^5 ] 7 6 [ Modular symbols space of level 56, weight 3, character $.3, and dimension 4 over Cyclotomic Field of order 12 and degree 4 ] q + a*q^3 + (-1/8*zeta_12^2*a^3 + 1/8*(6*zeta_12^2 - 3)*a^2 + 1/8*(17*zeta_12^2 - 17)*a + 1/8*(-39*zeta_12^2 + 78))*q^5 + (-1/8*a^3 + 1/8*(3*zeta_12^2 - 5)*a^2 + 25/8*zeta_12^2*a + 1/8*(10*zeta_12^2 + 47))*q^7 + (a^2 - 9*zeta_12^2)*q^9 + (-zeta_12^2*a^2 + (-2*zeta_12^2 + 1)*a + (13*zeta_12^2 - 13))*q^11 + O(q^12) Power series ring in q over Univariate Quotient Polynomial Algebra in a over Cyclotomic Field of order 12 and degree 4 with modulus a^4 - 28*zeta_12^2*a^2 + (-8*zeta_12^2 + 4)*a + 119*zeta_12^2 - 119 Total time: 0.700 seconds, Total memory usage: 4.94MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Tue Dec 6 03:57:20 2005 Input: G :=DirichletGroup(56,CyclotomicField(12)); G; X :=Elements(G); X; Y :=X[4]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; qEigenform(D[1],12);Parent($1); Output: Magma V2.11-10 Tue Dec 6 2005 03:57:19 on modular [Seed = 653276937] ------------------------------------- Group of Dirichlet characters of modulus 56 over Cyclotomic Field of order 12 and degree 4 [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3, $.3^2, $.1*$.3^2, $.2*$.3^2, $.1*$.2*$.3^2, $.3^3, $.1*$.3^3, $.2*$.3^3, $.1*$.2*$.3^3, $.3^4, $.1*$.3^4, $.2*$.3^4, $.1*$.2*$.3^4, $.3^5, $.1*$.3^5, $.2*$.3^5, $.1*$.2*$.3^5 ] 8 2 [ Modular symbols space of level 56, weight 3, character $.1*$.2, and dimension 4 over Cyclotomic Field of order 12 and degree 4, Modular symbols space of level 56, weight 3, character $.1*$.2, and dimension 8 over Cyclotomic Field of order 12 and degree 4 ] q + (55/26448*a^3 - 13/3306*a^2 + 7639/13224*a - 1251/1102)*q^2 + (-1/6612*a^3 + 31/3306*a^2 - 199/3306*a + 1979/551)*q^3 + (1/456*a^3 - 5/456*a^2 + 199/228*a - 443/76)*q^4 + (-1/13224*a^3 + 31/6612*a^2 - 3505/6612*a + 1979/1102)*q^5 + (7/1102*a^3 - 83/4408*a^2 + 2235/1102*a - 9035/2204)*q^6 + (-1/456*a^3 + 5/456*a^2 - 199/228*a + 215/76)*q^7 + (-47/13224*a^3 - 49/1653*a^2 - 6047/6612*a - 2257/551)*q^8 + (-1/1653*a^3 + 62/1653*a^2 - 398/1653*a + 1855/551)*q^9 + (-7/2204*a^3 + 317/4408*a^2 - 421/551*a + 11405/2204)*q^10 + (1/1102*a^3 - 31/551*a^2 + 199/551*a - 3058/551)*q^11 + O(q^12) Power series ring in q over Univariate Quotient Polynomial Algebra in a over Cyclotomic Field of order 12 and degree 4 with modulus a^4 - 8*a^3 + 356*a^2 - 2256*a + 6948 Total time: 0.640 seconds, Total memory usage: 4.85MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Tue Dec 6 03:57:07 2005 Input: G :=DirichletGroup(56,CyclotomicField(12)); G; X :=Elements(G); X; Y :=X[3]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; qEigenform(D[1],12);Parent($1); Output: Magma V2.11-10 Tue Dec 6 2005 03:57:06 on modular [Seed = 569847811] ------------------------------------- Group of Dirichlet characters of modulus 56 over Cyclotomic Field of order 12 and degree 4 [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3, $.3^2, $.1*$.3^2, $.2*$.3^2, $.1*$.2*$.3^2, $.3^3, $.1*$.3^3, $.2*$.3^3, $.1*$.2*$.3^3, $.3^4, $.1*$.3^4, $.2*$.3^4, $.1*$.2*$.3^4, $.3^5, $.1*$.3^5, $.2*$.3^5, $.1*$.2*$.3^5 ] 8 2 [] >> qEigenform(D[1],12);Parent($1); ^ Runtime error in '[]': Sequence element 1 not defined Set of null sequences Total time: 0.210 seconds, Total memory usage: 4.34MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Tue Dec 6 03:56:36 2005 Input: G :=DirichletGroup(56,CyclotomicField(12)); G; X :=Elements(G); X; Y :=X[2]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; qEigenform(D[1],12);Parent($1); Output: Magma V2.11-10 Tue Dec 6 2005 03:56:35 on modular [Seed = 365102102] ------------------------------------- Group of Dirichlet characters of modulus 56 over Cyclotomic Field of order 12 and degree 4 [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3, $.3^2, $.1*$.3^2, $.2*$.3^2, $.1*$.2*$.3^2, $.3^3, $.1*$.3^3, $.2*$.3^3, $.1*$.2*$.3^3, $.3^4, $.1*$.3^4, $.2*$.3^4, $.1*$.2*$.3^4, $.3^5, $.1*$.3^5, $.2*$.3^5, $.1*$.2*$.3^5 ] 4 2 [] >> qEigenform(D[1],12);Parent($1); ^ Runtime error in '[]': Sequence element 1 not defined Set of null sequences Total time: 0.770 seconds, Total memory usage: 4.75MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Tue Dec 6 03:55:51 2005 Input: G :=DirichletGroup(14); G; X :=Elements(G); X; Y :=X[2]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; qEigenform(D[1],12);Parent($1); Output: Magma V2.11-10 Tue Dec 6 2005 03:55:51 on modular [Seed = 249044825] ------------------------------------- Group of Dirichlet characters of modulus 14 over Rational Field [ 1, $.1 ] 7 2 [] >> qEigenform(D[1],12);Parent($1); ^ Runtime error in '[]': Sequence element 1 not defined Set of null sequences Total time: 0.270 seconds, Total memory usage: 4.50MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Tue Dec 6 03:55:42 2005 Input: G :=DirichletGroup(14); G; X :=Elements(G); X; Y :=X[4]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; qEigenform(D[1],12);Parent($1); Output: Magma V2.11-10 Tue Dec 6 2005 03:55:42 on modular [Seed = 165615687] ------------------------------------- Group of Dirichlet characters of modulus 14 over Rational Field [ 1, $.1 ] >> Y :=X[4]; Conductor(Y); Order(Y); ^ Runtime error in '[]': Sequence element 4 not defined >> Y :=X[4]; Conductor(Y); Order(Y); ^ User error: Identifier 'Y' has not been declared or assigned >> Y :=X[4]; Conductor(Y); Order(Y); ^ User error: Identifier 'Y' has not been declared or assigned >> M := ModularSymbols(Y, 3, 1); ^ User error: Identifier 'Y' has not been declared or assigned >> D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); ^ User error: Identifier 'M' has not been declared or assigned >> D; ^ User error: Identifier 'D' has not been declared or assigned >> qEigenform(D[1],12);Parent($1); ^ User error: Identifier 'D' has not been declared or assigned Set of sequences over Group of Dirichlet characters of modulus 14 over Rational Field Total time: 0.190 seconds, Total memory usage: 3.34MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Tue Dec 6 03:55:31 2005 Input: G :=DirichletGroup(28); G; X :=Elements(G); X; Y :=X[4]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; qEigenform(D[1],12);Parent($1); Output: Magma V2.11-10 Tue Dec 6 2005 03:55:31 on modular [Seed = 80876871] ------------------------------------- Group of Dirichlet characters of modulus 28 over Rational Field [ 1, $.1, $.2, $.1*$.2 ] 28 2 [] >> qEigenform(D[1],12);Parent($1); ^ Runtime error in '[]': Sequence element 1 not defined Set of null sequences Total time: 0.210 seconds, Total memory usage: 3.91MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Tue Dec 6 03:54:13 2005 Input: G :=DirichletGroup(28); G; X :=Elements(G); X; Y :=X[3]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; qEigenform(D[1],12);Parent($1); Output: Magma V2.11-10 Tue Dec 6 2005 03:54:13 on modular [Seed = 2141810058] ------------------------------------- Group of Dirichlet characters of modulus 28 over Rational Field [ 1, $.1, $.2, $.1*$.2 ] 7 2 [ Modular symbols space of level 28, weight 3, character $.2, and dimension 2 over Rational Field ] q + a*q^3 - a*q^5 + (-a + 5)*q^7 - 15*q^9 - 6*q^11 + O(q^12) Power series ring in q over Univariate Quotient Polynomial Algebra in a over Rational Field with modulus a^2 + 24 Total time: 0.330 seconds, Total memory usage: 4.69MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Tue Dec 6 03:53:28 2005 Input: G :=DirichletGroup(28); G; X :=Elements(G); X; Y :=X[2]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; qEigenform(D[1],12);Parent($1); Output: Magma V2.11-10 Tue Dec 6 2005 03:53:28 on modular [Seed = 2007066047] ------------------------------------- Group of Dirichlet characters of modulus 28 over Rational Field [ 1, $.1, $.2, $.1*$.2 ] 4 2 [ Modular symbols space of level 28, weight 3, character $.1, and dimension 6 over Rational Field ] q + (1/64*a^5 - 1/32*a^4 + 7/16*a^3 - 5/8*a^2 + 9/4*a - 3/2)*q^2 + a*q^3 + (1/64*a^5 - 1/32*a^4 + 5/16*a^3 - 9/8*a^2 + 1/4*a - 13/2)*q^4 + (1/8*a^4 + 3*a^2 + 10)*q^5 + (-1/32*a^5 - 1/16*a^4 - 5/8*a^3 - 7/4*a^2 - 3/2*a - 7)*q^6 + (-1/8*a^3 - 2*a)*q^7 + (-1/64*a^5 + 1/32*a^4 - 9/16*a^3 + 9/8*a^2 - 25/4*a + 9/2)*q^8 + (a^2 + 9)*q^9 + (1/8*a^4 + 1/4*a^3 + 3*a^2 + 5*a + 6)*q^10 + (-1/8*a^5 - 3*a^3 - 12*a)*q^11 + O(q^12) Power series ring in q over Univariate Quotient Polynomial Algebra in a over Rational Field with modulus a^6 + 32*a^4 + 256*a^2 + 448 Total time: 0.320 seconds, Total memory usage: 4.72MB '60.225.' ************** MAGMA ***************** Host 60.225.131.213 (60.225.131.213) Time: Tue Dec 6 03:50:42 2005 Input: LogIntegral(200); Output: Magma V2.11-10 Tue Dec 6 2005 03:50:42 on modular [Seed = 1874427085] ------------------------------------- 50.192171165963783809779121 Total time: 0.190 seconds, Total memory usage: 3.24MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Tue Dec 6 03:48:12 2005 Input: G :=DirichletGroup(56); G; X :=Elements(G); X; Y :=X[4]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; qEigenform(D[1],12);Parent($1); Output: Magma V2.11-10 Tue Dec 6 2005 03:48:11 on modular [Seed = 1739682924] ------------------------------------- Group of Dirichlet characters of modulus 56 over Rational Field [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3 ] 8 2 [ Modular symbols space of level 56, weight 3, character $.1*$.2, and dimension 4 over Rational Field, Modular symbols space of level 56, weight 3, character $.1*$.2, and dimension 8 over Rational Field ] q + (31/48768*a^3 - 7/12192*a^2 + 2425/6096*a - 275/508)*q^2 + (-1/32512*a^3 + 33/8128*a^2 - 111/4064*a + 3599/1016)*q^3 + (1/1524*a^3 - 5/1524*a^2 + 74/127*a - 1868/381)*q^4 + (-1/97536*a^3 + 11/8128*a^2 - 4175/12192*a + 3599/3048)*q^5 + (47/24384*a^3 - 9/2032*a^2 + 4201/3048*a - 1513/762)*q^6 + (-1/1524*a^3 + 5/1524*a^2 - 74/127*a + 725/381)*q^7 + (-7/6096*a^3 - 23/1524*a^2 - 523/762*a - 646/127)*q^8 + (-1/8128*a^3 + 33/2032*a^2 - 111/1016*a + 805/254)*q^9 + (-7/8128*a^3 + 185/6096*a^2 - 1315/3048*a + 3443/762)*q^10 + (3/16256*a^3 - 99/4064*a^2 + 333/2032*a - 2669/508)*q^11 + O(q^12) Power series ring in q over Univariate Quotient Polynomial Algebra in a over Rational Field with modulus a^4 - 8*a^3 + 776*a^2 - 5056*a + 24448 Total time: 0.410 seconds, Total memory usage: 4.64MB '60.225.' ************** MAGMA ***************** Host 60.225.131.213 (60.225.131.213) Time: Tue Dec 6 03:47:44 2005 Input: #[n: n in [100 .. 200] | IsSquare(n)]; Output: Magma V2.11-10 Tue Dec 6 2005 03:47:44 on modular [Seed = 1655992594] ------------------------------------- 5 Total time: 0.180 seconds, Total memory usage: 3.24MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Tue Dec 6 03:47:35 2005 Input: G :=DirichletGroup(56); G; X :=Elements(G); X; Y :=X[4]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; qEigenform(D[2],12);Parent($1); Output: Magma V2.11-10 Tue Dec 6 2005 03:47:34 on modular [Seed = 1569410619] ------------------------------------- Group of Dirichlet characters of modulus 56 over Rational Field [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3 ] 8 2 [ Modular symbols space of level 56, weight 3, character $.1*$.2, and dimension 4 over Rational Field, Modular symbols space of level 56, weight 3, character $.1*$.2, and dimension 8 over Rational Field ] q + (-14817/2874195968*a^7 - 26645/718548992*a^6 - 10231/89818624*a^5 - 9201/44909312*a^4 - 329307/25662464*a^3 - 524939/6415616*a^2 + 14383/168832*a + 634373/801952)*q^2 + (-1251/302546944*a^7 - 1723/75636736*a^6 - 1363/4727296*a^5 - 103/147728*a^4 - 72009/2701312*a^3 - 44577/675328*a^2 - 92401/337664*a - 57833/84416)*q^3 + (-331/179637248*a^7 + 1423/25662464*a^6 + 29587/89818624*a^5 + 138223/44909312*a^4 - 23141/3207808*a^3 + 87083/400976*a^2 + 13159/21104*a + 506731/100244)*q^4 + (-1251/302546944*a^7 - 1723/75636736*a^6 - 1363/4727296*a^5 - 103/147728*a^4 - 72009/2701312*a^3 - 44577/675328*a^2 - 430065/337664*a - 57833/84416)*q^5 + (473/44909312*a^7 + 11735/89818624*a^6 + 2511/6415616*a^5 + 138053/22454656*a^4 + 73027/1603904*a^3 + 12557/25061*a^2 + 4731/10552*a + 282483/50122)*q^6 + (195/44909312*a^7 + 3823/44909312*a^6 + 4421/11227328*a^5 + 33199/11227328*a^4 + 1065/50122*a^3 + 60859/200488*a^2 + 1209/1319*a + 83369/25061)*q^7 + (4975/179637248*a^7 + 4203/25662464*a^6 + 147831/89818624*a^5 + 266099/44909312*a^4 + 363411/3207808*a^3 + 242029/400976*a^2 + 66663/21104*a + 1007587/100244)*q^8 + (779/75636736*a^7 + 803/18909184*a^6 + 61/168832*a^5 - 459/73864*a^4 + 22977/675328*a^3 - 9895/168832*a^2 + 29737/84416*a - 182703/21104)*q^9 + (241/37818368*a^7 - 1587/9454592*a^6 - 855/1181824*a^5 - 3845/590912*a^4 + 891/337664*a^3 - 40149/84416*a^2 - 69053/42208*a - 23973/10552)*q^10 + (-7247/151273472*a^7 - 7687/37818368*a^6 - 4311/2363648*a^5 - 409/73864*a^4 - 231309/1350656*a^3 - 191429/337664*a^2 - 298965/168832*a - 406957/42208)*q^11 + O(q^12) Power series ring in q over Univariate Quotient Polynomial Algebra in a over Rational Field with modulus a^8 + 8*a^7 + 80*a^6 + 256*a^5 + 4944*a^4 + 24192*a^3 + 172928*a^2 + 250880*a + 1533952 Total time: 0.410 seconds, Total memory usage: 4.64MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Tue Dec 6 03:47:16 2005 Input: G :=DirichletGroup(56); G; X :=Elements(G); X; Y :=X[4]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; qEigenform(D[4],12);Parent($1); Output: Magma V2.11-10 Tue Dec 6 2005 03:47:15 on modular [Seed = 1485720331] ------------------------------------- Group of Dirichlet characters of modulus 56 over Rational Field [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3 ] 8 2 [ Modular symbols space of level 56, weight 3, character $.1*$.2, and dimension 4 over Rational Field, Modular symbols space of level 56, weight 3, character $.1*$.2, and dimension 8 over Rational Field ] >> qEigenform(D[4],12);Parent($1); ^ Runtime error in '[]': Sequence element 4 not defined Set of sequences over Power Structure of ModSym Total time: 0.400 seconds, Total memory usage: 4.45MB '60.225.' ************** MAGMA ***************** Host 60.225.131.213 (60.225.131.213) Time: Tue Dec 6 03:45:59 2005 Input: [n: n in [200 .. 625] | IsSquare(n)]; Output: Magma V2.11-10 Tue Dec 6 2005 03:45:58 on modular [Seed = 1384406740] ------------------------------------- [ 225, 256, 289, 324, 361, 400, 441, 484, 529, 576, 625 ] Total time: 0.180 seconds, Total memory usage: 3.24MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Tue Dec 6 03:45:54 2005 Input: G :=DirichletGroup(56); G; X :=Elements(G); X; Y :=X[7]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; qEigenform(D[4],12);Parent($1); Output: Magma V2.11-10 Tue Dec 6 2005 03:45:54 on modular [Seed = 1302020559] ------------------------------------- Group of Dirichlet characters of modulus 56 over Rational Field [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3 ] 56 2 [ Modular symbols space of level 56, weight 3, character $.2*$.3, and dimension 2 over Rational Field, Modular symbols space of level 56, weight 3, character $.2*$.3, and dimension 2 over Rational Field, Modular symbols space of level 56, weight 3, character $.2*$.3, and dimension 2 over Rational Field, Modular symbols space of level 56, weight 3, character $.2*$.3, and dimension 8 over Rational Field ] q + (-61/779607751680*a^7 + 19/13319658240*a^6 - 33617/151590396160*a^5 + 367/221994304*a^4 - 5068603/28423199280*a^3 - 163153/138746440*a^2 + 1166340923/42634798920*a - 445123/424734)*q^2 + (-54449/44567576471040*a^7 - 321481/742792941184*a^5 + 34597151/58030698530*a^3 - 382734991/1421159964*a)*q^3 + (118831/267405458826240*a^7 - 19/6659829120*a^6 + 2324557/2785473529440*a^5 - 367/110997152*a^4 + 295787251/928491176480*a^3 + 163153/69373220*a^2 - 86314273/3045342780*a - 191978/212367)*q^4 + (313/1856982352960*a^7 + 2368551/3713964705920*a^5 + 56897591/464245588240*a^3 - 877243639/2368599940*a)*q^5 + (-61961/89135152942080*a^7 + 185/1997948736*a^6 - 993989/1856982352960*a^5 + 174781/3329914560*a^4 + 219879617/928491176480*a^3 - 2885719/83247864*a^2 + 359027981/7105799820*a + 66335023/6371010)*q^6 + (-41/815489280*a^6 - 39/11326240*a^4 + 352067/8494680*a^2 - 40889569/3185505)*q^7 + (-8159/11141894117760*a^7 - 19/2219943040*a^6 - 13654757/11141894117760*a^5 - 1101/110997152*a^4 - 390638659/1392736764720*a^3 + 489459/69373220*a^2 + 4673211/2368599940*a + 303545/70789)*q^8 + (19/1664957280*a^6 + 367/27749288*a^4 - 163153/17343305*a^2 - 718657/212367)*q^9 + (-15397/17827030588416*a^7 + 361/7991794944*a^6 - 4356529/3713964705920*a^5 + 34865/665982912*a^4 + 21216887/185698235296*a^3 + 61459/13874644*a^2 + 1495379449/3552899910*a - 1936955/1274202)*q^10 + (15397/26740545882624*a^7 + 4356529/5570947058880*a^5 - 21216887/278547352944*a^3 + 281070506/5329349865*a)*q^11 + O(q^12) Power series ring in q over Univariate Quotient Polynomial Algebra in a over Rational Field with modulus a^8 + 1520*a^6 + 55296*a^4 - 331287040*a^2 + 140439061504 Total time: 0.350 seconds, Total memory usage: 4.54MB '60.225.' ************** MAGMA ***************** Host 60.225.131.213 (60.225.131.213) Time: Tue Dec 6 03:45:44 2005 Input: [n: n in [200 .. 500] | IsSquare(n)]; Output: Magma V2.11-10 Tue Dec 6 2005 03:45:44 on modular [Seed = 1218331824] ------------------------------------- [ 225, 256, 289, 324, 361, 400, 441, 484 ] Total time: 0.190 seconds, Total memory usage: 3.24MB '60.225.' ************** MAGMA ***************** Host 60.225.131.213 (60.225.131.213) Time: Tue Dec 6 03:45:37 2005 Input: [n: n in [300 .. 500] | IsSquare(n)]; Output: Magma V2.11-10 Tue Dec 6 2005 03:45:37 on modular [Seed = 1133858738] ------------------------------------- [ 324, 361, 400, 441, 484 ] Total time: 0.190 seconds, Total memory usage: 3.24MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Tue Dec 6 03:45:25 2005 Input: G :=DirichletGroup(56); G; X :=Elements(G); X; Y :=X[7]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; qEigenform(D[1],12);Parent($1); Output: Magma V2.11-10 Tue Dec 6 2005 03:45:24 on modular [Seed = 3211375450] ------------------------------------- Group of Dirichlet characters of modulus 56 over Rational Field [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3 ] 56 2 [ Modular symbols space of level 56, weight 3, character $.2*$.3, and dimension 2 over Rational Field, Modular symbols space of level 56, weight 3, character $.2*$.3, and dimension 2 over Rational Field, Modular symbols space of level 56, weight 3, character $.2*$.3, and dimension 2 over Rational Field, Modular symbols space of level 56, weight 3, character $.2*$.3, and dimension 8 over Rational Field ] q + (1/16*a + 3/2)*q^2 + (3/16*a + 1/2)*q^4 + 7*q^7 + (5/16*a - 9/2)*q^8 - 9*q^9 - a*q^11 + O(q^12) Power series ring in q over Univariate Quotient Polynomial Algebra in a over Rational Field with modulus a^2 + 448 Total time: 0.340 seconds, Total memory usage: 4.54MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Tue Dec 6 03:45:05 2005 Input: G :=DirichletGroup(56); G; X :=Elements(G); X; Y :=X[6]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; qEigenform(D[1],12);Parent($1); Output: Magma V2.11-10 Tue Dec 6 2005 03:45:04 on modular [Seed = 3094262889] ------------------------------------- Group of Dirichlet characters of modulus 56 over Rational Field [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3 ] 28 2 [] >> qEigenform(D[1],12);Parent($1); ^ Runtime error in '[]': Sequence element 1 not defined Set of null sequences Total time: 0.210 seconds, Total memory usage: 4.34MB '60.225.' ************** MAGMA ***************** Host 60.225.131.213 (60.225.131.213) Time: Tue Dec 6 03:44:37 2005 Input: [n: n in [100 .. 200] | IsSquare(n)]; Output: Magma V2.11-10 Tue Dec 6 2005 03:44:36 on modular [Seed = 3043211648] ------------------------------------- [ 100, 121, 144, 169, 196 ] Total time: 0.180 seconds, Total memory usage: 3.24MB '60.225.' ************** MAGMA ***************** Host 60.225.131.213 (60.225.131.213) Time: Tue Dec 6 03:43:19 2005 Input: [n: n in [100 .. 200] | IsPrime(n)]; Output: Magma V2.11-10 Tue Dec 6 2005 03:43:18 on modular [Seed = 2927141577] ------------------------------------- [ 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199 ] Total time: 0.190 seconds, Total memory usage: 3.24MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Tue Dec 6 03:43:00 2005 Input: G :=DirichletGroup(56); G; X :=Elements(G); X; Y :=X[5]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; qEigenform(D[1],12);Parent($1); Output: Magma V2.11-10 Tue Dec 6 2005 03:42:59 on modular [Seed = 2877138937] ------------------------------------- Group of Dirichlet characters of modulus 56 over Rational Field [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3 ] 7 2 [ Modular symbols space of level 56, weight 3, character $.3, and dimension 4 over Rational Field ] q + a*q^3 + (1/2*a^3 + 7*a)*q^5 + (-1/2*a^3 + 1/2*a^2 - 5*a + 5)*q^7 + (a^2 + 9)*q^9 + (-2*a^2 - 14)*q^11 + O(q^12) Power series ring in q over Univariate Quotient Polynomial Algebra in a over Rational Field with modulus a^4 + 16*a^2 + 32 Total time: 0.430 seconds, Total memory usage: 4.64MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Tue Dec 6 03:42:29 2005 Input: G :=DirichletGroup(56); G; X :=Elements(G); X; Y :=X[5]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; Output: Magma V2.11-10 Tue Dec 6 2005 03:42:29 on modular [Seed = 2758977808] ------------------------------------- Group of Dirichlet characters of modulus 56 over Rational Field [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3 ] 7 2 [ Modular symbols space of level 56, weight 3, character $.3, and dimension 4 over Rational Field ] Total time: 0.420 seconds, Total memory usage: 4.45MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Tue Dec 6 03:41:40 2005 Input: G :=DirichletGroup(56); G; X :=Elements(G); X; Y :=X[5]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; #qEigenform(D[2],12),11;Parent($1); Output: Magma V2.11-10 Tue Dec 6 2005 03:41:39 on modular [Seed = 2655556380] ------------------------------------- Group of Dirichlet characters of modulus 56 over Rational Field [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3 ] 7 2 [ Modular symbols space of level 56, weight 3, character $.3, and dimension 4 over Rational Field ] >> #qEigenform(D[2],12),11;Parent($1); ^ Runtime error in '[]': Sequence element 2 not defined Set of sequences over Power Structure of ModSym Total time: 0.420 seconds, Total memory usage: 4.45MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Tue Dec 6 03:39:14 2005 Input: G :=DirichletGroup(56); G; X :=Elements(G); X; Y :=X[4]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; qEigenform(D[2],12),11;Parent($1); Output: Magma V2.11-10 Tue Dec 6 2005 03:39:14 on modular [Seed = 2605553384] ------------------------------------- Group of Dirichlet characters of modulus 56 over Rational Field [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3 ] 8 2 [ Modular symbols space of level 56, weight 3, character $.1*$.2, and dimension 4 over Rational Field, Modular symbols space of level 56, weight 3, character $.1*$.2, and dimension 8 over Rational Field ] q + (-571500746715/1775060251923248128*a^7 - 388629337553/887530125961624064*a^6 - 1028834904749/11678027973179264*a^5 + 4918368039129/55470632872601504*a^4 - 4651621445267367/443765062980812032*a^3 + 6509171845645319/221882531490406016*a^2 - 25815516211983173/110941265745203008*a + 7150300887080843/2411766646634848)*q^2 + (-202925409/2919506993294816*a^7 + 833622367/729876748323704*a^6 - 11090313389/729876748323704*a^5 + 122109745219/364938374161852*a^4 - 3634150197939/729876748323704*a^3 + 3921605990478/91234593540463*a^2 - 39299581757551/182469187080926*a + 5869895733196/3966721458281)*q^3 + (299769622133/1775060251923248128*a^7 + 6555626450207/887530125961624064*a^6 + 1138059007615/11678027973179264*a^5 + 122284993648093/55470632872601504*a^4 + 6459555605921081/443765062980812032*a^3 + 66760393194614839/221882531490406016*a^2 + 95645252448681499/110941265745203008*a + 31634693493840411/2411766646634848)*q^4 + (-202925409/5839013986589632*a^7 + 833622367/1459753496647408*a^6 - 11090313389/1459753496647408*a^5 + 122109745219/729876748323704*a^4 - 3634150197939/1459753496647408*a^3 + 1960802995239/91234593540463*a^2 - 221768768838477/364938374161852*a + 2934947866598/3966721458281)*q^5 + (711085480935/887530125961624064*a^7 + 235869057485/443765062980812032*a^6 + 1200042577193/5839013986589632*a^5 + 911209802185/3466914554537594*a^4 + 6181260606368675/221882531490406016*a^3 + 9166844150804637/110941265745203008*a^2 + 52756368685049465/55470632872601504*a + 4498614222256633/1205883323317424)*q^6 + (17241/35534698624*a^7 + 183763/35534698624*a^6 + 35797/233780912*a^5 + 1454255/1110459332*a^4 + 190238637/8883674656*a^3 + 1587947027/8883674656*a^2 + 2428447959/2220918664*a + 13762616057/2220918664)*q^7 + (1226158137473/1775060251923248128*a^7 - 986896349357/887530125961624064*a^6 + 2866470193731/11678027973179264*a^5 - 27191192590141/55470632872601504*a^4 + 16767816630761589/443765062980812032*a^3 - 4166678493727589/221882531490406016*a^2 + 262678257101054031/110941265745203008*a + 8568098621402639/2411766646634848)*q^8 + (247894637/364938374161852*a^7 + 470257646/91234593540463*a^6 + 34730559821/182469187080926*a^5 - 7463559867/182469187080926*a^4 + 2215406259756/91234593540463*a^3 - 12060094738132/91234593540463*a^2 + 74854681732280/91234593540463*a - 57822477939285/3966721458281)*q^9 + (-15609054847/23356055946358528*a^7 - 241973420793/11678027973179264*a^6 - 784427920321/2919506993294816*a^5 - 8645457727271/1459753496647408*a^4 - 243983987349699/5839013986589632*a^3 - 2142335224733605/2919506993294816*a^2 - 3401476783420169/1459753496647408*a - 680021364237737/31733771666248)*q^10 + (-3247072067/2919506993294816*a^7 + 2525600507/182469187080926*a^6 - 113545926879/364938374161852*a^5 + 375687815222/91234593540463*a^4 - 32070491902875/729876748323704*a^3 + 48544544489462/91234593540463*a^2 - 284832833856747/182469187080926*a + 57382437550564/3966721458281)*q^11 + O(q^12) 11 Power series ring in q over Univariate Quotient Polynomial Algebra in a over Rational Field with modulus a^8 + 8*a^7 + 404*a^6 + 2032*a^5 + 70164*a^4 + 254976*a^3 + 5537696*a^2 + 8220416*a + 144702016 Total time: 0.410 seconds, Total memory usage: 4.64MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Tue Dec 6 03:28:53 2005 Input: G :=DirichletGroup(56); G; X :=Elements(G); X; Y :=X[4]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; qEigenform(D[1],12),11;Parent($1); Output: Magma V2.11-10 Tue Dec 6 2005 03:28:52 on modular [Seed = 2220002305] ------------------------------------- Group of Dirichlet characters of modulus 56 over Rational Field [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3 ] 8 2 [ Modular symbols space of level 56, weight 3, character $.1*$.2, and dimension 4 over Rational Field, Modular symbols space of level 56, weight 3, character $.1*$.2, and dimension 8 over Rational Field ] q + (7/480*a^3 - 1/24*a^2 + 61/60*a - 13/5)*q^2 + (-1/480*a^3 + 1/24*a^2 - 13/60*a + 19/5)*q^3 + (1/60*a^3 - 1/12*a^2 + 26/15*a - 42/5)*q^4 + (-1/480*a^3 + 1/24*a^2 - 73/60*a + 19/5)*q^5 + (11/240*a^3 - 1/6*a^2 + 113/30*a - 48/5)*q^6 + (-1/60*a^3 + 1/12*a^2 - 26/15*a + 27/5)*q^7 + (-1/48*a^3 - 1/12*a^2 - 7/6*a - 2)*q^8 + (-1/120*a^3 + 1/6*a^2 - 13/15*a + 21/5)*q^9 + (-7/240*a^3 + 1/3*a^2 - 61/30*a + 36/5)*q^10 + (1/80*a^3 - 1/4*a^2 + 13/10*a - 34/5)*q^11 + O(q^12) 11 Power series ring in q over Univariate Quotient Polynomial Algebra in a over Rational Field with modulus a^4 - 8*a^3 + 104*a^2 - 576*a + 1152 Total time: 0.400 seconds, Total memory usage: 4.64MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Tue Dec 6 03:28:23 2005 Input: G :=DirichletGroup(56); G; X :=Elements(G); X; Y :=X[4]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; qEigenform(D[1],12),11);Parent($1); Output: Magma V2.11-10 Tue Dec 6 2005 03:28:22 on modular [Seed = 2169997670] ------------------------------------- Group of Dirichlet characters of modulus 56 over Rational Field [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3 ] 8 2 [ Modular symbols space of level 56, weight 3, character $.1*$.2, and dimension 4 over Rational Field, Modular symbols space of level 56, weight 3, character $.1*$.2, and dimension 8 over Rational Field ] >> qEigenform(D[1],12),11);Parent($1); ^ User error: bad syntax Total time: 0.380 seconds, Total memory usage: 4.45MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Tue Dec 6 03:26:47 2005 Input: G :=DirichletGroup(56); G; X :=Elements(G); X; Y :=X[4]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; Output: Magma V2.11-10 Tue Dec 6 2005 03:26:47 on modular [Seed = 4214092797] ------------------------------------- Group of Dirichlet characters of modulus 56 over Rational Field [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3 ] 8 2 [ Modular symbols space of level 56, weight 3, character $.1*$.2, and dimension 4 over Rational Field, Modular symbols space of level 56, weight 3, character $.1*$.2, and dimension 8 over Rational Field ] Total time: 0.390 seconds, Total memory usage: 4.45MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Tue Dec 6 03:26:37 2005 Input: G :=DirichletGroup(56); G; X :=Elements(G); X; Y :=X[3]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; Output: Magma V2.11-10 Tue Dec 6 2005 03:26:36 on modular [Seed = 4112776585] ------------------------------------- Group of Dirichlet characters of modulus 56 over Rational Field [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3 ] 8 2 [] Total time: 0.210 seconds, Total memory usage: 4.34MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Tue Dec 6 03:26:24 2005 Input: G :=DirichletGroup(56); G; X :=Elements(G); X; Y :=X[2]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; Output: Magma V2.11-10 Tue Dec 6 2005 03:26:23 on modular [Seed = 3996706432] ------------------------------------- Group of Dirichlet characters of modulus 56 over Rational Field [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3 ] 4 2 [] Total time: 0.490 seconds, Total memory usage: 4.45MB '131.156' ************** MAGMA ***************** Host 131.156.146.48 (131.156.146.48) Time: Tue Dec 6 02:05:37 2005 Input: P:=PolynomialRing(RationalField(),10); I:=ideal; GroebnerBasis(EliminationIdeal(I,{x,a,b,c,d,e,f})); GroebnerBasis(EliminationIdeal(I,{y,a,b,c,d,e,f})); Output: Magma V2.11-10 Tue Dec 6 2005 02:05:37 on modular [Seed = 499869213] ------------------------------------- [ x^2 - x*a - x*f + a*f - b*e, x*a*c - x*c*f + x*d*e - x*e^2 - a*c*f + b*c*e + c*f^2 - d*e*f, x*a*f + x*b*d - x*b*e - x*f^2 - a^2*f - a*b*d + a*b*e + a*f^2 + b^2*c - b*e*f, x*b*c - x*e*f + a*e*f - b*c*f + b*d*e - b*e^2, a^2*c*f + a*b*c*d - a*b*c*e - 2*a*c*f^2 + a*d*e*f - a*e^2*f - b^2*c^2 - b*c*d*f + 3*b*c*e*f + b*d^2*e - 2*b*d*e^2 + b*e^3 + c*f^3 - d*e*f^2 ] [ y^2 - y*d - y*e - c*f + d*e, y*a*c - y*c*f + y*d*e - y*e^2 - a*c*d + b*c^2 + c*d*f - c*e*f - d^2*e + d*e^2, y*a*f + y*b*d - y*b*e - y*f^2 - a*e*f + b*c*f - b*d*e + b*e^2, y*b*c - y*e*f + a*c*f - b*c*e - c*f^2 + d*e*f, a^2*c*f + a*b*c*d - a*b*c*e - 2*a*c*f^2 + a*d*e*f - a*e^2*f - b^2*c^2 - b*c*d*f + 3*b*c*e*f + b*d^2*e - 2*b*d*e^2 + b*e^3 + c*f^3 - d*e*f^2 ] Total time: 0.200 seconds, Total memory usage: 3.34MB '131.156' ************** MAGMA ***************** Host 131.156.146.48 (131.156.146.48) Time: Tue Dec 6 02:02:24 2005 Input: P:=PolynomialRing(RationalField(),10); I:=ideal; EliminationIdeal(I,{x,a,b,c,d,e,f}); EliminationIdeal(I,{y,a,b,c,d,e,f}); Output: Magma V2.11-10 Tue Dec 6 2005 02:02:23 on modular [Seed = 381706792] ------------------------------------- Ideal of Polynomial ring of rank 10 over Rational Field Lexicographical Order Variables: xi, yi, x, y, a, b, c, d, e, f Basis: [ -x^2*c + 2*x*c*f - x*d*e + x*e^2 - c*f^2 + d*e*f, -x^2*d + x^2*e + x*a*d - x*a*e - x*b*c + x*d*f - a*d*f + b*c*f, -x^3 + 2*x^2*a - x*a^2 + x*b*d - a*b*d + b^2*c, -x^2 + x*a + x*f - a*f + b*e ] Ideal of Polynomial ring of rank 10 over Rational Field Lexicographical Order Variables: xi, yi, x, y, a, b, c, d, e, f Basis: [ -y^2*b - y*a*f + 2*y*b*e + y*f^2 + a*e*f - b*e^2, -y^2*e - y*a*c + y*c*f + 2*y*e^2 + a*c*d - b*c^2 - c*d*f + 2*c*e*f + d^2*e - 2*d*e^2, y*b*c - y*e*f + a*c*f - b*c*e - c*f^2 + d*e*f, -y^2 + y*d + y*e + c*f - d*e ] Total time: 0.200 seconds, Total memory usage: 3.34MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 22:09:33 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^5*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1400); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=20); //RationalPoints(J:Bound:=12); //Chabauty(P,7); //Order(Q);Order(P); //P+Q;P+2*Q; Output: Magma V2.11-10 Mon Dec 5 2005 22:09:31 on modular [Seed = 4197234989] ------------------------------------- 1 Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 to JacHyp: J given by a rule [no inverse] [ (x^2 - 4, 2*x + 4, 2) ] 0.917694999912313628709531669468297352510 4.486499277983286154053028928413021210016 3.465735902799726547086160607290882840364 Total time: 1.889 seconds, Total memory usage: 39.58MB '64.33.1' ************** MAGMA ***************** Host 64.33.185.47 (64.33.185.47) Time: Mon Dec 5 21:59:46 2005 Input: 67+56=x Output: Magma V2.11-10 Mon Dec 5 2005 21:59:46 on modular [Seed = 1251866580] ------------------------------------- >> 67+56=x; ^ User error: Identifier 'x' has not been declared or assigned Total time: 0.200 seconds, Total memory usage: 3.24MB '64.33.1' ************** MAGMA ***************** Host 64.33.185.47 (64.33.185.47) Time: Mon Dec 5 21:59:37 2005 Input: 67+56 Output: Magma V2.11-10 Mon Dec 5 2005 21:59:37 on modular [Seed = 1301999848] ------------------------------------- 123 Total time: 0.190 seconds, Total memory usage: 3.24MB '64.33.1' ************** MAGMA ***************** Host 64.33.185.47 (64.33.185.47) Time: Mon Dec 5 21:59:31 2005 Input: x=67+56 Output: Magma V2.11-10 Mon Dec 5 2005 21:59:30 on modular [Seed = 1351085566] ------------------------------------- >> x=67+56; ^ User error: Identifier 'x' has not been declared or assigned Total time: 0.190 seconds, Total memory usage: 3.24MB '64.33.1' ************** MAGMA ***************** Host 64.33.185.47 (64.33.185.47) Time: Mon Dec 5 21:59:18 2005 Input: x=67x+56 Output: Magma V2.11-10 Mon Dec 5 2005 21:59:17 on modular [Seed = 1535960802] ------------------------------------- >> x=67x+56; ^ User error: Invalid hexadecimal integer Total time: 0.190 seconds, Total memory usage: 3.24MB '64.33.1' ************** MAGMA ***************** Host 64.33.185.47 (64.33.185.47) Time: Mon Dec 5 21:59:00 2005 Input: x=67x+56 Output: Magma V2.11-10 Mon Dec 5 2005 21:59:00 on modular [Seed = 1990314814] ------------------------------------- >> x=67x+56; ^ User error: Invalid hexadecimal integer Total time: 0.190 seconds, Total memory usage: 3.24MB '64.33.1' ************** MAGMA ***************** Host 64.33.185.47 (64.33.185.47) Time: Mon Dec 5 21:58:53 2005 Input: x=67y+56 Output: Magma V2.11-10 Mon Dec 5 2005 21:58:53 on modular [Seed = 2041496616] ------------------------------------- >> x=67y+56; ^ User error: bad syntax Total time: 0.190 seconds, Total memory usage: 3.24MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 21:54:43 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^4*3^8; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1400); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=20); //RationalPoints(J:Bound:=12); Chabauty(P,7); //Order(Q);Order(P); //P+Q;P+2*Q; Output: ** WARNING: Computation time exceeded 20 seconds, so computation was terminated after 20 seconds. ** Magma V2.11-10 Mon Dec 5 2005 21:54:23 on modular [Seed = 64273760] ------------------------------------- 1 Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 to JacHyp: J given by a rule [no inverse] [ (x^2 - 9*x, 9*x + 324, 2) ] 1.13829905958751855639529004360387879793177840048426356187981 11.991636334011282544968217138124976622401 8.862903321989681361582647123238902219762 Errors: /bin/sh: line 1: 32575 Alarm clock nice -n 19 /usr/local/bin/magma '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 21:52:07 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^4*3^8; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1400); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=20); RationalPoints(J:Bound:=12); //Chabauty(P,19); //Order(Q);Order(P); //P+Q;P+2*Q; Output: Magma V2.11-10 Mon Dec 5 2005 21:51:49 on modular [Seed = 80985085] ------------------------------------- 1 Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 to JacHyp: J given by a rule [no inverse] [ (x^2 - 9*x, 9*x + 324, 2) ] 1.13829905958751855639529004360387879793177840048426356187981 11.991636334011282544968217138124976622401 8.862903321989681361582647123238902219762 {@ (1, 0, 0), (x^2, 324, 2), (x^2, -324, 2), (x, 324, 1), (x, -324, 1) @} Total time: 17.890 seconds, Total memory usage: 39.53MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 21:50:13 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^4*3^8; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1400); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=20); //RationalPoints(J:Bound:=12); //Chabauty(P,19); //Order(Q);Order(P); //P+Q;P+2*Q; Output: Magma V2.11-10 Mon Dec 5 2005 21:49:55 on modular [Seed = 314944893] ------------------------------------- 1 Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 to JacHyp: J given by a rule [no inverse] [ (x^2 - 9*x, 9*x + 324, 2) ] 1.13829905958751855639529004360387879793177840048426356187981 11.991636334011282544968217138124976622401 8.862903321989681361582647123238902219762 Total time: 18.010 seconds, Total memory usage: 39.53MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 21:49:22 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^4*3^7; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1400); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=20); //RationalPoints(J:Bound:=12); Chabauty(P,19); //Order(Q);Order(P); //P+Q;P+2*Q; Output: Magma V2.11-10 Mon Dec 5 2005 21:49:20 on modular [Seed = 365079065] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 + 9*x + 54, 9*x - 162, 2) ] 1.700873027587985562177599271502715613824 9.856962253562785472319813381511113821939 8.1304951295442749006524836319572184166598 {@ @} Total time: 1.940 seconds, Total memory usage: 39.39MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 21:49:13 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^4*3^7; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1400); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=20); //RationalPoints(J:Bound:=12); Chabauty(P,17); //Order(Q);Order(P); //P+Q;P+2*Q; Output: Magma V2.11-10 Mon Dec 5 2005 21:49:10 on modular [Seed = 416261890] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 + 9*x + 54, 9*x - 162, 2) ] 1.700873027587985562177599271502715613824 9.856962253562785472319813381511113821939 8.1304951295442749006524836319572184166598 {@ <29826, 1, 4, 1>, <82452, 1, 4, 1>, <31836, 1, 4, 1>, <83364, 1, 4, 1>, <51712, 1, 4, 1>, <22684, 1, 4, 1>, <17708, 1, 4, 1>, <3618, 1, 4, 1> @} Total time: 2.850 seconds, Total memory usage: 39.39MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 21:49:03 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^4*3^7; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1400); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=20); //RationalPoints(J:Bound:=12); Chabauty(P,13); //Order(Q);Order(P); //P+Q;P+2*Q; Output: Magma V2.11-10 Mon Dec 5 2005 21:49:01 on modular [Seed = 603233282] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 + 9*x + 54, 9*x - 162, 2) ] 1.700873027587985562177599271502715613824 9.856962253562785472319813381511113821939 8.1304951295442749006524836319572184166598 {@ <346, 1, 4, 1>, <15020, 1, 4, 1>, <12736, 1, 4, 1>, <0, 1, 4, 1>, <21425, 1, 4, 1> @} Total time: 2.419 seconds, Total memory usage: 39.39MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 21:48:50 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^4*3^7; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1400); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=20); //RationalPoints(J:Bound:=12); Chabauty(P,11); //Order(Q);Order(P); //P+Q;P+2*Q; Output: Magma V2.11-10 Mon Dec 5 2005 21:48:48 on modular [Seed = 653367562] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 + 9*x + 54, 9*x - 162, 2) ] 1.700873027587985562177599271502715613824 9.856962253562785472319813381511113821939 8.1304951295442749006524836319572184166598 {@ <0, 1, 4, 1> @} Total time: 1.940 seconds, Total memory usage: 39.39MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 21:48:39 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^4*3^7; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1400); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=20); //RationalPoints(J:Bound:=12); Chabauty(P,7); //Order(Q);Order(P); //P+Q;P+2*Q; Output: Magma V2.11-10 Mon Dec 5 2005 21:48:37 on modular [Seed = 704550416] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 + 9*x + 54, 9*x - 162, 2) ] 1.700873027587985562177599271502715613824 9.856962253562785472319813381511113821939 8.1304951295442749006524836319572184166598 {@ <536, 1, 4, 1>, <1258, 1, 4, 1>, <766, 1, 4, 1> @} Total time: 2.020 seconds, Total memory usage: 39.39MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 21:44:47 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^4*3^7; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1400); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=20); //RationalPoints(J:Bound:=12); //Chabauty(P,19); //Order(Q);Order(P); //P+Q;P+2*Q; Output: Magma V2.11-10 Mon Dec 5 2005 21:44:45 on modular [Seed = 937463574] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 + 9*x + 54, 9*x - 162, 2) ] 1.700873027587985562177599271502715613824 9.856962253562785472319813381511113821939 8.1304951295442749006524836319572184166598 Total time: 1.520 seconds, Total memory usage: 39.39MB '172.130' ************** MAGMA ***************** Host 172.130.251.86 (172.130.251.86) Time: Mon Dec 5 21:38:13 2005 Input: "Replace this by some code, then click [PARI] or [MAGMA]!" Output: Magma V2.11-10 Mon Dec 5 2005 21:38:13 on modular [Seed = 3306323821] ------------------------------------- Replace this by some code, then click [PARI] or [MAGMA]! Total time: 0.190 seconds, Total memory usage: 3.24MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 21:37:28 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^4*3^6; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1400); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=20); //RationalPoints(J:Bound:=12); //Chabauty(P,19); //Order(Q);Order(P); //P+Q;P+2*Q; Output: Magma V2.11-10 Mon Dec 5 2005 21:37:25 on modular [Seed = 3424351892] ------------------------------------- 1 Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 to JacHyp: J given by a rule [no inverse] [ (x^2 + 3*x + 36, 3*x + 72, 2) ] 2.6951401700620685956297684449255413869414734353341592770870 10.1142760525465394052441122341781093637312 7.398086937098868439722320140675534613557 Total time: 2.629 seconds, Total memory usage: 39.47MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 21:37:14 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^4*3^6; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1400); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=20); RationalPoints(J:Bound:=12); //Chabauty(P,19); //Order(Q);Order(P); //P+Q;P+2*Q; Output: Magma V2.11-10 Mon Dec 5 2005 21:37:11 on modular [Seed = 3506858939] ------------------------------------- 1 Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 to JacHyp: J given by a rule [no inverse] [ (x^2 + 3*x + 36, 3*x + 72, 2) ] 2.6951401700620685956297684449255413869414734353341592770870 10.1142760525465394052441122341781093637312 7.398086937098868439722320140675534613557 {@ (1, 0, 0), (x^2, 108, 2), (x^2, -108, 2), (x, 108, 1), (x, -108, 1) @} Total time: 2.750 seconds, Total memory usage: 39.47MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 21:36:33 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^4*3^6; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1400); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=20); //RationalPoints(J:Bound:=12); //Chabauty(P,19); //Order(Q);Order(P); //P+Q;P+2*Q; Output: Magma V2.11-10 Mon Dec 5 2005 21:36:30 on modular [Seed = 3675021768] ------------------------------------- 1 Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 to JacHyp: J given by a rule [no inverse] [ (x^2 + 3*x + 36, 3*x + 72, 2) ] 2.6951401700620685956297684449255413869414734353341592770870 10.1142760525465394052441122341781093637312 7.398086937098868439722320140675534613557 Total time: 2.629 seconds, Total memory usage: 39.47MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 21:34:16 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^4*3^3; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1400); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=20); //RationalPoints(J:Bound:=12); Chabauty(P,19); //Order(Q);Order(P); //P+Q;P+2*Q; Output: Magma V2.11-10 Mon Dec 5 2005 21:34:14 on modular [Seed = 3708446278] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x + 2, -20, 1) ] 0.887004017177288147926102307775262838447 6.501416595843100382592300278524636022103 5.2008623597626490569318296668304832042724 {@ <130319, 1, 4, 1> @} Total time: 2.089 seconds, Total memory usage: 39.45MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 21:34:09 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^4*3^3; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1400); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=20); //RationalPoints(J:Bound:=12); Chabauty(P,17); //Order(Q);Order(P); //P+Q;P+2*Q; Output: Magma V2.11-10 Mon Dec 5 2005 21:34:06 on modular [Seed = 3861994821] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x + 2, -20, 1) ] 0.887004017177288147926102307775262838447 6.501416595843100382592300278524636022103 5.2008623597626490569318296668304832042724 {@ <83519, 1, 4, 1>, <71129, 1, 4, 1>, <32700, 1, 4, 1>, <41477, 1, 4, 1>, <79888, 1, 4, 1>, <6059, 1, 4, 1>, <21532, 1, 4, 1> @} Total time: 2.790 seconds, Total memory usage: 39.45MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 21:34:01 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^4*3^3; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1400); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=20); //RationalPoints(J:Bound:=12); Chabauty(P,13); //Order(Q);Order(P); //P+Q;P+2*Q; Output: Magma V2.11-10 Mon Dec 5 2005 21:33:58 on modular [Seed = 3946598464] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x + 2, -20, 1) ] 0.887004017177288147926102307775262838447 6.501416595843100382592300278524636022103 5.2008623597626490569318296668304832042724 {@ <28559, 1, 4, 1>, <18034, 1, 4, 1>, <26, 1, 4, 1>, <0, 1, 4, 1>, <6858, 1, 4, 1>, <5292, 1, 4, 1>, <5644, 1, 4, 1> @} Total time: 2.450 seconds, Total memory usage: 39.45MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 21:32:22 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^4*3^3; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1400); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=20); //RationalPoints(J:Bound:=12); Chabauty(P,11); //Order(Q);Order(P); //P+Q;P+2*Q; Output: Magma V2.11-10 Mon Dec 5 2005 21:32:20 on modular [Seed = 3963312101] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x + 2, -20, 1) ] 0.887004017177288147926102307775262838447 6.501416595843100382592300278524636022103 5.2008623597626490569318296668304832042724 {@ <14639, 1, 4, 1>, <1835, 1, 4, 1> @} Total time: 1.919 seconds, Total memory usage: 39.45MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 21:29:53 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^3*3^9; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1400); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=20); //RationalPoints(J:Bound:=12); Chabauty(P,11); //Order(Q);Order(P); //P+Q;P+2*Q; Output: Magma V2.11-10 Mon Dec 5 2005 21:29:51 on modular [Seed = 4264118876] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 + 9*x + 81, -9*x + 324, 2) ] 3.8147298171104413244492568924472542117389814640423666477508 10.0853919220257459932592798346033758733630 9.1332133940617909495679892002151349774645 {@ @} Total time: 2.200 seconds, Total memory usage: 39.51MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 21:29:37 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^3*3^9; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1400); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=12); Chabauty(P,11); //Order(Q);Order(P); //P+Q;P+2*Q; Output: Magma V2.11-10 Mon Dec 5 2005 21:29:35 on modular [Seed = 2203334449] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 + 9*x + 81, -9*x + 324, 2) ] 3.8147298171104413244492568924472542117389814640423666477508 10.0853919220257459932592798346033758733630 9.1332133940617909495679892002151349774645 {@ @} Total time: 2.209 seconds, Total memory usage: 39.55MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 21:25:47 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^3*3^4; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1400); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=12); Chabauty(P,11); //Order(Q);Order(P); //P+Q;P+2*Q; Output: Magma V2.11-10 Mon Dec 5 2005 21:25:45 on modular [Seed = 2672306699] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 21/4*x - 63/4, 147/8*x + 117/8, 2) ] 4.5968876495584691807324090445792654211205124632896604980007 5.781716759018522400865294183991207198153 5.471172431834758644917171743806715961974 {@ @} Total time: 1.770 seconds, Total memory usage: 39.51MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 21:23:50 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^3*3^1; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1400); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=12); Chabauty(P,19); //Order(Q);Order(P); //P+Q;P+2*Q; Output: Magma V2.11-10 Mon Dec 5 2005 21:23:48 on modular [Seed = 2692158889] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x - 1, 5, 1) ] 1.10576830681247164656097111182671309904513044221613666621520 3.0583747280739988265523538592098468041222 3.273947854498539262126681269961664552701 {@ <1, 1, 4, 1>, <40452, 1, 4, 1> @} Total time: 2.049 seconds, Total memory usage: 39.27MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 21:23:19 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^3*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1400); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=12); Chabauty(P,19); //Order(Q);Order(P); //P+Q;P+2*Q; Output: Magma V2.11-10 Mon Dec 5 2005 21:23:17 on modular [Seed = 2775715981] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x - 1, 3, 1) ] 1.5252793887716762503213810181895103809213885083514362505150 2.908055924839945003811228631858578292786 2.541539662053132801196517778679980749598 {@ <1, 1, 4, 1> @} Total time: 1.919 seconds, Total memory usage: 39.27MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 21:23:11 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^3*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1400); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=12); Chabauty(P,17); //Order(Q);Order(P); //P+Q;P+2*Q; Output: Magma V2.11-10 Mon Dec 5 2005 21:23:08 on modular [Seed = 2927172511] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x - 1, 3, 1) ] 1.5252793887716762503213810181895103809213885083514362505150 2.908055924839945003811228631858578292786 2.541539662053132801196517778679980749598 {@ <1, 1, 4, 1>, <71940, 1, 4, 1>, <3394, 1, 4, 1>, <0, 1, 4, 1>, <51125, 1, 4, 1>, <27105, 1, 4, 1>, <16153, 1, 4, 1>, <64966, 1, 4, 1> @} Total time: 2.830 seconds, Total memory usage: 39.37MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 21:22:58 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^3*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1400); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=12); Chabauty(P,13); //Order(Q);Order(P); //P+Q;P+2*Q; Output: Magma V2.11-10 Mon Dec 5 2005 21:22:56 on modular [Seed = 2976253068] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x - 1, 3, 1) ] 1.5252793887716762503213810181895103809213885083514362505150 2.908055924839945003811228631858578292786 2.541539662053132801196517778679980749598 {@ <1, 1, 4, 1> @} Total time: 1.830 seconds, Total memory usage: 39.28MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 21:22:50 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^3*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1400); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=12); Chabauty(P,11); //Order(Q);Order(P); //P+Q;P+2*Q; Output: Magma V2.11-10 Mon Dec 5 2005 21:22:48 on modular [Seed = 3127708799] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x - 1, 3, 1) ] 1.5252793887716762503213810181895103809213885083514362505150 2.908055924839945003811228631858578292786 2.541539662053132801196517778679980749598 {@ <1, 1, 4, 1>, <6605, 1, 4, 1>, <12049, 1, 4, 1>, <10607, 1, 4, 1>, <13847, 1, 4, 1> @} Total time: 2.200 seconds, Total memory usage: 39.30MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 21:22:41 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^3*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1400); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=12); Chabauty(P,1); //Order(Q);Order(P); //P+Q;P+2*Q; Output: Magma V2.11-10 Mon Dec 5 2005 21:22:40 on modular [Seed = 3211266418] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x - 1, 3, 1) ] 1.5252793887716762503213810181895103809213885083514362505150 2.908055924839945003811228631858578292786 2.541539662053132801196517778679980749598 >> Chabauty(P,1); ^ Runtime error in 'Chabauty': p must be a prime number Total time: 1.429 seconds, Total memory usage: 39.31MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 21:21:46 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^3*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1400); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=12); Chabauty(P,7); //Order(Q);Order(P); //P+Q;P+2*Q; Output: Magma V2.11-10 Mon Dec 5 2005 21:21:44 on modular [Seed = 1285290697] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x - 1, 3, 1) ] 1.5252793887716762503213810181895103809213885083514362505150 2.908055924839945003811228631858578292786 2.541539662053132801196517778679980749598 {@ <1, 1, 4, 1>, <2238, 1, 4, 1>, <0, 1, 4, 1> @} Total time: 1.970 seconds, Total memory usage: 39.43MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 21:21:33 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^3*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1400); B:=ReducedBasis(V); B; P:=J!B[1]; Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=12); Chabauty(P,7); //Order(Q);Order(P); //P+Q;P+2*Q; Output: Magma V2.11-10 Mon Dec 5 2005 21:21:31 on modular [Seed = 1367795655] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x - 1, 3, 1) ] >> P:=J!B[1]; Q:=J![x,54]; ^ Runtime error in '!': Points specified by second polynomial are not on the curve 1.5252793887716762503213810181895103809213885083514362505150 2.908055924839945003811228631858578292786 2.541539662053132801196517778679980749598 {@ <1, 1, 4, 1>, <2238, 1, 4, 1>, <0, 1, 4, 1> @} Total time: 1.990 seconds, Total memory usage: 39.50MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 21:08:54 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^2*3^6; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1400); B:=ReducedBasis(V); B; P:=J!B[1]; Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=20); //RationalPoints(J:Bound:=12); Chabauty(P,11); //Order(Q);Order(P); P+Q;P+2*Q; Output: Magma V2.11-10 Mon Dec 5 2005 21:08:51 on modular [Seed = 1774115047] ------------------------------------- 1 Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 to JacHyp: J given by a rule [no inverse] [ (x^2 - 9/4*x - 27/2, -135/8*x + 27/4, 2) ] 2.367463309945994887046414980850866680181 8.937165595523057578067849122405558415787 6.473890696352274693832677312064632522802 {@ <0, 1, 4, 1> @} (x^2 - 10*x - 72, 44*x + 198, 2) (x^2 + 6*x + 36, -12*x + 18, 2) Total time: 3.009 seconds, Total memory usage: 39.47MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 21:08:18 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^2*3^6; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1400); B:=ReducedBasis(V); B; P:=J!B[1]; Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=12); Chabauty(P,11); //Order(Q);Order(P); P+Q;P+2*Q; Output: Magma V2.11-10 Mon Dec 5 2005 21:08:15 on modular [Seed = 1923466176] ------------------------------------- 1 Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 to JacHyp: J given by a rule [no inverse] [ (x^2 - 9/4*x - 27/2, -135/8*x + 27/4, 2) ] 2.367463309945994887046414980850866680181 8.937165595523057578067849122405558415787 6.473890696352274693832677312064632522802 {@ <0, 1, 4, 1> @} (x^2 - 10*x - 72, 44*x + 198, 2) (x^2 + 6*x + 36, -12*x + 18, 2) Total time: 3.049 seconds, Total memory usage: 39.47MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 21:07:23 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^2*3^6; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1400); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,2]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=12); Chabauty(P,11); //Order(Q);Order(P); //2*Q; Output: Magma V2.11-10 Mon Dec 5 2005 21:07:20 on modular [Seed = 1973601296] ------------------------------------- 1 Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 to JacHyp: J given by a rule [no inverse] [ (x^2 - 9/4*x - 27/2, -135/8*x + 27/4, 2) ] 2.367463309945994887046414980850866680181 8.937165595523057578067849122405558415787 6.473890696352274693832677312064632522802 {@ <0, 1, 4, 1> @} Total time: 3.020 seconds, Total memory usage: 39.47MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 21:07:15 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^2*3^6; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1400); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,2]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=12); Chabauty(P,11); //Order(Q);Order(P); //2*Q; Output: Magma V2.11-10 Mon Dec 5 2005 21:07:12 on modular [Seed = 2125057811] ------------------------------------- 1 Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 to JacHyp: J given by a rule [no inverse] [ (x^2 - 9/4*x - 27/2, -135/8*x + 27/4, 2) ] 2.367463309945994887046414980850866680181 8.937165595523057578067849122405558415787 6.473890696352274693832677312064632522802 {@ <0, 1, 4, 1> @} Total time: 3.060 seconds, Total memory usage: 39.47MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 21:06:16 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^2*3^5; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1400); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,2]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=12); Chabauty(P,29); //Order(Q);Order(P); //2*Q; Output: Magma V2.11-10 Mon Dec 5 2005 21:06:14 on modular [Seed = 14135595] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x + 3, -27, 1) ] 1.261435494443878535646756345939700208739 6.719594869637023362623462808509672914228 5.741482503906868232902513820782948719699 {@ <707278, 1, 4, 1> @} Total time: 2.220 seconds, Total memory usage: 39.60MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 21:06:06 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^2*3^5; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1400); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,2]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=12); Chabauty(P,23); //Order(Q);Order(P); //2*Q; Output: Magma V2.11-10 Mon Dec 5 2005 21:06:03 on modular [Seed = 97694771] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x + 3, -27, 1) ] 1.261435494443878535646756345939700208739 6.719594869637023362623462808509672914228 5.741482503906868232902513820782948719699 {@ <279838, 1, 4, 1>, <227084, 1, 4, 1>, <170618, 1, 4, 1>, <70671, 1, 4, 1>, <202525, 1, 4, 1>, <0, 1, 4, 1>, <172723, 1, 4, 1>, <131279, 1, 4, 1>, <278961, 1, 4, 1>, <88671, 1, 4, 1>, <119888, 1, 4, 1> @} Total time: 3.350 seconds, Total memory usage: 39.58MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 21:05:55 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^2*3^5; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1400); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,2]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=12); Chabauty(P,19); //Order(Q);Order(P); //2*Q; Output: Magma V2.11-10 Mon Dec 5 2005 21:05:53 on modular [Seed = 249151282] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x + 3, -27, 1) ] 1.261435494443878535646756345939700208739 6.719594869637023362623462808509672914228 5.741482503906868232902513820782948719699 {@ <130318, 1, 4, 1>, <25679, 1, 4, 1> @} Total time: 2.250 seconds, Total memory usage: 39.72MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 21:05:47 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^2*3^5; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1400); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,2]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=12); Chabauty(P,17); //Order(Q);Order(P); //2*Q; Output: Magma V2.11-10 Mon Dec 5 2005 21:05:45 on modular [Seed = 331653591] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x + 3, -27, 1) ] 1.261435494443878535646756345939700208739 6.719594869637023362623462808509672914228 5.741482503906868232902513820782948719699 {@ <83518, 1, 4, 1>, <68269, 1, 4, 1> @} Total time: 2.089 seconds, Total memory usage: 39.61MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 21:05:34 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^2*3^5; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1400); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,2]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=12); Chabauty(P,13); //Order(Q);Order(P); //2*Q; Output: Magma V2.11-10 Mon Dec 5 2005 21:05:32 on modular [Seed = 348366533] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x + 3, -27, 1) ] 1.261435494443878535646756345939700208739 6.719594869637023362623462808509672914228 5.741482503906868232902513820782948719699 {@ <28558, 1, 4, 1>, <16269, 1, 4, 1> @} Total time: 2.029 seconds, Total memory usage: 39.58MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 21:05:23 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^2*3^5; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1400); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,2]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=12); Chabauty(P,11); //Order(Q);Order(P); //2*Q; Output: Magma V2.11-10 Mon Dec 5 2005 21:05:19 on modular [Seed = 432974843] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x + 3, -27, 1) ] 1.261435494443878535646756345939700208739 6.719594869637023362623462808509672914228 5.741482503906868232902513820782948719699 {@ <14638, 1, 4, 1>, <10148, 1, 4, 1>, <8477, 1, 4, 1>, <7929, 1, 4, 1>, <1577, 1, 4, 1>, <12853, 1, 4, 1>, <3367, 1, 4, 1>, <0, 1, 4, 1>, <1495, 1, 4, 1>, <6129, 1, 4, 1> @} Total time: 3.020 seconds, Total memory usage: 39.62MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 21:03:05 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^2*3^5; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1400); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,2]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=12); //Chabauty(P,19); //Order(Q);Order(P); //2*Q; Output: Magma V2.11-10 Mon Dec 5 2005 21:03:03 on modular [Seed = 586518600] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x + 3, -27, 1) ] 1.261435494443878535646756345939700208739 6.719594869637023362623462808509672914228 5.741482503906868232902513820782948719699 Total time: 1.610 seconds, Total memory usage: 39.61MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 20:47:12 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^2*3^4; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1400); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,2]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=12); Chabauty(P,19); //Order(Q);Order(P); //2*Q; Output: Magma V2.11-10 Mon Dec 5 2005 20:47:06 on modular [Seed = 3457769578] ------------------------------------- 1 Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 to JacHyp: J given by a rule [no inverse] [ (x^2 + 2*x + 12, -4*x - 6, 2) ] 0.333877813949881714276569288505833032214 7.0893735547043793827827401012245389724008 5.00907431146146177197235032950126491659705 {@ <86443, 1, 4, 1>, <0, 1, 4, 1>, <130318, 1, 4, 1>, <45920, 1, 4, 1>, <6, 1, 4, 1> @} Total time: 5.870 seconds, Total memory usage: 39.53MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 20:45:41 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^2*3^4; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,2]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=12); Chabauty(P,19); //Order(Q);Order(P); //2*Q; Output: Magma V2.11-10 Mon Dec 5 2005 20:45:35 on modular [Seed = 3556997845] ------------------------------------- 1 Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 to JacHyp: J given by a rule [no inverse] [ (x^2 + 2*x + 12, -4*x - 6, 2) ] 0.333877813949881714276569288505833032214 7.0893735547043793827827401012245389724008 5.00907431146146177197235032950126491659705 {@ <86443, 1, 4, 1>, <0, 1, 4, 1>, <130318, 1, 4, 1>, <45920, 1, 4, 1>, <6, 1, 4, 1> @} Total time: 5.660 seconds, Total memory usage: 39.53MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 20:44:58 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^2*3^3; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,2]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=12); Chabauty(P,19); //Order(Q);Order(P); //2*Q; Output: Magma V2.11-10 Mon Dec 5 2005 20:44:56 on modular [Seed = 3675026941] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 53/4*x + 61/4, -379/8*x + 567/8, 2) ] 6.7361670782816389366391748079280207656859762715293480883850 4.886740291509942372599127687343020233174 4.276666119016055311042186838219581113494 {@ @} Total time: 1.659 seconds, Total memory usage: 39.39MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 20:44:51 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^2*3^3; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,2]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=12); Chabauty(P,17); //Order(Q);Order(P); //2*Q; Output: Magma V2.11-10 Mon Dec 5 2005 20:44:48 on modular [Seed = 3725160688] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 53/4*x + 61/4, -379/8*x + 567/8, 2) ] 6.7361670782816389366391748079280207656859762715293480883850 4.886740291509942372599127687343020233174 4.276666119016055311042186838219581113494 {@ <45766, 1, 4, 1>, <58628, 1, 4, 1>, <8022, 1, 4, 1>, <65546, 1, 4, 1>, <10803, 1, 4, 1>, <22990, 1, 4, 1>, <52424, 1, 4, 1>, <5012, 1, 4, 1> @} Total time: 2.640 seconds, Total memory usage: 39.38MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 20:44:41 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^2*3^3; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,2]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=12); Chabauty(P,13); //Order(Q);Order(P); //2*Q; Output: Magma V2.11-10 Mon Dec 5 2005 20:44:38 on modular [Seed = 3845278685] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 53/4*x + 61/4, -379/8*x + 567/8, 2) ] 6.7361670782816389366391748079280207656859762715293480883850 4.886740291509942372599127687343020233174 4.276666119016055311042186838219581113494 {@ <14058, 1, 4, 1>, <0, 1, 4, 1>, <24455, 1, 4, 1>, <12913, 1, 4, 1>, <14425, 1, 4, 1> @} Total time: 2.160 seconds, Total memory usage: 39.42MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 20:44:32 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^2*3^3; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,2]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=12); Chabauty(P,11); //Order(Q);Order(P); //2*Q; Output: Magma V2.11-10 Mon Dec 5 2005 20:44:31 on modular [Seed = 3896461097] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 53/4*x + 61/4, -379/8*x + 567/8, 2) ] 6.7361670782816389366391748079280207656859762715293480883850 4.886740291509942372599127687343020233174 4.276666119016055311042186838219581113494 {@ <0, 1, 5, 1> @} Total time: 1.600 seconds, Total memory usage: 39.41MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 20:44:24 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^2*3^3; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,2]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=12); Chabauty(P,7); //Order(Q);Order(P); //2*Q; Output: Magma V2.11-10 Mon Dec 5 2005 20:44:22 on modular [Seed = 4013441063] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 53/4*x + 61/4, -379/8*x + 567/8, 2) ] 6.7361670782816389366391748079280207656859762715293480883850 4.886740291509942372599127687343020233174 4.276666119016055311042186838219581113494 {@ <794, 1, 4, 1>, <1049, 1, 4, 1> @} Total time: 1.710 seconds, Total memory usage: 39.39MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 20:42:34 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^2*3^3; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,2]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=12); //Chabauty(P,41); //Order(Q);Order(P); //2*Q; Output: Magma V2.11-10 Mon Dec 5 2005 20:42:33 on modular [Seed = 4062534981] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 53/4*x + 61/4, -379/8*x + 567/8, 2) ] 6.7361670782816389366391748079280207656859762715293480883850 4.886740291509942372599127687343020233174 4.276666119016055311042186838219581113494 Total time: 1.229 seconds, Total memory usage: 39.38MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 20:42:26 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^2*3^3; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; //TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,2]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=12); //Chabauty(P,41); //Order(Q);Order(P); //2*Q; Output: Magma V2.11-10 Mon Dec 5 2005 20:42:25 on modular [Seed = 4180563526] ------------------------------------- 1 [ (x^2 - 53/4*x + 61/4, -379/8*x + 567/8, 2) ] 6.7361670782816389366391748079280207656859762715293480883850 4.886740291509942372599127687343020233174 4.276666119016055311042186838219581113494 Total time: 1.199 seconds, Total memory usage: 38.67MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 20:35:53 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^2*3^2; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; //TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,2]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=12); Chabauty(P,41); //Order(Q);Order(P); //2*Q; Output: Magma V2.11-10 Mon Dec 5 2005 20:35:48 on modular [Seed = 2354783558] ------------------------------------- 1 [ (x + 2, -2, 1) ] 1.11914558715778900555040229189072826373624994678060472287904 5.275565584207482041996250875230549956259 3.544257926570648850112023346937897310415 {@ <0, 1, 5, 1>, <115856199, 1, 5, 1>, <112103779, 1, 5, 1> @} Total time: 4.559 seconds, Total memory usage: 38.67MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 20:35:40 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^2*3^2; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; //TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,2]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=12); Chabauty(P,37); //Order(Q);Order(P); //2*Q; Output: Magma V2.11-10 Mon Dec 5 2005 20:35:34 on modular [Seed = 2470722123] ------------------------------------- 1 [ (x + 2, -2, 1) ] 1.11914558715778900555040229189072826373624994678060472287904 5.275565584207482041996250875230549956259 3.544257926570648850112023346937897310415 {@ <1165288, 1, 4, 1>, <1589581, 1, 4, 1>, <148749, 1, 4, 1>, <430481, 1, 4, 1>, <1363809, 1, 4, 1>, <0, 1, 4, 1>, <1874159, 1, 4, 1>, <187917, 1, 4, 1>, <564161, 1, 4, 1>, <1289506, 1, 4, 1>, <129135, 1, 4, 1>, <1403053, 1, 4, 1>, <1171182, 1, 4, 1>, <450502, 1, 4, 1>, <1721883, 1, 4, 1>, <369145, 1, 4, 1>, <1802688, 1, 4, 1> @} Total time: 5.910 seconds, Total memory usage: 38.67MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 20:35:25 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^2*3^2; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; //TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,2]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=12); Chabauty(P,31); //Order(Q);Order(P); //2*Q; Output: Magma V2.11-10 Mon Dec 5 2005 20:35:20 on modular [Seed = 2520858480] ------------------------------------- 1 [ (x + 2, -2, 1) ] 1.11914558715778900555040229189072826373624994678060472287904 5.275565584207482041996250875230549956259 3.544257926570648850112023346937897310415 {@ <0, 1, 4, 1>, <923519, 1, 4, 1>, <804103, 1, 4, 1>, <275093, 1, 4, 1>, <65871, 1, 4, 1>, <275737, 1, 4, 1>, <907036, 1, 4, 1>, <222533, 1, 4, 1>, <849946, 1, 4, 1>, <438850, 1, 4, 1>, <665474, 1, 4, 1> @} Total time: 4.589 seconds, Total memory usage: 38.67MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 20:35:09 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^2*3^2; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,2]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=12); Chabauty(P,23); //Order(Q);Order(P); //2*Q; Output: Magma V2.11-10 Mon Dec 5 2005 20:35:04 on modular [Seed = 2638885995] ------------------------------------- 1 Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 to JacHyp: J given by a rule [no inverse] [ (x + 2, -2, 1) ] 1.11914558715778900555040229189072826373624994678060472287904 5.275565584207482041996250875230549956259 3.544257926570648850112023346937897310415 {@ <279839, 1, 4, 1>, <183996, 1, 4, 1>, <156556, 1, 4, 1>, <0, 1, 4, 1>, <131380, 1, 4, 1>, <183336, 1, 4, 1>, <234426, 1, 4, 1>, <70852, 1, 4, 1>, <175397, 1, 4, 1>, <255331, 1, 4, 1> @} Total time: 4.879 seconds, Total memory usage: 39.47MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 20:34:53 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^2*3^2; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,2]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=12); Chabauty(P,29); //Order(Q);Order(P); //2*Q; Output: Magma V2.11-10 Mon Dec 5 2005 20:34:49 on modular [Seed = 2692155687] ------------------------------------- 1 Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 to JacHyp: J given by a rule [no inverse] [ (x + 2, -2, 1) ] 1.11914558715778900555040229189072826373624994678060472287904 5.275565584207482041996250875230549956259 3.544257926570648850112023346937897310415 {@ <481069, 1, 4, 1>, <707279, 1, 4, 1>, <0, 1, 4, 1>, <480287, 1, 4, 1>, <176530, 1, 4, 1>, <148044, 1, 4, 1>, <95583, 1, 4, 1> @} Total time: 4.389 seconds, Total memory usage: 39.47MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 20:34:39 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^2*3^2; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,2]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=12); Chabauty(P,19); //Order(Q);Order(P); //2*Q; Output: Magma V2.11-10 Mon Dec 5 2005 20:34:35 on modular [Seed = 2809138738] ------------------------------------- 1 Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 to JacHyp: J given by a rule [no inverse] [ (x + 2, -2, 1) ] 1.11914558715778900555040229189072826373624994678060472287904 5.275565584207482041996250875230549956259 3.544257926570648850112023346937897310415 {@ <6554, 1, 4, 1>, <130319, 1, 4, 1>, <0, 1, 4, 1>, <2044, 1, 4, 1>, <25657, 1, 4, 1> @} Total time: 4.250 seconds, Total memory usage: 39.47MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 20:34:27 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^2*3^2; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,2]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=12); Chabauty(P,17); //Order(Q);Order(P); //2*Q; Output: Magma V2.11-10 Mon Dec 5 2005 20:34:23 on modular [Seed = 2860319499] ------------------------------------- 1 Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 to JacHyp: J given by a rule [no inverse] [ (x + 2, -2, 1) ] 1.11914558715778900555040229189072826373624994678060472287904 5.275565584207482041996250875230549956259 3.544257926570648850112023346937897310415 {@ <279, 1, 4, 1>, <83519, 1, 4, 1>, <47775, 1, 4, 1>, <0, 1, 4, 1>, <38178, 1, 4, 1>, <31851, 1, 4, 1>, <64072, 1, 4, 1>, <3372, 1, 4, 1> @} Total time: 4.450 seconds, Total memory usage: 39.47MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 20:34:14 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^2*3^2; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,2]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=12); Chabauty(P,13); //Order(Q);Order(P); //2*Q; Output: Magma V2.11-10 Mon Dec 5 2005 20:34:10 on modular [Seed = 2976258077] ------------------------------------- 1 Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 to JacHyp: J given by a rule [no inverse] [ (x + 2, -2, 1) ] 1.11914558715778900555040229189072826373624994678060472287904 5.275565584207482041996250875230549956259 3.544257926570648850112023346937897310415 {@ <19662, 1, 4, 1>, <0, 1, 4, 1>, <21569, 1, 4, 1>, <28559, 1, 4, 1>, <14650, 1, 4, 1> @} Total time: 4.080 seconds, Total memory usage: 39.47MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 20:34:00 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^2*3^2; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,2]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=12); Chabauty(P,11); //Order(Q);Order(P); //2*Q; Output: Magma V2.11-10 Mon Dec 5 2005 20:33:56 on modular [Seed = 3026394382] ------------------------------------- 1 Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 to JacHyp: J given by a rule [no inverse] [ (x + 2, -2, 1) ] 1.11914558715778900555040229189072826373624994678060472287904 5.275565584207482041996250875230549956259 3.544257926570648850112023346937897310415 {@ <14639, 1, 4, 1>, <6873, 1, 4, 1>, <0, 1, 4, 1> @} Total time: 3.629 seconds, Total memory usage: 39.47MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 20:33:45 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^2*3^2; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,2]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=12); //Chabauty(P,11); //Order(Q);Order(P); //2*Q; Output: Magma V2.11-10 Mon Dec 5 2005 20:33:41 on modular [Seed = 3144422128] ------------------------------------- 1 Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 to JacHyp: J given by a rule [no inverse] [ (x + 2, -2, 1) ] 1.11914558715778900555040229189072826373624994678060472287904 5.275565584207482041996250875230549956259 3.544257926570648850112023346937897310415 Total time: 3.259 seconds, Total memory usage: 39.47MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 20:32:19 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^2*3^2; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,2]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=12); //Chabauty(P,19); //Order(Q);Order(P); //2*Q; Output: Magma V2.11-10 Mon Dec 5 2005 20:32:15 on modular [Seed = 3194558367] ------------------------------------- 1 Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 to JacHyp: J given by a rule [no inverse] [ (x + 2, -2, 1) ] 1.11914558715778900555040229189072826373624994678060472287904 5.275565584207482041996250875230549956259 3.544257926570648850112023346937897310415 Total time: 3.250 seconds, Total memory usage: 39.47MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 20:30:42 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^2*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,2]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=12); Chabauty(P,19); //Order(Q);Order(P); //2*Q; Output: Magma V2.11-10 Mon Dec 5 2005 20:30:38 on modular [Seed = 1150545604] ------------------------------------- 1 Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 to JacHyp: J given by a rule [no inverse] [ (x^2 - 2*x, 2*x + 2, 2) ] 0.8210515689040369205164469676161908314449811799266926438375 3.514883163091594117716474749077096897972 2.0794415416798359282516963643745297042213 {@ <2, 1, 4, 1>, <0, 1, 4, 1> @} Total time: 3.540 seconds, Total memory usage: 39.55MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 20:30:31 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^2*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,2]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=12); Chabauty(P,17); //Order(Q);Order(P); //2*Q; Output: Magma V2.11-10 Mon Dec 5 2005 20:30:26 on modular [Seed = 1268572615] ------------------------------------- 1 Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 to JacHyp: J given by a rule [no inverse] [ (x^2 - 2*x, 2*x + 2, 2) ] 0.8210515689040369205164469676161908314449811799266926438375 3.514883163091594117716474749077096897972 2.0794415416798359282516963643745297042213 {@ <37142, 1, 4, 1>, <7232, 1, 4, 1>, <16290, 1, 4, 1>, <2, 1, 4, 1>, <0, 1, 4, 1>, <63333, 1, 4, 1>, <12209, 1, 4, 1> @} Total time: 4.200 seconds, Total memory usage: 39.55MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 20:30:20 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^2*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,2]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=12); Chabauty(P,13); //Order(Q);Order(P); //2*Q; Output: Magma V2.11-10 Mon Dec 5 2005 20:30:16 on modular [Seed = 1318708440] ------------------------------------- 1 Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 to JacHyp: J given by a rule [no inverse] [ (x^2 - 2*x, 2*x + 2, 2) ] 0.8210515689040369205164469676161908314449811799266926438375 3.514883163091594117716474749077096897972 2.0794415416798359282516963643745297042213 {@ <21536, 1, 4, 1>, <2, 1, 4, 1>, <2144, 1, 4, 1>, <21325, 1, 4, 1>, <10144, 1, 4, 1>, <0, 1, 4, 1> @} Total time: 3.790 seconds, Total memory usage: 39.55MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 20:29:56 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^2*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,2]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=12); Chabauty(P,13); //Order(Q);Order(P); 2*Q; Output: Magma V2.11-10 Mon Dec 5 2005 20:29:52 on modular [Seed = 1434648571] ------------------------------------- 1 Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 to JacHyp: J given by a rule [no inverse] [ (x^2 - 2*x, 2*x + 2, 2) ] 0.8210515689040369205164469676161908314449811799266926438375 3.514883163091594117716474749077096897972 2.0794415416798359282516963643745297042213 {@ <21536, 1, 4, 1>, <2, 1, 4, 1>, <2144, 1, 4, 1>, <21325, 1, 4, 1>, <10144, 1, 4, 1>, <0, 1, 4, 1> @} >> 2*Q;; ^ User error: Identifier 'Q' has not been declared or assigned Total time: 3.790 seconds, Total memory usage: 39.55MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 20:29:46 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^2*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,2]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=12); Chabauty(P,11); //Order(Q);Order(P); 2*Q; Output: Magma V2.11-10 Mon Dec 5 2005 20:29:41 on modular [Seed = 1485828835] ------------------------------------- 1 Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 to JacHyp: J given by a rule [no inverse] [ (x^2 - 2*x, 2*x + 2, 2) ] 0.8210515689040369205164469676161908314449811799266926438375 3.514883163091594117716474749077096897972 2.0794415416798359282516963643745297042213 {@ <2, 1, 4, 1>, <7806, 1, 4, 1>, <14167, 1, 4, 1>, <12775, 1, 4, 1>, <8028, 1, 4, 1>, <12820, 1, 4, 1>, <10090, 1, 4, 1>, <8185, 1, 4, 1>, <0, 1, 4, 1>, <9523, 1, 4, 1> @} >> 2*Q;; ^ User error: Identifier 'Q' has not been declared or assigned Total time: 4.280 seconds, Total memory usage: 39.55MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 20:29:31 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^2*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,2]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=12); Chabauty(P,7); //Order(Q);Order(P); 2*Q; Output: Magma V2.11-10 Mon Dec 5 2005 20:29:25 on modular [Seed = 1602811156] ------------------------------------- 1 Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 to JacHyp: J given by a rule [no inverse] [ (x^2 - 2*x, 2*x + 2, 2) ] 0.8210515689040369205164469676161908314449811799266926438375 3.514883163091594117716474749077096897972 2.0794415416798359282516963643745297042213 {@ <2, 1, 4, 1>, <623, 1, 4, 1>, <0, 1, 4, 1>, <416, 1, 4, 1>, <0, 1, 0, 6> @} >> 2*Q;; ^ User error: Identifier 'Q' has not been declared or assigned Total time: 5.889 seconds, Total memory usage: 39.55MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 20:26:18 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^2*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; Q:=J![x,2]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=12); //Chabauty(P,11); //Order(Q);Order(P); 2*Q; Output: Magma V2.11-10 Mon Dec 5 2005 20:26:15 on modular [Seed = 1706217196] ------------------------------------- 1 Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 to JacHyp: J given by a rule [no inverse] [ (x^2 - 2*x, 2*x + 2, 2) ] 0.8210515689040369205164469676161908314449811799266926438375 3.514883163091594117716474749077096897972 2.0794415416798359282516963643745297042213 (x^2, 2, 2) Total time: 2.930 seconds, Total memory usage: 39.55MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 20:25:40 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^2*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-2,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); RationalPoints(J:Bound:=12); Chabauty(P,11); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 20:25:36 on modular [Seed = 1757397958] ------------------------------------- 1 Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 to JacHyp: J given by a rule [no inverse] [ (x^2 - 2*x, 2*x + 2, 2) ] 0.8210515689040369205164469676161908314449811799266926438375 3.514883163091594117716474749077096897972 2.0794415416798359282516963643745297042213 {@ (1, 0, 0), (x^2, 2, 2), (x^2, -2, 2), (x, 2, 1), (x, -2, 1), (x - 2, 6, 1), (x - 2, -6, 1), (x^2 + x + 2, x, 2), (x^2 + x + 2, -x, 2), (x^2 - 2*x - 4, 4*x + 6, 2), (x^2 - 2*x - 4, -4*x - 6, 2), (x^2 - 2*x, 4*x - 2, 2), (x^2 - 2*x, -4*x + 2, 2), (x^2 - 2*x, 2*x + 2, 2), (x^2 - 2*x, -2*x - 2, 2), (x^2 + 3/4*x - 3/4, 9/8*x + 11/8, 2), (x^2 + 3/4*x - 3/4, -9/8*x - 11/8, 2) @} {@ <2, 1, 4, 1>, <7806, 1, 4, 1>, <14167, 1, 4, 1>, <12775, 1, 4, 1>, <8028, 1, 4, 1>, <12820, 1, 4, 1>, <10090, 1, 4, 1>, <8185, 1, 4, 1>, <0, 1, 4, 1>, <9523, 1, 4, 1> @} Total time: 4.299 seconds, Total memory usage: 39.55MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 20:24:47 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^2*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-2,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=12); Chabauty(P,11); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 20:24:42 on modular [Seed = 1874379787] ------------------------------------- 1 Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 to JacHyp: J given by a rule [no inverse] [ (x^2 - 2*x, 2*x + 2, 2) ] 0.8210515689040369205164469676161908314449811799266926438375 3.514883163091594117716474749077096897972 2.0794415416798359282516963643745297042213 {@ <2, 1, 4, 1>, <7806, 1, 4, 1>, <14167, 1, 4, 1>, <12775, 1, 4, 1>, <8028, 1, 4, 1>, <12820, 1, 4, 1>, <10090, 1, 4, 1>, <8185, 1, 4, 1>, <0, 1, 4, 1>, <9523, 1, 4, 1> @} Total time: 4.240 seconds, Total memory usage: 39.55MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 20:21:43 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^1*3^7; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-2,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=12); Chabauty(P,11); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 20:21:41 on modular [Seed = 2007028716] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 9*x + 27, -9*x, 2) ] 2.1066332234882459981534596345198271520181 7.855157844852767304923448366250813574110 6.744200768424384281818019389040865280504 {@ @} Total time: 1.679 seconds, Total memory usage: 39.25MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 20:20:59 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^1*3^7; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=5000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-2,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=12); //Chabauty(P,19); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 20:20:54 on modular [Seed = 2058214962] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 9*x + 27, -9*x, 2) ] 2.1066332234882459981534596345198271520181 7.855157844852767304923448366250813574110 6.744200768424384281818019389040865280504 Total time: 5.450 seconds, Total memory usage: 40.47MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 20:19:38 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^1*3^7; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-2,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=12); //Chabauty(P,19); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 20:19:37 on modular [Seed = 30844182] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 9*x + 27, -9*x, 2) ] 2.1066332234882459981534596345198271520181 7.855157844852767304923448366250813574110 6.744200768424384281818019389040865280504 Total time: 1.459 seconds, Total memory usage: 39.47MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 20:19:35 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^1*3^7; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-2,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=12); //Chabauty(P,19); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 20:19:34 on modular [Seed = 80977948] ------------------------------------- TwoSelmerGroupData( J: Jacobian of Hyperelliptic Curve defined by y^2 = x^5 + 4374 ... ) TwoSelmerWork( C: Hyperelliptic Curve defined by y^2 = x^5 + 4374 over Rationa..., BoundType: Default, Bound: 0, UseUnits: true, Points: {}, Fields: {} ) LocalImageOdd( f: x^5 + 4374, p: 2, m1: Mapping from: Univariate Polynomial Ring in x over Rational ..., m2: Mapping from: Univariate Polynomial Ring in x over Rational ..., v: 6, Points: {} ) try( ) try( d: 2 ) SquarefreeRoots( F: $.1^4 + O(2^50)*$.1^3 - (462313490571*2^3 + O(2^53))*$.1^2 +... ) In file "/usr/local/magma/package/Geometry/Arith/sqfroots.m", line 84, column 15: >> return Roots(F:IsSquarefree); ^ Runtime error in 'Roots': Insufficient precision to find roots >> r:=TwoSelmerGroupData(J);r; ^ User error: Identifier 'r' has not been declared or assigned Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 9*x + 27, -9*x, 2) ] 2.1066332234882459981534596345198271520181 7.855157844852767304923448366250813574110 6.744200768424384281818019389040865280504 Total time: 1.449 seconds, Total memory usage: 39.66MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 20:19:32 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^1*3^7; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-2,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=12); //Chabauty(P,19); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 20:19:30 on modular [Seed = 199012119] ------------------------------------- TwoSelmerGroupData( J: Jacobian of Hyperelliptic Curve defined by y^2 = x^5 + 4374 ... ) TwoSelmerWork( C: Hyperelliptic Curve defined by y^2 = x^5 + 4374 over Rationa..., BoundType: Default, Bound: 0, UseUnits: true, Points: {}, Fields: {} ) LocalImageOdd( f: x^5 + 4374, p: 2, m1: Mapping from: Univariate Polynomial Ring in x over Rational ..., m2: Mapping from: Univariate Polynomial Ring in x over Rational ..., v: 6, Points: {} ) try( ) try( d: 2 ) SquarefreePart( F: $.1^4 + O(2^50)*$.1^3 + (4949974972277*2^3 + O(2^53))*$.1^2 ... ) In file "/usr/local/magma/package/Geometry/Arith/loclib.m", line 120, column 5: >> assert R eq 0; ^ Runtime error in assert: Assertion failed >> r:=TwoSelmerGroupData(J);r; ^ User error: Identifier 'r' has not been declared or assigned Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 9*x + 27, -9*x, 2) ] 2.1066332234882459981534596345198271520181 7.855157844852767304923448366250813574110 6.744200768424384281818019389040865280504 Total time: 1.820 seconds, Total memory usage: 40.50MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 20:19:28 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^1*3^7; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-2,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=12); //Chabauty(P,19); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 20:19:27 on modular [Seed = 249145903] ------------------------------------- TwoSelmerGroupData( J: Jacobian of Hyperelliptic Curve defined by y^2 = x^5 + 4374 ... ) TwoSelmerWork( C: Hyperelliptic Curve defined by y^2 = x^5 + 4374 over Rationa..., BoundType: Default, Bound: 0, UseUnits: true, Points: {}, Fields: {} ) LocalImageOdd( f: x^5 + 4374, p: 2, m1: Mapping from: Univariate Polynomial Ring in x over Rational ..., m2: Mapping from: Univariate Polynomial Ring in x over Rational ..., v: 6, Points: {} ) try( ) try( d: 2 ) SquarefreeRoots( F: $.1^4 + O(2^50)*$.1^3 + (1407999997813*2^3 + O(2^53))*$.1^2 ... ) In file "/usr/local/magma/package/Geometry/Arith/sqfroots.m", line 84, column 15: >> return Roots(F:IsSquarefree); ^ Runtime error in 'Roots': Insufficient precision to find roots >> r:=TwoSelmerGroupData(J);r; ^ User error: Identifier 'r' has not been declared or assigned Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 9*x + 27, -9*x, 2) ] 2.1066332234882459981534596345198271520181 7.855157844852767304923448366250813574110 6.744200768424384281818019389040865280504 Total time: 1.280 seconds, Total memory usage: 39.39MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 20:19:25 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^1*3^7; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-2,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=12); //Chabauty(P,19); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 20:19:24 on modular [Seed = 365080887] ------------------------------------- TwoSelmerGroupData( J: Jacobian of Hyperelliptic Curve defined by y^2 = x^5 + 4374 ... ) TwoSelmerWork( C: Hyperelliptic Curve defined by y^2 = x^5 + 4374 over Rationa..., BoundType: Default, Bound: 0, UseUnits: true, Points: {}, Fields: {} ) LocalImageOdd( f: x^5 + 4374, p: 2, m1: Mapping from: Univariate Polynomial Ring in x over Rational ..., m2: Mapping from: Univariate Polynomial Ring in x over Rational ..., v: 6, Points: {} ) try( ) try( d: 2 ) SquarefreeRoots( F: $.1^4 + O(2^50)*$.1^3 + (2845492574069*2^3 + O(2^53))*$.1^2 ... ) In file "/usr/local/magma/package/Geometry/Arith/sqfroots.m", line 84, column 15: >> return Roots(F:IsSquarefree); ^ Runtime error in 'Roots': Insufficient precision to find roots >> r:=TwoSelmerGroupData(J);r; ^ User error: Identifier 'r' has not been declared or assigned Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 9*x + 27, -9*x, 2) ] 2.1066332234882459981534596345198271520181 7.855157844852767304923448366250813574110 6.744200768424384281818019389040865280504 Total time: 1.129 seconds, Total memory usage: 39.12MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 20:19:21 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^1*3^7; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-2,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=12); //Chabauty(P,19); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 20:19:20 on modular [Seed = 416267326] ------------------------------------- TwoSelmerGroupData( J: Jacobian of Hyperelliptic Curve defined by y^2 = x^5 + 4374 ... ) TwoSelmerWork( C: Hyperelliptic Curve defined by y^2 = x^5 + 4374 over Rationa..., BoundType: Default, Bound: 0, UseUnits: true, Points: {}, Fields: {} ) LocalImageOdd( f: x^5 + 4374, p: 2, m1: Mapping from: Univariate Polynomial Ring in x over Rational ..., m2: Mapping from: Univariate Polynomial Ring in x over Rational ..., v: 6, Points: {} ) try( ) try( d: 2 ) SquarefreeRoots( F: $.1^4 + O(2^50)*$.1^3 + (588297271157*2^3 + O(2^53))*$.1^2 +... ) In file "/usr/local/magma/package/Geometry/Arith/sqfroots.m", line 84, column 15: >> return Roots(F:IsSquarefree); ^ Runtime error in 'Roots': Insufficient precision to find roots >> r:=TwoSelmerGroupData(J);r; ^ User error: Identifier 'r' has not been declared or assigned Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 9*x + 27, -9*x, 2) ] 2.1066332234882459981534596345198271520181 7.855157844852767304923448366250813574110 6.744200768424384281818019389040865280504 Total time: 1.159 seconds, Total memory usage: 39.16MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 20:18:13 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^1*3^5; C:=HyperellipticCurve(f); J:=Jacobian(C); //r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-2,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=12); Chabauty(P,19); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 20:18:11 on modular [Seed = 533248754] ------------------------------------- Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x - 3, 27, 1) ] 1.0964087875758887506838932751688111624269 6.368926785599472960071306705182515591862 5.279384383533571359957692406477497674322 {@ <3, 1, 4, 1> @} Total time: 1.530 seconds, Total memory usage: 38.12MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 20:18:06 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^1*3^5; C:=HyperellipticCurve(f); J:=Jacobian(C); //r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-2,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=12); Chabauty(P,17); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 20:18:04 on modular [Seed = 586516432] ------------------------------------- Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x - 3, 27, 1) ] 1.0964087875758887506838932751688111624269 6.368926785599472960071306705182515591862 5.279384383533571359957692406477497674322 {@ <3, 1, 4, 1>, <63266, 1, 4, 1>, <54395, 1, 4, 1>, <77867, 1, 4, 1>, <46987, 1, 4, 1>, <77545, 1, 4, 1>, <54827, 1, 4, 1>, <16840, 1, 4, 1> @} Total time: 2.410 seconds, Total memory usage: 38.12MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 20:17:56 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^1*3^5; C:=HyperellipticCurve(f); J:=Jacobian(C); //r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-2,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=12); Chabauty(P,13); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 20:17:54 on modular [Seed = 704548053] ------------------------------------- Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x - 3, 27, 1) ] 1.0964087875758887506838932751688111624269 6.368926785599472960071306705182515591862 5.279384383533571359957692406477497674322 {@ <3, 1, 4, 1>, <13181, 1, 4, 1>, <16533, 1, 4, 1>, <20252, 1, 4, 1>, <24913, 1, 4, 1>, <18857, 1, 4, 1> @} Total time: 1.940 seconds, Total memory usage: 38.12MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 20:17:48 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^1*3^5; C:=HyperellipticCurve(f); J:=Jacobian(C); //r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-2,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=12); Chabauty(P,11); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 20:17:45 on modular [Seed = 754682283] ------------------------------------- Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x - 3, 27, 1) ] 1.0964087875758887506838932751688111624269 6.368926785599472960071306705182515591862 5.279384383533571359957692406477497674322 {@ <3, 1, 4, 1>, <875, 1, 4, 1>, <9660, 1, 4, 1>, <2248, 1, 4, 1>, <6456, 1, 4, 1>, <5430, 1, 4, 1>, <13630, 1, 4, 1>, <2351, 1, 4, 1>, <636, 1, 4, 1> @} Total time: 2.359 seconds, Total memory usage: 38.12MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 20:17:41 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^1*3^5; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-2,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=12); Chabauty(P,11); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 20:17:38 on modular [Seed = 870618815] ------------------------------------- TwoSelmerGroupData( J: Jacobian of Hyperelliptic Curve defined by y^2 = x^5 + 486 o... ) TwoSelmerWork( C: Hyperelliptic Curve defined by y^2 = x^5 + 486 over Rational..., BoundType: Default, Bound: 0, UseUnits: true, Points: {}, Fields: {} ) LocalImageOdd( f: x^5 + 486, p: 2, m1: Mapping from: Univariate Polynomial Ring in x over Rational ..., m2: Mapping from: Univariate Polynomial Ring in x over Rational ..., v: 6, Points: {} ) try( ) try( d: 2 ) SquarefreeRoots( F: $.1^4 + O(2^50)*$.1^3 + (12817354456845*2^3 + O(2^53))*$.1^2... ) In file "/usr/local/magma/package/Geometry/Arith/sqfroots.m", line 84, column 15: >> return Roots(F:IsSquarefree); ^ Runtime error in 'Roots': Insufficient precision to find roots >> r:=TwoSelmerGroupData(J);r; ^ User error: Identifier 'r' has not been declared or assigned Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x - 3, 27, 1) ] 1.0964087875758887506838932751688111624269 6.368926785599472960071306705182515591862 5.279384383533571359957692406477497674322 {@ <3, 1, 4, 1>, <875, 1, 4, 1>, <9660, 1, 4, 1>, <2248, 1, 4, 1>, <6456, 1, 4, 1>, <5430, 1, 4, 1>, <13630, 1, 4, 1>, <2351, 1, 4, 1>, <636, 1, 4, 1> @} Total time: 2.810 seconds, Total memory usage: 39.43MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 20:17:26 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^1*3^5; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-2,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=12); Chabauty(P,7); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 20:17:24 on modular [Seed = 920753068] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x - 3, 27, 1) ] 1.0964087875758887506838932751688111624269 6.368926785599472960071306705182515591862 5.279384383533571359957692406477497674322 {@ <3, 1, 4, 1>, <2369, 1, 4, 1> @} Total time: 1.860 seconds, Total memory usage: 39.31MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 20:15:20 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^1*3^5; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-2,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=12); Chabauty(P,17); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 20:15:18 on modular [Seed = 3222758873] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x - 3, 27, 1) ] 1.0964087875758887506838932751688111624269 6.368926785599472960071306705182515591862 5.279384383533571359957692406477497674322 {@ <3, 1, 4, 1>, <63266, 1, 4, 1>, <54395, 1, 4, 1>, <77867, 1, 4, 1>, <46987, 1, 4, 1>, <77545, 1, 4, 1>, <54827, 1, 4, 1>, <16840, 1, 4, 1> @} Total time: 2.870 seconds, Total memory usage: 39.23MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 20:15:14 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^1*3^5; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-2,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=12); Chabauty(P,17); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 20:15:12 on modular [Seed = 3306315994] ------------------------------------- TwoSelmerGroupData( J: Jacobian of Hyperelliptic Curve defined by y^2 = x^5 + 486 o... ) TwoSelmerWork( C: Hyperelliptic Curve defined by y^2 = x^5 + 486 over Rational..., BoundType: Default, Bound: 0, UseUnits: true, Points: {}, Fields: {} ) LocalImageOdd( f: x^5 + 486, p: 2, m1: Mapping from: Univariate Polynomial Ring in x over Rational ..., m2: Mapping from: Univariate Polynomial Ring in x over Rational ..., v: 6, Points: {} ) try( ) try( d: 2 ) SquarefreeRoots( F: $.1^4 + O(2^50)*$.1^3 - (34471543027*2^3 + O(2^53))*$.1^2 + ... ) In file "/usr/local/magma/package/Geometry/Arith/sqfroots.m", line 84, column 15: >> return Roots(F:IsSquarefree); ^ Runtime error in 'Roots': Insufficient precision to find roots >> r:=TwoSelmerGroupData(J);r; ^ User error: Identifier 'r' has not been declared or assigned Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x - 3, 27, 1) ] 1.0964087875758887506838932751688111624269 6.368926785599472960071306705182515591862 5.279384383533571359957692406477497674322 {@ <3, 1, 4, 1>, <63266, 1, 4, 1>, <54395, 1, 4, 1>, <77867, 1, 4, 1>, <46987, 1, 4, 1>, <77545, 1, 4, 1>, <54827, 1, 4, 1>, <16840, 1, 4, 1> @} Total time: 2.680 seconds, Total memory usage: 39.11MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 20:15:02 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^1*3^5; C:=HyperellipticCurve(f); J:=Jacobian(C); //r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-2,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=12); Chabauty(P,17); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 20:15:00 on modular [Seed = 3390926817] ------------------------------------- Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x - 3, 27, 1) ] 1.0964087875758887506838932751688111624269 6.368926785599472960071306705182515591862 5.279384383533571359957692406477497674322 {@ <3, 1, 4, 1>, <63266, 1, 4, 1>, <54395, 1, 4, 1>, <77867, 1, 4, 1>, <46987, 1, 4, 1>, <77545, 1, 4, 1>, <54827, 1, 4, 1>, <16840, 1, 4, 1> @} Total time: 2.410 seconds, Total memory usage: 38.12MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 20:06:25 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^1*3^4; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-2,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=12); Chabauty(P,13); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 20:06:23 on modular [Seed = 3741874415] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 + 3/4*x - 9/4, -3/8*x - 99/8, 2) ] 3.10929166505257268510955302447913689669881188175885053195997 5.0871519665718709241496748503892215814574 4.546976191088164899027528915195813871219 {@ <2090, 1, 4, 1> @} Total time: 1.740 seconds, Total memory usage: 39.24MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 20:06:21 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^1*3^4; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-2,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=12); Chabauty(P,13); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 20:06:19 on modular [Seed = 3828564944] ------------------------------------- TwoSelmerGroupData( J: Jacobian of Hyperelliptic Curve defined by y^2 = x^5 + 162 o... ) TwoSelmerWork( C: Hyperelliptic Curve defined by y^2 = x^5 + 162 over Rational..., BoundType: Default, Bound: 0, UseUnits: true, Points: {}, Fields: {} ) LocalImageOdd( f: x^5 + 162, p: 2, m1: Mapping from: Univariate Polynomial Ring in x over Rational ..., m2: Mapping from: Univariate Polynomial Ring in x over Rational ..., v: 6, Points: {} ) try( ) try( d: 2 ) SquarefreeRoots( F: $.1^4 + O(2^50)*$.1^3 + (3086848360367*2^3 + O(2^53))*$.1^2 ... ) In file "/usr/local/magma/package/Geometry/Arith/sqfroots.m", line 84, column 15: >> return Roots(F:IsSquarefree); ^ Runtime error in 'Roots': Insufficient precision to find roots >> r:=TwoSelmerGroupData(J);r; ^ User error: Identifier 'r' has not been declared or assigned Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 + 3/4*x - 9/4, -3/8*x - 99/8, 2) ] 3.10929166505257268510955302447913689669881188175885053195997 5.0871519665718709241496748503892215814574 4.546976191088164899027528915195813871219 {@ <2090, 1, 4, 1> @} Total time: 2.020 seconds, Total memory usage: 40.38MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 20:06:12 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^1*3^4; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-2,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=12); Chabauty(P,13); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 20:06:10 on modular [Seed = 3913174740] ------------------------------------- TwoSelmerGroupData( J: Jacobian of Hyperelliptic Curve defined by y^2 = x^5 + 162 o... ) TwoSelmerWork( C: Hyperelliptic Curve defined by y^2 = x^5 + 162 over Rational..., BoundType: Default, Bound: 0, UseUnits: true, Points: {}, Fields: {} ) LocalImageOdd( f: x^5 + 162, p: 2, m1: Mapping from: Univariate Polynomial Ring in x over Rational ..., m2: Mapping from: Univariate Polynomial Ring in x over Rational ..., v: 6, Points: {} ) try( ) try( d: 2 ) SquarefreeRoots( F: $.1^4 + O(2^50)*$.1^3 + (6289834180527*2^3 + O(2^53))*$.1^2 ... ) In file "/usr/local/magma/package/Geometry/Arith/sqfroots.m", line 84, column 15: >> return Roots(F:IsSquarefree); ^ Runtime error in 'Roots': Insufficient precision to find roots >> r:=TwoSelmerGroupData(J);r; ^ User error: Identifier 'r' has not been declared or assigned Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 + 3/4*x - 9/4, -3/8*x - 99/8, 2) ] 3.10929166505257268510955302447913689669881188175885053195997 5.0871519665718709241496748503892215814574 4.546976191088164899027528915195813871219 {@ <2090, 1, 4, 1> @} Total time: 1.700 seconds, Total memory usage: 39.49MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 19:57:37 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^0*3^8; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-2,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=12); Chabauty(P,17); //Order(Q);Order(P); //2*P; Output: ** WARNING: Computation time exceeded 20 seconds, so computation was terminated after 20 seconds. ** Magma V2.11-10 Mon Dec 5 2005 19:57:17 on modular [Seed = 2992967776] ------------------------------------- 1 Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 to JacHyp: J given by a rule [no inverse] Errors: /bin/sh: line 1: 31885 Alarm clock nice -n 19 /usr/local/bin/magma '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 19:56:57 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^0*3^8; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-2,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=12); Chabauty(P,17); //Order(Q);Order(P); //2*P; Output: ** WARNING: Computation time exceeded 20 seconds, so computation was terminated after 20 seconds. ** Magma V2.11-10 Mon Dec 5 2005 19:56:36 on modular [Seed = 3094288963] ------------------------------------- 1 Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 to JacHyp: J given by a rule [no inverse] Errors: /bin/sh: line 1: 31879 Alarm clock nice -n 19 /usr/local/bin/magma '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 19:54:04 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^0*3^5; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-2,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=12); Chabauty(P,11); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 19:54:02 on modular [Seed = 3177848744] ------------------------------------- 1 Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 to JacHyp: J given by a rule [no inverse] [ (x^2 + 3*x + 9, -3*x + 9, 2) ] 2.26560762534307442838198741030006503529825419756590246025749 6.02129893666715068547855842318402917446829 4.817286263160274487012870992172046628945 {@ <0, 1, 4, 1> @} Total time: 1.929 seconds, Total memory usage: 39.68MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 19:53:16 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^0*3^2; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-2,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=12); Chabauty(P,11); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 19:53:13 on modular [Seed = 1117118852] ------------------------------------- 1 Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 to JacHyp: J given by a rule [no inverse] [ (x^2 - 2, -2*x - 1, 2) ] 1.5626377849846081444970869663211590195034831524679039584881 4.1571752748544702272869038549736433015765 2.620061685824055104222380518326995219648 {@ <0, 1, 5, 1> @} Total time: 2.779 seconds, Total memory usage: 39.48MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 19:51:40 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^0*3^2; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-2,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=12); //Chabauty(P,19); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 19:51:37 on modular [Seed = 1251863020] ------------------------------------- 1 Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/5 Defined on 1 generator Relations: 5*P[1] = 0 to JacHyp: J given by a rule [no inverse] [ (x^2 - 2, -2*x - 1, 2) ] 1.5626377849846081444970869663211590195034831524679039584881 4.1571752748544702272869038549736433015765 2.620061685824055104222380518326995219648 Total time: 2.629 seconds, Total memory usage: 39.48MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 19:50:42 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^0*3^1; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-2,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=12); Chabauty(P,19); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 19:50:40 on modular [Seed = 1335421968] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x - 1, 2, 1) ] 1.21306285522647802452835275718114319003158614249346456250448 2.0948660038325433522600978285640988496059 1.887653493378648643292217027045311416545 {@ <1, 1, 4, 1> @} Total time: 1.790 seconds, Total memory usage: 39.37MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 19:50:33 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^0*3^1; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-2,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=12); Chabauty(P,17); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 19:50:31 on modular [Seed = 1417934652] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x - 1, 2, 1) ] 1.21306285522647802452835275718114319003158614249346456250448 2.0948660038325433522600978285640988496059 1.887653493378648643292217027045311416545 {@ <1, 1, 4, 1>, <27726, 1, 4, 1>, <4450, 1, 4, 1>, <45574, 1, 4, 1>, <54682, 1, 4, 1>, <3462, 1, 4, 1>, <39256, 1, 4, 1>, <67968, 1, 4, 1> @} Total time: 2.529 seconds, Total memory usage: 39.38MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 19:50:24 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^0*3^1; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-2,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=12); Chabauty(P,13); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 19:50:21 on modular [Seed = 1502542376] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x - 1, 2, 1) ] 1.21306285522647802452835275718114319003158614249346456250448 2.0948660038325433522600978285640988496059 1.887653493378648643292217027045311416545 {@ <1, 1, 4, 1>, <419, 1, 4, 1>, <4227, 1, 4, 1>, <7898, 1, 4, 1>, <19914, 1, 4, 1>, <0, 1, 4, 1> @} Total time: 2.140 seconds, Total memory usage: 39.39MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 19:50:13 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^0*3^1; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-2,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=12); Chabauty(P,11); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 19:50:11 on modular [Seed = 1586101538] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x - 1, 2, 1) ] 1.21306285522647802452835275718114319003158614249346456250448 2.0948660038325433522600978285640988496059 1.887653493378648643292217027045311416545 {@ <1, 1, 4, 1>, <13410, 1, 4, 1> @} Total time: 1.679 seconds, Total memory usage: 39.39MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 19:50:03 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^0*3^1; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-2,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=12); Chabauty(P,7); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 19:50:02 on modular [Seed = 1672790118] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x - 1, 2, 1) ] 1.21306285522647802452835275718114319003158614249346456250448 2.0948660038325433522600978285640988496059 1.887653493378648643292217027045311416545 {@ <1, 1, 4, 1>, <1133, 1, 4, 1>, <2194, 1, 4, 1> @} Total time: 1.790 seconds, Total memory usage: 39.39MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 19:49:52 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^0*3^1; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-2,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); RationalPoints(J:Bound:=12); //Chabauty(P,7); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 19:49:51 on modular [Seed = 1757399928] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x - 1, 2, 1) ] 1.21306285522647802452835275718114319003158614249346456250448 2.0948660038325433522600978285640988496059 1.887653493378648643292217027045311416545 {@ (1, 0, 0), (x - 1, 2, 1), (x - 1, -2, 1) @} Total time: 1.350 seconds, Total memory usage: 39.43MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 19:49:17 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^0*3^1; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-2,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); RationalPoints(J:Bound:=12); //Chabauty(P,29); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 19:49:16 on modular [Seed = 1840959070] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x - 1, 2, 1) ] 1.21306285522647802452835275718114319003158614249346456250448 2.0948660038325433522600978285640988496059 1.887653493378648643292217027045311416545 {@ (1, 0, 0), (x - 1, 2, 1), (x - 1, -2, 1) @} Total time: 1.290 seconds, Total memory usage: 39.41MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 19:49:05 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^0*3^1; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=12); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-2,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); //Chabauty(P,29); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 19:49:04 on modular [Seed = 1923469645] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x - 1, 2, 1) ] 1.21306285522647802452835275718114319003158614249346456250448 2.0948660038325433522600978285640988496059 1.887653493378648643292217027045311416545 Total time: 0.950 seconds, Total memory usage: 39.38MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 19:47:23 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^0*3^1; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1300); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-2,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); //Chabauty(P,29); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 19:47:22 on modular [Seed = 2007034834] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x - 1, 2, 1) ] 1.21306285522647802452835275718114319003158614249346456250448 2.0948660038325433522600978285640988496059 1.887653493378648643292217027045311416545 Total time: 1.350 seconds, Total memory usage: 39.47MB '168.168' ************** MAGMA ***************** Host 168.168.67.112 (168.168.67.112) Time: Mon Dec 5 18:57:32 2005 Input: 1024^1024 Output: Magma V2.11-10 Mon Dec 5 2005 18:57:32 on modular [Seed = 754695423] ------------------------------------- 3524971412108382657134814839800281546439142134396647106039138260573107027685474\ 9365048330296473663862456968155395298373973259049475943113619888338673116133666\ 8147068707652719076562056460186083699855587212676703217390319386338332818891926\ 2015842653180692314423926972687639995196119198034802329170347230576378241039458\ 9758934585631111078120435303032688818751446435291371357171755632775362932694795\ 0763134366874696380043276893902467353218558306108568659249137608267637760032658\ 5171655733421064227734347575779978049902155982241243427508708431729345512957040\ 6707590002071704673135527533543217355987568107697577946785796412456048360072965\ 6168710248662446500810590681830381345185142229871868373945980198595129936003792\ 3619019757683890508073335998909468700899941624772202006199255993140187235737970\ 8488585003666965930609730430774107407494018065365845077094320534700692354400169\ 8241315783891536569167546822524255627428950268220861122361857689319404333240786\ 9238646364237802929158238455090401228426527712466745281698565933749758099159251\ 0201479766500877427834566619156314388107585743546289067551052434075678195345373\ 3639195713232101136226155117651343296272079557936053768928759383576728708813056\ 7930552129335997542780192199753489147409086811346735778435978338309108571710080\ 7228425031226776985197364359404683041506613943646666199454899363685801848776729\ 6858378032282161138338547424434092214804502325631304177096253207949716727377373\ 8598397552004773997816512490691685793196090240739784153665765037875801240915720\ 5939513085324282439290108909069036515430690359963152986587749930516880670326145\ 0369876070529616967815564185509662018228218579780200625368240156976209572227380\ 6553883218709740985950266919658902596119944875899737379297319172333554977239487\ 8874050854532785922475822836403793986623193174020931432381418437022760412682276\ 3829893548396254532412898071082609051342346791309548675704473545497601746910070\ 7852845274502799494385322948054451236883137876111968161671932763730814231510512\ 0528704683515182038320225078665313911731749364255621284434304945437214609406008\ 6405209720295099554355680948888157014704194108891565239711821728144232741409554\ 2807059432838166704828677197285770343552580354470783456777402720661414341998241\ 0109261930698311010857874866840743851472857645330929169548403751084494725893729\ 3554504737710599868010583420219027353676279009748723681378389963973798981614548\ 2597091073285820278128297393764284797338183867298069339903942934261300159514896\ 8082010016061022316242842367672741265405434553107296623559604413326352140529618\ 1711754506578842550993346187227316979201855824371823913976733011681606825166392\ 1470656698146596173137480894913174236475299307832636771411700140421093025153813\ 2442219335072672096865184691303027156962439777053707286583949764055151291816402\ 5464624527191347971790992102335775962779256460318241722748740845621134400433973\ 9519106547362071710425068604089658092870084259391917328384453147095220560087448\ 2302488523867074532907781264990865351844684807012208039108287564534854500486391\ 5388760636114766656202302948114683518353740720605302159079093112818161319422197\ 76 Total time: 0.190 seconds, Total memory usage: 3.24MB '168.168' ************** MAGMA ***************** Host 168.168.67.112 (168.168.67.112) Time: Mon Dec 5 18:57:21 2005 Input: 1024^1024^3 Output: Magma V2.11-10 Mon Dec 5 2005 18:57:20 on modular [Seed = 837198318] ------------------------------------- >> 1024^1024^3; ^ Runtime error in '^': Argument 2 (1073741824) is too large Total time: 0.190 seconds, Total memory usage: 3.24MB '168.168' ************** MAGMA ***************** Host 168.168.67.112 (168.168.67.112) Time: Mon Dec 5 18:56:56 2005 Input: 1024^9999 Output: Magma V2.11-10 Mon Dec 5 2005 18:56:56 on modular [Seed = 920757953] ------------------------------------- 9755879814593598710390944964160484286918253311075993083706340565069783520158763\ 1672606184700434536172184022895073813958009648404371642349247153040568823172609\ 3641565417350692699838908923155515553300808450316512571981016141330897477008661\ 2330908863206560463571340681293649220239094087378460205364049336725544698214590\ 9201484794690929739558378875956299966990925505720468315641797669143507857597160\ 4419853088273531816198589663932375536698956361797485279322445111455574275652932\ 7671029471425412924795787458748107524029918332158777143869477511729305162470255\ 2752303235936113628941152557352618706704888742055285829812888619353928394329586\ 3173179418709840035072036793204211427803584460006365175429561534101598040502501\ 7666603138064354062789905857966821056102473678761936386847047945271833319810447\ 7503950902186280963889085279623407172102513494899737825067814628282473651444032\ 2617545242496240500599602944609772575382316787435729974159614833623173066138322\ 0426985882367921437870420793230646877395269289154985002134710803579719476788410\ 2390060757338801904323415527970984630974829583492192943919286764438313175595233\ 3589430583448210275822044964144575769737220498571840505091794591971217290739243\ 6361877429217111702392889302251437611689388835919607588139768372048891791803022\ 8567765182201952551733390530747143987660297906559154257877865062766762051167774\ 6218775486739477296309358858615706192412506699581996109556224771136667212252436\ 0595410463388564544103778274164481157691821750247233900899592022905652709285383\ 4396490556886108721966456683024660217345266716745494327567429310992723897738458\ 7420266731157517138219781390453083323040983694346296626645359123811737827073045\ 2965269691176506806197174484735472789380793089194963999896938690529181761552481\ 3553870957258158700775280925160187788847596690939008281285519678409331944453085\ 9524471592551701272678752369339423192696880168707971275238849839315883327824522\ 2848931451445631364000438812267316806399748372353252229452317938743177037522765\ 3902394829054410908526989350803290551371114681520070289492348907695530745428526\ 3433537300827781190285175638771225486909034277877856114964271790557611172453444\ 2337486854381810963875592801401857505431669481789531299605978090741878188580804\ 7861013896015177359548022880835156346555736112072369979402060248160258183873280\ 7660676151327331128934498085301740830153797624302042783834813392958364166482870\ 2355090032520222559118221720886731305233792674873914094694210558257774640949217\ 1005109619949865042205863666480204948848602888865081893827824708235614601510675\ 8566829641344293578928104280653132324148128089203156113717150264041971124319613\ 6546357823187799405734305020694311704335168288744516592345277039485292030705080\ 6811825945806690751338881338734337320513242710423198925065438880617888956468820\ 6471478277838844293698821689581076693635750039949178450150802104619853715837070\ 1586162852020712629792275528912455039922421096052125864890480703215602647938607\ 9027509722726286975570669277130672553458073741321811754995512076165145407202008\ 3867274205270989683391527949614430395118383849453516793582995378301081103355587\ 6928207694434324068707221352372289781380971812077323527130828179739967269669070\ 6522053137296112837902191527435056474251142988305961119743114866393904889184427\ 8342012877885178104325116182820362751478648629375524387027790094238132767349333\ 4519656833404372401522241535503140243589764917186866224826413049821302458417321\ 7821477493782178721829032732776199406170133317152890102693633290861876140118655\ 9355983191275655956997001655212602128405873694151458006056712598958039497168494\ 6658741395315892722850258900956369809473625976793687664604779466252011645871059\ 8095682866046477965882265335662203237014873771357275732956272353781004397786800\ 0810574329385824384096724354485662493612626269037382423370093370967667864211762\ 6049065707780279534346847287823990826949318327931840395106460839379594473633943\ 4677379235657513558172548334069915372821380740296861178196294813751718738715600\ 4504888979791871100594222389407010720445002933865127223838445686818502957004745\ 6383136826722719007724340410107115393464862894669681577899293542961356005773585\ 6815666702570787582170385320692280682551186044799860931670051266577555634329397\ 4304708916491617529404093753536156129714372692380954845212855087568783998058997\ 1370680546782224059176070796587362817124396484730244457905255960284504798400698\ 4080609476712080607362580417220691879429220441442266487422860780678075569162144\ 0395535305737410849473113400859665412659301268489449735957262950057276274444139\ 9715480469952804604670669670015321790172054780675899876744444377829802599070467\ 1064119616864741746350122118761982858542111617385494944950291053490911884642177\ 1691937675034697108152327192407608377806002903574521244546530742722044212184301\ 6262459995112633158327103418416850574886303640492788332071094228627720597938560\ 0032753876902156908902475779158579799379117023156181680839740632717464319897952\ 5379695969250769880595471337813850335602928996806859257087590257107957734501685\ 0962960906546662581116919824031723279788894879158597575499875678106898978175478\ 5766894757984290459812062655287844718254055634222288739363605797790703057990324\ 9596682055296712033030929222341025236053486705094952004145316716438828394505191\ 5235607664378013016883775541561739675174323643534006098368664710519751530439237\ 7048509194756028215511060889428234474295596888031891413474583717885370305663458\ 1853401127091971965515995145061612299066529346869711839639645149370142955634874\ 5004045181799775205150051475516117753389513191498100173630726856938378865853676\ 4653595903933457117812751068676213418398771905666052852293530644521641274077353\ 3884842830990949242485413662828498195437558597834719660691918742959398020273298\ 2626518195667803644949463585467579859017324034691173822935803975147104589698490\ 5604253853530672859980608336188220100895440257117718037123901139014832332863025\ 1491095656871109897060601187574972016954849165134221189362896865829423830900094\ 2233978542037763152818939098071707641837543785793138596275906713129836292287407\ 5306771966136131077505386830385440741212602424090200448402341053834244298244196\ 3671555408547812253820848531388602271607398199384921156818705190617400122453090\ 9213227337578485407230286188044791319347144837099190972984881570076163973498916\ 7631786640815184786609877119914573753850525407237787941787166448811813952824999\ 4295545375085670498217686028636606783733396411513726690582254711049530484193086\ 9258073632469358233994560827410878750327287226854996690279485512390699164577114\ 5005567095258321802221615726902860527221798691148898344350736460767403701564180\ 8427314680423537831033964209658598138530623793994488003503210273593445806770619\ 6754982007518921329196770873160147655385705787892302772412995901407406640698813\ 9341415790680367034429953938744300957466706326608203626570060100472780395081576\ 0119674713763323330280099292037091730639112633097056053042437559615204355909061\ 2988578305253079568644263440244758517683628736387602121543573441638262073385937\ 6550283939078454934018186511424979373131557167273892308761948289919597694037631\ 5877815615976304227812569261862438544594436477607471394833145751247207497675142\ 4811831256087533423953258861820471211399853122476248908206880319115614209354680\ 0240177453569435856826123343246307979375813766690509536728620202081621295850196\ 2541831254555403648551120767979131957721208379151425723561221580473306962755239\ 1459835186191312075949538737984430846735621889057551327511101591187475199332105\ 1187499212912761662408998884017200816336949969155081624297784448308604605191553\ 3472808262417796544166850026204128391196220202719930643637789339378469529149558\ 1605219061019524877571582354465329114070718653290483584351415962646854946307779\ 3065053738228575496042254868240718850623875449564681564213374080821846976022464\ 9982722678299481767628137385922461866230814763117556962126647067976198770744325\ 4695448118047046692489546894678463214823790515767903723315328375863070655447732\ 0240825949343108098909984225700645349435643941861448323671407906021299467029023\ 2607380257854523040134103033193493283046851814946819869093392583510652398987043\ 9917346219655219746314612432521895307623299737663879887912246819301519720755646\ 6625515948213101115341109752861381445428976350612905893610505228571021997306405\ 5218395689792171512891812739908407582233848193441645444071272521436140387746948\ 7153591485020721052798955362757118097891309613171420906105465975779677817045504\ 8790987679684799764294431704906634615381339687973494328062115889902031431395291\ 2066514095238127038419701502459544498349735620840750057308594902817870522410032\ 8987255242955700393244733592487945592223392359091367529021872466372563739673064\ 7991224110456887055837846872631996921011190707675817915610209677682983337857576\ 5726602953112867513203300343617661780488437359017767020182510988491547494999142\ 3984006586450558900604672596455660800724786401761341465272010575392392778174724\ 3337807473203381274705102463957319943135378613551360235922597605510303395809603\ 7113622229776717641076896027194191158455095791143991929554442043771608833072449\ 5769407758825331577698496435330283195719296007423923278800226914831913409194387\ 1260228884995639709039228687228275264274913993356598245732698265076363134244085\ 2388719036324269055162805039514224014020036457310230294036869260026779672563150\ 8528926067211752016736146948601316085822480564833673847445524144546378177109801\ 2088354294635094315482598960185652388280845189254744772548220621673244373003291\ 4577524358929795506239059545033501080996217877502783939164045245006623034013493\ 3022641814718255481553675765646398875315843653703665831763627102743588870073054\ 3362226394196694473075748700714622109407799819372104467040018536220111845966571\ 07 ** WARNING: Output too long, hence truncated. '168.168' ************** MAGMA ***************** Host 168.168.67.112 (168.168.67.112) Time: Mon Dec 5 18:56:33 2005 Input: 1024^1024 Output: Magma V2.11-10 Mon Dec 5 2005 18:56:32 on modular [Seed = 1022078131] ------------------------------------- 3524971412108382657134814839800281546439142134396647106039138260573107027685474\ 9365048330296473663862456968155395298373973259049475943113619888338673116133666\ 8147068707652719076562056460186083699855587212676703217390319386338332818891926\ 2015842653180692314423926972687639995196119198034802329170347230576378241039458\ 9758934585631111078120435303032688818751446435291371357171755632775362932694795\ 0763134366874696380043276893902467353218558306108568659249137608267637760032658\ 5171655733421064227734347575779978049902155982241243427508708431729345512957040\ 6707590002071704673135527533543217355987568107697577946785796412456048360072965\ 6168710248662446500810590681830381345185142229871868373945980198595129936003792\ 3619019757683890508073335998909468700899941624772202006199255993140187235737970\ 8488585003666965930609730430774107407494018065365845077094320534700692354400169\ 8241315783891536569167546822524255627428950268220861122361857689319404333240786\ 9238646364237802929158238455090401228426527712466745281698565933749758099159251\ 0201479766500877427834566619156314388107585743546289067551052434075678195345373\ 3639195713232101136226155117651343296272079557936053768928759383576728708813056\ 7930552129335997542780192199753489147409086811346735778435978338309108571710080\ 7228425031226776985197364359404683041506613943646666199454899363685801848776729\ 6858378032282161138338547424434092214804502325631304177096253207949716727377373\ 8598397552004773997816512490691685793196090240739784153665765037875801240915720\ 5939513085324282439290108909069036515430690359963152986587749930516880670326145\ 0369876070529616967815564185509662018228218579780200625368240156976209572227380\ 6553883218709740985950266919658902596119944875899737379297319172333554977239487\ 8874050854532785922475822836403793986623193174020931432381418437022760412682276\ 3829893548396254532412898071082609051342346791309548675704473545497601746910070\ 7852845274502799494385322948054451236883137876111968161671932763730814231510512\ 0528704683515182038320225078665313911731749364255621284434304945437214609406008\ 6405209720295099554355680948888157014704194108891565239711821728144232741409554\ 2807059432838166704828677197285770343552580354470783456777402720661414341998241\ 0109261930698311010857874866840743851472857645330929169548403751084494725893729\ 3554504737710599868010583420219027353676279009748723681378389963973798981614548\ 2597091073285820278128297393764284797338183867298069339903942934261300159514896\ 8082010016061022316242842367672741265405434553107296623559604413326352140529618\ 1711754506578842550993346187227316979201855824371823913976733011681606825166392\ 1470656698146596173137480894913174236475299307832636771411700140421093025153813\ 2442219335072672096865184691303027156962439777053707286583949764055151291816402\ 5464624527191347971790992102335775962779256460318241722748740845621134400433973\ 9519106547362071710425068604089658092870084259391917328384453147095220560087448\ 2302488523867074532907781264990865351844684807012208039108287564534854500486391\ 5388760636114766656202302948114683518353740720605302159079093112818161319422197\ 76 Total time: 0.190 seconds, Total memory usage: 3.24MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 18:33:16 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^8*3^5; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1300); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-2,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); Chabauty(P,29); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 18:33:13 on modular [Seed = 3523567571] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x - 12, 432, 1) ] 0.602462028809127141440708856954530013567 11.1773194110379626886475632925362976571738 8.514071226146649470571442306615654991963 {@ <12, 1, 4, 1> @} Total time: 2.529 seconds, Total memory usage: 39.20MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 18:33:03 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^8*3^5; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1300); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-2,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); Chabauty(P,19); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 18:33:01 on modular [Seed = 3708440776] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x - 12, 432, 1) ] 0.602462028809127141440708856954530013567 11.1773194110379626886475632925362976571738 8.514071226146649470571442306615654991963 {@ <12, 1, 4, 1>, <117425, 1, 4, 1> @} Total time: 2.560 seconds, Total memory usage: 39.20MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 18:32:56 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^8*3^5; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1300); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-2,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); Chabauty(P,17); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 18:32:52 on modular [Seed = 3624883652] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x - 12, 432, 1) ] 0.602462028809127141440708856954530013567 11.1773194110379626886475632925362976571738 8.514071226146649470571442306615654991963 {@ <12, 1, 4, 1>, <25746, 1, 4, 1>, <41974, 1, 4, 1>, <66925, 1, 4, 1>, <1795, 1, 4, 1>, <83509, 1, 4, 1>, <5869, 1, 4, 1>, <44342, 1, 4, 1> @} Total time: 3.279 seconds, Total memory usage: 39.20MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 18:32:46 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^8*3^5; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1300); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-2,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); Chabauty(P,13); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 18:32:43 on modular [Seed = 3811864256] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x - 12, 432, 1) ] 0.602462028809127141440708856954530013567 11.1773194110379626886475632925362976571738 8.514071226146649470571442306615654991963 {@ <12, 1, 4, 1>, <6861, 1, 4, 1>, <16555, 1, 4, 1>, <25976, 1, 4, 1>, <0, 1, 4, 1> @} Total time: 2.830 seconds, Total memory usage: 39.20MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 18:32:38 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^8*3^5; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1300); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-2,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); Chabauty(P,11); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 18:32:35 on modular [Seed = 3996738501] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x - 12, 432, 1) ] 0.602462028809127141440708856954530013567 11.1773194110379626886475632925362976571738 8.514071226146649470571442306615654991963 {@ <12, 1, 4, 1>, <5637, 1, 4, 1>, <7726, 1, 4, 1>, <4854, 1, 4, 1>, <13605, 1, 4, 1> @} Total time: 2.759 seconds, Total memory usage: 39.20MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 18:32:29 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^8*3^5; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1300); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-2,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); Chabauty(P,7); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 18:32:26 on modular [Seed = 3913180378] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x - 12, 432, 1) ] 0.602462028809127141440708856954530013567 11.1773194110379626886475632925362976571738 8.514071226146649470571442306615654991963 {@ <12, 1, 4, 1>, <176, 1, 4, 1>, <0, 1, 4, 1> @} Total time: 2.430 seconds, Total memory usage: 39.20MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 18:26:00 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^8*3^5; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1300); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-2,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); //Chabauty(P,19); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 18:25:58 on modular [Seed = 4062543180] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x - 12, 432, 1) ] 0.602462028809127141440708856954530013567 11.1773194110379626886475632925362976571738 8.514071226146649470571442306615654991963 Total time: 1.970 seconds, Total memory usage: 39.21MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 18:25:47 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^8*3^5; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1300); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-2,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=100); //RationalPoints(J:Bound:=1000); //Chabauty(P,19); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 18:25:45 on modular [Seed = 4247417532] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x - 12, 432, 1) ] 0.602462028809127141440708856954530013567 11.1773194110379626886475632925362976571738 8.514071226146649470571442306615654991963 Total time: 1.960 seconds, Total memory usage: 39.20MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 18:25:34 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^8*3^5; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1300); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-2,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=10); //RationalPoints(J:Bound:=1000); //Chabauty(P,19); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 18:25:32 on modular [Seed = 4163860396] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x - 12, 432, 1) ] 0.602462028809127141440708856954530013567 11.1773194110379626886475632925362976571738 8.514071226146649470571442306615654991963 Total time: 1.980 seconds, Total memory usage: 39.20MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 18:25:22 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^8*3^5; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1300); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-2,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); //Chabauty(P,19); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 18:25:20 on modular [Seed = 2203323611] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x - 12, 432, 1) ] 0.602462028809127141440708856954530013567 11.1773194110379626886475632925362976571738 8.514071226146649470571442306615654991963 Total time: 1.960 seconds, Total memory usage: 39.20MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 18:23:55 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^8*3^1; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1300); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-2,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); Chabauty(P,19); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 18:23:52 on modular [Seed = 2388197822] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 + 16, 8*x + 16, 2) ] 0.1895600860537845105361496034702936767600 7.0795782111135686301214930155703845270839 5.584438456365023626850788341488919779599 {@ <4, 1, 4, 1> @} Total time: 2.970 seconds, Total memory usage: 39.15MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 18:23:47 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^8*3^1; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1300); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-2,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); Chabauty(P,17); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 18:23:43 on modular [Seed = 2304639310] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 + 16, 8*x + 16, 2) ] 0.1895600860537845105361496034702936767600 7.0795782111135686301214930155703845270839 5.584438456365023626850788341488919779599 {@ <4, 1, 4, 1>, <77093, 1, 4, 1>, <56288, 1, 4, 1>, <69699, 1, 4, 1>, <76710, 1, 4, 1>, <80434, 1, 4, 1>, <61220, 1, 4, 1>, <44724, 1, 4, 1> @} Total time: 3.830 seconds, Total memory usage: 39.15MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 18:23:39 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^8*3^1; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1300); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-2,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); Chabauty(P,13); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 18:23:36 on modular [Seed = 2487426132] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 + 16, 8*x + 16, 2) ] 0.1895600860537845105361496034702936767600 7.0795782111135686301214930155703845270839 5.584438456365023626850788341488919779599 {@ <12745, 1, 4, 1>, <4, 1, 4, 1> @} Total time: 2.960 seconds, Total memory usage: 39.15MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 18:23:27 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^8*3^1; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1300); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-2,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); Chabauty(P,11); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 18:23:23 on modular [Seed = 2672301900] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 + 16, 8*x + 16, 2) ] 0.1895600860537845105361496034702936767600 7.0795782111135686301214930155703845270839 5.584438456365023626850788341488919779599 {@ <4, 1, 4, 1>, <4975, 1, 4, 1>, <3147, 1, 4, 1>, <283, 1, 4, 1>, <12220, 1, 4, 1>, <13979, 1, 4, 1>, <7190, 1, 4, 1>, <10940, 1, 4, 1>, <8901, 1, 4, 1> @} Total time: 3.839 seconds, Total memory usage: 39.15MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 18:23:17 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^8*3^1; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1300); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-2,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); Chabauty(P,7); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 18:23:14 on modular [Seed = 2588742208] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 + 16, 8*x + 16, 2) ] 0.1895600860537845105361496034702936767600 7.0795782111135686301214930155703845270839 5.584438456365023626850788341488919779599 {@ <4, 1, 4, 1>, <0, 1, 4, 1> @} Total time: 2.910 seconds, Total memory usage: 39.15MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 18:22:09 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^8*3^1; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1300); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-2,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); //Chabauty(P,11); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 18:22:07 on modular [Seed = 2742299396] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 + 16, 8*x + 16, 2) ] 0.1895600860537845105361496034702936767600 7.0795782111135686301214930155703845270839 5.584438456365023626850788341488919779599 Total time: 2.480 seconds, Total memory usage: 39.15MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 18:21:51 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^8*3^1; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-2,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); RationalPoints(J:Bound:=1300); //Chabauty(P,11); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 18:21:48 on modular [Seed = 2927175194] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 + 16, 8*x + 16, 2) ] 0.1895600860537845105361496034702936767600 7.0795782111135686301214930155703845270839 5.584438456365023626850788341488919779599 {@ (1, 0, 0), (x^2 - 8*x + 16, 40*x - 144, 2), (x^2 - 8*x + 16, -40*x + 144, 2), (x - 4, 16, 1), (x - 4, -16, 1), (x^2 + 3*x + 12, 11*x + 12, 2), (x^2 + 3*x + 12, -11*x - 12, 2), (x^2 + 16, 8*x + 16, 2), (x^2 + 16, -8*x - 16, 2), (x^2 - 56/25*x + 448/25, 196/125*x + 2832/125, 2), (x^2 - 56/25*x + 448/25, -196/125*x - 2832/125, 2) @} Total time: 2.669 seconds, Total memory usage: 39.44MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 18:20:03 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^8*3^1; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-2,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); RationalPoints(J:Bound:=1000); //Chabauty(P,11); //Order(Q);Order(P); 2*P; Output: Magma V2.11-10 Mon Dec 5 2005 18:20:00 on modular [Seed = 2826903728] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 + 16, 8*x + 16, 2) ] 0.1895600860537845105361496034702936767600 7.0795782111135686301214930155703845270839 5.584438456365023626850788341488919779599 {@ (1, 0, 0), (x - 4, 16, 1), (x - 4, -16, 1), (x^2 + 3*x + 12, 11*x + 12, 2), (x^2 + 3*x + 12, -11*x - 12, 2), (x^2 + 16, 8*x + 16, 2), (x^2 + 16, -8*x - 16, 2), (x^2 - 56/25*x + 448/25, 196/125*x + 2832/125, 2), (x^2 - 56/25*x + 448/25, -196/125*x - 2832/125, 2) @} (x - 4, 16, 1) Total time: 2.390 seconds, Total memory usage: 39.06MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 18:19:32 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^8*3^1; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-2,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); RationalPoints(J:Bound:=1000); //Chabauty(P,11); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Mon Dec 5 2005 18:19:30 on modular [Seed = 3009690505] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 + 16, 8*x + 16, 2) ] 0.1895600860537845105361496034702936767600 7.0795782111135686301214930155703845270839 5.584438456365023626850788341488919779599 {@ (1, 0, 0), (x - 4, 16, 1), (x - 4, -16, 1), (x^2 + 3*x + 12, 11*x + 12, 2), (x^2 + 3*x + 12, -11*x - 12, 2), (x^2 + 16, 8*x + 16, 2), (x^2 + 16, -8*x - 16, 2), (x^2 - 56/25*x + 448/25, 196/125*x + 2832/125, 2), (x^2 - 56/25*x + 448/25, -196/125*x - 2832/125, 2) @} Total time: 2.370 seconds, Total memory usage: 39.06MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 18:15:52 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^8*3^1; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-2,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); //Chabauty(P,11); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Mon Dec 5 2005 18:15:49 on modular [Seed = 3127718823] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 + 16, 8*x + 16, 2) ] 0.1895600860537845105361496034702936767600 7.0795782111135686301214930155703845270839 5.584438456365023626850788341488919779599 Total time: 2.370 seconds, Total memory usage: 39.06MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 18:03:35 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^7*3^2; C:=HyperellipticCurve(f); J:=Jacobian(C); //r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-2,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); Chabauty(P,11); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Mon Dec 5 2005 18:03:33 on modular [Seed = 1622664221] ------------------------------------- Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 4*x + 24, -4*x + 48, 2) ] 1.8303250705036937366598913071602697543760764146097835184960 7.2287971751661735539333571091849363786360 5.854748528437133214836130418465152537301 {@ @} Total time: 1.139 seconds, Total memory usage: 38.43MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 18:03:28 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^7*3^2; C:=HyperellipticCurve(f); J:=Jacobian(C); //r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-2,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); Chabauty(P,7); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Mon Dec 5 2005 18:03:27 on modular [Seed = 1807544172] ------------------------------------- Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 4*x + 24, -4*x + 48, 2) ] 1.8303250705036937366598913071602697543760764146097835184960 7.2287971751661735539333571091849363786360 5.854748528437133214836130418465152537301 {@ <1721, 1, 4, 1>, <738, 1, 4, 1> @} Total time: 1.300 seconds, Total memory usage: 38.43MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 18:03:20 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^7*3^2; C:=HyperellipticCurve(f); J:=Jacobian(C); //r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-2,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); //Chabauty(P,7); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Mon Dec 5 2005 18:03:19 on modular [Seed = 1990324857] ------------------------------------- Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 4*x + 24, -4*x + 48, 2) ] 1.8303250705036937366598913071602697543760764146097835184960 7.2287971751661735539333571091849363786360 5.854748528437133214836130418465152537301 Total time: 0.820 seconds, Total memory usage: 38.43MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 18:02:35 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^7*3^2; C:=HyperellipticCurve(f); J:=Jacobian(C); //r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-2,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); //Chabauty(P,11); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Mon Dec 5 2005 18:02:34 on modular [Seed = 1906767171] ------------------------------------- Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 4*x + 24, -4*x + 48, 2) ] 1.8303250705036937366598913071602697543760764146097835184960 7.2287971751661735539333571091849363786360 5.854748528437133214836130418465152537301 Total time: 0.830 seconds, Total memory usage: 38.43MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 18:02:27 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^7*3^2; C:=HyperellipticCurve(f); J:=Jacobian(C); //r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=3000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-2,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); //Chabauty(P,11); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Mon Dec 5 2005 18:02:24 on modular [Seed = 2091644996] ------------------------------------- Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 4*x + 24, -4*x + 48, 2) ] 1.8303250705036937366598913071602697543760764146097835184960 7.2287971751661735539333571091849363786360 5.854748528437133214836130418465152537301 Total time: 2.229 seconds, Total memory usage: 38.52MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 17:51:32 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^7*3^2; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-2,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); //Chabauty(P,11); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Mon Dec 5 2005 17:51:30 on modular [Seed = 955236210] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 4*x + 24, -4*x + 48, 2) ] 1.8303250705036937366598913071602697543760764146097835184960 7.2287971751661735539333571091849363786360 5.854748528437133214836130418465152537301 Total time: 1.360 seconds, Total memory usage: 39.39MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 17:51:29 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^7*3^2; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-2,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); //Chabauty(P,11); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Mon Dec 5 2005 17:51:27 on modular [Seed = 3256173972] ------------------------------------- TwoSelmerGroupData( J: Jacobian of Hyperelliptic Curve defined by y^2 = x^5 - 1152 ... ) TwoSelmerWork( C: Hyperelliptic Curve defined by y^2 = x^5 - 1152 over Rationa..., BoundType: Default, Bound: 0, UseUnits: true, Points: {}, Fields: {} ) LocalImageOdd( f: x^5 - 1152, p: 2, m1: Mapping from: Univariate Polynomial Ring in x over Rational ..., m2: Mapping from: Univariate Polynomial Ring in x over Rational ..., v: 30, Points: {} ) try( ) try( d: 2 ) SquarefreeRoots( F: $.1^4 + O(2^50)*$.1^3 - (14185465184247*2^3 + O(2^53))*$.1^2... ) In file "/usr/local/magma/package/Geometry/Arith/sqfroots.m", line 84, column 15: >> return Roots(F:IsSquarefree); ^ Runtime error in 'Roots': Insufficient precision to find roots >> r:=TwoSelmerGroupData(J);r; ^ User error: Identifier 'r' has not been declared or assigned Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 4*x + 24, -4*x + 48, 2) ] 1.8303250705036937366598913071602697543760764146097835184960 7.2287971751661735539333571091849363786360 5.854748528437133214836130418465152537301 Total time: 1.159 seconds, Total memory usage: 39.39MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 17:51:20 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^7*3^2; C:=HyperellipticCurve(f); J:=Jacobian(C); //r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-2,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); //Chabauty(P,11); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Mon Dec 5 2005 17:51:19 on modular [Seed = 3474475153] ------------------------------------- Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 4*x + 24, -4*x + 48, 2) ] 1.8303250705036937366598913071602697543760764146097835184960 7.2287971751661735539333571091849363786360 5.854748528437133214836130418465152537301 Total time: 0.820 seconds, Total memory usage: 38.43MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 17:49:50 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^7*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); //r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-2,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); Chabauty(P,11); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Mon Dec 5 2005 17:49:49 on modular [Seed = 3357495292] ------------------------------------- Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 + 4*x + 8, 4*x, 2) ] 1.6140195870701401484033759978634435179079021417836051436947 5.393385283934348998903993780164142041461 4.389932143546320292975803435901784931143 {@ @} Total time: 1.129 seconds, Total memory usage: 38.54MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 17:49:40 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^7*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); //r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-2,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); Chabauty(P,7); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Mon Dec 5 2005 17:49:39 on modular [Seed = 3573699311] ------------------------------------- Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 + 4*x + 8, 4*x, 2) ] 1.6140195870701401484033759978634435179079021417836051436947 5.393385283934348998903993780164142041461 4.389932143546320292975803435901784931143 {@ <1311, 1, 4, 1>, <769, 1, 4, 1>, <1594, 1, 4, 1> @} Total time: 1.270 seconds, Total memory usage: 38.54MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 17:47:22 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^7*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-2,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); //Chabauty(P,17); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Mon Dec 5 2005 17:47:20 on modular [Seed = 3725155999] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 + 4*x + 8, 4*x, 2) ] 1.6140195870701401484033759978634435179079021417836051436947 5.393385283934348998903993780164142041461 4.389932143546320292975803435901784931143 Total time: 1.360 seconds, Total memory usage: 39.61MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 17:47:19 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^7*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-2,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); //Chabauty(P,17); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Mon Dec 5 2005 17:47:16 on modular [Seed = 3675020182] ------------------------------------- TwoSelmerGroupData( J: Jacobian of Hyperelliptic Curve defined by y^2 = x^5 - 128 o... ) TwoSelmerWork( C: Hyperelliptic Curve defined by y^2 = x^5 - 128 over Rational..., BoundType: Default, Bound: 0, UseUnits: true, Points: {}, Fields: {} ) LocalImageOdd( f: x^5 - 128, p: 2, m1: Mapping from: Univariate Polynomial Ring in x over Rational ..., m2: Mapping from: Univariate Polynomial Ring in x over Rational ..., v: 30, Points: {} ) try( ) try( d: 2 ) SquarefreeRoots( F: $.1^4 + O(2^50)*$.1^3 - (137394841321471*2^3 + O(2^53))*$.1^... ) In file "/usr/local/magma/package/Geometry/Arith/sqfroots.m", line 84, column 15: >> return Roots(F:IsSquarefree); ^ Runtime error in 'Roots': Insufficient precision to find roots >> r:=TwoSelmerGroupData(J);r; ^ User error: Identifier 'r' has not been declared or assigned Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 + 4*x + 8, 4*x, 2) ] 1.6140195870701401484033759978634435179079021417836051436947 5.393385283934348998903993780164142041461 4.389932143546320292975803435901784931143 Total time: 2.069 seconds, Total memory usage: 40.90MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 17:47:14 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^7*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-2,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); //Chabauty(P,17); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Mon Dec 5 2005 17:47:13 on modular [Seed = 3828573356] ------------------------------------- TwoSelmerGroupData( J: Jacobian of Hyperelliptic Curve defined by y^2 = x^5 - 128 o... ) TwoSelmerWork( C: Hyperelliptic Curve defined by y^2 = x^5 - 128 over Rational..., BoundType: Default, Bound: 0, UseUnits: true, Points: {}, Fields: {} ) LocalImageOdd( f: x^5 - 128, p: 2, m1: Mapping from: Univariate Polynomial Ring in x over Rational ..., m2: Mapping from: Univariate Polynomial Ring in x over Rational ..., v: 30, Points: {} ) try( ) try( d: 2 ) SquarefreeRoots( F: $.1^4 + O(2^50)*$.1^3 + (278930146590721*2^3 + O(2^53))*$.1^... ) In file "/usr/local/magma/package/Geometry/Arith/sqfroots.m", line 84, column 15: >> return Roots(F:IsSquarefree); ^ Runtime error in 'Roots': Insufficient precision to find roots >> r:=TwoSelmerGroupData(J);r; ^ User error: Identifier 'r' has not been declared or assigned Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 + 4*x + 8, 4*x, 2) ] 1.6140195870701401484033759978634435179079021417836051436947 5.393385283934348998903993780164142041461 4.389932143546320292975803435901784931143 Total time: 1.169 seconds, Total memory usage: 39.57MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 17:46:57 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^7*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); //r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-2,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); //Chabauty(P,17); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Mon Dec 5 2005 17:46:56 on modular [Seed = 3778437560] ------------------------------------- Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 + 4*x + 8, 4*x, 2) ] 1.6140195870701401484033759978634435179079021417836051436947 5.393385283934348998903993780164142041461 4.389932143546320292975803435901784931143 Total time: 0.790 seconds, Total memory usage: 38.54MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 17:38:14 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^6*3^5; C:=HyperellipticCurve(f); J:=Jacobian(C); //r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-2,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); Chabauty(P,17); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Mon Dec 5 2005 17:38:12 on modular [Seed = 2220033769] ------------------------------------- Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 4*x + 48, -4*x - 24, 2) ] 1.802849529668960157376835690894363395849 9.531058522619703794561201943405304565704 7.589874985400055724681799478004752901208 {@ @} Total time: 1.320 seconds, Total memory usage: 38.13MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 17:38:05 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^6*3^5; C:=HyperellipticCurve(f); J:=Jacobian(C); //r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-2,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); Chabauty(P,13); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Mon Dec 5 2005 17:38:03 on modular [Seed = 2169897971] ------------------------------------- Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 4*x + 48, -4*x - 24, 2) ] 1.802849529668960157376835690894363395849 9.531058522619703794561201943405304565704 7.589874985400055724681799478004752901208 {@ <6462, 1, 4, 1>, <18387, 1, 4, 1>, <25670, 1, 4, 1>, <0, 1, 4, 1>, <25738, 1, 4, 1>, <22629, 1, 4, 1> @} Total time: 1.790 seconds, Total memory usage: 38.13MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 17:37:57 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^6*3^5; C:=HyperellipticCurve(f); J:=Jacobian(C); //r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-2,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); Chabauty(P,11); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Mon Dec 5 2005 17:37:55 on modular [Seed = 2287930098] ------------------------------------- Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 4*x + 48, -4*x - 24, 2) ] 1.802849529668960157376835690894363395849 9.531058522619703794561201943405304565704 7.589874985400055724681799478004752901208 {@ <4765, 1, 4, 1>, <13493, 1, 4, 1>, <8772, 1, 4, 1>, <7875, 1, 4, 1>, <5842, 1, 4, 1>, <2126, 1, 4, 1>, <3119, 1, 4, 1>, <12152, 1, 4, 1>, <4354, 1, 4, 1> @} Total time: 2.250 seconds, Total memory usage: 38.13MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 17:37:49 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^6*3^5; C:=HyperellipticCurve(f); J:=Jacobian(C); //r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-2,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); Chabauty(P,7); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Mon Dec 5 2005 17:37:47 on modular [Seed = 2504136599] ------------------------------------- Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 4*x + 48, -4*x - 24, 2) ] 1.802849529668960157376835690894363395849 9.531058522619703794561201943405304565704 7.589874985400055724681799478004752901208 {@ <727, 1, 4, 1>, <0, 1, 4, 1> @} Total time: 1.389 seconds, Total memory usage: 38.13MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 17:33:57 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^6*3^5; C:=HyperellipticCurve(f); J:=Jacobian(C); //r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-2,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); Chabauty(P,19); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Mon Dec 5 2005 17:33:56 on modular [Seed = 2672303346] ------------------------------------- Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 4*x + 48, -4*x - 24, 2) ] 1.802849529668960157376835690894363395849 9.531058522619703794561201943405304565704 7.589874985400055724681799478004752901208 {@ <0, 1, 4, 1> @} Total time: 1.419 seconds, Total memory usage: 38.13MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 17:32:53 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^6*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); //r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-2,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); Chabauty(P,19); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Mon Dec 5 2005 17:32:52 on modular [Seed = 2555321269] ------------------------------------- Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 + 8, 4*x + 8, 2) ] 1.0352414493635452559799411644881992927384 5.284155864092557547639414967205314398943 3.927834023173023420030982021596333885754 {@ @} Total time: 1.330 seconds, Total memory usage: 38.23MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 17:32:46 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^6*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); //r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-2,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); Chabauty(P,17); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Mon Dec 5 2005 17:32:44 on modular [Seed = 2775721616] ------------------------------------- Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 + 8, 4*x + 8, 2) ] 1.0352414493635452559799411644881992927384 5.284155864092557547639414967205314398943 3.927834023173023420030982021596333885754 {@ <48522, 1, 4, 1>, <54152, 1, 4, 1> @} Total time: 1.540 seconds, Total memory usage: 38.23MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 17:32:38 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^6*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); //r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-2,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); Chabauty(P,13); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Mon Dec 5 2005 17:32:36 on modular [Seed = 2927177107] ------------------------------------- Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 + 8, 4*x + 8, 2) ] 1.0352414493635452559799411644881992927384 5.284155864092557547639414967205314398943 3.927834023173023420030982021596333885754 {@ <20536, 1, 4, 1>, <0, 1, 4, 1>, <28317, 1, 4, 1>, <21380, 1, 4, 1>, <23888, 1, 4, 1> @} Total time: 1.800 seconds, Total memory usage: 38.23MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 17:32:29 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^6*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); //r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-2,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); Chabauty(P,11); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Mon Dec 5 2005 17:32:27 on modular [Seed = 2877052777] ------------------------------------- Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 + 8, 4*x + 8, 2) ] 1.0352414493635452559799411644881992927384 5.284155864092557547639414967205314398943 3.927834023173023420030982021596333885754 {@ <9360, 1, 4, 1>, <10998, 1, 4, 1>, <1096, 1, 4, 1>, <10363, 1, 4, 1>, <9477, 1, 4, 1>, <3855, 1, 4, 1>, <11310, 1, 4, 1>, <12235, 1, 4, 1>, <9156, 1, 4, 1> @} Total time: 2.250 seconds, Total memory usage: 38.23MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 17:32:12 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^6*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-2,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); Chabauty(P,7); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Mon Dec 5 2005 17:32:10 on modular [Seed = 3026405479] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 + 8, 4*x + 8, 2) ] 1.0352414493635452559799411644881992927384 5.284155864092557547639414967205314398943 3.927834023173023420030982021596333885754 {@ <1706, 1, 4, 1>, <1622, 1, 4, 1>, <1354, 1, 4, 1>, <557, 1, 4, 1> @} Total time: 2.319 seconds, Total memory usage: 39.80MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 17:32:07 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^6*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-2,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); Chabauty(P,7); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Mon Dec 5 2005 17:32:04 on modular [Seed = 2976271724] ------------------------------------- TwoSelmerGroupData( J: Jacobian of Hyperelliptic Curve defined by y^2 = x^5 - 64 ov... ) TwoSelmerWork( C: Hyperelliptic Curve defined by y^2 = x^5 - 64 over Rational ..., BoundType: Default, Bound: 0, UseUnits: true, Points: {}, Fields: {} ) LocalImageOdd( f: x^5 - 64, p: 2, m1: Mapping from: Univariate Polynomial Ring in x over Rational ..., m2: Mapping from: Univariate Polynomial Ring in x over Rational ..., v: 26, Points: {} ) try( ) try( d: 2 ) SquarefreeRoots( F: $.1^4 + O(2^50)*$.1^3 - (559154624462847*2^2 + O(2^52))*$.1^... ) In file "/usr/local/magma/package/Geometry/Arith/sqfroots.m", line 84, column 15: >> return Roots(F:IsSquarefree); ^ Runtime error in 'Roots': Insufficient precision to find roots >> r:=TwoSelmerGroupData(J);r; ^ User error: Identifier 'r' has not been declared or assigned Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 + 8, 4*x + 8, 2) ] 1.0352414493635452559799411644881992927384 5.284155864092557547639414967205314398943 3.927834023173023420030982021596333885754 {@ <1706, 1, 4, 1>, <1622, 1, 4, 1>, <1354, 1, 4, 1>, <557, 1, 4, 1> @} Total time: 2.240 seconds, Total memory usage: 40.02MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 17:29:31 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^6*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-2,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); //Chabauty(P,7); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Mon Dec 5 2005 17:29:29 on modular [Seed = 3144435659] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 + 8, 4*x + 8, 2) ] 1.0352414493635452559799411644881992927384 5.284155864092557547639414967205314398943 3.927834023173023420030982021596333885754 Total time: 1.429 seconds, Total memory usage: 39.25MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 17:28:31 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^7*3^3; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); //V:=RationalPoints(J:Bound:=1000); //B:=ReducedBasis(V); //B; //P:=J!B[1]; //Q:=J![x-2,0]; //P1:=J![x^2+6*x+36,12*x-18]; //Height(P); //HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); //Chabauty(P,7); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Mon Dec 5 2005 17:28:29 on modular [Seed = 1150531696] ------------------------------------- 0 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] Total time: 1.449 seconds, Total memory usage: 6.55MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 17:28:14 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^7*3^3; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); //V:=RationalPoints(J:Bound:=1000); //B:=ReducedBasis(V); //B; //P:=J!B[1]; //Q:=J![x-2,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); //Chabauty(P,7); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Mon Dec 5 2005 17:28:13 on modular [Seed = 1100397701] ------------------------------------- 0 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] >> Height(P); ^ User error: Identifier 'P' has not been declared or assigned 7.790638689351186835930798024215308990709 6.587156720882539675766293909746836340427 Total time: 1.080 seconds, Total memory usage: 5.96MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 17:28:12 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^7*3^3; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); //V:=RationalPoints(J:Bound:=1000); //B:=ReducedBasis(V); //B; //P:=J!B[1]; //Q:=J![x-2,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); //Chabauty(P,7); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Mon Dec 5 2005 17:28:10 on modular [Seed = 1251850121] ------------------------------------- TwoSelmerGroupData( J: Jacobian of Hyperelliptic Curve defined by y^2 = x^5 - 3456 ... ) TwoSelmerWork( C: Hyperelliptic Curve defined by y^2 = x^5 - 3456 over Rationa..., BoundType: Default, Bound: 0, UseUnits: true, Points: {}, Fields: {} ) LocalImageOdd( f: x^5 - 3456, p: 2, m1: Mapping from: Univariate Polynomial Ring in x over Rational ..., m2: Mapping from: Univariate Polynomial Ring in x over Rational ..., v: 30, Points: {} ) try( ) try( d: 2 ) SquarefreeRoots( F: $.1^4 + O(2^50)*$.1^3 + (110447995125787*2^3 + O(2^53))*$.1^... ) In file "/usr/local/magma/package/Geometry/Arith/sqfroots.m", line 84, column 15: >> return Roots(F:IsSquarefree); ^ Runtime error in 'Roots': Insufficient precision to find roots >> r:=TwoSelmerGroupData(J);r; ^ User error: Identifier 'r' has not been declared or assigned Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] >> Height(P); ^ User error: Identifier 'P' has not been declared or assigned 7.790638689351186835930798024215308990709 6.587156720882539675766293909746836340427 Total time: 1.540 seconds, Total memory usage: 6.92MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 17:28:09 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^7*3^3; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); //V:=RationalPoints(J:Bound:=1000); //B:=ReducedBasis(V); //B; //P:=J!B[1]; //Q:=J![x-2,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); //Chabauty(P,7); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Mon Dec 5 2005 17:28:08 on modular [Seed = 1468050070] ------------------------------------- TwoSelmerGroupData( J: Jacobian of Hyperelliptic Curve defined by y^2 = x^5 - 3456 ... ) TwoSelmerWork( C: Hyperelliptic Curve defined by y^2 = x^5 - 3456 over Rationa..., BoundType: Default, Bound: 0, UseUnits: true, Points: {}, Fields: {} ) LocalImageOdd( f: x^5 - 3456, p: 2, m1: Mapping from: Univariate Polynomial Ring in x over Rational ..., m2: Mapping from: Univariate Polynomial Ring in x over Rational ..., v: 30, Points: {} ) try( ) try( d: 2 ) SquarefreeRoots( F: $.1^4 + O(2^50)*$.1^3 - (479125739405285*2^3 + O(2^53))*$.1^... ) In file "/usr/local/magma/package/Geometry/Arith/sqfroots.m", line 84, column 15: >> return Roots(F:IsSquarefree); ^ Runtime error in 'Roots': Insufficient precision to find roots >> r:=TwoSelmerGroupData(J);r; ^ User error: Identifier 'r' has not been declared or assigned Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] >> Height(P); ^ User error: Identifier 'P' has not been declared or assigned 7.790638689351186835930798024215308990709 6.587156720882539675766293909746836340427 Total time: 0.750 seconds, Total memory usage: 5.79MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 17:28:04 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^7*3^3; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); //V:=RationalPoints(J:Bound:=1000); //B:=ReducedBasis(V); //B; //P:=J!B[1]; //Q:=J![x-2,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); //Chabauty(P,7); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Mon Dec 5 2005 17:28:03 on modular [Seed = 1351069082] ------------------------------------- TwoSelmerGroupData( J: Jacobian of Hyperelliptic Curve defined by y^2 = x^5 - 3456 ... ) TwoSelmerWork( C: Hyperelliptic Curve defined by y^2 = x^5 - 3456 over Rationa..., BoundType: Default, Bound: 0, UseUnits: true, Points: {}, Fields: {} ) LocalImageOdd( f: x^5 - 3456, p: 2, m1: Mapping from: Univariate Polynomial Ring in x over Rational ..., m2: Mapping from: Univariate Polynomial Ring in x over Rational ..., v: 30, Points: {} ) try( ) try( d: 2 ) SquarefreeRoots( F: $.1^4 + O(2^50)*$.1^3 + (363636024410139*2^3 + O(2^53))*$.1^... ) In file "/usr/local/magma/package/Geometry/Arith/sqfroots.m", line 84, column 15: >> return Roots(F:IsSquarefree); ^ Runtime error in 'Roots': Insufficient precision to find roots >> r:=TwoSelmerGroupData(J);r; ^ User error: Identifier 'r' has not been declared or assigned Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] >> Height(P); ^ User error: Identifier 'P' has not been declared or assigned 7.790638689351186835930798024215308990709 6.587156720882539675766293909746836340427 Total time: 0.670 seconds, Total memory usage: 5.69MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 17:28:01 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^7*3^3; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); //V:=RationalPoints(J:Bound:=1000); //B:=ReducedBasis(V); //B; //P:=J!B[1]; //Q:=J![x-2,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); //Chabauty(P,7); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Mon Dec 5 2005 17:28:00 on modular [Seed = 1569366164] ------------------------------------- TwoSelmerGroupData( J: Jacobian of Hyperelliptic Curve defined by y^2 = x^5 - 3456 ... ) TwoSelmerWork( C: Hyperelliptic Curve defined by y^2 = x^5 - 3456 over Rationa..., BoundType: Default, Bound: 0, UseUnits: true, Points: {}, Fields: {} ) LocalImageOdd( f: x^5 - 3456, p: 2, m1: Mapping from: Univariate Polynomial Ring in x over Rational ..., m2: Mapping from: Univariate Polynomial Ring in x over Rational ..., v: 30, Points: {} ) try( ) try( d: 2 ) SquarefreeRoots( F: $.1^4 + O(2^50)*$.1^3 - (532686333542373*2^3 + O(2^53))*$.1^... ) In file "/usr/local/magma/package/Geometry/Arith/sqfroots.m", line 84, column 15: >> return Roots(F:IsSquarefree); ^ Runtime error in 'Roots': Insufficient precision to find roots >> r:=TwoSelmerGroupData(J);r; ^ User error: Identifier 'r' has not been declared or assigned Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] >> Height(P); ^ User error: Identifier 'P' has not been declared or assigned 7.790638689351186835930798024215308990709 6.587156720882539675766293909746836340427 Total time: 1.229 seconds, Total memory usage: 6.36MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 17:21:47 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^5*3^8; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-2,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); //Chabauty(P,7); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Mon Dec 5 2005 17:21:37 on modular [Seed = 1656069161] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x - 18, 1296, 1) ] 1.386675839719476251203965940701820868253 11.892918185334237139088150509239494492736 9.325001442362978234527468537544353265163 Total time: 9.410 seconds, Total memory usage: 39.32MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 16:49:26 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^5*3^2; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-2,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); Chabauty(P,19); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Mon Dec 5 2005 16:49:24 on modular [Seed = 920764651] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 + 12, 6*x + 12, 2) ] 1.9005600093018022353148889820639309934627735633382760835424 6.297987188505878218328823948992073015360 4.930552287690539468946487589854250446546 {@ @} Total time: 1.740 seconds, Total memory usage: 39.32MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 16:49:18 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^5*3^2; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-2,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); Chabauty(P,17); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Mon Dec 5 2005 16:49:15 on modular [Seed = 870630870] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 + 12, 6*x + 12, 2) ] 1.9005600093018022353148889820639309934627735633382760835424 6.297987188505878218328823948992073015360 4.930552287690539468946487589854250446546 {@ <34002, 1, 4, 1>, <41826, 1, 4, 1>, <7056, 1, 4, 1>, <37191, 1, 4, 1>, <3354, 1, 4, 1>, <0, 1, 4, 1>, <68570, 1, 4, 1>, <39624, 1, 4, 1> @} Total time: 2.660 seconds, Total memory usage: 39.32MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 16:49:10 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^5*3^2; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-2,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); Chabauty(P,13); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Mon Dec 5 2005 16:49:08 on modular [Seed = 1022085841] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 + 12, 6*x + 12, 2) ] 1.9005600093018022353148889820639309934627735633382760835424 6.297987188505878218328823948992073015360 4.930552287690539468946487589854250446546 {@ <12435, 1, 4, 1>, <5943, 1, 4, 1>, <27236, 1, 4, 1>, <12180, 1, 4, 1>, <21368, 1, 4, 1>, <8039, 1, 4, 1> @} Total time: 2.209 seconds, Total memory usage: 39.32MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 16:48:54 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^5*3^2; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-2,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); Chabauty(P,11); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Mon Dec 5 2005 16:48:53 on modular [Seed = 971952069] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 + 12, 6*x + 12, 2) ] 1.9005600093018022353148889820639309934627735633382760835424 6.297987188505878218328823948992073015360 4.930552287690539468946487589854250446546 {@ <0, 1, 5, 1> @} Total time: 1.600 seconds, Total memory usage: 39.32MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 16:48:45 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^5*3^2; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-2,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); Chabauty(P,7); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Mon Dec 5 2005 16:48:43 on modular [Seed = 3239458624] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 + 12, 6*x + 12, 2) ] 1.9005600093018022353148889820639309934627735633382760835424 6.297987188505878218328823948992073015360 4.930552287690539468946487589854250446546 {@ <866, 1, 4, 1>, <1342, 1, 4, 1>, <920, 1, 4, 1>, <648, 1, 4, 1> @} Total time: 1.870 seconds, Total memory usage: 39.32MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 16:45:22 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^5*3^2; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-2,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); //Chabauty(P,29); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Mon Dec 5 2005 16:45:21 on modular [Seed = 3474470916] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 + 12, 6*x + 12, 2) ] 1.9005600093018022353148889820639309934627735633382760835424 6.297987188505878218328823948992073015360 4.930552287690539468946487589854250446546 Total time: 1.350 seconds, Total memory usage: 39.32MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 16:42:19 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^5*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-2,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); Chabauty(P,29); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Mon Dec 5 2005 16:42:17 on modular [Seed = 3573686998] ------------------------------------- 1 Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 to JacHyp: J given by a rule [no inverse] [ (x - 6, -88, 1) ] 2.45262499493363818734406374479496930832252395894370636996241 4.486499277983286154053028928413021210016 3.465735902799726547086160607290882840364 {@ <6, 1, 4, 1> @} Total time: 2.069 seconds, Total memory usage: 39.58MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 16:42:10 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^5*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-2,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); Chabauty(P,23); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Mon Dec 5 2005 16:42:07 on modular [Seed = 3490129861] ------------------------------------- 1 Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 to JacHyp: J given by a rule [no inverse] [ (x - 6, -88, 1) ] 2.45262499493363818734406374479496930832252395894370636996241 4.486499277983286154053028928413021210016 3.465735902799726547086160607290882840364 {@ <6, 1, 4, 1>, <219276, 1, 4, 1>, <12216, 1, 4, 1>, <184709, 1, 4, 1>, <77609, 1, 4, 1>, <50827, 1, 4, 1>, <88495, 1, 4, 1>, <199560, 1, 4, 1>, <1347, 1, 4, 1>, <216043, 1, 4, 1>, <181837, 1, 4, 1> @} Total time: 3.319 seconds, Total memory usage: 39.58MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 16:41:56 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^5*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-2,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); Chabauty(P,19); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Mon Dec 5 2005 16:41:54 on modular [Seed = 3741853895] ------------------------------------- 1 Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 to JacHyp: J given by a rule [no inverse] [ (x - 6, -88, 1) ] 2.45262499493363818734406374479496930832252395894370636996241 4.486499277983286154053028928413021210016 3.465735902799726547086160607290882840364 {@ <6, 1, 4, 1>, <104373, 1, 4, 1> @} Total time: 2.009 seconds, Total memory usage: 39.58MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 16:41:46 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^5*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-2,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); Chabauty(P,17); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Mon Dec 5 2005 16:41:43 on modular [Seed = 3845264072] ------------------------------------- 1 Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 to JacHyp: J given by a rule [no inverse] [ (x - 6, -88, 1) ] 2.45262499493363818734406374479496930832252395894370636996241 4.486499277983286154053028928413021210016 3.465735902799726547086160607290882840364 {@ <6, 1, 4, 1>, <0, 1, 4, 1>, <31710, 1, 4, 1>, <70973, 1, 4, 1>, <40187, 1, 4, 1>, <15766, 1, 4, 1>, <76595, 1, 4, 1> @} Total time: 2.819 seconds, Total memory usage: 39.58MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 16:41:36 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^5*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-2,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); Chabauty(P,13); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Mon Dec 5 2005 16:41:34 on modular [Seed = 3761706964] ------------------------------------- 1 Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 to JacHyp: J given by a rule [no inverse] [ (x - 6, -88, 1) ] 2.45262499493363818734406374479496930832252395894370636996241 4.486499277983286154053028928413021210016 3.465735902799726547086160607290882840364 {@ <6, 1, 4, 1>, <9054, 1, 4, 1> @} Total time: 1.940 seconds, Total memory usage: 39.58MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 16:40:59 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^5*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; Q:=J![x-2,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); Chabauty(P,11); //Order(Q);Order(P); Order(Q); Output: Magma V2.11-10 Mon Dec 5 2005 16:40:57 on modular [Seed = 4079223584] ------------------------------------- 1 Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 to JacHyp: J given by a rule [no inverse] [ (x - 6, -88, 1) ] 2.45262499493363818734406374479496930832252395894370636996241 4.486499277983286154053028928413021210016 3.465735902799726547086160607290882840364 {@ <0, 1, 0, 1> @} 2 Total time: 1.810 seconds, Total memory usage: 39.58MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 16:39:57 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^5*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-2,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); Chabauty(P,11); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Mon Dec 5 2005 16:39:56 on modular [Seed = 2520868352] ------------------------------------- 1 Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 to JacHyp: J given by a rule [no inverse] [ (x - 6, -88, 1) ] 2.45262499493363818734406374479496930832252395894370636996241 4.486499277983286154053028928413021210016 3.465735902799726547086160607290882840364 {@ <0, 1, 0, 1> @} Total time: 1.780 seconds, Total memory usage: 39.58MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 16:39:27 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^5*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-2,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); Chabauty(P,7); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Mon Dec 5 2005 16:39:24 on modular [Seed = 2638900523] ------------------------------------- 1 Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 to JacHyp: J given by a rule [no inverse] [ (x - 6, -88, 1) ] 2.45262499493363818734406374479496930832252395894370636996241 4.486499277983286154053028928413021210016 3.465735902799726547086160607290882840364 {@ <6, 1, 4, 1>, <43, 1, 4, 1>, <63414557, 1, 10, 1>, <6923584, 1, 10, 1> @} Total time: 2.129 seconds, Total memory usage: 39.58MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 16:39:01 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^5*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; Q:=J![x-2,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); RationalPoints(J:Bound:=1000); //Chabauty(P,19); //Order(Q);Order(P); P+Q; Output: Magma V2.11-10 Mon Dec 5 2005 16:38:59 on modular [Seed = 2893765861] ------------------------------------- 1 Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 to JacHyp: J given by a rule [no inverse] [ (x - 6, -88, 1) ] 2.45262499493363818734406374479496930832252395894370636996241 4.486499277983286154053028928413021210016 3.465735902799726547086160607290882840364 {@ (1, 0, 0), (x - 2, 0, 1), (x - 6, 88, 1), (x - 6, -88, 1), (x^2 - 8*x + 12, 22*x - 44, 2), (x^2 - 8*x + 12, -22*x + 44, 2) @} (x^2 - 8*x + 12, -22*x + 44, 2) Total time: 1.500 seconds, Total memory usage: 39.58MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 16:38:47 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^5*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; Q:=J![x-2,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); RationalPoints(J:Bound:=1000); //Chabauty(P,19); //Order(Q);Order(P); //P+Q; Output: Magma V2.11-10 Mon Dec 5 2005 16:38:45 on modular [Seed = 2877053940] ------------------------------------- 1 Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 to JacHyp: J given by a rule [no inverse] [ (x - 6, -88, 1) ] 2.45262499493363818734406374479496930832252395894370636996241 4.486499277983286154053028928413021210016 3.465735902799726547086160607290882840364 {@ (1, 0, 0), (x - 2, 0, 1), (x - 6, 88, 1), (x - 6, -88, 1), (x^2 - 8*x + 12, 22*x - 44, 2), (x^2 - 8*x + 12, -22*x + 44, 2) @} Total time: 1.500 seconds, Total memory usage: 39.58MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 16:36:23 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^5*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); RationalPoints(J:Bound:=1000); //Chabauty(P,19); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 16:36:22 on modular [Seed = 3009692255] ------------------------------------- 1 Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 to JacHyp: J given by a rule [no inverse] [ (x - 6, -88, 1) ] 2.45262499493363818734406374479496930832252395894370636996241 4.486499277983286154053028928413021210016 3.465735902799726547086160607290882840364 {@ (1, 0, 0), (x - 2, 0, 1), (x - 6, 88, 1), (x - 6, -88, 1), (x^2 - 8*x + 12, 22*x - 44, 2), (x^2 - 8*x + 12, -22*x + 44, 2) @} Total time: 1.490 seconds, Total memory usage: 39.58MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 16:35:42 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^5*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); //Chabauty(P,19); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 16:35:41 on modular [Seed = 3194572585] ------------------------------------- 1 Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 to JacHyp: J given by a rule [no inverse] [ (x - 6, -88, 1) ] 2.45262499493363818734406374479496930832252395894370636996241 4.486499277983286154053028928413021210016 3.465735902799726547086160607290882840364 Total time: 1.490 seconds, Total memory usage: 39.58MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 16:32:44 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^4*3^5; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); Chabauty(P,19); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 16:32:43 on modular [Seed = 1167243140] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 4*x + 12, -22*x + 48, 2) ] 2.9580260296990585038462787524616012507096372251349783007517 8.583595559989264458353773970777672753291 6.665678744653461978792156649393850810454 {@ @} Total time: 1.770 seconds, Total memory usage: 39.24MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 16:32:31 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^4*3^5; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); Chabauty(P,17); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 16:32:29 on modular [Seed = 1083683479] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 4*x + 12, -22*x + 48, 2) ] 2.9580260296990585038462787524616012507096372251349783007517 8.583595559989264458353773970777672753291 6.665678744653461978792156649393850810454 {@ @} Total time: 1.740 seconds, Total memory usage: 39.19MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 16:32:23 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^4*3^5; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); Chabauty(P,13); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 16:32:20 on modular [Seed = 1335410065] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 4*x + 12, -22*x + 48, 2) ] 2.9580260296990585038462787524616012507096372251349783007517 8.583595559989264458353773970777672753291 6.665678744653461978792156649393850810454 {@ <12692, 1, 4, 1>, <13291, 1, 4, 1> @} Total time: 1.840 seconds, Total memory usage: 39.21MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 16:32:15 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^4*3^5; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); Chabauty(P,11); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 16:32:12 on modular [Seed = 1251850412] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 4*x + 12, -22*x + 48, 2) ] 2.9580260296990585038462787524616012507096372251349783007517 8.583595559989264458353773970777672753291 6.665678744653461978792156649393850810454 {@ <13950, 1, 4, 1>, <4406, 1, 4, 1>, <11263, 1, 4, 1>, <3142, 1, 4, 1>, <12460, 1, 4, 1> @} Total time: 2.149 seconds, Total memory usage: 39.17MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 16:32:05 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^4*3^5; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); Chabauty(P,7); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 16:32:03 on modular [Seed = 1367778232] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 4*x + 12, -22*x + 48, 2) ] 2.9580260296990585038462787524616012507096372251349783007517 8.583595559989264458353773970777672753291 6.665678744653461978792156649393850810454 {@ <157, 1, 4, 1>, <0, 1, 4, 1> @} Total time: 1.810 seconds, Total memory usage: 39.18MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 16:27:30 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^4*3^5; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); //Chabauty(P,19); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 16:27:29 on modular [Seed = 1602792612] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 4*x + 12, -22*x + 48, 2) ] 2.9580260296990585038462787524616012507096372251349783007517 8.583595559989264458353773970777672753291 6.665678744653461978792156649393850810454 Total time: 1.340 seconds, Total memory usage: 39.20MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 16:24:58 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^4*3^4; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1804); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); Chabauty(P,19); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 16:24:56 on modular [Seed = 1706202743] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 3*x + 18, 3*x - 18, 2) ] 1.7439335091906151447535514932141656497618318953142006958004 7.0577094035575824236560705668075205893059 5.933270552208055517861993158112167007351 {@ @} Total time: 2.029 seconds, Total memory usage: 39.09MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 16:23:43 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^4*3^4; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=100); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); Chabauty(P,19); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 16:23:41 on modular [Seed = 1622641799] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 3*x + 18, 3*x - 18, 2) ] 1.7439335091906151447535514932141656497618318953142006958004 7.0577094035575824236560705668075205893059 5.933270552208055517861993158112167007351 {@ @} Total time: 1.500 seconds, Total memory usage: 38.99MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 16:23:35 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^4*3^4; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=100); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=20); //RationalPoints(J:Bound:=1000); Chabauty(P,19); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 16:23:34 on modular [Seed = 1874363267] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 3*x + 18, 3*x - 18, 2) ] 1.7439335091906151447535514932141656497618318953142006958004 7.0577094035575824236560705668075205893059 5.933270552208055517861993158112167007351 {@ @} Total time: 1.500 seconds, Total memory usage: 38.99MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 16:23:22 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^4*3^4; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=100); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=10); //RationalPoints(J:Bound:=1000); Chabauty(P,19); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 16:23:20 on modular [Seed = 1790805656] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 3*x + 18, 3*x - 18, 2) ] 1.7439335091906151447535514932141656497618318953142006958004 7.0577094035575824236560705668075205893059 5.933270552208055517861993158112167007351 {@ @} Total time: 1.500 seconds, Total memory usage: 38.99MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 16:22:27 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^4*3^4; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=100); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); Chabauty(P,19); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 16:22:25 on modular [Seed = 1940168664] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 3*x + 18, 3*x - 18, 2) ] 1.7439335091906151447535514932141656497618318953142006958004 7.0577094035575824236560705668075205893059 5.933270552208055517861993158112167007351 {@ @} Total time: 1.500 seconds, Total memory usage: 38.99MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 16:22:18 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^4*3^4; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=100); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); Chabauty(P,17); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 16:22:16 on modular [Seed = 2125042380] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 3*x + 18, 3*x - 18, 2) ] 1.7439335091906151447535514932141656497618318953142006958004 7.0577094035575824236560705668075205893059 5.933270552208055517861993158112167007351 {@ <47711, 1, 4, 1>, <33808, 1, 4, 1>, <21178, 1, 4, 1>, <46068, 1, 4, 1>, <0, 1, 4, 1>, <32331, 1, 4, 1>, <14664, 1, 4, 1>, <48079, 1, 4, 1> @} Total time: 2.500 seconds, Total memory usage: 38.99MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 16:22:09 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^4*3^4; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=100); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); Chabauty(P,13); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 16:22:07 on modular [Seed = 97708352] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 3*x + 18, 3*x - 18, 2) ] 1.7439335091906151447535514932141656497618318953142006958004 7.0577094035575824236560705668075205893059 5.933270552208055517861993158112167007351 {@ <132, 1, 4, 1>, <23152, 1, 4, 1>, <21272, 1, 4, 1>, <307, 1, 4, 1>, <0, 1, 4, 1> @} Total time: 2.020 seconds, Total memory usage: 38.99MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 16:22:01 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^4*3^4; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=100); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); Chabauty(P,11); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 16:21:58 on modular [Seed = 14150732] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 3*x + 18, 3*x - 18, 2) ] 1.7439335091906151447535514932141656497618318953142006958004 7.0577094035575824236560705668075205893059 5.933270552208055517861993158112167007351 {@ <5956, 1, 4, 1>, <13803, 1, 4, 1>, <10084, 1, 4, 1>, <728, 1, 4, 1>, <5209, 1, 4, 1>, <8231, 1, 4, 1>, <8488, 1, 4, 1>, <10891, 1, 4, 1>, <13875, 1, 4, 1>, <10460, 1, 4, 1> @} Total time: 2.450 seconds, Total memory usage: 38.99MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 16:21:51 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^4*3^4; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=100); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); Chabauty(P,7); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 16:21:49 on modular [Seed = 265872216] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 3*x + 18, 3*x - 18, 2) ] 1.7439335091906151447535514932141656497618318953142006958004 7.0577094035575824236560705668075205893059 5.933270552208055517861993158112167007351 {@ <1090, 1, 4, 1> @} Total time: 1.409 seconds, Total memory usage: 38.99MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 16:20:18 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^4*3^2; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=100); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); Chabauty(P,7); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 16:20:16 on modular [Seed = 381811260] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x - 10, 316, 1) ] 3.42486350460119354543892077010836727047869921855726085241610 5.586784795732860417404869965061177273340 4.468454167317242596001666175548799401169 {@ <10, 1, 4, 1>, <381, 1, 4, 1> @} Total time: 1.459 seconds, Total memory usage: 39.30MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 16:20:01 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^4*3^2; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=100); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); Chabauty(P,11); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 16:20:00 on modular [Seed = 499839271] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x - 10, 316, 1) ] 3.42486350460119354543892077010836727047869921855726085241610 5.586784795732860417404869965061177273340 4.468454167317242596001666175548799401169 {@ <10, 1, 5, 1> @} Total time: 1.409 seconds, Total memory usage: 39.30MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 16:17:29 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^4*3^2; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=100); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); //Chabauty(P,11); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 16:17:28 on modular [Seed = 636668023] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x - 10, 316, 1) ] 3.42486350460119354543892077010836727047869921855726085241610 5.586784795732860417404869965061177273340 4.468454167317242596001666175548799401169 Total time: 1.120 seconds, Total memory usage: 39.30MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 16:15:36 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^4*3^1; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=100); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); Chabauty(P,11); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 16:15:34 on modular [Seed = 754696928] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 12*x + 28, 38*x - 104, 2) ] 3.5469721771777763679248622062281245897778981929410244076538 4.547022304274841971998782326210084579449 3.736045974871836135071502684267115598090 {@ @} Total time: 1.449 seconds, Total memory usage: 39.10MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 16:15:25 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^4*3^1; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=100); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); Chabauty(P,7); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 16:15:23 on modular [Seed = 737986042] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 12*x + 28, 38*x - 104, 2) ] 3.5469721771777763679248622062281245897778981929410244076538 4.547022304274841971998782326210084579449 3.736045974871836135071502684267115598090 {@ <848, 1, 4, 1>, <677, 1, 4, 1>, <0, 1, 4, 1> @} Total time: 1.570 seconds, Total memory usage: 39.08MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 16:12:29 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^4*3^1; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=100); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); //Chabauty(P,29); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 16:12:27 on modular [Seed = 920770615] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 12*x + 28, 38*x - 104, 2) ] 3.5469721771777763679248622062281245897778981929410244076538 4.547022304274841971998782326210084579449 3.736045974871836135071502684267115598090 Total time: 1.100 seconds, Total memory usage: 39.08MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 16:09:43 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^4*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=100); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); Chabauty(P,29); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 16:09:42 on modular [Seed = 1022088080] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x - 2, 4, 1) ] 0.6968251359498457086204281863462458917705803822461774257029 4.1563368162387419397237618820788837781216 3.00363778242642967414133919298543179498766 {@ <2, 1, 4, 1> @} Total time: 1.740 seconds, Total memory usage: 39.10MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 16:09:36 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^4*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=100); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); Chabauty(P,23); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 16:09:32 on modular [Seed = 1005376653] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x - 2, 4, 1) ] 0.6968251359498457086204281863462458917705803822461774257029 4.1563368162387419397237618820788837781216 3.00363778242642967414133919298543179498766 {@ <2, 1, 4, 1>, <149367, 1, 4, 1>, <249855, 1, 4, 1>, <165386, 1, 4, 1>, <259234, 1, 4, 1>, <128789, 1, 4, 1>, <137229, 1, 4, 1>, <155916, 1, 4, 1>, <63763, 1, 4, 1>, <67400, 1, 4, 1>, <127439, 1, 4, 1> @} Total time: 3.029 seconds, Total memory usage: 39.10MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 16:09:22 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^4*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=100); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); Chabauty(P,19); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 16:09:21 on modular [Seed = 3339724925] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x - 2, 4, 1) ] 0.6968251359498457086204281863462458917705803822461774257029 4.1563368162387419397237618820788837781216 3.00363778242642967414133919298543179498766 {@ <2, 1, 4, 1>, <62911, 1, 4, 1> @} Total time: 1.710 seconds, Total memory usage: 39.10MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 16:09:12 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^4*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=100); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); Chabauty(P,17); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 16:09:10 on modular [Seed = 3424330625] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x - 2, 4, 1) ] 0.6968251359498457086204281863462458917705803822461774257029 4.1563368162387419397237618820788837781216 3.00363778242642967414133919298543179498766 {@ <2, 1, 4, 1>, <8607, 1, 4, 1>, <0, 1, 4, 1>, <6846, 1, 4, 1>, <10605, 1, 4, 1>, <13759, 1, 4, 1>, <61626, 1, 4, 1> @} Total time: 2.529 seconds, Total memory usage: 39.10MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 16:09:03 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^4*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=100); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); Chabauty(P,13); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 16:09:01 on modular [Seed = 3374196383] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x - 2, 4, 1) ] 0.6968251359498457086204281863462458917705803822461774257029 4.1563368162387419397237618820788837781216 3.00363778242642967414133919298543179498766 {@ <2, 1, 4, 1>, <20355, 1, 4, 1>, <0, 1, 4, 1>, <7240, 1, 4, 1>, <24204, 1, 4, 1> @} Total time: 2.060 seconds, Total memory usage: 39.10MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 16:08:53 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^4*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=100); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); Chabauty(P,11); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 16:08:51 on modular [Seed = 3658298754] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x - 2, 4, 1) ] 0.6968251359498457086204281863462458917705803822461774257029 4.1563368162387419397237618820788837781216 3.00363778242642967414133919298543179498766 {@ <2, 1, 4, 1>, <10898, 1, 4, 1>, <5044, 1, 4, 1>, <3222, 1, 4, 1> @} Total time: 1.929 seconds, Total memory usage: 39.10MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 16:08:46 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^4*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=100); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); Chabauty(P,7); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 16:08:44 on modular [Seed = 3878684800] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x - 2, 4, 1) ] 0.6968251359498457086204281863462458917705803822461774257029 4.1563368162387419397237618820788837781216 3.00363778242642967414133919298543179498766 {@ <2, 1, 4, 1>, <1811, 1, 4, 1>, <153, 1, 4, 1>, <930, 1, 4, 1> @} Total time: 1.669 seconds, Total memory usage: 39.10MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 16:07:43 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^4*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=100); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); Chabauty(P,19); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 16:07:42 on modular [Seed = 3996713419] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x - 2, 4, 1) ] 0.6968251359498457086204281863462458917705803822461774257029 4.1563368162387419397237618820788837781216 3.00363778242642967414133919298543179498766 {@ <2, 1, 4, 1>, <62911, 1, 4, 1> @} Total time: 1.700 seconds, Total memory usage: 39.10MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 16:07:36 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^4*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=100); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); Chabauty(P,17); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 16:07:33 on modular [Seed = 3946579141] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x - 2, 4, 1) ] 0.6968251359498457086204281863462458917705803822461774257029 4.1563368162387419397237618820788837781216 3.00363778242642967414133919298543179498766 {@ <2, 1, 4, 1>, <8607, 1, 4, 1>, <0, 1, 4, 1>, <6846, 1, 4, 1>, <10605, 1, 4, 1>, <13759, 1, 4, 1>, <61626, 1, 4, 1> @} Total time: 2.500 seconds, Total memory usage: 39.10MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 16:07:25 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^4*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=100); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); Chabauty(P,13); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 16:07:22 on modular [Seed = 4029095873] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x - 2, 4, 1) ] 0.6968251359498457086204281863462458917705803822461774257029 4.1563368162387419397237618820788837781216 3.00363778242642967414133919298543179498766 {@ <2, 1, 4, 1>, <20355, 1, 4, 1>, <0, 1, 4, 1>, <7240, 1, 4, 1>, <24204, 1, 4, 1> @} Total time: 2.089 seconds, Total memory usage: 39.10MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 16:07:17 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^4*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=100); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); Chabauty(P,11); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 16:07:15 on modular [Seed = 4247392965] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x - 2, 4, 1) ] 0.6968251359498457086204281863462458917705803822461774257029 4.1563368162387419397237618820788837781216 3.00363778242642967414133919298543179498766 {@ <2, 1, 4, 1>, <10898, 1, 4, 1>, <5044, 1, 4, 1>, <3222, 1, 4, 1> @} Total time: 1.940 seconds, Total memory usage: 39.10MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 16:07:07 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^4*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=100); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); Chabauty(P,7); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 16:07:05 on modular [Seed = 4197257784] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x - 2, 4, 1) ] 0.6968251359498457086204281863462458917705803822461774257029 4.1563368162387419397237618820788837781216 3.00363778242642967414133919298543179498766 {@ <2, 1, 4, 1>, <1811, 1, 4, 1>, <153, 1, 4, 1>, <930, 1, 4, 1> @} Total time: 1.659 seconds, Total memory usage: 39.10MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 16:04:03 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^4*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=100); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); //Chabauty(P,31); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 16:04:01 on modular [Seed = 2203347961] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x - 2, 4, 1) ] 0.6968251359498457086204281863462458917705803822461774257029 4.1563368162387419397237618820788837781216 3.00363778242642967414133919298543179498766 Total time: 1.169 seconds, Total memory usage: 39.10MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 16:03:36 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^4*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=100); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; //Height(P); //HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); //Chabauty(P,31); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 16:03:35 on modular [Seed = 2287952020] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x - 2, 4, 1) ] Total time: 1.030 seconds, Total memory usage: 39.10MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 15:51:33 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^3*3^8; C:=HyperellipticCurve(f); J:=Jacobian(C); //r:=TwoSelmerGroupData(J);r; //TorsionSubgroup(J); V:=RationalPoints(J:Bound:=100); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; //Height(P); //HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); Chabauty(P,31); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 15:51:22 on modular [Seed = 653380951] ------------------------------------- [ (x - 9, -81, 1) ] {@ <9, 1, 4, 1>, <178093, 1, 4, 1>, <402744, 1, 4, 1>, <210384, 1, 4, 1>, <381074, 1, 4, 1>, <456991, 1, 4, 1>, <597056, 1, 4, 1>, <517834, 1, 4, 1>, <218317, 1, 4, 1>, <486504, 1, 4, 1>, <48365, 1, 4, 1>, <210439, 1, 4, 1>, <310588, 1, 4, 1>, <784202, 1, 4, 1>, <530206, 1, 4, 1> @} Total time: 11.279 seconds, Total memory usage: 37.70MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 15:51:15 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^3*3^8; C:=HyperellipticCurve(f); J:=Jacobian(C); //r:=TwoSelmerGroupData(J);r; //TorsionSubgroup(J); V:=RationalPoints(J:Bound:=100); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; //Height(P); //HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); Chabauty(P,29); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 15:51:05 on modular [Seed = 603244577] ------------------------------------- [ (x - 9, -81, 1) ] {@ <9, 1, 4, 1>, <683665, 1, 4, 1> @} Total time: 9.890 seconds, Total memory usage: 37.70MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 15:50:54 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^3*3^8; C:=HyperellipticCurve(f); J:=Jacobian(C); //r:=TwoSelmerGroupData(J);r; //TorsionSubgroup(J); V:=RationalPoints(J:Bound:=100); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; //Height(P); //HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); Chabauty(P,23); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 15:50:43 on modular [Seed = 687850288] ------------------------------------- [ (x - 9, -81, 1) ] {@ <9, 1, 4, 1>, <143999, 1, 4, 1>, <42845, 1, 4, 1>, <106066, 1, 4, 1>, <19302, 1, 4, 1>, <53815, 1, 4, 1>, <5678, 1, 4, 1>, <30257, 1, 4, 1>, <15766, 1, 4, 1>, <31409, 1, 4, 1>, <110347, 1, 4, 1>, <263785, 1, 4, 1> @} Total time: 11.130 seconds, Total memory usage: 37.70MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 15:50:33 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^3*3^8; C:=HyperellipticCurve(f); J:=Jacobian(C); //r:=TwoSelmerGroupData(J);r; //TorsionSubgroup(J); V:=RationalPoints(J:Bound:=100); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; //Height(P); //HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); Chabauty(P,19); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 15:50:23 on modular [Seed = 904060417] ------------------------------------- [ (x - 9, -81, 1) ] {@ <9, 1, 4, 1> @} Total time: 9.689 seconds, Total memory usage: 37.70MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 15:50:09 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^3*3^8; C:=HyperellipticCurve(f); J:=Jacobian(C); //r:=TwoSelmerGroupData(J);r; //TorsionSubgroup(J); V:=RationalPoints(J:Bound:=100); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); Chabauty(P,17); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 15:49:57 on modular [Seed = 853923580] ------------------------------------- [ (x - 9, -81, 1) ] 1.6642840212335934079377833369579033923280931581487693872132 10.2429006144863725275999473024470300199948 8.400805201616384488637825708933451174361 {@ <9, 1, 4, 1> @} Total time: 11.599 seconds, Total memory usage: 37.70MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 15:49:51 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^3*3^8; C:=HyperellipticCurve(f); J:=Jacobian(C); //r:=TwoSelmerGroupData(J);r; //TorsionSubgroup(J); V:=RationalPoints(J:Bound:=100); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); Chabauty(P,13); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 15:49:39 on modular [Seed = 1072225255] ------------------------------------- [ (x - 9, -81, 1) ] 1.6642840212335934079377833369579033923280931581487693872132 10.2429006144863725275999473024470300199948 8.400805201616384488637825708933451174361 {@ <9, 1, 4, 1>, <9591, 1, 4, 1>, <21818, 1, 4, 1>, <21730, 1, 4, 1>, <23727, 1, 4, 1>, <14172, 1, 4, 1>, <3128, 1, 4, 1> @} Total time: 12.009 seconds, Total memory usage: 37.70MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 15:49:24 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^3*3^8; C:=HyperellipticCurve(f); J:=Jacobian(C); //r:=TwoSelmerGroupData(J);r; //TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); Chabauty(P,13); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 15:49:12 on modular [Seed = 3407621943] ------------------------------------- [ (x - 9, -81, 1) ] 1.6642840212335934079377833369579033923280931581487693872132 10.2429006144863725275999473024470300199948 8.400805201616384488637825708933451174361 {@ <9, 1, 4, 1>, <9591, 1, 4, 1>, <21818, 1, 4, 1>, <21730, 1, 4, 1>, <23727, 1, 4, 1>, <14172, 1, 4, 1>, <3128, 1, 4, 1> @} Total time: 12.160 seconds, Total memory usage: 37.70MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 15:49:04 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^3*3^8; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); Chabauty(P,13); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 15:48:48 on modular [Seed = 3691722840] ------------------------------------- TwoSelmerGroupData( J: Jacobian of Hyperelliptic Curve defined by y^2 = x^5 - 52488... ) TwoSelmerWork( C: Hyperelliptic Curve defined by y^2 = x^5 - 52488 over Ration..., BoundType: Default, Bound: 0, UseUnits: true, Points: {}, Fields: {} ) LocalImageOdd( f: x^5 - 52488, p: 2, m1: Mapping from: Univariate Polynomial Ring in x over Rational ..., m2: Mapping from: Univariate Polynomial Ring in x over Rational ..., v: 14, Points: {} ) try( ) try( d: 2 ) SquarefreeRoots( F: $.1^4 + O(2^50)*$.1^3 - (467387893081695*2^3 + O(2^53))*$.1^... ) In file "/usr/local/magma/package/Geometry/Arith/sqfroots.m", line 84, column 15: >> return Roots(F:IsSquarefree); ^ Runtime error in 'Roots': Insufficient precision to find roots >> r:=TwoSelmerGroupData(J);r; ^ User error: Identifier 'r' has not been declared or assigned Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x - 9, -81, 1) ] 1.6642840212335934079377833369579033923280931581487693872132 10.2429006144863725275999473024470300199948 8.400805201616384488637825708933451174361 {@ <9, 1, 4, 1>, <9591, 1, 4, 1>, <21818, 1, 4, 1>, <21730, 1, 4, 1>, <23727, 1, 4, 1>, <14172, 1, 4, 1>, <3128, 1, 4, 1> @} Total time: 12.449 seconds, Total memory usage: 39.41MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 15:48:36 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^3*3^8; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); Chabauty(P,11); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 15:48:22 on modular [Seed = 3607114613] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x - 9, -81, 1) ] 1.6642840212335934079377833369579033923280931581487693872132 10.2429006144863725275999473024470300199948 8.400805201616384488637825708933451174361 {@ <9, 1, 4, 1>, <8071, 1, 4, 1>, <10913, 1, 4, 1>, <2668, 1, 4, 1>, <0, 1, 4, 1>, <409, 1, 4, 1>, <1657, 1, 4, 1>, <13764, 1, 4, 1>, <4218, 1, 4, 1>, <1539, 1, 4, 1> @} Total time: 13.169 seconds, Total memory usage: 39.40MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 15:48:04 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^3*3^8; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=10); //RationalPoints(J:Bound:=1000); Chabauty(P,7); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 15:47:52 on modular [Seed = 3913158053] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x - 9, -81, 1) ] 1.6642840212335934079377833369579033923280931581487693872132 10.2429006144863725275999473024470300199948 8.400805201616384488637825708933451174361 {@ <9, 1, 4, 1>, <1615, 1, 4, 1>, <2148, 1, 4, 1> @} Total time: 12.080 seconds, Total memory usage: 39.39MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 15:41:55 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^3*3^8; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=10); //RationalPoints(J:Bound:=1000); //Chabauty(P,19); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 15:41:44 on modular [Seed = 2270192315] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x - 9, -81, 1) ] 1.6642840212335934079377833369579033923280931581487693872132 10.2429006144863725275999473024470300199948 8.400805201616384488637825708933451174361 Total time: 11.609 seconds, Total memory usage: 39.40MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 15:41:12 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^3*3^8; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); //Chabauty(P,19); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 15:41:00 on modular [Seed = 2588769394] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x - 9, -81, 1) ] 1.6642840212335934079377833369579033923280931581487693872132 10.2429006144863725275999473024470300199948 8.400805201616384488637825708933451174361 Total time: 11.609 seconds, Total memory usage: 39.42MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 15:40:12 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^3*3^4; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); Chabauty(P,19); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 15:40:10 on modular [Seed = 2638904067] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 3*x + 9, 9*x - 18, 2) ] 2.5166999406878115072977293072304979085489602407477620799018 6.514124951463928861795457675272891001233 5.471172431834758644917171743806715961974 {@ @} Total time: 1.960 seconds, Total memory usage: 39.52MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 15:40:03 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^3*3^4; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); Chabauty(P,17); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 15:40:00 on modular [Seed = 2420602372] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 3*x + 9, 9*x - 18, 2) ] 2.5166999406878115072977293072304979085489602407477620799018 6.514124951463928861795457675272891001233 5.471172431834758644917171743806715961974 {@ <11307, 1, 4, 1> @} Total time: 2.100 seconds, Total memory usage: 39.43MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 15:39:54 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^3*3^4; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); Chabauty(P,13); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 15:39:52 on modular [Seed = 2877050119] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 3*x + 9, 9*x - 18, 2) ] 2.5166999406878115072977293072304979085489602407477620799018 6.514124951463928861795457675272891001233 5.471172431834758644917171743806715961974 {@ <16183, 1, 4, 1>, <1782, 1, 4, 1>, <6337, 1, 4, 1>, <2738, 1, 4, 1>, <15695, 1, 4, 1>, <18082, 1, 4, 1> @} Total time: 2.430 seconds, Total memory usage: 39.39MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 15:39:45 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^3*3^4; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); Chabauty(P,11); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 15:39:42 on modular [Seed = 2927183892] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 3*x + 9, 9*x - 18, 2) ] 2.5166999406878115072977293072304979085489602407477620799018 6.514124951463928861795457675272891001233 5.471172431834758644917171743806715961974 {@ <0, 1, 4, 1> @} Total time: 1.909 seconds, Total memory usage: 39.42MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 15:39:36 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^3*3^4; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); Chabauty(P,7); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 15:39:34 on modular [Seed = 2708883219] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 3*x + 9, 9*x - 18, 2) ] 2.5166999406878115072977293072304979085489602407477620799018 6.514124951463928861795457675272891001233 5.471172431834758644917171743806715961974 {@ <2150, 1, 4, 1>, <535, 1, 4, 1>, <125, 1, 4, 1> @} Total time: 1.960 seconds, Total memory usage: 39.41MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 15:38:41 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^3*3^4; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1275); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); //Chabauty(P,11); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 15:38:40 on modular [Seed = 2959562452] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 3*x + 9, 9*x - 18, 2) ] 2.5166999406878115072977293072304979085489602407477620799018 6.514124951463928861795457675272891001233 5.471172431834758644917171743806715961974 Total time: 1.649 seconds, Total memory usage: 39.48MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 15:37:34 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^3*3^4; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); //Chabauty(P,11); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 15:37:33 on modular [Seed = 1235136368] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 3*x + 9, 9*x - 18, 2) ] 2.5166999406878115072977293072304979085489602407477620799018 6.514124951463928861795457675272891001233 5.471172431834758644917171743806715961974 Total time: 1.500 seconds, Total memory usage: 39.40MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 15:37:26 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^3*3^4; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=10); //RationalPoints(J:Bound:=1000); //Chabauty(P,11); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 15:37:24 on modular [Seed = 1285270140] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 3*x + 9, 9*x - 18, 2) ] 2.5166999406878115072977293072304979085489602407477620799018 6.514124951463928861795457675272891001233 5.471172431834758644917171743806715961974 Total time: 1.560 seconds, Total memory usage: 39.50MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 15:37:16 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^3*3^4; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=1000); //Chabauty(P,11); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 15:37:14 on modular [Seed = 1200660617] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 3*x + 9, 9*x - 18, 2) ] 2.5166999406878115072977293072304979085489602407477620799018 6.514124951463928861795457675272891001233 5.471172431834758644917171743806715961974 Total time: 1.490 seconds, Total memory usage: 39.38MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 15:35:16 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^3*3^4; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=1); //RationalPoints(J:Bound:=1000); //Chabauty(P,11); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 15:35:13 on modular [Seed = 1502527212] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 3*x + 9, 9*x - 18, 2) ] 2.5166999406878115072977293072304979085489602407477620799018 8.803102712951222835961923441572148585428 5.471172431834758644917171743806715961974 Total time: 1.470 seconds, Total memory usage: 39.47MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 15:24:29 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^3*3^1; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=1); //RationalPoints(J:Bound:=1000); Chabauty(P,11); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 15:24:28 on modular [Seed = 1384493210] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 43/9*x + 73/9, -251/27*x + 554/27, 2) ] 5.16057625047072430311345959953099031630616226408707241792455 5.713556585783996031183901371149946289857 3.273947854498539262126681269961664552701 {@ @} Total time: 1.429 seconds, Total memory usage: 39.50MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 15:24:22 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^3*3^1; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=1); //RationalPoints(J:Bound:=1000); Chabauty(P,7); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 15:24:20 on modular [Seed = 1434630037] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 43/9*x + 73/9, -251/27*x + 554/27, 2) ] 5.16057625047072430311345959953099031630616226408707241792455 5.713556585783996031183901371149946289857 3.273947854498539262126681269961664552701 {@ <0, 1, 0, 1>, <1935, 1, 4, 1>, <0, 1, 4, 1>, <268, 1, 4, 1> @} Total time: 1.679 seconds, Total memory usage: 39.49MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 15:23:30 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^3*3^1; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1111); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=1); //RationalPoints(J:Bound:=1000); //Chabauty(P,11); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 15:23:29 on modular [Seed = 1757383325] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 43/9*x + 73/9, -251/27*x + 554/27, 2) ] 5.16057625047072430311345959953099031630616226408707241792455 5.713556585783996031183901371149946289857 3.273947854498539262126681269961664552701 Total time: 1.229 seconds, Total memory usage: 39.49MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 15:20:54 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^3*3^1; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=1); //RationalPoints(J:Bound:=1000); //Chabauty(P,11); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 15:20:53 on modular [Seed = 148892185] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 43/9*x + 73/9, -251/27*x + 554/27, 2) ] 5.16057625047072430311345959953099031630616226408707241792455 5.713556585783996031183901371149946289857 3.273947854498539262126681269961664552701 Total time: 1.240 seconds, Total memory usage: 39.56MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 15:17:09 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^3*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=1); //RationalPoints(J:Bound:=1000); Chabauty(P,11); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 15:17:07 on modular [Seed = 14148471] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 + x + 9, 5*x + 8, 2) ] 3.47426304826961249980683698951257553773742512428567231010826 4.830829590104535747512612652516993975442 2.541539662053132801196517778679980749598 {@ @} Total time: 1.520 seconds, Total memory usage: 39.56MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 15:16:58 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^3*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=1); //RationalPoints(J:Bound:=1000); Chabauty(P,7); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 15:16:56 on modular [Seed = 416283412] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 + x + 9, 5*x + 8, 2) ] 3.47426304826961249980683698951257553773742512428567231010826 4.830829590104535747512612652516993975442 2.541539662053132801196517778679980749598 {@ <626, 1, 4, 1>, <2399, 1, 4, 1>, <95, 1, 4, 1> @} Total time: 1.639 seconds, Total memory usage: 39.50MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 15:13:19 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^3*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=1); //RationalPoints(J:Bound:=1000); //Chabauty(P,19); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 15:13:18 on modular [Seed = 314953336] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 + x + 9, 5*x + 8, 2) ] 3.47426304826961249980683698951257553773742512428567231010826 4.830829590104535747512612652516993975442 2.541539662053132801196517778679980749598 Total time: 1.149 seconds, Total memory usage: 39.50MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 15:12:10 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^3*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; //Height(P); //HeightConstant(J:Effort:=1); //RationalPoints(J:Bound:=1000); //Chabauty(P,19); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 15:12:09 on modular [Seed = 603258638] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 + x + 9, 5*x + 8, 2) ] Total time: 1.060 seconds, Total memory usage: 39.48MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Mon Dec 5 15:11:55 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^3*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; //Height(P); //HeightConstant(J:Effort:=1); //RationalPoints(J:Bound:=1000); //Chabauty(P,19); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Mon Dec 5 2005 15:11:53 on modular [Seed = 653392411] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x - 1, 3, 1) ] Total time: 1.169 seconds, Total memory usage: 39.18MB '128.139' ************** MAGMA ***************** Host 128.139.226.37 (128.139.226.37) Time: Mon Dec 5 14:51:38 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); B:=Matrix(K,3,3,[[b,1,1],[0,a,1],[0,0,c]]); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*A*Transpose(P)-B)); G:=GroebnerBasis(I); print Factorization(G[10]); Output: Magma V2.11-10 Mon Dec 5 2005 14:51:37 on modular [Seed = 3323011002] ------------------------------------- [ <$.8^2 + (a - 2*b*c + b + c - 1)/(a*b + b^2 - b)*$.8*$.9 + (-2*b*c + c)/(a*b + b^2 - b)*$.8 + (a*c + c^2 - c)/(a*b + b^2 - b)*$.9^2 + (2*c^2 - c)/(a*b + b^2 - b)*$.9 + (-a*c + c^2)/(a*b + b^2 - b), 1>, <$.8^2 + (a - 2*b*c + b + c - 1)/(a*b + b^2 - b)*$.8*$.9 + (2*b*c - c)/(a*b + b^2 - b)*$.8 + (a*c + c^2 - c)/(a*b + b^2 - b)*$.9^2 + (-2*c^2 + c)/(a*b + b^2 - b)*$.9 + (-a*c + c^2)/(a*b + b^2 - b), 1> ] Total time: 0.980 seconds, Total memory usage: 4.27MB '128.139' ************** MAGMA ***************** Host 128.139.226.36 (128.139.226.36) Time: Mon Dec 5 14:35:21 2005 Input: K:=FunctionField(RationalField(),3); A:=Matrix(K,3,3,[[a,1,1],[0,b,1],[0,0,c]]); B:=Matrix(K,3,3,[[b,1,1],[0,a,1],[0,0,c]]); R:=PolynomialRing(K,9); P:=Matrix(R,3,3,[R.1,R.2,R.3,R.4,R.5,R.6,R.7,R.8,R.9]); I:=Ideal(Eltseq(P*A*Transpose(P)-B)); print GroebnerBasis(I); Output: Magma V2.11-10 Mon Dec 5 2005 14:35:19 on modular [Seed = 3896458666] ------------------------------------- [ $.1 + (-a*b + b*c - b - c + 1)/(a*c + b*c - c)*$.7 + (b^2 - b*c - 2*b + 1)/(a*c + b*c - c)*$.8 + (-2*b + 1)/(a + b - 1)*$.9, $.2 + (a*b + a*c - a)/(a*c + b*c - c)*$.7 + (-a - b^2 + b*c + b)/(a*c + b*c - c)*$.8 + (-a + b)/(a + b - 1)*$.9, $.3 + b/c*$.8, $.4 + (a^2 - a*c - a - b + 1)/(a*c + b*c - c)*$.7 + (-a*b + a - b*c)/(a*c + b*c - c)*$.8 + (a - b)/(a + b - 1)*$.9, $.5 + (-a^2 + a*c)/(a*c + b*c - c)*$.7 + (a*b - a*c - 2*a - b + c + 1)/(a*c + b*c - c)*$.8 + (-2*a + 1)/(a + b - 1)*$.9, $.6 + a/c*$.7 + 1/c*$.8, $.7^2 + (b + c - 1)/(a*b - 1/2*a)*$.7*$.9 + (a*b^2 - a*b + 1/2*b^2 - b + 1/2)/(a^2*b - 1/2*a^2)*$.8^2 + (a*b - a - b*c + c)/(a^2*b - 1/2*a^2)*$.8*$.9 + (a*b*c - a*c + 1/2*c^2)/(a^2*b - 1/2*a^2)*$.9^2 + (-a*b*c + a*c - 1/2*c^2)/(a^2*b - 1/2*a^2), $.7*$.8 + (-c + 1/2)/(b - 1/2)*$.7*$.9 + (1/2*a*b - 1/2*b^2 + b - 1/2)/(a*b - 1/2*a)*$.8^2 + (1/2*a + b*c - c)/(a*b - 1/2*a)*$.8*$.9 + (1/2*a*c - 1/2*c^2)/(a*b - 1/2*a)*$.9^2 + (-1/2*a*c + 1/2*c^2)/(a*b - 1/2*a), $.7*$.9^2 + (-a*b^2*c + a*b*c - 1/4*a*c + b^2*c^2 - b*c^2 + 1/4*c^2)/(a*b^2*c + a*b*c^2 - a*b*c - 1/4*a*b - 1/4*a*c + 1/4*a - 1/4*b^2 - 1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4)*$.7 + (-1/2*a^2*b^3 + 1/4*a^2*b^2 - a*b^4 + 3/2*a*b^3 - 1/2*a*b^2 - 1/2*b^5 + 5/4*b^4 - b^3 + 1/4*b^2)/(a^2*b^2*c + a^2*b*c^2 - a^2*b*c - 1/4*a^2*b - 1/4*a^2*c + 1/4*a^2 - 1/4*a*b^2 - 1/2*a*b*c + 1/2*a*b - 1/4*a*c^2 + 1/2*a*c - 1/4*a)*$.8^3 + (-1/2*a^2*b^2*c - 3/4*a^2*b^2 + 1/2*a^2*b + a*b^3*c - 3/2*a*b^3 - a*b^2*c + 5/2*a*b^2 + 1/2*a*b*c - a*b + 3/2*b^4*c - 3/4*b^4 - 3*b^3*c + 2*b^3 + 2*b^2*c - 7/4*b^2 - 1/2*b*c + 1/2*b)/(a^2*b^2*c + a^2*b*c^2 - a^2*b*c - 1/4*a^2*b - 1/4*a^2*c + 1/4*a^2 - 1/4*a*b^2 - 1/2*a*b*c + 1/2*a*b - 1/4*a*c^2 + 1/2*a*c - 1/4*a)*$.8^2*$.9 + (-1/2*a^2*b^2*c - 1/4*a^2*b*c - 1/4*a^2*b + 1/4*a^2 - 3/2*a*b^3*c - 1/2*a*b^2*c^2 + 15/4*a*b^2*c - 1/4*a*b^2 + 3/4*a*b*c^2 - 2*a*b*c + 1/2*a*b + 1/4*a*c - 1/4*a - 3/2*b^3*c^2 + 3/2*b^3*c + 9/4*b^2*c^2 - 5/2*b^2*c - b*c^2 + b*c)/(a^2*b^2*c + a^2*b*c^2 - a^2*b*c - 1/4*a^2*b - 1/4*a^2*c + 1/4*a^2 - 1/4*a*b^2 - 1/2*a*b*c + 1/2*a*b - 1/4*a*c^2 + 1/2*a*c - 1/4*a)*$.8*$.9^2 + (1/2*a^2*b^2*c - 1/4*a^2*b*c + 3/2*a*b^3*c - 1/2*a*b^2*c^2 - 11/4*a*b^2*c + 1/4*a*b*c^2 + 3/2*a*b*c - 1/4*a*c + 1/2*b^3*c^2 - 1/4*b^2*c^2)/(a^2*b^2*c + a^2*b*c^2 - a^2*b*c - 1/4*a^2*b - 1/4*a^2*c + 1/4*a^2 - 1/4*a*b^2 - 1/2*a*b*c + 1/2*a*b - 1/4*a*c^2 + 1/2*a*c - 1/4*a)*$.8 + (-1/2*a^2*b*c^2 - 1/4*a^2*b*c + 1/4*a^2*c + 3/2*a*b^2*c^2 - 1/4*a*b^2*c + 1/2*a*b*c^3 - 5/4*a*b*c^2 + 1/2*a*b*c + 1/4*a*c^2 - 1/4*a*c + 1/2*b^2*c^3 - 3/4*b^2*c^2 - 1/2*b*c^3 + 1/2*b*c^2)/(a^2*b^2*c + a^2*b*c^2 - a^2*b*c - 1/4*a^2*b - 1/4*a^2*c + 1/4*a^2 - 1/4*a*b^2 - 1/2*a*b*c + 1/2*a*b - 1/4*a*c^2 + 1/2*a*c - 1/4*a)*$.9^3 + (1/2*a^2*b*c^2 + 1/4*a^2*b*c - 1/4*a^2*c - 3/2*a*b^2*c^2 + 1/4*a*b^2*c - 1/2*a*b*c^3 + 5/4*a*b*c^2 - 1/2*a*b*c - 1/4*a*c^2 + 1/4*a*c - 1/2*b^2*c^3 + 3/4*b^2*c^2 + 1/2*b*c^3 - 1/2*b*c^2)/(a^2*b^2*c + a^2*b*c^2 - a^2*b*c - 1/4*a^2*b - 1/4*a^2*c + 1/4*a^2 - 1/4*a*b^2 - 1/2*a*b*c + 1/2*a*b - 1/4*a*c^2 + 1/2*a*c - 1/4*a)*$.9, $.8^4 + (2*a - 4*b*c + 2*b + 2*c - 2)/(a*b + b^2 - b)*$.8^3*$.9 + (2*a^2*b*c + a^2 + 2*a*b^2*c + 2*a*b*c^2 - 8*a*b*c + 2*a*b + 2*a*c - 2*a + 6*b^2*c^2 - 6*b^2*c + b^2 - 6*b*c^2 + 8*b*c - 2*b + c^2 - 2*c + 1)/(a^2*b^2 + 2*a*b^3 - 2*a*b^2 + b^4 - 2*b^3 + b^2)*$.8^2*$.9^2 + (-2*a^2*b*c - 2*a*b^2*c + 2*a*b*c^2 + 2*a*b*c - 2*b^2*c^2 + 2*b*c^2 - c^2)/(a^2*b^2 + 2*a*b^3 - 2*a*b^2 + b^4 - 2*b^3 + b^2)*$.8^2 + (2*a^2*c - 4*a*b*c^2 + 2*a*b*c + 4*a*c^2 - 4*a*c - 4*b*c^3 + 6*b*c^2 - 2*b*c + 2*c^3 - 4*c^2 + 2*c)/(a^2*b^2 + 2*a*b^3 - 2*a*b^2 + b^4 - 2*b^3 + b^2)*$.8*$.9^3 + (-2*a^2*c + 4*a*b*c^2 - 2*a*b*c + 2*a*c + 4*b*c^3 - 2*b*c^2 - 2*c^3)/(a^2*b^2 + 2*a*b^3 - 2*a*b^2 + b^4 - 2*b^3 + b^2)*$.8*$.9 + (a^2*c^2 + 2*a*c^3 - 2*a*c^2 + c^4 - 2*c^3 + c^2)/(a^2*b^2 + 2*a*b^3 - 2*a*b^2 + b^4 - 2*b^3 + b^2)*$.9^4 + (-2*a^2*c^2 + 2*a*c^2 - 2*c^4 + 2*c^3 - c^2)/(a^2*b^2 + 2*a*b^3 - 2*a*b^2 + b^4 - 2*b^3 + b^2)*$.9^2 + (a^2*c^2 - 2*a*c^3 + c^4)/(a^2*b^2 + 2*a*b^3 - 2*a*b^2 + b^4 - 2*b^3 + b^2) ] Total time: 0.980 seconds, Total memory usage: 4.30MB '80.240.' ************** MAGMA ***************** Host 80.240.228.94 (80.240.228.94) Time: Mon Dec 5 11:14:42 2005 Input: 5! Output: Magma V2.11-10 Mon Dec 5 2005 11:14:41 on modular [Seed = 2588786397] ------------------------------------- >> 5!; ^ User error: bad syntax Total time: 0.180 seconds, Total memory usage: 3.24MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Mon Dec 5 06:18:27 2005 Input: G :=DirichletGroup(1680); G; X :=Elements(G); X; Y :=X[30]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; Output: ** WARNING: Computation used more memory than allowed. ** Magma V2.11-10 Mon Dec 5 2005 06:18:21 on modular [Seed = 4230716976] ------------------------------------- Group of Dirichlet characters of modulus 1680 over Rational Field [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3, $.4, $.1*$.4, $.2*$.4, $.1*$.2*$.4, $.3*$.4, $.1*$.3*$.4, $.2*$.3*$.4, $.1*$.2*$.3*$.4, $.5, $.1*$.5, $.2*$.5, $.1*$.2*$.5, $.3*$.5, $.1*$.3*$.5, $.2*$.3*$.5, $.1*$.2*$.3*$.5, $.4*$.5, $.1*$.4*$.5, $.2*$.4*$.5, $.1*$.2*$.4*$.5, $.3*$.4*$.5, $.1*$.3*$.4*$.5, $.2*$.3*$.4*$.5, $.1*$.2*$.3*$.4*$.5 ] 420 2 Current total memory usage: 91.9MB, failed memory request: 81.0MB System Error: User memory limit has been reached >> D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); ^ Runtime error: Variable 'M' has not been initialized >> D; ^ User error: Identifier 'D' has not been declared or assigned Total time: 5.740 seconds, Total memory usage: 91.93MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Mon Dec 5 05:44:54 2005 Input: G :=DirichletGroup(1680); G; X :=Elements(G); X; Y :=X[2]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; Output: ** WARNING: Computation used more memory than allowed. ** Magma V2.11-10 Mon Dec 5 2005 05:44:48 on modular [Seed = 1602750030] ------------------------------------- Group of Dirichlet characters of modulus 1680 over Rational Field [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3, $.4, $.1*$.4, $.2*$.4, $.1*$.2*$.4, $.3*$.4, $.1*$.3*$.4, $.2*$.3*$.4, $.1*$.2*$.3*$.4, $.5, $.1*$.5, $.2*$.5, $.1*$.2*$.5, $.3*$.5, $.1*$.3*$.5, $.2*$.3*$.5, $.1*$.2*$.3*$.5, $.4*$.5, $.1*$.4*$.5, $.2*$.4*$.5, $.1*$.2*$.4*$.5, $.3*$.4*$.5, $.1*$.3*$.4*$.5, $.2*$.3*$.4*$.5, $.1*$.2*$.3*$.4*$.5 ] 4 2 Current total memory usage: 92.0MB, failed memory request: 80.9MB System Error: User memory limit has been reached >> D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); ^ Runtime error: Variable 'M' has not been initialized >> D; ^ User error: Identifier 'D' has not been declared or assigned Total time: 5.679 seconds, Total memory usage: 92.02MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Mon Dec 5 04:29:30 2005 Input: G :=DirichletGroup(840); G; X :=Elements(G); X; Y :=X[2]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; Output: ** WARNING: Computation time exceeded 20 seconds, so computation was terminated after 20 seconds. ** Magma V2.11-10 Mon Dec 5 2005 04:29:09 on modular [Seed = 670067827] ------------------------------------- Group of Dirichlet characters of modulus 840 over Rational Field [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3, $.4, $.1*$.4, $.2*$.4, $.1*$.2*$.4, $.3*$.4, $.1*$.3*$.4, $.2*$.3*$.4, $.1*$.2*$.3*$.4, $.5, $.1*$.5, $.2*$.5, $.1*$.2*$.5, $.3*$.5, $.1*$.3*$.5, $.2*$.3*$.5, $.1*$.2*$.3*$.5, $.4*$.5, $.1*$.4*$.5, $.2*$.4*$.5, $.1*$.2*$.4*$.5, $.3*$.4*$.5, $.1*$.3*$.4*$.5, $.2*$.3*$.4*$.5, $.1*$.2*$.3*$.4*$.5 ] 4 2 Errors: /bin/sh: line 1: 29206 Alarm clock nice -n 19 /usr/local/bin/magma '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Mon Dec 5 04:26:49 2005 Input: G :=DirichletGroup(210); G; X :=Elements(G); X; Y :=X[2]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; Output: Magma V2.11-10 Mon Dec 5 2005 04:26:46 on modular [Seed = 553084103] ------------------------------------- Group of Dirichlet characters of modulus 210 over Rational Field [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3 ] 3 2 [ Modular symbols space of level 210, weight 3, character $.1, and dimension 16 over Rational Field ] Total time: 2.799 seconds, Total memory usage: 9.98MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Mon Dec 5 04:26:30 2005 Input: G :=DirichletGroup(210); G; X :=Elements(G); X; Y :=X[8]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; Output: Magma V2.11-10 Mon Dec 5 2005 04:26:30 on modular [Seed = 988643021] ------------------------------------- Group of Dirichlet characters of modulus 210 over Rational Field [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3 ] 105 2 [] Total time: 0.220 seconds, Total memory usage: 4.59MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Mon Dec 5 04:26:18 2005 Input: G :=DirichletGroup(210); G; X :=Elements(G); X; Y :=X[7]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; Output: Magma V2.11-10 Mon Dec 5 2005 04:26:16 on modular [Seed = 937456574] ------------------------------------- Group of Dirichlet characters of modulus 210 over Rational Field [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3 ] 35 2 [ Modular symbols space of level 210, weight 3, character $.2*$.3, and dimension 16 over Rational Field ] Total time: 2.299 seconds, Total memory usage: 9.98MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Mon Dec 5 04:26:05 2005 Input: G :=DirichletGroup(210); G; X :=Elements(G); X; Y :=X[6]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; Output: Magma V2.11-10 Mon Dec 5 2005 04:26:05 on modular [Seed = 820475052] ------------------------------------- Group of Dirichlet characters of modulus 210 over Rational Field [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3 ] 21 2 [] Total time: 0.220 seconds, Total memory usage: 4.60MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Mon Dec 5 04:21:55 2005 Input: G :=DirichletGroup(210); G; X :=Elements(G); X; Y :=X[5]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; qEigenform(D[1],20);Parent($1); Output: Magma V2.11-10 Mon Dec 5 2005 04:21:52 on modular [Seed = 3357510986] ------------------------------------- Group of Dirichlet characters of modulus 210 over Rational Field [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3 ] 7 2 [ Modular symbols space of level 210, weight 3, character $.3, and dimension 8 over Rational Field ] q + (-415353/140631020042240*a^7 - 301369/10817770772480*a^6 - 100645827/70315510021120*a^5 - 1687471593/70315510021120*a^4 - 6165419051/7031551002112*a^3 - 7544009613/1004507286016*a^2 - 2395504268061/17578877505280*a + 952079037763/2511268215040)*q^2 + (61/270466846720*a^7 + 1051/19319060480*a^6 + 102729/135233423360*a^5 + 1445303/67616711680*a^4 + 8801395/13523342336*a^3 + 101272359/6761671168*a^2 + 6674090427/33808355840*a + 6266199807/2414882560)*q^3 + 2*q^4 + (105129/67123253550080*a^7 + 38499/368809085440*a^6 + 65880289/33561626775040*a^5 + 985815917/16780813387520*a^4 + 19244054491/16780813387520*a^3 + 248612660747/8390406693760*a^2 + 339284794093/1198629527680*a + 2679931711057/599314763840)*q^5 + (337287/154526132773376*a^7 + 357451/11886625597952*a^6 + 102999019/77263066386688*a^5 + 2547290899/77263066386688*a^4 + 26135542317/38631533193344*a^3 + 82937289347/5518790456192*a^2 + 718594671095/2759395228096*a + 5318504874287/2759395228096)*q^6 + (73866797867/35134297756416954368*a^7 + 42074836351/241306990085281280*a^6 + 340062766719647/87835744391042385920*a^5 + 386359492218157/4391787219552119296*a^4 + 65227685421317269/43917872195521192960*a^3 + 9493755569266281/171554188263754660*a^2 + 1783706007473175811/3136990871108656640*a + 5859779321633194923/784247717777164160)*q^7 + (-415353/70315510021120*a^7 - 301369/5408885386240*a^6 - 100645827/35157755010560*a^5 - 1687471593/35157755010560*a^4 - 6165419051/3515775501056*a^3 - 7544009613/502253643008*a^2 - 2395504268061/8789438752640*a + 952079037763/1255634107520)*q^8 - 3*q^9 + (7209/3315131348480*a^7 + 656157/13260525393920*a^6 + 7380803/3315131348480*a^5 + 36675163/947180385280*a^4 + 454198953/414391418560*a^3 + 40618382071/3315131348480*a^2 + 302621545967/828782837120*a + 300249322681/236795096320)*q^10 + (-2473932893/320568410186286080*a^7 - 68024963/3522729782266880*a^6 - 796914927311/160284205093143040*a^5 - 9589847773573/160284205093143040*a^4 - 233568053516403/80142102546571520*a^3 - 2084080830459627/80142102546571520*a^2 - 3440859888022621/8014210254657152*a - 15675074276252681/5724435896183680)*q\ ^11 + (61/135233423360*a^7 + 1051/9659530240*a^6 + 102729/67616711680*a^5 + 1445303/33808355840*a^4 + 8801395/6761671168*a^3 + 101272359/3380835584*a^2 + 6674090427/16904177920*a + 6266199807/1207441280)*q^12 + (2473932893/641136820372572160*a^7 + 68024963/7045459564533760*a^6 + 796914927311/320568410186286080*a^5 + 9589847773573/320568410186286080*a^4 + 233568053516403/160284205093143040*a^3 + 2084080830459627/160284205093143040*a^2 + 11455070142679773/16028420509314304*a + 15675074276252681/11448871792367360)*q^13 + (2703342821/182610695199672320*a^7 + 1159611501/3260905271422720*a^6 + 875460355197/91305347599836160*a^5 + 4664601406499/22826336899959040*a^4 + 266625016194911/45652673799918080*a^3 + 104881037503867/1426646056247440*a^2 + 987000415519317/652181054284544*a + 7102726963661553/815226317855680)*q^1\ 4 + (121/44910161920*a^7 - 53/6045598720*a^6 + 231887/157185566720*a^5 - 284741/19648195840*a^4 + 3017419/11227540480*a^3 - 77204177/19648195840*a^2 + 574080053/39296391680*a - 1458574873/350860640)*q^15 + 4*q^16 + (336932919/66744486619333120*a^7 + 37905946599/71878677897743360*a^6 + 61112850011/8343060827416640*a^5 + 128501335337131/467211406335331840*a^4 + 547642963535037/116802851583832960*a^3 + 35806771762695601/233605703167665920*a^2 + 28268746826087251/29200712895958240*a + 405372914402092591/16686121654833280)*q^17 + (1246059/140631020042240*a^7 + 904107/10817770772480*a^6 + 301937481/70315510021120*a^5 + 5062414779/70315510021120*a^4 + 18496257153/7031551002112*a^3 + 22632028839/1004507286016*a^2 + 7186512804183/17578877505280*a - 2856237113289/2511268215040)*q^18 + (113207027013/17567148878208477184*a^7 - 89451024127/211143616324621120*a^6 - 22617155986209/6273981742217313280*a^5 - 202900838349135/2195893609776059648*a^4 - 80258710845285461/21958936097760596480*a^3 - 187840457305308673/2744867012220074560*a^2 - 7223781603714594173/10979468048880298240*a - 5315748266184103957/392123858888582080)*q^19 + O(q^20) Power series ring in q over Univariate Quotient Polynomial Algebra in a over Rational Field with modulus a^8 + 16*a^7 + 760*a^6 + 13600*a^5 + 432984*a^4 + 6066368*a^3 + 133021408*a^2 + 750049664*a + 14968881424 Total time: 2.970 seconds, Total memory usage: 9.99MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Mon Dec 5 04:21:07 2005 Input: G :=DirichletGroup(210); G; X :=Elements(G); X; Y :=X[5]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; Output: Magma V2.11-10 Mon Dec 5 2005 04:21:04 on modular [Seed = 3323036532] ------------------------------------- Group of Dirichlet characters of modulus 210 over Rational Field [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3 ] 7 2 [ Modular symbols space of level 210, weight 3, character $.3, and dimension 8 over Rational Field ] Total time: 2.950 seconds, Total memory usage: 9.99MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Mon Dec 5 04:20:54 2005 Input: G :=DirichletGroup(210); G; X :=Elements(G); X; Y :=X[4]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; Output: Magma V2.11-10 Mon Dec 5 2005 04:20:51 on modular [Seed = 3708459164] ------------------------------------- Group of Dirichlet characters of modulus 210 over Rational Field [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3 ] 15 2 [ Modular symbols space of level 210, weight 3, character $.1*$.2, and dimension 24 over Rational Field ] Total time: 2.299 seconds, Total memory usage: 9.99MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Mon Dec 5 04:20:43 2005 Input: G :=DirichletGroup(210); G; X :=Elements(G); X; Y :=X[3]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; Output: Magma V2.11-10 Mon Dec 5 2005 04:20:43 on modular [Seed = 3624902017] ------------------------------------- Group of Dirichlet characters of modulus 210 over Rational Field [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3 ] 5 2 [] Total time: 0.220 seconds, Total memory usage: 4.63MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Mon Dec 5 04:20:31 2005 Input: G :=DirichletGroup(210); G; X :=Elements(G); X; Y :=X[2]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; Output: Magma V2.11-10 Mon Dec 5 2005 04:20:28 on modular [Seed = 3607139001] ------------------------------------- Group of Dirichlet characters of modulus 210 over Rational Field [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3 ] 3 2 [ Modular symbols space of level 210, weight 3, character $.1, and dimension 16 over Rational Field ] Total time: 2.810 seconds, Total memory usage: 9.98MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Mon Dec 5 04:19:55 2005 Input: G :=DirichletGroup(105); G; X :=Elements(G); X; Y :=X[2]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; Output: Magma V2.11-10 Mon Dec 5 2005 04:19:54 on modular [Seed = 3523581914] ------------------------------------- Group of Dirichlet characters of modulus 105 over Rational Field [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3 ] 3 2 [ Modular symbols space of level 105, weight 3, character $.1, and dimension 16 over Rational Field ] Total time: 0.740 seconds, Total memory usage: 4.81MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Mon Dec 5 04:19:45 2005 Input: G :=DirichletGroup(105); G; X :=Elements(G); X; Y :=X[3]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; Output: Magma V2.11-10 Mon Dec 5 2005 04:19:45 on modular [Seed = 3913198826] ------------------------------------- Group of Dirichlet characters of modulus 105 over Rational Field [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3 ] 5 2 [] Total time: 0.210 seconds, Total memory usage: 4.50MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Mon Dec 5 04:19:32 2005 Input: G :=DirichletGroup(105); G; X :=Elements(G); X; Y :=X[4]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; Output: Magma V2.11-10 Mon Dec 5 2005 04:19:32 on modular [Seed = 3828589042] ------------------------------------- Group of Dirichlet characters of modulus 105 over Rational Field [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3 ] 15 2 [ Modular symbols space of level 105, weight 3, character $.1*$.2, and dimension 24 over Rational Field ] Total time: 0.600 seconds, Total memory usage: 4.81MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Mon Dec 5 04:19:22 2005 Input: G :=DirichletGroup(105); G; X :=Elements(G); X; Y :=X[5]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; Output: Magma V2.11-10 Mon Dec 5 2005 04:19:21 on modular [Seed = 3811878600] ------------------------------------- Group of Dirichlet characters of modulus 105 over Rational Field [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3 ] 7 2 [ Modular symbols space of level 105, weight 3, character $.3, and dimension 12 over Rational Field ] Total time: 0.740 seconds, Total memory usage: 4.62MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Mon Dec 5 04:19:01 2005 Input: G :=DirichletGroup(105); G; X :=Elements(G); X; Y :=X[6]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; Output: Magma V2.11-10 Mon Dec 5 2005 04:19:01 on modular [Seed = 4264147913] ------------------------------------- Group of Dirichlet characters of modulus 105 over Rational Field [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3 ] 21 2 [] Total time: 0.220 seconds, Total memory usage: 4.38MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Mon Dec 5 04:18:45 2005 Input: G :=DirichletGroup(105); G; X :=Elements(G); X; Y :=X[7]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; Output: Magma V2.11-10 Mon Dec 5 2005 04:18:44 on modular [Seed = 4112692019] ------------------------------------- Group of Dirichlet characters of modulus 105 over Rational Field [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3 ] 35 2 [ Modular symbols space of level 105, weight 3, character $.2*$.3, and dimension 16 over Rational Field ] Total time: 0.590 seconds, Total memory usage: 4.62MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Mon Dec 5 04:18:34 2005 Input: G :=DirichletGroup(105); G; X :=Elements(G); X; Y :=X[8]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; Output: Magma V2.11-10 Mon Dec 5 2005 04:18:33 on modular [Seed = 4029134379] ------------------------------------- Group of Dirichlet characters of modulus 105 over Rational Field [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3 ] 105 2 [] Total time: 0.210 seconds, Total memory usage: 3.91MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Mon Dec 5 04:18:20 2005 Input: G :=DirichletGroup(105); G; X :=Elements(G); X; Y :=X[16]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; Output: Magma V2.11-10 Mon Dec 5 2005 04:18:20 on modular [Seed = 2404960609] ------------------------------------- Group of Dirichlet characters of modulus 105 over Rational Field [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3 ] >> Y :=X[16]; Conductor(Y); Order(Y); ^ Runtime error in '[]': Sequence element 16 not defined >> Y :=X[16]; Conductor(Y); Order(Y); ^ User error: Identifier 'Y' has not been declared or assigned >> Y :=X[16]; Conductor(Y); Order(Y); ^ User error: Identifier 'Y' has not been declared or assigned >> M := ModularSymbols(Y, 3, 1); ^ User error: Identifier 'Y' has not been declared or assigned >> D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); ^ User error: Identifier 'M' has not been declared or assigned >> D; ^ User error: Identifier 'D' has not been declared or assigned Total time: 0.190 seconds, Total memory usage: 3.34MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Mon Dec 5 03:54:20 2005 Input: G :=DirichletGroup(420); G; X :=Elements(G); X; Y :=X[16]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; Output: ** WARNING: Computation time exceeded 20 seconds, so computation was terminated after 20 seconds. ** Magma V2.11-10 Mon Dec 5 2005 03:54:00 on modular [Seed = 1367741003] ------------------------------------- Group of Dirichlet characters of modulus 420 over Rational Field [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3, $.4, $.1*$.4, $.2*$.4, $.1*$.2*$.4, $.3*$.4, $.1*$.3*$.4, $.2*$.3*$.4, $.1*$.2*$.3*$.4 ] 420 2 Errors: /bin/sh: line 1: 29027 Alarm clock nice -n 19 /usr/local/bin/magma '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Mon Dec 5 03:53:50 2005 Input: G :=DirichletGroup(420); G; X :=Elements(G); X; Y :=X[1]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; Output: Magma V2.11-10 Mon Dec 5 2005 03:53:49 on modular [Seed = 1824206659] ------------------------------------- Group of Dirichlet characters of modulus 420 over Rational Field [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3, $.4, $.1*$.4, $.2*$.4, $.1*$.2*$.4, $.3*$.4, $.1*$.3*$.4, $.2*$.3*$.4, $.1*$.2*$.3*$.4 ] 1 1 [] Total time: 0.230 seconds, Total memory usage: 4.82MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Mon Dec 5 03:53:32 2005 Input: G :=DirichletGroup(420); G; X :=Elements(G); X; Y :=X[2]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; Output: Magma V2.11-10 Mon Dec 5 2005 03:53:21 on modular [Seed = 1807493208] ------------------------------------- Group of Dirichlet characters of modulus 420 over Rational Field [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3, $.4, $.1*$.4, $.2*$.4, $.1*$.2*$.4, $.3*$.4, $.1*$.3*$.4, $.2*$.3*$.4, $.1*$.2*$.3*$.4 ] 4 2 [ Modular symbols space of level 420, weight 3, character $.1, and dimension 48 over Rational Field ] Total time: 11.220 seconds, Total memory usage: 25.76MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Mon Dec 5 03:53:00 2005 Input: G :=DirichletGroup(420); G; X :=Elements(G); X; Y :=X[3]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; Output: Magma V2.11-10 Mon Dec 5 2005 03:52:51 on modular [Seed = 1622203569] ------------------------------------- Group of Dirichlet characters of modulus 420 over Rational Field [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3, $.4, $.1*$.4, $.2*$.4, $.1*$.2*$.4, $.3*$.4, $.1*$.3*$.4, $.2*$.3*$.4, $.1*$.2*$.3*$.4 ] 3 2 [ Modular symbols space of level 420, weight 3, character $.2, and dimension 16 over Rational Field ] Total time: 8.779 seconds, Total memory usage: 25.76MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Mon Dec 5 03:52:39 2005 Input: G :=DirichletGroup(420); G; X :=Elements(G); X; Y :=X[4]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; Output: Magma V2.11-10 Mon Dec 5 2005 03:52:39 on modular [Seed = 2142225306] ------------------------------------- Group of Dirichlet characters of modulus 420 over Rational Field [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3, $.4, $.1*$.4, $.2*$.4, $.1*$.2*$.4, $.3*$.4, $.1*$.3*$.4, $.2*$.3*$.4, $.1*$.2*$.3*$.4 ] 12 2 [] Total time: 0.220 seconds, Total memory usage: 4.47MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Mon Dec 5 03:52:14 2005 Input: G :=DirichletGroup(420); G; X :=Elements(G); X; Y :=X[5]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; Output: Magma V2.11-10 Mon Dec 5 2005 03:52:13 on modular [Seed = 2058012405] ------------------------------------- Group of Dirichlet characters of modulus 420 over Rational Field [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3, $.4, $.1*$.4, $.2*$.4, $.1*$.2*$.4, $.3*$.4, $.1*$.3*$.4, $.2*$.3*$.4, $.1*$.2*$.3*$.4 ] 5 2 [] Total time: 0.220 seconds, Total memory usage: 4.77MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Mon Dec 5 03:51:57 2005 Input: G :=DirichletGroup(420); G; X :=Elements(G); X; Y :=X[6]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; Output: Magma V2.11-10 Mon Dec 5 2005 03:51:47 on modular [Seed = 1906429408] ------------------------------------- Group of Dirichlet characters of modulus 420 over Rational Field [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3, $.4, $.1*$.4, $.2*$.4, $.1*$.2*$.4, $.3*$.4, $.1*$.3*$.4, $.2*$.3*$.4, $.1*$.2*$.3*$.4 ] 20 2 [ Modular symbols space of level 420, weight 3, character $.1*$.3, and dimension 72 over Rational Field ] Total time: 9.310 seconds, Total memory usage: 25.76MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Mon Dec 5 03:51:35 2005 Input: G :=DirichletGroup(420); G; X :=Elements(G); X; Y :=X[7]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; Output: Magma V2.11-10 Mon Dec 5 2005 03:51:27 on modular [Seed = 266318770] ------------------------------------- Group of Dirichlet characters of modulus 420 over Rational Field [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3, $.4, $.1*$.4, $.2*$.4, $.1*$.2*$.4, $.3*$.4, $.1*$.3*$.4, $.2*$.3*$.4, $.1*$.2*$.3*$.4 ] 15 2 [ Modular symbols space of level 420, weight 3, character $.2*$.3, and dimension 24 over Rational Field ] Total time: 7.230 seconds, Total memory usage: 25.77MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Mon Dec 5 03:51:15 2005 Input: G :=DirichletGroup(420); G; X :=Elements(G); X; Y :=X[8]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; Output: Magma V2.11-10 Mon Dec 5 2005 03:51:15 on modular [Seed = 182105787] ------------------------------------- Group of Dirichlet characters of modulus 420 over Rational Field [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3, $.4, $.1*$.4, $.2*$.4, $.1*$.2*$.4, $.3*$.4, $.1*$.3*$.4, $.2*$.3*$.4, $.1*$.2*$.3*$.4 ] 60 2 [] Total time: 0.210 seconds, Total memory usage: 4.43MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Mon Dec 5 03:50:35 2005 Input: G :=DirichletGroup(420); G; X :=Elements(G); X; Y :=X[9]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; Output: Magma V2.11-10 Mon Dec 5 2005 03:50:26 on modular [Seed = 30522721] ------------------------------------- Group of Dirichlet characters of modulus 420 over Rational Field [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3, $.4, $.1*$.4, $.2*$.4, $.1*$.2*$.4, $.3*$.4, $.1*$.3*$.4, $.2*$.3*$.4, $.1*$.2*$.3*$.4 ] 7 2 [ Modular symbols space of level 420, weight 3, character $.4, and dimension 12 over Rational Field ] Total time: 9.199 seconds, Total memory usage: 25.75MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Mon Dec 5 03:50:12 2005 Input: G :=DirichletGroup(420); G; X :=Elements(G); X; Y :=X[10]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; Output: Magma V2.11-10 Mon Dec 5 2005 03:50:12 on modular [Seed = 483172458] ------------------------------------- Group of Dirichlet characters of modulus 420 over Rational Field [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3, $.4, $.1*$.4, $.2*$.4, $.1*$.2*$.4, $.3*$.4, $.1*$.3*$.4, $.2*$.3*$.4, $.1*$.2*$.3*$.4 ] 28 2 [] Total time: 0.220 seconds, Total memory usage: 4.59MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Mon Dec 5 03:49:59 2005 Input: G :=DirichletGroup(420); G; X :=Elements(G); X; Y :=X[11]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; Output: Magma V2.11-10 Mon Dec 5 2005 03:49:59 on modular [Seed = 466331422] ------------------------------------- Group of Dirichlet characters of modulus 420 over Rational Field [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3, $.4, $.1*$.4, $.2*$.4, $.1*$.2*$.4, $.3*$.4, $.1*$.3*$.4, $.2*$.3*$.4, $.1*$.2*$.3*$.4 ] 21 2 [] Total time: 0.220 seconds, Total memory usage: 4.67MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Mon Dec 5 03:49:32 2005 Input: G :=DirichletGroup(420); G; X :=Elements(G); X; Y :=X[12]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; Output: Magma V2.11-10 Mon Dec 5 2005 03:49:21 on modular [Seed = 382118453] ------------------------------------- Group of Dirichlet characters of modulus 420 over Rational Field [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3, $.4, $.1*$.4, $.2*$.4, $.1*$.2*$.4, $.3*$.4, $.1*$.3*$.4, $.2*$.3*$.4, $.1*$.2*$.3*$.4 ] 84 2 [ Modular symbols space of level 420, weight 3, character $.1*$.2*$.4, and dimension 128 over Rational Field ] Total time: 11.460 seconds, Total memory usage: 33.97MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Mon Dec 5 03:48:19 2005 Input: G :=DirichletGroup(420); G; X :=Elements(G); X; Y :=X[13]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; Output: Magma V2.11-10 Mon Dec 5 2005 03:48:12 on modular [Seed = 754749906] ------------------------------------- Group of Dirichlet characters of modulus 420 over Rational Field [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3, $.4, $.1*$.4, $.2*$.4, $.1*$.2*$.4, $.3*$.4, $.1*$.3*$.4, $.2*$.3*$.4, $.1*$.2*$.3*$.4 ] 35 2 [ Modular symbols space of level 420, weight 3, character $.3*$.4, and dimension 16 over Rational Field ] Total time: 7.750 seconds, Total memory usage: 25.76MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Mon Dec 5 03:48:13 2005 Input: G :=DirichletGroup(420); G; X :=Elements(G); X; Y :=X[13]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; Output: Magma V2.11-10 Mon Dec 5 2005 03:48:05 on modular [Seed = 704222430] ------------------------------------- Group of Dirichlet characters of modulus 420 over Rational Field [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3, $.4, $.1*$.4, $.2*$.4, $.1*$.2*$.4, $.3*$.4, $.1*$.3*$.4, $.2*$.3*$.4, $.1*$.2*$.3*$.4 ] 35 2 [ Modular symbols space of level 420, weight 3, character $.3*$.4, and dimension 16 over Rational Field ] Total time: 7.710 seconds, Total memory usage: 25.76MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Mon Dec 5 03:47:55 2005 Input: G :=DirichletGroup(420); G; X :=Elements(G); X; Y :=X[14]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; Output: Magma V2.11-10 Mon Dec 5 2005 03:47:54 on modular [Seed = 620009900] ------------------------------------- Group of Dirichlet characters of modulus 420 over Rational Field [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3, $.4, $.1*$.4, $.2*$.4, $.1*$.2*$.4, $.3*$.4, $.1*$.3*$.4, $.2*$.3*$.4, $.1*$.2*$.3*$.4 ] 140 2 [] Total time: 0.220 seconds, Total memory usage: 4.54MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Mon Dec 5 03:47:37 2005 Input: G :=DirichletGroup(420); G; X :=Elements(G); X; Y :=X[15]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; Output: Magma V2.11-10 Mon Dec 5 2005 03:47:37 on modular [Seed = 603168440] ------------------------------------- Group of Dirichlet characters of modulus 420 over Rational Field [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3, $.4, $.1*$.4, $.2*$.4, $.1*$.2*$.4, $.3*$.4, $.1*$.3*$.4, $.2*$.3*$.4, $.1*$.2*$.3*$.4 ] 105 2 [] Total time: 0.220 seconds, Total memory usage: 4.62MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Mon Dec 5 03:45:45 2005 Input: G :=DirichletGroup(420); G; X :=Elements(G); X; Y :=X[16]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; Output: ** WARNING: Computation time exceeded 20 seconds, so computation was terminated after 20 seconds. ** Magma V2.11-10 Mon Dec 5 2005 03:45:24 on modular [Seed = 1005291053] ------------------------------------- Group of Dirichlet characters of modulus 420 over Rational Field [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3, $.4, $.1*$.4, $.2*$.4, $.1*$.2*$.4, $.3*$.4, $.1*$.3*$.4, $.2*$.3*$.4, $.1*$.2*$.3*$.4 ] 420 2 Errors: /bin/sh: line 1: 28937 Alarm clock nice -n 19 /usr/local/bin/magma '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Mon Dec 5 03:45:36 2005 Input: G :=DirichletGroup(420); G; X :=Elements(G); X; Y :=X[16]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; Output: ** WARNING: Computation time exceeded 20 seconds, so computation was terminated after 20 seconds. ** Magma V2.11-10 Mon Dec 5 2005 03:45:16 on modular [Seed = 887392538] ------------------------------------- Group of Dirichlet characters of modulus 420 over Rational Field [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3, $.4, $.1*$.4, $.2*$.4, $.1*$.2*$.4, $.3*$.4, $.1*$.3*$.4, $.2*$.3*$.4, $.1*$.2*$.3*$.4 ] 420 2 Errors: /bin/sh: line 1: 28934 Alarm clock nice -n 19 /usr/local/bin/magma '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Mon Dec 5 03:45:29 2005 Input: G :=DirichletGroup(420); G; X :=Elements(G); X; Y :=X[16]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; Output: ** WARNING: Computation time exceeded 20 seconds, so computation was terminated after 20 seconds. ** Magma V2.11-10 Mon Dec 5 2005 03:45:08 on modular [Seed = 836865047] ------------------------------------- Group of Dirichlet characters of modulus 420 over Rational Field [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3, $.4, $.1*$.4, $.2*$.4, $.1*$.2*$.4, $.3*$.4, $.1*$.3*$.4, $.2*$.3*$.4, $.1*$.2*$.3*$.4 ] 420 2 Errors: /bin/sh: line 1: 28931 Alarm clock nice -n 19 /usr/local/bin/magma '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Mon Dec 5 02:37:53 2005 Input: G :=DirichletGroup(168); G; X :=Elements(G); X; Y :=X[16]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; qEigenform(D[5],20);Parent($1); Output: Magma V2.11-10 Mon Dec 5 2005 02:37:52 on modular [Seed = 281066862] ------------------------------------- Group of Dirichlet characters of modulus 168 over Rational Field [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3, $.4, $.1*$.4, $.2*$.4, $.1*$.2*$.4, $.3*$.4, $.1*$.3*$.4, $.2*$.3*$.4, $.1*$.2*$.3*$.4 ] 168 2 [ Modular symbols space of level 168, weight 3, character $.1*$.2*$.3*$.4, and dimension 1 over Rational Field, Modular symbols space of level 168, weight 3, character $.1*$.2*$.3*$.4, and dimension 1 over Rational Field, Modular symbols space of level 168, weight 3, character $.1*$.2*$.3*$.4, and dimension 1 over Rational Field, Modular symbols space of level 168, weight 3, character $.1*$.2*$.3*$.4, and dimension 1 over Rational Field, Modular symbols space of level 168, weight 3, character $.1*$.2*$.3*$.4, and dimension 8 over Rational Field, Modular symbols space of level 168, weight 3, character $.1*$.2*$.3*$.4, and dimension 48 over Rational Field ] q + (-9799/5016464987212800*a^7 - 661127/4459079988633600*a^6 + 5969/573638077440*a^5 + 16758991/65574705715200*a^4 - 603775387/13934624964480*a^3 + 14610182343/30965833254400*a^2 + 65387258923/11612187470400*a - 19476177313999/7741458313600)*q^2 + (-266142103/19906741199783116800*a^7 + 149817191/311042831246611200*a^6 + 173029313/2845446140620800*a^5 - 22099254151/5184047187443520*a^4 - 9483376629317/138241258331827200*a^3 + 334575867229/43200393228696*a^2 - 246875651449831/2304020972197120*a - 235675861161782/45000409613225)*q^3 + (1/58023161856*a^7 - 43/348138971136*a^6 - 41/439569408*a^5 + 25841/9670526976*a^4 + 70519/402938624*a^3 - 4371163/805877248*a^2 + 2739201/100734656*a + 1629936657/201469312)*q^4 + (-1399/21560302961664*a^7 + 6661/2695037870208*a^6 + 85841/305098626816*a^5 - 215599/13210969952*a^4 - 535770697/1347518935104*a^3 + 1685781297/56146622296*a^2 - 17695656925/112293244592*a - 307734345945/14036655574)*q^5 + (380413753/15129123311835168768*a^7 - 121715656291/94557020698969804800*a^6 - 1531167499/18021158890598400*a^5 + 6353210557501/875527969434905600*a^4 + 4581423310679/154504935782630400*a^3 - 7802577696362677/656645977076179200*a^2 + 55036834534786031/218881992358726400*a + 376624072046688339/54720498089681600)*q^6 + (4273/569634332748800*a^7 - 2640299/13671223985971200*a^6 - 8790163/233032227033600*a^5 + 1161831817/539653578393600*a^4 + 120073871/8900536449200*a^3 - 205016482211/44971131532800*a^2 + 4504955976967/17801072898400*a + 30549650870213/14240858318720)*q^7 + (-32959/627058123401600*a^7 + 5657203/6688619982950400*a^6 + 329969/1434095193600*a^5 - 6274717/728607841280*a^4 - 15539903573/34836562411200*a^3 + 137223833609/9289749976320*a^2 - 28854144049/232243749408*a - 70644368356359/3870729156800)*q^8 + (-2141/29336052556800*a^7 + 50267/29336052556800*a^6 + 3949/12579782400*a^5 - 1967903/143804179200*a^4 - 109833567/203722587200*a^3 + 4225583033/203722587200*a^2 - 10351853423/50930646800*a - 232538089563/10186129360)*q^9 + (984493/15380126984217600*a^7 - 22965527/10253417989478400*a^6 - 33425539/93212890813440*a^5 + 57365051147/2563354497369600*a^4 + 24398075569/42722574956160*a^3 - 3408111602657/71204291593600*a^2 + 15923119891739/35602145796800*a + 599172767199101/17801072898400)*q^10 + (2141/44004078835200*a^7 - 50267/44004078835200*a^6 - 3949/18869673600*a^5 + 1967903/215706268800*a^4 + 36611189/101861293600*a^3 - 4225583033/305583880800*a^2 + 10351853423/76395970200*a + 67326567161/5093064680)*q^11 + (86939533/42025342532875468800*a^7 + 6590802397/23639255174742451200*a^6 - 746191843/54063476671795200*a^5 - 8095928922437/1969937931228537600*a^4 + 405858460990847/2626583908304716800*a^3 + 1207679029054651/164161494269044800*a^2 - 102554420323546167/218881992358726400*a - 24688973293588019/13680124522420400)*q^12 + (11741/603142234675200*a^7 + 62981/452356676006400*a^6 - 1522997/13707778060800*a^5 + 39145307/37696389667200*a^4 + 1353058567/4188487740800*a^3 - 5436188881/3141365805600*a^2 - 473047637677/1047121935200*a + 308381226277/52356096760)*q^13 + (-1547688293/45387369935505506304*a^7 + 237574576913/189114041397939609600*a^6 + 11306818819/162190430015385600*a^5 - 132731396923/16263677450803200*a^4 + 1064613417972889/7879751724914150400*a^3 + 264583332386069/69120629165913600*a^2 - 388090243377087737/656645977076179200*a + 22152894165870723/109440996179363200)*q^14 + (-180143/3344309991475200*a^7 - 6134779/2229539994316800*a^6 + 59051/159343910400*a^5 + 26703019/37158999905280*a^4 - 23941714277/23224374940800*a^3 + 10816103935/619316665088*a^2 + 41676710529/387072915680*a - 158322600633729/3870729156800)*q^15 + (1/9670526976*a^7 - 43/58023161856*a^6 - 41/73261568*a^5 + 25841/1611754496*a^4 + 211557/201469312*a^3 - 13113489/402938624*a^2 + 8217603/50367328*a + 3278055475/100734656)*q^16 + (44257/1951641294096384*a^7 - 37219/27106129084672*a^6 - 11017/836895923712*a^5 + 53766363/6776532271168*a^4 - 11728498927/40659193627008*a^3 - 10618010781/1694133067792*a^2 + 2047200601843/3388266135584*a + 27085206645/423533266948)*q^17 + (365423/3344309991475200*a^7 - 1321169/4459079988633600*a^6 - 112979/191212692480*a^5 + 5672739703/371589999052800*a^4 + 2740611011/2322437494080*a^3 - 1051602176679/30965833254400*a^2 + 566135531627/3870729156800*a + 248415428686947/7741458313600)*q^18 + (-7/332230809600*a^7 + 103/166115404800*a^6 + 91/769052800*a^5 - 18113/2768590080*a^4 - 482923/2307158400*a^3 + 642433/46143168*a^2 - 1216047/19226320*a - 1005382699/96131600)*q^19 + O(q^20) Power series ring in q over Univariate Quotient Polynomial Algebra in a over Rational Field with modulus a^8 - 5328*a^6 + 96768*a^5 + 13200480*a^4 - 183472128*a^3 - 7304649984*a^2 + 365769105408*a + 8863960211712 Total time: 1.110 seconds, Total memory usage: 6.62MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Mon Dec 5 02:37:17 2005 Input: G :=DirichletGroup(168); G; X :=Elements(G); X; Y :=X[16]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; qEigenform(D[5],44);Parent($1); Output: Magma V2.11-10 Mon Dec 5 2005 02:37:16 on modular [Seed = 398965269] ------------------------------------- Group of Dirichlet characters of modulus 168 over Rational Field [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3, $.4, $.1*$.4, $.2*$.4, $.1*$.2*$.4, $.3*$.4, $.1*$.3*$.4, $.2*$.3*$.4, $.1*$.2*$.3*$.4 ] 168 2 [ Modular symbols space of level 168, weight 3, character $.1*$.2*$.3*$.4, and dimension 1 over Rational Field, Modular symbols space of level 168, weight 3, character $.1*$.2*$.3*$.4, and dimension 1 over Rational Field, Modular symbols space of level 168, weight 3, character $.1*$.2*$.3*$.4, and dimension 1 over Rational Field, Modular symbols space of level 168, weight 3, character $.1*$.2*$.3*$.4, and dimension 1 over Rational Field, Modular symbols space of level 168, weight 3, character $.1*$.2*$.3*$.4, and dimension 8 over Rational Field, Modular symbols space of level 168, weight 3, character $.1*$.2*$.3*$.4, and dimension 48 over Rational Field ] q + (-366161/49852747597086720*a^7 - 2797469/49852747597086720*a^6 + 158391289/2492637379854336*a^5 + 8901662149/12463186899271680*a^4 - 96803494055/623159344963584*a^3 - 12765707507471/3115796724817920*a^2 + 9452998877071/778949181204480*a + 799379725775941/259649727068160)*q^2 + (-133704061/3556162661925519360*a^7 - 722881927/1778081330962759680*a^6 + 20493849955/59269377698758656*a^5 + 115827613183/49391148082298880*a^4 - 4414687931579/4939114808229888*a^3 - 318224077585391/37043361061724160*a^2 + 5196394685395351/55565041592586240*a + 136453130538773749/27782520796293120)*q^3 + (-1/42785832960*a^7 - 17/21392916480*a^6 + 461/2139291648*a^5 + 31537/5348229120*a^4 - 305647/534822912*a^3 - 21474203/1337057280*a^2 + 15212051/668528640*a + 879076513/111421440)*q^4 + (106591/806589313440768*a^7 + 40379/50411832090048*a^6 - 76382663/67215776120064*a^5 - 8079881/1400328669168*a^4 + 5136411875/1867104892224*a^3 + 33800872687/1050246501876*a^2 - 60881185195/12602958022512*a - 12267009810503/787684876407)*q^5 + (-45302249/533424399288827904*a^7 - 646673957/1066848798577655808*a^6 + 33290024101/44452033274068992*a^5 + 120943483541/29634688849379328*a^4 - 2320917648437/1234778702057472*a^3 - 450321482240941/22226016637034496*a^2 + 700967914085105/8334756238887936*a + 174868732364074967/16669512477775872)*q^6 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514334695/96171558912*a^3 + 34216014163/1081930037760*a^2 - 47465191101 ** WARNING: Output too long, hence truncated. '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Mon Dec 5 02:36:32 2005 Input: G :=DirichletGroup(168); G; X :=Elements(G); X; Y :=X[16]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; qEigenform(D[5],90);Parent($1); Output: Magma V2.11-10 Mon Dec 5 2005 02:36:31 on modular [Seed = 500020172] ------------------------------------- Group of Dirichlet characters of modulus 168 over Rational Field [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3, $.4, $.1*$.4, $.2*$.4, $.1*$.2*$.4, $.3*$.4, $.1*$.3*$.4, $.2*$.3*$.4, $.1*$.2*$.3*$.4 ] 168 2 [ Modular symbols space of level 168, weight 3, character $.1*$.2*$.3*$.4, and dimension 1 over Rational Field, Modular symbols space of level 168, weight 3, character $.1*$.2*$.3*$.4, and dimension 1 over Rational Field, Modular symbols space of level 168, weight 3, character $.1*$.2*$.3*$.4, and dimension 1 over Rational Field, Modular symbols space of level 168, weight 3, character $.1*$.2*$.3*$.4, and dimension 1 over Rational Field, Modular symbols space of level 168, weight 3, character $.1*$.2*$.3*$.4, and dimension 8 over Rational Field, Modular symbols space of level 168, weight 3, character $.1*$.2*$.3*$.4, and dimension 48 over Rational Field ] q + (317/173203430400*a^7 - 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147/19676800*a^5 + 2853/39353600*a^4 + 58849/19676800*a^3 + 222341/9838400*a^2 + 10248889/7378800*a - 15283/17200)*q^19 + (144803/3972132003840*a^7 + 441/2149422080*a^6 - 2120691/66202200064*a^5 + 673813/5910910720*a^4 + 1273273011/82752750080*a^3 + 407651341/1477727680*a^2 + 66517853527/8866366080*a + 712507313/33584720)*q^20 + (-19247/472872857600*a^7 - 29929/354654643200*a^6 + 5034671/118218214400*a^5 - 24527361/59109107200*a^4 - 119652341/5910910720*a^3 - 349257509/3694319200*a^2 - 43458287687/7388638400*a - 1218787157/1007541600)*q^21 + (1/46126080*a^7 + 1/13178880*a^6 - 53/2306304*a^5 + 613/3294720*a^4 + 32111/2882880*a^3 + 63211/823680*a^2 + 308719/102960*a + 409/160)*q^22 + (-1/605606400*a^7 + 11/173030400*a^6 + 101/75700800*a^5 - 2239/43257600*a^4 + 13879/37850400*a^3 - 171883/10814400*a^2 + 5041/168975*a + 4437783/300400)*q^23 + (9467/794426400768*a^7 - 956143/4965165004800*a^6 - 22940193/1655055001600*a^5 + 160175209/413763750400*a^4 + 1370346727/413763750400*a^3 - 11793248219/103440937600*a^2 + 50196143149/44331830400*a - 26243615287/1007541600)*q^24 - 59*q^25 + (307511/11916396011520*a^7 - 65579/212792785920*a^6 - 1564845/66202200064*a^5 + 10404001/17732732160*a^4 + 1687795847/248258250240*a^3 - 252223303/4433183040*a^2 + 74671076059/26599098240*a - 8232535577/302262480)*q^26 + (292471/13240440012800*a^7 - 56689/206881875200*a^6 - 71564517/3310110003200*a^5 + 28429461/51720468800*a^4 + 4822759341/827527500800*a^3 - 277065927/3232529300*a^2 + 68854988087/29554553600*a - 23333182/807325)*q^27 + (193/13222809600*a^7 - 691/3777945600*a^6 - 6359/413212800*a^5 + 354649/944486400*a^4 + 2953637/826425600*a^3 - 20432513/236121600*a^2 + 41789051/14757600*a - 2467087/82560)*q^28 + (1/242242560*a^7 - 11/69212160*a^6 - 101/30280320*a^5 + 2239/17303040*a^4 - 13879/15140160*a^3 + 171883/4325760*a^2 - 5041/67590*a - 4437783/120160)*q^29 + (-547/5498521600*a^7 - 219/1099704320*a^6 + 13753/137463040*a^5 - 28863/34365760*a^4 - 16308011/343657600*a^3 - 119331099/343657600*a^2 - 55323101/4295720*a + 143667699/3905200)*q^30 + (-1/305141760*a^7 + 7/54489600*a^6 + 183/42380800*a^5 - 791/4540800*a^4 - 30941/31785600*a^3 + 81791/1135200*a^2 - 497557/3405600*a + 1076131/77400)*q^31 + (139/1968220800*a^7 + 2243/31491532800*a^6 - 271937/3936441600*a^5 + 4741733/7872883200*a^4 + 15797837/492055200*a^3 + 97513441/393644160*a^2 + 308292737/35146800*a + 27393021/7810400)*q^32 + (-102653/4965165004800*a^7 + 153017/1241291251200*a^6 + 770937/37614886400*a^5 - 3574881/9403721600*a^4 - 62978211/9403721600*a^3 + 480839973/25860234400*a^2 - 32243595151/11082957600*a + 319324289/19375800)*q^33 + (-53/2479276800*a^7 + 31/141672960*a^6 + 1299/55095040*a^5 - 1165/2361216*a^4 - 819031/103303200*a^3 + 1459543/14757600*a^2 - 247133/80496*a + 2289809/77400)*q^34 + (-27737/189149143040*a^7 + 36731/177327321600*a^6 + 33781299/236436428800*a^5 - 11901339/7388638400*a^4 - 3676303591/59109107200*a^3 - 1390742347/3694319200*a^2 - 250432670383/14777276800*a + 1043336056/31485675)*q^35 + (5561/76979302400*a^7 + 28411/153958604800*a^6 - 107033/1480371200*a^5 + 22164511/38489651200*a^4 + 168499519/4811206400*a^3 + 179789341/740185600*a^2 + 1625176379/171828800*a + 55626669/6248320)*q^36 + (1/9225216*a^7 + 1/2635776*a^6 - 265/2306304*a^5 + 613/658944*a^4 + 32111/576576*a^3 + 63211/164736*a^2 + 308719/20592*a - 231/32)*q^37 + (-15277/1083308728320*a^7 - 21739/1489549501440*a^6 + 829461/66202200064*a^5 - 14446879/124129125120*a^4 - 1332490559/248258250240*a^3 - 2193684743/31032281280*a^2 - 67682599603/26599098240*a - 656085151/302262480)*q^38 + (43/687315200*a^7 + 111/343657600*a^6 - 46009/687315200*a^5 + 330273/687315200*a^4 + 2918131/85914400*a^3 + 17731503/85914400*a^2 + 388335583/42957200*a + 5355567/300400)*q^39 + (-1/28173600*a^7 - 1099/1416729600*a^6 + 16917/550950400*a^5 + 42917/118060800*a^4 - 3319819/206606400*a^3 - 15301991/29515200*a^2 - 398112133/44272800*a - 9798481/154800)*q^40 + (-447/25462384640*a^7 + 1541/3182798080*a^6 + 21213/1273119232*a^5 - 561357/795699520*a^4 - 3416397/1591399040*a^3 + 3460881/18084080*a^2 - 22398963/56835680*a + 28489169/645860)*q^41 + (3009109/39721320038400*a^7 - 170821/2837237145600*a^6 - 238842831/3310110003200*a^5 + 14512241/18187417600*a^4 + 25473257153/827527500800*a^3 + 12039490211/59109107200*a^2 + 880428043283/88663660800*a + 71585213903/4030166400)*q^42 + 24*q^43 + (-101/2706303600*a^7 - 14737/173203430400*a^6 + 794803/21650428800*a^5 - 12595207/43300857600*a^4 - 48115219/2706303600*a^3 - 1337570791/10825214400*a^2 - 929072323/193307400*a - 9885609/781040)*q^44 + (-2237/82752750080*a^7 - 20497/29554553600*a^6 + 37251/2350930400*a^5 + 349053/671694400*a^4 - 25385811/2350930400*a^3 - 1101878883/1847159600*a^2 - 3165829207/461789900*a - 3312508801/41980900)*q^45 + (1/46126080*a^7 + 1/13178880*a^6 - 53/2306304*a^5 + 613/3294720*a^4 + 32111/2882880*a^3 + 63211/823680*a^2 + 308719/102960*a - 4711/160)*q^46 + (4637/74477475072*a^7 - 137/1329954912*a^6 - 115173/2068818752*a^5 + 77659/110829576*a^4 + 34422593/1551614064*a^3 + 606 ** WARNING: Output too long, hence truncated. '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Mon Dec 5 02:35:21 2005 Input: G :=DirichletGroup(168); G; X :=Elements(G); X; Y :=X[16]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; qEigenform(D[1],90);Parent($1); Output: Magma V2.11-10 Mon Dec 5 2005 02:35:20 on modular [Seed = 13676182] ------------------------------------- Group of Dirichlet characters of modulus 168 over Rational Field [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3, $.4, $.1*$.4, $.2*$.4, $.1*$.2*$.4, $.3*$.4, $.1*$.3*$.4, $.2*$.3*$.4, $.1*$.2*$.3*$.4 ] 168 2 [ Modular symbols space of level 168, weight 3, character $.1*$.2*$.3*$.4, and dimension 1 over Rational Field, Modular symbols space of level 168, weight 3, character $.1*$.2*$.3*$.4, and dimension 1 over Rational Field, Modular symbols space of level 168, weight 3, character $.1*$.2*$.3*$.4, and dimension 1 over Rational Field, Modular symbols space of level 168, weight 3, character $.1*$.2*$.3*$.4, and dimension 1 over Rational Field, Modular symbols space of level 168, weight 3, character $.1*$.2*$.3*$.4, and dimension 8 over Rational Field, Modular symbols space of level 168, weight 3, character $.1*$.2*$.3*$.4, and dimension 48 over Rational Field ] q - 2*q^2 - 3*q^3 + 4*q^4 + 6*q^6 - 7*q^7 - 8*q^8 + 9*q^9 - 12*q^12 - 2*q^13 + 14*q^14 + 16*q^16 + 22*q^17 - 18*q^18 + 21*q^21 + 38*q^23 + 24*q^24 + 25*q^25 + 4*q^26 - 27*q^27 - 28*q^28 + 26*q^29 + 34*q^31 - 32*q^32 - 44*q^34 + 36*q^36 + 6*q^39 - 26*q^41 - 42*q^42 - 82*q^43 - 76*q^46 - 48*q^48 + 49*q^49 - 50*q^50 - 66*q^51 - 8*q^52 - 22*q^53 + 54*q^54 + 56*q^56 - 52*q^58 + 106*q^59 + 94*q^61 - 68*q^62 - 63*q^63 + 64*q^64 - 34*q^67 + 88*q^68 - 114*q^69 - 58*q^71 - 72*q^72 - 75*q^75 - 12*q^78 + 81*q^81 + 52*q^82 + 58*q^83 + 84*q^84 + 164*q^86 - 78*q^87 - 122*q^89 + O(q^90) Power series ring in q over Rational Field Total time: 1.090 seconds, Total memory usage: 6.62MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Mon Dec 5 02:32:56 2005 Input: G :=DirichletGroup(168); G; X :=Elements(G); X; Y :=X[16]; Conductor(Y); Order(Y); M := ModularSymbols(Y, 3, 1); D := NewformDecomposition(NewSubspace(CuspidalSubspace(M))); D; Norm(Coefficient(qEigenform(D[1],12),11)); Norm(Coefficient(qEigenform(D[2],12),11)); Norm(Coefficient(qEigenform(D[3],12),11)); Norm(Coefficient(qEigenform(D[4],12),11)); Norm(Coefficient(qEigenform(D[5],12),11)); Norm(Coefficient(qEigenform(D[6],12),11)); Norm(Coefficient(qEigenform(D[7],12),11)); Norm(Coefficient(qEigenform(D[8],12),11)); Norm(Coefficient(qEigenform(D[9],12),11)); Norm(Coefficient(qEigenform(D[10],12),11)); Norm(Coefficient(qEigenform(D[11],12),11)); Norm(Coefficient(qEigenform(D[12],12),11)); Norm(Coefficient(qEigenform(D[13],12),11)); Norm(Coefficient(qEigenform(D[14],12),11)); Norm(Coefficient(qEigenform(D[15],12),11)); Norm(Coefficient(qEigenform(D[16],12),11)); Norm(Coefficient(qEigenform(D[17],12),11)); Norm(Coefficient(qEigenform(D[18],12),11)); Norm(Coefficient(qEigenform(D[19],12),11)); Output: ** WARNING: Computation time exceeded 20 seconds, so computation was terminated after 20 seconds. ** Magma V2.11-10 Mon Dec 5 2005 02:32:35 on modular [Seed = 131574074] ------------------------------------- Group of Dirichlet characters of modulus 168 over Rational Field [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3, $.4, $.1*$.4, $.2*$.4, $.1*$.2*$.4, $.3*$.4, $.1*$.3*$.4, $.2*$.3*$.4, $.1*$.2*$.3*$.4 ] 168 2 [ Modular symbols space of level 168, weight 3, character $.1*$.2*$.3*$.4, and dimension 1 over Rational Field, Modular symbols space of level 168, weight 3, character $.1*$.2*$.3*$.4, and dimension 1 over Rational Field, Modular symbols space of level 168, weight 3, character $.1*$.2*$.3*$.4, and dimension 1 over Rational Field, Modular symbols space of level 168, weight 3, character $.1*$.2*$.3*$.4, and dimension 1 over Rational Field, Modular symbols space of level 168, weight 3, character $.1*$.2*$.3*$.4, and dimension 8 over Rational Field, Modular symbols space of level 168, weight 3, character $.1*$.2*$.3*$.4, and dimension 48 over Rational Field ] 0 0 0 0 1048576 Errors: /bin/sh: line 1: 27945 Alarm clock nice -n 19 /usr/local/bin/magma '60.225.' ************** MAGMA ***************** Host 60.225.131.213 (60.225.131.213) Time: Sun Dec 4 21:21:42 2005 Input: Factorization(777); Factorization(777777); Output: Magma V2.11-10 Sun Dec 4 2005 21:21:41 on modular [Seed = 3077001384] ------------------------------------- [ <3, 1>, <7, 1>, <37, 1> ] [ <3, 1>, <7, 2>, <11, 1>, <13, 1>, <37, 1> ] Total time: 0.190 seconds, Total memory usage: 3.24MB '60.225.' ************** MAGMA ***************** Host 60.225.131.213 (60.225.131.213) Time: Sun Dec 4 21:20:37 2005 Input: Factorization(4444); Output: Magma V2.11-10 Sun Dec 4 2005 21:20:36 on modular [Seed = 2975945068] ------------------------------------- [ <2, 2>, <11, 1>, <101, 1> ] Total time: 0.190 seconds, Total memory usage: 3.24MB '128.135' ************** MAGMA ***************** Host 128.135.197.33 (128.135.197.33) Time: Sun Dec 4 18:22:47 2005 Input: 3*7*691/(98*288); Output: Magma V2.11-10 Sun Dec 4 2005 18:22:47 on modular [Seed = 2927531971] ------------------------------------- 691/1344 Total time: 0.190 seconds, Total memory usage: 3.24MB '128.135' ************** MAGMA ***************** Host 128.135.197.33 (128.135.197.33) Time: Sun Dec 4 18:20:02 2005 Input: Factorization (945); Output: Magma V2.11-10 Sun Dec 4 2005 18:20:02 on modular [Seed = 1417965276] ------------------------------------- [ <3, 3>, <5, 1>, <7, 1> ] Total time: 0.180 seconds, Total memory usage: 3.24MB '128.135' ************** MAGMA ***************** Host 128.135.197.33 (128.135.197.33) Time: Sun Dec 4 18:19:48 2005 Input: Factorization (691); Output: Magma V2.11-10 Sun Dec 4 2005 18:19:48 on modular [Seed = 1401121261] ------------------------------------- [ <691, 1> ] Total time: 0.190 seconds, Total memory usage: 3.24MB '128.135' ************** MAGMA ***************** Host 128.135.197.33 (128.135.197.33) Time: Sun Dec 4 18:19:24 2005 Input: Factorization (638512875); Output: Magma V2.11-10 Sun Dec 4 2005 18:19:23 on modular [Seed = 1586390031] ------------------------------------- [ <3, 6>, <5, 3>, <7, 2>, <11, 1>, <13, 1> ] Total time: 0.190 seconds, Total memory usage: 3.24MB '128.135' ************** MAGMA ***************** Host 128.135.197.33 (128.135.197.33) Time: Sun Dec 4 18:19:18 2005 Input: Factorize (638512875); Output: Magma V2.11-10 Sun Dec 4 2005 18:19:17 on modular [Seed = 1167425306] ------------------------------------- >> Factorize (638512875);; ^ User error: Identifier 'Factorize' has not been declared or assigned Total time: 0.190 seconds, Total memory usage: 3.24MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Sun Dec 4 18:17:48 2005 Input: G :=DirichletGroup(1176); G; X :=Elements(G); X; Output: Magma V2.11-10 Sun Dec 4 2005 18:17:47 on modular [Seed = 1990609991] ------------------------------------- Group of Dirichlet characters of modulus 1176 over Rational Field [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3, $.4, $.1*$.4, $.2*$.4, $.1*$.2*$.4, $.3*$.4, $.1*$.3*$.4, $.2*$.3*$.4, $.1*$.2*$.3*$.4 ] Total time: 0.200 seconds, Total memory usage: 3.34MB '69.165.' ************** MAGMA ***************** Host 69.165.160.197 (69.165.160.197) Time: Sun Dec 4 12:38:06 2005 Input: 233*7 Output: Magma V2.11-10 Sun Dec 4 2005 12:38:05 on modular [Seed = 4281308580] ------------------------------------- 1631 Total time: 0.190 seconds, Total memory usage: 3.24MB '82.32.2' ************** MAGMA ***************** Host 82.32.210.219 (82.32.210.219) Time: Sun Dec 4 08:46:05 2005 Input: SetVerbose("BestCode",2); BKLC(GF(2),144,72); Output: Magma V2.11-10 Sun Dec 4 2005 08:46:04 on modular [Seed = 1451746925] ------------------------------------- Construction of a [ 144 , 72 , 22 ] Code: [1]: [15, 11, 3] "BCH code (d = 3, b = 1)" Cyclic Linear Code over GF(2) BCHCode with parameters 15 3 [2]: [11, 7, 3] Linear Code over GF(2) Shortening of [1] at { 12 .. 15 } [3]: [8, 7, 2] Linear Code over GF(2) Dual of the RepetitionCode of length 8 [4]: [127, 70] "BCH code (d = 20, b = 127)" Cyclic Linear Code over GF(2) BCHCode with parameters 127 20 0 [5]: [127, 70] "BCH code (d = 18, b = 125)" Cyclic Linear Code over GF(2) BCHCode with parameters 127 18 125 [6]: [127, 77] "BCH code (d = 16, b = 127)" Cyclic Linear Code over GF(2) BCHCode with parameters 127 16 0 [7]: [146, 77, 21] Linear Code over GF(2) ConstructionXX using [6] [5] [4] [3] and [2] [8]: [147, 77, 22] Linear Code over GF(2) ExtendCode [7] by 1 [9]: [144, 74, 22] Linear Code over GF(2) Shortening of [8] at { 145 .. 147 } [10]: [144, 72, 22] Linear Code over GF(2) Subcode of [9] [144, 72, 22] Linear Code over GF(2) Generator matrix: [1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 1 1 0 0 0 0 1 0 0 1 0 1 1 0 0 0 1 1 0 0 1 1 1 0 0 0 0 0 0 0 1 1 1 1 0 0 0 0 0 1 0 1 1 0 0 1 0 0 1 0 1 0 0 0 1 1 1 1 0 0 0 1 0 0 1] [0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 1 0 1 0 1 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 1 1 1 1 0 0 1 0 0 1 1 1 1 0 1 1 1 1 1 0 1 1 1 0 1 0 1 0 1 0 0 1 1 1 0 1 0 1 1 0 0 0] [0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 1 1 0 1 0 1 1 1 0 1 0 1 0 0 0 1 1 1 0 0 1 1 1 0 1 1 1 1 0 0 0 1 1 0 1 1 1 0 1 0 1 0 0 0 1 0 1 0 0 0 1 0 0 0 0 1 1 0 0 1 0 0 1 0 1] [0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 0 1 0 0 0 1 1 1 0 1 0 1 0 0 0 1 0 1 1 1 1 1 0 0 1 1 1 0 1 1 0 0 1 0 0 0 0 0 0 0 1 1 1 1 1 0 0 1 0 0 0 1 1 0 1 1 0 1 0 0 0 0 1 1 1] [0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 1 0 1 0 1 0 1 1 0 0 1 1 1 0 0 0 0 1 0 1 1 0 1 0 0 1 0 0 0 0 1 0 0 1 0 1 1 0 0 1 1 1 0 1 1 1 1 0 1 0 1 1 0 0 0 0 1 0 1 0 1 1 0] [0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 0 1 1 0 1 0 1 1 1 0 0 1 1 1 1 1 1 1 1 0 0 0 1 0 1 0 1 0 0 1 0 1 1 1 0 0 0 1 0 1 0 0 1 0 1 1 1 1 1 1 1 1 1 0 1 0 1 1 1 0 1 0 0 0 1 0] [0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 1 1 0 0 1 1 1 0 0 0 0 0 0 0 0 0 1 1 0 1 0 0 0 1 1 1 0 1 0 1 1 1 0 1 0 0 1 1 1 0 1 0 0 1 1 0 0 0 0 1 1 1 1 0 1 1 0 1 1 0 1 0 0 0 1] [0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 1 0 1 1 0 0 0 0 1 1 1 1 1 0 0 0 0 0 1 1 1 0 0 0 0 0 1 1 1 0 1 0 1 1 0 0 0 0 0 0 1 0 0 0 1 1 0 0 1 1 1 1 1 1 1 1 1 1 1 0 1 0 0] [0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 1 1 0 0 1 1 1 0 0 0 0 0 0 0 0 0 1 1 0 1 0 0 0 1 1 1 0 1 0 1 1 1 0 1 0 0 1 1 1 0 1 0 0 1 1 0 1 0 1 1 0 1 0 0 0 1 1 1 1 1 0 1 0] [0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 1 1 0 1 1 1 1 0 0 0 0 0 0 1 1 1 1 1 1 0 0 1 1 1 0 1 0 0 1 0 1 1 1 0 1 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 1 1 1 0 0 1 1 1 1 1 1 0 1 0 0] [0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 1 0 0 0 1 0 0 0 1 1 0 0 0 1 0 0 0 0 0 0 1 1 1 0 1 1 0 1 0 1 0 0 1 0 0 1 1 0 1 1 0 1 0 1 0 0 1 1 0 1 1 1 0 1 0 0 1 0 1 1 1 0 0 1 1] [0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 1 0 1 0 1 0 1 0 1 1 0 0 0 0 1 1 0 1 0 1 0 0 1 0 1 1 1 1 1 1 0 1 1 1 0 1 0 0 0 1 0 1 1 1 0 0 0 1 1 0 0 1 0 0 1 1 1 0 0 1 0 0 1 0 1] [0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 1 1 0 1 1 1 1 0 0 0 0 0 0 1 1 1 1 1 1 0 0 1 1 1 0 1 0 0 1 0 1 1 1 0 1 0 0 0 1 1 0 0 0 0 0 1 1 1 0 0 0 0 1 1 0 0 0 0 1 1 1 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 1 1 0 1 1 1 1 0 0 0 0 0 0 1 1 1 1 1 1 0 0 1 1 1 0 1 0 0 1 0 1 1 1 0 1 0 0 0 1 1 0 0 0 0 1 1 0 1 1 0 1 1 0 1 0 0 0 0 1 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 1 0 0 0 1 0 1 1 1 0 0 1 1 1 0 0 1 0 1 1 0 0 1 1 1 0 1 1 1 1 1 0 1 0 0 1 0 1 0 0 1 1 1 0 1 1 1 1 1 0 0 0 1 0 1 1 1 1 0 1 1 1 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 1 0 1 1 0 1 0 1 1 1 0 0 0 0 0 0 1 1 0 0 0 0 1 0 0 0 0 0 0 1 0 0 1 0 1 1 1 0 0 1 1 0 0 1 1 0 0 0 0 0 1 1 0 1 0 0 1 0 1 1 1 1 1 0 1 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 0 1 1 0 1 0 0 1 0 0 0 0 0 0 1 1 1 0 0 1 1 1 1 1 1 0 1 1 1 0 1 0 1 1 1 1 1 0 0 1 1 0 1 0 0 1 0 1 1 1 0 0 1 0 1 1 0 1 1 1 1 1 0 1 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 0 0 1 1 0 0 1 0 0 1 1 1 0 0 1 0 1 0 1 1 0 0 1 1 0 0 1 0 0 0 0 1 1 1 0 0 1 0 0 0 0 1 0 0 1 0 1 1 1 0 0 0 0 1 0 1 1 1 1 1 0 1 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 1 1 0 0 0 0 0 0 1 1 0 0 0 1 0 0 1 1 0 1 0 1 0 1 1 0 0 0 1 0 1 0 1 0 1 1 1 0 0 1 1 1 1 0 1 1 0 1 0 1 1 0 1 0 0 1 0 1 1 1 1 1 0 1 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 1 1 0 1 0 0 1 1 1 0 0 0 0 0 0 1 1 0 0 1 0 1 0 0 0 1 1 1 1 1 0 1 1 1 1 1 1 0 0 1 1 0 0 1 1 1 1 1 0 0 1 1 0 0 0 0 0 1 0 1 1 1 0 0 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 1 1 1 1 1 1 1 1 0 1 1 1 1 0 0 0 0 0 0 0 1 1 1 1 1 1 1 0 1 0 0 1 1 0 1 0 0 1 0 0 0 0 1 0 0 0 1 1 0 0 1 1 1 0 1 0 0 1 0 1 0 1 1 0 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 1 1 0 1 1 0 1 0 0 0 1 0 0 0 1 1 0 0 1 1 0 1 0 0 1 0 0 0 0 1 1 0 0 1 0 1 1 0 0 1 1 0 1 0 0 1 1 0 1 0 1 1 0 1 1 1 0 0 1 0 1 0 1 1 0] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 1 1 0 1 1 0 1 0 0 0 1 0 0 0 1 1 0 0 1 1 0 1 0 0 1 0 0 0 0 1 1 0 0 1 0 1 1 0 0 1 1 0 1 0 0 1 1 0 1 0 1 1 0 1 0 0 0 0 1 0 1 0 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 1 1 0 0 0 0 0 1 0 1 0 0 1 1 0 0 1 1 0 0 0 1 0 1 1 0 1 1 0 0 0 0 0 1 0 1 0 1 0 1 1 0 0 0 1 1 0 0 0 0 1 0 0 1 1 0 0 0 1 0 0 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 1 1 0 1 0 0 1 0 1 0 1 1 1 0 0 1 0 0 1 1 0 0 1 0 0 1 0 1 1 1 1 0 1 1 1 0 1 0 0 1 1 0 1 0 1 0 0 1 1 0 1 0 0 1 0 0 0 0 1 1 0 1 0 0 0 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 1 0 1 0 1 1 0 1 1 1 1 0 1 1 1 1 1 1 0 0 1 1 0 1 0 1 0 1 1 1 0 0 0 0 0 1 1 1 1 1 1 1 0 1 0 0 0 1 0 0 0 1 0 1 1 1 0 0 0 1 1 1 1 1 0 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 1 1 0 0 0 0 0 0 1 0 0 0 1 0 1 1 0 0 1 0 1 0 1 1 0 1 1 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 1 1 0 1 1 0 1 0 0 0 1 1 1 0 1 0 0 0 1 0 1 0 1 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 1 0 1 1 1 0 0 0 1 0 1 0 1 0 0 1 1 1 1 1 1 0 1 1 1 1 0 0 1 1 1 0 1 1 1 1 0 1 1 1 1 0 1 1 1 0 0 0 0 0 1 1 0 0 1 1 0 0 0 1 0 0 1] [0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 1 0 0 0 0 1 0 1 0 1 1 0 1 0 1 1 0 0 1 1 0 0 1 1 1 0 0 1 1 0 0 1 0 0 1 1 0 0 1 1 0 0 0 1 0 1 0 1 1 1 1 1 0 1 0 1 1 0 0 0] [0 0 0 0 0 0 0 0 0 0 ** WARNING: Output too long, hence truncated. '132.70.' ************** MAGMA ***************** Host 132.70.50.117 (132.70.50.117) Time: Sun Dec 4 05:39:50 2005 Input: Br:=BraidGroup(4); a:=Br.1;b:=Br.2;c:=Br.3; D:=(a*b*c*a*b*a);F:=(a*b*c)^2; P:=IsConjugate(D,F); Output: Magma V2.11-10 Sun Dec 4 2005 05:39:50 on modular [Seed = 620077168] ------------------------------------- Total time: 0.190 seconds, Total memory usage: 3.24MB '132.70.' ************** MAGMA ***************** Host 132.70.50.117 (132.70.50.117) Time: Sun Dec 4 05:33:36 2005 Input: Br:=BraidGroup(4); a:=Br.1;b:=Br.2;c:=Br.3; D:=(a*b*c*a*b*a);F:=(a*b*c)^2; IsConjugate(D,F); Output: Magma V2.11-10 Sun Dec 4 2005 05:33:36 on modular [Seed = 3506459890] ------------------------------------- true Total time: 0.190 seconds, Total memory usage: 3.24MB '132.70.' ************** MAGMA ***************** Host 132.70.50.117 (132.70.50.117) Time: Sun Dec 4 05:32:42 2005 Input: Br:=BraidGroup(4); a:=Br.1;b:=Br.2;c:=Br.3; D:=(a*b*c*a*b*a);F:=(a*b*c)^2; [T,T1]:=IsConjugate(D,F); Output: Magma V2.11-10 Sun Dec 4 2005 05:32:42 on modular [Seed = 3458156583] ------------------------------------- >> [T,T1]:=IsConjugate(D,F); ^ User error: Illegal left hand side of an assignment statement Total time: 0.190 seconds, Total memory usage: 3.24MB '132.70.' ************** MAGMA ***************** Host 132.70.50.117 (132.70.50.117) Time: Sun Dec 4 05:28:42 2005 Input: Br:=BraidGroup(4); a:=Br.1;b:=Br.2;c:=Br.3; D:=(a*b*c*a*b*a);F:=(a*b*c)^2; T:=IsConjugate(D,F); WhatType(T) ; Output: Magma V2.11-10 Sun Dec 4 2005 05:28:42 on modular [Seed = 3289727805] ------------------------------------- >> WhatType(T) ^ User error: Identifier 'WhatType' has not been declared or assigned Total time: 0.190 seconds, Total memory usage: 3.24MB '132.70.' ************** MAGMA ***************** Host 132.70.50.117 (132.70.50.117) Time: Sun Dec 4 05:28:30 2005 Input: Br:=BraidGroup(4); a:=Br.1;b:=Br.2;c:=Br.3; D:=(a*b*c*a*b*a);F:=(a*b*c)^2; T:=IsConjugate(D,F); whattype(T) ; Output: Magma V2.11-10 Sun Dec 4 2005 05:28:30 on modular [Seed = 3239069228] ------------------------------------- >> whattype(T) ^ User error: Identifier 'whattype' has not been declared or assigned Total time: 0.190 seconds, Total memory usage: 3.24MB '132.70.' ************** MAGMA ***************** Host 132.70.50.117 (132.70.50.117) Time: Sun Dec 4 05:28:10 2005 Input: Br:=BraidGroup(4); a:=Br.1;b:=Br.2;c:=Br.3; D:=(a*b*c*a*b*a);F:=(a*b*c)^2; T:=IsConjugate(D,F); T[2]; Output: Magma V2.11-10 Sun Dec 4 2005 05:28:10 on modular [Seed = 4264524135] ------------------------------------- >> T[2];; ^ Runtime error in '[]': Bad argument types Total time: 0.190 seconds, Total memory usage: 3.24MB '132.70.' ************** MAGMA ***************** Host 132.70.50.117 (132.70.50.117) Time: Sun Dec 4 05:26:26 2005 Input: Br:=BraidGroup(4); a:=Br.1;b:=Br.2;c:=Br.3; D:=(a*b*c*a*b*a);F:=(a*b*c)^2; T:=IsConjugate(D,F); T; Output: Magma V2.11-10 Sun Dec 4 2005 05:26:26 on modular [Seed = 4096095109] ------------------------------------- true Total time: 0.190 seconds, Total memory usage: 3.24MB '132.70.' ************** MAGMA ***************** Host 132.70.50.117 (132.70.50.117) Time: Sun Dec 4 05:26:04 2005 Input: Br:=BraidGroup(4); a:=Br.1;b:=Br.2;c:=Br.3; D:=(a*b*c*a*b*a);F:=(a*b*c)^2; T:=isConjugate(D,F); T; Output: Magma V2.11-10 Sun Dec 4 2005 05:26:03 on modular [Seed = 4045436649] ------------------------------------- >> T:=isConjugate(D,F); ^ User error: Identifier 'isConjugate' has not been declared or assigned >> T;; ^ User error: Identifier 'T' has not been declared or assigned Total time: 0.190 seconds, Total memory usage: 3.24MB '132.70.' ************** MAGMA ***************** Host 132.70.50.117 (132.70.50.117) Time: Sun Dec 4 05:25:06 2005 Input: Br:=BraidGroup(4); a:=Br.1;b:=Br.2;c:=Br.3; D:=(a*b*c*a*b*a);F:=(a*b*c)^2; IsConjugate(D,F); Output: Magma V2.11-10 Sun Dec 4 2005 05:25:05 on modular [Seed = 3997133260] ------------------------------------- true Total time: 0.190 seconds, Total memory usage: 3.24MB '132.70.' ************** MAGMA ***************** Host 132.70.50.117 (132.70.50.117) Time: Sun Dec 4 05:23:28 2005 Input: Br:=BraidGroup(4); a:=Br.1;b:=Br.2;c:=Br.3; D:=(a*b*c*a*b*a);F:=(a*b*c)^2; PositiveConjugates(F); Output: Magma V2.11-10 Sun Dec 4 2005 05:23:28 on modular [Seed = 3828704359] ------------------------------------- {@ Br.3 * Br.2 * Br.1 * Br.3 * Br.2 * Br.3 @} Total time: 0.190 seconds, Total memory usage: 3.24MB '132.70.' ************** MAGMA ***************** Host 132.70.50.117 (132.70.50.117) Time: Sun Dec 4 05:22:41 2005 Input: Br:=BraidGroup(4); a:=Br.1;b:=Br.2;c:=Br.3; D:=(a*b*c*a*b*a);F:=(a*b*c)^2 PositiveConjugates(F); Output: Magma V2.11-10 Sun Dec 4 2005 05:22:41 on modular [Seed = 3811730353] ------------------------------------- >> PositiveConjugates(F);; ^ User error: bad syntax Total time: 0.190 seconds, Total memory usage: 3.24MB '132.70.' ************** MAGMA ***************** Host 132.70.50.117 (132.70.50.117) Time: Sun Dec 4 05:20:29 2005 Input: Br:=BraidGroup(4); a:=Br.1;b:=Br.2;c:=Br.3; D:=(a*b*c*a*b*a);F:=(a*b*c)^2 Output: Magma V2.11-10 Sun Dec 4 2005 05:20:27 on modular [Seed = 2588613458] ------------------------------------- Total time: 0.210 seconds, Total memory usage: 3.24MB '204.210' ************** MAGMA ***************** Host 204.210.35.48 (204.210.35.48) Time: Sun Dec 4 03:52:23 2005 Input: G :=DirichletGroup(1575); G; X :=Elements(G); X; Output: Magma V2.11-10 Sun Dec 4 2005 03:52:23 on modular [Seed = 1022218584] ------------------------------------- Group of Dirichlet characters of modulus 1575 over Rational Field [ 1, $.1, $.2, $.1*$.2, $.3, $.1*$.3, $.2*$.3, $.1*$.2*$.3 ] Total time: 0.190 seconds, Total memory usage: 3.34MB '161.246' ************** MAGMA ***************** Host 161.246.1.33 (161.246.1.33) Time: Sun Dec 4 00:04:53 2005 Input: 2+3 Output: Magma V2.11-10 Sun Dec 4 2005 00:04:53 on modular [Seed = 3963475716] ------------------------------------- 5 Total time: 0.190 seconds, Total memory usage: 3.24MB '24.158.' ************** MAGMA ***************** Host 24.158.163.183 (24.158.163.183) Time: Sat Dec 3 18:38:20 2005 Input: PR := PolynomialRing(RationalField(), 3); I := ideal; Groebner(I); I; Output: Magma V2.11-10 Sat Dec 3 2005 18:38:16 on modular [Seed = 4146501764] ------------------------------------- Ideal of Polynomial ring of rank 3 over Rational Field Lexicographical Order Variables: t, x, y Groebner basis: [ t^2 - 881468632448381366573236683482557819679940281233931949107407297230523\ 37978943342401657863550929248859858595742780052545132851789925614198409\ 65890428329870646111856827881383710690971227741142932604027197561650409\ 25125402837936120450078542683149044830295192323970601909312468153468633\ 17346114209676594823743330674408214872353594733791189252806684672260108\ 65314395800559554891665664695515627595412817451848634505655029814098086\ 92447418381951844984510128713578056169221035834151173057293367433802441\ 97075996481196562838811518958219547140650952459081317638824504086936336\ 55152613270231007054672967701333022471880356816564488615430985913568499\ 98570945175162427123089113181839479329642472288509258049451274606899357\ 28414398652671582064162705276077004712071357172940983407119239602167142\ 12999714874773499806153074422110353722732488435357865689492709594950307\ 30808404071291471503917195000430875596466072590511545554397895074227701\ 46034403785570756512070180529374318768027967388400817352217972292078597\ 91696334544813285540938583462482240808871396102407074256550837427950467\ 09735082904846880928527285442476134233199439150000434248805256638931115\ 89774991874125772137238696716377221916929512319307870907621750917716496\ 37139211239427677846623309126743035897594089798452080331446824730256827\ 08452218475563544058655156301436681367830088748581810382619892722065167\ 07830552127579233741552908333263803841217520198941798485221423490302726\ 79540838347047859896944806667811403078660627987895478387587446126966013\ 71874460248114450040247006389038947283901274967378884778607074651170457\ 99586587910168448769709071451982042527742478665959276648202933181800673\ 94479353222423158444939898338951071566402247484018736009761036912959334\ 32134487204849408042423399572163349123228744124467011646920896987768583\ 22597174914422331486208611697295360000000000000000000000000/63822184766\ 98146611041098502826768428796825644905358012592084487394159829130713495\ 94963064187513259274115849765916400834353984800651053031725085689974625\ 03184090809290171475661354908761031599878047109000389326662909672526579\ 27867767871688853654694467344956205338778533868535733150953120661302700\ 37207745689687723470660814991612589582045557321511428493321119529764850\ 98154590941344244780254538650948294910196862766695014577658780179988149\ 54966984908957311683888793606532081614327151239575683487577938491305483\ 68336288304810842410690685663348623309539171423763405981496755410688780\ 17955920029370648645824037653907268279005576593653515599440687072737018\ 73744729172045035750061750500622902635389741405423640332711573610711384\ 89077177988972000701819365903901035006980304514091608256038370125552216\ 86312348001218791640380256499773842719118987572373430757265348912296056\ 52110089873220366581669884277300336820032041916273777027755211082257840\ 17895697857374573792883671174985035120831738171863261544320376370166358\ 96094952321554437633716797099229735263889237468336374605438940120956401\ 99604989730451052911730057502689676856848690110587740550720575460663894\ 13908243001521415535883777575546378252090031231133833516190912070289361\ 07328206503316814241801578251068311358187148224259453418539700840613495\ 66241580552102911114782410893058457300090010395044006701700164671381111\ 68105729379047870965348755551196342848778469476921284915700483724259215\ 12547056707888135359641449136674161620347169187357029120681787923062733\ 18557569854864613756414163939228945179085520320433491949650508195075923\ 35630935291353352230042736857770910023520647057744049846769505208490963\ 76006341993878304540398659341962407354877354735783814430380291546166019\ 41887484408009637009000276945891737381414902602785939809168697925848270\ 9464085301804841363301053624633610152767772169833911*t*y^27 + 16145893244588019376321463759147101976054935655561059823010489353963009\ 95857280876045537358073066293646780347476751437396739060058377866891614\ 56958696459339321764451415803140190302531452764875464930045906699429392\ 93322197711290533346477885997704462519826824238395287070127382054334825\ 12474154678496780636450590167845652712038994355838278984888207707401136\ 25793101228875854173302043860918791935103024875827289424685640482585888\ 58527872842853282182062781647541561385354893199020090099382245944071376\ 17512817248742682206013705633610232054322954082279994063896441253448449\ 85346152522942705767309622364519937441451848751162340430863262042777544\ 81613073723888803675696796843708677322713387739455901933990027946795462\ 03413907973996855287381187409138404706105632484713921107131351330810291\ 30882218866024701152824647805101133029445682306957060509514785413913927\ 02975991009369511431508267835319244302544318312708912057705591174954092\ 56754545807622466250514459683223972238742211592013008987780034165014005\ 20762734024413972758306388541659428870424914285449729061264865229054252\ 38843378207817525535905207426582748723137040977502926977634751768186698\ 49581209374388602372388115673748962542027564878465454029491751240952477\ 34612791431978069681060826807695029958268938415100506426028189406456810\ 47151753004300858542276117825497020675952427736778915349393009710275069\ 48984179044091308012297661610223584745791147499482444973386598700462218\ 22606684483968114187424005870778280377989242995451388532501416851196355\ 01833124158085873190081333983264290540365607375139933318620476903777326\ 80107573193720458309561560284258689641907871095592925772353970575124512\ 12475404115735794731369583795917400222525265781613763422453149992626793\ 68082207701480540682384020108812882191767165651145406631366951396502112\ 07997057420021592274019931910213468160000000000000000000000/63822184766\ 98146611041098502826768428796825644905358012592084487394159829130713495\ 94963064187513259274115849765916400834353984800651053031725085689974625\ 03184090809290171475661354908761031599878047109000389326662909672526579\ 27867767871688853654694467344956205338778533868535733150953120661302700\ 37207745689687723470660814991612589582045557321511428493321119529764850\ 98154590941344244780254538650948294910196862766695014577658780179988149\ 54966984908957311683888793606532081614327151239575683487577938491305483\ 68336288304810842410690685663348623309539171423763405981496755410688780\ 17955920029370648645824037653907268279005576593653515599440687072737018\ 73744729172045035750061750500622902635389741405423640332711573610711384\ 89077177988972000701819365903901035006980304514091608256038370125552216\ 86312348001218791640380256499773842719118987572373430757265348912296056\ 52110089873220366581669884277300336820032041916273777027755211082257840\ 17895697857374573792883671174985035120831738171863261544320376370166358\ 96094952321554437633716797099229735263889237468336374605438940120956401\ 99604989730451052911730057502689676856848690110587740550720575460663894\ 13908243001521415535883777575546378252090031231133833516190912070289361\ 07328206503316814241801578251068311358187148224259453418539700840613495\ 66241580552102911114782410893058457300090010395044006701700164671381111\ 68105729379047870965348755551196342848778469476921284915700483724259215\ 12547056707888135359641449136674161620347169187357029120681787923062733\ 18557569854864613756414163939228945179085520320433491949650508195075923\ 35630935291353352230042736857770910023520647057744049846769505208490963\ 76006341993878304540398659341962407354877354735783814430380291546166019\ 41887484408009637009000276945891737381414902602785939809168697925848270\ 9464085301804841363301053624633610152767772169833911*t*y^26 - 14189569054767755841931853466183156132446347954453444573569145856951694\ 97594498858653122916161244563504680207863736830617783868905678588623263\ 58149999570175871115229328171082776321630175215040653265091559032547797\ 31430174135118414507556058351671949932506942478301266480284529031139437\ 41948941211604196291522896965189054469466259155503783241197069335082147\ 73147912721291181077513656885507356175832838774546070403733244183896350\ 95446525802377648670029602668889796922504929178979896629976088075503016\ 66367885038783727071910279103789092599463229739996561041273969939960124\ 90504451432947103273023964424762583695568813765769913557391797055569260\ 03266758660192688706865894970120436086965615126630219052628167651298978\ 43492852102361245459176793134050457018607409028604703270481579914732705\ 81011795630745919129853669628298786502591595974677665329154884586273146\ 96521801683268382183066190908691653150183855602060625946786621458606013\ 92698455171727954285104993759171409390867910757794957084321809070738001\ 46478087269697971501738074571113565736127070688036023597767204985658452\ 17140623587337134427872758514246847269965626360127204036526844463226741\ 6141667452863041499240889640760710761960227057997382487953994 ** WARNING: Output too long, hence truncated. '24.158.' ************** MAGMA ***************** Host 24.158.163.183 (24.158.163.183) Time: Sat Dec 3 18:37:19 2005 Input: PR := PolynomialRing(RationalField(), 3); I := ideal; Groebner(I); Output: Magma V2.11-10 Sat Dec 3 2005 18:37:17 on modular [Seed = 4264530376] ------------------------------------- Total time: 1.690 seconds, Total memory usage: 7.06MB '24.158.' ************** MAGMA ***************** Host 24.158.163.183 (24.158.163.183) Time: Sat Dec 3 18:37:03 2005 Input: PR := PolynomialRing(RationalField(), 3); I := ideal; Groebner(I); Output: Magma V2.11-10 Sat Dec 3 2005 18:37:01 on modular [Seed = 4213871869] ------------------------------------- Total time: 1.700 seconds, Total memory usage: 6.97MB '24.158.' ************** MAGMA ***************** Host 24.158.163.183 (24.158.163.183) Time: Sat Dec 3 18:35:08 2005 Input: PR := PolynomialRing(RationalField(), 3); I := ideal; time Groebner(I); Output: Magma V2.11-10 Sat Dec 3 2005 18:35:07 on modular [Seed = 3239094641] ------------------------------------- Time: 0.010 Total time: 0.200 seconds, Total memory usage: 3.43MB '24.158.' ************** MAGMA ***************** Host 24.158.163.183 (24.158.163.183) Time: Sat Dec 3 18:34:34 2005 Input: PR := PolynomialRing(RationalField(), 3); I := ideal; time Groebner(I); Output: Magma V2.11-10 Sat Dec 3 2005 18:34:34 on modular [Seed = 3624513609] ------------------------------------- Time: 0.010 Total time: 0.200 seconds, Total memory usage: 3.43MB '128.196' ************** MAGMA ***************** Host 128.196.20.120 (128.196.20.120) Time: Sat Dec 3 15:19:02 2005 Input: rand(23984793287429832938) Output: Magma V2.11-10 Sat Dec 3 2005 15:19:02 on modular [Seed = 47234208] ------------------------------------- >> rand(23984793287429832938); ^ User error: Identifier 'rand' has not been declared or assigned Total time: 0.200 seconds, Total memory usage: 3.24MB '128.196' ************** MAGMA ***************** Host 128.196.20.120 (128.196.20.120) Time: Sat Dec 3 15:18:50 2005 Input: fft(1283129847219873192873192873) Output: Magma V2.11-10 Sat Dec 3 2005 15:18:49 on modular [Seed = 98024376] ------------------------------------- >> fft(1283129847219873192873192873) ^ User error: Identifier 'fft' has not been declared or assigned Total time: 0.190 seconds, Total memory usage: 3.24MB '193.249' ************** MAGMA ***************** Host 193.249.239.125 (193.249.239.125) Time: Sat Dec 3 13:23:45 2005 Input: P := PolynomialRing(RationalField(), 4); I:=IdealWithFixedBasis([-16*y*x^2-48*x*y+4*x^2+144*y^2+16*x^2*y^2+96*x*y^2, 288*x*z^2-144*x*z-64*x^2*z+64*x^2*z^2+16*x^2+324*z^2, 36*z^2-64*y^2*z+48*z^2*y-96*y*z+16*y^2*z^2+144*x^2*a^2+64*y^2+576*a^2+96*x*a+576*x*a^2+4*x^2+48*x^2*a, 36*z^2+256*a^2+128*a^2*z+192*z*a+16*z^2*a^2+48*z^2*a, 64*a^2+64*y^2*a^2+64*y*a+128*a^2*y+16*y^2+64*y^2*a, x^2+2*y^2+3*z^2+4*a^2-10, x^2+y^2+z^2+a^2+3*x+y+2*z-a-15]); C := Coordinates(I, P!1); Output: ** WARNING: Computation time exceeded 20 seconds, so computation was terminated after 20 seconds. ** Magma V2.11-10 Sat Dec 3 2005 13:23:24 on modular [Seed = 1268378305] ------------------------------------- Errors: /bin/sh: line 1: 21853 Alarm clock nice -n 19 /usr/local/bin/magma '193.250' ************** MAGMA ***************** Host 193.250.127.92 (193.250.127.92) Time: Sat Dec 3 12:43:19 2005 Input: Q := RationalField(); P := PolynomialRing(Q, 4); I:=IdealWithFixedBasis([-16*y*x^2-48*x*y+4*x^2+144*y^2+16*x^2*y^2+96*x*y^2, 288*x*z^2-144*x*z-64*x^2*z+64*x^2*z^2+16*x^2+324*z^2, 36*z^2-64*y^2*z+48*z^2*y-96*y*z+16*y^2*z^2+144*x^2*a^2+64*y^2+576*a^2+96*x*a+576*x*a^2+4*x^2+48*x^2*a, 36*z^2+256*a^2+128*a^2*z+192*z*a+16*z^2*a^2+48*z^2*a, 64*a^2+64*y^2*a^2+64*y*a+128*a^2*y+16*y^2+64*y^2*a, x^2+2*y^2+3*z^2+4*a^2-10, x^2+y^2+z^2+a^2+3*x+y+2*z-a-15]); g:=1; C := Coordinates(I, -16*y*x^2-48*x*y+4*x^2+144*y^2+16*x^2*y^2+96*x*y^2); Output: ** WARNING: Computation time exceeded 20 seconds, so computation was terminated after 20 seconds. ** Magma V2.11-10 Sat Dec 3 2005 12:42:59 on modular [Seed = 1022273033] ------------------------------------- Errors: /bin/sh: line 1: 21802 Alarm clock nice -n 19 /usr/local/bin/magma '193.250' ************** MAGMA ***************** Host 193.250.127.92 (193.250.127.92) Time: Sat Dec 3 12:42:28 2005 Input: Q := RationalField(); P := PolynomialRing(Q, 4); I:=IdealWithFixedBasis([-16*y*x^2-48*x*y+4*x^2+144*y^2+16*x^2*y^2+96*x*y^2, 288*x*z^2-144*x*z-64*x^2*z+64*x^2*z^2+16*x^2+324*z^2, 36*z^2-64*y^2*z+48*z^2*y-96*y*z+16*y^2*z^2+144*x^2*a^2+64*y^2+576*a^2+96*x*a+576*x*a^2+4*x^2+48*x^2*a, 36*z^2+256*a^2+128*a^2*z+192*z*a+16*z^2*a^2+48*z^2*a, 64*a^2+64*y^2*a^2+64*y*a+128*a^2*y+16*y^2+64*y^2*a, x^2+2*y^2+3*z^2+4*a^2-10, x^2+y^2+z^2+a^2+3*x+y+2*z-a-15]); g:=1; C := Coordinates(I, g); Output: Magma V2.11-10 Sat Dec 3 2005 12:42:27 on modular [Seed = 870426918] ------------------------------------- >> C := Coordinates(I, g);; ^ Runtime error in 'Coordinates': Bad argument types Argument types given: RngMPol, RngIntElt Total time: 0.190 seconds, Total memory usage: 3.34MB '193.250' ************** MAGMA ***************** Host 193.250.127.92 (193.250.127.92) Time: Sat Dec 3 12:40:57 2005 Input: Q := RationalField(); P := PolynomialRing(Q, 4); I:=IdealWithFixedBasis([-16*y*x^2-48*x*y+4*x^2+144*y^2+16*x^2*y^2+96*x*y^2, 288*x*z^2-144*x*z-64*x^2*z+64*x^2*z^2+16*x^2+324*z^2, 36*z^2-64*y^2*z+48*z^2*y-96*y*z+16*y^2*z^2+144*x^2*a^2+64*y^2+576*a^2+96*x*a+576*x*a^2+4*x^2+48*x^2*a, 36*z^2+256*a^2+128*a^2*z+192*z*a+16*z^2*a^2+48*z^2*a, 64*a^2+64*y^2*a^2+64*y*a+128*a^2*y+16*y^2+64*y^2*a, x^2+2*y^2+3*z^2+4*a^2-10, x^2+y^2+z^2+a^2+3*x+y+2*z-a-15]); I; Output: Magma V2.11-10 Sat Dec 3 2005 12:40:56 on modular [Seed = 232756915] ------------------------------------- Ideal of Polynomial ring of rank 4 over Rational Field Lexicographical Order Variables: x, y, z, a User basis: [ 16*x^2*y^2 - 16*x^2*y + 4*x^2 + 96*x*y^2 - 48*x*y + 144*y^2, 64*x^2*z^2 - 64*x^2*z + 16*x^2 + 288*x*z^2 - 144*x*z + 324*z^2, 144*x^2*a^2 + 48*x^2*a + 4*x^2 + 576*x*a^2 + 96*x*a + 16*y^2*z^2 - 64*y^2*z + 64*y^2 + 48*y*z^2 - 96*y*z + 36*z^2 + 576*a^2, 16*z^2*a^2 + 48*z^2*a + 36*z^2 + 128*z*a^2 + 192*z*a + 256*a^2, 64*y^2*a^2 + 64*y^2*a + 16*y^2 + 128*y*a^2 + 64*y*a + 64*a^2, x^2 + 2*y^2 + 3*z^2 + 4*a^2 - 10, x^2 + 3*x + y^2 + y + z^2 + 2*z + a^2 - a - 15 ] Basis: [ 16*x^2*y^2 - 16*x^2*y + 4*x^2 + 96*x*y^2 - 48*x*y + 144*y^2, 64*x^2*z^2 - 64*x^2*z + 16*x^2 + 288*x*z^2 - 144*x*z + 324*z^2, 144*x^2*a^2 + 48*x^2*a + 4*x^2 + 576*x*a^2 + 96*x*a + 16*y^2*z^2 - 64*y^2*z + 64*y^2 + 48*y*z^2 - 96*y*z + 36*z^2 + 576*a^2, 16*z^2*a^2 + 48*z^2*a + 36*z^2 + 128*z*a^2 + 192*z*a + 256*a^2, 64*y^2*a^2 + 64*y^2*a + 16*y^2 + 128*y*a^2 + 64*y*a + 64*a^2, x^2 + 2*y^2 + 3*z^2 + 4*a^2 - 10, x^2 + 3*x + y^2 + y + z^2 + 2*z + a^2 - a - 15 ] Total time: 0.190 seconds, Total memory usage: 3.34MB '193.250' ************** MAGMA ***************** Host 193.250.127.92 (193.250.127.92) Time: Sat Dec 3 12:40:35 2005 Input: Q := RationalField(); P := PolynomialRing(Q, 4); I:=IdealWithFixedBasis([-16*y*x^2-48*x*y+4*x^2+144*y^2+16*x^2*y^2+96*x*y^2, 288*x*z^2-144*x*z-64*x^2*z+64*x^2*z^2+16*x^2+324*z^2, 36*z^2-64*y^2*z+48*z^2*y-96*y*z+16*y^2*z^2+144*x^2*a^2+64*y^2+576*a^2+96*x*a+576*x*a^2+4*x^2+48*x^2*a, 36*z^2+256*a^2+128*a^2*z+192*z*a+16*z^2*a^2+48*z^2*a, 64*a^2+64*y^2*a^2+64*y*a+128*a^2*y+16*y^2+64*y^2*a, x^2+2*y^2+3*z^2+4*a^2-10, x^2+y^2+z^2+a^2+3*x+y+2*z-a-15]); Coordinates(I,1); Output: Magma V2.11-10 Sat Dec 3 2005 12:40:34 on modular [Seed = 13802372] ------------------------------------- >> Coordinates(I,1);; ^ Runtime error in 'Coordinates': Bad argument types Argument types given: RngMPol, RngIntElt Total time: 0.190 seconds, Total memory usage: 3.34MB '193.250' ************** MAGMA ***************** Host 193.250.127.92 (193.250.127.92) Time: Sat Dec 3 12:40:13 2005 Input: Q := RationalField(); P := PolynomialRing(Q, 4); I:=IdealWithFixedBasis([-16*y*x^2-48*x*y+4*x^2+144*y^2+16*x^2*y^2+96*x*y^2, 288*x*z^2-144*x*z-64*x^2*z+64*x^2*z^2+16*x^2+324*z^2, 36*z^2-64*y^2*z+48*z^2*y-96*y*z+16*y^2*z^2+144*x^2*a^2+64*y^2+576*a^2+96*x*a+576*x*a^2+4*x^2+48*x^2*a, 36*z^2+256*a^2+128*a^2*z+192*z*a+16*z^2*a^2+48*z^2*a, 64*a^2+64*y^2*a^2+64*y*a+128*a^2*y+16*y^2+64*y^2*a, x^2+2*y^2+3*z^2+4*a^2-10, x^2+y^2+z^2+a^2+3*x+y+2*z-a-15]); Output: Magma V2.11-10 Sat Dec 3 2005 12:40:13 on modular [Seed = 131439724] ------------------------------------- Total time: 0.190 seconds, Total memory usage: 3.34MB '193.250' ************** MAGMA ***************** Host 193.250.127.92 (193.250.127.92) Time: Sat Dec 3 12:39:52 2005 Input: Q := RationalField(); P := PolynomialRing(Q, 4); I:=IdealWithFixedBasis([-16*y*x^2-48*x*y+4*x^2+144*y^2+16*x^2*y^2+96*x*y^2, 288*x*z^2-144*x*z-64*x^2*z+64*x^2*z^2+16*x^2+324*z^2, 36*z^2-64*y^2*z+48*z^2*y-96*y*z+16*y^2*z^2+144*x^2*a^2+64*y^2+576*a^2+96*x*a+576*x*a^2+4*x^2+48*x^2*a, 36*z^2+256*a^2+128*a^2*z+192*z*a+16*z^2*a^2+48*z^2*a, 64*a^2+64*y^2*a^2+64*y*a+128*a^2*y+16*y^2+64*y^2*a, x^2+2*y^2+3*z^2+4*a^2-10, x^2+y^2+z^2+a^2+3*x+y+2*z-a-15]); Coordinates(I, -16*y*x^2-48*x*y+4*x^2+144*y^2+16*x^2*y^2+96*x*y^2); Output: ** WARNING: Computation time exceeded 20 seconds, so computation was terminated after 20 seconds. ** Magma V2.11-10 Sat Dec 3 2005 12:39:31 on modular [Seed = 500139608] ------------------------------------- Errors: /bin/sh: line 1: 21771 Alarm clock nice -n 19 /usr/local/bin/magma '193.250' ************** MAGMA ***************** Host 193.250.127.92 (193.250.127.92) Time: Sat Dec 3 12:38:18 2005 Input: Q := RationalField(); P := PolynomialRing(Q, 4); I:=IdealWithFixedBasis([-16*y*x^2-48*x*y+4*x^2+144*y^2+16*x^2*y^2+96*x*y^2, 288*x*z^2-144*x*z-64*x^2*z+64*x^2*z^2+16*x^2+324*z^2, 36*z^2-64*y^2*z+48*z^2*y-96*y*z+16*y^2*z^2+144*x^2*a^2+64*y^2+576*a^2+96*x*a+576*x*a^2+4*x^2+48*x^2*a, 36*z^2+256*a^2+128*a^2*z+192*z*a+16*z^2*a^2+48*z^2*a, 64*a^2+64*y^2*a^2+64*y*a+128*a^2*y+16*y^2+64*y^2*a, x^2+2*y^2+3*z^2+4*a^2-10, x^2+y^2+z^2+a^2+3*x+y+2*z-a-15]); Coordinates(I, 1); Output: Magma V2.11-10 Sat Dec 3 2005 12:38:18 on modular [Seed = 4013782258] ------------------------------------- >> Coordinates(I, 1);; ^ Runtime error in 'Coordinates': Bad argument types Argument types given: RngMPol, RngIntElt Total time: 0.190 seconds, Total memory usage: 3.34MB '193.250' ************** MAGMA ***************** Host 193.250.127.92 (193.250.127.92) Time: Sat Dec 3 12:36:46 2005 Input: Q := RationalField(); P := PolynomialRing(Q, 4); I:=IdealWithFixedBasis; Coordinates(I, 1); Output: Magma V2.11-10 Sat Dec 3 2005 12:36:46 on modular [Seed = 3862196831] ------------------------------------- >> I:=IdealWithFixedBasis> Coordinates(I, 1);; ^ User error: Identifier 'I' has not been declared or assigned Total time: 0.200 seconds, Total memory usage: 3.24MB '193.250' ************** MAGMA ***************** Host 193.250.127.92 (193.250.127.92) Time: Sat Dec 3 12:34:53 2005 Input: Q := RationalField(); P := PolynomialRing(Q, 4); I:=ideal; Coordinates(I, 1); Output: Magma V2.11-10 Sat Dec 3 2005 12:34:52 on modular [Seed = 4079052760] ------------------------------------- >> Coordinates(I, 1);; ^ Runtime error in 'Coordinates': Bad argument types Argument types given: RngMPol, RngIntElt Total time: 0.190 seconds, Total memory usage: 3.34MB '193.250' ************** MAGMA ***************** Host 193.250.127.92 (193.250.127.92) Time: Sat Dec 3 12:31:45 2005 Input: Q := RationalField(); P := PolynomialRing(Q, 4); Coordinates(ideal, 1); Output: Magma V2.11-10 Sat Dec 3 2005 12:31:44 on modular [Seed = 3222426967] ------------------------------------- >> Coordinates(ideal := PolynomialRing(Q, 4); Coordinates([-16*y*x^2-48*x*y+4*x^2+144*y^2+16*x^2*y^2+96*x*y^2, 288*x*z^2-144*x*z-64*x^2*z+64*x^2*z^2+16*x^2+324*z^2, 36*z^2-64*y^2*z+48*z^2*y-96*y*z+16*y^2*z^2+144*x^2*a^2+64*y^2+576*a^2+96*x*a+576*x*a^2+4*x^2+48*x^2*a, 36*z^2+256*a^2+128*a^2*z+192*z*a+16*z^2*a^2+48*z^2*a, 64*a^2+64*y^2*a^2+64*y*a+128*a^2*y+16*y^2+64*y^2*a, x^2+2*y^2+3*z^2+4*a^2-10, x^2+y^2+z^2+a^2+3*x+y+2*z-a-15], 1); Output: Magma V2.11-10 Sat Dec 3 2005 12:29:47 on modular [Seed = 2691839420] ------------------------------------- >> Coordinates([-16*y*x^2-48*x*y+4*x^2+144*y^2+16*x^2*y^2+96*x*y^2, 288*x*z^2- ^ Runtime error in 'Coordinates': Bad argument types Argument types given: SeqEnum[RngMPolElt], RngIntElt Total time: 0.200 seconds, Total memory usage: 3.34MB '193.250' ************** MAGMA ***************** Host 193.250.127.15 (193.250.127.15) Time: Sat Dec 3 12:26:32 2005 Input: Q := RationalField(); P := PolynomialRing(Q, 4); Coordinates([x^2+36*y^2-4*x^2*y+4*x^2*y^2+24*y^2*x-12*x*y, 9*z^2+128*z*a^2+48*a*z^2+64*a^2*z^2+64*a^2+48*z*a, 81*z^2-16*x^2*z+16*x^2*z^2-36*x*z+4*x^2+72*z^2*x, 16*y^2+9*z^2+24*x*a+144*a^2+16*x^2*a-16*y^2*z+x^2+64*x^2*a^2+4*y^2*z^2+12*z^2*y+192*x*a^2-24*y*z, 16*a^2+64*a^2*y^2+64*a^2*y+32*a*y^2+16*a*y+4*y^2, x^2+2*y^2+3*z^2+4*a^2-10, x^2+y^2+z^2+t^2+3*x+y+2*z-a-15], 1); Output: Magma V2.11-10 Sat Dec 3 2005 12:26:32 on modular [Seed = 704364406] ------------------------------------- >> +64*a^2*y+32*a*y^2+16*a*y+4*y^2, x^2+2*y^2+3*z^2+4*a^2-10, x^2+y^2+z^2+t^2+ ^ User error: Identifier 't' has not been declared or assigned Total time: 0.190 seconds, Total memory usage: 3.34MB '193.250' ************** MAGMA ***************** Host 193.250.127.15 (193.250.127.15) Time: Sat Dec 3 12:24:49 2005 Input: Coordinates([x^2+36*y^2-4*x^2*y+4*x^2*y^2+24*y^2*x-12*x*y, 9*z^2+128*z*a^2+48*a*z^2+64*a^2*z^2+64*a^2+48*z*a, 81*z^2-16*x^2*z+16*x^2*z^2-36*x*z+4*x^2+72*z^2*x, 16*y^2+9*z^2+24*x*a+144*a^2+16*x^2*a-16*y^2*z+x^2+64*x^2*a^2+4*y^2*z^2+12*z^2*y+192*x*a^2-24*y*z, 16*a^2+64*a^2*y^2+64*a^2*y+32*a*y^2+16*a*y+4*y^2, x^2+2*y^2+3*z^2+4*a^2-10, x^2+y^2+z^2+t^2+3*x+y+2*z-a-15], 1); Output: Magma V2.11-10 Sat Dec 3 2005 12:24:49 on modular [Seed = 805157081] ------------------------------------- >> Coordinates([x^2+36*y^2-4*x^2*y+4*x^2*y^2+24*y^2*x-12*x*y, 9*z^2+128*z*a^2+ ^ User error: Identifier 'x' has not been declared or assigned Total time: 0.190 seconds, Total memory usage: 3.24MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Fri Dec 2 17:47:01 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^2*3^1; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-1,0]; //P1:=J![x^2+6*x+36,12*x-18]; //Height(P); //HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=3335); Chabauty(P,17); //Order(Q);Order(P); //P+Q;2*P;2*P+Q;3*P; Output: Magma V2.11-10 Fri Dec 2 2005 17:46:59 on modular [Seed = 3239201833] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 2*x + 4, -2*x + 6, 2) ] {@ @} Total time: 1.639 seconds, Total memory usage: 39.40MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Fri Dec 2 17:46:53 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^2*3^1; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-1,0]; //P1:=J![x^2+6*x+36,12*x-18]; //Height(P); //HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=3335); Chabauty(P,11); //Order(Q);Order(P); //P+Q;2*P;2*P+Q;3*P; Output: Magma V2.11-10 Fri Dec 2 2005 17:46:52 on modular [Seed = 3458024217] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 2*x + 4, -2*x + 6, 2) ] {@ <116034, 1, 5, 1>, <57537, 1, 5, 1> @} Total time: 1.570 seconds, Total memory usage: 39.37MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Fri Dec 2 17:46:45 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^2*3^1; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-1,0]; //P1:=J![x^2+6*x+36,12*x-18]; //Height(P); //HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=3335); Chabauty(P,7); //Order(Q);Order(P); //P+Q;2*P;2*P+Q;3*P; Output: Magma V2.11-10 Fri Dec 2 2005 17:46:44 on modular [Seed = 3407365652] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 2*x + 4, -2*x + 6, 2) ] {@ <1322, 1, 4, 1>, <0, 1, 4, 1>, <1047, 1, 4, 1> @} Total time: 1.669 seconds, Total memory usage: 39.37MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Fri Dec 2 17:43:39 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^2*3^1; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-1,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=3335); //Chabauty(P,11); //Order(Q);Order(P); //P+Q;2*P;2*P+Q;3*P; Output: Magma V2.11-10 Fri Dec 2 2005 17:43:37 on modular [Seed = 2976005283] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 2*x + 4, -2*x + 6, 2) ] 1.44778617658396039921843136573580142519427894255151861457351 3.654849499758574762261498916478491203095 2.811849734125242389181859855656213507312 Total time: 1.330 seconds, Total memory usage: 39.37MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Fri Dec 2 17:43:28 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^2*3^1; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-1,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=1); //RationalPoints(J:Bound:=3335); //Chabauty(P,11); //Order(Q);Order(P); //P+Q;2*P;2*P+Q;3*P; Output: Magma V2.11-10 Fri Dec 2 2005 17:43:27 on modular [Seed = 3194827706] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 2*x + 4, -2*x + 6, 2) ] 1.44778617658396039921843136573580142519427894255151861457351 4.923736996348681348058582185502237552710 2.811849734125242389181859855656213507312 Total time: 1.330 seconds, Total memory usage: 39.37MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Fri Dec 2 17:43:20 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^2*3^1; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; //P:=J!B[1]; //Q:=J![x-1,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=1); //RationalPoints(J:Bound:=3335); //Chabauty(P,11); //Order(Q);Order(P); //P+Q;2*P;2*P+Q;3*P; Output: Magma V2.11-10 Fri Dec 2 2005 17:43:18 on modular [Seed = 3144169142] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 2*x + 4, -2*x + 6, 2) ] >> Height(P); ^ User error: Identifier 'P' has not been declared or assigned 4.923736996348681348058582185502237552710 2.811849734125242389181859855656213507312 Total time: 1.280 seconds, Total memory usage: 39.40MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Fri Dec 2 17:42:18 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^2*3^1; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; //P:=J!B[1]; //Q:=J![x-1,0]; //P1:=J![x^2+6*x+36,12*x-18]; //Height(P); //HeightConstant(J:Effort:=10); //RationalPoints(J:Bound:=3335); //Chabauty(P,11); //Order(Q);Order(P); //P+Q;2*P;2*P+Q;3*P; Output: Magma V2.11-10 Fri Dec 2 2005 17:42:17 on modular [Seed = 2691903074] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 2*x + 4, -2*x + 6, 2) ] Total time: 1.229 seconds, Total memory usage: 39.38MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Fri Dec 2 17:37:35 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^1*3^9; C:=HyperellipticCurve(f); J:=Jacobian(C); //r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=3000); B:=ReducedBasis(V); B; //P:=J!B[1]; //Q:=J![x-1,0]; //P1:=J![x^2+6*x+36,12*x-18]; //Height(P); //HeightConstant(J:Effort:=10); //RationalPoints(J:Bound:=3335); Chabauty(P,11); //Order(Q);Order(P); //P+Q;2*P;2*P+Q;3*P; Output: Magma V2.11-10 Fri Dec 2 2005 17:37:33 on modular [Seed = 2639139387] ------------------------------------- Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [] >> Chabauty(P,11); ^ User error: Identifier 'P' has not been declared or assigned Total time: 1.870 seconds, Total memory usage: 38.58MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Fri Dec 2 17:36:48 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^1*3^7; C:=HyperellipticCurve(f); J:=Jacobian(C); //r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-1,0]; //P1:=J![x^2+6*x+36,12*x-18]; //Height(P); //HeightConstant(J:Effort:=10); //RationalPoints(J:Bound:=3335); Chabauty(P,11); //Order(Q);Order(P); //P+Q;2*P;2*P+Q;3*P; Output: Magma V2.11-10 Fri Dec 2 2005 17:36:46 on modular [Seed = 2588481007] ------------------------------------- Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 9*x + 135, -9*x + 486, 2) ] {@ @} Total time: 1.040 seconds, Total memory usage: 38.49MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Fri Dec 2 17:36:40 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^1*3^7; C:=HyperellipticCurve(f); J:=Jacobian(C); //r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-1,0]; //P1:=J![x^2+6*x+36,12*x-18]; //Height(P); //HeightConstant(J:Effort:=10); //RationalPoints(J:Bound:=3335); Chabauty(P,7); //Order(Q);Order(P); //P+Q;2*P;2*P+Q;3*P; Output: Magma V2.11-10 Fri Dec 2 2005 17:36:39 on modular [Seed = 2270431483] ------------------------------------- Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 9*x + 135, -9*x + 486, 2) ] {@ <0, 1, 4, 1> @} Total time: 1.020 seconds, Total memory usage: 38.49MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Fri Dec 2 17:35:47 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^1*3^7; C:=HyperellipticCurve(f); J:=Jacobian(C); //r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=4381); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-1,0]; //P1:=J![x^2+6*x+36,12*x-18]; //Height(P); //HeightConstant(J:Effort:=10); //RationalPoints(J:Bound:=3335); //Chabauty(P,13); //Order(Q);Order(P); //P+Q;2*P;2*P+Q;3*P; Output: Magma V2.11-10 Fri Dec 2 2005 17:35:43 on modular [Seed = 2371748537] ------------------------------------- Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 9*x + 135, -9*x + 486, 2) ] Total time: 3.879 seconds, Total memory usage: 38.58MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Fri Dec 2 17:34:32 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^1*3^7; C:=HyperellipticCurve(f); J:=Jacobian(C); //r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-1,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=10); //RationalPoints(J:Bound:=3335); //Chabauty(P,13); //Order(Q);Order(P); //P+Q;2*P;2*P+Q;3*P; Output: Magma V2.11-10 Fri Dec 2 2005 17:34:31 on modular [Seed = 2321089142] ------------------------------------- Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 9*x + 135, -9*x + 486, 2) ] 2.0994830387457112732428474216976235481166 7.855157844852767304923448366250813574110 6.744200768424384281818019389040865280504 Total time: 0.900 seconds, Total memory usage: 38.49MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Fri Dec 2 17:34:13 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^1*3^7; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-1,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=10); //RationalPoints(J:Bound:=3335); //Chabauty(P,13); //Order(Q);Order(P); //P+Q;2*P;2*P+Q;3*P; Output: Magma V2.11-10 Fri Dec 2 2005 17:34:12 on modular [Seed = 1889794207] ------------------------------------- TwoSelmerGroupData( J: Jacobian of Hyperelliptic Curve defined by y^2 = x^5 - 4374 ... ) TwoSelmerWork( C: Hyperelliptic Curve defined by y^2 = x^5 - 4374 over Rationa..., BoundType: Default, Bound: 0, UseUnits: true, Points: {}, Fields: {} ) LocalImageOdd( f: x^5 - 4374, p: 2, m1: Mapping from: Univariate Polynomial Ring in x over Rational ..., m2: Mapping from: Univariate Polynomial Ring in x over Rational ..., v: 6, Points: {} ) try( ) try( d: 2 ) SquarefreePart( F: $.1^4 + O(2^50)*$.1^3 + (2866242652299*2^3 + O(2^53))*$.1^2 ... ) In file "/usr/local/magma/package/Geometry/Arith/loclib.m", line 120, column 5: >> assert R eq 0; ^ Runtime error in assert: Assertion failed >> r:=TwoSelmerGroupData(J);r; ^ User error: Identifier 'r' has not been declared or assigned Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 9*x + 135, -9*x + 486, 2) ] 2.0994830387457112732428474216976235481166 7.855157844852767304923448366250813574110 6.744200768424384281818019389040865280504 Total time: 1.159 seconds, Total memory usage: 39.48MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Fri Dec 2 17:34:10 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^1*3^7; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-1,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=10); //RationalPoints(J:Bound:=3335); //Chabauty(P,13); //Order(Q);Order(P); //P+Q;2*P;2*P+Q;3*P; Output: Magma V2.11-10 Fri Dec 2 2005 17:34:08 on modular [Seed = 2108618134] ------------------------------------- TwoSelmerGroupData( J: Jacobian of Hyperelliptic Curve defined by y^2 = x^5 - 4374 ... ) TwoSelmerWork( C: Hyperelliptic Curve defined by y^2 = x^5 - 4374 over Rationa..., BoundType: Default, Bound: 0, UseUnits: true, Points: {}, Fields: {} ) LocalImageOdd( f: x^5 - 4374, p: 2, m1: Mapping from: Univariate Polynomial Ring in x over Rational ..., m2: Mapping from: Univariate Polynomial Ring in x over Rational ..., v: 6, Points: {} ) try( ) try( d: 2 ) SquarefreeRoots( F: $.1^4 + O(2^50)*$.1^3 + (11336353931*2^3 + O(2^53))*$.1^2 + ... ) In file "/usr/local/magma/package/Geometry/Arith/sqfroots.m", line 84, column 15: >> return Roots(F:IsSquarefree); ^ Runtime error in 'Roots': Insufficient precision to find roots >> r:=TwoSelmerGroupData(J);r; ^ User error: Identifier 'r' has not been declared or assigned Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 9*x + 135, -9*x + 486, 2) ] 2.0994830387457112732428474216976235481166 7.855157844852767304923448366250813574110 6.744200768424384281818019389040865280504 Total time: 1.229 seconds, Total memory usage: 39.65MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Fri Dec 2 17:34:06 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^1*3^7; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-1,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=10); //RationalPoints(J:Bound:=3335); //Chabauty(P,13); //Order(Q);Order(P); //P+Q;2*P;2*P+Q;3*P; Output: Magma V2.11-10 Fri Dec 2 2005 17:34:04 on modular [Seed = 2057958061] ------------------------------------- TwoSelmerGroupData( J: Jacobian of Hyperelliptic Curve defined by y^2 = x^5 - 4374 ... ) TwoSelmerWork( C: Hyperelliptic Curve defined by y^2 = x^5 - 4374 over Rationa..., BoundType: Default, Bound: 0, UseUnits: true, Points: {}, Fields: {} ) LocalImageOdd( f: x^5 - 4374, p: 2, m1: Mapping from: Univariate Polynomial Ring in x over Rational ..., m2: Mapping from: Univariate Polynomial Ring in x over Rational ..., v: 6, Points: {} ) try( ) try( d: 2 ) SquarefreeRoots( F: $.1^4 + O(2^50)*$.1^3 + (2431040096395*2^3 + O(2^53))*$.1^2 ... ) In file "/usr/local/magma/package/Geometry/Arith/sqfroots.m", line 84, column 15: >> return Roots(F:IsSquarefree); ^ Runtime error in 'Roots': Insufficient precision to find roots >> r:=TwoSelmerGroupData(J);r; ^ User error: Identifier 'r' has not been declared or assigned Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 9*x + 135, -9*x + 486, 2) ] 2.0994830387457112732428474216976235481166 7.855157844852767304923448366250813574110 6.744200768424384281818019389040865280504 Total time: 1.389 seconds, Total memory usage: 39.89MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Fri Dec 2 17:33:15 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^1*3^7; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-1,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=3335); //Chabauty(P,13); //Order(Q);Order(P); //P+Q;2*P;2*P+Q;3*P; Output: Magma V2.11-10 Fri Dec 2 2005 17:33:14 on modular [Seed = 1739911110] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 9*x + 135, -9*x + 486, 2) ] 2.0994830387457112732428474216976235481166 7.855157844852767304923448366250813574110 6.744200768424384281818019389040865280504 Total time: 1.360 seconds, Total memory usage: 39.62MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Fri Dec 2 17:33:12 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^1*3^7; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-1,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=3335); //Chabauty(P,13); //Order(Q);Order(P); //P+Q;2*P;2*P+Q;3*P; Output: Magma V2.11-10 Fri Dec 2 2005 17:33:10 on modular [Seed = 1824515265] ------------------------------------- TwoSelmerGroupData( J: Jacobian of Hyperelliptic Curve defined by y^2 = x^5 - 4374 ... ) TwoSelmerWork( C: Hyperelliptic Curve defined by y^2 = x^5 - 4374 over Rationa..., BoundType: Default, Bound: 0, UseUnits: true, Points: {}, Fields: {} ) LocalImageOdd( f: x^5 - 4374, p: 2, m1: Mapping from: Univariate Polynomial Ring in x over Rational ..., m2: Mapping from: Univariate Polynomial Ring in x over Rational ..., v: 6, Points: {} ) try( ) try( d: 2 ) SquarefreeRoots( F: $.1^4 + O(2^50)*$.1^3 + (4481794443403*2^3 + O(2^53))*$.1^2 ... ) In file "/usr/local/magma/package/Geometry/Arith/sqfroots.m", line 84, column 15: >> return Roots(F:IsSquarefree); ^ Runtime error in 'Roots': Insufficient precision to find roots >> r:=TwoSelmerGroupData(J);r; ^ User error: Identifier 'r' has not been declared or assigned Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 9*x + 135, -9*x + 486, 2) ] 2.0994830387457112732428474216976235481166 7.855157844852767304923448366250813574110 6.744200768424384281818019389040865280504 Total time: 1.250 seconds, Total memory usage: 39.65MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Fri Dec 2 17:32:13 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^1*3^7; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-1,0]; //P1:=J![x^2+6*x+36,12*x-18]; //Height(P); //HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=3335); //Chabauty(P,13); //Order(Q);Order(P); //P+Q;2*P;2*P+Q;3*P; Output: Magma V2.11-10 Fri Dec 2 2005 17:32:11 on modular [Seed = 1773854985] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 9*x + 135, -9*x + 486, 2) ] Total time: 1.389 seconds, Total memory usage: 39.94MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Fri Dec 2 17:32:09 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^1*3^7; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-1,0]; //P1:=J![x^2+6*x+36,12*x-18]; //Height(P); //HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=3335); //Chabauty(P,13); //Order(Q);Order(P); //P+Q;2*P;2*P+Q;3*P; Output: Magma V2.11-10 Fri Dec 2 2005 17:32:08 on modular [Seed = 1451613749] ------------------------------------- TwoSelmerGroupData( J: Jacobian of Hyperelliptic Curve defined by y^2 = x^5 - 4374 ... ) TwoSelmerWork( C: Hyperelliptic Curve defined by y^2 = x^5 - 4374 over Rationa..., BoundType: Default, Bound: 0, UseUnits: true, Points: {}, Fields: {} ) LocalImageOdd( f: x^5 - 4374, p: 2, m1: Mapping from: Univariate Polynomial Ring in x over Rational ..., m2: Mapping from: Univariate Polynomial Ring in x over Rational ..., v: 6, Points: {} ) try( ) try( d: 2 ) SquarefreeRoots( F: $.1^4 + O(2^50)*$.1^3 + (3136045451403*2^3 + O(2^53))*$.1^2 ... ) In file "/usr/local/magma/package/Geometry/Arith/sqfroots.m", line 84, column 15: >> return Roots(F:IsSquarefree); ^ Runtime error in 'Roots': Insufficient precision to find roots >> r:=TwoSelmerGroupData(J);r; ^ User error: Identifier 'r' has not been declared or assigned Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 9*x + 135, -9*x + 486, 2) ] Total time: 0.950 seconds, Total memory usage: 39.48MB '134.219' ************** MAGMA ***************** Host 134.219.148.42 (134.219.148.42) Time: Fri Dec 2 17:19:56 2005 Input: X_1 := Matrix(2, [-195,-5488, 7,197]); X_1; X_2 := Matrix(2, [-244,-8575, 7,246]); X_2; X_3 := Matrix(2, [-293,-12348, 7,295]); X_3; X_4 := Matrix(2, [-342,-16807, 7,344]); X_4; Y_1 := X_1*X_2*X_3*X_4*X_1*X_3*X_4*X_2*X_1; Y_1; Y_2 := X_2*X_3*X_4*X_2*X_1*X_4*X_1*X_2*X_3; Y_2; Y_3 := X_3*X_4*X_1*X_2*X_3*X_3*X_4*X_2*X_1; Y_3; Y_4 := X_4*X_2*X_3*X_3*X_4*X_2*X_1; Y_4; M_h := Y_2*Y_2*Y_3*Y_3*Y_4*Y_2*Y_1*Y_1*Y_4*Y_1*Y_2*Y_3*Y_3*Y_4*Y_2*Y_1; M_h; Output: Magma V2.11-10 Fri Dec 2 2005 17:19:56 on modular [Seed = 3590675806] ------------------------------------- [ -195 -5488] [ 7 197] [ -244 -8575] [ 7 246] [ -293 -12348] [ 7 295] [ -342 -16807] [ 7 344] [ 69345188618985448 1951377454536301239] [ -2489586226940967 -70057094531380844] [ 131106034502394869 5525581561639693421] [ -3761566848742780 -158534612851161791] [ -2218671813420922 -62433547906440305] [ 53009660061589 1491694774747402] [ -787183141527 -22151377270120] [ 16111464649 453377508977] [174801202447173743443223007264396063127947174680137917788974928763633474431825\ 923127812764804103556846639940324358013520109641947732704660271041589584499\ 909560268503525396803490530790881045657521939684848324648124706619281356934\ 214616009 49189155336418359001571089536228203226645682500819731343070818030\ 762449294395637712226450360083199785715870402344870712023063267842815329883\ 387972888749152671310857554893494316673702451267702799899272127655005791718\ 09906473046765588053498] [-50152261163360352587873686556143650085611238910779610910078001486474383017486\ 837524441779375963989882962618294275101840528321534728771536504184072523470\ 599840842222559888346896245267584454214056781638316184724344125170913032348\ 59664640 -14112874112423136298000268676695965974621382401792280023681233324\ 114068232525766449879999838138754649564291888818063752268661389688076235120\ 718716711020836096141088071210225042276346867667474085720005819433945928496\ 5878498467511643917191] Total time: 0.190 seconds, Total memory usage: 3.24MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Fri Dec 2 17:19:25 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^1*3^4; C:=HyperellipticCurve(f); J:=Jacobian(C); //r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-1,0]; //P1:=J![x^2+6*x+36,12*x-18]; //Height(P); //HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=3335); Chabauty(P,13); //Order(Q);Order(P); //P+Q;2*P;2*P+Q;3*P; Output: Magma V2.11-10 Fri Dec 2 2005 17:19:23 on modular [Seed = 3306573482] ------------------------------------- Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x - 3, -9, 1) ] {@ <3, 1, 4, 1> @} Total time: 1.090 seconds, Total memory usage: 38.43MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Fri Dec 2 17:19:13 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^1*3^4; C:=HyperellipticCurve(f); J:=Jacobian(C); //r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-1,0]; //P1:=J![x^2+6*x+36,12*x-18]; //Height(P); //HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=3335); Chabauty(P,11); //Order(Q);Order(P); //P+Q;2*P;2*P+Q;3*P; Output: Magma V2.11-10 Fri Dec 2 2005 17:19:12 on modular [Seed = 3026665424] ------------------------------------- Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x - 3, -9, 1) ] {@ <3, 1, 4, 1>, <12367, 1, 4, 1> @} Total time: 1.110 seconds, Total memory usage: 38.43MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Fri Dec 2 17:19:02 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^1*3^4; C:=HyperellipticCurve(f); J:=Jacobian(C); //r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-1,0]; //P1:=J![x^2+6*x+36,12*x-18]; //Height(P); //HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=3335); Chabauty(P,7); //Order(Q);Order(P); //P+Q;2*P;2*P+Q;3*P; Output: Magma V2.11-10 Fri Dec 2 2005 17:19:01 on modular [Seed = 2944148734] ------------------------------------- Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x - 3, -9, 1) ] {@ <3, 1, 4, 1>, <1600, 1, 4, 1>, <348, 1, 4, 1> @} Total time: 1.139 seconds, Total memory usage: 38.43MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Fri Dec 2 17:16:03 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^1*3^4; C:=HyperellipticCurve(f); J:=Jacobian(C); //r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-1,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=3335); //Chabauty(P,17); //Order(Q);Order(P); //P+Q;2*P;2*P+Q;3*P; Output: Magma V2.11-10 Fri Dec 2 2005 17:16:03 on modular [Seed = 1603066872] ------------------------------------- Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x - 3, -9, 1) ] 1.17935651881148701436661436842904543928154423678172645308610 5.819560159017277385079838341670905384560 4.546976191088164899027528915195813871219 Total time: 0.840 seconds, Total memory usage: 38.43MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Fri Dec 2 17:15:53 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^1*3^4; C:=HyperellipticCurve(f); J:=Jacobian(C); //r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-1,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=1); //RationalPoints(J:Bound:=3335); //Chabauty(P,17); //Order(Q);Order(P); //P+Q;2*P;2*P+Q;3*P; Output: Magma V2.11-10 Fri Dec 2 2005 17:15:52 on modular [Seed = 836941229] ------------------------------------- Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x - 3, -9, 1) ] 1.17935651881148701436661436842904543928154423678172645308610 7.415390739944626049829071986511986400701 4.546976191088164899027528915195813871219 Total time: 0.800 seconds, Total memory usage: 38.43MB '134.219' ************** MAGMA ***************** Host 134.219.148.42 (134.219.148.42) Time: Fri Dec 2 17:15:49 2005 Input: X_1 := Matrix(2, [-195,-5488, 7,197]); X_1; X_2 := Matrix(2, [-244,-8575, 7,246]); X_2; X_3 := Matrix(2, [-293,-12348, 7,295]); X_3; X_4 := Matrix(2, [-342,-16807, 7,344]); X_4; Y_1 := X_1*X_2*X_3*X_4*X_1*X_3*X_4*X_2*X_1; Y_1; Y_2 := X_2*X_3*X_4*X_2*X_1*X_4*X_1*X_2*X_3; Y_2; Y_3 := X_3*X_4*X_1*X_2*X_3*X_3*X_4*X_2*X_1; Y_3; Y_4 := X_4*X_2*X_3*X_3*X_4*X_2*X_1; Y_4; M_h := Y_2*Y_2*Y_3*Y_3*Y_4*Y_2*Y_1*Y_1*Y_4*Y_1*Y_2*Y_3*Y_3*Y_4*Y_2*Y_1; M_h; Output: Magma V2.11-10 Fri Dec 2 2005 17:15:49 on modular [Seed = 887599269] ------------------------------------- [ -195 -5488] [ 7 197] [ -244 -8575] [ 7 246] [ -293 -12348] [ 7 295] [ -342 -16807] [ 7 344] [ 69345188618985448 1951377454536301239] [ -2489586226940967 -70057094531380844] [ 131106034502394869 5525581561639693421] [ -3761566848742780 -158534612851161791] [ -2218671813420922 -62433547906440305] [ 53009660061589 1491694774747402] [ -787183141527 -22151377270120] [ 16111464649 453377508977] [174801202447173743443223007264396063127947174680137917788974928763633474431825\ 923127812764804103556846639940324358013520109641947732704660271041589584499\ 909560268503525396803490530790881045657521939684848324648124706619281356934\ 214616009 49189155336418359001571089536228203226645682500819731343070818030\ 762449294395637712226450360083199785715870402344870712023063267842815329883\ 387972888749152671310857554893494316673702451267702799899272127655005791718\ 09906473046765588053498] [-50152261163360352587873686556143650085611238910779610910078001486474383017486\ 837524441779375963989882962618294275101840528321534728771536504184072523470\ 599840842222559888346896245267584454214056781638316184724344125170913032348\ 59664640 -14112874112423136298000268676695965974621382401792280023681233324\ 114068232525766449879999838138754649564291888818063752268661389688076235120\ 718716711020836096141088071210225042276346867667474085720005819433945928496\ 5878498467511643917191] Total time: 0.190 seconds, Total memory usage: 3.24MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Fri Dec 2 17:15:45 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^1*3^4; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-1,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=1); //RationalPoints(J:Bound:=3335); //Chabauty(P,17); //Order(Q);Order(P); //P+Q;2*P;2*P+Q;3*P; Output: Magma V2.11-10 Fri Dec 2 2005 17:15:44 on modular [Seed = 805088187] ------------------------------------- TwoSelmerGroupData( J: Jacobian of Hyperelliptic Curve defined by y^2 = x^5 - 162 o... ) TwoSelmerWork( C: Hyperelliptic Curve defined by y^2 = x^5 - 162 over Rational..., BoundType: Default, Bound: 0, UseUnits: true, Points: {}, Fields: {} ) LocalImageOdd( f: x^5 - 162, p: 2, m1: Mapping from: Univariate Polynomial Ring in x over Rational ..., m2: Mapping from: Univariate Polynomial Ring in x over Rational ..., v: 6, Points: {} ) try( ) try( d: 2 ) SquarefreeRoots( F: $.1^4 + O(2^50)*$.1^3 - (275398131631*2^3 + O(2^53))*$.1^2 +... ) In file "/usr/local/magma/package/Geometry/Arith/sqfroots.m", line 84, column 15: >> return Roots(F:IsSquarefree); ^ Runtime error in 'Roots': Insufficient precision to find roots >> r:=TwoSelmerGroupData(J);r; ^ User error: Identifier 'r' has not been declared or assigned Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x - 3, -9, 1) ] 1.17935651881148701436661436842904543928154423678172645308610 7.415390739944626049829071986511986400701 4.546976191088164899027528915195813871219 Total time: 1.090 seconds, Total memory usage: 39.48MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Fri Dec 2 17:14:49 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^1*3^4; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-1,0]; //P1:=J![x^2+6*x+36,12*x-18]; //Height(P); //HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=3335); //Chabauty(P,17); //Order(Q);Order(P); //P+Q;2*P;2*P+Q;3*P; Output: Magma V2.11-10 Fri Dec 2 2005 17:14:48 on modular [Seed = 4045567404] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x - 3, -9, 1) ] Total time: 1.139 seconds, Total memory usage: 39.41MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Fri Dec 2 17:14:46 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^1*3^4; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-1,0]; //P1:=J![x^2+6*x+36,12*x-18]; //Height(P); //HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=3335); //Chabauty(P,17); //Order(Q);Order(P); //P+Q;2*P;2*P+Q;3*P; Output: Magma V2.11-10 Fri Dec 2 2005 17:14:45 on modular [Seed = 3929632442] ------------------------------------- TwoSelmerGroupData( J: Jacobian of Hyperelliptic Curve defined by y^2 = x^5 - 162 o... ) TwoSelmerWork( C: Hyperelliptic Curve defined by y^2 = x^5 - 162 over Rational..., BoundType: Default, Bound: 0, UseUnits: true, Points: {}, Fields: {} ) LocalImageOdd( f: x^5 - 162, p: 2, m1: Mapping from: Univariate Polynomial Ring in x over Rational ..., m2: Mapping from: Univariate Polynomial Ring in x over Rational ..., v: 6, Points: {} ) try( ) try( d: 2 ) SquarefreeRoots( F: $.1^4 + O(2^50)*$.1^3 + (3747911827537*2^3 + O(2^53))*$.1^2 ... ) In file "/usr/local/magma/package/Geometry/Arith/sqfroots.m", line 84, column 15: >> return Roots(F:IsSquarefree); ^ Runtime error in 'Roots': Insufficient precision to find roots >> r:=TwoSelmerGroupData(J);r; ^ User error: Identifier 'r' has not been declared or assigned Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x - 3, -9, 1) ] Total time: 1.090 seconds, Total memory usage: 39.62MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Fri Dec 2 17:14:43 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^1*3^4; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-1,0]; //P1:=J![x^2+6*x+36,12*x-18]; //Height(P); //HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=3335); //Chabauty(P,17); //Order(Q);Order(P); //P+Q;2*P;2*P+Q;3*P; Output: Magma V2.11-10 Fri Dec 2 2005 17:14:42 on modular [Seed = 4013714360] ------------------------------------- TwoSelmerGroupData( J: Jacobian of Hyperelliptic Curve defined by y^2 = x^5 - 162 o... ) TwoSelmerWork( C: Hyperelliptic Curve defined by y^2 = x^5 - 162 over Rational..., BoundType: Default, Bound: 0, UseUnits: true, Points: {}, Fields: {} ) LocalImageOdd( f: x^5 - 162, p: 2, m1: Mapping from: Univariate Polynomial Ring in x over Rational ..., m2: Mapping from: Univariate Polynomial Ring in x over Rational ..., v: 6, Points: {} ) try( ) try( d: 2 ) SquarefreeRoots( F: $.1^4 + O(2^50)*$.1^3 + (265353691217*2^3 + O(2^53))*$.1^2 +... ) In file "/usr/local/magma/package/Geometry/Arith/sqfroots.m", line 84, column 15: >> return Roots(F:IsSquarefree); ^ Runtime error in 'Roots': Insufficient precision to find roots >> r:=TwoSelmerGroupData(J);r; ^ User error: Identifier 'r' has not been declared or assigned Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x - 3, -9, 1) ] Total time: 1.000 seconds, Total memory usage: 39.44MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Fri Dec 2 17:14:39 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^1*3^4; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-1,0]; //P1:=J![x^2+6*x+36,12*x-18]; //Height(P); //HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=3335); //Chabauty(P,17); //Order(Q);Order(P); //P+Q;2*P;2*P+Q;3*P; Output: Magma V2.11-10 Fri Dec 2 2005 17:14:38 on modular [Seed = 3761464491] ------------------------------------- TwoSelmerGroupData( J: Jacobian of Hyperelliptic Curve defined by y^2 = x^5 - 162 o... ) TwoSelmerWork( C: Hyperelliptic Curve defined by y^2 = x^5 - 162 over Rational..., BoundType: Default, Bound: 0, UseUnits: true, Points: {}, Fields: {} ) LocalImageOdd( f: x^5 - 162, p: 2, m1: Mapping from: Univariate Polynomial Ring in x over Rational ..., m2: Mapping from: Univariate Polynomial Ring in x over Rational ..., v: 6, Points: {} ) try( ) try( d: 2 ) SquarefreeRoots( F: $.1^4 + O(2^50)*$.1^3 + (1028919001169*2^3 + O(2^53))*$.1^2 ... ) In file "/usr/local/magma/package/Geometry/Arith/sqfroots.m", line 84, column 15: >> return Roots(F:IsSquarefree); ^ Runtime error in 'Roots': Insufficient precision to find roots >> r:=TwoSelmerGroupData(J);r; ^ User error: Identifier 'r' has not been declared or assigned Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x - 3, -9, 1) ] Total time: 1.179 seconds, Total memory usage: 39.78MB '134.219' ************** MAGMA ***************** Host 134.219.148.42 (134.219.148.42) Time: Fri Dec 2 17:14:27 2005 Input: X_1 := Matrix(2, [-195,-5488, 7,197]); X_1; X_2 := Matrix(2, [-244,-8575, 7,246]); X_2; X_3 := Matrix(2, [-293,-12348, 7,295]); X_3; X_4 := Matrix(2, [-342,-16807, 7,344]); X_4; Y_1 := X_1*X_2*X_3*X_4*X_1*X_3*X_4*X_2*X_1; Y_1; Y_2 := X_2*X_3*X_4*X_2*X_1*X_4*X_1*X_2*X_3; Y_2; Y_3 := X_3*X_4*X_1*X_2*X_3*X_3*X_4*X_2*X_1; Y_3; Y_4 := X_4*X_2*X_3*X_3*X_4*X_2*X_1; Y_4; M_h := Y_2*Y_2*Y_3*Y_3*Y_4*Y_2*Y_1*Y_1*Y_4*Y_1*Y_2*Y_3*Y_3*Y_4*Y_2*Y_1; M_h; Output: Magma V2.11-10 Fri Dec 2 2005 17:14:27 on modular [Seed = 3357232270] ------------------------------------- [ -195 -5488] [ 7 197] [ -244 -8575] [ 7 246] [ -293 -12348] [ 7 295] [ -342 -16807] [ 7 344] [ 69345188618985448 1951377454536301239] [ -2489586226940967 -70057094531380844] [ 131106034502394869 5525581561639693421] [ -3761566848742780 -158534612851161791] [ -2218671813420922 -62433547906440305] [ 53009660061589 1491694774747402] [ -787183141527 -22151377270120] [ 16111464649 453377508977] [174801202447173743443223007264396063127947174680137917788974928763633474431825\ 923127812764804103556846639940324358013520109641947732704660271041589584499\ 909560268503525396803490530790881045657521939684848324648124706619281356934\ 214616009 49189155336418359001571089536228203226645682500819731343070818030\ 762449294395637712226450360083199785715870402344870712023063267842815329883\ 387972888749152671310857554893494316673702451267702799899272127655005791718\ 09906473046765588053498] [-50152261163360352587873686556143650085611238910779610910078001486474383017486\ 837524441779375963989882962618294275101840528321534728771536504184072523470\ 599840842222559888346896245267584454214056781638316184724344125170913032348\ 59664640 -14112874112423136298000268676695965974621382401792280023681233324\ 114068232525766449879999838138754649564291888818063752268661389688076235120\ 718716711020836096141088071210225042276346867667474085720005819433945928496\ 5878498467511643917191] Total time: 0.190 seconds, Total memory usage: 3.24MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Fri Dec 2 17:14:20 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^0*3^9; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-1,0]; //P1:=J![x^2+6*x+36,12*x-18]; //Height(P); //HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=3335); //Chabauty(P,17); //Order(Q);Order(P); //P+Q;2*P;2*P+Q;3*P; Output: Magma V2.11-10 Fri Dec 2 2005 17:14:19 on modular [Seed = 3441314291] ------------------------------------- 2 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [] >> P:=J!B[1]; //Q:=J![x-1,0]; ^ Runtime error in '[]': Sequence element 1 not defined Total time: 0.890 seconds, Total memory usage: 39.09MB '134.219' ************** MAGMA ***************** Host 134.219.148.42 (134.219.148.42) Time: Fri Dec 2 17:13:22 2005 Input: X_1 := Matrix(2, [-195,-5488, 7,197]); X_1; X_2 := Matrix(2, [-244,-8575, 7,246]); X_2; X_3 := Matrix(2, [-293,-12348, 7,295]); X_3; X_4 := Matrix(2, [-342,-16807, 7,344]); X_4; Y_1 := X_1*X_2*X_3*X_4*X_1*X_3*X_4*X_2*X_1; Y_1; Y_2 := X_2*X_3*X_4*X_2*X_1*X_4*X_1*X_2*X_3; Y_2; Y_3 := X_3*X_4*X_1*X_2*X_3*X_3*X_4*X_2*X_1; Y_3; Y_4 := X_4*X_2*X_3*X_3*X_4*X_2*X_1; Y_4; Output: Magma V2.11-10 Fri Dec 2 2005 17:13:22 on modular [Seed = 2186348165] ------------------------------------- [ -195 -5488] [ 7 197] [ -244 -8575] [ 7 246] [ -293 -12348] [ 7 295] [ -342 -16807] [ 7 344] [ 69345188618985448 1951377454536301239] [ -2489586226940967 -70057094531380844] [ 131106034502394869 5525581561639693421] [ -3761566848742780 -158534612851161791] [ -2218671813420922 -62433547906440305] [ 53009660061589 1491694774747402] [ -787183141527 -22151377270120] [ 16111464649 453377508977] Total time: 0.180 seconds, Total memory usage: 3.24MB '134.219' ************** MAGMA ***************** Host 134.219.148.42 (134.219.148.42) Time: Fri Dec 2 17:12:53 2005 Input: X_1 := Matrix(2, [-195,-5488, 7,197]); X_1; X_2 := Matrix(2, [-244,-8575, 7,246]); X_2; X_3 := Matrix(2, [-293,-12348, 7,295]); X_3; X_4 := Matrix(2, [-342,-16807, 7,344]); X_4; Y_1 := X_1*X_2*X_3*X_4*X_1*X_3*X_4*X_2*X_1; Y_1; Y_2 := X_2*X_3*X_4*X_2*X_1*X_4*X_1*X_2*X_3; Y_2; Y_3 := X_3*X_4*X_1*X_2*X_3*X_3*X_4*X_2*X_1; Y_3; X_4*X_2*X_3*X_3*X_4*X_2*X_1; Output: Magma V2.11-10 Fri Dec 2 2005 17:12:53 on modular [Seed = 1552931260] ------------------------------------- [ -195 -5488] [ 7 197] [ -244 -8575] [ 7 246] [ -293 -12348] [ 7 295] [ -342 -16807] [ 7 344] [ 69345188618985448 1951377454536301239] [ -2489586226940967 -70057094531380844] [ 131106034502394869 5525581561639693421] [ -3761566848742780 -158534612851161791] [ -2218671813420922 -62433547906440305] [ 53009660061589 1491694774747402] [ -787183141527 -22151377270120] [ 16111464649 453377508977] Total time: 0.190 seconds, Total memory usage: 3.24MB '134.219' ************** MAGMA ***************** Host 134.219.148.42 (134.219.148.42) Time: Fri Dec 2 17:12:27 2005 Input: X_1 := Matrix(2, [-195,-5488, 7,197]); X_1; X_2 := Matrix(2, [-244,-8575, 7,246]); X_2; X_3 := Matrix(2, [-293,-12348, 7,295]); X_3; X_4 := Matrix(2, [-342,-16807, 7,344]); X_4; Y_1 := X_1*X_2*X_3*X_4*X_1*X_3*X_4*X_2*X_1; Y_1; Y_2 := X_2*X_3*X_4*X_2*X_1*X_4*X_1*X_2*X_3; Y_2; X_3*X_4*X_1*X_2*X_3*X_3*X_4*X_2*X_1; X_4*X_2*X_3*X_3*X_4*X_2*X_1; Output: Magma V2.11-10 Fri Dec 2 2005 17:12:26 on modular [Seed = 1005108981] ------------------------------------- [ -195 -5488] [ 7 197] [ -244 -8575] [ 7 246] [ -293 -12348] [ 7 295] [ -342 -16807] [ 7 344] [ 69345188618985448 1951377454536301239] [ -2489586226940967 -70057094531380844] [ 131106034502394869 5525581561639693421] [ -3761566848742780 -158534612851161791] [ -2218671813420922 -62433547906440305] [ 53009660061589 1491694774747402] [ -787183141527 -22151377270120] [ 16111464649 453377508977] Total time: 0.190 seconds, Total memory usage: 3.24MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Fri Dec 2 17:09:57 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^0*3^7; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-1,0]; //P1:=J![x^2+6*x+36,12*x-18]; //Height(P); //HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=3335); Chabauty(P,17); //Order(Q);Order(P); //P+Q;2*P;2*P+Q;3*P; Output: Magma V2.11-10 Fri Dec 2 2005 17:09:56 on modular [Seed = 2992715648] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 + 9*x + 27, 9*x + 81, 2) ] {@ @} Total time: 1.490 seconds, Total memory usage: 39.07MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Fri Dec 2 17:09:50 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^0*3^7; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-1,0]; //P1:=J![x^2+6*x+36,12*x-18]; //Height(P); //HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=3335); Chabauty(P,13); //Order(Q);Order(P); //P+Q;2*P;2*P+Q;3*P; Output: Magma V2.11-10 Fri Dec 2 2005 17:09:48 on modular [Seed = 2792694668] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 + 9*x + 27, 9*x + 81, 2) ] {@ <24013, 1, 4, 1>, <16235, 1, 4, 1>, <26370, 1, 4, 1>, <0, 1, 4, 1>, <20134, 1, 4, 1>, <1104, 1, 4, 1> @} Total time: 2.080 seconds, Total memory usage: 39.08MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Fri Dec 2 17:09:41 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^0*3^7; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-1,0]; //P1:=J![x^2+6*x+36,12*x-18]; //Height(P); //HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=3335); Chabauty(P,11); //Order(Q);Order(P); //P+Q;2*P;2*P+Q;3*P; Output: Magma V2.11-10 Fri Dec 2 2005 17:09:38 on modular [Seed = 2639141049] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 + 9*x + 27, 9*x + 81, 2) ] {@ <1599, 1, 4, 1>, <8504, 1, 4, 1>, <8684, 1, 4, 1>, <10365, 1, 4, 1>, <12748, 1, 4, 1>, <13396, 1, 4, 1>, <14385, 1, 4, 1>, <5722, 1, 4, 1>, <5694, 1, 4, 1> @} Total time: 2.419 seconds, Total memory usage: 39.11MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Fri Dec 2 17:09:27 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^0*3^7; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-1,0]; //P1:=J![x^2+6*x+36,12*x-18]; //Height(P); //HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=3335); Chabauty(P,7); //Order(Q);Order(P); //P+Q;2*P;2*P+Q;3*P; Output: Magma V2.11-10 Fri Dec 2 2005 17:09:25 on modular [Seed = 2355038623] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 + 9*x + 27, 9*x + 81, 2) ] {@ <1459, 1, 4, 1>, <541, 1, 4, 1>, <0, 1, 4, 1> @} Total time: 1.560 seconds, Total memory usage: 39.07MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Fri Dec 2 17:08:40 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^0*3^7; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=5065); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-1,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=3335); //Chabauty(P,11); //Order(Q);Order(P); //P+Q;2*P;2*P+Q;3*P; Output: Magma V2.11-10 Fri Dec 2 2005 17:08:33 on modular [Seed = 1451610289] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 + 9*x + 27, 9*x + 81, 2) ] 2.2371134717500031028052074162261463081860 7.961423061998688543831823619116705236375 6.282102648051087408873197974735414235103 Total time: 6.070 seconds, Total memory usage: 39.16MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Fri Dec 2 17:05:08 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^0*3^7; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-1,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=5); //RationalPoints(J:Bound:=3335); //Chabauty(P,11); //Order(Q);Order(P); //P+Q;2*P;2*P+Q;3*P; Output: Magma V2.11-10 Fri Dec 2 2005 17:05:07 on modular [Seed = 466132238] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 + 9*x + 27, 9*x + 81, 2) ] 2.2371134717500031028052074162261463081860 7.961423061998688543831823619116705236375 6.282102648051087408873197974735414235103 Total time: 1.280 seconds, Total memory usage: 39.10MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Fri Dec 2 17:04:57 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^0*3^7; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-1,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=1); //RationalPoints(J:Bound:=3335); //Chabauty(P,11); //Order(Q);Order(P); //P+Q;2*P;2*P+Q;3*P; Output: Magma V2.11-10 Fri Dec 2 2005 17:04:56 on modular [Seed = 516790288] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 + 9*x + 27, 9*x + 81, 2) ] 2.2371134717500031028052074162261463081860 9.962718751034201590559070379422135389092 6.282102648051087408873197974735414235103 Total time: 1.189 seconds, Total memory usage: 39.08MB '128.112' ************** MAGMA ***************** Host 128.112.37.143 (128.112.37.143) Time: Fri Dec 2 17:01:24 2005 Input: (x^4+y^4)^10000 Output: Magma V2.11-10 Fri Dec 2 2005 17:01:24 on modular [Seed = 198741049] ------------------------------------- >> (x^4+y^4)^10000; ^ User error: Identifier 'x' has not been declared or assigned Total time: 0.200 seconds, Total memory usage: 3.24MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Fri Dec 2 16:57:02 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^0*3^7; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-1,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=1); //RationalPoints(J:Bound:=3335); //Chabauty(P,11); //Order(Q);Order(P); //P+Q;2*P;2*P+Q;3*P; Output: Magma V2.11-10 Fri Dec 2 2005 16:57:00 on modular [Seed = 3658046193] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 + 9*x + 27, 9*x + 81, 2) ] 2.2371134717500031028052074162261463081860 9.962718751034201590559070379422135389092 6.282102648051087408873197974735414235103 Total time: 1.209 seconds, Total memory usage: 39.10MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Fri Dec 2 16:55:48 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^0*3^6; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=10000); B:=ReducedBasis(V); B; //P:=J!B[1]; //Q:=J![x-1,0]; //P1:=J![x^2+6*x+36,12*x-18]; //Height(P); //HeightConstant(J:Effort:=1); //RationalPoints(J:Bound:=3335); //Chabauty(P,11); //Order(Q);Order(P); //P+Q;2*P;2*P+Q;3*P; Output: ** WARNING: Computation time exceeded 20 seconds, so computation was terminated after 20 seconds. ** Magma V2.11-10 Fri Dec 2 2005 16:55:27 on modular [Seed = 3489878296] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] Errors: /bin/sh: line 1: 17623 Alarm clock nice -n 19 /usr/local/bin/magma '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Fri Dec 2 16:49:33 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^0*3^4; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-1,0]; //P1:=J![x^2+6*x+36,12*x-18]; //Height(P); //HeightConstant(J:Effort:=1); //RationalPoints(J:Bound:=3335); Chabauty(P,11); //Order(Q);Order(P); //P+Q;2*P;2*P+Q;3*P; Output: Magma V2.11-10 Fri Dec 2 2005 16:49:32 on modular [Seed = 3407366560] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 54/25*x + 18/5, 717/125*x - 189/25, 2) ] {@ @} Total time: 1.409 seconds, Total memory usage: 39.12MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Fri Dec 2 16:49:28 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^0*3^4; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-1,0]; //P1:=J![x^2+6*x+36,12*x-18]; //Height(P); //HeightConstant(J:Effort:=1); //RationalPoints(J:Bound:=3335); Chabauty(P,11); //Order(Q);Order(P); //P+Q;2*P;2*P+Q;3*P; Output: Magma V2.11-10 Fri Dec 2 2005 16:49:26 on modular [Seed = 3222487713] ------------------------------------- TwoSelmerGroupData( J: Jacobian of Hyperelliptic Curve defined by y^2 = x^5 - 81 ov... ) TwoSelmerWork( C: Hyperelliptic Curve defined by y^2 = x^5 - 81 over Rational ..., BoundType: Default, Bound: 0, UseUnits: true, Points: {}, Fields: {} ) LocalImageOdd( f: x^5 - 81, p: 2, m1: Mapping from: Univariate Polynomial Ring in x over Rational ..., m2: Mapping from: Univariate Polynomial Ring in x over Rational ..., v: 2, Points: {} ) try( ) try( d: 2 ) SquarefreeRoots( F: $.1^4 + O(2^50)*$.1^3 - (1909325743*2^2 + O(2^52))*$.1^2 + O... ) In file "/usr/local/magma/package/Geometry/Arith/sqfroots.m", line 84, column 15: >> return Roots(F:IsSquarefree); ^ Runtime error in 'Roots': Insufficient precision to find roots >> r:=TwoSelmerGroupData(J);r; ^ User error: Identifier 'r' has not been declared or assigned Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 54/25*x + 18/5, 717/125*x - 189/25, 2) ] {@ @} Total time: 1.330 seconds, Total memory usage: 39.24MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Fri Dec 2 16:49:19 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^0*3^4; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-1,0]; //P1:=J![x^2+6*x+36,12*x-18]; //Height(P); //HeightConstant(J:Effort:=1); //RationalPoints(J:Bound:=3335); Chabauty(P,7); //Order(Q);Order(P); //P+Q;2*P;2*P+Q;3*P; Output: Magma V2.11-10 Fri Dec 2 2005 16:49:17 on modular [Seed = 3306571679] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 54/25*x + 18/5, 717/125*x - 189/25, 2) ] {@ <325, 1, 4, 1> @} Total time: 1.370 seconds, Total memory usage: 39.06MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Fri Dec 2 16:48:09 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^0*3^4; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-1,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=1); RationalPoints(J:Bound:=3335); //Chabauty(P,17); //Order(Q);Order(P); //P+Q;2*P;2*P+Q;3*P; Output: Magma V2.11-10 Fri Dec 2 2005 16:48:06 on modular [Seed = 3144169898] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 54/25*x + 18/5, 717/125*x - 189/25, 2) ] 5.049935286802904695717596589467749543594443514617086990154667 6.842606509147086978356785189932877290677 4.0848780707148680260827075008903628258424 {@ (1, 0, 0), (x^2 - 54/25*x + 18/5, 717/125*x - 189/25, 2), (x^2 - 54/25*x + 18/5, -717/125*x + 189/25, 2) @} Total time: 2.830 seconds, Total memory usage: 39.17MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Fri Dec 2 16:46:49 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^0*3^4; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-1,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J:Effort:=1); RationalPoints(J:Bound:=1000); //Chabauty(P,17); //Order(Q);Order(P); //P+Q;2*P;2*P+Q;3*P; Output: Magma V2.11-10 Fri Dec 2 2005 16:46:47 on modular [Seed = 2959291248] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 54/25*x + 18/5, 717/125*x - 189/25, 2) ] 5.049935286802904695717596589467749543594443514617086990154667 6.842606509147086978356785189932877290677 4.0848780707148680260827075008903628258424 {@ (1, 0, 0), (x^2 - 54/25*x + 18/5, 717/125*x - 189/25, 2), (x^2 - 54/25*x + 18/5, -717/125*x + 189/25, 2) @} Total time: 1.240 seconds, Total memory usage: 39.13MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Fri Dec 2 16:38:29 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^0*3^4; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-1,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J); RationalPoints(J:Bound:=1000); //Chabauty(P,17); //Order(Q);Order(P); //P+Q;2*P;2*P+Q;3*P; Output: Magma V2.11-10 Fri Dec 2 2005 16:38:28 on modular [Seed = 2742050416] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 54/25*x + 18/5, 717/125*x - 189/25, 2) ] 5.049935286802904695717596589467749543594443514617086990154667 10.478993790995409841346196964902895694514 4.0848780707148680260827075008903628258424 {@ (1, 0, 0), (x^2 - 54/25*x + 18/5, 717/125*x - 189/25, 2), (x^2 - 54/25*x + 18/5, -717/125*x + 189/25, 2) @} Total time: 1.120 seconds, Total memory usage: 39.10MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Fri Dec 2 16:38:20 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^0*3^4; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-1,0]; //P1:=J![x^2+6*x+36,12*x-18]; //Height(P); //HeightConstant(J); RationalPoints(J:Bound:=1000); //Chabauty(P,17); //Order(Q);Order(P); //P+Q;2*P;2*P+Q;3*P; Output: Magma V2.11-10 Fri Dec 2 2005 16:38:18 on modular [Seed = 2555077973] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 54/25*x + 18/5, 717/125*x - 189/25, 2) ] {@ (1, 0, 0), (x^2 - 54/25*x + 18/5, 717/125*x - 189/25, 2), (x^2 - 54/25*x + 18/5, -717/125*x + 189/25, 2) @} Total time: 1.050 seconds, Total memory usage: 39.09MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Fri Dec 2 16:36:28 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^0*3^3; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-1,0]; //P1:=J![x^2+6*x+36,12*x-18]; //Height(P); //HeightConstant(J); RationalPoints(J:Bound:=1000); Chabauty(P,17); //Order(Q);Order(P); //P+Q;2*P;2*P+Q;3*P; Output: Magma V2.11-10 Fri Dec 2 2005 16:36:26 on modular [Seed = 2672583362] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 14*x + 28, 49*x - 111, 2) ] {@ (1, 0, 0), (x^2 - 14*x + 28, 49*x - 111, 2), (x^2 - 14*x + 28, -49*x + 111, 2) @} {@ @} Total time: 1.490 seconds, Total memory usage: 39.08MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Fri Dec 2 16:36:21 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^0*3^3; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-1,0]; //P1:=J![x^2+6*x+36,12*x-18]; //Height(P); //HeightConstant(J); RationalPoints(J:Bound:=1000); Chabauty(P,13); //Order(Q);Order(P); //P+Q;2*P;2*P+Q;3*P; Output: Magma V2.11-10 Fri Dec 2 2005 16:36:19 on modular [Seed = 2487708096] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 14*x + 28, 49*x - 111, 2) ] {@ (1, 0, 0), (x^2 - 14*x + 28, 49*x - 111, 2), (x^2 - 14*x + 28, -49*x + 111, 2) @} {@ <28450, 1, 4, 1>, <915, 1, 4, 1>, <19153, 1, 4, 1>, <13553, 1, 4, 1>, <0, 1, 4, 1> @} Total time: 2.009 seconds, Total memory usage: 39.10MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Fri Dec 2 16:36:12 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^0*3^3; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-1,0]; //P1:=J![x^2+6*x+36,12*x-18]; //Height(P); //HeightConstant(J); RationalPoints(J:Bound:=1000); Chabauty(P,11); //Order(Q);Order(P); //P+Q;2*P;2*P+Q;3*P; Output: Magma V2.11-10 Fri Dec 2 2005 16:36:10 on modular [Seed = 2321119175] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 14*x + 28, 49*x - 111, 2) ] {@ (1, 0, 0), (x^2 - 14*x + 28, 49*x - 111, 2), (x^2 - 14*x + 28, -49*x + 111, 2) @} {@ <8830, 1, 4, 1>, <14431, 1, 4, 1>, <14126, 1, 4, 1>, <13415, 1, 4, 1> @} Total time: 1.909 seconds, Total memory usage: 39.09MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Fri Dec 2 16:35:49 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^0*3^3; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-1,0]; //P1:=J![x^2+6*x+36,12*x-18]; //Height(P); //HeightConstant(J); RationalPoints(J:Bound:=1000); Chabauty(P,7); //Order(Q);Order(P); //P+Q;2*P;2*P+Q;3*P; Output: Magma V2.11-10 Fri Dec 2 2005 16:35:48 on modular [Seed = 2405200572] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 14*x + 28, 49*x - 111, 2) ] {@ (1, 0, 0), (x^2 - 14*x + 28, 49*x - 111, 2), (x^2 - 14*x + 28, -49*x + 111, 2) @} {@ <0, 1, 4, 1> @} Total time: 1.429 seconds, Total memory usage: 39.19MB '134.219' ************** MAGMA ***************** Host 134.219.148.42 (134.219.148.42) Time: Fri Dec 2 16:33:06 2005 Input: X_1 := Matrix(2, [-195,-5488, 7,197]); X_1; X_2 := Matrix(2, [-244,-8575, 7,246]); X_2; X_3 := Matrix(2, [-293,-12348, 7,295]); X_3; X_4 := Matrix(2, [-342,-16807, 7,344]); X_4; X_1*X_2*X_3*X_4*X_1*X_3*X_4*X_2*X_1; X_2*X_3*X_4*X_2*X_1*X_4*X_1*X_2*X_3; X_3*X_4*X_1*X_2*X_3*X_3*X_4*X_2*X_1; X_4*X_2*X_3*X_3*X_4*X_2*X_1; Output: Magma V2.11-10 Fri Dec 2 2005 16:33:06 on modular [Seed = 2220325393] ------------------------------------- [ -195 -5488] [ 7 197] [ -244 -8575] [ 7 246] [ -293 -12348] [ 7 295] [ -342 -16807] [ 7 344] [ 69345188618985448 1951377454536301239] [ -2489586226940967 -70057094531380844] [ 131106034502394869 5525581561639693421] [ -3761566848742780 -158534612851161791] [ -2218671813420922 -62433547906440305] [ 53009660061589 1491694774747402] [ -787183141527 -22151377270120] [ 16111464649 453377508977] Total time: 0.190 seconds, Total memory usage: 3.24MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Fri Dec 2 16:30:44 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^0*3^3; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-1,0]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J); RationalPoints(J:Bound:=1000); //Chabauty(P,11); //Order(Q);Order(P); //P+Q;2*P;2*P+Q;3*P; Output: Magma V2.11-10 Fri Dec 2 2005 16:30:42 on modular [Seed = 2057929855] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 14*x + 28, 49*x - 111, 2) ] 4.8248798986869435073842048284586441146816125911687293600645 8.848223363362808836218352143025897569490 3.352469878269461565152544009608679022739 {@ (1, 0, 0), (x^2 - 14*x + 28, 49*x - 111, 2), (x^2 - 14*x + 28, -49*x + 111, 2) @} Total time: 1.139 seconds, Total memory usage: 39.11MB '134.219' ************** MAGMA ***************** Host 134.219.148.42 (134.219.148.42) Time: Fri Dec 2 16:30:05 2005 Input: X_1 := Matrix(2, [-195,-5488, 7,197]); X_1; X_2 := Matrix(2, [-244,-8575, 7,246]); X_2; X_3 := Matrix(2, [-293,-12348, 7,295]); X_3; X_4 := Matrix(2, [-342,-16807, 7,344]); X_4; X_1*X_2*X_3*X_4*X_1*X_3*X_4*X_2*X_1; X_2*X_3*X_4*X_2*X_1*X_4*X_1*X_2*X_3; X_3*X_4*X_1*X_2*X_3*X_3*X_4*X_2*X_1; Output: Magma V2.11-10 Fri Dec 2 2005 16:30:05 on modular [Seed = 2142011557] ------------------------------------- [ -195 -5488] [ 7 197] [ -244 -8575] [ 7 246] [ -293 -12348] [ 7 295] [ -342 -16807] [ 7 344] [ 69345188618985448 1951377454536301239] [ -2489586226940967 -70057094531380844] [ 131106034502394869 5525581561639693421] [ -3761566848742780 -158534612851161791] [ -2218671813420922 -62433547906440305] [ 53009660061589 1491694774747402] Total time: 0.190 seconds, Total memory usage: 3.24MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Fri Dec 2 16:30:01 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^0*3^3; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-1,0]; //P1:=J![x^2+6*x+36,12*x-18]; //Height(P); //HeightConstant(J); RationalPoints(J:Bound:=1000); //Chabauty(P,11); //Order(Q);Order(P); //P+Q;2*P;2*P+Q;3*P; Output: Magma V2.11-10 Fri Dec 2 2005 16:30:00 on modular [Seed = 1957137323] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 14*x + 28, 49*x - 111, 2) ] {@ (1, 0, 0), (x^2 - 14*x + 28, 49*x - 111, 2), (x^2 - 14*x + 28, -49*x + 111, 2) @} Total time: 1.080 seconds, Total memory usage: 39.07MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Fri Dec 2 16:29:40 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^0*3^3; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x-1,0]; //P1:=J![x^2+6*x+36,12*x-18]; //Height(P); //HeightConstant(J); //RationalPoints(J:Bound:=1000); Chabauty(P,11); //Order(Q);Order(P); //P+Q;2*P;2*P+Q;3*P; Output: Magma V2.11-10 Fri Dec 2 2005 16:29:38 on modular [Seed = 1773835948] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 14*x + 28, 49*x - 111, 2) ] {@ <8830, 1, 4, 1>, <14431, 1, 4, 1>, <14126, 1, 4, 1>, <13415, 1, 4, 1> @} Total time: 1.980 seconds, Total memory usage: 39.16MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Fri Dec 2 16:29:29 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^0*3^3; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; Q:=J![x-1,0]; //P1:=J![x^2+6*x+36,12*x-18]; //Height(P); //HeightConstant(J); //RationalPoints(J:Bound:=1000); Chabauty(P,11); //Order(Q);Order(P); //P+Q;2*P;2*P+Q;3*P; Output: Magma V2.11-10 Fri Dec 2 2005 16:29:27 on modular [Seed = 1857917406] ------------------------------------- 1 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [ (x^2 - 14*x + 28, 49*x - 111, 2) ] >> P:=J!B[1]; Q:=J![x-1,0]; ^ Runtime error in '!': Points specified by second polynomial are not on the curve {@ <8830, 1, 4, 1>, <14431, 1, 4, 1>, <14126, 1, 4, 1>, <13415, 1, 4, 1> @} Total time: 1.980 seconds, Total memory usage: 39.11MB '134.219' ************** MAGMA ***************** Host 134.219.148.42 (134.219.148.42) Time: Fri Dec 2 16:26:56 2005 Input: X_1 := Matrix(2, [-195,-5488, 7,197]); X_1; X_2 := Matrix(2, [-244,-8575, 7,246]); X_2; X_3 := Matrix(2, [-293,-12348, 7,295]); X_3; X_4 := Matrix(2, [-342,-16807, 7,344]); X_4; X_1*X_2*X_3*X_4*X_1*X_3*X_4*X_2*X_1; X_2*X_3*X_4*X_2*X_1*X_4*X_1*X_2*X_3; Output: Magma V2.11-10 Fri Dec 2 2005 16:26:56 on modular [Seed = 1672518724] ------------------------------------- [ -195 -5488] [ 7 197] [ -244 -8575] [ 7 246] [ -293 -12348] [ 7 295] [ -342 -16807] [ 7 344] [ 69345188618985448 1951377454536301239] [ -2489586226940967 -70057094531380844] [ 131106034502394869 5525581561639693421] [ -3761566848742780 -158534612851161791] Total time: 0.190 seconds, Total memory usage: 3.24MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Fri Dec 2 16:26:44 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^0*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; Q:=J![x-1,0]; //P1:=J![x^2+6*x+36,12*x-18]; //Height(P); //HeightConstant(J); //RationalPoints(J:Bound:=1000); Chabauty(P,11); //Order(Q);Order(P); //P+Q;2*P;2*P+Q;3*P; Output: Magma V2.11-10 Fri Dec 2 2005 16:26:42 on modular [Seed = 1485547341] ------------------------------------- 1 Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 to JacHyp: J given by a rule [no inverse] [ (x^2 - x + 1, x - 1, 2) ] {@ @} Total time: 1.919 seconds, Total memory usage: 39.78MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Fri Dec 2 16:26:28 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^0*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; Q:=J![x-1,0]; //P1:=J![x^2+6*x+36,12*x-18]; //Height(P); //HeightConstant(J); //RationalPoints(J:Bound:=1000); Chabauty(P,7); //Order(Q);Order(P); //P+Q;2*P;2*P+Q;3*P; Output: Magma V2.11-10 Fri Dec 2 2005 16:26:26 on modular [Seed = 1569628760] ------------------------------------- 1 Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 to JacHyp: J given by a rule [no inverse] [ (x^2 - x + 1, x - 1, 2) ] {@ <822, 1, 4, 1>, <1951, 1, 4, 1>, <235, 1, 4, 1> @} Total time: 1.980 seconds, Total memory usage: 39.63MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Fri Dec 2 16:23:58 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^0*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; Q:=J![x-1,0]; //P1:=J![x^2+6*x+36,12*x-18]; //Height(P); //HeightConstant(J); RationalPoints(J:Bound:=1000); //Chabauty(P,19); //Order(Q);Order(P); P+Q;2*P;2*P+Q;3*P; Output: Magma V2.11-10 Fri Dec 2 2005 16:23:56 on modular [Seed = 1400941368] ------------------------------------- 1 Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 to JacHyp: J given by a rule [no inverse] [ (x^2 - x + 1, x - 1, 2) ] {@ (1, 0, 0), (x - 1, 0, 1), (x^2 + 2*x + 2, x - 1, 2), (x^2 + 2*x + 2, -x + 1, 2), (x^2 - x + 1, x - 1, 2), (x^2 - x + 1, -x + 1, 2), (x^2 + 2/9*x + 5/9, 79/54*x + 13/54, 2), (x^2 + 2/9*x + 5/9, -79/54*x - 13/54, 2), (x^2 - 9/64*x + 125/64, 571/512*x + 809/512, 2), (x^2 - 9/64*x + 125/64, -571/512*x - 809/512, 2) @} (x^2 + 2*x + 2, -x + 1, 2) (x^2 + 2/9*x + 5/9, -79/54*x - 13/54, 2) (x^2 - 9/64*x + 125/64, 571/512*x + 809/512, 2) (x^2 - 6097/81*x + 12905/81, 469619/729*x - 1028707/729, 2) Total time: 1.520 seconds, Total memory usage: 39.63MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Fri Dec 2 16:23:02 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^0*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; Q:=J![x-1,0]; //P1:=J![x^2+6*x+36,12*x-18]; //Height(P); //HeightConstant(J); RationalPoints(J:Bound:=1000); //Chabauty(P,19); //Order(Q);Order(P); P+Q;2*P;2*P+Q; Output: Magma V2.11-10 Fri Dec 2 2005 16:23:00 on modular [Seed = 1251586681] ------------------------------------- 1 Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 to JacHyp: J given by a rule [no inverse] [ (x^2 - x + 1, x - 1, 2) ] {@ (1, 0, 0), (x - 1, 0, 1), (x^2 + 2*x + 2, x - 1, 2), (x^2 + 2*x + 2, -x + 1, 2), (x^2 - x + 1, x - 1, 2), (x^2 - x + 1, -x + 1, 2), (x^2 + 2/9*x + 5/9, 79/54*x + 13/54, 2), (x^2 + 2/9*x + 5/9, -79/54*x - 13/54, 2), (x^2 - 9/64*x + 125/64, 571/512*x + 809/512, 2), (x^2 - 9/64*x + 125/64, -571/512*x - 809/512, 2) @} (x^2 + 2*x + 2, -x + 1, 2) (x^2 + 2/9*x + 5/9, -79/54*x - 13/54, 2) (x^2 - 9/64*x + 125/64, 571/512*x + 809/512, 2) Total time: 1.510 seconds, Total memory usage: 39.64MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Fri Dec 2 16:20:12 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^0*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; Height(P); HeightConstant(J); RationalPoints(J:Bound:=1000); //Chabauty(P,19); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Fri Dec 2 2005 16:20:10 on modular [Seed = 1083423181] ------------------------------------- 1 Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 to JacHyp: J given by a rule [no inverse] [ (x^2 - x + 1, x - 1, 2) ] 1.20771088607746974821221747684835952376521486877031981730180 4.0495988164421265487168416984396464580923 1.1552453009332421823620535357636276134549 {@ (1, 0, 0), (x - 1, 0, 1), (x^2 + 2*x + 2, x - 1, 2), (x^2 + 2*x + 2, -x + 1, 2), (x^2 - x + 1, x - 1, 2), (x^2 - x + 1, -x + 1, 2), (x^2 + 2/9*x + 5/9, 79/54*x + 13/54, 2), (x^2 + 2/9*x + 5/9, -79/54*x - 13/54, 2), (x^2 - 9/64*x + 125/64, 571/512*x + 809/512, 2), (x^2 - 9/64*x + 125/64, -571/512*x - 809/512, 2) @} Total time: 1.590 seconds, Total memory usage: 39.63MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Fri Dec 2 16:18:22 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^0*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; //Height(P); //HeightConstant(J); RationalPoints(J:Bound:=1000); //Chabauty(P,19); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Fri Dec 2 2005 16:18:21 on modular [Seed = 1184218972] ------------------------------------- 1 Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 to JacHyp: J given by a rule [no inverse] [ (x^2 - x + 1, x - 1, 2) ] {@ (1, 0, 0), (x - 1, 0, 1), (x^2 + 2*x + 2, x - 1, 2), (x^2 + 2*x + 2, -x + 1, 2), (x^2 - x + 1, x - 1, 2), (x^2 - x + 1, -x + 1, 2), (x^2 + 2/9*x + 5/9, 79/54*x + 13/54, 2), (x^2 + 2/9*x + 5/9, -79/54*x - 13/54, 2), (x^2 - 9/64*x + 125/64, 571/512*x + 809/512, 2), (x^2 - 9/64*x + 125/64, -571/512*x - 809/512, 2) @} Total time: 1.530 seconds, Total memory usage: 39.64MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Fri Dec 2 16:18:00 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^0*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; //Height(P); //HeightConstant(J); RationalPoints(J:Bound:=1000); //Chabauty(P,19); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Fri Dec 2 2005 16:17:57 on modular [Seed = 1005109817] ------------------------------------- 1 Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 to JacHyp: J given by a rule [no inverse] [ (x^2 - x + 1, x - 1, 2) ] {@ (1, 0, 0), (x - 1, 0, 1), (x^2 + 2*x + 2, x - 1, 2), (x^2 + 2*x + 2, -x + 1, 2), (x^2 - x + 1, x - 1, 2), (x^2 - x + 1, -x + 1, 2), (x^2 + 2/9*x + 5/9, 79/54*x + 13/54, 2), (x^2 + 2/9*x + 5/9, -79/54*x - 13/54, 2), (x^2 - 9/64*x + 125/64, 571/512*x + 809/512, 2), (x^2 - 9/64*x + 125/64, -571/512*x - 809/512, 2) @} Total time: 1.520 seconds, Total memory usage: 39.60MB '134.219' ************** MAGMA ***************** Host 134.219.148.42 (134.219.148.42) Time: Fri Dec 2 16:17:37 2005 Input: X_1 := Matrix(2, [-195,-5488, 7,197]); X_1; X_2 := Matrix(2, [-244,-8575, 7,246]); X_2; X_3 := Matrix(2, [-293,-12348, 7,295]); X_3; X_4 := Matrix(2, [-342,-16807, 7,344]); X_4; X_1*X_2*X_3*X_4*X_1*X_3*X_4*X_2*X_1; Output: Magma V2.11-10 Fri Dec 2 2005 16:17:37 on modular [Seed = 820234950] ------------------------------------- [ -195 -5488] [ 7 197] [ -244 -8575] [ 7 246] [ -293 -12348] [ 7 295] [ -342 -16807] [ 7 344] [ 69345188618985448 1951377454536301239] [ -2489586226940967 -70057094531380844] Total time: 0.200 seconds, Total memory usage: 3.24MB '134.219' ************** MAGMA ***************** Host 134.219.148.42 (134.219.148.42) Time: Fri Dec 2 16:16:01 2005 Input: X_1 := Matrix(2, [-195,-5488, 7,197]); X_1; X_2 := Matrix(2, [-244,-8575, 7,246]); X_2; X_3 := Matrix(2, [-293,-12348, 7,295]); X_3; X_4 := Matrix(2, [-342,-16807, 7,344]); X_4; Y_1 :=X_1*X_2*X_3*X_4*X_1*X_3*X_4*X_2*X_1; Y_1; Output: Magma V2.11-10 Fri Dec 2 2005 16:16:01 on modular [Seed = 704304063] ------------------------------------- [ -195 -5488] [ 7 197] [ -244 -8575] [ 7 246] [ -293 -12348] [ 7 295] [ -342 -16807] [ 7 344] [ 69345188618985448 1951377454536301239] [ -2489586226940967 -70057094531380844] Total time: 0.190 seconds, Total memory usage: 3.24MB '134.219' ************** MAGMA ***************** Host 134.219.148.42 (134.219.148.42) Time: Fri Dec 2 16:15:30 2005 Input: X_1 := Matrix(2, [-195,-5488, 7,197]); X_1; X_2 := Matrix(2, [-244,-8575, 7,246]); X_2; X_3 := Matrix(2, [-293,-12348, 7,295]); X_3; X_4 := Matrix(2, [-342,-16807, 7,344]); X_4; X_1*X_2*X_3*X_4*X_1*X_3*X_4*X_2*X_1; Output: Magma V2.11-10 Fri Dec 2 2005 16:15:30 on modular [Seed = 788386907] ------------------------------------- [ -195 -5488] [ 7 197] [ -244 -8575] [ 7 246] [ -293 -12348] [ 7 295] [ -342 -16807] [ 7 344] [ 69345188618985448 1951377454536301239] [ -2489586226940967 -70057094531380844] Total time: 0.190 seconds, Total memory usage: 3.24MB '134.219' ************** MAGMA ***************** Host 134.219.148.42 (134.219.148.42) Time: Fri Dec 2 16:14:21 2005 Input: X_1 := Matrix(2, [-195,-5488, 7,197]); X_1; X_2 := Matrix(2, [-244,-8575, 7,246]); X_2; X_3 := Matrix(2, [-293,-12348, 7,295]); X_3; X_4 := Matrix(2, [-342,-16807, 7,344]); X_4; Output: Magma V2.11-10 Fri Dec 2 2005 16:14:21 on modular [Seed = 602986761] ------------------------------------- [ -195 -5488] [ 7 197] [ -244 -8575] [ 7 246] [ -293 -12348] [ 7 295] [ -342 -16807] [ 7 344] Total time: 0.200 seconds, Total memory usage: 3.24MB '134.219' ************** MAGMA ***************** Host 134.219.148.42 (134.219.148.42) Time: Fri Dec 2 16:13:21 2005 Input: X_1 := Matrix(2, [-195,-5488, 7,197]); X_1; X_2 := Matrix(2, [-244,-8575, 7,246]); X_2; X_3 := Matrix(2, [-293,-12348, 7,295]); X_3; Output: Magma V2.11-10 Fri Dec 2 2005 16:13:21 on modular [Seed = 416015342] ------------------------------------- [ -195 -5488] [ 7 197] [ -244 -8575] [ 7 246] [ -293 -12348] [ 7 295] Total time: 0.200 seconds, Total memory usage: 3.24MB '134.219' ************** MAGMA ***************** Host 134.219.148.42 (134.219.148.42) Time: Fri Dec 2 16:12:29 2005 Input: X_1 := Matrix(2, [-195,-5488, 7,197]); X_1; X_2 := Matrix(2, [-244,-8575, 7,246]); X_2; Output: Magma V2.11-10 Fri Dec 2 2005 16:12:28 on modular [Seed = 500098726] ------------------------------------- [ -195 -5488] [ 7 197] [ -244 -8575] [ 7 246] Total time: 0.190 seconds, Total memory usage: 3.24MB '134.219' ************** MAGMA ***************** Host 134.219.148.42 (134.219.148.42) Time: Fri Dec 2 16:11:32 2005 Input: X_1 := Matrix(2, [-195,-5488, 7,197]); X_1; Output: Magma V2.11-10 Fri Dec 2 2005 16:11:32 on modular [Seed = 314698129] ------------------------------------- [ -195 -5488] [ 7 197] Total time: 0.190 seconds, Total memory usage: 3.24MB '134.219' ************** MAGMA ***************** Host 134.219.148.42 (134.219.148.42) Time: Fri Dec 2 16:09:16 2005 Input: X_1 := Matrix(2, [1,2, 3,4]); X_1; Output: Magma V2.11-10 Fri Dec 2 2005 16:09:16 on modular [Seed = 148634867] ------------------------------------- [1 2] [3 4] Total time: 0.190 seconds, Total memory usage: 3.24MB '134.219' ************** MAGMA ***************** Host 134.219.148.42 (134.219.148.42) Time: Fri Dec 2 16:05:58 2005 Input: X_1 := Matrix(2, [-195,-5488, 7,197]); X_1; B := Matrix(2, [-244,-8575, 7,197]); B; C := Matrix(2, [-293,-12348, 7,295]); C; A*B*C; Output: Magma V2.11-10 Fri Dec 2 2005 16:05:57 on modular [Seed = 4213706702] ------------------------------------- [ -195 -5488] [ 7 197] [ -244 -8575] [ 7 197] [ -293 -12348] [ 7 295] >> A*B*C; ^ User error: Identifier 'A' has not been declared or assigned Total time: 0.190 seconds, Total memory usage: 3.24MB '134.219' ************** MAGMA ***************** Host 134.219.148.42 (134.219.148.42) Time: Fri Dec 2 16:04:57 2005 Input: A := Matrix(2, [-195,-5488, 7,197]); A; B := Matrix(2, [-244,-8575, 7,197]); B; A*B; C := Matrix(2, [-293,-12348, 7,295]); C; A*B*C; Output: Magma V2.11-10 Fri Dec 2 2005 16:04:57 on modular [Seed = 4129620751] ------------------------------------- [ -195 -5488] [ 7 197] [ -244 -8575] [ 7 197] [ 9164 590989] [ -329 -21216] [ -293 -12348] [ 7 295] [ 1451871 61184683] [ -52115 -2196228] Total time: 0.190 seconds, Total memory usage: 3.24MB '134.219' ************** MAGMA ***************** Host 134.219.148.42 (134.219.148.42) Time: Fri Dec 2 16:03:42 2005 Input: A := Matrix(2, [-195,-5488, 7,197]); A; B := Matrix(2, [-244,-8575, 7,197]); B; A*B; C := Matrix(2, [-293,-12348, 7,295]); C; Output: Magma V2.11-10 Fri Dec 2 2005 16:03:42 on modular [Seed = 4045538399] ------------------------------------- [ -195 -5488] [ 7 197] [ -244 -8575] [ 7 197] [ 9164 590989] [ -329 -21216] [ -293 -12348] [ 7 295] Total time: 0.190 seconds, Total memory usage: 3.24MB '134.219' ************** MAGMA ***************** Host 134.219.148.42 (134.219.148.42) Time: Fri Dec 2 16:03:24 2005 Input: A := Matrix(2, [-195,-5488, 7,197]); A; B := Matrix(2, [-244,-8575, 7,197]); B; A*B; C := Matrix(2, [-293,-12348, 7,295]); Output: Magma V2.11-10 Fri Dec 2 2005 16:03:23 on modular [Seed = 3963558263] ------------------------------------- [ -195 -5488] [ 7 197] [ -244 -8575] [ 7 197] [ 9164 590989] [ -329 -21216] Total time: 0.190 seconds, Total memory usage: 3.24MB '134.219' ************** MAGMA ***************** Host 134.219.148.42 (134.219.148.42) Time: Fri Dec 2 16:03:00 2005 Input: A := Matrix(2, [-195,-5488, 7,197]); A; B := Matrix(2, [-244,-8575, 7,197]); B; A*B; C := Matrix(2, [-293,-12348, 7,295]); C; Output: Magma V2.11-10 Fri Dec 2 2005 16:03:00 on modular [Seed = 3878949475] ------------------------------------- [ -195 -5488] [ 7 197] [ -244 -8575] [ 7 197] [ 9164 590989] [ -329 -21216] [ -293 -12348] [ 7 295] Total time: 0.190 seconds, Total memory usage: 3.24MB '134.219' ************** MAGMA ***************** Host 134.219.148.42 (134.219.148.42) Time: Fri Dec 2 16:00:49 2005 Input: A := Matrix(2, [-195,-5488, 7,197]); A; B := Matrix(2, [-244,-8575, 7,197]); B; A*B; Output: Magma V2.11-10 Fri Dec 2 2005 16:00:49 on modular [Seed = 3778155744] ------------------------------------- [ -195 -5488] [ 7 197] [ -244 -8575] [ 7 197] [ 9164 590989] [ -329 -21216] Total time: 0.190 seconds, Total memory usage: 3.24MB '134.219' ************** MAGMA ***************** Host 134.219.148.42 (134.219.148.42) Time: Fri Dec 2 15:59:08 2005 Input: A := Matrix(2, [-195,-5488, 7,197]); A; B := Matrix(2, [-244,-8575, 7,197]); B; Output: Magma V2.11-10 Fri Dec 2 2005 15:59:08 on modular [Seed = 2655873026] ------------------------------------- [ -195 -5488] [ 7 197] [ -244 -8575] [ 7 197] Total time: 0.200 seconds, Total memory usage: 3.24MB '134.219' ************** MAGMA ***************** Host 134.219.148.42 (134.219.148.42) Time: Fri Dec 2 15:58:11 2005 Input: A := Matrix(2, [-195,-5488, 7,197]); A; Output: Magma V2.11-10 Fri Dec 2 2005 15:58:10 on modular [Seed = 2571792379] ------------------------------------- [ -195 -5488] [ 7 197] Total time: 0.190 seconds, Total memory usage: 3.24MB '134.219' ************** MAGMA ***************** Host 134.219.148.42 (134.219.148.42) Time: Fri Dec 2 15:57:59 2005 Input: X := Matrix(2, [-195,-5488, 7,197]); X; Output: Magma V2.11-10 Fri Dec 2 2005 15:57:58 on modular [Seed = 2487706306] ------------------------------------- [ -195 -5488] [ 7 197] Total time: 0.190 seconds, Total memory usage: 3.24MB '134.219' ************** MAGMA ***************** Host 134.219.148.42 (134.219.148.42) Time: Fri Dec 2 15:25:43 2005 Input: A := Matrix(IntegerRing(), 2, 2, [1,2, 3,4]); A; B := Matrix(IntegerRing(), 2, 2, [1,n, 0,1]); B; C := Matrix(2, [1,2, 3,4]); C; Output: Magma V2.11-10 Fri Dec 2 2005 15:25:42 on modular [Seed = 1973853107] ------------------------------------- [1 2] [3 4] >> B := Matrix(IntegerRing(), 2, 2, [1,n, 0,1]); ^ User error: Identifier 'n' has not been declared or assigned >> B; ^ User error: Identifier 'B' has not been declared or assigned [1 2] [3 4] Total time: 0.190 seconds, Total memory usage: 3.24MB '134.219' ************** MAGMA ***************** Host 134.219.148.42 (134.219.148.42) Time: Fri Dec 2 15:22:48 2005 Input: A := Matrix(IntegerRing(), 2, 2, [1,2, 3,4]); A; B := Matrix(IntegerRing(), 2, 2, [1,n, 0,1]); B; Output: Magma V2.11-10 Fri Dec 2 2005 15:22:47 on modular [Seed = 1757120127] ------------------------------------- [1 2] [3 4] >> B := Matrix(IntegerRing(), 2, 2, [1,n, 0,1]); ^ User error: Identifier 'n' has not been declared or assigned >> B;; ^ User error: Identifier 'B' has not been declared or assigned Total time: 0.190 seconds, Total memory usage: 3.24MB '134.219' ************** MAGMA ***************** Host 134.219.148.42 (134.219.148.42) Time: Fri Dec 2 15:22:12 2005 Input: A := Matrix(IntegerRing(), 2, 2, [1,2, 3,4]); A; Output: Magma V2.11-10 Fri Dec 2 2005 15:22:11 on modular [Seed = 1672514704] ------------------------------------- [1 2] [3 4] Total time: 0.190 seconds, Total memory usage: 3.24MB '134.219' ************** MAGMA ***************** Host 134.219.148.42 (134.219.148.42) Time: Fri Dec 2 15:22:07 2005 Input: A := Matrix(IntegerRing(), 2, 2, [1,2, 3,4]); A; Output: Magma V2.11-10 Fri Dec 2 2005 15:22:06 on modular [Seed = 1586344368] ------------------------------------- [1 2] [3 4] Total time: 0.200 seconds, Total memory usage: 3.24MB '134.219' ************** MAGMA ***************** Host 134.219.148.42 (134.219.148.42) Time: Fri Dec 2 15:21:50 2005 Input: X := Matrix(IntegerRing(), 2, 2, [1,2, 3,4]); X; Output: Magma V2.11-10 Fri Dec 2 2005 15:21:50 on modular [Seed = 1502262919] ------------------------------------- [1 2] [3 4] Total time: 0.190 seconds, Total memory usage: 3.24MB '134.219' ************** MAGMA ***************** Host 134.219.148.42 (134.219.148.42) Time: Fri Dec 2 12:35:09 2005 Input: ? matrix linear algebra Output: Magma V2.11-10 Fri Dec 2 2005 12:35:09 on modular [Seed = 382062810] ------------------------------------- 207 matches: 1 O /magma/ring-field-algebra 2 O /system/database/matgps 3 S /magma/combinatorial-geometrical-incidence/graph/\ matrix-vector-space 4 S /magma/group/matrix 5 S /magma/group/matrix/creation-general-linear-group/\ creation-general-matrix-group 6 S /magma/group/matrix/soluble-matrix-group 7 S /magma/module-lattice/general/creation-module/matrix-module 8 S /magma/module-lattice/mat/creation/matrix 9 S /magma/module-lattice/quadratic/form-action-matrix 10 S /magma/module-lattice/smat/creation/matrix 11 S /magma/module-lattice/smat/creation/matrix 12 S /magma/ring-field-algebra/diffrings/related-matrix 13 S /magma/ring-field-algebra/invariant-ring/group-action/matrix 14 S /magma/ring-field-algebra/matrix 15 I /magma/combinatorial-geometrical-incidence/alg_sym/matrices/\ matrices-from-elem/ElementaryToHomogeneousMatrix 16 I /magma/combinatorial-geometrical-incidence/alg_sym/matrices/\ matrices-from-elem/ElementaryToMonomialMatrix 17 I /magma/combinatorial-geometrical-incidence/alg_sym/matrices/\ matrices-from-elem/ElementaryToPowerSumMatrix 18 I /magma/combinatorial-geometrical-incidence/alg_sym/matrices/\ matrices-from-elem/ElementaryToSchurMatrix 19 I /magma/combinatorial-geometrical-incidence/alg_sym/matrices/\ matrices-from-homo/HomogeneousToElementaryMatrix 20 I /magma/combinatorial-geometrical-incidence/alg_sym/matrices/\ matrices-from-homo/HomogeneousToMonomialMatrix 21 I /magma/combinatorial-geometrical-incidence/alg_sym/matrices/\ matrices-from-homo/HomogeneousToPowerSumMatrix 22 I /magma/combinatorial-geometrical-incidence/alg_sym/matrices/\ matrices-from-homo/HomogeneousToSchurMatrix 23 I /magma/combinatorial-geometrical-incidence/alg_sym/matrices/\ matrices-from-monomial/MonomialToElementaryMatrix 24 I /magma/combinatorial-geometrical-incidence/alg_sym/matrices/\ matrices-from-monomial/MonomialToHomogeneousMatrix 25 I /magma/combinatorial-geometrical-incidence/alg_sym/matrices/\ matrices-from-monomial/MonomialToPowerSumMatrix 26 I /magma/combinatorial-geometrical-incidence/alg_sym/matrices/\ matrices-from-monomial/MonomialToSchurMatrix 27 I /magma/combinatorial-geometrical-incidence/alg_sym/matrices/\ matrices-from-power/PowerSumToElementaryMatrix 28 I /magma/combinatorial-geometrical-incidence/alg_sym/matrices/\ matrices-from-power/PowerSumToHomogeneousMatrix 29 I /magma/combinatorial-geometrical-incidence/alg_sym/matrices/\ matrices-from-power/PowerSumToMonomialMatrix 30 I /magma/combinatorial-geometrical-incidence/alg_sym/matrices/\ matrices-from-power/PowerSumToSchurMatrix 31 I /magma/combinatorial-geometrical-incidence/alg_sym/matrices/\ matrices-from-schur/SchurToElementaryMatrix 32 I /magma/combinatorial-geometrical-incidence/alg_sym/matrices/\ matrices-from-schur/SchurToHomogeneousMatrix 33 I /magma/combinatorial-geometrical-incidence/alg_sym/matrices/\ matrices-from-schur/SchurToMonomialMatrix 34 I /magma/combinatorial-geometrical-incidence/alg_sym/matrices/\ matrices-from-schur/SchurToPowerSumMatrix 35 I /magma/combinatorial-geometrical-incidence/\ error-correcting-additive-code/access/code-subspace/\ GeneratorMatrix 36 I /magma/combinatorial-geometrical-incidence/\ error-correcting-additive-code/access/dual-space/ParityCheckMatrix 37 I /magma/combinatorial-geometrical-incidence/\ error-correcting-linear-code/access/code-subspace/GeneratorMatrix 38 I /magma/combinatorial-geometrical-incidence/\ error-correcting-linear-code/access/code-subspace/GeneratorMatrix 39 I /magma/combinatorial-geometrical-incidence/\ error-correcting-linear-code/access/dual-space/ParityCheckMatrix 40 I /magma/combinatorial-geometrical-incidence/\ error-correcting-linear-code/access/GeneratorMatrix 41 I /magma/combinatorial-geometrical-incidence/\ error-correcting-linear-code/access/ParityCheckMatrix 42 I /magma/combinatorial-geometrical-incidence/graph/\ matrix-vector-space/AdjacencyMatrix 43 I /magma/combinatorial-geometrical-incidence/graph/\ matrix-vector-space/DistanceMatrix 44 I /magma/combinatorial-geometrical-incidence/graph/\ matrix-vector-space/IncidenceMatrix 45 I /magma/combinatorial-geometrical-incidence/graph/\ matrix-vector-space/IntersectionMatrix 46 I /magma/combinatorial-geometrical-incidence/graph/\ matrix-vector-space/IntersectionMatrix 47 I /magma/combinatorial-geometrical-incidence/\ incidence-structure-design/invariant/IncidenceMatrix 48 I /magma/combinatorial-geometrical-incidence/\ incidence-structure-hadamard/hadamard-database/Matrix 49 I /magma/combinatorial-geometrical-incidence/\ incidence-structure-hadamard/hadamard/HadamardMatrixFromInteger 50 I /magma/combinatorial-geometrical-incidence/\ incidence-structure-hadamard/hadamard/HadamardMatrixToInteger 51 I /magma/combinatorial-geometrical-incidence/plane/access/\ IncidenceMatrix 52 I /magma/combinatorial-geometrical-incidence/rsgraph/\ resolution-graphs/numerical-data/CartanMatrix 53 I /magma/combinatorial-geometrical-incidence/tableaux-chap/\ tableaux/tableaux-rsk/InverseRSKCorrespondenceMatrix 54 I /magma/geometry/algebraic-curve/curves/curves-attributes/\ HessianMatrix 55 I /magma/geometry/algebraic-curve/curves/curves-attributes/\ JacobianMatrix 56 I /magma/geometry/brandt-modules/ModBrdt:brandt-modules/\ Associated-Structures/GramMatrix 57 I /magma/geometry/brandt-modules/ModBrdt:brandt-modules/\ Associated-Structures/InnerProductMatrix 58 I /magma/geometry/conic/crvgen0-sec:isoms/crvgen0-subsec:isoms/\ ParametrizationMatrix 59 I /magma/geometry/hyperelliptic-curve/analytic/creation-access/\ BigPeriodMatrix 60 I /magma/geometry/hyperelliptic-curve/analytic/creation-access/\ SmallPeriodMatrix 61 I /magma/geometry/hyperelliptic-curve/mordell-weil-heights-hyp/\ HeightPairingMatrix 62 I /magma/geometry/modular-forms/homspace/\ subsec_matrix_and_module_structure/MatrixAlgebra 63 I /magma/geometry/modular-forms/morphisms/subsec_matrix_structure/\ IntegralMatrix 64 I /magma/geometry/modular-forms/morphisms/subsec_matrix_structure/\ IntegralMatrixOverQ 65 I /magma/geometry/modular-forms/morphisms/subsec_matrix_structure/\ Matrix 66 I /magma/geometry/modular-forms/morphisms/subsec_matrix_structure/\ RealMatrix 67 I /magma/geometry/modular-symbol/changing-level/DegeneracyMatrix 68 I /magma/geometry/schemes/maps/maps-automorphisms/\ projective-automorphisms/Matrix 69 I /magma/geometry/schemes/schemes-basic/equations/HessianMatrix 70 I /magma/geometry/schemes/schemes-basic/equations/JacobianMatrix 71 I /magma/group/abelian/conjugate/ClassMatrix 72 I /magma/group/databases-overview/ATLAS-database/\ constructing-representations/MatrixGroup 73 I /magma/group/databases-overview/quatgps/\ QuaternionicMatrixGroupDatabase 74 I /magma/group/databases-overview/ratgps/RationalMatrixGroupDatabase 75 I /magma/group/fiurg/action/rootrefl/ReflectionMatrix 76 I /magma/group/fiurg/create-group-refl-complex/RootSystemMatrix 77 I /magma/group/fiurg/operations/CartanMatrix 78 I /magma/group/fiurg/operations/CoxeterMatrix 79 I /magma/group/group-overview/creation-general/magma/\ PermutationGroup 80 I /magma/group/groups-lie-type/operations/CartanMatrix 81 I /magma/group/groups-lie-type/operations/CoxeterMatrix 82 I /magma/group/matrix/character-representation/ChangeOfBasisMatrix 83 I /magma/group/matrix/creation-general-linear-group/\ creation-general-matrix-group/MatrixGroup 84 I /magma/group/matrix/large-degree/semilinearity/CentralisingMatrix 85 I /magma/group/root-data/operators-root-dtm/CartanMatrix 86 I /magma/group/root-data/operators-root-dtm/CoxeterMatrix 87 I /magma/group/root-data/roots-coroots-weights/rootrefl/\ ReflectionMatrix 88 I /magma/group/soluble/subgroups/subgroup-classes/BurnsideMatrix 89 I /magma/group/soluble/subgroups/subgroup-classes/\ DisplayBurnsideMatrix 90 I /magma/lie-theory/coxeter-description/cartanmat/CartanMatrix 91 I /magma/lie-theory/coxeter-description/cartanmat/CartanMatrix 92 I /magma/lie-theory/coxeter-description/cartanmat/IsCartanMatrix 93 I /magma/lie-theory/coxeter-description/coxetermat/CoxeterMatrix 94 I /magma/lie-theory/coxeter-description/coxetermat/CoxeterMatrix 95 I /magma/lie-theory/coxeter-description/coxetermat/IsCoxeterMatrix 96 I /magma/lie-theory/coxeter-description/finiteaffine/CartanMatrix 97 I /magma/lie-theory/coxeter-description/finiteaffine/CoxeterMatrix 98 I /magma/lie-theory/coxeter-description/finiteaffine/ ** WARNING: Output too long, hence truncated. '134.219' ************** MAGMA ***************** Host 134.219.148.42 (134.219.148.42) Time: Fri Dec 2 12:33:00 2005 Input: ? matrix description ? 14 S Output: Magma V2.11-10 Fri Dec 2 2005 12:33:00 on modular [Seed = 331404638] ------------------------------------- 207 matches: 1 O /magma/ring-field-algebra 2 O /system/database/matgps 3 S /magma/combinatorial-geometrical-incidence/graph/\ matrix-vector-space 4 S /magma/group/matrix 5 S /magma/group/matrix/creation-general-linear-group/\ creation-general-matrix-group 6 S /magma/group/matrix/soluble-matrix-group 7 S /magma/module-lattice/general/creation-module/matrix-module 8 S /magma/module-lattice/mat/creation/matrix 9 S /magma/module-lattice/quadratic/form-action-matrix 10 S /magma/module-lattice/smat/creation/matrix 11 S /magma/module-lattice/smat/creation/matrix 12 S /magma/ring-field-algebra/diffrings/related-matrix 13 S /magma/ring-field-algebra/invariant-ring/group-action/matrix 14 S /magma/ring-field-algebra/matrix 15 I /magma/combinatorial-geometrical-incidence/alg_sym/matrices/\ matrices-from-elem/ElementaryToHomogeneousMatrix 16 I /magma/combinatorial-geometrical-incidence/alg_sym/matrices/\ matrices-from-elem/ElementaryToMonomialMatrix 17 I /magma/combinatorial-geometrical-incidence/alg_sym/matrices/\ matrices-from-elem/ElementaryToPowerSumMatrix 18 I /magma/combinatorial-geometrical-incidence/alg_sym/matrices/\ matrices-from-elem/ElementaryToSchurMatrix 19 I /magma/combinatorial-geometrical-incidence/alg_sym/matrices/\ matrices-from-homo/HomogeneousToElementaryMatrix 20 I /magma/combinatorial-geometrical-incidence/alg_sym/matrices/\ matrices-from-homo/HomogeneousToMonomialMatrix 21 I /magma/combinatorial-geometrical-incidence/alg_sym/matrices/\ matrices-from-homo/HomogeneousToPowerSumMatrix 22 I /magma/combinatorial-geometrical-incidence/alg_sym/matrices/\ matrices-from-homo/HomogeneousToSchurMatrix 23 I /magma/combinatorial-geometrical-incidence/alg_sym/matrices/\ matrices-from-monomial/MonomialToElementaryMatrix 24 I /magma/combinatorial-geometrical-incidence/alg_sym/matrices/\ matrices-from-monomial/MonomialToHomogeneousMatrix 25 I /magma/combinatorial-geometrical-incidence/alg_sym/matrices/\ matrices-from-monomial/MonomialToPowerSumMatrix 26 I /magma/combinatorial-geometrical-incidence/alg_sym/matrices/\ matrices-from-monomial/MonomialToSchurMatrix 27 I /magma/combinatorial-geometrical-incidence/alg_sym/matrices/\ matrices-from-power/PowerSumToElementaryMatrix 28 I /magma/combinatorial-geometrical-incidence/alg_sym/matrices/\ matrices-from-power/PowerSumToHomogeneousMatrix 29 I /magma/combinatorial-geometrical-incidence/alg_sym/matrices/\ matrices-from-power/PowerSumToMonomialMatrix 30 I /magma/combinatorial-geometrical-incidence/alg_sym/matrices/\ matrices-from-power/PowerSumToSchurMatrix 31 I /magma/combinatorial-geometrical-incidence/alg_sym/matrices/\ matrices-from-schur/SchurToElementaryMatrix 32 I /magma/combinatorial-geometrical-incidence/alg_sym/matrices/\ matrices-from-schur/SchurToHomogeneousMatrix 33 I /magma/combinatorial-geometrical-incidence/alg_sym/matrices/\ matrices-from-schur/SchurToMonomialMatrix 34 I /magma/combinatorial-geometrical-incidence/alg_sym/matrices/\ matrices-from-schur/SchurToPowerSumMatrix 35 I /magma/combinatorial-geometrical-incidence/\ error-correcting-additive-code/access/code-subspace/\ GeneratorMatrix 36 I /magma/combinatorial-geometrical-incidence/\ error-correcting-additive-code/access/dual-space/ParityCheckMatrix 37 I /magma/combinatorial-geometrical-incidence/\ error-correcting-linear-code/access/code-subspace/GeneratorMatrix 38 I /magma/combinatorial-geometrical-incidence/\ error-correcting-linear-code/access/code-subspace/GeneratorMatrix 39 I /magma/combinatorial-geometrical-incidence/\ error-correcting-linear-code/access/dual-space/ParityCheckMatrix 40 I /magma/combinatorial-geometrical-incidence/\ error-correcting-linear-code/access/GeneratorMatrix 41 I /magma/combinatorial-geometrical-incidence/\ error-correcting-linear-code/access/ParityCheckMatrix 42 I /magma/combinatorial-geometrical-incidence/graph/\ matrix-vector-space/AdjacencyMatrix 43 I /magma/combinatorial-geometrical-incidence/graph/\ matrix-vector-space/DistanceMatrix 44 I /magma/combinatorial-geometrical-incidence/graph/\ matrix-vector-space/IncidenceMatrix 45 I /magma/combinatorial-geometrical-incidence/graph/\ matrix-vector-space/IntersectionMatrix 46 I /magma/combinatorial-geometrical-incidence/graph/\ matrix-vector-space/IntersectionMatrix 47 I /magma/combinatorial-geometrical-incidence/\ incidence-structure-design/invariant/IncidenceMatrix 48 I /magma/combinatorial-geometrical-incidence/\ incidence-structure-hadamard/hadamard-database/Matrix 49 I /magma/combinatorial-geometrical-incidence/\ incidence-structure-hadamard/hadamard/HadamardMatrixFromInteger 50 I /magma/combinatorial-geometrical-incidence/\ incidence-structure-hadamard/hadamard/HadamardMatrixToInteger 51 I /magma/combinatorial-geometrical-incidence/plane/access/\ IncidenceMatrix 52 I /magma/combinatorial-geometrical-incidence/rsgraph/\ resolution-graphs/numerical-data/CartanMatrix 53 I /magma/combinatorial-geometrical-incidence/tableaux-chap/\ tableaux/tableaux-rsk/InverseRSKCorrespondenceMatrix 54 I /magma/geometry/algebraic-curve/curves/curves-attributes/\ HessianMatrix 55 I /magma/geometry/algebraic-curve/curves/curves-attributes/\ JacobianMatrix 56 I /magma/geometry/brandt-modules/ModBrdt:brandt-modules/\ Associated-Structures/GramMatrix 57 I /magma/geometry/brandt-modules/ModBrdt:brandt-modules/\ Associated-Structures/InnerProductMatrix 58 I /magma/geometry/conic/crvgen0-sec:isoms/crvgen0-subsec:isoms/\ ParametrizationMatrix 59 I /magma/geometry/hyperelliptic-curve/analytic/creation-access/\ BigPeriodMatrix 60 I /magma/geometry/hyperelliptic-curve/analytic/creation-access/\ SmallPeriodMatrix 61 I /magma/geometry/hyperelliptic-curve/mordell-weil-heights-hyp/\ HeightPairingMatrix 62 I /magma/geometry/modular-forms/homspace/\ subsec_matrix_and_module_structure/MatrixAlgebra 63 I /magma/geometry/modular-forms/morphisms/subsec_matrix_structure/\ IntegralMatrix 64 I /magma/geometry/modular-forms/morphisms/subsec_matrix_structure/\ IntegralMatrixOverQ 65 I /magma/geometry/modular-forms/morphisms/subsec_matrix_structure/\ Matrix 66 I /magma/geometry/modular-forms/morphisms/subsec_matrix_structure/\ RealMatrix 67 I /magma/geometry/modular-symbol/changing-level/DegeneracyMatrix 68 I /magma/geometry/schemes/maps/maps-automorphisms/\ projective-automorphisms/Matrix 69 I /magma/geometry/schemes/schemes-basic/equations/HessianMatrix 70 I /magma/geometry/schemes/schemes-basic/equations/JacobianMatrix 71 I /magma/group/abelian/conjugate/ClassMatrix 72 I /magma/group/databases-overview/ATLAS-database/\ constructing-representations/MatrixGroup 73 I /magma/group/databases-overview/quatgps/\ QuaternionicMatrixGroupDatabase 74 I /magma/group/databases-overview/ratgps/RationalMatrixGroupDatabase 75 I /magma/group/fiurg/action/rootrefl/ReflectionMatrix 76 I /magma/group/fiurg/create-group-refl-complex/RootSystemMatrix 77 I /magma/group/fiurg/operations/CartanMatrix 78 I /magma/group/fiurg/operations/CoxeterMatrix 79 I /magma/group/group-overview/creation-general/magma/\ PermutationGroup 80 I /magma/group/groups-lie-type/operations/CartanMatrix 81 I /magma/group/groups-lie-type/operations/CoxeterMatrix 82 I /magma/group/matrix/character-representation/ChangeOfBasisMatrix 83 I /magma/group/matrix/creation-general-linear-group/\ creation-general-matrix-group/MatrixGroup 84 I /magma/group/matrix/large-degree/semilinearity/CentralisingMatrix 85 I /magma/group/root-data/operators-root-dtm/CartanMatrix 86 I /magma/group/root-data/operators-root-dtm/CoxeterMatrix 87 I /magma/group/root-data/roots-coroots-weights/rootrefl/\ ReflectionMatrix 88 I /magma/group/soluble/subgroups/subgroup-classes/BurnsideMatrix 89 I /magma/group/soluble/subgroups/subgroup-classes/\ DisplayBurnsideMatrix 90 I /magma/lie-theory/coxeter-description/cartanmat/CartanMatrix 91 I /magma/lie-theory/coxeter-description/cartanmat/CartanMatrix 92 I /magma/lie-theory/coxeter-description/cartanmat/IsCartanMatrix 93 I /magma/lie-theory/coxeter-description/coxetermat/CoxeterMatrix 94 I /magma/lie-theory/coxeter-description/coxetermat/CoxeterMatrix 95 I /magma/lie-theory/coxeter-description/coxetermat/IsCoxeterMatrix 96 I /magma/lie-theory/coxeter-description/finiteaffine/CartanMatrix 97 I /magma/lie-theory/coxeter-description/finiteaffine/CoxeterMatrix 98 I /magma/lie-theory/coxeter-description/finiteaffine/ ** WARNING: Output too long, hence truncated. '134.219' ************** MAGMA ***************** Host 134.219.148.42 (134.219.148.42) Time: Fri Dec 2 12:32:35 2005 Input: ? matrix description ? 4 S Output: Magma V2.11-10 Fri Dec 2 2005 12:32:35 on modular [Seed = 215992894] ------------------------------------- 207 matches: 1 O /magma/ring-field-algebra 2 O /system/database/matgps 3 S /magma/combinatorial-geometrical-incidence/graph/\ matrix-vector-space 4 S /magma/group/matrix 5 S /magma/group/matrix/creation-general-linear-group/\ creation-general-matrix-group 6 S /magma/group/matrix/soluble-matrix-group 7 S /magma/module-lattice/general/creation-module/matrix-module 8 S /magma/module-lattice/mat/creation/matrix 9 S /magma/module-lattice/quadratic/form-action-matrix 10 S /magma/module-lattice/smat/creation/matrix 11 S /magma/module-lattice/smat/creation/matrix 12 S /magma/ring-field-algebra/diffrings/related-matrix 13 S /magma/ring-field-algebra/invariant-ring/group-action/matrix 14 S /magma/ring-field-algebra/matrix 15 I /magma/combinatorial-geometrical-incidence/alg_sym/matrices/\ matrices-from-elem/ElementaryToHomogeneousMatrix 16 I /magma/combinatorial-geometrical-incidence/alg_sym/matrices/\ matrices-from-elem/ElementaryToMonomialMatrix 17 I /magma/combinatorial-geometrical-incidence/alg_sym/matrices/\ matrices-from-elem/ElementaryToPowerSumMatrix 18 I /magma/combinatorial-geometrical-incidence/alg_sym/matrices/\ matrices-from-elem/ElementaryToSchurMatrix 19 I /magma/combinatorial-geometrical-incidence/alg_sym/matrices/\ matrices-from-homo/HomogeneousToElementaryMatrix 20 I /magma/combinatorial-geometrical-incidence/alg_sym/matrices/\ matrices-from-homo/HomogeneousToMonomialMatrix 21 I /magma/combinatorial-geometrical-incidence/alg_sym/matrices/\ matrices-from-homo/HomogeneousToPowerSumMatrix 22 I /magma/combinatorial-geometrical-incidence/alg_sym/matrices/\ matrices-from-homo/HomogeneousToSchurMatrix 23 I /magma/combinatorial-geometrical-incidence/alg_sym/matrices/\ matrices-from-monomial/MonomialToElementaryMatrix 24 I /magma/combinatorial-geometrical-incidence/alg_sym/matrices/\ matrices-from-monomial/MonomialToHomogeneousMatrix 25 I /magma/combinatorial-geometrical-incidence/alg_sym/matrices/\ matrices-from-monomial/MonomialToPowerSumMatrix 26 I /magma/combinatorial-geometrical-incidence/alg_sym/matrices/\ matrices-from-monomial/MonomialToSchurMatrix 27 I /magma/combinatorial-geometrical-incidence/alg_sym/matrices/\ matrices-from-power/PowerSumToElementaryMatrix 28 I /magma/combinatorial-geometrical-incidence/alg_sym/matrices/\ matrices-from-power/PowerSumToHomogeneousMatrix 29 I /magma/combinatorial-geometrical-incidence/alg_sym/matrices/\ matrices-from-power/PowerSumToMonomialMatrix 30 I /magma/combinatorial-geometrical-incidence/alg_sym/matrices/\ matrices-from-power/PowerSumToSchurMatrix 31 I /magma/combinatorial-geometrical-incidence/alg_sym/matrices/\ matrices-from-schur/SchurToElementaryMatrix 32 I /magma/combinatorial-geometrical-incidence/alg_sym/matrices/\ matrices-from-schur/SchurToHomogeneousMatrix 33 I /magma/combinatorial-geometrical-incidence/alg_sym/matrices/\ matrices-from-schur/SchurToMonomialMatrix 34 I /magma/combinatorial-geometrical-incidence/alg_sym/matrices/\ matrices-from-schur/SchurToPowerSumMatrix 35 I /magma/combinatorial-geometrical-incidence/\ error-correcting-additive-code/access/code-subspace/\ GeneratorMatrix 36 I /magma/combinatorial-geometrical-incidence/\ error-correcting-additive-code/access/dual-space/ParityCheckMatrix 37 I /magma/combinatorial-geometrical-incidence/\ error-correcting-linear-code/access/code-subspace/GeneratorMatrix 38 I /magma/combinatorial-geometrical-incidence/\ error-correcting-linear-code/access/code-subspace/GeneratorMatrix 39 I /magma/combinatorial-geometrical-incidence/\ error-correcting-linear-code/access/dual-space/ParityCheckMatrix 40 I /magma/combinatorial-geometrical-incidence/\ error-correcting-linear-code/access/GeneratorMatrix 41 I /magma/combinatorial-geometrical-incidence/\ error-correcting-linear-code/access/ParityCheckMatrix 42 I /magma/combinatorial-geometrical-incidence/graph/\ matrix-vector-space/AdjacencyMatrix 43 I /magma/combinatorial-geometrical-incidence/graph/\ matrix-vector-space/DistanceMatrix 44 I /magma/combinatorial-geometrical-incidence/graph/\ matrix-vector-space/IncidenceMatrix 45 I /magma/combinatorial-geometrical-incidence/graph/\ matrix-vector-space/IntersectionMatrix 46 I /magma/combinatorial-geometrical-incidence/graph/\ matrix-vector-space/IntersectionMatrix 47 I /magma/combinatorial-geometrical-incidence/\ incidence-structure-design/invariant/IncidenceMatrix 48 I /magma/combinatorial-geometrical-incidence/\ incidence-structure-hadamard/hadamard-database/Matrix 49 I /magma/combinatorial-geometrical-incidence/\ incidence-structure-hadamard/hadamard/HadamardMatrixFromInteger 50 I /magma/combinatorial-geometrical-incidence/\ incidence-structure-hadamard/hadamard/HadamardMatrixToInteger 51 I /magma/combinatorial-geometrical-incidence/plane/access/\ IncidenceMatrix 52 I /magma/combinatorial-geometrical-incidence/rsgraph/\ resolution-graphs/numerical-data/CartanMatrix 53 I /magma/combinatorial-geometrical-incidence/tableaux-chap/\ tableaux/tableaux-rsk/InverseRSKCorrespondenceMatrix 54 I /magma/geometry/algebraic-curve/curves/curves-attributes/\ HessianMatrix 55 I /magma/geometry/algebraic-curve/curves/curves-attributes/\ JacobianMatrix 56 I /magma/geometry/brandt-modules/ModBrdt:brandt-modules/\ Associated-Structures/GramMatrix 57 I /magma/geometry/brandt-modules/ModBrdt:brandt-modules/\ Associated-Structures/InnerProductMatrix 58 I /magma/geometry/conic/crvgen0-sec:isoms/crvgen0-subsec:isoms/\ ParametrizationMatrix 59 I /magma/geometry/hyperelliptic-curve/analytic/creation-access/\ BigPeriodMatrix 60 I /magma/geometry/hyperelliptic-curve/analytic/creation-access/\ SmallPeriodMatrix 61 I /magma/geometry/hyperelliptic-curve/mordell-weil-heights-hyp/\ HeightPairingMatrix 62 I /magma/geometry/modular-forms/homspace/\ subsec_matrix_and_module_structure/MatrixAlgebra 63 I /magma/geometry/modular-forms/morphisms/subsec_matrix_structure/\ IntegralMatrix 64 I /magma/geometry/modular-forms/morphisms/subsec_matrix_structure/\ IntegralMatrixOverQ 65 I /magma/geometry/modular-forms/morphisms/subsec_matrix_structure/\ Matrix 66 I /magma/geometry/modular-forms/morphisms/subsec_matrix_structure/\ RealMatrix 67 I /magma/geometry/modular-symbol/changing-level/DegeneracyMatrix 68 I /magma/geometry/schemes/maps/maps-automorphisms/\ projective-automorphisms/Matrix 69 I /magma/geometry/schemes/schemes-basic/equations/HessianMatrix 70 I /magma/geometry/schemes/schemes-basic/equations/JacobianMatrix 71 I /magma/group/abelian/conjugate/ClassMatrix 72 I /magma/group/databases-overview/ATLAS-database/\ constructing-representations/MatrixGroup 73 I /magma/group/databases-overview/quatgps/\ QuaternionicMatrixGroupDatabase 74 I /magma/group/databases-overview/ratgps/RationalMatrixGroupDatabase 75 I /magma/group/fiurg/action/rootrefl/ReflectionMatrix 76 I /magma/group/fiurg/create-group-refl-complex/RootSystemMatrix 77 I /magma/group/fiurg/operations/CartanMatrix 78 I /magma/group/fiurg/operations/CoxeterMatrix 79 I /magma/group/group-overview/creation-general/magma/\ PermutationGroup 80 I /magma/group/groups-lie-type/operations/CartanMatrix 81 I /magma/group/groups-lie-type/operations/CoxeterMatrix 82 I /magma/group/matrix/character-representation/ChangeOfBasisMatrix 83 I /magma/group/matrix/creation-general-linear-group/\ creation-general-matrix-group/MatrixGroup 84 I /magma/group/matrix/large-degree/semilinearity/CentralisingMatrix 85 I /magma/group/root-data/operators-root-dtm/CartanMatrix 86 I /magma/group/root-data/operators-root-dtm/CoxeterMatrix 87 I /magma/group/root-data/roots-coroots-weights/rootrefl/\ ReflectionMatrix 88 I /magma/group/soluble/subgroups/subgroup-classes/BurnsideMatrix 89 I /magma/group/soluble/subgroups/subgroup-classes/\ DisplayBurnsideMatrix 90 I /magma/lie-theory/coxeter-description/cartanmat/CartanMatrix 91 I /magma/lie-theory/coxeter-description/cartanmat/CartanMatrix 92 I /magma/lie-theory/coxeter-description/cartanmat/IsCartanMatrix 93 I /magma/lie-theory/coxeter-description/coxetermat/CoxeterMatrix 94 I /magma/lie-theory/coxeter-description/coxetermat/CoxeterMatrix 95 I /magma/lie-theory/coxeter-description/coxetermat/IsCoxeterMatrix 96 I /magma/lie-theory/coxeter-description/finiteaffine/CartanMatrix 97 I /magma/lie-theory/coxeter-description/finiteaffine/CoxeterMatrix 98 I /magma/lie-theory/coxeter-description/finiteaffine/ ** WARNING: Output too long, hence truncated. '134.219' ************** MAGMA ***************** Host 134.219.148.42 (134.219.148.42) Time: Fri Dec 2 12:31:56 2005 Input: ? matrix description ? 4 S Output: Magma V2.11-10 Fri Dec 2 2005 12:31:56 on modular [Seed = 148622853] ------------------------------------- 207 matches: 1 O /magma/ring-field-algebra 2 O /system/database/matgps 3 S /magma/combinatorial-geometrical-incidence/graph/\ matrix-vector-space 4 S /magma/group/matrix 5 S /magma/group/matrix/creation-general-linear-group/\ creation-general-matrix-group 6 S /magma/group/matrix/soluble-matrix-group 7 S /magma/module-lattice/general/creation-module/matrix-module 8 S /magma/module-lattice/mat/creation/matrix 9 S /magma/module-lattice/quadratic/form-action-matrix 10 S /magma/module-lattice/smat/creation/matrix 11 S /magma/module-lattice/smat/creation/matrix 12 S /magma/ring-field-algebra/diffrings/related-matrix 13 S /magma/ring-field-algebra/invariant-ring/group-action/matrix 14 S /magma/ring-field-algebra/matrix 15 I /magma/combinatorial-geometrical-incidence/alg_sym/matrices/\ matrices-from-elem/ElementaryToHomogeneousMatrix 16 I /magma/combinatorial-geometrical-incidence/alg_sym/matrices/\ matrices-from-elem/ElementaryToMonomialMatrix 17 I /magma/combinatorial-geometrical-incidence/alg_sym/matrices/\ matrices-from-elem/ElementaryToPowerSumMatrix 18 I /magma/combinatorial-geometrical-incidence/alg_sym/matrices/\ matrices-from-elem/ElementaryToSchurMatrix 19 I /magma/combinatorial-geometrical-incidence/alg_sym/matrices/\ matrices-from-homo/HomogeneousToElementaryMatrix 20 I /magma/combinatorial-geometrical-incidence/alg_sym/matrices/\ matrices-from-homo/HomogeneousToMonomialMatrix 21 I /magma/combinatorial-geometrical-incidence/alg_sym/matrices/\ matrices-from-homo/HomogeneousToPowerSumMatrix 22 I /magma/combinatorial-geometrical-incidence/alg_sym/matrices/\ matrices-from-homo/HomogeneousToSchurMatrix 23 I /magma/combinatorial-geometrical-incidence/alg_sym/matrices/\ matrices-from-monomial/MonomialToElementaryMatrix 24 I /magma/combinatorial-geometrical-incidence/alg_sym/matrices/\ matrices-from-monomial/MonomialToHomogeneousMatrix 25 I /magma/combinatorial-geometrical-incidence/alg_sym/matrices/\ matrices-from-monomial/MonomialToPowerSumMatrix 26 I /magma/combinatorial-geometrical-incidence/alg_sym/matrices/\ matrices-from-monomial/MonomialToSchurMatrix 27 I /magma/combinatorial-geometrical-incidence/alg_sym/matrices/\ matrices-from-power/PowerSumToElementaryMatrix 28 I /magma/combinatorial-geometrical-incidence/alg_sym/matrices/\ matrices-from-power/PowerSumToHomogeneousMatrix 29 I /magma/combinatorial-geometrical-incidence/alg_sym/matrices/\ matrices-from-power/PowerSumToMonomialMatrix 30 I /magma/combinatorial-geometrical-incidence/alg_sym/matrices/\ matrices-from-power/PowerSumToSchurMatrix 31 I /magma/combinatorial-geometrical-incidence/alg_sym/matrices/\ matrices-from-schur/SchurToElementaryMatrix 32 I /magma/combinatorial-geometrical-incidence/alg_sym/matrices/\ matrices-from-schur/SchurToHomogeneousMatrix 33 I /magma/combinatorial-geometrical-incidence/alg_sym/matrices/\ matrices-from-schur/SchurToMonomialMatrix 34 I /magma/combinatorial-geometrical-incidence/alg_sym/matrices/\ matrices-from-schur/SchurToPowerSumMatrix 35 I /magma/combinatorial-geometrical-incidence/\ error-correcting-additive-code/access/code-subspace/\ GeneratorMatrix 36 I /magma/combinatorial-geometrical-incidence/\ error-correcting-additive-code/access/dual-space/ParityCheckMatrix 37 I /magma/combinatorial-geometrical-incidence/\ error-correcting-linear-code/access/code-subspace/GeneratorMatrix 38 I /magma/combinatorial-geometrical-incidence/\ error-correcting-linear-code/access/code-subspace/GeneratorMatrix 39 I /magma/combinatorial-geometrical-incidence/\ error-correcting-linear-code/access/dual-space/ParityCheckMatrix 40 I /magma/combinatorial-geometrical-incidence/\ error-correcting-linear-code/access/GeneratorMatrix 41 I /magma/combinatorial-geometrical-incidence/\ error-correcting-linear-code/access/ParityCheckMatrix 42 I /magma/combinatorial-geometrical-incidence/graph/\ matrix-vector-space/AdjacencyMatrix 43 I /magma/combinatorial-geometrical-incidence/graph/\ matrix-vector-space/DistanceMatrix 44 I /magma/combinatorial-geometrical-incidence/graph/\ matrix-vector-space/IncidenceMatrix 45 I /magma/combinatorial-geometrical-incidence/graph/\ matrix-vector-space/IntersectionMatrix 46 I /magma/combinatorial-geometrical-incidence/graph/\ matrix-vector-space/IntersectionMatrix 47 I /magma/combinatorial-geometrical-incidence/\ incidence-structure-design/invariant/IncidenceMatrix 48 I /magma/combinatorial-geometrical-incidence/\ incidence-structure-hadamard/hadamard-database/Matrix 49 I /magma/combinatorial-geometrical-incidence/\ incidence-structure-hadamard/hadamard/HadamardMatrixFromInteger 50 I /magma/combinatorial-geometrical-incidence/\ incidence-structure-hadamard/hadamard/HadamardMatrixToInteger 51 I /magma/combinatorial-geometrical-incidence/plane/access/\ IncidenceMatrix 52 I /magma/combinatorial-geometrical-incidence/rsgraph/\ resolution-graphs/numerical-data/CartanMatrix 53 I /magma/combinatorial-geometrical-incidence/tableaux-chap/\ tableaux/tableaux-rsk/InverseRSKCorrespondenceMatrix 54 I /magma/geometry/algebraic-curve/curves/curves-attributes/\ HessianMatrix 55 I /magma/geometry/algebraic-curve/curves/curves-attributes/\ JacobianMatrix 56 I /magma/geometry/brandt-modules/ModBrdt:brandt-modules/\ Associated-Structures/GramMatrix 57 I /magma/geometry/brandt-modules/ModBrdt:brandt-modules/\ Associated-Structures/InnerProductMatrix 58 I /magma/geometry/conic/crvgen0-sec:isoms/crvgen0-subsec:isoms/\ ParametrizationMatrix 59 I /magma/geometry/hyperelliptic-curve/analytic/creation-access/\ BigPeriodMatrix 60 I /magma/geometry/hyperelliptic-curve/analytic/creation-access/\ SmallPeriodMatrix 61 I /magma/geometry/hyperelliptic-curve/mordell-weil-heights-hyp/\ HeightPairingMatrix 62 I /magma/geometry/modular-forms/homspace/\ subsec_matrix_and_module_structure/MatrixAlgebra 63 I /magma/geometry/modular-forms/morphisms/subsec_matrix_structure/\ IntegralMatrix 64 I /magma/geometry/modular-forms/morphisms/subsec_matrix_structure/\ IntegralMatrixOverQ 65 I /magma/geometry/modular-forms/morphisms/subsec_matrix_structure/\ Matrix 66 I /magma/geometry/modular-forms/morphisms/subsec_matrix_structure/\ RealMatrix 67 I /magma/geometry/modular-symbol/changing-level/DegeneracyMatrix 68 I /magma/geometry/schemes/maps/maps-automorphisms/\ projective-automorphisms/Matrix 69 I /magma/geometry/schemes/schemes-basic/equations/HessianMatrix 70 I /magma/geometry/schemes/schemes-basic/equations/JacobianMatrix 71 I /magma/group/abelian/conjugate/ClassMatrix 72 I /magma/group/databases-overview/ATLAS-database/\ constructing-representations/MatrixGroup 73 I /magma/group/databases-overview/quatgps/\ QuaternionicMatrixGroupDatabase 74 I /magma/group/databases-overview/ratgps/RationalMatrixGroupDatabase 75 I /magma/group/fiurg/action/rootrefl/ReflectionMatrix 76 I /magma/group/fiurg/create-group-refl-complex/RootSystemMatrix 77 I /magma/group/fiurg/operations/CartanMatrix 78 I /magma/group/fiurg/operations/CoxeterMatrix 79 I /magma/group/group-overview/creation-general/magma/\ PermutationGroup 80 I /magma/group/groups-lie-type/operations/CartanMatrix 81 I /magma/group/groups-lie-type/operations/CoxeterMatrix 82 I /magma/group/matrix/character-representation/ChangeOfBasisMatrix 83 I /magma/group/matrix/creation-general-linear-group/\ creation-general-matrix-group/MatrixGroup 84 I /magma/group/matrix/large-degree/semilinearity/CentralisingMatrix 85 I /magma/group/root-data/operators-root-dtm/CartanMatrix 86 I /magma/group/root-data/operators-root-dtm/CoxeterMatrix 87 I /magma/group/root-data/roots-coroots-weights/rootrefl/\ ReflectionMatrix 88 I /magma/group/soluble/subgroups/subgroup-classes/BurnsideMatrix 89 I /magma/group/soluble/subgroups/subgroup-classes/\ DisplayBurnsideMatrix 90 I /magma/lie-theory/coxeter-description/cartanmat/CartanMatrix 91 I /magma/lie-theory/coxeter-description/cartanmat/CartanMatrix 92 I /magma/lie-theory/coxeter-description/cartanmat/IsCartanMatrix 93 I /magma/lie-theory/coxeter-description/coxetermat/CoxeterMatrix 94 I /magma/lie-theory/coxeter-description/coxetermat/CoxeterMatrix 95 I /magma/lie-theory/coxeter-description/coxetermat/IsCoxeterMatrix 96 I /magma/lie-theory/coxeter-description/finiteaffine/CartanMatrix 97 I /magma/lie-theory/coxeter-description/finiteaffine/CoxeterMatrix 98 I /magma/lie-theory/coxeter-description/finiteaffine/ ** WARNING: Output too long, hence truncated. '134.219' ************** MAGMA ***************** Host 134.219.148.42 (134.219.148.42) Time: Fri Dec 2 12:31:38 2005 Input: ? matrix description Output: Magma V2.11-10 Fri Dec 2 2005 12:31:37 on modular [Seed = 97960701] ------------------------------------- 207 matches: 1 O /magma/ring-field-algebra 2 O /system/database/matgps 3 S /magma/combinatorial-geometrical-incidence/graph/\ matrix-vector-space 4 S /magma/group/matrix 5 S /magma/group/matrix/creation-general-linear-group/\ creation-general-matrix-group 6 S /magma/group/matrix/soluble-matrix-group 7 S /magma/module-lattice/general/creation-module/matrix-module 8 S /magma/module-lattice/mat/creation/matrix 9 S /magma/module-lattice/quadratic/form-action-matrix 10 S /magma/module-lattice/smat/creation/matrix 11 S /magma/module-lattice/smat/creation/matrix 12 S /magma/ring-field-algebra/diffrings/related-matrix 13 S /magma/ring-field-algebra/invariant-ring/group-action/matrix 14 S /magma/ring-field-algebra/matrix 15 I /magma/combinatorial-geometrical-incidence/alg_sym/matrices/\ matrices-from-elem/ElementaryToHomogeneousMatrix 16 I /magma/combinatorial-geometrical-incidence/alg_sym/matrices/\ matrices-from-elem/ElementaryToMonomialMatrix 17 I /magma/combinatorial-geometrical-incidence/alg_sym/matrices/\ matrices-from-elem/ElementaryToPowerSumMatrix 18 I /magma/combinatorial-geometrical-incidence/alg_sym/matrices/\ matrices-from-elem/ElementaryToSchurMatrix 19 I /magma/combinatorial-geometrical-incidence/alg_sym/matrices/\ matrices-from-homo/HomogeneousToElementaryMatrix 20 I /magma/combinatorial-geometrical-incidence/alg_sym/matrices/\ matrices-from-homo/HomogeneousToMonomialMatrix 21 I /magma/combinatorial-geometrical-incidence/alg_sym/matrices/\ matrices-from-homo/HomogeneousToPowerSumMatrix 22 I /magma/combinatorial-geometrical-incidence/alg_sym/matrices/\ matrices-from-homo/HomogeneousToSchurMatrix 23 I /magma/combinatorial-geometrical-incidence/alg_sym/matrices/\ matrices-from-monomial/MonomialToElementaryMatrix 24 I /magma/combinatorial-geometrical-incidence/alg_sym/matrices/\ matrices-from-monomial/MonomialToHomogeneousMatrix 25 I /magma/combinatorial-geometrical-incidence/alg_sym/matrices/\ matrices-from-monomial/MonomialToPowerSumMatrix 26 I /magma/combinatorial-geometrical-incidence/alg_sym/matrices/\ matrices-from-monomial/MonomialToSchurMatrix 27 I /magma/combinatorial-geometrical-incidence/alg_sym/matrices/\ matrices-from-power/PowerSumToElementaryMatrix 28 I /magma/combinatorial-geometrical-incidence/alg_sym/matrices/\ matrices-from-power/PowerSumToHomogeneousMatrix 29 I /magma/combinatorial-geometrical-incidence/alg_sym/matrices/\ matrices-from-power/PowerSumToMonomialMatrix 30 I /magma/combinatorial-geometrical-incidence/alg_sym/matrices/\ matrices-from-power/PowerSumToSchurMatrix 31 I /magma/combinatorial-geometrical-incidence/alg_sym/matrices/\ matrices-from-schur/SchurToElementaryMatrix 32 I /magma/combinatorial-geometrical-incidence/alg_sym/matrices/\ matrices-from-schur/SchurToHomogeneousMatrix 33 I /magma/combinatorial-geometrical-incidence/alg_sym/matrices/\ matrices-from-schur/SchurToMonomialMatrix 34 I /magma/combinatorial-geometrical-incidence/alg_sym/matrices/\ matrices-from-schur/SchurToPowerSumMatrix 35 I /magma/combinatorial-geometrical-incidence/\ error-correcting-additive-code/access/code-subspace/\ GeneratorMatrix 36 I /magma/combinatorial-geometrical-incidence/\ error-correcting-additive-code/access/dual-space/ParityCheckMatrix 37 I /magma/combinatorial-geometrical-incidence/\ error-correcting-linear-code/access/code-subspace/GeneratorMatrix 38 I /magma/combinatorial-geometrical-incidence/\ error-correcting-linear-code/access/code-subspace/GeneratorMatrix 39 I /magma/combinatorial-geometrical-incidence/\ error-correcting-linear-code/access/dual-space/ParityCheckMatrix 40 I /magma/combinatorial-geometrical-incidence/\ error-correcting-linear-code/access/GeneratorMatrix 41 I /magma/combinatorial-geometrical-incidence/\ error-correcting-linear-code/access/ParityCheckMatrix 42 I /magma/combinatorial-geometrical-incidence/graph/\ matrix-vector-space/AdjacencyMatrix 43 I /magma/combinatorial-geometrical-incidence/graph/\ matrix-vector-space/DistanceMatrix 44 I /magma/combinatorial-geometrical-incidence/graph/\ matrix-vector-space/IncidenceMatrix 45 I /magma/combinatorial-geometrical-incidence/graph/\ matrix-vector-space/IntersectionMatrix 46 I /magma/combinatorial-geometrical-incidence/graph/\ matrix-vector-space/IntersectionMatrix 47 I /magma/combinatorial-geometrical-incidence/\ incidence-structure-design/invariant/IncidenceMatrix 48 I /magma/combinatorial-geometrical-incidence/\ incidence-structure-hadamard/hadamard-database/Matrix 49 I /magma/combinatorial-geometrical-incidence/\ incidence-structure-hadamard/hadamard/HadamardMatrixFromInteger 50 I /magma/combinatorial-geometrical-incidence/\ incidence-structure-hadamard/hadamard/HadamardMatrixToInteger 51 I /magma/combinatorial-geometrical-incidence/plane/access/\ IncidenceMatrix 52 I /magma/combinatorial-geometrical-incidence/rsgraph/\ resolution-graphs/numerical-data/CartanMatrix 53 I /magma/combinatorial-geometrical-incidence/tableaux-chap/\ tableaux/tableaux-rsk/InverseRSKCorrespondenceMatrix 54 I /magma/geometry/algebraic-curve/curves/curves-attributes/\ HessianMatrix 55 I /magma/geometry/algebraic-curve/curves/curves-attributes/\ JacobianMatrix 56 I /magma/geometry/brandt-modules/ModBrdt:brandt-modules/\ Associated-Structures/GramMatrix 57 I /magma/geometry/brandt-modules/ModBrdt:brandt-modules/\ Associated-Structures/InnerProductMatrix 58 I /magma/geometry/conic/crvgen0-sec:isoms/crvgen0-subsec:isoms/\ ParametrizationMatrix 59 I /magma/geometry/hyperelliptic-curve/analytic/creation-access/\ BigPeriodMatrix 60 I /magma/geometry/hyperelliptic-curve/analytic/creation-access/\ SmallPeriodMatrix 61 I /magma/geometry/hyperelliptic-curve/mordell-weil-heights-hyp/\ HeightPairingMatrix 62 I /magma/geometry/modular-forms/homspace/\ subsec_matrix_and_module_structure/MatrixAlgebra 63 I /magma/geometry/modular-forms/morphisms/subsec_matrix_structure/\ IntegralMatrix 64 I /magma/geometry/modular-forms/morphisms/subsec_matrix_structure/\ IntegralMatrixOverQ 65 I /magma/geometry/modular-forms/morphisms/subsec_matrix_structure/\ Matrix 66 I /magma/geometry/modular-forms/morphisms/subsec_matrix_structure/\ RealMatrix 67 I /magma/geometry/modular-symbol/changing-level/DegeneracyMatrix 68 I /magma/geometry/schemes/maps/maps-automorphisms/\ projective-automorphisms/Matrix 69 I /magma/geometry/schemes/schemes-basic/equations/HessianMatrix 70 I /magma/geometry/schemes/schemes-basic/equations/JacobianMatrix 71 I /magma/group/abelian/conjugate/ClassMatrix 72 I /magma/group/databases-overview/ATLAS-database/\ constructing-representations/MatrixGroup 73 I /magma/group/databases-overview/quatgps/\ QuaternionicMatrixGroupDatabase 74 I /magma/group/databases-overview/ratgps/RationalMatrixGroupDatabase 75 I /magma/group/fiurg/action/rootrefl/ReflectionMatrix 76 I /magma/group/fiurg/create-group-refl-complex/RootSystemMatrix 77 I /magma/group/fiurg/operations/CartanMatrix 78 I /magma/group/fiurg/operations/CoxeterMatrix 79 I /magma/group/group-overview/creation-general/magma/\ PermutationGroup 80 I /magma/group/groups-lie-type/operations/CartanMatrix 81 I /magma/group/groups-lie-type/operations/CoxeterMatrix 82 I /magma/group/matrix/character-representation/ChangeOfBasisMatrix 83 I /magma/group/matrix/creation-general-linear-group/\ creation-general-matrix-group/MatrixGroup 84 I /magma/group/matrix/large-degree/semilinearity/CentralisingMatrix 85 I /magma/group/root-data/operators-root-dtm/CartanMatrix 86 I /magma/group/root-data/operators-root-dtm/CoxeterMatrix 87 I /magma/group/root-data/roots-coroots-weights/rootrefl/\ ReflectionMatrix 88 I /magma/group/soluble/subgroups/subgroup-classes/BurnsideMatrix 89 I /magma/group/soluble/subgroups/subgroup-classes/\ DisplayBurnsideMatrix 90 I /magma/lie-theory/coxeter-description/cartanmat/CartanMatrix 91 I /magma/lie-theory/coxeter-description/cartanmat/CartanMatrix 92 I /magma/lie-theory/coxeter-description/cartanmat/IsCartanMatrix 93 I /magma/lie-theory/coxeter-description/coxetermat/CoxeterMatrix 94 I /magma/lie-theory/coxeter-description/coxetermat/CoxeterMatrix 95 I /magma/lie-theory/coxeter-description/coxetermat/IsCoxeterMatrix 96 I /magma/lie-theory/coxeter-description/finiteaffine/CartanMatrix 97 I /magma/lie-theory/coxeter-description/finiteaffine/CoxeterMatrix 98 I /magma/lie-theory/coxeter-description/finiteaffine/\ ** WARNING: Output too long, hence truncated. '147.96.' ************** MAGMA ***************** Host 147.96.19.40 (147.96.19.40) Time: Fri Dec 2 08:16:34 2005 Input: P:=PolynomialRing(GF(2),3,"grevlex"); S:=[x*z+y*z+z,x*y+x*z+y*z+x,y+z+1]; GroebnerBasis(S); I:=ideal; S:=Append(S,z^3+z^2+x*z); GroebnerBasis(S); Output: Magma V2.11-10 Fri Dec 2 2005 08:16:33 on modular [Seed = 198705478] ------------------------------------- [ x*z + z, z^2 + z, y + z + 1 ] [ y + 1, z ] Total time: 0.190 seconds, Total memory usage: 3.24MB '147.96.' ************** MAGMA ***************** Host 147.96.19.40 (147.96.19.40) Time: Fri Dec 2 08:15:35 2005 Input: P:=PolynomialRing(GF(2),3,"grevlex"); S:=[x*z+y*z+z,x*y+x*z+y*z+x,y+z+1]; GroebnerBasis(S); I:=ideal; Coordinates(P,z^3+z^2+x*z); Output: Magma V2.11-10 Fri Dec 2 2005 08:15:34 on modular [Seed = 232654471] ------------------------------------- [ x*z + z, z^2 + z, y + z + 1 ] [ z^3 + x*z + z^2 ] Total time: 0.190 seconds, Total memory usage: 3.24MB '147.96.' ************** MAGMA ***************** Host 147.96.19.40 (147.96.19.40) Time: Fri Dec 2 08:15:05 2005 Input: P:=PolynomialRing(GF(2),3,"grevlex"); S:=[x*z+y*z+z,x*y+x*z+y*z+x,y+z+1]; GroebnerBasis(S); I:=ideal; Coordinates(z^3+z^2+x*z,P); Output: Magma V2.11-10 Fri Dec 2 2005 08:15:04 on modular [Seed = 281222068] ------------------------------------- [ x*z + z, z^2 + z, y + z + 1 ] >> Coordinates(z^3+z^2+x*z,P);; ^ Runtime error in 'Coordinates': Bad argument types Argument types given: RngMPolElt, RngMPol Total time: 0.190 seconds, Total memory usage: 3.24MB '147.96.' ************** MAGMA ***************** Host 147.96.19.40 (147.96.19.40) Time: Fri Dec 2 08:14:26 2005 Input: P:=PolynomialRing(GF(2),3,"grevlex"); S:=[x*z+y*z+z,x*y+x*z+y*z+x,y+z+1]; GroebnerBasis(S); I:=ideal; Output: Magma V2.11-10 Fri Dec 2 2005 08:14:25 on modular [Seed = 466096342] ------------------------------------- [ x*z + z, z^2 + z, y + z + 1 ] Total time: 0.190 seconds, Total memory usage: 3.24MB '147.96.' ************** MAGMA ***************** Host 147.96.19.40 (147.96.19.40) Time: Fri Dec 2 08:13:17 2005 Input: P:=PolynomialRing(GF(2),3,"grevlex"); S:=[x*z+y*z+z,x*y+x*z+y*z+x,y+z+1]; GroebnerBasis(S); Output: Magma V2.11-10 Fri Dec 2 2005 08:13:17 on modular [Seed = 516756751] ------------------------------------- [ x*z + z, z^2 + z, y + z + 1 ] Total time: 0.190 seconds, Total memory usage: 3.24MB '159.149' ************** MAGMA ***************** Host 159.149.2.252 (159.149.2.252) Time: Fri Dec 2 05:09:15 2005 Input: a Output: Magma V2.11-10 Fri Dec 2 2005 05:09:14 on modular [Seed = 1502218093] ------------------------------------- >> a; ^ User error: Identifier 'a' has not been declared or assigned Total time: 0.190 seconds, Total memory usage: 3.24MB '60.225.' ************** MAGMA ***************** Host 60.225.131.213 (60.225.131.213) Time: Fri Dec 2 04:09:27 2005 Input: Factorization(2941825781) Output: Magma V2.11-10 Fri Dec 2 2005 04:09:27 on modular [Seed = 398744367] ------------------------------------- [ <1487, 1>, <1978363, 1> ] Total time: 0.190 seconds, Total memory usage: 3.24MB '60.225.' ************** MAGMA ***************** Host 60.225.131.213 (60.225.131.213) Time: Fri Dec 2 04:09:07 2005 Input: Factorization(2941825781) Output: Magma V2.11-10 Fri Dec 2 2005 04:09:06 on modular [Seed = 483348561] ------------------------------------- [ <1487, 1>, <1978363, 1> ] Total time: 0.190 seconds, Total memory usage: 3.24MB '60.225.' ************** MAGMA ***************** Host 60.225.131.213 (60.225.131.213) Time: Fri Dec 2 04:09:01 2005 Input: Factorization(294182578u1) Output: Magma V2.11-10 Fri Dec 2 2005 04:09:00 on modular [Seed = 771645822] ------------------------------------- >> Factorization(294182578u1); ^ User error: bad syntax Total time: 0.190 seconds, Total memory usage: 3.24MB '60.225.' ************** MAGMA ***************** Host 60.225.131.213 (60.225.131.213) Time: Fri Dec 2 04:08:48 2005 Input: Factorization(29418257yyyyyyyyyyyyyyyyyyyyyyyyyyyyyi8u1) Output: Magma V2.11-10 Fri Dec 2 2005 04:08:47 on modular [Seed = 1055748702] ------------------------------------- >> Factorization(29418257yyyyyyyyyyyyyyyyyyyyyyyyyyyyyi8u1); ^ User error: bad syntax Total time: 0.200 seconds, Total memory usage: 3.24MB '60.225.' ************** MAGMA ***************** Host 60.225.131.213 (60.225.131.213) Time: Fri Dec 2 04:08:48 2005 Input: Factorization(29418257yyyyyyyyyyyyyyyyyyyyyyyyyyyyyi8u1) Output: Magma V2.11-10 Fri Dec 2 2005 04:08:47 on modular [Seed = 1100090657] ------------------------------------- >> Factorization(29418257yyyyyyyyyyyyyyyyyyyyyyyyyyyyyi8u1); ^ User error: bad syntax Total time: 0.200 seconds, Total memory usage: 3.24MB '60.225.' ************** MAGMA ***************** Host 60.225.131.213 (60.225.131.213) Time: Fri Dec 2 04:08:47 2005 Input: Factorization(29418257yyyyyyyyyyyyyyyyyyyyyyyyyyyyyi8u1) Output: Magma V2.11-10 Fri Dec 2 2005 04:08:47 on modular [Seed = 1116802594] ------------------------------------- >> Factorization(29418257yyyyyyyyyyyyyyyyyyyyyyyyyyyyyi8u1); ^ User error: bad syntax Total time: 0.200 seconds, Total memory usage: 3.24MB '60.225.' ************** MAGMA ***************** Host 60.225.131.213 (60.225.131.213) Time: Fri Dec 2 04:08:47 2005 Input: Factorization(29418257yyyyyyyyyyyyyyyyyyyyyyyyyyyyyi8u1) Output: Magma V2.11-10 Fri Dec 2 2005 04:08:46 on modular [Seed = 1200884518] ------------------------------------- >> Factorization(29418257yyyyyyyyyyyyyyyyyyyyyyyyyyyyyi8u1); ^ User error: bad syntax Total time: 0.200 seconds, Total memory usage: 3.24MB '60.225.' ************** MAGMA ***************** Host 60.225.131.213 (60.225.131.213) Time: Fri Dec 2 04:08:47 2005 Input: Factorization(29418257yyyyyyyyyyyyyyyyyyyyyyyyyyyyyi8u1) Output: Magma V2.11-10 Fri Dec 2 2005 04:08:46 on modular [Seed = 1218118697] ------------------------------------- >> Factorization(29418257yyyyyyyyyyyyyyyyyyyyyyyyyyyyyi8u1); ^ User error: bad syntax Total time: 0.200 seconds, Total memory usage: 3.24MB '60.225.' ************** MAGMA ***************** Host 60.225.131.213 (60.225.131.213) Time: Fri Dec 2 04:08:47 2005 Input: Factorization(29418257yyyyyyyyyyyyyyyyyyyyyyyyyyyyyi8u1) Output: Magma V2.11-10 Fri Dec 2 2005 04:08:46 on modular [Seed = 1318912559] ------------------------------------- >> Factorization(29418257yyyyyyyyyyyyyyyyyyyyyyyyyyyyyi8u1); ^ User error: bad syntax Total time: 0.190 seconds, Total memory usage: 3.24MB '60.225.' ************** MAGMA ***************** Host 60.225.131.213 (60.225.131.213) Time: Fri Dec 2 04:08:47 2005 Input: Factorization(29418257yyyyyyyyyyyyyyyyyyyyyyyyyyyyyi8u1) Output: Magma V2.11-10 Fri Dec 2 2005 04:08:47 on modular [Seed = 1150748708] ------------------------------------- >> Factorization(29418257yyyyyyyyyyyyyyyyyyyyyyyyyyyyyi8u1); ^ User error: bad syntax Total time: 0.190 seconds, Total memory usage: 3.24MB '60.225.' ************** MAGMA ***************** Host 60.225.131.213 (60.225.131.213) Time: Fri Dec 2 04:08:47 2005 Input: Factorization(29418257yyyyyyyyyyyyyyyyyyyyyyyyyyyyyi8u1) Output: Magma V2.11-10 Fri Dec 2 2005 04:08:46 on modular [Seed = 1400905522] ------------------------------------- >> Factorization(29418257yyyyyyyyyyyyyyyyyyyyyyyyyyyyyi8u1); ^ User error: bad syntax Total time: 0.190 seconds, Total memory usage: 3.24MB '60.225.' ************** MAGMA ***************** Host 60.225.131.213 (60.225.131.213) Time: Fri Dec 2 04:08:47 2005 Input: Factorization(29418257yyyyyyyyyyyyyyyyyyyyyyyyyyyyyi8u1) Output: Magma V2.11-10 Fri Dec 2 2005 04:08:46 on modular [Seed = 1586303551] ------------------------------------- >> Factorization(29418257yyyyyyyyyyyyyyyyyyyyyyyyyyyyyi8u1); ^ User error: bad syntax Total time: 0.190 seconds, Total memory usage: 3.24MB '60.225.' ************** MAGMA ***************** Host 60.225.131.213 (60.225.131.213) Time: Fri Dec 2 04:08:47 2005 Input: Factorization(29418257yyyyyyyyyyyyyyyyyyyyyyyyyyyyyi8u1) Output: Magma V2.11-10 Fri Dec 2 2005 04:08:46 on modular [Seed = 1434851636] ------------------------------------- >> Factorization(29418257yyyyyyyyyyyyyyyyyyyyyyyyyyyyyi8u1); ^ User error: bad syntax Total time: 0.190 seconds, Total memory usage: 3.24MB '60.225.' ************** MAGMA ***************** Host 60.225.131.213 (60.225.131.213) Time: Fri Dec 2 04:08:47 2005 Input: Factorization(29418257yyyyyyyyyyyyyyyyyyyyyyyyyyyyyi8u1) Output: Magma V2.11-10 Fri Dec 2 2005 04:08:46 on modular [Seed = 1485509689] ------------------------------------- >> Factorization(29418257yyyyyyyyyyyyyyyyyyyyyyyyyyyyyi8u1); ^ User error: bad syntax Total time: 0.200 seconds, Total memory usage: 3.24MB '60.225.' ************** MAGMA ***************** Host 60.225.131.213 (60.225.131.213) Time: Fri Dec 2 04:08:46 2005 Input: Factorization(29418257yyyyyyyyyyyyyyyyyyyyyyyyyyyyyi8u1) Output: Magma V2.11-10 Fri Dec 2 2005 04:08:46 on modular [Seed = 1518933563] ------------------------------------- >> Factorization(29418257yyyyyyyyyyyyyyyyyyyyyyyyyyyyyi8u1); ^ User error: bad syntax Total time: 0.200 seconds, Total memory usage: 3.24MB '60.225.' ************** MAGMA ***************** Host 60.225.131.213 (60.225.131.213) Time: Fri Dec 2 04:08:46 2005 Input: Factorization(29418257yyyyyyyyyyyyyyyyyyyyyyyyyyyyyi8u1) Output: Magma V2.11-10 Fri Dec 2 2005 04:08:45 on modular [Seed = 1706436871] ------------------------------------- >> Factorization(29418257yyyyyyyyyyyyyyyyyyyyyyyyyyyyyi8u1); ^ User error: bad syntax Total time: 0.180 seconds, Total memory usage: 3.24MB '60.225.' ************** MAGMA ***************** Host 60.225.131.213 (60.225.131.213) Time: Fri Dec 2 04:08:45 2005 Input: Factorization(29418257yyyyyyyyyyyyyyyyyyyyyyyyyyyyyi8u1) Output: Magma V2.11-10 Fri Dec 2 2005 04:08:45 on modular [Seed = 1773806859] ------------------------------------- >> Factorization(29418257yyyyyyyyyyyyyyyyyyyyyyyyyyyyyi8u1); ^ User error: bad syntax Total time: 0.190 seconds, Total memory usage: 3.24MB '60.225.' ************** MAGMA ***************** Host 60.225.131.213 (60.225.131.213) Time: Fri Dec 2 04:08:10 2005 Input: Factorization(21) Output: Magma V2.11-10 Fri Dec 2 2005 04:08:10 on modular [Seed = 1874600682] ------------------------------------- [ <3, 1>, <7, 1> ] Total time: 0.180 seconds, Total memory usage: 3.24MB '60.225.' ************** MAGMA ***************** Host 60.225.131.213 (60.225.131.213) Time: Fri Dec 2 03:53:44 2005 Input: Factorization(2795413680) Output: Magma V2.11-10 Fri Dec 2 2005 03:53:43 on modular [Seed = 3708720434] ------------------------------------- [ <2, 4>, <3, 5>, <5, 1>, <143797, 1> ] Total time: 0.180 seconds, Total memory usage: 3.24MB '60.225.' ************** MAGMA ***************** Host 60.225.131.213 (60.225.131.213) Time: Fri Dec 2 03:53:18 2005 Input: Factorization 2795413680 Output: Magma V2.11-10 Fri Dec 2 2005 03:53:18 on modular [Seed = 3624636638] ------------------------------------- >> Factorization 2795413680; ^ User error: bad syntax Total time: 0.190 seconds, Total memory usage: 3.24MB '130.130' ************** MAGMA ***************** Host 130.130.37.6 (130.130.37.6) Time: Fri Dec 2 00:08:46 2005 Input: Pi Output: Magma V2.11-10 Fri Dec 2 2005 00:08:46 on modular [Seed = 3128307486] ------------------------------------- Intrinsic 'Pi' Signatures: ( R) -> FldReElt ( R) -> FldReElt ( R) -> FldPrElt ( R) -> NFldReElt Create pi to the precision of R as a real number. ( R, p) -> FldPrElt Create pi to the precision p as an element of R. Total time: 0.190 seconds, Total memory usage: 3.24MB '139.182' ************** MAGMA ***************** Host 139.182.146.132 (139.182.146.132) Time: Thu Dec 1 20:01:46 2005 Input: G:=Group; H:=Group; f,G1,k:=CosetAction(G,sub); f,H1,k:=CosetAction(H,sub); IsIsomorphic(G1,H1); Output: Magma V2.11-10 Thu Dec 1 2005 20:01:46 on modular [Seed = 652933545] ------------------------------------- false Total time: 0.230 seconds, Total memory usage: 18.57MB '139.182' ************** MAGMA ***************** Host 139.182.146.132 (139.182.146.132) Time: Thu Dec 1 20:00:52 2005 Input: G:=Group; H:=Group; f,G1,k:=CosetAction(G,sub); f,H1,k:=CosetAction(H,sub); IsIsomorphic(G1,H1); Output: Magma V2.11-10 Thu Dec 1 2005 20:00:52 on modular [Seed = 821882753] ------------------------------------- false Total time: 0.240 seconds, Total memory usage: 18.57MB '139.182' ************** MAGMA ***************** Host 139.182.146.132 (139.182.146.132) Time: Thu Dec 1 19:59:10 2005 Input: G:=Group; H:=Group; f,G1,k:=CosetAction(G,sub); f,H1,k:=CosetAction(H,sub); IsIsomorphic(G1,H1); Output: Magma V2.11-10 Thu Dec 1 2005 19:59:10 on modular [Seed = 2336707747] ------------------------------------- false Total time: 0.240 seconds, Total memory usage: 18.57MB '139.182' ************** MAGMA ***************** Host 139.182.146.132 (139.182.146.132) Time: Thu Dec 1 19:58:47 2005 Input: G:=Group; H:=Group; Output: Magma V2.11-10 Thu Dec 1 2005 19:58:47 on modular [Seed = 2386710975] ------------------------------------- Total time: 0.190 seconds, Total memory usage: 3.24MB '139.182' ************** MAGMA ***************** Host 139.182.146.132 (139.182.146.132) Time: Thu Dec 1 19:48:20 2005 Input: G:=Group; H:=Group; f,G1,k:=CosetAction(G,sub); f,H1,k:=CosetAction(H,sub); IsIsomorphic(G1,H1); Output: Magma V2.11-10 Thu Dec 1 2005 19:48:19 on modular [Seed = 2572501820] ------------------------------------- true Homomorphism of GrpPerm: G1, Degree 100, Order 2^2 * 5^2 into GrpPerm: H1, Degree 100, Order 2^2 * 5^2 induced by (1, 2, 6, 13, 21, 29, 37, 45, 53, 61, 69, 77, 85, 92, 84, 76, 68, 60, 52, 44, 36, 28, 20, 12, 5)(3, 7, 14, 22, 30, 38, 46, 54, 62, 70, 78, 86, 93, 97, 90, 82, 74, 66, 58, 50, 42, 34, 26, 18, 10)(4, 11, 19, 27, 35, 43, 51, 59, 67, 75, 83, 91, 98, 94, 87, 79, 71, 63, 55, 47, 39, 31, 23, 15, 8)(9, 17, 25, 33, 41, 49, 57, 65, 73, 81, 89, 96, 100, 99, 95, 88, 80, 72, 64, 56, 48, 40, 32, 24, 16) |--> (1, 21, 53, 85, 68, 36, 5, 13, 45, 77, 76, 44, 12, 6, 37, 69, 84, 52, 20, 2, 29, 61, 92, 60, 28)(3, 33, 65, 96, 70, 38, 7, 25, 57, 89, 78, 46, 14, 17, 49, 81, 86, 54, 22, 9, 41, 73, 93, 62, 30)(4, 35, 67, 98, 71, 39, 8, 27, 59, 91, 79, 47, 15, 19, 51, 83, 87, 55, 23, 11, 43, 75, 94, 63, 31)(10, 40, 72, 99, 82, 50, 18, 32, 64, 95, 90, 58, 26, 24, 56, 88, 97, 66, 34, 16, 48, 80, 100, 74, 42) (1, 3)(2, 7)(4, 9)(5, 10)(6, 14)(8, 16)(11, 17)(12, 18)(13, 22)(15, 24)(19, 25)(20, 26)(21, 30)(23, 32)(27, 33)(28, 34)(29, 38)(31, 40)(35, 41)(36, 42)(37, 46)(39, 48)(43, 49)(44, 50)(45, 54)(47, 56)(51, 57)(52, 58)(53, 62)(55, 64)(59, 65)(60, 66)(61, 70)(63, 72)(67, 73)(68, 74)(69, 78)(71, 80)(75, 81)(76, 82)(77, 86)(79, 88)(83, 89)(84, 90)(85, 93)(87, 95)(91, 96)(92, 97)(94, 99)(98, 100) |--> (1, 10)(2, 16)(3, 4)(5, 18)(6, 24)(7, 8)(9, 11)(12, 26)(13, 32)(14, 15)(17, 19)(20, 34)(21, 40)(22, 23)(25, 27)(28, 42)(29, 48)(30, 31)(33, 35)(36, 50)(37, 56)(38, 39)(41, 43)(44, 58)(45, 64)(46, 47)(49, 51)(52, 66)(53, 72)(54, 55)(57, 59)(60, 74)(61, 80)(62, 63)(65, 67)(68, 82)(69, 88)(70, 71)(73, 75)(76, 90)(77, 95)(78, 79)(81, 83)(84, 97)(85, 99)(86, 87)(89, 91)(92, 100)(93, 94)(96, 98) (1, 4)(2, 8)(3, 9)(5, 11)(6, 15)(7, 16)(10, 17)(12, 19)(13, 23)(14, 24)(18, 25)(20, 27)(21, 31)(22, 32)(26, 33)(28, 35)(29, 39)(30, 40)(34, 41)(36, 43)(37, 47)(38, 48)(42, 49)(44, 51)(45, 55)(46, 56)(50, 57)(52, 59)(53, 63)(54, 64)(58, 65)(60, 67)(61, 71)(62, 72)(66, 73)(68, 75)(69, 79)(70, 80)(74, 81)(76, 83)(77, 87)(78, 88)(82, 89)(84, 91)(85, 94)(86, 95)(90, 96)(92, 98)(93, 99)(97, 100) |--> (1, 3)(2, 7)(4, 10)(5, 9)(6, 14)(8, 16)(11, 18)(12, 17)(13, 22)(15, 24)(19, 26)(20, 25)(21, 30)(23, 32)(27, 34)(28, 33)(29, 38)(31, 40)(35, 42)(36, 41)(37, 46)(39, 48)(43, 50)(44, 49)(45, 54)(47, 56)(51, 58)(52, 57)(53, 62)(55, 64)(59, 66)(60, 65)(61, 70)(63, 72)(67, 74)(68, 73)(69, 78)(71, 80)(75, 82)(76, 81)(77, 86)(79, 88)(83, 90)(84, 89)(85, 93)(87, 95)(91, 97)(92, 96)(94, 99)(98, 100) Total time: 0.300 seconds, Total memory usage: 18.57MB '139.182' ************** MAGMA ***************** Host 139.182.146.132 (139.182.146.132) Time: Thu Dec 1 19:48:12 2005 Input: G:=Group; H:=Group; f,G1,k:=CosetAction(G,sub); f,H1,k:=CosetAction(H,sub); IsIsomorphic(G1,H1); Output: Magma V2.11-10 Thu Dec 1 2005 19:48:12 on modular [Seed = 2622506534] ------------------------------------- true Homomorphism of GrpPerm: G1, Degree 100, Order 2^2 * 5^2 into GrpPerm: H1, Degree 100, Order 2^2 * 5^2 induced by (1, 2, 6, 13, 21, 29, 37, 45, 53, 61, 69, 77, 85, 92, 84, 76, 68, 60, 52, 44, 36, 28, 20, 12, 5)(3, 7, 14, 22, 30, 38, 46, 54, 62, 70, 78, 86, 93, 97, 90, 82, 74, 66, 58, 50, 42, 34, 26, 18, 10)(4, 11, 19, 27, 35, 43, 51, 59, 67, 75, 83, 91, 98, 94, 87, 79, 71, 63, 55, 47, 39, 31, 23, 15, 8)(9, 17, 25, 33, 41, 49, 57, 65, 73, 81, 89, 96, 100, 99, 95, 88, 80, 72, 64, 56, 48, 40, 32, 24, 16) |--> (1, 21, 53, 85, 68, 36, 5, 13, 45, 77, 76, 44, 12, 6, 37, 69, 84, 52, 20, 2, 29, 61, 92, 60, 28)(3, 33, 65, 96, 70, 38, 7, 25, 57, 89, 78, 46, 14, 17, 49, 81, 86, 54, 22, 9, 41, 73, 93, 62, 30)(4, 35, 67, 98, 71, 39, 8, 27, 59, 91, 79, 47, 15, 19, 51, 83, 87, 55, 23, 11, 43, 75, 94, 63, 31)(10, 40, 72, 99, 82, 50, 18, 32, 64, 95, 90, 58, 26, 24, 56, 88, 97, 66, 34, 16, 48, 80, 100, 74, 42) (1, 3)(2, 7)(4, 9)(5, 10)(6, 14)(8, 16)(11, 17)(12, 18)(13, 22)(15, 24)(19, 25)(20, 26)(21, 30)(23, 32)(27, 33)(28, 34)(29, 38)(31, 40)(35, 41)(36, 42)(37, 46)(39, 48)(43, 49)(44, 50)(45, 54)(47, 56)(51, 57)(52, 58)(53, 62)(55, 64)(59, 65)(60, 66)(61, 70)(63, 72)(67, 73)(68, 74)(69, 78)(71, 80)(75, 81)(76, 82)(77, 86)(79, 88)(83, 89)(84, 90)(85, 93)(87, 95)(91, 96)(92, 97)(94, 99)(98, 100) |--> (1, 10)(2, 16)(3, 4)(5, 18)(6, 24)(7, 8)(9, 11)(12, 26)(13, 32)(14, 15)(17, 19)(20, 34)(21, 40)(22, 23)(25, 27)(28, 42)(29, 48)(30, 31)(33, 35)(36, 50)(37, 56)(38, 39)(41, 43)(44, 58)(45, 64)(46, 47)(49, 51)(52, 66)(53, 72)(54, 55)(57, 59)(60, 74)(61, 80)(62, 63)(65, 67)(68, 82)(69, 88)(70, 71)(73, 75)(76, 90)(77, 95)(78, 79)(81, 83)(84, 97)(85, 99)(86, 87)(89, 91)(92, 100)(93, 94)(96, 98) (1, 4)(2, 8)(3, 9)(5, 11)(6, 15)(7, 16)(10, 17)(12, 19)(13, 23)(14, 24)(18, 25)(20, 27)(21, 31)(22, 32)(26, 33)(28, 35)(29, 39)(30, 40)(34, 41)(36, 43)(37, 47)(38, 48)(42, 49)(44, 51)(45, 55)(46, 56)(50, 57)(52, 59)(53, 63)(54, 64)(58, 65)(60, 67)(61, 71)(62, 72)(66, 73)(68, 75)(69, 79)(70, 80)(74, 81)(76, 83)(77, 87)(78, 88)(82, 89)(84, 91)(85, 94)(86, 95)(90, 96)(92, 98)(93, 99)(97, 100) |--> (1, 3)(2, 7)(4, 10)(5, 9)(6, 14)(8, 16)(11, 18)(12, 17)(13, 22)(15, 24)(19, 26)(20, 25)(21, 30)(23, 32)(27, 34)(28, 33)(29, 38)(31, 40)(35, 42)(36, 41)(37, 46)(39, 48)(43, 50)(44, 49)(45, 54)(47, 56)(51, 58)(52, 57)(53, 62)(55, 64)(59, 66)(60, 65)(61, 70)(63, 72)(67, 74)(68, 73)(69, 78)(71, 80)(75, 82)(76, 81)(77, 86)(79, 88)(83, 90)(84, 89)(85, 93)(87, 95)(91, 97)(92, 96)(94, 99)(98, 100) Total time: 0.300 seconds, Total memory usage: 18.57MB '139.182' ************** MAGMA ***************** Host 139.182.146.132 (139.182.146.132) Time: Thu Dec 1 19:48:06 2005 Input: G:=Group; H:=Group; f,G1,k:=CosetAction(G,sub); f,H1,k:=CosetAction(H,sub); IsIsomorphic(G1,H1); Output: Magma V2.11-10 Thu Dec 1 2005 19:48:05 on modular [Seed = 2472495395] ------------------------------------- true Homomorphism of GrpPerm: G1, Degree 100, Order 2^2 * 5^2 into GrpPerm: H1, Degree 100, Order 2^2 * 5^2 induced by (1, 2, 6, 13, 21, 29, 37, 45, 53, 61, 69, 77, 85, 92, 84, 76, 68, 60, 52, 44, 36, 28, 20, 12, 5)(3, 7, 14, 22, 30, 38, 46, 54, 62, 70, 78, 86, 93, 97, 90, 82, 74, 66, 58, 50, 42, 34, 26, 18, 10)(4, 11, 19, 27, 35, 43, 51, 59, 67, 75, 83, 91, 98, 94, 87, 79, 71, 63, 55, 47, 39, 31, 23, 15, 8)(9, 17, 25, 33, 41, 49, 57, 65, 73, 81, 89, 96, 100, 99, 95, 88, 80, 72, 64, 56, 48, 40, 32, 24, 16) |--> (1, 21, 53, 85, 68, 36, 5, 13, 45, 77, 76, 44, 12, 6, 37, 69, 84, 52, 20, 2, 29, 61, 92, 60, 28)(3, 33, 65, 96, 70, 38, 7, 25, 57, 89, 78, 46, 14, 17, 49, 81, 86, 54, 22, 9, 41, 73, 93, 62, 30)(4, 35, 67, 98, 71, 39, 8, 27, 59, 91, 79, 47, 15, 19, 51, 83, 87, 55, 23, 11, 43, 75, 94, 63, 31)(10, 40, 72, 99, 82, 50, 18, 32, 64, 95, 90, 58, 26, 24, 56, 88, 97, 66, 34, 16, 48, 80, 100, 74, 42) (1, 3)(2, 7)(4, 9)(5, 10)(6, 14)(8, 16)(11, 17)(12, 18)(13, 22)(15, 24)(19, 25)(20, 26)(21, 30)(23, 32)(27, 33)(28, 34)(29, 38)(31, 40)(35, 41)(36, 42)(37, 46)(39, 48)(43, 49)(44, 50)(45, 54)(47, 56)(51, 57)(52, 58)(53, 62)(55, 64)(59, 65)(60, 66)(61, 70)(63, 72)(67, 73)(68, 74)(69, 78)(71, 80)(75, 81)(76, 82)(77, 86)(79, 88)(83, 89)(84, 90)(85, 93)(87, 95)(91, 96)(92, 97)(94, 99)(98, 100) |--> (1, 10)(2, 16)(3, 4)(5, 18)(6, 24)(7, 8)(9, 11)(12, 26)(13, 32)(14, 15)(17, 19)(20, 34)(21, 40)(22, 23)(25, 27)(28, 42)(29, 48)(30, 31)(33, 35)(36, 50)(37, 56)(38, 39)(41, 43)(44, 58)(45, 64)(46, 47)(49, 51)(52, 66)(53, 72)(54, 55)(57, 59)(60, 74)(61, 80)(62, 63)(65, 67)(68, 82)(69, 88)(70, 71)(73, 75)(76, 90)(77, 95)(78, 79)(81, 83)(84, 97)(85, 99)(86, 87)(89, 91)(92, 100)(93, 94)(96, 98) (1, 4)(2, 8)(3, 9)(5, 11)(6, 15)(7, 16)(10, 17)(12, 19)(13, 23)(14, 24)(18, 25)(20, 27)(21, 31)(22, 32)(26, 33)(28, 35)(29, 39)(30, 40)(34, 41)(36, 43)(37, 47)(38, 48)(42, 49)(44, 51)(45, 55)(46, 56)(50, 57)(52, 59)(53, 63)(54, 64)(58, 65)(60, 67)(61, 71)(62, 72)(66, 73)(68, 75)(69, 79)(70, 80)(74, 81)(76, 83)(77, 87)(78, 88)(82, 89)(84, 91)(85, 94)(86, 95)(90, 96)(92, 98)(93, 99)(97, 100) |--> (1, 3)(2, 7)(4, 10)(5, 9)(6, 14)(8, 16)(11, 18)(12, 17)(13, 22)(15, 24)(19, 26)(20, 25)(21, 30)(23, 32)(27, 34)(28, 33)(29, 38)(31, 40)(35, 42)(36, 41)(37, 46)(39, 48)(43, 50)(44, 49)(45, 54)(47, 56)(51, 58)(52, 57)(53, 62)(55, 64)(59, 66)(60, 65)(61, 70)(63, 72)(67, 74)(68, 73)(69, 78)(71, 80)(75, 82)(76, 81)(77, 86)(79, 88)(83, 90)(84, 89)(85, 93)(87, 95)(91, 97)(92, 96)(94, 99)(98, 100) Total time: 0.300 seconds, Total memory usage: 18.57MB '139.182' ************** MAGMA ***************** Host 139.182.146.132 (139.182.146.132) Time: Thu Dec 1 19:44:44 2005 Input: G:=Group; H:=Group; f,G1,k:=CosetAction(G,sub); f,H1,k:=CosetAction(H,sub); IsIsomorphic(G1,H1); Output: Magma V2.11-10 Thu Dec 1 2005 19:44:43 on modular [Seed = 2841997860] ------------------------------------- true Homomorphism of GrpPerm: G1, Degree 100, Order 2^2 * 5^2 into GrpPerm: H1, Degree 100, Order 2^2 * 5^2 induced by (1, 2, 6, 13, 21, 29, 37, 45, 53, 61, 69, 77, 85, 92, 84, 76, 68, 60, 52, 44, 36, 28, 20, 12, 5)(3, 7, 14, 22, 30, 38, 46, 54, 62, 70, 78, 86, 93, 97, 90, 82, 74, 66, 58, 50, 42, 34, 26, 18, 10)(4, 11, 19, 27, 35, 43, 51, 59, 67, 75, 83, 91, 98, 94, 87, 79, 71, 63, 55, 47, 39, 31, 23, 15, 8)(9, 17, 25, 33, 41, 49, 57, 65, 73, 81, 89, 96, 100, 99, 95, 88, 80, 72, 64, 56, 48, 40, 32, 24, 16) |--> (1, 21, 53, 85, 68, 36, 5, 13, 45, 77, 76, 44, 12, 6, 37, 69, 84, 52, 20, 2, 29, 61, 92, 60, 28)(3, 33, 65, 96, 70, 38, 7, 25, 57, 89, 78, 46, 14, 17, 49, 81, 86, 54, 22, 9, 41, 73, 93, 62, 30)(4, 31, 63, 94, 75, 43, 11, 23, 55, 87, 83, 51, 19, 15, 47, 79, 91, 59, 27, 8, 39, 71, 98, 67, 35)(10, 42, 74, 100, 80, 48, 16, 34, 66, 97, 88, 56, 24, 26, 58, 90, 95, 64, 32, 18, 50, 82, 99, 72, 40) (1, 3)(2, 7)(4, 9)(5, 10)(6, 14)(8, 16)(11, 17)(12, 18)(13, 22)(15, 24)(19, 25)(20, 26)(21, 30)(23, 32)(27, 33)(28, 34)(29, 38)(31, 40)(35, 41)(36, 42)(37, 46)(39, 48)(43, 49)(44, 50)(45, 54)(47, 56)(51, 57)(52, 58)(53, 62)(55, 64)(59, 65)(60, 66)(61, 70)(63, 72)(67, 73)(68, 74)(69, 78)(71, 80)(75, 81)(76, 82)(77, 86)(79, 88)(83, 89)(84, 90)(85, 93)(87, 95)(91, 96)(92, 97)(94, 99)(98, 100) |--> (1, 4)(2, 8)(3, 10)(5, 11)(6, 15)(7, 16)(9, 18)(12, 19)(13, 23)(14, 24)(17, 26)(20, 27)(21, 31)(22, 32)(25, 34)(28, 35)(29, 39)(30, 40)(33, 42)(36, 43)(37, 47)(38, 48)(41, 50)(44, 51)(45, 55)(46, 56)(49, 58)(52, 59)(53, 63)(54, 64)(57, 66)(60, 67)(61, 71)(62, 72)(65, 74)(68, 75)(69, 79)(70, 80)(73, 82)(76, 83)(77, 87)(78, 88)(81, 90)(84, 91)(85, 94)(86, 95)(89, 97)(92, 98)(93, 99)(96, 100) (1, 4)(2, 8)(3, 9)(5, 11)(6, 15)(7, 16)(10, 17)(12, 19)(13, 23)(14, 24)(18, 25)(20, 27)(21, 31)(22, 32)(26, 33)(28, 35)(29, 39)(30, 40)(34, 41)(36, 43)(37, 47)(38, 48)(42, 49)(44, 51)(45, 55)(46, 56)(50, 57)(52, 59)(53, 63)(54, 64)(58, 65)(60, 67)(61, 71)(62, 72)(66, 73)(68, 75)(69, 79)(70, 80)(74, 81)(76, 83)(77, 87)(78, 88)(82, 89)(84, 91)(85, 94)(86, 95)(90, 96)(92, 98)(93, 99)(97, 100) |--> (1, 3)(2, 7)(4, 10)(5, 9)(6, 14)(8, 16)(11, 18)(12, 17)(13, 22)(15, 24)(19, 26)(20, 25)(21, 30)(23, 32)(27, 34)(28, 33)(29, 38)(31, 40)(35, 42)(36, 41)(37, 46)(39, 48)(43, 50)(44, 49)(45, 54)(47, 56)(51, 58)(52, 57)(53, 62)(55, 64)(59, 66)(60, 65)(61, 70)(63, 72)(67, 74)(68, 73)(69, 78)(71, 80)(75, 82)(76, 81)(77, 86)(79, 88)(83, 90)(84, 89)(85, 93)(87, 95)(91, 97)(92, 96)(94, 99)(98, 100) Total time: 0.290 seconds, Total memory usage: 18.57MB '139.182' ************** MAGMA ***************** Host 139.182.146.132 (139.182.146.132) Time: Thu Dec 1 19:43:11 2005 Input: G:=Group; H:=Group; f,G1,k:=CosetAction(G,sub); f,H1,k:=CosetAction(H,sub); IsIsomorphic(G1,H1); Output: Magma V2.11-10 Thu Dec 1 2005 19:43:11 on modular [Seed = 2892003141] ------------------------------------- false Total time: 0.250 seconds, Total memory usage: 18.57MB '139.182' ************** MAGMA ***************** Host 139.182.146.132 (139.182.146.132) Time: Thu Dec 1 19:43:07 2005 Input: G:=Group; H:=Group; f,G1,k:=CosetAction(G,sub); f,H1,k:=CosetAction(H,sub); IsIsomorphic(G1,H1); Output: Magma V2.11-10 Thu Dec 1 2005 19:43:07 on modular [Seed = 2741991502] ------------------------------------- false Total time: 0.260 seconds, Total memory usage: 18.57MB '139.182' ************** MAGMA ***************** Host 139.182.146.132 (139.182.146.132) Time: Thu Dec 1 19:40:14 2005 Input: G:=Group; H:=Group; f,G1,k:=CosetAction(G,sub); f,H1,k:=CosetAction(H,sub); IsIsomorphic(G1,H1); Output: Magma V2.11-10 Thu Dec 1 2005 19:40:14 on modular [Seed = 3111480609] ------------------------------------- false Total time: 0.260 seconds, Total memory usage: 18.57MB '139.182' ************** MAGMA ***************** Host 139.182.146.132 (139.182.146.132) Time: Thu Dec 1 19:37:15 2005 Input: G:=Group; H:=Group; f,G1,k:=CosetAction(G,sub); f,H1,k:=CosetAction(H,sub); IsIsomorphic(G1,H1); Output: Magma V2.11-10 Thu Dec 1 2005 19:37:14 on modular [Seed = 3389398662] ------------------------------------- false Total time: 0.260 seconds, Total memory usage: 18.57MB '139.182' ************** MAGMA ***************** Host 139.182.146.132 (139.182.146.132) Time: Thu Dec 1 19:35:14 2005 Input: G:=Group; Order(G) Output: Magma V2.11-10 Thu Dec 1 2005 19:35:14 on modular [Seed = 3222017396] ------------------------------------- 100 Total time: 0.220 seconds, Total memory usage: 18.57MB '139.182' ************** MAGMA ***************** Host 139.182.146.132 (139.182.146.132) Time: Thu Dec 1 19:32:40 2005 Input: G:=Group; Order(G) Output: Magma V2.11-10 Thu Dec 1 2005 19:32:40 on modular [Seed = 3339391641] ------------------------------------- 4 Total time: 0.210 seconds, Total memory usage: 18.57MB '139.182' ************** MAGMA ***************** Host 139.182.146.132 (139.182.146.132) Time: Thu Dec 1 19:32:32 2005 Input: G:=Group; Output: Magma V2.11-10 Thu Dec 1 2005 19:32:32 on modular [Seed = 3658878860] ------------------------------------- Total time: 0.180 seconds, Total memory usage: 3.24MB '139.182' ************** MAGMA ***************** Host 139.182.146.132 (139.182.146.132) Time: Thu Dec 1 19:23:51 2005 Input: for i in [1..25] do for j in [1..25] do G:=Group; if Order(G) eq 100 then print i,j; end if; end for; end for; Output: Magma V2.11-10 Thu Dec 1 2005 19:23:44 on modular [Seed = 3542025397] ------------------------------------- 1 1 1 24 24 1 24 24 Total time: 6.980 seconds, Total memory usage: 18.57MB '139.182' ************** MAGMA ***************** Host 139.182.146.132 (139.182.146.132) Time: Thu Dec 1 19:22:42 2005 Input: for i in [1..25] do for j in [1..25] do G:=Group; if Order(G) eq 100 then print i,j; end if; end for; end for; Output: Magma V2.11-10 Thu Dec 1 2005 19:22:34 on modular [Seed = 3945216855] ------------------------------------- 1 1 1 24 24 1 24 24 Total time: 7.160 seconds, Total memory usage: 18.57MB '139.182' ************** MAGMA ***************** Host 139.182.146.132 (139.182.146.132) Time: Thu Dec 1 19:07:26 2005 Input: G:=Group; H:=Group; f,G1,k:=CosetAction(G,sub); f,H1,k:=CosetAction(H,sub); IsIsomorphic(G1,H1); Output: Magma V2.11-10 Thu Dec 1 2005 19:07:26 on modular [Seed = 197646403] ------------------------------------- true Homomorphism of GrpPerm: G1, Degree 100, Order 2^2 * 5^2 into GrpPerm: H1, Degree 100, Order 2^2 * 5^2 induced by (1, 2, 6, 17, 41, 81, 67, 31, 54, 86, 93, 59, 88, 100, 66, 90, 95, 61, 27, 51, 84, 73, 35, 13, 4)(3, 9, 25, 57, 72, 53, 22, 7, 20, 49, 71, 34, 70, 46, 18, 44, 69, 33, 12, 32, 68, 42, 65, 30, 11)(5, 15, 39, 79, 43, 78, 38, 14, 37, 77, 48, 19, 47, 76, 36, 75, 56, 24, 8, 23, 55, 74, 80, 40, 16)(10, 28, 62, 96, 99, 83, 50, 26, 60, 94, 89, 64, 98, 87, 58, 92, 85, 52, 29, 63, 97, 91, 82, 45, 21) |--> (1, 17, 61, 84, 90, 88, 31, 73, 4, 6, 81, 51, 66, 59, 67, 86, 13, 2, 41, 27, 99, 93, 100, 54, 35)(3, 57, 33, 68, 44, 70, 7, 72, 11, 25, 69, 32, 18, 34, 22, 49, 30, 9, 42, 12, 46, 71, 53, 20, 65)(5, 79, 24, 55, 75, 47, 14, 43, 16, 39, 56, 23, 36, 19, 38, 77, 40, 15, 74, 8, 76, 48, 78, 37, 80)(10, 95, 52, 96, 92, 97, 26, 98, 21, 62, 85, 63, 58, 64, 50, 94, 45, 28, 91, 29, 87, 89, 83, 60, 82) (1, 3, 10, 5)(2, 7, 21, 8)(4, 12, 28, 14)(6, 18, 45, 19)(9, 26, 16, 27)(11, 29, 15, 31)(13, 34, 62, 36)(17, 42, 82, 43)(20, 50, 24, 51)(22, 52, 23, 54)(25, 58, 40, 59)(30, 64, 39, 66)(32, 60, 38, 61)(33, 63, 37, 67)(35, 72, 96, 74)(41, 57, 91, 80)(44, 83, 48, 84)(46, 85, 47, 86)(49, 87, 56, 88)(53, 89, 55, 90)(65, 99, 79, 73)(68, 92, 78, 93)(69, 98, 77, 100)(70, 94, 76, 95)(71, 97, 75, 81) |--> (1, 5, 10, 3)(2, 8, 21, 7)(4, 14, 28, 12)(6, 19, 45, 18)(9, 27, 16, 26)(11, 31, 15, 29)(13, 36, 62, 34)(17, 43, 82, 42)(20, 51, 24, 50)(22, 54, 23, 52)(25, 59, 40, 58)(30, 66, 39, 64)(32, 61, 38, 60)(33, 67, 37, 63)(35, 74, 95, 72)(41, 79, 98, 65)(44, 84, 48, 83)(46, 86, 47, 85)(49, 88, 56, 87)(53, 90, 55, 89)(57, 73, 80, 91)(68, 93, 78, 92)(69, 99, 77, 97)(70, 81, 76, 94)(71, 100, 75, 96) Total time: 0.310 seconds, Total memory usage: 19.20MB '139.182' ************** MAGMA ***************** Host 139.182.146.132 (139.182.146.132) Time: Thu Dec 1 19:06:42 2005 Input: G:=Group; H:=Group; f,G1,k:=CosetAction(G,sub); f,H1,k:=CosetAction(H,sub); IsIsomorphic(G1,H1); Output: Magma V2.11-10 Thu Dec 1 2005 19:06:42 on modular [Seed = 247651690] ------------------------------------- true Homomorphism of GrpPerm: G1, Degree 100, Order 2^2 * 5^2 into GrpPerm: H1, Degree 100, Order 2^2 * 5^2 induced by (1, 2, 6, 17, 41, 81, 67, 31, 54, 86, 93, 59, 88, 100, 66, 90, 95, 61, 27, 51, 84, 73, 35, 13, 4)(3, 9, 25, 57, 72, 53, 22, 7, 20, 49, 71, 34, 70, 46, 18, 44, 69, 33, 12, 32, 68, 42, 65, 30, 11)(5, 15, 39, 79, 43, 78, 38, 14, 37, 77, 48, 19, 47, 76, 36, 75, 56, 24, 8, 23, 55, 74, 80, 40, 16)(10, 28, 62, 96, 99, 83, 50, 26, 60, 94, 89, 64, 98, 87, 58, 92, 85, 52, 29, 63, 97, 91, 82, 45, 21) |--> (1, 17, 61, 84, 90, 88, 31, 73, 4, 6, 81, 51, 66, 59, 67, 86, 13, 2, 41, 27, 99, 93, 100, 54, 35)(3, 57, 33, 68, 44, 70, 7, 72, 11, 25, 69, 32, 18, 34, 22, 49, 30, 9, 42, 12, 46, 71, 53, 20, 65)(5, 79, 24, 55, 75, 47, 14, 43, 16, 39, 56, 23, 36, 19, 38, 77, 40, 15, 74, 8, 76, 48, 78, 37, 80)(10, 95, 52, 96, 92, 97, 26, 98, 21, 62, 85, 63, 58, 64, 50, 94, 45, 28, 91, 29, 87, 89, 83, 60, 82) (1, 3, 10, 5)(2, 7, 21, 8)(4, 12, 28, 14)(6, 18, 45, 19)(9, 26, 16, 27)(11, 29, 15, 31)(13, 34, 62, 36)(17, 42, 82, 43)(20, 50, 24, 51)(22, 52, 23, 54)(25, 58, 40, 59)(30, 64, 39, 66)(32, 60, 38, 61)(33, 63, 37, 67)(35, 72, 96, 74)(41, 57, 91, 80)(44, 83, 48, 84)(46, 85, 47, 86)(49, 87, 56, 88)(53, 89, 55, 90)(65, 99, 79, 73)(68, 92, 78, 93)(69, 98, 77, 100)(70, 94, 76, 95)(71, 97, 75, 81) |--> (1, 5, 10, 3)(2, 8, 21, 7)(4, 14, 28, 12)(6, 19, 45, 18)(9, 27, 16, 26)(11, 31, 15, 29)(13, 36, 62, 34)(17, 43, 82, 42)(20, 51, 24, 50)(22, 54, 23, 52)(25, 59, 40, 58)(30, 66, 39, 64)(32, 61, 38, 60)(33, 67, 37, 63)(35, 74, 95, 72)(41, 79, 98, 65)(44, 84, 48, 83)(46, 86, 47, 85)(49, 88, 56, 87)(53, 90, 55, 89)(57, 73, 80, 91)(68, 93, 78, 92)(69, 99, 77, 97)(70, 81, 76, 94)(71, 100, 75, 96) Total time: 0.300 seconds, Total memory usage: 19.20MB '139.182' ************** MAGMA ***************** Host 139.182.146.132 (139.182.146.132) Time: Thu Dec 1 19:05:41 2005 Input: G:=Group; H:=Group; f,G1,k:=CosetAction(G,sub); f,H1,k:=CosetAction(H,sub); IsIsomorphic(G1,H1); Output: Magma V2.11-10 Thu Dec 1 2005 19:05:40 on modular [Seed = 80267430] ------------------------------------- false Total time: 0.280 seconds, Total memory usage: 18.57MB '139.182' ************** MAGMA ***************** Host 139.182.146.132 (139.182.146.132) Time: Thu Dec 1 19:04:39 2005 Input: G:=Group; Order(G) Output: Magma V2.11-10 Thu Dec 1 2005 19:04:38 on modular [Seed = 467130363] ------------------------------------- 20 Total time: 0.210 seconds, Total memory usage: 18.56MB '139.182' ************** MAGMA ***************** Host 139.182.146.132 (139.182.146.132) Time: Thu Dec 1 19:04:29 2005 Input: G:=Group; order(G) Output: Magma V2.11-10 Thu Dec 1 2005 19:04:29 on modular [Seed = 517133029] ------------------------------------- >> order(G); ^ User error: Identifier 'order' has not been declared or assigned Total time: 0.180 seconds, Total memory usage: 3.24MB '139.182' ************** MAGMA ***************** Host 139.182.146.132 (139.182.146.132) Time: Thu Dec 1 19:03:27 2005 Input: G:=Group; H:=Group; f,G1,k:=CosetAction(G,sub); f,H1,k:=CosetAction(H,sub); IsIsomorphic(G1,H1); Output: Magma V2.11-10 Thu Dec 1 2005 19:03:27 on modular [Seed = 366598444] ------------------------------------- false Total time: 0.240 seconds, Total memory usage: 18.57MB '139.182' ************** MAGMA ***************** Host 139.182.146.132 (139.182.146.132) Time: Thu Dec 1 19:02:34 2005 Input: G:=Group; H:=Group; f,G1,k:=CosetAction(G,sub); f,H1,k:=CosetAction(H,sub); IsIsomorphic(G1,H1); Output: Magma V2.11-10 Thu Dec 1 2005 19:02:34 on modular [Seed = 736618311] ------------------------------------- false Total time: 0.290 seconds, Total memory usage: 18.57MB '139.182' ************** MAGMA ***************** Host 139.182.146.132 (139.182.146.132) Time: Thu Dec 1 19:02:29 2005 Input: G:=Group; H:=Group; f,G1,k:=CosetAction(G,sub); f,H1,k:=CosetAction(H,sub); IsIsomorphic(G1,H1); Output: Magma V2.11-10 Thu Dec 1 2005 19:02:28 on modular [Seed = 786621004] ------------------------------------- false Total time: 0.290 seconds, Total memory usage: 18.57MB '139.182' ************** MAGMA ***************** Host 139.182.146.132 (139.182.146.132) Time: Thu Dec 1 18:55:10 2005 Input: G:=Group; H:=Group; f,G1,k:=CosetAction(G,sub); f,H1,k:=CosetAction(H,sub); IsIsomorphic(G1,H1); Output: Magma V2.11-10 Thu Dec 1 2005 18:55:10 on modular [Seed = 669772047] ------------------------------------- false Total time: 0.280 seconds, Total memory usage: 18.57MB '139.182' ************** MAGMA ***************** Host 139.182.146.132 (139.182.146.132) Time: Thu Dec 1 18:49:00 2005 Input: for i in [1..25] do G:=Group; if Order(G) eq 100 then print i; end if; end for; end for; Output: Magma V2.11-10 Thu Dec 1 2005 18:48:59 on modular [Seed = 1072953250] ------------------------------------- 1 7 18 24 >> if Order(G) eq 100 then print i; end if; end for; end for; ^ User error: bad syntax Total time: 0.490 seconds, Total memory usage: 18.56MB '139.182' ************** MAGMA ***************** Host 139.182.146.132 (139.182.146.132) Time: Thu Dec 1 18:48:51 2005 Input: for i in [1..25] do G:=Group; if Order(G) eq 100 then print i,j; end if; end for; end for; Output: Magma V2.11-10 Thu Dec 1 2005 18:48:50 on modular [Seed = 855571110] ------------------------------------- >> if Order(G) eq 100 then print i,j; end if; end for; end for; ^ User error: Identifier 'j' has not been declared or assigned >> if Order(G) eq 100 then print i,j; end if; end for; end for; ^ User error: bad syntax Total time: 0.190 seconds, Total memory usage: 3.24MB '139.182' ************** MAGMA ***************** Host 139.182.146.132 (139.182.146.132) Time: Thu Dec 1 18:27:30 2005 Input: G:=Group; H:=Group; f,G1,k:=CosetAction(G,sub); f,H1,k:=CosetAction(H,sub); IsIsomorphic(G1,H1); Output: Magma V2.11-10 Thu Dec 1 2005 18:27:30 on modular [Seed = 1519820750] ------------------------------------- false Total time: 0.290 seconds, Total memory usage: 18.57MB '139.182' ************** MAGMA ***************** Host 139.182.146.132 (139.182.146.132) Time: Thu Dec 1 18:26:57 2005 Input: G:=Group; H:=Group; f,G1,k:=CosetAction(G,sub); f,H1,k:=CosetAction(H,sub); Output: Magma V2.11-10 Thu Dec 1 2005 18:26:57 on modular [Seed = 1369281766] ------------------------------------- Total time: 0.240 seconds, Total memory usage: 18.57MB '139.182' ************** MAGMA ***************** Host 139.182.146.132 (139.182.146.132) Time: Thu Dec 1 18:26:55 2005 Input: G:=Group; H:=Group; f,G1,k:=CosetAction(G,sub); f,H1,k:=CosetAction(H,sub); Output: Magma V2.11-10 Thu Dec 1 2005 18:26:54 on modular [Seed = 1419284892] ------------------------------------- Total time: 0.230 seconds, Total memory usage: 18.57MB '139.182' ************** MAGMA ***************** Host 139.182.146.132 (139.182.146.132) Time: Thu Dec 1 18:26:42 2005 Input: G:=Group; H:=Group; f,G1,k:=CosetAction(G,sub); f,H1,k:=CosetAction(H,sub); Output: Magma V2.11-10 Thu Dec 1 2005 18:26:42 on modular [Seed = 1806151343] ------------------------------------- Total time: 0.240 seconds, Total memory usage: 18.57MB '139.182' ************** MAGMA ***************** Host 139.182.146.132 (139.182.146.132) Time: Thu Dec 1 18:26:41 2005 Input: G:=Group; H:=Group; f,G1,k:=CosetAction(G,sub); f,H1,k:=CosetAction(H,sub); Output: Magma V2.11-10 Thu Dec 1 2005 18:26:41 on modular [Seed = 1856154537] ------------------------------------- Total time: 0.240 seconds, Total memory usage: 18.57MB '139.182' ************** MAGMA ***************** Host 139.182.146.132 (139.182.146.132) Time: Thu Dec 1 18:26:39 2005 Input: G:=Group; H:=Group; f,G1,k:=CosetAction(G,sub); f,H1,k:=CosetAction(H,sub); Output: Magma V2.11-10 Thu Dec 1 2005 18:26:38 on modular [Seed = 1705615549] ------------------------------------- Total time: 0.230 seconds, Total memory usage: 18.57MB '139.182' ************** MAGMA ***************** Host 139.182.146.132 (139.182.146.132) Time: Thu Dec 1 18:26:31 2005 Input: G:=Group; H:=Group; f,G1,k:=CosetAction(G,sub); f,H1,k:=CosetAction(H,sub); Output: Magma V2.11-10 Thu Dec 1 2005 18:26:30 on modular [Seed = 2025110952] ------------------------------------- Total time: 0.230 seconds, Total memory usage: 18.57MB '139.182' ************** MAGMA ***************** Host 139.182.146.132 (139.182.146.132) Time: Thu Dec 1 18:26:18 2005 Input: G:=Group; H:=Group; f,G1,k:=CosetAction(G,sub); f,H1,k:=CosetAction(H,sub); Output: Magma V2.11-10 Thu Dec 1 2005 18:26:18 on modular [Seed = 2142485139] ------------------------------------- Total time: 0.240 seconds, Total memory usage: 18.57MB '139.182' ************** MAGMA ***************** Host 139.182.146.132 (139.182.146.132) Time: Thu Dec 1 18:20:03 2005 Input: for i in [1..2] do for j in [1..2] do G:=Group; if Order(G) eq 100 then print i,j; end if; end for; end for; Output: Magma V2.11-10 Thu Dec 1 2005 18:20:00 on modular [Seed = 1991946272] ------------------------------------- Total time: 2.250 seconds, Total memory usage: 23.67MB '139.182' ************** MAGMA ***************** Host 139.182.146.132 (139.182.146.132) Time: Thu Dec 1 18:19:10 2005 Input: for i in [1..5] do for j in [1..5] do G:=Group; if Order(G) eq 100 then print i,j; end if; end for; end for; Output: Magma V2.11-10 Thu Dec 1 2005 18:18:49 on modular [Seed = 2353023778] ------------------------------------- Total time: 20.000 seconds, Total memory usage: 20.85MB '139.182' ************** MAGMA ***************** Host 139.182.146.132 (139.182.146.132) Time: Thu Dec 1 18:16:08 2005 Input: for i in [1..25] do for j in [1..25] do G:=Group; if Order(G) eq 100 then print i,j; end if; end for; end for; Output: Magma V2.11-10 Thu Dec 1 2005 18:16:01 on modular [Seed = 2219328450] ------------------------------------- 1 1 1 24 24 1 24 24 Total time: 7.049 seconds, Total memory usage: 18.57MB '139.182' ************** MAGMA ***************** Host 139.182.146.132 (139.182.146.132) Time: Thu Dec 1 18:15:50 2005 Input: for i in [1..25] do for j in [1..25] do for|for> G:=Group; for|for> if Order(G) eq 100 then print i,j; end if; end for; end for; Output: Magma V2.11-10 Thu Dec 1 2005 18:15:50 on modular [Seed = 2589351371] ------------------------------------- >> for|for> G:=Group; ^ User error: bad syntax >> for|for> if Order(G) eq 100 then print i,j; end if; end for; end for; ^ User error: bad syntax Total time: 0.190 seconds, Total memory usage: 3.24MB '139.182' ************** MAGMA ***************** Host 139.182.146.132 (139.182.146.132) Time: Thu Dec 1 18:13:39 2005 Input: G:=Group; Order(G) Output: Magma V2.11-10 Thu Dec 1 2005 18:13:38 on modular [Seed = 3128326767] ------------------------------------- 100 Total time: 0.210 seconds, Total memory usage: 18.56MB '139.182' ************** MAGMA ***************** Host 139.182.146.132 (139.182.146.132) Time: Thu Dec 1 18:12:40 2005 Input: G:=Group; Order(G) Output: Magma V2.11-10 Thu Dec 1 2005 18:12:39 on modular [Seed = 3238855121] ------------------------------------- 100 Total time: 0.210 seconds, Total memory usage: 18.56MB '139.182' ************** MAGMA ***************** Host 139.182.146.132 (139.182.146.132) Time: Thu Dec 1 18:12:15 2005 Input: G:=Group; H:=Group; J:=Group; K:=Group; Output: Magma V2.11-10 Thu Dec 1 2005 18:12:15 on modular [Seed = 3742572519] ------------------------------------- >> H:=Group; ^ User error: Identifier 'y' has not been declared or assigned >> J:=Group; ^ User error: Identifier 'z' has not been declared or assigned >> K:=Group; ^ User error: Identifier 'v' has not been declared or assigned Total time: 0.180 seconds, Total memory usage: 3.24MB '139.182' ************** MAGMA ***************** Host 139.182.146.132 (139.182.146.132) Time: Thu Dec 1 18:12:12 2005 Input: G:=Group; H:=Group; J:=Group; K:=Group; Output: Magma V2.11-10 Thu Dec 1 2005 18:12:12 on modular [Seed = 3558348015] ------------------------------------- >> H:=Group; ^ User error: Identifier 'y' has not been declared or assigned >> J:=Group; ^ User error: Identifier 'z' has not been declared or assigned >> K:=Group; ^ User error: Identifier 'v' has not been declared or assigned Total time: 0.180 seconds, Total memory usage: 3.24MB '139.182' ************** MAGMA ***************** Host 139.182.146.132 (139.182.146.132) Time: Thu Dec 1 18:11:54 2005 Input: G:=Group; H:=Group; J:=Group; K:=Group; Output: Magma V2.11-10 Thu Dec 1 2005 18:11:54 on modular [Seed = 3794674351] ------------------------------------- >> H:=Group; ^ User error: Identifier 'y' has not been declared or assigned >> J:=Group; ^ User error: Identifier 'z' has not been declared or assigned >> K:=Group; ^ User error: Identifier 'v' has not been declared or assigned Total time: 0.190 seconds, Total memory usage: 3.24MB '139.182' ************** MAGMA ***************** Host 139.182.146.132 (139.182.146.132) Time: Thu Dec 1 18:10:43 2005 Input: G:=Group; H:=Group; J:=Group; K:=Group; Order(G) Output: Magma V2.11-10 Thu Dec 1 2005 18:10:43 on modular [Seed = 180800398] ------------------------------------- 100 Total time: 0.210 seconds, Total memory usage: 18.57MB '139.182' ************** MAGMA ***************** Host 139.182.146.132 (139.182.146.132) Time: Thu Dec 1 18:09:48 2005 Input: G:=Group; H:=Group; J:=Group; K:=Group; Order(G,H,J,K) Output: Magma V2.11-10 Thu Dec 1 2005 18:09:47 on modular [Seed = 299749597] ------------------------------------- >> Order(G,H,J,K) ^ Runtime error in 'Order': Bad argument types Argument types given: GrpFP, GrpFP, GrpFP, GrpFP Total time: 0.190 seconds, Total memory usage: 3.24MB '139.182' ************** MAGMA ***************** Host 139.182.146.132 (139.182.146.132) Time: Thu Dec 1 18:09:34 2005 Input: G:=Group; H:=Group; J:=Group; K:=Group; Order(G,H,J,k) Output: Magma V2.11-10 Thu Dec 1 2005 18:09:34 on modular [Seed = 383440847] ------------------------------------- >> Order(G,H,J,k) ^ User error: Identifier 'k' has not been declared or assigned Total time: 0.190 seconds, Total memory usage: 3.24MB '139.182' ************** MAGMA ***************** Host 139.182.146.132 (139.182.146.132) Time: Thu Dec 1 18:09:10 2005 Input: G:=Group; H:=Group; J:=Group; K:=Group; Order(G) Order(H) Order(J) Order(K) Output: Magma V2.11-10 Thu Dec 1 2005 18:09:10 on modular [Seed = 736619210] ------------------------------------- >> Order(H) ^ User error: bad syntax >> Order(K); ^ User error: bad syntax Total time: 0.190 seconds, Total memory usage: 3.24MB '139.182' ************** MAGMA ***************** Host 139.182.146.132 (139.182.146.132) Time: Thu Dec 1 18:05:13 2005 Input: G:=Group; Order(G) Output: Magma V2.11-10 Thu Dec 1 2005 18:05:12 on modular [Seed = 3355707920] ------------------------------------- 100 Total time: 0.210 seconds, Total memory usage: 18.56MB '139.182' ************** MAGMA ***************** Host 139.182.146.132 (139.182.146.132) Time: Thu Dec 1 18:03:26 2005 Input: for i in [1..25] do G:=Group; if Order(G) eq 100 then print i; end if; end for; Output: Magma V2.11-10 Thu Dec 1 2005 18:03:25 on modular [Seed = 3945212326] ------------------------------------- 1 7 18 24 Total time: 0.470 seconds, Total memory usage: 18.56MB '139.182' ************** MAGMA ***************** Host 139.182.146.132 (139.182.146.132) Time: Thu Dec 1 17:54:04 2005 Input: for i in [1..25] do G:=Group; if Order(G) eq 100 then print i; end if; end for; Output: Magma V2.11-10 Thu Dec 1 2005 17:54:04 on modular [Seed = 702879061] ------------------------------------- 1 7 18 24 Total time: 0.480 seconds, Total memory usage: 18.56MB '139.182' ************** MAGMA ***************** Host 139.182.146.132 (139.182.146.132) Time: Thu Dec 1 17:53:54 2005 Input: for i in [1..25] do G:=Group; if Order(G) eq 100 then print i,j; end if; end for; Output: Magma V2.11-10 Thu Dec 1 2005 17:53:54 on modular [Seed = 619190358] ------------------------------------- >> if Order(G) eq 100 then print i,j; end if; end for;; ^ User error: Identifier 'j' has not been declared or assigned Total time: 0.180 seconds, Total memory usage: 3.24MB '139.182' ************** MAGMA ***************** Host 139.182.146.132 (139.182.146.132) Time: Thu Dec 1 17:52:09 2005 Input: for i in [1..24] do G:=Group; if Order(G) eq 100 then print i; end if; end for; end for; Output: ** WARNING: Computation time exceeded 20 seconds, so computation was terminated after 20 seconds. ** Magma V2.11-10 Thu Dec 1 2005 17:51:49 on modular [Seed = 1039204552] ------------------------------------- Errors: /bin/sh: line 1: 13133 Alarm clock nice -n 19 /usr/local/bin/magma '139.182' ************** MAGMA ***************** Host 139.182.146.132 (139.182.146.132) Time: Thu Dec 1 17:51:38 2005 Input: for i in [1..24] do G:=Group; if Order(G) eq 100 then print i; end if; end for; end for; Output: Magma V2.11-10 Thu Dec 1 2005 17:51:37 on modular [Seed = 955516209] ------------------------------------- >> G:=Group; ^ User error: Identifier 'x' has not been declared or assigned >> if Order(G) eq 100 then print i; end if; end for; end for;; ^ User error: bad syntax Total time: 0.180 seconds, Total memory usage: 3.24MB '60.225.' ************** MAGMA ***************** Host 60.225.131.213 (60.225.131.213) Time: Thu Dec 1 17:49:32 2005 Input: 6009/241 Output: Magma V2.11-10 Thu Dec 1 2005 17:49:31 on modular [Seed = 1082949323] ------------------------------------- 6009/241 Total time: 0.190 seconds, Total memory usage: 3.24MB '60.225.' ************** MAGMA ***************** Host 60.225.131.213 (60.225.131.213) Time: Thu Dec 1 17:49:18 2005 Input: 241*83*3 Output: Magma V2.11-10 Thu Dec 1 2005 17:49:18 on modular [Seed = 1536648645] ------------------------------------- 60009 Total time: 0.180 seconds, Total memory usage: 3.24MB '60.225.' ************** MAGMA ***************** Host 60.225.131.213 (60.225.131.213) Time: Thu Dec 1 17:49:06 2005 Input: 6009/(241*83*3) Output: Magma V2.11-10 Thu Dec 1 2005 17:49:05 on modular [Seed = 1839289340] ------------------------------------- 2003/20003 Total time: 0.190 seconds, Total memory usage: 3.24MB '60.225.' ************** MAGMA ***************** Host 60.225.131.213 (60.225.131.213) Time: Thu Dec 1 17:48:39 2005 Input: Factorization(60009); Output: Magma V2.11-10 Thu Dec 1 2005 17:48:39 on modular [Seed = 1755600607] ------------------------------------- [ <3, 1>, <83, 1>, <241, 1> ] Total time: 0.190 seconds, Total memory usage: 3.24MB '60.225.' ************** MAGMA ***************** Host 60.225.131.213 (60.225.131.213) Time: Thu Dec 1 17:48:23 2005 Input: 17582637/293 Output: Magma V2.11-10 Thu Dec 1 2005 17:48:23 on modular [Seed = 1672437156] ------------------------------------- 60009 Total time: 0.190 seconds, Total memory usage: 3.24MB '60.225.' ************** MAGMA ***************** Host 60.225.131.213 (60.225.131.213) Time: Thu Dec 1 17:48:03 2005 Input: Factorization(17582637); Output: Magma V2.11-10 Thu Dec 1 2005 17:48:03 on modular [Seed = 1975077798] ------------------------------------- [ <3, 1>, <83, 1>, <241, 1>, <293, 1> ] Total time: 0.190 seconds, Total memory usage: 3.24MB '60.225.' ************** MAGMA ***************** Host 60.225.131.213 (60.225.131.213) Time: Thu Dec 1 17:47:47 2005 Input: 190983499808487/10862051 Output: Magma V2.11-10 Thu Dec 1 2005 17:47:46 on modular [Seed = 1891389074] ------------------------------------- 17582637 Total time: 0.200 seconds, Total memory usage: 3.24MB '60.225.' ************** MAGMA ***************** Host 60.225.131.213 (60.225.131.213) Time: Thu Dec 1 17:47:21 2005 Input: Factorization(190983499808487); Output: Magma V2.11-10 Thu Dec 1 2005 17:47:21 on modular [Seed = 2319889538] ------------------------------------- [ <3, 1>, <83, 1>, <241, 1>, <293, 1>, <10862051, 1> ] Total time: 0.190 seconds, Total memory usage: 3.24MB '139.182' ************** MAGMA ***************** Host 139.182.146.132 (139.182.146.132) Time: Thu Dec 1 17:47:18 2005 Input: for i in [1..24] do for j in [1..24] do G:=Group; if Order(G) eq 100 then print i,j; end if; end for; end for; Output: Magma V2.11-10 Thu Dec 1 2005 17:47:11 on modular [Seed = 2152513182] ------------------------------------- 1 1 1 24 24 1 24 24 Total time: 6.799 seconds, Total memory usage: 18.57MB '60.225.' ************** MAGMA ***************** Host 60.225.131.213 (60.225.131.213) Time: Thu Dec 1 17:47:15 2005 Input: Factorization(190983499808489709827); Output: Magma V2.11-10 Thu Dec 1 2005 17:47:15 on modular [Seed = 2236201863] ------------------------------------- [ <11, 1>, <79, 1>, <5639, 1>, <547021, 1>, <71247557, 1> ] Total time: 0.190 seconds, Total memory usage: 3.24MB '60.225.' ************** MAGMA ***************** Host 60.225.131.213 (60.225.131.213) Time: Thu Dec 1 17:47:08 2005 Input: Factorization(1909834998084897098709873749827); Output: Magma V2.11-10 Thu Dec 1 2005 17:47:08 on modular [Seed = 2639373184] ------------------------------------- [ <4649, 1>, <184369, 1>, <225809, 1>, <1912121, 1>, <5160500003, 1> ] Total time: 0.200 seconds, Total memory usage: 3.24MB '60.225.' ************** MAGMA ***************** Host 60.225.131.213 (60.225.131.213) Time: Thu Dec 1 17:46:58 2005 Input: Factorization(190983499808489709870987398740982709374894749827); Output: Magma V2.11-10 Thu Dec 1 2005 17:46:58 on modular [Seed = 2488838791] ------------------------------------- [ <3, 1>, <13, 1>, <19, 1>, <47, 1>, <1607, 1>, <11587613, 1>, <122763733, 1>, <135498912413, 1>, <17703699416059, 1> ] Total time: 0.550 seconds, Total memory usage: 3.43MB '139.182' ************** MAGMA ***************** Host 139.182.146.132 (139.182.146.132) Time: Thu Dec 1 17:46:46 2005 Input: for i in [1..24] do for j in [20..24] do G:=Group; if Order(G) eq 100 then print i,j; end if; end for; end for; Output: Magma V2.11-10 Thu Dec 1 2005 17:46:44 on modular [Seed = 2438836148] ------------------------------------- 1 24 24 24 Total time: 1.350 seconds, Total memory usage: 18.57MB '60.225.' ************** MAGMA ***************** Host 60.225.131.213 (60.225.131.213) Time: Thu Dec 1 17:46:42 2005 Input: Factorization(1909834998084897098709873987409827093748948709128749012749827); Output: Magma V2.11-10 Thu Dec 1 2005 17:46:41 on modular [Seed = 2858849418] ------------------------------------- [ <165261827, 1>, <53012766743, 1>, <1008225912887, 1>, <1747295644667, 1>, <123742412152451683, 1> ] Total time: 0.840 seconds, Total memory usage: 3.34MB '60.225.' ************** MAGMA ***************** Host 60.225.131.213 (60.225.131.213) Time: Thu Dec 1 17:46:32 2005 Input: IsPrime(1909834998084897098709873987409827093748948709128749012749827); Output: Magma V2.11-10 Thu Dec 1 2005 17:46:32 on modular [Seed = 2808846750] ------------------------------------- false Total time: 0.190 seconds, Total memory usage: 3.24MB '60.225.' ************** MAGMA ***************** Host 60.225.131.213 (60.225.131.213) Time: Thu Dec 1 17:46:24 2005 Input: IsPrime(19098349980848970987098739874098270937489327091879812749873948709128749012749827); Output: Magma V2.11-10 Thu Dec 1 2005 17:46:23 on modular [Seed = 2691473134] ------------------------------------- false Total time: 0.180 seconds, Total memory usage: 3.24MB '60.225.' ************** MAGMA ***************** Host 60.225.131.213 (60.225.131.213) Time: Thu Dec 1 17:46:19 2005 Input: IsPrime(1909834998084897098709873987409827093748932709187981274987394870912874901274982749874290749012734983729); Output: Magma V2.11-10 Thu Dec 1 2005 17:46:18 on modular [Seed = 2994638073] ------------------------------------- false Total time: 0.190 seconds, Total memory usage: 3.24MB '60.225.' ************** MAGMA ***************** Host 60.225.131.213 (60.225.131.213) Time: Thu Dec 1 17:46:10 2005 Input: IsPrime(19098349980848970987098739874098270937489327091879812749873948709128749012749827498742907490127349837290); Output: Magma V2.11-10 Thu Dec 1 2005 17:46:10 on modular [Seed = 3288922759] ------------------------------------- false Total time: 0.190 seconds, Total memory usage: 3.24MB '60.225.' ************** MAGMA ***************** Host 60.225.131.213 (60.225.131.213) Time: Thu Dec 1 17:46:05 2005 Input: IsPrime(190983499808489709870987398740982709374893270918798127498739487091287490127498274987429074901273498372909); Output: Magma V2.11-10 Thu Dec 1 2005 17:46:04 on modular [Seed = 3708937088] ------------------------------------- false Total time: 0.190 seconds, Total memory usage: 3.24MB '60.225.' ************** MAGMA ***************** Host 60.225.131.213 (60.225.131.213) Time: Thu Dec 1 17:45:55 2005 Input: IsPrime(190983499808489709870987398740982709374893270918798127498739487091287490127498274987429074901273498372908); Output: Magma V2.11-10 Thu Dec 1 2005 17:45:55 on modular [Seed = 4012102077] ------------------------------------- false Total time: 0.180 seconds, Total memory usage: 3.24MB '60.225.' ************** MAGMA ***************** Host 60.225.131.213 (60.225.131.213) Time: Thu Dec 1 17:45:46 2005 Input: IsPrime(190983499808489709870987398740982709374893270918798127498739487091287490127498274987429074901273498372907); Output: Magma V2.11-10 Thu Dec 1 2005 17:45:45 on modular [Seed = 4264738966] ------------------------------------- false Total time: 0.190 seconds, Total memory usage: 3.24MB '139.182' ************** MAGMA ***************** Host 139.182.146.132 (139.182.146.132) Time: Thu Dec 1 17:45:27 2005 Input: for i in [1..24] do for j in [20..24] do G:=Group; if Order(G) eq 100 then print i,j; end if; end for; end for; Output: Magma V2.11-10 Thu Dec 1 2005 17:45:10 on modular [Seed = 163919502] ------------------------------------- Total time: 16.519 seconds, Total memory usage: 20.76MB '139.182' ************** MAGMA ***************** Host 139.182.146.132 (139.182.146.132) Time: Thu Dec 1 17:44:40 2005 Input: for i in [10..20] do for j in [10..20] do G:=Group; if Order(G) eq 100 then print i,j; end if; end for; end for; Output: Magma V2.11-10 Thu Dec 1 2005 17:44:30 on modular [Seed = 467084452] ------------------------------------- Total time: 9.410 seconds, Total memory usage: 19.13MB '139.182' ************** MAGMA ***************** Host 139.182.146.132 (139.182.146.132) Time: Thu Dec 1 17:44:05 2005 Input: for i in [1..10] do for j in [1..10] do G:=Group; if Order(G) eq 100 then print i,j; end if; end for; end for; Output: Magma V2.11-10 Thu Dec 1 2005 17:43:57 on modular [Seed = 349711304] ------------------------------------- Total time: 7.070 seconds, Total memory usage: 20.84MB '139.182' ************** MAGMA ***************** Host 139.182.146.132 (139.182.146.132) Time: Thu Dec 1 17:41:38 2005 Input: F5:=FiniteField(5); G:=GL(2,5); Order(G) Output: Magma V2.11-10 Thu Dec 1 2005 17:41:37 on modular [Seed = 299707969] ------------------------------------- 480 Total time: 0.180 seconds, Total memory usage: 3.24MB '139.182' ************** MAGMA ***************** Host 139.182.146.132 (139.182.146.132) Time: Thu Dec 1 17:36:03 2005 Input: G:=Group; H:=Group; f,G1,k:=CosetAction(G,sub); f,H1,k:=CosetAction(H,sub); IsIsomorphic(G1,H1); Output: Magma V2.11-10 Thu Dec 1 2005 17:36:03 on modular [Seed = 1039205535] ------------------------------------- false Total time: 0.280 seconds, Total memory usage: 18.57MB '139.182' ************** MAGMA ***************** Host 139.182.146.132 (139.182.146.132) Time: Thu Dec 1 17:35:52 2005 Input: G:=Group; H:=Group; f,G1,k:=CosetAction(G,sub); f,H1,k:=CosetAction(H,sub); IsIsomorphic(G1,H1); Output: Magma V2.11-10 Thu Dec 1 2005 17:35:52 on modular [Seed = 922357647] ------------------------------------- false Total time: 0.280 seconds, Total memory usage: 18.57MB '139.182' ************** MAGMA ***************** Host 139.182.146.132 (139.182.146.132) Time: Thu Dec 1 17:33:55 2005 Input: G:=Group; H:=Group; f,G1,k:=CosetAction(G,sub); f,H1,k:=CosetAction(H,sub); IsIsomorphic(G1,H1); Output: Magma V2.11-10 Thu Dec 1 2005 17:33:54 on modular [Seed = 821823763] ------------------------------------- true Homomorphism of GrpPerm: G1, Degree 100, Order 2^2 * 5^2 into GrpPerm: H1, Degree 100, Order 2^2 * 5^2 induced by (1, 2, 6, 17, 41, 81, 67, 31, 54, 86, 93, 59, 88, 100, 66, 90, 95, 61, 27, 51, 84, 73, 35, 13, 4)(3, 9, 25, 57, 72, 53, 22, 7, 20, 49, 71, 34, 70, 46, 18, 44, 69, 33, 12, 32, 68, 42, 65, 30, 11)(5, 15, 39, 79, 43, 78, 38, 14, 37, 77, 48, 19, 47, 76, 36, 75, 56, 24, 8, 23, 55, 74, 80, 40, 16)(10, 28, 62, 96, 99, 83, 50, 26, 60, 94, 89, 64, 98, 87, 58, 92, 85, 52, 29, 63, 97, 91, 82, 45, 21) |--> (1, 17, 61, 84, 90, 88, 31, 73, 4, 6, 81, 51, 66, 59, 67, 86, 13, 2, 41, 27, 99, 93, 100, 54, 35)(3, 57, 33, 68, 44, 70, 7, 72, 11, 25, 69, 32, 18, 34, 22, 49, 30, 9, 42, 12, 46, 71, 53, 20, 65)(5, 79, 24, 55, 75, 47, 14, 43, 16, 39, 56, 23, 36, 19, 38, 77, 40, 15, 74, 8, 76, 48, 78, 37, 80)(10, 95, 52, 96, 92, 97, 26, 98, 21, 62, 85, 63, 58, 64, 50, 94, 45, 28, 91, 29, 87, 89, 83, 60, 82) (1, 3, 10, 5)(2, 7, 21, 8)(4, 12, 28, 14)(6, 18, 45, 19)(9, 26, 16, 27)(11, 29, 15, 31)(13, 34, 62, 36)(17, 42, 82, 43)(20, 50, 24, 51)(22, 52, 23, 54)(25, 58, 40, 59)(30, 64, 39, 66)(32, 60, 38, 61)(33, 63, 37, 67)(35, 72, 96, 74)(41, 57, 91, 80)(44, 83, 48, 84)(46, 85, 47, 86)(49, 87, 56, 88)(53, 89, 55, 90)(65, 99, 79, 73)(68, 92, 78, 93)(69, 98, 77, 100)(70, 94, 76, 95)(71, 97, 75, 81) |--> (1, 5, 10, 3)(2, 8, 21, 7)(4, 14, 28, 12)(6, 19, 45, 18)(9, 27, 16, 26)(11, 31, 15, 29)(13, 36, 62, 34)(17, 43, 82, 42)(20, 51, 24, 50)(22, 54, 23, 52)(25, 59, 40, 58)(30, 66, 39, 64)(32, 61, 38, 60)(33, 67, 37, 63)(35, 74, 95, 72)(41, 79, 98, 65)(44, 84, 48, 83)(46, 86, 47, 85)(49, 88, 56, 87)(53, 90, 55, 89)(57, 73, 80, 91)(68, 93, 78, 92)(69, 99, 77, 97)(70, 81, 76, 94)(71, 100, 75, 96) Total time: 0.300 seconds, Total memory usage: 19.20MB '139.182' ************** MAGMA ***************** Host 139.182.146.132 (139.182.146.132) Time: Thu Dec 1 17:33:38 2005 Input: G:=Group; H:=Group; f,G1,k:=CosetAction(G,sub); f,H1,k:=CosetAction(H,sub); IsIsomorphic(G1,H1); Output: Magma V2.11-10 Thu Dec 1 2005 17:33:37 on modular [Seed = 1300329597] ------------------------------------- false Total time: 0.290 seconds, Total memory usage: 18.57MB '139.182' ************** MAGMA ***************** Host 139.182.146.132 (139.182.146.132) Time: Thu Dec 1 17:33:29 2005 Input: G:=Group; H:=Group; f,G1,k:=CosetAction(G,sub); f,H1,k:=CosetAction(H,sub); IsIsomorphic(G1,H1); Output: Magma V2.11-10 Thu Dec 1 2005 17:33:29 on modular [Seed = 1183481614] ------------------------------------- false Total time: 0.280 seconds, Total memory usage: 18.57MB '139.182' ************** MAGMA ***************** Host 139.182.146.132 (139.182.146.132) Time: Thu Dec 1 17:33:08 2005 Input: G:=Group; H:=Group; f,G1,k:=CosetAction(G,sub); f,H1,k:=CosetAction(H,sub); IsIsomorphic(G1,H1); Output: Magma V2.11-10 Thu Dec 1 2005 17:33:08 on modular [Seed = 1133476374] ------------------------------------- true Homomorphism of GrpPerm: G1, Degree 100, Order 2^2 * 5^2 into GrpPerm: H1, Degree 100, Order 2^2 * 5^2 induced by (1, 2, 6, 17, 41, 81, 67, 31, 54, 86, 93, 59, 88, 100, 66, 90, 95, 61, 27, 51, 84, 73, 35, 13, 4)(3, 9, 25, 57, 72, 53, 22, 7, 20, 49, 71, 34, 70, 46, 18, 44, 69, 33, 12, 32, 68, 42, 65, 30, 11)(5, 15, 39, 79, 43, 78, 38, 14, 37, 77, 48, 19, 47, 76, 36, 75, 56, 24, 8, 23, 55, 74, 80, 40, 16)(10, 28, 62, 96, 99, 83, 50, 26, 60, 94, 89, 64, 98, 87, 58, 92, 85, 52, 29, 63, 97, 91, 82, 45, 21) |--> (1, 17, 61, 84, 90, 88, 31, 73, 4, 6, 81, 51, 66, 59, 67, 86, 13, 2, 41, 27, 99, 93, 100, 54, 35)(3, 57, 33, 68, 44, 70, 7, 72, 11, 25, 69, 32, 18, 34, 22, 49, 30, 9, 42, 12, 46, 71, 53, 20, 65)(5, 79, 24, 55, 75, 47, 14, 43, 16, 39, 56, 23, 36, 19, 38, 77, 40, 15, 74, 8, 76, 48, 78, 37, 80)(10, 95, 52, 96, 92, 97, 26, 98, 21, 62, 85, 63, 58, 64, 50, 94, 45, 28, 91, 29, 87, 89, 83, 60, 82) (1, 3, 10, 5)(2, 7, 21, 8)(4, 12, 28, 14)(6, 18, 45, 19)(9, 26, 16, 27)(11, 29, 15, 31)(13, 34, 62, 36)(17, 42, 82, 43)(20, 50, 24, 51)(22, 52, 23, 54)(25, 58, 40, 59)(30, 64, 39, 66)(32, 60, 38, 61)(33, 63, 37, 67)(35, 72, 96, 74)(41, 57, 91, 80)(44, 83, 48, 84)(46, 85, 47, 86)(49, 87, 56, 88)(53, 89, 55, 90)(65, 99, 79, 73)(68, 92, 78, 93)(69, 98, 77, 100)(70, 94, 76, 95)(71, 97, 75, 81) |--> (1, 5, 10, 3)(2, 8, 21, 7)(4, 14, 28, 12)(6, 19, 45, 18)(9, 27, 16, 26)(11, 31, 15, 29)(13, 36, 62, 34)(17, 43, 82, 42)(20, 51, 24, 50)(22, 54, 23, 52)(25, 59, 40, 58)(30, 66, 39, 64)(32, 61, 38, 60)(33, 67, 37, 63)(35, 74, 95, 72)(41, 79, 98, 65)(44, 84, 48, 83)(46, 86, 47, 85)(49, 88, 56, 87)(53, 90, 55, 89)(57, 73, 80, 91)(68, 93, 78, 92)(69, 99, 77, 97)(70, 81, 76, 94)(71, 100, 75, 96) Total time: 0.320 seconds, Total memory usage: 19.20MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Thu Dec 1 17:11:45 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^9*3^9; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; //P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; //Height(P); //HeightConstant(J); RationalPoints(J:Bound:=1000); //Chabauty(P,19); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Thu Dec 1 2005 17:11:44 on modular [Seed = 3339448725] ------------------------------------- 0 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [] {@ (1, 0, 0) @} Total time: 1.090 seconds, Total memory usage: 39.45MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Thu Dec 1 17:11:38 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^9*3^8; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; //P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; //Height(P); //HeightConstant(J); RationalPoints(J:Bound:=1000); //Chabauty(P,19); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Thu Dec 1 2005 17:11:37 on modular [Seed = 3255760041] ------------------------------------- 0 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [] {@ (1, 0, 0) @} Total time: 1.100 seconds, Total memory usage: 39.44MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Thu Dec 1 17:11:30 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^9*3^7; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; //P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; //Height(P); //HeightConstant(J); RationalPoints(J:Bound:=1000); //Chabauty(P,19); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Thu Dec 1 2005 17:11:29 on modular [Seed = 3642093498] ------------------------------------- 0 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [] {@ (1, 0, 0) @} Total time: 1.159 seconds, Total memory usage: 39.51MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Thu Dec 1 17:11:22 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^9*3^4; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; //P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; //Height(P); //HeightConstant(J); RationalPoints(J:Bound:=1000); //Chabauty(P,19); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Thu Dec 1 2005 17:11:21 on modular [Seed = 3558400687] ------------------------------------- 0 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [] {@ (1, 0, 0) @} Total time: 1.330 seconds, Total memory usage: 39.65MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Thu Dec 1 17:11:16 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^9*3^2; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; //P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; //Height(P); //HeightConstant(J); RationalPoints(J:Bound:=1000); //Chabauty(P,19); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Thu Dec 1 2005 17:11:15 on modular [Seed = 3542082978] ------------------------------------- 0 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [] {@ (1, 0, 0) @} Total time: 1.199 seconds, Total memory usage: 39.45MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Thu Dec 1 17:11:09 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^9*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; //P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; //Height(P); //HeightConstant(J); RationalPoints(J:Bound:=1000); //Chabauty(P,19); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Thu Dec 1 2005 17:11:08 on modular [Seed = 3995261280] ------------------------------------- 0 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [] {@ (1, 0, 0) @} Total time: 1.060 seconds, Total memory usage: 39.48MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Thu Dec 1 17:11:02 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^8*3^9; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; //P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; //Height(P); //HeightConstant(J); RationalPoints(J:Bound:=1000); //Chabauty(P,19); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Thu Dec 1 2005 17:11:00 on modular [Seed = 3844723299] ------------------------------------- 0 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [] {@ (1, 0, 0) @} Total time: 1.540 seconds, Total memory usage: 39.28MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Thu Dec 1 17:10:55 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^8*3^8; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; //P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; //Height(P); //HeightConstant(J); RationalPoints(J:Bound:=1000); //Chabauty(P,19); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Thu Dec 1 2005 17:10:54 on modular [Seed = 3761035132] ------------------------------------- 0 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [] {@ (1, 0, 0) @} Total time: 0.990 seconds, Total memory usage: 39.08MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Thu Dec 1 17:10:49 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^8*3^6; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; //P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; //Height(P); //HeightConstant(J); RationalPoints(J:Bound:=1000); //Chabauty(P,19); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Thu Dec 1 2005 17:10:48 on modular [Seed = 4281583717] ------------------------------------- 0 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [] {@ (1, 0, 0) @} Total time: 0.980 seconds, Total memory usage: 39.15MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Thu Dec 1 17:10:42 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^8*3^4; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; //P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; //Height(P); //HeightConstant(J); RationalPoints(J:Bound:=1000); //Chabauty(P,19); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Thu Dec 1 2005 17:10:41 on modular [Seed = 4114203238] ------------------------------------- 0 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [] {@ (1, 0, 0) @} Total time: 1.060 seconds, Total memory usage: 39.45MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Thu Dec 1 17:10:35 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^8*3^3; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; //P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; //Height(P); //HeightConstant(J); RationalPoints(J:Bound:=1000); //Chabauty(P,19); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Thu Dec 1 2005 17:10:34 on modular [Seed = 4030515032] ------------------------------------- 0 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [] {@ (1, 0, 0) @} Total time: 1.320 seconds, Total memory usage: 38.99MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Thu Dec 1 17:10:28 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^8*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; //P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; //Height(P); //HeightConstant(J); RationalPoints(J:Bound:=1000); //Chabauty(P,19); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Thu Dec 1 2005 17:10:27 on modular [Seed = 264452270] ------------------------------------- 0 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [] {@ (1, 0, 0) @} Total time: 0.970 seconds, Total memory usage: 39.19MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Thu Dec 1 17:10:14 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^7*3^8; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; //P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; //Height(P); //HeightConstant(J); RationalPoints(J:Bound:=1000); //Chabauty(P,19); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Thu Dec 1 2005 17:10:13 on modular [Seed = 180764093] ------------------------------------- 0 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [] {@ (1, 0, 0) @} Total time: 1.010 seconds, Total memory usage: 39.40MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Thu Dec 1 17:10:11 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^7*3^8; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; //P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; //Height(P); //HeightConstant(J); RationalPoints(J:Bound:=1000); //Chabauty(P,19); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Thu Dec 1 2005 17:10:10 on modular [Seed = 30226097] ------------------------------------- TwoSelmerGroupData( J: Jacobian of Hyperelliptic Curve defined by y^2 = x^5 - 83980... ) TwoSelmerWork( C: Hyperelliptic Curve defined by y^2 = x^5 - 839808 over Ratio..., BoundType: Default, Bound: 0, UseUnits: true, Points: {}, Fields: {} ) LocalImageOdd( f: x^5 - 839808, p: 2, m1: Mapping from: Univariate Polynomial Ring in x over Rational ..., m2: Mapping from: Univariate Polynomial Ring in x over Rational ..., v: 30, Points: {} ) try( ) try( d: 2 ) SquarefreeRoots( F: $.1^4 + O(2^50)*$.1^3 + (476095097870753*2^3 + O(2^53))*$.1^... ) In file "/usr/local/magma/package/Geometry/Arith/sqfroots.m", line 84, column 15: >> return Roots(F:IsSquarefree); ^ Runtime error in 'Roots': Insufficient precision to find roots >> r:=TwoSelmerGroupData(J);r; ^ User error: Identifier 'r' has not been declared or assigned Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [] {@ (1, 0, 0) @} Total time: 1.310 seconds, Total memory usage: 40.03MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Thu Dec 1 17:10:08 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^7*3^8; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; //P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; //Height(P); //HeightConstant(J); RationalPoints(J:Bound:=1000); //Chabauty(P,19); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Thu Dec 1 2005 17:10:07 on modular [Seed = 483404753] ------------------------------------- TwoSelmerGroupData( J: Jacobian of Hyperelliptic Curve defined by y^2 = x^5 - 83980... ) TwoSelmerWork( C: Hyperelliptic Curve defined by y^2 = x^5 - 839808 over Ratio..., BoundType: Default, Bound: 0, UseUnits: true, Points: {}, Fields: {} ) LocalImageOdd( f: x^5 - 839808, p: 2, m1: Mapping from: Univariate Polynomial Ring in x over Rational ..., m2: Mapping from: Univariate Polynomial Ring in x over Rational ..., v: 30, Points: {} ) try( ) try( d: 2 ) SquarefreeRoots( F: $.1^4 + O(2^50)*$.1^3 - (157074761311839*2^3 + O(2^53))*$.1^... ) In file "/usr/local/magma/package/Geometry/Arith/sqfroots.m", line 84, column 15: >> return Roots(F:IsSquarefree); ^ Runtime error in 'Roots': Insufficient precision to find roots >> r:=TwoSelmerGroupData(J);r; ^ User error: Identifier 'r' has not been declared or assigned Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [] {@ (1, 0, 0) @} Total time: 1.530 seconds, Total memory usage: 40.49MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Thu Dec 1 17:10:05 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^7*3^8; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; //P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; //Height(P); //HeightConstant(J); RationalPoints(J:Bound:=1000); //Chabauty(P,19); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Thu Dec 1 2005 17:10:03 on modular [Seed = 467086546] ------------------------------------- TwoSelmerGroupData( J: Jacobian of Hyperelliptic Curve defined by y^2 = x^5 - 83980... ) TwoSelmerWork( C: Hyperelliptic Curve defined by y^2 = x^5 - 839808 over Ratio..., BoundType: Default, Bound: 0, UseUnits: true, Points: {}, Fields: {} ) LocalImageOdd( f: x^5 - 839808, p: 2, m1: Mapping from: Univariate Polynomial Ring in x over Rational ..., m2: Mapping from: Univariate Polynomial Ring in x over Rational ..., v: 30, Points: {} ) try( ) try( d: 2 ) SquarefreeRoots( F: $.1^4 + O(2^50)*$.1^3 + (193871502907809*2^3 + O(2^53))*$.1^... ) In file "/usr/local/magma/package/Geometry/Arith/sqfroots.m", line 84, column 15: >> return Roots(F:IsSquarefree); ^ Runtime error in 'Roots': Insufficient precision to find roots >> r:=TwoSelmerGroupData(J);r; ^ User error: Identifier 'r' has not been declared or assigned Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [] {@ (1, 0, 0) @} Total time: 1.940 seconds, Total memory usage: 40.94MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Thu Dec 1 17:10:01 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^7*3^8; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; //P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; //Height(P); //HeightConstant(J); RationalPoints(J:Bound:=1000); //Chabauty(P,19); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Thu Dec 1 2005 17:09:59 on modular [Seed = 383394259] ------------------------------------- TwoSelmerGroupData( J: Jacobian of Hyperelliptic Curve defined by y^2 = x^5 - 83980... ) TwoSelmerWork( C: Hyperelliptic Curve defined by y^2 = x^5 - 839808 over Ratio..., BoundType: Default, Bound: 0, UseUnits: true, Points: {}, Fields: {} ) LocalImageOdd( f: x^5 - 839808, p: 2, m1: Mapping from: Univariate Polynomial Ring in x over Rational ..., m2: Mapping from: Univariate Polynomial Ring in x over Rational ..., v: 30, Points: {} ) try( ) try( d: 2 ) SquarefreeRoots( F: $.1^4 + O(2^50)*$.1^3 + (20685491280289*2^3 + O(2^53))*$.1^2... ) In file "/usr/local/magma/package/Geometry/Arith/sqfroots.m", line 84, column 15: >> return Roots(F:IsSquarefree); ^ Runtime error in 'Roots': Insufficient precision to find roots >> r:=TwoSelmerGroupData(J);r; ^ User error: Identifier 'r' has not been declared or assigned Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [] {@ (1, 0, 0) @} Total time: 2.100 seconds, Total memory usage: 41.19MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Thu Dec 1 17:09:58 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^7*3^8; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; //P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; //Height(P); //HeightConstant(J); RationalPoints(J:Bound:=1000); //Chabauty(P,19); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Thu Dec 1 2005 17:09:57 on modular [Seed = 769727210] ------------------------------------- TwoSelmerGroupData( J: Jacobian of Hyperelliptic Curve defined by y^2 = x^5 - 83980... ) TwoSelmerWork( C: Hyperelliptic Curve defined by y^2 = x^5 - 839808 over Ratio..., BoundType: Default, Bound: 0, UseUnits: true, Points: {}, Fields: {} ) LocalImageOdd( f: x^5 - 839808, p: 2, m1: Mapping from: Univariate Polynomial Ring in x over Rational ..., m2: Mapping from: Univariate Polynomial Ring in x over Rational ..., v: 30, Points: {} ) try( ) try( d: 2 ) SquarefreeRoots( F: $.1^4 + O(2^50)*$.1^3 + (422959695731105*2^3 + O(2^53))*$.1^... ) In file "/usr/local/magma/package/Geometry/Arith/sqfroots.m", line 84, column 15: >> return Roots(F:IsSquarefree); ^ Runtime error in 'Roots': Insufficient precision to find roots >> r:=TwoSelmerGroupData(J);r; ^ User error: Identifier 'r' has not been declared or assigned Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [] {@ (1, 0, 0) @} Total time: 0.850 seconds, Total memory usage: 39.45MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Thu Dec 1 17:09:55 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^7*3^8; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; //P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; //Height(P); //HeightConstant(J); RationalPoints(J:Bound:=1000); //Chabauty(P,19); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Thu Dec 1 2005 17:09:54 on modular [Seed = 686039016] ------------------------------------- TwoSelmerGroupData( J: Jacobian of Hyperelliptic Curve defined by y^2 = x^5 - 83980... ) TwoSelmerWork( C: Hyperelliptic Curve defined by y^2 = x^5 - 839808 over Ratio..., BoundType: Default, Bound: 0, UseUnits: true, Points: {}, Fields: {} ) LocalImageOdd( f: x^5 - 839808, p: 2, m1: Mapping from: Univariate Polynomial Ring in x over Rational ..., m2: Mapping from: Univariate Polynomial Ring in x over Rational ..., v: 30, Points: {} ) try( ) try( d: 2 ) SquarefreeRoots( F: $.1^4 + O(2^50)*$.1^3 - (65742955365983*2^3 + O(2^53))*$.1^2... ) In file "/usr/local/magma/package/Geometry/Arith/sqfroots.m", line 84, column 15: >> return Roots(F:IsSquarefree); ^ Runtime error in 'Roots': Insufficient precision to find roots >> r:=TwoSelmerGroupData(J);r; ^ User error: Identifier 'r' has not been declared or assigned Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [] {@ (1, 0, 0) @} Total time: 0.940 seconds, Total memory usage: 39.55MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Thu Dec 1 17:09:52 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^7*3^8; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; //P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; //Height(P); //HeightConstant(J); RationalPoints(J:Bound:=1000); //Chabauty(P,19); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Thu Dec 1 2005 17:09:51 on modular [Seed = 669716730] ------------------------------------- TwoSelmerGroupData( J: Jacobian of Hyperelliptic Curve defined by y^2 = x^5 - 83980... ) TwoSelmerWork( C: Hyperelliptic Curve defined by y^2 = x^5 - 839808 over Ratio..., BoundType: Default, Bound: 0, UseUnits: true, Points: {}, Fields: {} ) LocalImageOdd( f: x^5 - 839808, p: 2, m1: Mapping from: Univariate Polynomial Ring in x over Rational ..., m2: Mapping from: Univariate Polynomial Ring in x over Rational ..., v: 30, Points: {} ) try( ) try( d: 2 ) SquarefreeRoots( F: $.1^4 + O(2^50)*$.1^3 - (87987361015391*2^3 + O(2^53))*$.1^2... ) In file "/usr/local/magma/package/Geometry/Arith/sqfroots.m", line 84, column 15: >> return Roots(F:IsSquarefree); ^ Runtime error in 'Roots': Insufficient precision to find roots >> r:=TwoSelmerGroupData(J);r; ^ User error: Identifier 'r' has not been declared or assigned Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [] {@ (1, 0, 0) @} Total time: 0.790 seconds, Total memory usage: 39.36MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Thu Dec 1 17:09:49 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^7*3^8; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; //P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; //Height(P); //HeightConstant(J); RationalPoints(J:Bound:=1000); //Chabauty(P,19); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Thu Dec 1 2005 17:09:48 on modular [Seed = 586028540] ------------------------------------- TwoSelmerGroupData( J: Jacobian of Hyperelliptic Curve defined by y^2 = x^5 - 83980... ) TwoSelmerWork( C: Hyperelliptic Curve defined by y^2 = x^5 - 839808 over Ratio..., BoundType: Default, Bound: 0, UseUnits: true, Points: {}, Fields: {} ) LocalImageOdd( f: x^5 - 839808, p: 2, m1: Mapping from: Univariate Polynomial Ring in x over Rational ..., m2: Mapping from: Univariate Polynomial Ring in x over Rational ..., v: 30, Points: {} ) try( ) try( d: 2 ) SquarefreeRoots( F: $.1^4 + O(2^50)*$.1^3 + (248690918693281*2^3 + O(2^53))*$.1^... ) In file "/usr/local/magma/package/Geometry/Arith/sqfroots.m", line 84, column 15: >> return Roots(F:IsSquarefree); ^ Runtime error in 'Roots': Insufficient precision to find roots >> r:=TwoSelmerGroupData(J);r; ^ User error: Identifier 'r' has not been declared or assigned Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [] {@ (1, 0, 0) @} Total time: 1.310 seconds, Total memory usage: 40.03MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Thu Dec 1 17:09:34 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^7*3^5; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; //P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; //Height(P); //HeightConstant(J); RationalPoints(J:Bound:=1000); //Chabauty(P,19); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Thu Dec 1 2005 17:09:33 on modular [Seed = 972361430] ------------------------------------- 0 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [] {@ (1, 0, 0) @} Total time: 0.980 seconds, Total memory usage: 39.57MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Thu Dec 1 17:09:31 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^7*3^5; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; //P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; //Height(P); //HeightConstant(J); RationalPoints(J:Bound:=1000); //Chabauty(P,19); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Thu Dec 1 2005 17:09:30 on modular [Seed = 888669148] ------------------------------------- TwoSelmerGroupData( J: Jacobian of Hyperelliptic Curve defined by y^2 = x^5 - 31104... ) TwoSelmerWork( C: Hyperelliptic Curve defined by y^2 = x^5 - 31104 over Ration..., BoundType: Default, Bound: 0, UseUnits: true, Points: {}, Fields: {} ) LocalImageOdd( f: x^5 - 31104, p: 2, m1: Mapping from: Univariate Polynomial Ring in x over Rational ..., m2: Mapping from: Univariate Polynomial Ring in x over Rational ..., v: 30, Points: {} ) try( ) try( d: 2 ) SquarefreeRoots( F: $.1^4 + O(2^50)*$.1^3 + (360351121932531*2^3 + O(2^53))*$.1^... ) In file "/usr/local/magma/package/Geometry/Arith/sqfroots.m", line 84, column 15: >> return Roots(F:IsSquarefree); ^ Runtime error in 'Roots': Insufficient precision to find roots >> r:=TwoSelmerGroupData(J);r; ^ User error: Identifier 'r' has not been declared or assigned Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [] {@ (1, 0, 0) @} Total time: 0.950 seconds, Total memory usage: 39.64MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Thu Dec 1 17:09:28 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^7*3^5; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; //P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; //Height(P); //HeightConstant(J); RationalPoints(J:Bound:=1000); //Chabauty(P,19); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Thu Dec 1 2005 17:09:27 on modular [Seed = 1300331176] ------------------------------------- TwoSelmerGroupData( J: Jacobian of Hyperelliptic Curve defined by y^2 = x^5 - 31104... ) TwoSelmerWork( C: Hyperelliptic Curve defined by y^2 = x^5 - 31104 over Ration..., BoundType: Default, Bound: 0, UseUnits: true, Points: {}, Fields: {} ) LocalImageOdd( f: x^5 - 31104, p: 2, m1: Mapping from: Univariate Polynomial Ring in x over Rational ..., m2: Mapping from: Univariate Polynomial Ring in x over Rational ..., v: 30, Points: {} ) try( ) try( d: 2 ) SquarefreeRoots( F: $.1^4 + O(2^50)*$.1^3 + (223198192861427*2^3 + O(2^53))*$.1^... ) In file "/usr/local/magma/package/Geometry/Arith/sqfroots.m", line 84, column 15: >> return Roots(F:IsSquarefree); ^ Runtime error in 'Roots': Insufficient precision to find roots >> r:=TwoSelmerGroupData(J);r; ^ User error: Identifier 'r' has not been declared or assigned Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [] {@ (1, 0, 0) @} Total time: 0.790 seconds, Total memory usage: 39.45MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Thu Dec 1 17:09:26 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^7*3^5; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; //P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; //Height(P); //HeightConstant(J); RationalPoints(J:Bound:=1000); //Chabauty(P,19); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Thu Dec 1 2005 17:09:24 on modular [Seed = 1216642990] ------------------------------------- TwoSelmerGroupData( J: Jacobian of Hyperelliptic Curve defined by y^2 = x^5 - 31104... ) TwoSelmerWork( C: Hyperelliptic Curve defined by y^2 = x^5 - 31104 over Ration..., BoundType: Default, Bound: 0, UseUnits: true, Points: {}, Fields: {} ) LocalImageOdd( f: x^5 - 31104, p: 2, m1: Mapping from: Univariate Polynomial Ring in x over Rational ..., m2: Mapping from: Univariate Polynomial Ring in x over Rational ..., v: 30, Points: {} ) try( ) try( d: 2 ) SquarefreeRoots( F: $.1^4 + O(2^50)*$.1^3 + (322914534490355*2^3 + O(2^53))*$.1^... ) In file "/usr/local/magma/package/Geometry/Arith/sqfroots.m", line 84, column 15: >> return Roots(F:IsSquarefree); ^ Runtime error in 'Roots': Insufficient precision to find roots >> r:=TwoSelmerGroupData(J);r; ^ User error: Identifier 'r' has not been declared or assigned Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [] {@ (1, 0, 0) @} Total time: 1.179 seconds, Total memory usage: 40.06MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Thu Dec 1 17:09:23 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^7*3^5; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; //P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; //Height(P); //HeightConstant(J); RationalPoints(J:Bound:=1000); //Chabauty(P,19); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Thu Dec 1 2005 17:09:20 on modular [Seed = 1200320677] ------------------------------------- TwoSelmerGroupData( J: Jacobian of Hyperelliptic Curve defined by y^2 = x^5 - 31104... ) TwoSelmerWork( C: Hyperelliptic Curve defined by y^2 = x^5 - 31104 over Ration..., BoundType: Default, Bound: 0, UseUnits: true, Points: {}, Fields: {} ) LocalImageOdd( f: x^5 - 31104, p: 2, m1: Mapping from: Univariate Polynomial Ring in x over Rational ..., m2: Mapping from: Univariate Polynomial Ring in x over Rational ..., v: 30, Points: {} ) try( ) try( d: 2 ) SquarefreeRoots( F: $.1^4 + O(2^50)*$.1^3 + (244790732652787*2^3 + O(2^53))*$.1^... ) In file "/usr/local/magma/package/Geometry/Arith/sqfroots.m", line 84, column 15: >> return Roots(F:IsSquarefree); ^ Runtime error in 'Roots': Insufficient precision to find roots >> r:=TwoSelmerGroupData(J);r; ^ User error: Identifier 'r' has not been declared or assigned Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [] {@ (1, 0, 0) @} Total time: 2.069 seconds, Total memory usage: 41.31MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Thu Dec 1 17:09:18 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^7*3^5; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; //P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; //Height(P); //HeightConstant(J); RationalPoints(J:Bound:=1000); //Chabauty(P,19); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Thu Dec 1 2005 17:09:17 on modular [Seed = 1116632509] ------------------------------------- TwoSelmerGroupData( J: Jacobian of Hyperelliptic Curve defined by y^2 = x^5 - 31104... ) TwoSelmerWork( C: Hyperelliptic Curve defined by y^2 = x^5 - 31104 over Ration..., BoundType: Default, Bound: 0, UseUnits: true, Points: {}, Fields: {} ) LocalImageOdd( f: x^5 - 31104, p: 2, m1: Mapping from: Univariate Polynomial Ring in x over Rational ..., m2: Mapping from: Univariate Polynomial Ring in x over Rational ..., v: 30, Points: {} ) try( ) try( d: 2 ) SquarefreeRoots( F: $.1^4 + O(2^50)*$.1^3 - (392528053362445*2^3 + O(2^53))*$.1^... ) In file "/usr/local/magma/package/Geometry/Arith/sqfroots.m", line 84, column 15: >> return Roots(F:IsSquarefree); ^ Runtime error in 'Roots': Insufficient precision to find roots >> r:=TwoSelmerGroupData(J);r; ^ User error: Identifier 'r' has not been declared or assigned Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [] {@ (1, 0, 0) @} Total time: 0.800 seconds, Total memory usage: 39.46MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Thu Dec 1 17:09:08 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^6*3^7; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; //P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; //Height(P); //HeightConstant(J); RationalPoints(J:Bound:=1000); //Chabauty(P,19); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Thu Dec 1 2005 17:09:07 on modular [Seed = 1502965416] ------------------------------------- 0 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [] {@ (1, 0, 0) @} Total time: 0.930 seconds, Total memory usage: 39.12MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Thu Dec 1 17:09:00 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^6*3^4; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; //P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; //Height(P); //HeightConstant(J); RationalPoints(J:Bound:=1000); //Chabauty(P,19); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Thu Dec 1 2005 17:08:59 on modular [Seed = 1755602785] ------------------------------------- 0 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [] {@ (1, 0, 0) @} Total time: 1.060 seconds, Total memory usage: 39.20MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Thu Dec 1 17:08:57 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^6*3^4; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; //P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; //Height(P); //HeightConstant(J); RationalPoints(J:Bound:=1000); //Chabauty(P,19); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Thu Dec 1 2005 17:08:56 on modular [Seed = 1739280493] ------------------------------------- TwoSelmerGroupData( J: Jacobian of Hyperelliptic Curve defined by y^2 = x^5 - 5184 ... ) TwoSelmerWork( C: Hyperelliptic Curve defined by y^2 = x^5 - 5184 over Rationa..., BoundType: Default, Bound: 0, UseUnits: true, Points: {}, Fields: {} ) LocalImageOdd( f: x^5 - 5184, p: 2, m1: Mapping from: Univariate Polynomial Ring in x over Rational ..., m2: Mapping from: Univariate Polynomial Ring in x over Rational ..., v: 26, Points: {} ) try( ) try( d: 2 ) SquarefreeRoots( F: $.1^4 + O(2^50)*$.1^3 - (460650123493295*2^2 + O(2^52))*$.1^... ) In file "/usr/local/magma/package/Geometry/Arith/sqfroots.m", line 84, column 15: >> return Roots(F:IsSquarefree); ^ Runtime error in 'Roots': Insufficient precision to find roots >> r:=TwoSelmerGroupData(J);r; ^ User error: Identifier 'r' has not been declared or assigned Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [] {@ (1, 0, 0) @} Total time: 0.940 seconds, Total memory usage: 39.31MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Thu Dec 1 17:08:53 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^6*3^4; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; //P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; //Height(P); //HeightConstant(J); RationalPoints(J:Bound:=1000); //Chabauty(P,19); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Thu Dec 1 2005 17:08:52 on modular [Seed = 1655592292] ------------------------------------- TwoSelmerGroupData( J: Jacobian of Hyperelliptic Curve defined by y^2 = x^5 - 5184 ... ) TwoSelmerWork( C: Hyperelliptic Curve defined by y^2 = x^5 - 5184 over Rationa..., BoundType: Default, Bound: 0, UseUnits: true, Points: {}, Fields: {} ) LocalImageOdd( f: x^5 - 5184, p: 2, m1: Mapping from: Univariate Polynomial Ring in x over Rational ..., m2: Mapping from: Univariate Polynomial Ring in x over Rational ..., v: 26, Points: {} ) try( ) try( d: 2 ) SquarefreeRoots( F: $.1^4 + O(2^50)*$.1^3 + (423969681244241*2^2 + O(2^52))*$.1^... ) In file "/usr/local/magma/package/Geometry/Arith/sqfroots.m", line 84, column 15: >> return Roots(F:IsSquarefree); ^ Runtime error in 'Roots': Insufficient precision to find roots >> r:=TwoSelmerGroupData(J);r; ^ User error: Identifier 'r' has not been declared or assigned Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [] {@ (1, 0, 0) @} Total time: 1.070 seconds, Total memory usage: 39.50MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Thu Dec 1 17:08:48 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^6*3^3; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; //P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; //Height(P); //HeightConstant(J); RationalPoints(J:Bound:=1000); //Chabauty(P,19); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Thu Dec 1 2005 17:08:47 on modular [Seed = 2041925220] ------------------------------------- 0 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [] {@ (1, 0, 0) @} Total time: 0.960 seconds, Total memory usage: 39.37MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Thu Dec 1 17:08:37 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^6*3^2; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; //P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; //Height(P); //HeightConstant(J); RationalPoints(J:Bound:=1000); //Chabauty(P,19); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Thu Dec 1 2005 17:08:36 on modular [Seed = 1958232930] ------------------------------------- 0 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [] {@ (1, 0, 0) @} Total time: 1.110 seconds, Total memory usage: 39.37MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Thu Dec 1 17:08:28 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^6*3^1; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; //P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; //Height(P); //HeightConstant(J); RationalPoints(J:Bound:=1000); //Chabauty(P,19); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Thu Dec 1 2005 17:08:27 on modular [Seed = 1941914714] ------------------------------------- 0 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [] {@ (1, 0, 0) @} Total time: 0.920 seconds, Total memory usage: 39.09MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Thu Dec 1 17:08:18 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^5*3^9; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; //P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; //Height(P); //HeightConstant(J); RationalPoints(J:Bound:=1000); //Chabauty(P,19); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Thu Dec 1 2005 17:08:17 on modular [Seed = 2403580077] ------------------------------------- 0 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [] {@ (1, 0, 0) @} Total time: 1.000 seconds, Total memory usage: 39.38MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Thu Dec 1 17:08:11 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^5*3^1; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; //P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; //Height(P); //HeightConstant(J); RationalPoints(J:Bound:=1000); //Chabauty(P,19); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Thu Dec 1 2005 17:08:10 on modular [Seed = 2219357117] ------------------------------------- 0 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [] {@ (1, 0, 0) @} Total time: 0.940 seconds, Total memory usage: 39.38MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Thu Dec 1 17:00:29 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^4*3^9; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; //P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; //Height(P); //HeightConstant(J); RationalPoints(J:Bound:=1000); //Chabauty(P,19); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Thu Dec 1 2005 17:00:28 on modular [Seed = 3995262945] ------------------------------------- 0 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [] {@ (1, 0, 0) @} Total time: 0.910 seconds, Total memory usage: 39.12MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Thu Dec 1 17:00:24 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^4*3^8; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; //P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; //Height(P); //HeightConstant(J); RationalPoints(J:Bound:=1000); //Chabauty(P,19); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Thu Dec 1 2005 17:00:23 on modular [Seed = 3945257711] ------------------------------------- 0 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [] {@ (1, 0, 0) @} Total time: 0.880 seconds, Total memory usage: 39.06MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Thu Dec 1 17:00:19 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^4*3^7; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; //P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; //Height(P); //HeightConstant(J); RationalPoints(J:Bound:=1000); //Chabauty(P,19); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Thu Dec 1 2005 17:00:18 on modular [Seed = 3761033697] ------------------------------------- 0 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [] {@ (1, 0, 0) @} Total time: 0.930 seconds, Total memory usage: 39.37MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Thu Dec 1 16:58:59 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^4*3^3; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; //P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; //Height(P); //HeightConstant(J); RationalPoints(J:Bound:=1000); //Chabauty(P,19); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Thu Dec 1 2005 16:58:58 on modular [Seed = 4247899308] ------------------------------------- 0 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [] {@ (1, 0, 0) @} Total time: 0.910 seconds, Total memory usage: 39.16MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Thu Dec 1 16:58:06 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^3*3^7; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; //P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; //Height(P); //HeightConstant(J); RationalPoints(J:Bound:=1000); //Chabauty(P,19); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Thu Dec 1 2005 16:58:05 on modular [Seed = 4197894244] ------------------------------------- 0 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [] {@ (1, 0, 0) @} Total time: 0.920 seconds, Total memory usage: 39.45MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Thu Dec 1 16:58:02 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^3*3^6; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; //P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; //Height(P); //HeightConstant(J); RationalPoints(J:Bound:=1000); //Chabauty(P,19); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Thu Dec 1 2005 16:58:01 on modular [Seed = 4147889517] ------------------------------------- 0 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [] {@ (1, 0, 0) @} Total time: 0.920 seconds, Total memory usage: 39.50MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Thu Dec 1 16:56:57 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^3*3^5; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; //P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; //Height(P); //HeightConstant(J); RationalPoints(J:Bound:=1000); //Chabauty(P,19); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Thu Dec 1 2005 16:56:56 on modular [Seed = 213925591] ------------------------------------- 0 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [] {@ (1, 0, 0) @} Total time: 0.890 seconds, Total memory usage: 39.57MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Thu Dec 1 16:56:20 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^3*3^2; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; //P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; //Height(P); //HeightConstant(J); RationalPoints(J:Bound:=1000); //Chabauty(P,19); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Thu Dec 1 2005 16:56:18 on modular [Seed = 163920889] ------------------------------------- 0 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [] {@ (1, 0, 0) @} Total time: 0.950 seconds, Total memory usage: 39.41MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Thu Dec 1 16:55:35 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^2*3^9; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; //P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; //Height(P); //HeightConstant(J); RationalPoints(J:Bound:=1000); //Chabauty(P,19); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Thu Dec 1 2005 16:55:34 on modular [Seed = 130757922] ------------------------------------- 0 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [] {@ (1, 0, 0) @} Total time: 0.860 seconds, Total memory usage: 39.13MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Thu Dec 1 16:54:55 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^2*3^6; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; //P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; //Height(P); //HeightConstant(J); RationalPoints(J:Bound:=1000); //Chabauty(P,19); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Thu Dec 1 2005 16:54:54 on modular [Seed = 483405377] ------------------------------------- 0 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [] {@ (1, 0, 0) @} Total time: 0.920 seconds, Total memory usage: 39.39MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Thu Dec 1 16:54:49 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^2*3^5; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; //P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; //Height(P); //HeightConstant(J); RationalPoints(J:Bound:=1000); //Chabauty(P,19); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Thu Dec 1 2005 16:54:48 on modular [Seed = 433400642] ------------------------------------- 0 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [] {@ (1, 0, 0) @} Total time: 0.860 seconds, Total memory usage: 39.19MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Thu Dec 1 16:53:09 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^2*3^4; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; //P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; //Height(P); //HeightConstant(J); RationalPoints(J:Bound:=1000); //Chabauty(P,19); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Thu Dec 1 2005 16:53:08 on modular [Seed = 383395233] ------------------------------------- 0 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [] {@ (1, 0, 0) @} Total time: 0.870 seconds, Total memory usage: 39.14MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Thu Dec 1 16:53:03 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^2*3^3; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; //P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; //Height(P); //HeightConstant(J); RationalPoints(J:Bound:=1000); //Chabauty(P,19); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Thu Dec 1 2005 16:53:02 on modular [Seed = 333390046] ------------------------------------- 0 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [] {@ (1, 0, 0) @} Total time: 0.910 seconds, Total memory usage: 39.09MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Thu Dec 1 16:52:58 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^2*3^2; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; //P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; //Height(P); //HeightConstant(J); RationalPoints(J:Bound:=1000); //Chabauty(P,19); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Thu Dec 1 2005 16:52:57 on modular [Seed = 719722976] ------------------------------------- 0 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [] {@ (1, 0, 0) @} Total time: 0.880 seconds, Total memory usage: 39.11MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Thu Dec 1 16:51:32 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^2*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; //P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; //Height(P); //HeightConstant(J); RationalPoints(J:Bound:=1000); //Chabauty(P,19); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Thu Dec 1 2005 16:51:31 on modular [Seed = 669717559] ------------------------------------- 0 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [] {@ (1, 0, 0) @} Total time: 0.860 seconds, Total memory usage: 39.19MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Thu Dec 1 16:50:51 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^1*3^8; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; //P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; //Height(P); //HeightConstant(J); RationalPoints(J:Bound:=1000); //Chabauty(P,19); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Thu Dec 1 2005 16:50:50 on modular [Seed = 1022365557] ------------------------------------- 0 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [] {@ (1, 0, 0) @} Total time: 1.129 seconds, Total memory usage: 39.70MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Thu Dec 1 16:49:45 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^1*3^3; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; //P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; //Height(P); //HeightConstant(J); RationalPoints(J:Bound:=1000); //Chabauty(P,19); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Thu Dec 1 2005 16:49:44 on modular [Seed = 989206193] ------------------------------------- 2 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [] {@ (1, 0, 0) @} Total time: 1.040 seconds, Total memory usage: 39.67MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Thu Dec 1 16:48:32 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^1*3^1; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; //P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; //Height(P); //HeightConstant(J); RationalPoints(J:Bound:=1000); //Chabauty(P,19); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Thu Dec 1 2005 16:48:31 on modular [Seed = 939203563] ------------------------------------- 0 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [] {@ (1, 0, 0) @} Total time: 1.020 seconds, Total memory usage: 39.82MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Thu Dec 1 16:47:41 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^1*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; //P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; //Height(P); //HeightConstant(J); RationalPoints(J:Bound:=1000); //Chabauty(P,19); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Thu Dec 1 2005 16:47:40 on modular [Seed = 1283481667] ------------------------------------- 0 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [] {@ (1, 0, 0) @} Total time: 0.960 seconds, Total memory usage: 39.82MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Thu Dec 1 16:47:38 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^1*3^0; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; //P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; //Height(P); //HeightConstant(J); RationalPoints(J:Bound:=1000); //Chabauty(P,19); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Thu Dec 1 2005 16:47:37 on modular [Seed = 1250321984] ------------------------------------- TwoSelmerGroupData( J: Jacobian of Hyperelliptic Curve defined by y^2 = x^5 - 2 ove... ) TwoSelmerWork( C: Hyperelliptic Curve defined by y^2 = x^5 - 2 over Rational F..., BoundType: Default, Bound: 0, UseUnits: true, Points: {}, Fields: {} ) LocalImageOdd( f: x^5 - 2, p: 2, m1: Mapping from: Univariate Polynomial Ring in x over Rational ..., m2: Mapping from: Univariate Polynomial Ring in x over Rational ..., v: 6, Points: {} ) try( ) try( d: 2 ) SquarefreeRoots( F: $.1^4 + O(2^50)*$.1^3 + (342432677889*2^3 + O(2^53))*$.1^2 +... ) In file "/usr/local/magma/package/Geometry/Arith/sqfroots.m", line 84, column 15: >> return Roots(F:IsSquarefree); ^ Runtime error in 'Roots': Insufficient precision to find roots >> r:=TwoSelmerGroupData(J);r; ^ User error: Identifier 'r' has not been declared or assigned Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [] {@ (1, 0, 0) @} Total time: 0.810 seconds, Total memory usage: 39.80MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Thu Dec 1 16:46:58 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^0*3^8; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; //P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; //Height(P); //HeightConstant(J); RationalPoints(J:Bound:=1000); //Chabauty(P,19); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Thu Dec 1 2005 16:46:57 on modular [Seed = 1082944610] ------------------------------------- 0 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [] {@ (1, 0, 0) @} Total time: 0.860 seconds, Total memory usage: 39.14MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Thu Dec 1 16:45:42 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^0*3^5; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; //P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; //Height(P); //HeightConstant(J); RationalPoints(J:Bound:=1000); //Chabauty(P,19); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Thu Dec 1 2005 16:45:41 on modular [Seed = 1586655784] ------------------------------------- 0 Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 Mapping from: Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*P[1] = 0 to JacHyp: J given by a rule [no inverse] [] {@ (1, 0, 0), (x - 3, 0, 1) @} Total time: 0.870 seconds, Total memory usage: 39.65MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Thu Dec 1 16:44:32 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^0*3^2; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; //P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; //Height(P); //HeightConstant(J); RationalPoints(J:Bound:=1000); //Chabauty(P,19); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Thu Dec 1 2005 16:44:31 on modular [Seed = 1536652951] ------------------------------------- 0 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [] {@ (1, 0, 0) @} Total time: 0.840 seconds, Total memory usage: 39.08MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Thu Dec 1 16:44:26 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5-2^0*3^1; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; //P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; //Height(P); //HeightConstant(J); RationalPoints(J:Bound:=1000); //Chabauty(P,19); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Thu Dec 1 2005 16:44:25 on modular [Seed = 1352433050] ------------------------------------- 0 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [] {@ (1, 0, 0) @} Total time: 0.830 seconds, Total memory usage: 39.17MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Thu Dec 1 16:41:24 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^9*3^9; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; //P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; //Height(P); //HeightConstant(J); RationalPoints(J:Bound:=1000); //Chabauty(P,19); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Thu Dec 1 2005 16:41:23 on modular [Seed = 1839284253] ------------------------------------- 0 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [] {@ (1, 0, 0) @} Total time: 1.070 seconds, Total memory usage: 39.08MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Thu Dec 1 16:41:17 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^9*3^8; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; //P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; //Height(P); //HeightConstant(J); RationalPoints(J:Bound:=1000); //Chabauty(P,19); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Thu Dec 1 2005 16:41:16 on modular [Seed = 1789281509] ------------------------------------- 0 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [] {@ (1, 0, 0) @} Total time: 1.080 seconds, Total memory usage: 39.14MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Thu Dec 1 16:41:11 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^9*3^7; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; //P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; //Height(P); //HeightConstant(J); RationalPoints(J:Bound:=1000); //Chabauty(P,19); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Thu Dec 1 2005 16:41:10 on modular [Seed = 1739278818] ------------------------------------- 0 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [] {@ (1, 0, 0) @} Total time: 1.189 seconds, Total memory usage: 39.46MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Thu Dec 1 16:38:51 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^9*3^4; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; //P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; //Height(P); //HeightConstant(J); RationalPoints(J:Bound:=1000); //Chabauty(P,19); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Thu Dec 1 2005 16:38:50 on modular [Seed = 1908236136] ------------------------------------- 0 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [] {@ (1, 0, 0) @} Total time: 1.159 seconds, Total memory usage: 39.18MB '24.80.1' ************** MAGMA ***************** Host 24.80.117.231 (24.80.117.231) Time: Thu Dec 1 16:37:42 2005 Input: _:=PolynomialRing(Rationals()); f:=x^5+2^9*3^2; C:=HyperellipticCurve(f); J:=Jacobian(C); r:=TwoSelmerGroupData(J);r; TorsionSubgroup(J); V:=RationalPoints(J:Bound:=1000); B:=ReducedBasis(V); B; //P:=J!B[1]; //Q:=J![x,54]; //P1:=J![x^2+6*x+36,12*x-18]; //Height(P); //HeightConstant(J); RationalPoints(J:Bound:=1000); //Chabauty(P,19); //Order(Q);Order(P); //2*P; Output: Magma V2.11-10 Thu Dec 1 2005 16:37:41 on modular [Seed = 2539370974] ------------------------------------- 0 Abelian Group of order 1 Mapping from: Abelian Group of order 1 to JacHyp: J given by a rule [no inverse] [] {@ (1, 0, 0) @} Total time: 1.240 seconds, Total memory usage: 39.51MB '139.182' ************** MAGMA ***************** Host 139.182.146.132 (139.182.146.132) Time: Thu Dec 1 16:37:20 2005 Input: G:=Group