CoCalc Shared Fileswww / calc / calc / logfile_magma.txt
Author: William A. Stein

'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<x,y,z,u,v>:=PolynomialRing(RationalField(),5); I:=ideal<P |
-x + (y^2+1),
-y + (z^2+1),
-z + (u^2+1),
-u + (v^2+1),
-v + (x^2+1) >;

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
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<x,y,z,u,v>:=PolynomialRing(RationalField(),5); I:=ideal<P |
-x + (y^2+y+1),
-y + (z^2+z+1),
-z + (u^2+u+1),
-u + (v^2+v+1),
-v + (x^2+x+1) >;

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
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<x,y,z,u,v>:=PolynomialRing(RationalField(),3); I:=ideal<P |
-x + (y^2+y+1),
-y + (z^2+z+1),
-z + (u^2+u+1),
-u + (v^2+v+1),
-v + (x^2+x+1) >;

Output: Magma V2.11-10    Tue Dec 20 2005 09:24:05 on modular  [Seed = 3176590049]
-------------------------------------

>> P<x,y,z,u,v>:=PolynomialRing(RationalField(),3); I:=ideal<P |
^
Runtime error in 'AssignNames': Argument 2 should have length at most 3

>> -x + (y^2+y+1),
^
User error: Identifier 'x' has not been declared or assigned

^
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<x,y,z>:=PolynomialRing(RationalField(),3); I:=ideal<P |
-x + (y^2+y+1),
-y + (z^2+z+1),
-z + (x^2+x+1) >;

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
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<x,y,z>:=PolynomialRing(RationalField(),3); I:=ideal<P |
-x + y^2+y+1),
-y + z^2+z+1),
-z + (x^2+x+1) >;

Output: Magma V2.11-10    Tue Dec 20 2005 09:18:42 on modular  [Seed = 2625096967]
-------------------------------------

>> -x + y^2+y+1),
^

>> -y + z^2+z+1),
^

>> -z + (x^2+x+1) >;
^

^
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;
^

>> b := 00000009 8e59a5bf 2e0edbab 1487fcce ba428afa 855dfd12;
^

>> 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;
^

>> b := 000000098e59a5bf2e0edbab1487fcceba428afa855dfd12;
^

>> 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\
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0121988303345502026704739350854447208331960688620845925110719853167768327430021\
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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!;
^

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<x> := PolynomialRing(Integers());
> > f :=x^4-26*x^2+1;
> > K<y> := NumberField(f);
> > O := MaximalOrder(K);
> > I := IntegralBasis(K);
> > print I;
p:=3;
q:=7;
n:=p*q;
> >NormEquation(O, 4*n^4);
K<a> := 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<x> := PolynomialRing(Integers());
> > f :=x^4-26*x^2+1;
> > K<y> := NumberField(f);
> > O := MaximalOrder(K);
> > I := IntegralBasis(K);
> > print I;
p:=3;
q:=7;
n:=p*q;
> >NormEquation(O, 4*n^4);
K<a> := 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<x> := PolynomialRing(Integers());
> > f :=x^4-34*x^2+121;
> > K<y> := NumberField(f);
> > O := MaximalOrder(K);
> > I := IntegralBasis(K);
> > print I;
p:=3;
q:=7;
n:=p*q;
> >NormEquation(O, 4*n^4);
K<a> := 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<x> := PolynomialRing(Integers());
> > f :=x^4-34*x^2+121;
> > K<y> := NumberField(f);
> > O := MaximalOrder(K);
> > I := IntegralBasis(K);
> > print I;
p:=3;
q:=7;
n:=p*q;
> >NormEquation(O, 4*n^4);
K<a> := 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<x> := PolynomialRing(Integers());
> > f :=x^4-34*x^2+121;
> > K<y> := NumberField(f);
> > O := MaximalOrder(K);
> > I := IntegralBasis(K);
> > print I;
p:=3;
q:=7;
n:=p*q;
> >NormEquation(O, 4*n^4);
K<a> := 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<x> := PolynomialRing(Integers());
> > f :=x^4-20*x^2+16;
> > K<y> := NumberField(f);
> > O := MaximalOrder(K);
> > I := IntegralBasis(K);
> > print I;
p:=3;
q:=7;
n:=p*q;
> >NormEquation(O, 4*n^4);
K<a> := 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<x> := PolynomialRing(Integers());
> > f :=x^4-20*x^2+16;
> > K<y> := NumberField(f);
> > O := MaximalOrder(K);
> > I := IntegralBasis(K);
> > print I;
p:=3;
q:=7;
n:=p*q;
> >NormEquation(O, 4*n^4);
K<a> := 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<x> := PolynomialRing(Integers());
> > f :=x^4-20*x^2+16;
> > K<y> := NumberField(f);
> > O := MaximalOrder(K);
> > I := IntegralBasis(K);
> > print I;
p:=3;
q:=7;
n:=p*q;
> >NormEquation(O, 4*n^4);
K<a> := 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<x> := PolynomialRing(Integers());
> > f :=x^4-20*x^2+16;
> > K<y> := NumberField(f);
> > O := MaximalOrder(K);
> > I := IntegralBasis(K);
> > print I;
p:=3;
q:=7;
n:=p*q;
> >NormEquation(O, 4*n^4);
K<a> := 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<x> := PolynomialRing(Integers());
> > f :=x^4-20*x^2+16;
> > K<y> := NumberField(f);
> > O := MaximalOrder(K);
> > I := IntegralBasis(K);
> > print I;
p:=3;
q:=7;
n:=p*q;
> >NormEquation(O, 4*n^4);
K<a> := 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<x> := PolynomialRing(Integers());
> > f :=x^4-20*x^2+16;
> > K<y> := NumberField(f);
> > O := MaximalOrder(K);
> > I := IntegralBasis(K);
> > print I;
p:=3;
q:=7;
n:=p*q;
> >NormEquation(O, 4*n^4);
K<a> := 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));;
^

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<x> := PolynomialRing(Integers());
> > f :=x^4-20*x^2+16;
> > K<y> := NumberField(f);
> > O := MaximalOrder(K);
> > I := IntegralBasis(K);
> > print I;
p:=3;
q:=7;
n:=p*q;
> >NormEquation(O, 4*n^4);
K<a> := 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;
^

>> u1:=56-105*z-63z^2+49*z^3;
^

>> UnitEquation(a*u1,a*u1,2*a*n));;
^

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<x> := PolynomialRing(Integers());
> > f :=x^4-20*x^2+16;
> > K<y> := 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<x> := PolynomialRing(Integers());
> > f :=x^4-4*x^2+2;
> > K<y> := 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<x> := PolynomialRing(Integers());
> > f :=x^4-4*x^2+2;
> > K<y> := 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<x,y,z>:=PolynomialRing(RationalField(),3); I:=ideal<P |
-x + (-6*y^2+7*y+5),
-y + (-6*z^2+7*z+5),
-z^9 + (-6*x^2+7*x+5) >;

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
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<x,y,z>:=PolynomialRing(RationalField(),3); I:=ideal<P |
-x + (-6*y^2+7*y+5),
-y + (-6*z^2+7*z+5),
-z^3 + (-6*x^2+7*x+5) >;

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
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<x,y,z>:=PolynomialRing(RationalField(),3); I:=ideal<P |
-x^3 + (-6*y^2+7*y+5),
-y^3 + (-6*z^2+7*z+5),
-z^3 + (-6*x^2+7*x+5) >;

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
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<x,y,z>:=PolynomialRing(RationalField(),3); I:=ideal<P |
-x^8 + (-6*y^3+7*y+5),
-y^8 + (-6*z^3+7*z+5),
-z^8 + (-6*x^3+7*x+5) >;

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<x,y,z>:=PolynomialRing(RationalField(),3); I:=ideal<P |
-x^6 + (-6*y^3+7*y+5),
-y^7 + (-6*z^3+7*z+5),
-z^8 + (-6*x^3+7*x+5) >;

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<x,y,z>:=PolynomialRing(RationalField(),3); I:=ideal<P |
-x^2 + (-6*y^3+7*y+5),
-y^2 + (-6*z^3+7*z+5),
-z^2 + (-6*x^3+7*x+5) >;

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
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 -
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<K, 56 |   [1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0],[0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0],[0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1],[0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,1,1,1,0,1,0,0,0,1,1,1,0,1,1,0,0,1,1,1,0],[0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,1,1,1,0,1,1,0,0,1,1,1,0,0,1,0,0,1,1,1],[0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,1,0,0,1,1,1,0,0,1,0,0,1,1,1,1,0,1,0,0,1,1],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,1,0,0,1,1,1,1,1,0,1,0,0,1,0,0,1,1,1,0,1],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,1,0,1,0,0,1,1,1,1,1,0,1,0,0,1,0,0,1,1,1,0],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,1,1,0,1,1,1,0,1,0,0,1,0,1,1,1,0,1,0,0,1,0,0,1,1,1],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,0,1,0,0,1,1,0,1,0,0,1,1,1,0,1,1,1,0,1,0],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,1,1,0,1,0,0,1,1,0,1,0,0,1,1,0,0,1,1,1,0,1],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,1,1,1,0,1,0,0,1,1,0,1,0,0,1,1,0,0,1,1,1,0],[1,1,1,0,0,1,0,1,1,1,0,0,1,0,1,1,0,0,1,0,1,0,0,1,0,1,1,1,1,0,0,1,0,1,1,0,0,0,0,0,0,0,1,0,0,1,0,1,1,1,1,1,0,0,1,0],[0,1,1,1,0,0,1,0,1,1,1,0,0,1,1,1,1,0,0,1,0,1,0,0,1,0,1,1,1,1,0,0,1,0,1,0,0,0,0,0,0,0,1,1,0,0,1,0,1,0,1,1,1,0,0,1],[1,0,1,1,1,0,0,1,0,1,1,1,0,0,0,1,1,1,0,0,1,1,1,0,0,1,0,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0,1,1,1,0,0,1,0,1,0,1,1,1,0,0],[1,1,1,0,0,1,0,0,1,0,1,1,1,0,0,0,1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0,0,0,0,1,0,0,1,0,1,1,1,0,0,1,0,1,1,1,1,1,0,0,1,0],[0,1,1,1,0,0,1,0,0,1,0,1,1,1,1,0,0,1,0,1,1,0,0,1,0,1,1,1,0,0,0,0,0,0,0,1,1,0,0,1,0,1,1,1,0,0,1,0,1,0,1,1,1,0,0,1],[1,0,1,1,1,0,0,1,0,0,1,0,1,1,1,1,0,0,1,0,1,1,0,0,1,0,1,1,0,0,0,0,0,0,0,1,1,1,0,0,1,0,1,1,1,0,0,1,0,1,0,1,1,1,0,0],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,1,1,1,1,1],[1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]>;
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<K, 56 |   [1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0],[0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0],[0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1],[0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,1,1,1,0,1,0,0,0,1,1,1,0,1,1,0,0,1,1,1,0],[0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,1,1,1,0,1,1,0,0,1,1,1,0,0,1,0,0,1,1,1],[0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,1,0,0,1,1,1,0,0,1,0,0,1,1,1,1,0,1,0,0,1,1],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,1,0,0,1,1,1,1,1,0,1,0,0,1,0,0,1,1,1,0,1],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,1,0,1,0,0,1,1,1,1,1,0,1,0,0,1,0,0,1,1,1,0],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,1,1,0,1,1,1,0,1,0,0,1,0,1,1,1,0,1,0,0,1,0,0,1,1,1],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,0,1,0,0,1,1,0,1,0,0,1,1,1,0,1,1,1,0,1,0],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,1,1,0,1,0,0,1,1,0,1,0,0,1,1,0,0,1,1,1,0,1],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,1,1,1,0,1,0,0,1,1,0,1,0,0,1,1,0,0,1,1,1,0],[1,1,1,0,0,1,0,1,1,1,0,0,1,0,1,1,0,0,1,0,1,0,0,1,0,1,1,1,1,0,0,1,0,1,1,0,0,0,0,0,0,0,1,0,0,1,0,1,1,1,1,1,0,0,1,0],[0,1,1,1,0,0,1,0,1,1,1,0,0,1,1,1,1,0,0,1,0,1,0,0,1,0,1,1,1,1,0,0,1,0,1,0,0,0,0,0,0,0,1,1,0,0,1,0,1,0,1,1,1,0,0,1],[1,0,1,1,1,0,0,1,0,1,1,1,0,0,0,1,1,1,0,0,1,1,1,0,0,1,0,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0,1,1,1,0,0,1,0,1,0,1,1,1,0,0],[1,1,1,0,0,1,0,0,1,0,1,1,1,0,0,0,1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0,0,0,0,1,0,0,1,0,1,1,1,0,0,1,0,1,1,1,1,1,0,0,1,0],[0,1,1,1,0,0,1,0,0,1,0,1,1,1,1,0,0,1,0,1,1,0,0,1,0,1,1,1,0,0,0,0,0,0,0,1,1,0,0,1,0,1,1,1,0,0,1,0,1,0,1,1,1,0,0,1],[1,0,1,1,1,0,0,1,0,0,1,0,1,1,1,1,0,0,1,0,1,1,0,0,1,0,1,1,0,0,0,0,0,0,0,1,1,1,0,0,1,0,1,1,1,0,0,1,0,1,0,1,1,1,0,0],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,1,1,1,1,1],[1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]>;
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<K, 56 |   [1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0],[0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0],[0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1],[0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,1,1,1,0,1,0,0,0,1,1,1,0,1,1,0,0,1,1,1,0],[0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,1,1,1,0,1,1,0,0,1,1,1,0,0,1,0,0,1,1,1],[0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,1,0,0,1,1,1,0,0,1,0,0,1,1,1,1,0,1,0,0,1,1],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,1,0,0,1,1,1,1,1,0,1,0,0,1,0,0,1,1,1,0,1],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,1,0,1,0,0,1,1,1,1,1,0,1,0,0,1,0,0,1,1,1,0],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,1,1,0,1,1,1,0,1,0,0,1,0,1,1,1,0,1,0,0,1,0,0,1,1,1],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,0,1,0,0,1,1,0,1,0,0,1,1,1,0,1,1,1,0,1,0],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,1,1,0,1,0,0,1,1,0,1,0,0,1,1,0,0,1,1,1,0,1],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,1,1,1,0,1,0,0,1,1,0,1,0,0,1,1,0,0,1,1,1,0],[1,1,1,0,0,1,0,1,1,1,0,0,1,0,1,1,0,0,1,0,1,0,0,1,0,1,1,1,1,0,0,1,0,1,1,0,0,0,0,0,0,0,1,0,0,1,0,1,1,1,1,1,0,0,1,0],[0,1,1,1,0,0,1,0,1,1,1,0,0,1,1,1,1,0,0,1,0,1,0,0,1,0,1,1,1,1,0,0,1,0,1,0,0,0,0,0,0,0,1,1,0,0,1,0,1,0,1,1,1,0,0,1],[1,0,1,1,1,0,0,1,0,1,1,1,0,0,0,1,1,1,0,0,1,1,1,0,0,1,0,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0,1,1,1,0,0,1,0,1,0,1,1,1,0,0],[1,1,1,0,0,1,0,0,1,0,1,1,1,0,0,0,1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0,0,0,0,1,0,0,1,0,1,1,1,0,0,1,0,1,1,1,1,1,0,0,1,0],[0,1,1,1,0,0,1,0,0,1,0,1,1,1,1,0,0,1,0,1,1,0,0,1,0,1,1,1,0,0,0,0,0,0,0,1,1,0,0,1,0,1,1,1,0,0,1,0,1,0,1,1,1,0,0,1],[1,0,1,1,1,0,0,1,0,0,1,0,1,1,1,1,0,0,1,0,1,1,0,0,1,0,1,1,0,0,0,0,0,0,0,1,1,1,0,0,1,0,1,1,1,0,0,1,0,1,0,1,1,1,0,0],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,1,1,1,1,1],[1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]>;
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<K, 56 |   [1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0],[0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0],[0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1],[0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,1,1,1,0,1,0,0,0,1,1,1,0,1,1,0,0,1,1,1,0],[0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,1,1,1,0,1,1,0,0,1,1,1,0,0,1,0,0,1,1,1],[0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,1,0,0,1,1,1,0,0,1,0,0,1,1,1,1,0,1,0,0,1,1],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,1,0,0,1,1,1,1,1,0,1,0,0,1,0,0,1,1,1,0,1],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,1,0,1,0,0,1,1,1,1,1,0,1,0,0,1,0,0,1,1,1,0],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,1,1,0,1,1,1,0,1,0,0,1,0,1,1,1,0,1,0,0,1,0,0,1,1,1],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,0,1,0,0,1,1,0,1,0,0,1,1,1,0,1,1,1,0,1,0],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,1,1,0,1,0,0,1,1,0,1,0,0,1,1,0,0,1,1,1,0,1],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,1,1,1,0,1,0,0,1,1,0,1,0,0,1,1,0,0,1,1,1,0],[1,1,1,0,0,1,0,1,1,1,0,0,1,0,1,1,0,0,1,0,1,0,0,1,0,1,1,1,1,0,0,1,0,1,1,0,0,0,0,0,0,0,1,0,0,1,0,1,1,1,1,1,0,0,1,0],[0,1,1,1,0,0,1,0,1,1,1,0,0,1,1,1,1,0,0,1,0,1,0,0,1,0,1,1,1,1,0,0,1,0,1,0,0,0,0,0,0,0,1,1,0,0,1,0,1,0,1,1,1,0,0,1],[1,0,1,1,1,0,0,1,0,1,1,1,0,0,0,1,1,1,0,0,1,1,1,0,0,1,0,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0,1,1,1,0,0,1,0,1,0,1,1,1,0,0],[1,1,1,0,0,1,0,0,1,0,1,1,1,0,0,0,1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0,0,0,0,1,0,0,1,0,1,1,1,0,0,1,0,1,1,1,1,1,0,0,1,0],[0,1,1,1,0,0,1,0,0,1,0,1,1,1,1,0,0,1,0,1,1,0,0,1,0,1,1,1,0,0,0,0,0,0,0,1,1,0,0,1,0,1,1,1,0,0,1,0,1,0,1,1,1,0,0,1],[1,0,1,1,1,0,0,1,0,0,1,0,1,1,1,1,0,0,1,0,1,1,0,0,1,0,1,1,0,0,0,0,0,0,0,1,1,1,0,0,1,0,1,1,1,0,0,1,0,1,0,1,1,1,0,0],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,1,1,1,1,1],[1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]>;
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<K, 56 |   [1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0],[0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0],[0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1],[0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,1,1,1,0,1,0,0,0,1,1,1,0,1,1,0,0,1,1,1,0],[0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,1,1,1,0,1,1,0,0,1,1,1,0,0,1,0,0,1,1,1],[0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,1,0,0,1,1,1,0,0,1,0,0,1,1,1,1,0,1,0,0,1,1],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,1,0,0,1,1,1,1,1,0,1,0,0,1,0,0,1,1,1,0,1],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,1,0,1,0,0,1,1,1,1,1,0,1,0,0,1,0,0,1,1,1,0],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,1,1,0,1,1,1,0,1,0,0,1,0,1,1,1,0,1,0,0,1,0,0,1,1,1],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,0,1,0,0,1,1,0,1,0,0,1,1,1,0,1,1,1,0,1,0],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,1,1,0,1,0,0,1,1,0,1,0,0,1,1,0,0,1,1,1,0,1],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,1,1,1,0,1,0,0,1,1,0,1,0,0,1,1,0,0,1,1,1,0],[1,1,1,0,0,1,0,1,1,1,0,0,1,0,1,1,0,0,1,0,1,0,0,1,0,1,1,1,1,0,0,1,0,1,1,0,0,0,0,0,0,0,1,0,0,1,0,1,1,1,1,1,0,0,1,0],[0,1,1,1,0,0,1,0,1,1,1,0,0,1,1,1,1,0,0,1,0,1,0,0,1,0,1,1,1,1,0,0,1,0,1,0,0,0,0,0,0,0,1,1,0,0,1,0,1,0,1,1,1,0,0,1],[1,0,1,1,1,0,0,1,0,1,1,1,0,0,0,1,1,1,0,0,1,1,1,0,0,1,0,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0,1,1,1,0,0,1,0,1,0,1,1,1,0,0],[1,1,1,0,0,1,0,0,1,0,1,1,1,0,0,0,1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0,0,0,0,1,0,0,1,0,1,1,1,0,0,1,0,1,1,1,1,1,0,0,1,0],[0,1,1,1,0,0,1,0,0,1,0,1,1,1,1,0,0,1,0,1,1,0,0,1,0,1,1,1,0,0,0,0,0,0,0,1,1,0,0,1,0,1,1,1,0,0,1,0,1,0,1,1,1,0,0,1],[1,0,1,1,1,0,0,1,0,0,1,0,1,1,1,1,0,0,1,0,1,1,0,0,1,0,1,1,0,0,0,0,0,0,0,1,1,1,0,0,1,0,1,1,1,0,0,1,0,1,0,1,1,1,0,0],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,1,1,1,1,1],[1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]>;

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<x,y,z>:=PolynomialRing(RationalField(),3); I:=ideal<P |
-x + (-6*y^3+7*y+5),
-y + (-6*z^3+7*z+5),
-z + (-6*x^3+7*x+5) >;

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
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<x,y,z>:=PolynomialRing(RationalField(),3); I:=ideal<P |
-x + (-6*y^2-4*y+5),
-y + (-6*z^2-4*z+5),
-z + (-6*x^2-4*x+5) >;

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
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;
^

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<t1,t2,t3,t4>:=PolynomialRing(RationalField(),4); I:=ideal<P |
-2 + t2*t3 + t1*t4,
-2 + t1*t2*t3 + t4*t2*t3 + t1*t3*t4 + t1*t2*t4,
6 +  (t2*t3)^2 + (t1*t4)^2 - t1*t2*t3*t4 - (t1*t2*t3^2 + t4*t3*t2^2 + t1*t3*t4^2 + t1^2*t4*t2),
-6 + t1*t2^2*t3^3/t4   + t1^2*t3*t4^3/t2  +  t1^3*t2*t4^2/t3 + t4*t2^3*t3^2/t1 - 5*(t2*t3 - t1*t4)*(t1*t2*t3 + t4*t2*t3 - t1*t3*t4 - t1*t2*t4)  >;

Output: Magma V2.11-10    Sat Dec 17 2005 13:52:56 on modular  [Seed = 250155959]
-------------------------------------

>> P<t1,t2,t3,t4>:=PolynomialRing(RationalField(),4); I:=ideal<P |
^
Runtime error in ideal< ... >: Rhs argument 4 is invalid for this constructor

^
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<m1,m2,m3,m4>:=PolynomialRing(RationalField(),4); I:=ideal<P |
-1 + m1 + m2*m3/(1+m4^2)^(1/2) + (m3 + m3/(1+m4^2)^(1/2))^(1/2),
-4+ m1 + m2*m3/(1+m4^2)^(1/2) - (m3 + m3/(1+m4^2)^(1/2))^(1/2),
-2 + m1 - m2*m3/(1+m4^2)^(1/2) + (m3 - m3/(1+m4^2)^(1/2))^(1/2),
-3+ m1 - m2*m3/(1+m4^2)^(1/2) - (m3 - m3/(1+m4^2)^(1/2))^(1/2)  >;

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

^
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<m1,m2,m3,m4>:=PolynomialRing(RationalField(),4); I:=ideal<P |
-t1 + m1 + m2*m3/(1+m4^2)^(1/2) + (m3 + m3/(1+m4^2)^(1/2))^(1/2),
-t4+ m1 + m2*m3/(1+m4^2)^(1/2) - (m3 + m3/(1+m4^2)^(1/2))^(1/2),
-t2 + m1 - m2*m3/(1+m4^2)^(1/2) + (m3 - m3/(1+m4^2)^(1/2))^(1/2),
-t3 + m1 - m2*m3/(1+m4^2)^(1/2) - (m3 - m3/(1+m4^2)^(1/2))^(1/2)  >;

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

^
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<m1,m2,m3,m4>:=PolynomialRing(RationalField(),4); I:=ideal<P |
1 = m1 + m2*m3/(1+m4^2)^(1/2) + (m3 + m3/(1+m4^2)^(1/2))^(1/2),
4= m1 + m2*m3/(1+m4^2)^(1/2) - (m3 + m3/(1+m4^2)^(1/2))^(1/2),
2 = m1 - m2*m3/(1+m4^2)^(1/2) + (m3 - m3/(1+m4^2)^(1/2))^(1/2),
3 = m1 - m2*m3/(1+m4^2)^(1/2) - (m3 - m3/(1+m4^2)^(1/2))^(1/2)  >;

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

^
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<m1,m2,m3,m4>:=PolynomialRing(RationalField(),4); I:=ideal<P |
t1 = m1 + m2*m3/(1+m4^2)^(1/2) + (m3 + m3/(1+m4^2)^(1/2))^(1/2),
t4 = m1 + m2*m3/(1+m4^2)^(1/2) - (m3 + m3/(1+m4^2)^(1/2))^(1/2),
t2 = m1 - m2*m3/(1+m4^2)^(1/2) + (m3 - m3/(1+m4^2)^(1/2))^(1/2),
t3 = m1 - m2*m3/(1+m4^2)^(1/2) - (m3 - m3/(1+m4^2)^(1/2))^(1/2)  >;

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

^
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<y>:=PolynomialRing(RationalField(),1); I:=ideal<P |
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  >;

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
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<x>:=PolynomialRing(RationalField(),1); I:=ideal<P |
x^7 - 2*x^6 - x^5 + x^4 + x^3 + x^2 - x - 1  >;

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
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;
^

>> 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;
^

>> 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;
^

>> 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;
^

>> 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<x,y,z>:=PolynomialRing(RationalField(),3); I:=ideal<P |
-100 + (-6*y^2-144*y*z+27*z^2),
-1 + (-18/5*z^2-27/4*y*z-12*x*z-45/16*y^2-9*x*y-6*x^2),
-10 + (-1215/112*z^2-81/4*y*z-567/20*x*z-81/10*y^2-81/4*x*y-45/4*x^2) >;

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
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<x> := PolynomialRing(Integers());
> > f :=x^4-4*x^2+2;
> > K<y> := 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<x> := PolynomialRing(Integers());
> > f :=x^4-4*x^2+2;
> > K<y> := 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<x> := PolynomialRing(Integers());
> > f :=x^4-4*x^2+2;
> > K<y> := 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<x> := PolynomialRing(Integers());
> > f :=x^4-4*x^2+2;
> > K<y> := 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!;
^

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<x> := PolynomialRing(Integers());
> > f :=x^4-4*x^2+2;
> > K<y> := 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<x,y,z>:=PolynomialRing(RationalField(),3);  I:=ideal<P |
-100 + (-6*y^2-144*y*z+27*z^2),
-1  + (-18/5*z^2-27/4*y*z-12*x*z-45/16*y^2-9*x*y-6*x^2),
-10+(-1215/112*z^2-81/4*y*z-567/20*x*z-81/10*y^2-81/4*x*y-45/4*x^2)>;

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
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<x,y,z>:=PolynomialRing(RationalField(),3);  I:=ideal<P |
-100 + (-6*y^2-144*y*z+27*z^2),
-1  + (-18/5*z^2-27/4*y*z-12*x*z-45/16*y^2-9*x*y-6*x^2),
-10+(-1215/112*z^2-81/4*y*z-567/20*x*z-81/10*y^2-81/4*x*y-45/4*x^2)>;
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<x,y,z>:=PolynomialRing(RationalField(),3);  I:=ideal<P |
-100 + (-6*y^2-144*y*z+27*z^2),
-1  + (-18/5*z^2-27/4*y*z-12*x*z-45/16*y^2-9*x*y-6*x^2),
-10+(-1215/112*z^2-81/4*y*z-567/20*x*z-81/10*y^2-81/4*x*y-45/4*x^2)>;
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<x> := 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);;
^

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<x> := 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<x> := 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<x> := 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<x> := 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<x> := 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<x> := 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<x> := 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<x> := 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<K, 56 |   [1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0],[0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0],[0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1],[0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,1,1,1,0,1,0,0,0,1,1,1,0,1,1,0,0,1,1,1,0],[0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,1,1,1,0,1,1,0,0,1,1,1,0,0,1,0,0,1,1,1],[0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,1,0,0,1,1,1,0,0,1,0,0,1,1,1,1,0,1,0,0,1,1],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,1,0,0,1,1,1,1,1,0,1,0,0,1,0,0,1,1,1,0,1],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,1,0,1,0,0,1,1,1,1,1,0,1,0,0,1,0,0,1,1,1,0],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,1,1,0,1,1,1,0,1,0,0,1,0,1,1,1,0,1,0,0,1,0,0,1,1,1],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,0,1,0,0,1,1,0,1,0,0,1,1,1,0,1,1,1,0,1,0],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,1,1,0,1,0,0,1,1,0,1,0,0,1,1,0,0,1,1,1,0,1],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,1,1,1,0,1,0,0,1,1,0,1,0,0,1,1,0,0,1,1,1,0],[1,1,1,0,0,1,0,1,1,1,0,0,1,0,1,1,0,0,1,0,1,0,0,1,0,1,1,1,1,0,0,1,0,1,1,0,0,0,0,0,0,0,1,0,0,1,0,1,1,1,1,1,0,0,1,0],[0,1,1,1,0,0,1,0,1,1,1,0,0,1,1,1,1,0,0,1,0,1,0,0,1,0,1,1,1,1,0,0,1,0,1,0,0,0,0,0,0,0,1,1,0,0,1,0,1,0,1,1,1,0,0,1],[1,0,1,1,1,0,0,1,0,1,1,1,0,0,0,1,1,1,0,0,1,1,1,0,0,1,0,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0,1,1,1,0,0,1,0,1,0,1,1,1,0,0],[1,1,1,0,0,1,0,0,1,0,1,1,1,0,0,0,1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0,0,0,0,1,0,0,1,0,1,1,1,0,0,1,0,1,1,1,1,1,0,0,1,0],[0,1,1,1,0,0,1,0,0,1,0,1,1,1,1,0,0,1,0,1,1,0,0,1,0,1,1,1,0,0,0,0,0,0,0,1,1,0,0,1,0,1,1,1,0,0,1,0,1,0,1,1,1,0,0,1],[1,0,1,1,1,0,0,1,0,0,1,0,1,1,1,1,0,0,1,0,1,1,0,0,1,0,1,1,0,0,0,0,0,0,0,1,1,1,0,0,1,0,1,1,1,0,0,1,0,1,0,1,1,1,0,0],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,1,1,1,1,1],[1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]>;
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<K, 35 |L>;
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<K, 56 |   [1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0],[0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0],[0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1],[0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,1,1,1,0,1,0,0,0,1,1,1,0,1,1,0,0,1,1,1,0],[0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,1,1,1,0,1,1,0,0,1,1,1,0,0,1,0,0,1,1,1],[0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,1,0,0,1,1,1,0,0,1,0,0,1,1,1,1,0,1,0,0,1,1],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,1,0,0,1,1,1,1,1,0,1,0,0,1,0,0,1,1,1,0,1],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,1,0,1,0,0,1,1,1,1,1,0,1,0,0,1,0,0,1,1,1,0],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,1,1,0,1,1,1,0,1,0,0,1,0,1,1,1,0,1,0,0,1,0,0,1,1,1],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,0,1,0,0,1,1,0,1,0,0,1,1,1,0,1,1,1,0,1,0],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,1,1,0,1,0,0,1,1,0,1,0,0,1,1,0,0,1,1,1,0,1],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,1,1,1,0,1,0,0,1,1,0,1,0,0,1,1,0,0,1,1,1,0],[1,1,1,0,0,1,0,1,1,1,0,0,1,0,1,1,0,0,1,0,1,0,0,1,0,1,1,1,1,0,0,1,0,1,1,0,0,0,0,0,0,0,1,0,0,1,0,1,1,1,1,1,0,0,1,0],[0,1,1,1,0,0,1,0,1,1,1,0,0,1,1,1,1,0,0,1,0,1,0,0,1,0,1,1,1,1,0,0,1,0,1,0,0,0,0,0,0,0,1,1,0,0,1,0,1,0,1,1,1,0,0,1],[1,0,1,1,1,0,0,1,0,1,1,1,0,0,0,1,1,1,0,0,1,1,1,0,0,1,0,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0,1,1,1,0,0,1,0,1,0,1,1,1,0,0],[1,1,1,0,0,1,0,0,1,0,1,1,1,0,0,0,1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0,0,0,0,1,0,0,1,0,1,1,1,0,0,1,0,1,1,1,1,1,0,0,1,0],[0,1,1,1,0,0,1,0,0,1,0,1,1,1,1,0,0,1,0,1,1,0,0,1,0,1,1,1,0,0,0,0,0,0,0,1,1,0,0,1,0,1,1,1,0,0,1,0,1,0,1,1,1,0,0,1],[1,0,1,1,1,0,0,1,0,0,1,0,1,1,1,1,0,0,1,0,1,1,0,0,1,0,1,1,0,0,0,0,0,0,0,1,1,1,0,0,1,0,1,1,1,0,0,1,0,1,0,1,1,1,0,0],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,1,1,1,1,1],[1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]>;
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<K, 35 |L>;
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<K, 56 |   [1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0],[0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0],[0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1],[0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,1,1,1,0,1,0,0,0,1,1,1,0,1,1,0,0,1,1,1,0],[0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,1,1,1,0,1,1,0,0,1,1,1,0,0,1,0,0,1,1,1],[0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,1,0,0,1,1,1,0,0,1,0,0,1,1,1,1,0,1,0,0,1,1],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,1,0,0,1,1,1,1,1,0,1,0,0,1,0,0,1,1,1,0,1],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,1,0,1,0,0,1,1,1,1,1,0,1,0,0,1,0,0,1,1,1,0],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,1,1,0,1,1,1,0,1,0,0,1,0,1,1,1,0,1,0,0,1,0,0,1,1,1],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,0,1,0,0,1,1,0,1,0,0,1,1,1,0,1,1,1,0,1,0],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,1,1,0,1,0,0,1,1,0,1,0,0,1,1,0,0,1,1,1,0,1],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,1,1,1,0,1,0,0,1,1,0,1,0,0,1,1,0,0,1,1,1,0],[1,1,1,0,0,1,0,1,1,1,0,0,1,0,1,1,0,0,1,0,1,0,0,1,0,1,1,1,1,0,0,1,0,1,1,0,0,0,0,0,0,0,1,0,0,1,0,1,1,1,1,1,0,0,1,0],[0,1,1,1,0,0,1,0,1,1,1,0,0,1,1,1,1,0,0,1,0,1,0,0,1,0,1,1,1,1,0,0,1,0,1,0,0,0,0,0,0,0,1,1,0,0,1,0,1,0,1,1,1,0,0,1],[1,0,1,1,1,0,0,1,0,1,1,1,0,0,0,1,1,1,0,0,1,1,1,0,0,1,0,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0,1,1,1,0,0,1,0,1,0,1,1,1,0,0],[1,1,1,0,0,1,0,0,1,0,1,1,1,0,0,0,1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0,0,0,0,1,0,0,1,0,1,1,1,0,0,1,0,1,1,1,1,1,0,0,1,0],[0,1,1,1,0,0,1,0,0,1,0,1,1,1,1,0,0,1,0,1,1,0,0,1,0,1,1,1,0,0,0,0,0,0,0,1,1,0,0,1,0,1,1,1,0,0,1,0,1,0,1,1,1,0,0,1],[1,0,1,1,1,0,0,1,0,0,1,0,1,1,1,1,0,0,1,0,1,1,0,0,1,0,1,1,0,0,0,0,0,0,0,1,1,1,0,0,1,0,1,1,1,0,0,1,0,1,0,1,1,1,0,0],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,1,1,1,1,1],[1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]>;
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<K, 56 |   [1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0],[0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0],[0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1],[0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,1,1,1,0,1,0,0,0,1,1,1,0,1,1,0,0,1,1,1,0],[0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,1,1,1,0,1,1,0,0,1,1,1,0,0,1,0,0,1,1,1],[0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,1,0,0,1,1,1,0,0,1,0,0,1,1,1,1,0,1,0,0,1,1],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,1,0,0,1,1,1,1,1,0,1,0,0,1,0,0,1,1,1,0,1],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,1,0,1,0,0,1,1,1,1,1,0,1,0,0,1,0,0,1,1,1,0],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,1,1,0,1,1,1,0,1,0,0,1,0,1,1,1,0,1,0,0,1,0,0,1,1,1],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,0,1,0,0,1,1,0,1,0,0,1,1,1,0,1,1,1,0,1,0],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,1,1,0,1,0,0,1,1,0,1,0,0,1,1,0,0,1,1,1,0,1],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,1,1,1,0,1,0,0,1,1,0,1,0,0,1,1,0,0,1,1,1,0],[1,1,1,0,0,1,0,1,1,1,0,0,1,0,1,1,0,0,1,0,1,0,0,1,0,1,1,1,1,0,0,1,0,1,1,0,0,0,0,0,0,0,1,0,0,1,0,1,1,1,1,1,0,0,1,0],[0,1,1,1,0,0,1,0,1,1,1,0,0,1,1,1,1,0,0,1,0,1,0,0,1,0,1,1,1,1,0,0,1,0,1,0,0,0,0,0,0,0,1,1,0,0,1,0,1,0,1,1,1,0,0,1],[1,0,1,1,1,0,0,1,0,1,1,1,0,0,0,1,1,1,0,0,1,1,1,0,0,1,0,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0,1,1,1,0,0,1,0,1,0,1,1,1,0,0],[1,1,1,0,0,1,0,0,1,0,1,1,1,0,0,0,1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0,0,0,0,1,0,0,1,0,1,1,1,0,0,1,0,1,1,1,1,1,0,0,1,0],[0,1,1,1,0,0,1,0,0,1,0,1,1,1,1,0,0,1,0,1,1,0,0,1,0,1,1,1,0,0,0,0,0,0,0,1,1,0,0,1,0,1,1,1,0,0,1,0,1,0,1,1,1,0,0,1],[1,0,1,1,1,0,0,1,0,0,1,0,1,1,1,1,0,0,1,0,1,1,0,0,1,0,1,1,0,0,0,0,0,0,0,1,1,1,0,0,1,0,1,1,1,0,0,1,0,1,0,1,1,1,0,0],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,1,1,1,1,1],[1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]>;
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<K, 56 |   [1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0],[0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0],[0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1],[0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,1,1,1,0,1,0,0,0,1,1,1,0,1,1,0,0,1,1,1,0],[0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,1,1,1,0,1,1,0,0,1,1,1,0,0,1,0,0,1,1,1],[0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,1,0,0,1,1,1,0,0,1,0,0,1,1,1,1,0,1,0,0,1,1],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,1,0,0,1,1,1,1,1,0,1,0,0,1,0,0,1,1,1,0,1],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,1,0,1,0,0,1,1,1,1,1,0,1,0,0,1,0,0,1,1,1,0],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,1,1,0,1,1,1,0,1,0,0,1,0,1,1,1,0,1,0,0,1,0,0,1,1,1],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,0,1,0,0,1,1,0,1,0,0,1,1,1,0,1,1,1,0,1,0],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,1,1,0,1,0,0,1,1,0,1,0,0,1,1,0,0,1,1,1,0,1],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,1,1,1,0,1,0,0,1,1,0,1,0,0,1,1,0,0,1,1,1,0],[1,1,1,0,0,1,0,1,1,1,0,0,1,0,1,1,0,0,1,0,1,0,0,1,0,1,1,1,1,0,0,1,0,1,1,0,0,0,0,0,0,0,1,0,0,1,0,1,1,1,1,1,0,0,1,0],[0,1,1,1,0,0,1,0,1,1,1,0,0,1,1,1,1,0,0,1,0,1,0,0,1,0,1,1,1,1,0,0,1,0,1,0,0,0,0,0,0,0,1,1,0,0,1,0,1,0,1,1,1,0,0,1],[1,0,1,1,1,0,0,1,0,1,1,1,0,0,0,1,1,1,0,0,1,1,1,0,0,1,0,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0,1,1,1,0,0,1,0,1,0,1,1,1,0,0],[1,1,1,0,0,1,0,0,1,0,1,1,1,0,0,0,1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0,0,0,0,1,0,0,1,0,1,1,1,0,0,1,0,1,1,1,1,1,0,0,1,0],[0,1,1,1,0,0,1,0,0,1,0,1,1,1,1,0,0,1,0,1,1,0,0,1,0,1,1,1,0,0,0,0,0,0,0,1,1,0,0,1,0,1,1,1,0,0,1,0,1,0,1,1,1,0,0,1],[1,0,1,1,1,0,0,1,0,0,1,0,1,1,1,1,0,0,1,0,1,1,0,0,1,0,1,1,0,0,0,0,0,0,0,1,1,1,0,0,1,0,1,1,1,0,0,1,0,1,0,1,1,1,0,0],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,1,1,1,1,1],[1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]>;
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<K, 56 |   [1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0],[0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0],[0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1],[0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,1,1,1,0,1,0,0,0,1,1,1,0,1,1,0,0,1,1,1,0],[0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,1,1,1,0,1,1,0,0,1,1,1,0,0,1,0,0,1,1,1],[0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,1,0,0,1,1,1,0,0,1,0,0,1,1,1,1,0,1,0,0,1,1],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,1,0,0,1,1,1,1,1,0,1,0,0,1,0,0,1,1,1,0,1],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,1,0,1,0,0,1,1,1,1,1,0,1,0,0,1,0,0,1,1,1,0],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,1,1,0,1,1,1,0,1,0,0,1,0,1,1,1,0,1,0,0,1,0,0,1,1,1],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,0,1,0,0,1,1,0,1,0,0,1,1,1,0,1,1,1,0,1,0],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,1,1,0,1,0,0,1,1,0,1,0,0,1,1,0,0,1,1,1,0,1],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,1,1,1,0,1,0,0,1,1,0,1,0,0,1,1,0,0,1,1,1,0],[1,1,1,0,0,1,0,1,1,1,0,0,1,0,1,1,0,0,1,0,1,0,0,1,0,1,1,1,1,0,0,1,0,1,1,0,0,0,0,0,0,0,1,0,0,1,0,1,1,1,1,1,0,0,1,0],[0,1,1,1,0,0,1,0,1,1,1,0,0,1,1,1,1,0,0,1,0,1,0,0,1,0,1,1,1,1,0,0,1,0,1,0,0,0,0,0,0,0,1,1,0,0,1,0,1,0,1,1,1,0,0,1],[1,0,1,1,1,0,0,1,0,1,1,1,0,0,0,1,1,1,0,0,1,1,1,0,0,1,0,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0,1,1,1,0,0,1,0,1,0,1,1,1,0,0],[1,1,1,0,0,1,0,0,1,0,1,1,1,0,0,0,1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0,0,0,0,1,0,0,1,0,1,1,1,0,0,1,0,1,1,1,1,1,0,0,1,0],[0,1,1,1,0,0,1,0,0,1,0,1,1,1,1,0,0,1,0,1,1,0,0,1,0,1,1,1,0,0,0,0,0,0,0,1,1,0,0,1,0,1,1,1,0,0,1,0,1,0,1,1,1,0,0,1],[1,0,1,1,1,0,0,1,0,0,1,0,1,1,1,1,0,0,1,0,1,1,0,0,1,0,1,1,0,0,0,0,0,0,0,1,1,1,0,0,1,0,1,1,1,0,0,1,0,1,0,1,1,1,0,0],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,1,1,1,1,1],[1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]>;
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<K, 56 |   [1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0],[0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0],[0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1],[0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,1,1,1,0,1,0,0,0,1,1,1,0,1,1,0,0,1,1,1,0],[0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,1,1,1,0,1,1,0,0,1,1,1,0,0,1,0,0,1,1,1],[0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,1,0,0,1,1,1,0,0,1,0,0,1,1,1,1,0,1,0,0,1,1],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,1,0,0,1,1,1,1,1,0,1,0,0,1,0,0,1,1,1,0,1],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,1,0,1,0,0,1,1,1,1,1,0,1,0,0,1,0,0,1,1,1,0],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,1,1,0,1,1,1,0,1,0,0,1,0,1,1,1,0,1,0,0,1,0,0,1,1,1],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,0,1,0,0,1,1,0,1,0,0,1,1,1,0,1,1,1,0,1,0],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,1,1,0,1,0,0,1,1,0,1,0,0,1,1,0,0,1,1,1,0,1],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,1,1,1,0,1,0,0,1,1,0,1,0,0,1,1,0,0,1,1,1,0],[1,1,1,0,0,1,0,1,1,1,0,0,1,0,1,1,0,0,1,0,1,0,0,1,0,1,1,1,1,0,0,1,0,1,1,0,0,0,0,0,0,0,1,0,0,1,0,1,1,1,1,1,0,0,1,0],[0,1,1,1,0,0,1,0,1,1,1,0,0,1,1,1,1,0,0,1,0,1,0,0,1,0,1,1,1,1,0,0,1,0,1,0,0,0,0,0,0,0,1,1,0,0,1,0,1,0,1,1,1,0,0,1],[1,0,1,1,1,0,0,1,0,1,1,1,0,0,0,1,1,1,0,0,1,1,1,0,0,1,0,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0,1,1,1,0,0,1,0,1,0,1,1,1,0,0],[1,1,1,0,0,1,0,0,1,0,1,1,1,0,0,0,1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0,0,0,0,1,0,0,1,0,1,1,1,0,0,1,0,1,1,1,1,1,0,0,1,0],[0,1,1,1,0,0,1,0,0,1,0,1,1,1,1,0,0,1,0,1,1,0,0,1,0,1,1,1,0,0,0,0,0,0,0,1,1,0,0,1,0,1,1,1,0,0,1,0,1,0,1,1,1,0,0,1],[1,0,1,1,1,0,0,1,0,0,1,0,1,1,1,1,0,0,1,0,1,1,0,0,1,0,1,1,0,0,0,0,0,0,0,1,1,1,0,0,1,0,1,1,1,0,0,1,0,1,0,1,1,1,0,0],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,1,1,1,1,1],[1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]>:
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]>:
^

>> 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<K, 56 |   [1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0],[0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0],[0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1],[0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,1,1,1,0,1,0,0,0,1,1,1,0,1,1,0,0,1,1,1,0],[0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,1,1,1,0,1,1,0,0,1,1,1,0,0,1,0,0,1,1,1],[0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,1,0,0,1,1,1,0,0,1,0,0,1,1,1,1,0,1,0,0,1,1],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,1,0,0,1,1,1,1,1,0,1,0,0,1,0,0,1,1,1,0,1],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,1,0,1,0,0,1,1,1,1,1,0,1,0,0,1,0,0,1,1,1,0],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,1,1,0,1,1,1,0,1,0,0,1,0,1,1,1,0,1,0,0,1,0,0,1,1,1],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,0,1,0,0,1,1,0,1,0,0,1,1,1,0,1,1,1,0,1,0],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,1,1,0,1,0,0,1,1,0,1,0,0,1,1,0,0,1,1,1,0,1],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,1,1,1,0,1,0,0,1,1,0,1,0,0,1,1,0,0,1,1,1,0],[1,1,1,0,0,1,0,1,1,1,0,0,1,0,1,1,0,0,1,0,1,0,0,1,0,1,1,1,1,0,0,1,0,1,1,0,0,0,0,0,0,0,1,0,0,1,0,1,1,1,1,1,0,0,1,0],[0,1,1,1,0,0,1,0,1,1,1,0,0,1,1,1,1,0,0,1,0,1,0,0,1,0,1,1,1,1,0,0,1,0,1,0,0,0,0,0,0,0,1,1,0,0,1,0,1,0,1,1,1,0,0,1],[1,0,1,1,1,0,0,1,0,1,1,1,0,0,0,1,1,1,0,0,1,1,1,0,0,1,0,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0,1,1,1,0,0,1,0,1,0,1,1,1,0,0],[1,1,1,0,0,1,0,0,1,0,1,1,1,0,0,0,1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0,0,0,0,1,0,0,1,0,1,1,1,0,0,1,0,1,1,1,1,1,0,0,1,0],[0,1,1,1,0,0,1,0,0,1,0,1,1,1,1,0,0,1,0,1,1,0,0,1,0,1,1,1,0,0,0,0,0,0,0,1,1,0,0,1,0,1,1,1,0,0,1,0,1,0,1,1,1,0,0,1],[1,0,1,1,1,0,0,1,0,0,1,0,1,1,1,1,0,0,1,0,1,1,0,0,1,0,1,1,0,0,0,0,0,0,0,1,1,1,0,0,1,0,1,1,1,0,0,1,0,1,0,1,1,1,0,0],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,1,1,1,1,1],[1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]>;
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 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 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 1 0 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 1 0 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 1 0 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 1 0 0 0 0 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 1 0 0 0 0 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 1 0 0 0 0 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 1 0 0 0 0 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 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 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 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 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 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 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 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]
[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
** WARNING: Output too long, hence truncated.

'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<K, 56 |   [1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0],[0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0],[0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1],[0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,1,1,1,0,1,0,0,0,1,1,1,0,1,1,0,0,1,1,1,0],[0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,1,1,1,0,1,1,0,0,1,1,1,0,0,1,0,0,1,1,1],[0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,1,0,0,1,1,1,0,0,1,0,0,1,1,1,1,0,1,0,0,1,1],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,1,0,0,1,1,1,1,1,0,1,0,0,1,0,0,1,1,1,0,1],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,1,0,1,0,0,1,1,1,1,1,0,1,0,0,1,0,0,1,1,1,0],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,1,1,0,1,1,1,0,1,0,0,1,0,1,1,1,0,1,0,0,1,0,0,1,1,1],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,0,1,0,0,1,1,0,1,0,0,1,1,1,0,1,1,1,0,1,0],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,1,1,0,1,0,0,1,1,0,1,0,0,1,1,0,0,1,1,1,0,1],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,1,1,1,0,1,0,0,1,1,0,1,0,0,1,1,0,0,1,1,1,0],[1,1,1,0,0,1,0,1,1,1,0,0,1,0,1,1,0,0,1,0,1,0,0,1,0,1,1,1,1,0,0,1,0,1,1,0,0,0,0,0,0,0,1,0,0,1,0,1,1,1,1,1,0,0,1,0],[0,1,1,1,0,0,1,0,1,1,1,0,0,1,1,1,1,0,0,1,0,1,0,0,1,0,1,1,1,1,0,0,1,0,1,0,0,0,0,0,0,0,1,1,0,0,1,0,1,0,1,1,1,0,0,1],[1,0,1,1,1,0,0,1,0,1,1,1,0,0,0,1,1,1,0,0,1,1,1,0,0,1,0,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0,1,1,1,0,0,1,0,1,0,1,1,1,0,0],[1,1,1,0,0,1,0,0,1,0,1,1,1,0,0,0,1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0,0,0,0,1,0,0,1,0,1,1,1,0,0,1,0,1,1,1,1,1,0,0,1,0],[0,1,1,1,0,0,1,0,0,1,0,1,1,1,1,0,0,1,0,1,1,0,0,1,0,1,1,1,0,0,0,0,0,0,0,1,1,0,0,1,0,1,1,1,0,0,1,0,1,0,1,1,1,0,0,1],[1,0,1,1,1,0,0,1,0,0,1,0,1,1,1,1,0,0,1,0,1,1,0,0,1,0,1,1,0,0,0,0,0,0,0,1,1,1,0,0,1,0,1,1,1,0,0,1,0,1,0,1,1,1,0,0],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,1,1,1,1,1],[1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]>;
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 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 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 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 1 0 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 1 0 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 1 0 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 1 0 0 0 0 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 1 0 0 0 0 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 1 0 0 0 0 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 1 0 0 0 0 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 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 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 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 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 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 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 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: 3.34MB

'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<K, 56 |   [1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0],[0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0],[0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1],[0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,1,1,1,0,1,0,0,0,1,1,1,0,1,1,0,0,1,1,1,0],[0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,1,1,1,0,1,1,0,0,1,1,1,0,0,1,0,0,1,1,1],[0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,1,0,0,1,1,1,0,0,1,0,0,1,1,1,1,0,1,0,0,1,1],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,1,0,0,1,1,1,1,1,0,1,0,0,1,0,0,1,1,1,0,1],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,1,0,1,0,0,1,1,1,1,1,0,1,0,0,1,0,0,1,1,1,0],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,1,1,0,1,1,1,0,1,0,0,1,0,1,1,1,0,1,0,0,1,0,0,1,1,1],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,0,1,0,0,1,1,0,1,0,0,1,1,1,0,1,1,1,0,1,0],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,1,1,0,1,0,0,1,1,0,1,0,0,1,1,0,0,1,1,1,0,1],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,1,1,1,0,1,0,0,1,1,0,1,0,0,1,1,0,0,1,1,1,0],[1,1,1,0,0,1,0,1,1,1,0,0,1,0,1,1,0,0,1,0,1,0,0,1,0,1,1,1,1,0,0,1,0,1,1,0,0,0,0,0,0,0,1,0,0,1,0,1,1,1,1,1,0,0,1,0],[0,1,1,1,0,0,1,0,1,1,1,0,0,1,1,1,1,0,0,1,0,1,0,0,1,0,1,1,1,1,0,0,1,0,1,0,0,0,0,0,0,0,1,1,0,0,1,0,1,0,1,1,1,0,0,1],[1,0,1,1,1,0,0,1,0,1,1,1,0,0,0,1,1,1,0,0,1,1,1,0,0,1,0,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0,1,1,1,0,0,1,0,1,0,1,1,1,0,0],[1,1,1,0,0,1,0,0,1,0,1,1,1,0,0,0,1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0,0,0,0,1,0,0,1,0,1,1,1,0,0,1,0,1,1,1,1,1,0,0,1,0],[0,1,1,1,0,0,1,0,0,1,0,1,1,1,1,0,0,1,0,1,1,0,0,1,0,1,1,1,0,0,0,0,0,0,0,1,1,0,0,1,0,1,1,1,0,0,1,0,1,0,1,1,1,0,0,1],[1,0,1,1,1,0,0,1,0,0,1,0,1,1,1,1,0,0,1,0,1,1,0,0,1,0,1,1,0,0,0,0,0,0,0,1,1,1,0,0,1,0,1,1,1,0,0,1,0,1,0,1,1,1,0,0],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,1,1,1,1,1],[1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]>;
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<K, 56 |   [1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0],[0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0],[0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1],[0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,1,1,1,0,1,0,0,0,1,1,1,0,1,1,0,0,1,1,1,0],[0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,1,1,1,0,1,1,0,0,1,1,1,0,0,1,0,0,1,1,1],[0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,1,0,0,1,1,1,0,0,1,0,0,1,1,1,1,0,1,0,0,1,1],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,1,0,0,1,1,1,1,1,0,1,0,0,1,0,0,1,1,1,0,1],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,1,0,1,0,0,1,1,1,1,1,0,1,0,0,1,0,0,1,1,1,0],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,1,1,0,1,1,1,0,1,0,0,1,0,1,1,1,0,1,0,0,1,0,0,1,1,1],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,0,1,0,0,1,1,0,1,0,0,1,1,1,0,1,1,1,0,1,0],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,1,1,0,1,0,0,1,1,0,1,0,0,1,1,0,0,1,1,1,0,1],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,1,1,1,0,1,0,0,1,1,0,1,0,0,1,1,0,0,1,1,1,0],[1,1,1,0,0,1,0,1,1,1,0,0,1,0,1,1,0,0,1,0,1,0,0,1,0,1,1,1,1,0,0,1,0,1,1,0,0,0,0,0,0,0,1,0,0,1,0,1,1,1,1,1,0,0,1,0],[0,1,1,1,0,0,1,0,1,1,1,0,0,1,1,1,1,0,0,1,0,1,0,0,1,0,1,1,1,1,0,0,1,0,1,0,0,0,0,0,0,0,1,1,0,0,1,0,1,0,1,1,1,0,0,1],[1,0,1,1,1,0,0,1,0,1,1,1,0,0,0,1,1,1,0,0,1,1,1,0,0,1,0,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0,1,1,1,0,0,1,0,1,0,1,1,1,0,0],[1,1,1,0,0,1,0,0,1,0,1,1,1,0,0,0,1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0,0,0,0,1,0,0,1,0,1,1,1,0,0,1,0,1,1,1,1,1,0,0,1,0],[0,1,1,1,0,0,1,0,0,1,0,1,1,1,1,0,0,1,0,1,1,0,0,1,0,1,1,1,0,0,0,0,0,0,0,1,1,0,0,1,0,1,1,1,0,0,1,0,1,0,1,1,1,0,0,1],[1,0,1,1,1,0,0,1,0,0,1,0,1,1,1,1,0,0,1,0,1,1,0,0,1,0,1,1,0,0,0,0,0,0,0,1,1,1,0,0,1,0,1,1,1,0,0,1,0,1,0,1,1,1,0,0],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,1,1,1,1,1],[1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]>;
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<K, 56 |   [1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0],[0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0],[0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1],[0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,1,1,1,0,1,0,0,0,1,1,1,0,1,1,0,0,1,1,1,0],[0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,1,1,1,0,1,1,0,0,1,1,1,0,0,1,0,0,1,1,1],[0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,1,0,0,1,1,1,0,0,1,0,0,1,1,1,1,0,1,0,0,1,1],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,1,0,0,1,1,1,1,1,0,1,0,0,1,0,0,1,1,1,0,1],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,1,0,1,0,0,1,1,1,1,1,0,1,0,0,1,0,0,1,1,1,0],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,1,1,0,1,1,1,0,1,0,0,1,0,1,1,1,0,1,0,0,1,0,0,1,1,1],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,0,1,0,0,1,1,0,1,0,0,1,1,1,0,1,1,1,0,1,0],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,1,1,0,1,0,0,1,1,0,1,0,0,1,1,0,0,1,1,1,0,1],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,1,1,1,0,1,0,0,1,1,0,1,0,0,1,1,0,0,1,1,1,0],[1,1,1,0,0,1,0,1,1,1,0,0,1,0,1,1,0,0,1,0,1,0,0,1,0,1,1,1,1,0,0,1,0,1,1,0,0,0,0,0,0,0,1,0,0,1,0,1,1,1,1,1,0,0,1,0],[0,1,1,1,0,0,1,0,1,1,1,0,0,1,1,1,1,0,0,1,0,1,0,0,1,0,1,1,1,1,0,0,1,0,1,0,0,0,0,0,0,0,1,1,0,0,1,0,1,0,1,1,1,0,0,1],[1,0,1,1,1,0,0,1,0,1,1,1,0,0,0,1,1,1,0,0,1,1,1,0,0,1,0,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0,1,1,1,0,0,1,0,1,0,1,1,1,0,0],[1,1,1,0,0,1,0,0,1,0,1,1,1,0,0,0,1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0,0,0,0,1,0,0,1,0,1,1,1,0,0,1,0,1,1,1,1,1,0,0,1,0],[0,1,1,1,0,0,1,0,0,1,0,1,1,1,1,0,0,1,0,1,1,0,0,1,0,1,1,1,0,0,0,0,0,0,0,1,1,0,0,1,0,1,1,1,0,0,1,0,1,0,1,1,1,0,0,1],[1,0,1,1,1,0,0,1,0,0,1,0,1,1,1,1,0,0,1,0,1,1,0,0,1,0,1,1,0,0,0,0,0,0,0,1,1,1,0,0,1,0,1,1,1,0,0,1,0,1,0,1,1,1,0,0],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,1,1,1,1,1],[1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]>;
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<K, 56 |   [1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0],
[0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0],
[0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1],
[0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,1,1,1,0,1,0,0,0,1,1,1,0,1,1,0,0,1,1,1,0],
[0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,1,1,1,0,1,1,0,0,1,1,1,0,0,1,0,0,1,1,1],
[0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,1,0,0,1,1,1,0,0,1,0,0,1,1,1,1,0,1,0,0,1,1],
[0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,1,0,0,1,1,1,1,1,0,1,0,0,1,0,0,1,1,1,0,1],
[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,1,0,1,0,0,1,1,1,1,1,0,1,0,0,1,0,0,1,1,1,0],
[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,1,1,0,1,1,1,0,1,0,0,1,0,1,1,1,0,1,0,0,1,0,0,1,1,1],
[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,0,1,0,0,1,1,0,1,0,0,1,1,1,0,1,1,1,0,1,0],
[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,1,1,0,1,0,0,1,1,0,1,0,0,1,1,0,0,1,1,1,0,1],
[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,1,1,1,0,1,0,0,1,1,0,1,0,0,1,1,0,0,1,1,1,0],
[1,1,1,0,0,1,0,1,1,1,0,0,1,0,1,1,0,0,1,0,1,0,0,1,0,1,1,1,1,0,0,1,0,1,1,0,0,0,0,0,0,0,1,0,0,1,0,1,1,1,1,1,0,0,1,0],
[0,1,1,1,0,0,1,0,1,1,1,0,0,1,1,1,1,0,0,1,0,1,0,0,1,0,1,1,1,1,0,0,1,0,1,0,0,0,0,0,0,0,1,1,0,0,1,0,1,0,1,1,1,0,0,1],
[1,0,1,1,1,0,0,1,0,1,1,1,0,0,0,1,1,1,0,0,1,1,1,0,0,1,0,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0,1,1,1,0,0,1,0,1,0,1,1,1,0,0],
[1,1,1,0,0,1,0,0,1,0,1,1,1,0,0,0,1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0,0,0,0,1,0,0,1,0,1,1,1,0,0,1,0,1,1,1,1,1,0,0,1,0],
[0,1,1,1,0,0,1,0,0,1,0,1,1,1,1,0,0,1,0,1,1,0,0,1,0,1,1,1,0,0,0,0,0,0,0,1,1,0,0,1,0,1,1,1,0,0,1,0,1,0,1,1,1,0,0,1],
[1,0,1,1,1,0,0,1,0,0,1,0,1,1,1,1,0,0,1,0,1,1,0,0,1,0,1,1,0,0,0,0,0,0,0,1,1,1,0,0,1,0,1,1,1,0,0,1,0,1,0,1,1,1,0,0],
[0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0],
[0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,1,1,1,1,1],
[1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]>;
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),", "));
^

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;
^

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<P|[g1,g2,g3]>;

T1:=ideal<P|0>;
T2:=ideal<P|x[1]*x[3]>;
T3:=ideal<P|[x[1]*x[3],x[1]*x[2]]>;
T4:=ideal<P|[x[1],x[2]]>;
T5:=ideal<P|[x[2],x[3]]>;

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<alpha>:=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:;
^

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<K, 56 |
[1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0],[0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0],[0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1],[0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,1,1,1,0,1,0,0,0,1,1,1,0,1,1,0,0,1,1,1,0],[0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,1,1,1,0,1,1,0,0,1,1,1,0,0,1,0,0,1,1,1],[0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,1,0,0,1,1,1,0,0,1,0,0,1,1,1,1,0,1,0,0,1,1],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,1,0,0,1,1,1,1,1,0,1,0,0,1,0,0,1,1,1,0,1],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,1,0,1,0,0,1,1,1,1,1,0,1,0,0,1,0,0,1,1,1,0],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,1,1,0,1,1,1,0,1,0,0,1,0,1,1,1,0,1,0,0,1,0,0,1,1,1],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,0,1,0,0,1,1,0,1,0,0,1,1,1,0,1,1,1,0,1,0],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,1,1,0,1,0,0,1,1,0,1,0,0,1,1,0,0,1,1,1,0,1],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,1,1,1,0,1,0,0,1,1,0,1,0,0,1,1,0,0,1,1,1,0],[1,1,1,0,0,1,0,1,1,1,0,0,1,0,1,1,0,0,1,0,1,0,0,1,0,1,1,1,1,0,0,1,0,1,1,0,0,0,0,0,0,0,1,0,0,1,0,1,1,1,1,1,0,0,1,0],[0,1,1,1,0,0,1,0,1,1,1,0,0,1,1,1,1,0,0,1,0,1,0,0,1,0,1,1,1,1,0,0,1,0,1,0,0,0,0,0,0,0,1,1,0,0,1,0,1,0,1,1,1,0,0,1],[1,0,1,1,1,0,0,1,0,1,1,1,0,0,0,1,1,1,0,0,1,1,1,0,0,1,0,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0,1,1,1,0,0,1,0,1,0,1,1,1,0,0],[1,1,1,0,0,1,0,0,1,0,1,1,1,0,0,0,1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0,0,0,0,1,0,0,1,0,1,1,1,0,0,1,0,1,1,1,1,1,0,0,1,0],[0,1,1,1,0,0,1,0,0,1,0,1,1,1,1,0,0,1,0,1,1,0,0,1,0,1,1,1,0,0,0,0,0,0,0,1,1,0,0,1,0,1,1,1,0,0,1,0,1,0,1,1,1,0,0,1],[1,0,1,1,1,0,0,1,0,0,1,0,1,1,1,1,0,0,1,0,1,1,0,0,1,0,1,1,0,0,0,0,0,0,0,1,1,1,0,0,1,0,1,1,1,0,0,1,0,1,0,1,1,1,0,0],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,1,1,1,1,1],[1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]>;
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<K, 35 | (L)>;
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<K, 56 |
[1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0],[0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0],[0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1],[0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,1,1,1,0,1,0,0,0,1,1,1,0,1,1,0,0,1,1,1,0],[0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,1,1,1,0,1,1,0,0,1,1,1,0,0,1,0,0,1,1,1],[0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,1,0,0,1,1,1,0,0,1,0,0,1,1,1,1,0,1,0,0,1,1],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,1,0,0,1,1,1,1,1,0,1,0,0,1,0,0,1,1,1,0,1],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,1,0,1,0,0,1,1,1,1,1,0,1,0,0,1,0,0,1,1,1,0],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,1,1,0,1,1,1,0,1,0,0,1,0,1,1,1,0,1,0,0,1,0,0,1,1,1],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,0,1,0,0,1,1,0,1,0,0,1,1,1,0,1,1,1,0,1,0],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,1,1,0,1,0,0,1,1,0,1,0,0,1,1,0,0,1,1,1,0,1],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,1,1,1,0,1,0,0,1,1,0,1,0,0,1,1,0,0,1,1,1,0],[1,1,1,0,0,1,0,1,1,1,0,0,1,0,1,1,0,0,1,0,1,0,0,1,0,1,1,1,1,0,0,1,0,1,1,0,0,0,0,0,0,0,1,0,0,1,0,1,1,1,1,1,0,0,1,0],[0,1,1,1,0,0,1,0,1,1,1,0,0,1,1,1,1,0,0,1,0,1,0,0,1,0,1,1,1,1,0,0,1,0,1,0,0,0,0,0,0,0,1,1,0,0,1,0,1,0,1,1,1,0,0,1],[1,0,1,1,1,0,0,1,0,1,1,1,0,0,0,1,1,1,0,0,1,1,1,0,0,1,0,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0,1,1,1,0,0,1,0,1,0,1,1,1,0,0],[1,1,1,0,0,1,0,0,1,0,1,1,1,0,0,0,1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0,0,0,0,1,0,0,1,0,1,1,1,0,0,1,0,1,1,1,1,1,0,0,1,0],[0,1,1,1,0,0,1,0,0,1,0,1,1,1,1,0,0,1,0,1,1,0,0,1,0,1,1,1,0,0,0,0,0,0,0,1,1,0,0,1,0,1,1,1,0,0,1,0,1,0,1,1,1,0,0,1],[1,0,1,1,1,0,0,1,0,0,1,0,1,1,1,1,0,0,1,0,1,1,0,0,1,0,1,1,0,0,0,0,0,0,0,1,1,1,0,0,1,0,1,1,1,0,0,1,0,1,0,1,1,1,0,0],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,1,1,1,1,1],[1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]>;
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<K, 35 | (L)>;
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<K, 56 |
[1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0],[0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0],[0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1],[0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,1,1,1,0,1,0,0,0,1,1,1,0,1,1,0,0,1,1,1,0],[0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,1,1,1,0,1,1,0,0,1,1,1,0,0,1,0,0,1,1,1],[0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,1,0,0,1,1,1,0,0,1,0,0,1,1,1,1,0,1,0,0,1,1],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,1,0,0,1,1,1,1,1,0,1,0,0,1,0,0,1,1,1,0,1],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,1,0,1,0,0,1,1,1,1,1,0,1,0,0,1,0,0,1,1,1,0],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,1,1,0,1,1,1,0,1,0,0,1,0,1,1,1,0,1,0,0,1,0,0,1,1,1],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,0,1,0,0,1,1,0,1,0,0,1,1,1,0,1,1,1,0,1,0],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,1,1,0,1,0,0,1,1,0,1,0,0,1,1,0,0,1,1,1,0,1],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,1,1,1,0,1,0,0,1,1,0,1,0,0,1,1,0,0,1,1,1,0],[1,1,1,0,0,1,0,1,1,1,0,0,1,0,1,1,0,0,1,0,1,0,0,1,0,1,1,1,1,0,0,1,0,1,1,0,0,0,0,0,0,0,1,0,0,1,0,1,1,1,1,1,0,0,1,0],[0,1,1,1,0,0,1,0,1,1,1,0,0,1,1,1,1,0,0,1,0,1,0,0,1,0,1,1,1,1,0,0,1,0,1,0,0,0,0,0,0,0,1,1,0,0,1,0,1,0,1,1,1,0,0,1],[1,0,1,1,1,0,0,1,0,1,1,1,0,0,0,1,1,1,0,0,1,1,1,0,0,1,0,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0,1,1,1,0,0,1,0,1,0,1,1,1,0,0],[1,1,1,0,0,1,0,0,1,0,1,1,1,0,0,0,1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0,0,0,0,1,0,0,1,0,1,1,1,0,0,1,0,1,1,1,1,1,0,0,1,0],[0,1,1,1,0,0,1,0,0,1,0,1,1,1,1,0,0,1,0,1,1,0,0,1,0,1,1,1,0,0,0,0,0,0,0,1,1,0,0,1,0,1,1,1,0,0,1,0,1,0,1,1,1,0,0,1],[1,0,1,1,1,0,0,1,0,0,1,0,1,1,1,1,0,0,1,0,1,1,0,0,1,0,1,1,0,0,0,0,0,0,0,1,1,1,0,0,1,0,1,1,1,0,0,1,0,1,0,1,1,1,0,0],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,1,1,1,1,1],[1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]>;
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<K, 11 |
(L)>;
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<K, 11 |
^
Runtime error in 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<i> := ExtensionField<Fp, i | i^2 + 1>;
mu := i + 1;
while IsSquare(mu) do
mu +:= 1;
end while;
"mu =", mu;
xi := 1/(lambda^2*mu^3);
Fp12<z> := ExtensionField<Fp2, z | z^6 - xi>;
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<i> := ExtensionField<Fp, i | i^2 + 1>;
mu := i + 1;
while IsSquare(mu) do
mu +:= 1;
end while;
"mu =", mu;
xi := 1/(lambda^2*mu^3);
Fp12<z> := ExtensionField<Fp2, z | z^6 - xi>;
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<t> := FunctionField(GF(5,2));
A<x, y, w> := AffineAlgebra<F, x, y, w | w^5+w-x*t*(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), x^5+x - y/t, y^5+y - t>;
P<z> := 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<t> := FunctionField(GF(5,2));
A<x, y, w> := AffineAlgebra<F, x, y, w | w^5+w-x*t, x^5+x - y/t, y^5+y - t>;
P<z> := 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<t> := FunctionField(GF(5,2));
A<x, y, w> := AffineAlgebra<F, x, y, w | w^5+w-x*y, x^5+x - y/t, y^5+y - t>;
P<z> := 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<t> := FunctionField(GF(5,2));
A<x, y, w> := AffineAlgebra<F, x, y, w | w^5+w-x*t, x^5+x - y/t, y^5+y - t>;
P<z> := 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<t> := FunctionField(GF(5,2));
A<x, y, w> := AffineAlgebra<F, x, y, w | w^5+w-x, x^5+x - y/t, y^5+y - t>;
P<z> := 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<t> := FunctionField(GF(5,2));
A<x, y, w> := AffineAlgebra<F, x, y, w | w^5+w-x, x^5+x - y/t, y^5+y - t>;
P<z> := 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<t> := FunctionField(GF(5,2));
A<x, y, w> := AffineAlgebra<F, x, y, w | w^5+w-x, x^5+x - y/t, y^5+y - t>;
P<z> := 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<t> := FunctionField(GF(5,2));
A<x, y, w> := AffineAlgebra<F, x, y, w | w^5-t, x^5+x - y/t, y^5+y - t>;
P<z> := 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<t> := FunctionField(GF(5,2));
A<x, y, w> := AffineAlgebra<F, x, y | w^5-t, x^5+x - y/t, y^5+y - t>;
P<z> := PolynomialRing(F);
y^-1;

> MinimalPolynomial(x);

Output: Magma V2.11-10    Tue Dec 13 2005 21:55:52 on modular  [Seed = 1790353644]
-------------------------------------

>> A<x, y, w> := AffineAlgebra<F, x, y | w^5-t, x^5+x - y/t, y^5+y - t>;
^
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<i> := ExtensionField<Fp, i | i^2 + 1>;
mu := i + 1;
while IsSquare(mu) do
mu +:= 1;
end while;
"mu =", mu;
xi := 1/(lambda^2*mu^3);
Fp12<z> := ExtensionField<Fp2, z | z^6 - xi>;
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<t> := FunctionField(GF(5,2));
A<x, y> := AffineAlgebra<F, x, y | x^5+x - y/t, y^5+y - t>;
Q<w> := PolynomialRing(A);
B<w> := ext<Q | w^5+w-x*t/y>;
P<z> := 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<w> := ext<Q | w^5+w-x*t/y>;
^
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<t> := FunctionField(GF(5,2));
A<x, y> := AffineAlgebra<F, x, y | x^5+x - y/t, y^5+y - t>;
Q<w> := PolynomialRing(A);
B<w> := ext<A | w^5+w-x*t/y>;
P<z> := 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<w> := ext<A | w^5+w-x*t/y>;
^
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<t> := FunctionField(GF(5,2));
A<x, y> := AffineAlgebra<F, x, y | x^5+x - y/t, y^5+y - t>;
B<w> := ext<A | w^5+w-x*t/y>;
P<z> := 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<w> := ext<A | w^5+w-x*t/y>;
^
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<t> := FunctionField(GF(5,2));
A<x, y> := AffineAlgebra<F, x, y | x^5+x - y/t, y^5+y - t>;
B<w> := ext<A, w | w^5+w-x*t/y>;
P<z> := 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<w> := ext<A, w | w^5+w-x*t/y>;
^
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<t> := FunctionField(GF(5,2));
A<x, y> := AffineAlgebra<F, x, y | x^5+x - y/t, y^5+y - t>;
B<w> := Ext<A, w | w^5+w-x*t/y>;
P<z> := 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<w> := Ext<A, w | w^5+w-x*t/y>;
^
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<t> := FunctionField(GF(5,2));
A<x, y> := AffineAlgebra<F, x, y | x^5+x - y/t, y^5+y - t>;
B<x,y,w> := Ext<A, w | w^5+w-x*t/y>;
P<z> := 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<x,y,w> := Ext<A, w | w^5+w-x*t/y>;
^
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<t> := FunctionField(GF(5,2));
A<x, y> := AffineAlgebra<F, x, y | x^5+x - y/t, y^5+y - t>;
B<x,y,w> := AffineAlgebra<F, x, y, w | w^5+w-x*y^4-x, x^5+x - y/t, y^5+y - t>;
P<z> := 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<t> := FunctionField(GF(5,2));
A<x, y> := AffineAlgebra<F, x, y | x^5+x - y/t, y^5+y - t>;
B<x,y,w> := AffineAlgebra<F, x, y, w | w^5+w-x*y^4+x, x^5+x - y/t, y^5+y - t>;
P<z> := 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<t> := FunctionField(GF(5,2));
A<x, y> := AffineAlgebra<F, x, y | x^5+x - y/t, y^5+y - t>;
B<x,y,w> := AffineAlgebra<F, x, y, w | w^5+w-x*(y^4+1), x^5+x - y/t, y^5+y - t>;
P<z> := 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<t> := FunctionField(GF(5,2));
A<x, y> := AffineAlgebra<F, x, y | x^5+x - y/t, y^5+y - t>;
B<x,y,w> := AffineAlgebra<F, x, y, w | w^5+w-x*t*(y^-1), x^5+x - y/t, y^5+y - t>;
P<z> := 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<x,y,w> := AffineAlgebra<F, x, y, w | w^5+w-x*t*(y^-1), x^5+x - y/t, y^5+y
^
Runtime error in '^': Argument 2 (-1) should be >= 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<t> := FunctionField(GF(5,2));
A<x, y> := AffineAlgebra<F, x, y | x^5+x - y/t, y^5+y - t>;
B<x,y,w> := AffineAlgebra<F, x, y | w^5+w-x*t*(y^-1), x^5+x - y/t, y^5+y - t>;
P<z> := 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<x,y,w> := AffineAlgebra<F, x, y | w^5+w-x*t*(y^-1), x^5+x - y/t, y^5+y -
^
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 21:39:31 2005

Input: F<t> := FunctionField(GF(5,2));
A<x, y> := AffineAlgebra<F, x, y | x^5+x - y/t, y^5+y - t>;
B<x,y,w> := AffineAlgebra<A<x,y>, w | w^5+w-x*t/y>;
P<z> := 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<x,y,w> := AffineAlgebra<A<x,y>, w | w^5+w-x*t/y>;
^
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<t> := FunctionField(GF(5,2));
A<x, y> := AffineAlgebra<F, x, y | x^5+x - y/t, y^5+y - t>;
B<x,y,w> := AffineAlgebra<A, w | w^5+w-x*t/y>;
P<z> := 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<x,y,w> := AffineAlgebra<A, w | w^5+w-x*t/y>;
^
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<t> := FunctionField(GF(5,2));
A<x, y> := AffineAlgebra<F, x, y | x^5+x - y/t, y^5+y - t>;
B<w> := AffineAlgebra<A, w | w^5+w-x*t/y>;
P<z> := 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<t> := FunctionField(GF(5,2));
A<x, y> := AffineAlgebra<F, x, y | x^5+x - y/t, y^5+y - t>;
B<w> := AffineAlgebra<A, w | w^5+w-x*t/y>;
P<z> := 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<t> := FunctionField(GF(5,2));
A<x, y> := AffineAlgebra<F, x, y | x^5+x - y/t, y^5+y - t>;
B<w> := AffineAlgebra<A, w | w^5+w-x*t>;
P<z> := 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<t> := FunctionField(GF(5,2));
A<x, y> := AffineAlgebra<F, x, y | x^5+x - y/t, y^5+y - t>;
B<x,y,w> := AffineAlgebra<F, x, y, w | w^5+w-x*t , x^5+x - y/t, y^5+y - t>;
P<z> := 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<i> := ExtensionField<Fp, i | i^2 + 1>;
mu := i + 1;
while IsSquare(mu) do
mu +:= 1;
end while;
"mu =", mu;
xi := 1/(lambda^2*mu^3);
Fp12<z> := ExtensionField<Fp2, z | z^6 - xi>;
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<t> := FunctionField(GF(5,2));
A<x, y> := AffineAlgebra<F, x, y | x^5+x - y/t, y^5+y - t>;
B<x,y,w> := AffineAlgebra<F, x, y, w | w^5+w-x*t , x^5+x - y/t, y^5+y - t>;
P<z> := 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<t> := FunctionField(GF(5,2));
A<x, y> := AffineAlgebra<F, x, y | x^5+x - y/t, y^5+y - t>;
B<x,y,w> := AffineAlgebra<F, x, y, w | w^5+w-x*t/y , x^5+x - y/t, y^5+y - t>;
P<z> := PolynomialRing(F);
x^-1;

> MinimalPolynomial(x-y);

Output: Magma V2.11-10    Tue Dec 13 2005 21:35:19 on modular  [Seed = 4029429544]
-------------------------------------

>> B<x,y,w> := AffineAlgebra<F, x, y, w | w^5+w-x*t/y , x^5+x - y/t, y^5+y - t
^
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: 0.190 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<t> := FunctionField(GF(5,2));
A<x, y> := AffineAlgebra<F, x, y | x^5+x - y/t, y^5+y - t>;
B<x,y,w> := AffineAlgebra<F,x,y,w | w^5+w-x*t/y , x^5+x - y/t, y^5+y - t>;
P<z> := PolynomialRing(F);
x^-1;

> MinimalPolynomial(x-y);

Output: Magma V2.11-10    Tue Dec 13 2005 21:34:52 on modular  [Seed = 4113117702]
-------------------------------------

>> B<x,y,w> := AffineAlgebra<F,x,y,w | w^5+w-x*t/y , x^5+x - y/t, y^5+y - t>;
^
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<i> := ExtensionField<Fp, i | i^2 + 1>;
mu := i + 1;
while IsSquare(mu) do
mu +:= 1;
end while;
"mu =", mu;
xi := 1/(lambda^2*mu^3);
Fp12<z> := ExtensionField<Fp2, z | z^6 - xi>;
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<i> := ExtensionField<Fp, i | i^2 + 1>;
mu := i + 1;
while IsSquare(mu) do
mu +:= 1;
end while;
"mu =", mu;
xi := 1/(lambda^2*mu^3);
Fp12<z> := ExtensionField<Fp2, z | z^6 - xi>;
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<t> := FunctionField(GF(5,2));
A<x, y> := AffineAlgebra<F, x, y | x^5+x - y/t, y^5+y - t>;
P<z> := 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<i> := ExtensionField<Fp, i | i^2 + 1>;
mu := i + 1;
while IsSquare(mu) do
mu +:= 1;
end while;
"mu =", mu;
xi := 1/(lambda^2*mu^3);
Fp12<z> := ExtensionField<Fp2, z | z^6 - xi>;
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<i> := ExtensionField<Fp, i | i^2 + 1>;
mu := i + 1;
while IsSquare(mu) do
mu +:= 1;
end while;
"mu =", mu;
xi := 1/(lambda^2*mu^3);
Fp12<z> := ExtensionField<Fp2, z | z^6 - xi>;
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<t> := FunctionField(GF(5,2));
A<x, y> := AffineAlgebra<F, x, y | x^5+x - y/t, y^5+y - t>;
P<z> := 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<t> := FunctionField(GF(5,2));
A<x, y> := AffineAlgebra<F, x, y | x^5+x - y/t, y^5+y - t>;
P<z> := 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<t> := FunctionField(GF(5,2));
A<x, y> := AffineAlgebra<F, x, y | t*x^2 - y^2 + t + 1, y^3 - t>;
P<z> := 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<t> := FunctionField(IntegerRing());
A<x, y> := AffineAlgebra<F, x, y | t*x^2 - y^2 + t + 1, y^3 - t>;
P<z> := 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<t> := FunctionField(IntegerRing());
A<x, y> := AffineAlgebra<F, x, y | t*x^2 - y^2 + t + 1, y^3 - t>;
P<z> := 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<t> := FunctionField(IntegerRing());
A<x, y> := AffineAlgebra<F, x, y | t*x^2 - y^2 + t + 1, y^3 - t>;
P<z> := 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<t> := FunctionField(IntegerRing());
A<x, y> := AffineAlgebra<F, x, y | t*x^2 - y^2 + t + 1, y^3 - t>;
P<z> := 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<i> := ExtensionField<Fp, i | i^2 + 1>;
mu := i + 1;
while IsSquare(mu) do
mu +:= 1;
end while;
"mu =", mu;
xi := 1/(lambda^2*mu^3);
Fp12<z> := ExtensionField<Fp2, z | z^6 - xi>;
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<x> := FunctionField(GF(5,2));
P<y> := PolynomialRing(R);
F<alpha> := ext< R | y^5+y - x >;
F;
Q<z> := PolynomialRing(P);
Q;
G<beta> := ext<F | z^5+z - y/x>;
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<i> := ExtensionField<Fp, i | i^2 + 1>;
mu := i + 1;
while IsSquare(mu) do
mu +:= 1;
end while;
"mu =", mu;
xi := 1/(lambda^2*mu^3);
Fp12<z> := ExtensionField<Fp2, z | z^6 - xi>;
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<x> := FunctionField(GF(5,2));
P<y> := PolynomialRing(R);
F<alpha> := ext< R | y^5+y - x >;
F;
Q<z> := PolynomialRing(P);
G<beta> := ext<F | z^5+z - y/x>;
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<x> := FunctionField(GF(5,2));
P<y> := PolynomialRing(R);
F<alpha> := ext< R | y^5+y - x >;
F;
Q<z> := PolynomialRing(P);
G<beta> := ext<F | z^5+z - alpha/x>;
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<beta> := ext<F | z^5+z - alpha/x>;
^
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<x> := FunctionField(GF(5,2));
P<y> := PolynomialRing(R);
F<alpha> := ext< R | y^5+y - x >;
F;
Q<z> := PolynomialRing(P);
G<beta> := ext<F | z^5+z^1-alpha/x>;
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<beta> := ext<F | z^5+z^1-alpha/x>;
^
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<i> := ExtensionField<Fp, i | i^2 + 1>;
mu := i + 1;
while IsSquare(mu) do
mu +:= 1;
end while;
"mu =", mu;
xi := 1/(lambda^2*mu^3);
Fp12<z> := ExtensionField<Fp2, z | z^6 - xi>;
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<x> := FunctionField(GF(5,2));
P<y> := PolynomialRing(R);
F<alpha> := ext< R | y^5+y - x >;
F;
Q<z> := PolynomialRing(P);
G<beta> := ext<F | z^5+z-alpha/x>;
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<beta> := ext<F | z^5+z-alpha/x>;
^
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<x> := FunctionField(GF(5));
P<y> := PolynomialRing(R);
F<alpha> := ext< R | y^2 - 1/x >;
F;
Q<z> := PolynomialRing(P);
G<beta> := ext<F | z^3-y-x>;
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<x> := FunctionField(GF(5)); P<y> := PolynomialRing(R); F<alpha> := 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<x> := FunctionField(GF(5)); F<alpha> := ext< R | y^2 - 1/x >; F; Output: Magma V2.11-10 Tue Dec 13 2005 20:37:26 on modular [Seed = 570234184] ------------------------------------- >> F<alpha> := 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<x> := FunctionField(GF(5)); P<y> := PolynomialRing(R); F<alpha> := 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<x> := FunctionField(GF(5,2)); P<y> := PolynomialRing(R); F<z> := 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<x> := FunctionField(GF(5,2)); P<y> := PolynomialRing(R); F<alpha> := 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<x> := FunctionField(GF(5)); P<y> := PolynomialRing(R); F<alpha> := 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<i> := ExtensionField<Fp, i | i^2 + 1>; mu := i + 1; while IsSquare(mu) do mu +:= 1; end while; "mu =", mu; xi := 1/(lambda^2*mu^3); Fp12<z> := ExtensionField<Fp2, z | z^6 - xi>; 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<i> := ExtensionField<Fp, i | i^2 + 1>; mu := i + 1; while IsSquare(mu) do mu +:= 1; end while; "mu =", mu; xi := 1/(lambda^2*mu^3); Fp12<z> := ExtensionField<Fp2, z | z^6 - xi>; 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<i> := ExtensionField<Fp, i | i^2 + 1>; mu := i + 1; while IsSquare(mu) do mu +:= 1; end while; "mu =", mu; xi := 1/(lambda^2*mu^3); Fp12<z> := ExtensionField<Fp2, z | z^6 - xi>; 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<i> := ExtensionField<Fp, i | i^2 + 1>; mu := i + 1; while IsSquare(mu) do mu +:= 1; end while; "mu =", mu; xi := 1/(lambda^2*mu^3); Fp12<z> := ExtensionField<Fp2, z | z^6 - xi>; 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<i> := ExtensionField<Fp, i | i^2 + 1>; mu := i + 1; while IsSquare(mu) do mu +:= 1; end while; "mu =", mu; xi := 1/(lambda^2*mu^3); Fp12<z> := ExtensionField<Fp2, z | z^6 - xi>; 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<w> := 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<i> := ExtensionField<Fp, i | i^2 + 1>; mu := i + 1; while IsSquare(mu) do mu +:= 1; end while; "mu =", mu; xi := 1/(lambda^2*mu^3); Fp12<z> := ExtensionField<Fp2, z | z^6 - xi>; 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<z> := 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<i> := ExtensionField<Fp, i | i^2 + 1>; mu := i + 1; while IsSquare(mu) do mu +:= 1; end while; "mu =", mu; xi := 1/(lambda^2*mu^3); Fp12<z> := ExtensionField<Fp2, z | z^6 - xi>; 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<z> := 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<i> := ExtensionField<Fp, i | i^2 + 1>; mu := i + 1; while IsSquare(mu) do mu +:= 1; end while; "mu =", mu; xi := 1/(lambda^2*mu^3); Fp12<z> := ExtensionField<Fp2, z | z^6 - xi>; 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<z> := 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<i> := ExtensionField<Fp, i | i^2 + 1>; mu := i + 1; while IsSquare(mu) do mu +:= 1; end while; "mu =", mu; xi := 1/(lambda^2*mu^3); Fp12<z> := ExtensionField<Fp2, z | z^6 - xi>; 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<z> := 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<i> := ExtensionField<Fp, i | i^2 + 1>; mu := i + 1; while IsSquare(mu) do mu +:= 1; end while; "mu =", mu; xi := 1/(lambda^2*mu^3); Fp12<z> := ExtensionField<Fp2, z | z^6 - xi>; 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<z> := 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<i> := ExtensionField<Fp, i | i^2 + 1>; mu := i + 1; while IsSquare(mu) do mu +:= 1; end while; "mu =", mu; xi := 1/(lambda^2*mu^3); Fp12<z> := ExtensionField<Fp2, z | z^6 - xi>; 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<K, 56 | [1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0],[0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0],[0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1],[0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,1,1,1,0,1,0,0,0,1,1,1,0,1,1,0,0,1,1,1,0],[0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,1,1,1,0,1,1,0,0,1,1,1,0,0,1,0,0,1,1,1],[0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,1,0,0,1,1,1,0,0,1,0,0,1,1,1,1,0,1,0,0,1,1],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,1,0,0,1,1,1,1,1,0,1,0,0,1,0,0,1,1,1,0,1],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,1,0,1,0,0,1,1,1,1,1,0,1,0,0,1,0,0,1,1,1,0],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,1,1,0,1,1,1,0,1,0,0,1,0,1,1,1,0,1,0,0,1,0,0,1,1,1],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,0,1,0,0,1,1,0,1,0,0,1,1,1,0,1,1,1,0,1,0],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,1,1,0,1,0,0,1,1,0,1,0,0,1,1,0,0,1,1,1,0,1],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,1,1,1,0,1,0,0,1,1,0,1,0,0,1,1,0,0,1,1,1,0],[1,1,1,0,0,1,0,1,1,1,0,0,1,0,1,1,0,0,1,0,1,0,0,1,0,1,1,1,1,0,0,1,0,1,1,0,0,0,0,0,0,0,1,0,0,1,0,1,1,1,1,1,0,0,1,0],[0,1,1,1,0,0,1,0,1,1,1,0,0,1,1,1,1,0,0,1,0,1,0,0,1,0,1,1,1,1,0,0,1,0,1,0,0,0,0,0,0,0,1,1,0,0,1,0,1,0,1,1,1,0,0,1],[1,0,1,1,1,0,0,1,0,1,1,1,0,0,0,1,1,1,0,0,1,1,1,0,0,1,0,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0,1,1,1,0,0,1,0,1,0,1,1,1,0,0],[1,1,1,0,0,1,0,0,1,0,1,1,1,0,0,0,1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0,0,0,0,1,0,0,1,0,1,1,1,0,0,1,0,1,1,1,1,1,0,0,1,0],[0,1,1,1,0,0,1,0,0,1,0,1,1,1,1,0,0,1,0,1,1,0,0,1,0,1,1,1,0,0,0,0,0,0,0,1,1,0,0,1,0,1,1,1,0,0,1,0,1,0,1,1,1,0,0,1],[1,0,1,1,1,0,0,1,0,0,1,0,1,1,1,1,0,0,1,0,1,1,0,0,1,0,1,1,0,0,0,0,0,0,0,1,1,1,0,0,1,0,1,1,1,0,0,1,0,1,0,1,1,1,0,0],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,1,1,1,1,1],[1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]>; 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<K, 56 | [1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0],[0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0],[0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1],[0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,1,1,1,0,1,0,0,0,1,1,1,0,1,1,0,0,1,1,1,0],[0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,1,1,1,0,1,1,0,0,1,1,1,0,0,1,0,0,1,1,1],[0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,1,0,0,1,1,1,0,0,1,0,0,1,1,1,1,0,1,0,0,1,1],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,1,0,0,1,1,1,1,1,0,1,0,0,1,0,0,1,1,1,0,1],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,1,0,1,0,0,1,1,1,1,1,0,1,0,0,1,0,0,1,1,1,0],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,1,1,0,1,1,1,0,1,0,0,1,0,1,1,1,0,1,0,0,1,0,0,1,1,1],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,0,1,0,0,1,1,0,1,0,0,1,1,1,0,1,1,1,0,1,0],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,1,1,0,1,0,0,1,1,0,1,0,0,1,1,0,0,1,1,1,0,1],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,1,1,1,0,1,0,0,1,1,0,1,0,0,1,1,0,0,1,1,1,0],[1,1,1,0,0,1,0,1,1,1,0,0,1,0,1,1,0,0,1,0,1,0,0,1,0,1,1,1,1,0,0,1,0,1,1,0,0,0,0,0,0,0,1,0,0,1,0,1,1,1,1,1,0,0,1,0],[0,1,1,1,0,0,1,0,1,1,1,0,0,1,1,1,1,0,0,1,0,1,0,0,1,0,1,1,1,1,0,0,1,0,1,0,0,0,0,0,0,0,1,1,0,0,1,0,1,0,1,1,1,0,0,1],[1,0,1,1,1,0,0,1,0,1,1,1,0,0,0,1,1,1,0,0,1,1,1,0,0,1,0,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0,1,1,1,0,0,1,0,1,0,1,1,1,0,0],[1,1,1,0,0,1,0,0,1,0,1,1,1,0,0,0,1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0,0,0,0,1,0,0,1,0,1,1,1,0,0,1,0,1,1,1,1,1,0,0,1,0],[0,1,1,1,0,0,1,0,0,1,0,1,1,1,1,0,0,1,0,1,1,0,0,1,0,1,1,1,0,0,0,0,0,0,0,1,1,0,0,1,0,1,1,1,0,0,1,0,1,0,1,1,1,0,0,1],[1,0,1,1,1,0,0,1,0,0,1,0,1,1,1,1,0,0,1,0,1,1,0,0,1,0,1,1,0,0,0,0,0,0,0,1,1,1,0,0,1,0,1,1,1,0,0,1,0,1,0,1,1,1,0,0],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,1,1,1,1,1],[1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]>; 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<K, 56 | [1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0],[0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0],[0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1],[0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,1,1,1,0,1,0,0,0,1,1,1,0,1,1,0,0,1,1,1,0],[0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,1,1,1,0,1,1,0,0,1,1,1,0,0,1,0,0,1,1,1],[0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,1,0,0,1,1,1,0,0,1,0,0,1,1,1,1,0,1,0,0,1,1],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,1,0,0,1,1,1,1,1,0,1,0,0,1,0,0,1,1,1,0,1],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,1,0,1,0,0,1,1,1,1,1,0,1,0,0,1,0,0,1,1,1,0],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,1,1,0,1,1,1,0,1,0,0,1,0,1,1,1,0,1,0,0,1,0,0,1,1,1],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,0,1,0,0,1,1,0,1,0,0,1,1,1,0,1,1,1,0,1,0],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,1,1,0,1,0,0,1,1,0,1,0,0,1,1,0,0,1,1,1,0,1],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,1,1,1,0,1,0,0,1,1,0,1,0,0,1,1,0,0,1,1,1,0],[1,1,1,0,0,1,0,1,1,1,0,0,1,0,1,1,0,0,1,0,1,0,0,1,0,1,1,1,1,0,0,1,0,1,1,0,0,0,0,0,0,0,1,0,0,1,0,1,1,1,1,1,0,0,1,0],[0,1,1,1,0,0,1,0,1,1,1,0,0,1,1,1,1,0,0,1,0,1,0,0,1,0,1,1,1,1,0,0,1,0,1,0,0,0,0,0,0,0,1,1,0,0,1,0,1,0,1,1,1,0,0,1],[1,0,1,1,1,0,0,1,0,1,1,1,0,0,0,1,1,1,0,0,1,1,1,0,0,1,0,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0,1,1,1,0,0,1,0,1,0,1,1,1,0,0],[1,1,1,0,0,1,0,0,1,0,1,1,1,0,0,0,1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0,0,0,0,1,0,0,1,0,1,1,1,0,0,1,0,1,1,1,1,1,0,0,1,0],[0,1,1,1,0,0,1,0,0,1,0,1,1,1,1,0,0,1,0,1,1,0,0,1,0,1,1,1,0,0,0,0,0,0,0,1,1,0,0,1,0,1,1,1,0,0,1,0,1,0,1,1,1,0,0,1],[1,0,1,1,1,0,0,1,0,0,1,0,1,1,1,1,0,0,1,0,1,1,0,0,1,0,1,1,0,0,0,0,0,0,0,1,1,1,0,0,1,0,1,1,1,0,0,1,0,1,0,1,1,1,0,0],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,1,1,1,1,1],[1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]>; 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<K, 56 | [1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0],[0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0],[0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1],[0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,1,1,1,0,1,0,0,0,1,1,1,0,1,1,0,0,1,1,1,0],[0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,1,1,1,0,1,1,0,0,1,1,1,0,0,1,0,0,1,1,1],[0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,1,0,0,1,1,1,0,0,1,0,0,1,1,1,1,0,1,0,0,1,1],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,1,0,0,1,1,1,1,1,0,1,0,0,1,0,0,1,1,1,0,1],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,1,0,1,0,0,1,1,1,1,1,0,1,0,0,1,0,0,1,1,1,0],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,1,1,0,1,1,1,0,1,0,0,1,0,1,1,1,0,1,0,0,1,0,0,1,1,1],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,0,1,0,0,1,1,0,1,0,0,1,1,1,0,1,1,1,0,1,0],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,1,1,0,1,0,0,1,1,0,1,0,0,1,1,0,0,1,1,1,0,1],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,1,1,1,0,1,0,0,1,1,0,1,0,0,1,1,0,0,1,1,1,0],[1,1,1,0,0,1,0,1,1,1,0,0,1,0,1,1,0,0,1,0,1,0,0,1,0,1,1,1,1,0,0,1,0,1,1,0,0,0,0,0,0,0,1,0,0,1,0,1,1,1,1,1,0,0,1,0],[0,1,1,1,0,0,1,0,1,1,1,0,0,1,1,1,1,0,0,1,0,1,0,0,1,0,1,1,1,1,0,0,1,0,1,0,0,0,0,0,0,0,1,1,0,0,1,0,1,0,1,1,1,0,0,1],[1,0,1,1,1,0,0,1,0,1,1,1,0,0,0,1,1,1,0,0,1,1,1,0,0,1,0,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0,1,1,1,0,0,1,0,1,0,1,1,1,0,0],[1,1,1,0,0,1,0,0,1,0,1,1,1,0,0,0,1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0,0,0,0,1,0,0,1,0,1,1,1,0,0,1,0,1,1,1,1,1,0,0,1,0],[0,1,1,1,0,0,1,0,0,1,0,1,1,1,1,0,0,1,0,1,1,0,0,1,0,1,1,1,0,0,0,0,0,0,0,1,1,0,0,1,0,1,1,1,0,0,1,0,1,0,1,1,1,0,0,1],[1,0,1,1,1,0,0,1,0,0,1,0,1,1,1,1,0,0,1,0,1,1,0,0,1,0,1,1,0,0,0,0,0,0,0,1,1,1,0,0,1,0,1,1,1,0,0,1,0,1,0,1,1,1,0,0],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,1,1,1,1,1],[1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]>; 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<K, 56 | [1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0],[0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0],[0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1],[0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,1,1,1,0,1,0,0,0,1,1,1,0,1,1,0,0,1,1,1,0],[0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,1,1,1,0,1,1,0,0,1,1,1,0,0,1,0,0,1,1,1],[0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,1,0,0,1,1,1,0,0,1,0,0,1,1,1,1,0,1,0,0,1,1],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,1,0,0,1,1,1,1,1,0,1,0,0,1,0,0,1,1,1,0,1],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,1,0,1,0,0,1,1,1,1,1,0,1,0,0,1,0,0,1,1,1,0],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,1,1,0,1,1,1,0,1,0,0,1,0,1,1,1,0,1,0,0,1,0,0,1,1,1],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,0,1,0,0,1,1,0,1,0,0,1,1,1,0,1,1,1,0,1,0],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,1,1,0,1,0,0,1,1,0,1,0,0,1,1,0,0,1,1,1,0,1],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,1,1,1,0,1,0,0,1,1,0,1,0,0,1,1,0,0,1,1,1,0],[1,1,1,0,0,1,0,1,1,1,0,0,1,0,1,1,0,0,1,0,1,0,0,1,0,1,1,1,1,0,0,1,0,1,1,0,0,0,0,0,0,0,1,0,0,1,0,1,1,1,1,1,0,0,1,0],[0,1,1,1,0,0,1,0,1,1,1,0,0,1,1,1,1,0,0,1,0,1,0,0,1,0,1,1,1,1,0,0,1,0,1,0,0,0,0,0,0,0,1,1,0,0,1,0,1,0,1,1,1,0,0,1],[1,0,1,1,1,0,0,1,0,1,1,1,0,0,0,1,1,1,0,0,1,1,1,0,0,1,0,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0,1,1,1,0,0,1,0,1,0,1,1,1,0,0],[1,1,1,0,0,1,0,0,1,0,1,1,1,0,0,0,1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0,0,0,0,1,0,0,1,0,1,1,1,0,0,1,0,1,1,1,1,1,0,0,1,0],[0,1,1,1,0,0,1,0,0,1,0,1,1,1,1,0,0,1,0,1,1,0,0,1,0,1,1,1,0,0,0,0,0,0,0,1,1,0,0,1,0,1,1,1,0,0,1,0,1,0,1,1,1,0,0,1],[1,0,1,1,1,0,0,1,0,0,1,0,1,1,1,1,0,0,1,0,1,1,0,0,1,0,1,1,0,0,0,0,0,0,0,1,1,1,0,0,1,0,1,1,1,0,0,1,0,1,0,1,1,1,0,0],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,1,1,1,1,1],[1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]>; 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<K, 56 | [1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0],[0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0],[0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1],[0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,1,1,1,0,1,0,0,0,1,1,1,0,1,1,0,0,1,1,1,0],[0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,1,1,1,0,1,1,0,0,1,1,1,0,0,1,0,0,1,1,1],[0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,1,0,0,1,1,1,0,0,1,0,0,1,1,1,1,0,1,0,0,1,1],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,1,0,0,1,1,1,1,1,0,1,0,0,1,0,0,1,1,1,0,1],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,1,0,1,0,0,1,1,1,1,1,0,1,0,0,1,0,0,1,1,1,0],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,1,1,0,1,1,1,0,1,0,0,1,0,1,1,1,0,1,0,0,1,0,0,1,1,1],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,0,1,0,0,1,1,0,1,0,0,1,1,1,0,1,1,1,0,1,0],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,1,1,0,1,0,0,1,1,0,1,0,0,1,1,0,0,1,1,1,0,1],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,1,1,1,0,1,0,0,1,1,0,1,0,0,1,1,0,0,1,1,1,0],[1,1,1,0,0,1,0,1,1,1,0,0,1,0,1,1,0,0,1,0,1,0,0,1,0,1,1,1,1,0,0,1,0,1,1,0,0,0,0,0,0,0,1,0,0,1,0,1,1,1,1,1,0,0,1,0],[0,1,1,1,0,0,1,0,1,1,1,0,0,1,1,1,1,0,0,1,0,1,0,0,1,0,1,1,1,1,0,0,1,0,1,0,0,0,0,0,0,0,1,1,0,0,1,0,1,0,1,1,1,0,0,1],[1,0,1,1,1,0,0,1,0,1,1,1,0,0,0,1,1,1,0,0,1,1,1,0,0,1,0,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0,1,1,1,0,0,1,0,1,0,1,1,1,0,0],[1,1,1,0,0,1,0,0,1,0,1,1,1,0,0,0,1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0,0,0,0,1,0,0,1,0,1,1,1,0,0,1,0,1,1,1,1,1,0,0,1,0],[0,1,1,1,0,0,1,0,0,1,0,1,1,1,1,0,0,1,0,1,1,0,0,1,0,1,1,1,0,0,0,0,0,0,0,1,1,0,0,1,0,1,1,1,0,0,1,0,1,0,1,1,1,0,0,1],[1,0,1,1,1,0,0,1,0,0,1,0,1,1,1,1,0,0,1,0,1,1,0,0,1,0,1,1,0,0,0,0,0,0,0,1,1,1,0,0,1,0,1,1,1,0,0,1,0,1,0,1,1,1,0,0],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,1,1,1,1,1],[1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]>; 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<K, 56 | [1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0],[0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0],[0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1],[0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,1,1,1,0,1,0,0,0,1,1,1,0,1,1,0,0,1,1,1,0],[0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,1,1,1,0,1,1,0,0,1,1,1,0,0,1,0,0,1,1,1],[0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,1,0,0,1,1,1,0,0,1,0,0,1,1,1,1,0,1,0,0,1,1],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,1,0,0,1,1,1,1,1,0,1,0,0,1,0,0,1,1,1,0,1],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,1,0,1,0,0,1,1,1,1,1,0,1,0,0,1,0,0,1,1,1,0],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,1,1,0,1,1,1,0,1,0,0,1,0,1,1,1,0,1,0,0,1,0,0,1,1,1],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,0,1,0,0,1,1,0,1,0,0,1,1,1,0,1,1,1,0,1,0],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,1,1,0,1,0,0,1,1,0,1,0,0,1,1,0,0,1,1,1,0,1],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,1,1,1,0,1,0,0,1,1,0,1,0,0,1,1,0,0,1,1,1,0],[1,1,1,0,0,1,0,1,1,1,0,0,1,0,1,1,0,0,1,0,1,0,0,1,0,1,1,1,1,0,0,1,0,1,1,0,0,0,0,0,0,0,1,0,0,1,0,1,1,1,1,1,0,0,1,0],[0,1,1,1,0,0,1,0,1,1,1,0,0,1,1,1,1,0,0,1,0,1,0,0,1,0,1,1,1,1,0,0,1,0,1,0,0,0,0,0,0,0,1,1,0,0,1,0,1,0,1,1,1,0,0,1],[1,0,1,1,1,0,0,1,0,1,1,1,0,0,0,1,1,1,0,0,1,1,1,0,0,1,0,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0,1,1,1,0,0,1,0,1,0,1,1,1,0,0],[1,1,1,0,0,1,0,0,1,0,1,1,1,0,0,0,1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0,0,0,0,1,0,0,1,0,1,1,1,0,0,1,0,1,1,1,1,1,0,0,1,0],[0,1,1,1,0,0,1,0,0,1,0,1,1,1,1,0,0,1,0,1,1,0,0,1,0,1,1,1,0,0,0,0,0,0,0,1,1,0,0,1,0,1,1,1,0,0,1,0,1,0,1,1,1,0,0,1],[1,0,1,1,1,0,0,1,0,0,1,0,1,1,1,1,0,0,1,0,1,1,0,0,1,0,1,1,0,0,0,0,0,0,0,1,1,1,0,0,1,0,1,1,1,0,0,1,0,1,0,1,1,1,0,0],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,1,1,1,1,1],[1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]>; 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<K, 56 | [1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0],[0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0],[0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1],[0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,1,1,1,0,1,0,0,0,1,1,1,0,1,1,0,0,1,1,1,0],[0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,1,1,1,0,1,1,0,0,1,1,1,0,0,1,0,0,1,1,1],[0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,1,0,0,1,1,1,0,0,1,0,0,1,1,1,1,0,1,0,0,1,1],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,1,0,0,1,1,1,1,1,0,1,0,0,1,0,0,1,1,1,0,1],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,1,0,1,0,0,1,1,1,1,1,0,1,0,0,1,0,0,1,1,1,0],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,1,1,0,1,1,1,0,1,0,0,1,0,1,1,1,0,1,0,0,1,0,0,1,1,1],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,0,1,0,0,1,1,0,1,0,0,1,1,1,0,1,1,1,0,1,0],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,1,1,0,1,0,0,1,1,0,1,0,0,1,1,0,0,1,1,1,0,1],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,1,1,1,0,1,0,0,1,1,0,1,0,0,1,1,0,0,1,1,1,0],[1,1,1,0,0,1,0,1,1,1,0,0,1,0,1,1,0,0,1,0,1,0,0,1,0,1,1,1,1,0,0,1,0,1,1,0,0,0,0,0,0,0,1,0,0,1,0,1,1,1,1,1,0,0,1,0],[0,1,1,1,0,0,1,0,1,1,1,0,0,1,1,1,1,0,0,1,0,1,0,0,1,0,1,1,1,1,0,0,1,0,1,0,0,0,0,0,0,0,1,1,0,0,1,0,1,0,1,1,1,0,0,1],[1,0,1,1,1,0,0,1,0,1,1,1,0,0,0,1,1,1,0,0,1,1,1,0,0,1,0,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0,1,1,1,0,0,1,0,1,0,1,1,1,0,0],[1,1,1,0,0,1,0,0,1,0,1,1,1,0,0,0,1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0,0,0,0,1,0,0,1,0,1,1,1,0,0,1,0,1,1,1,1,1,0,0,1,0],[0,1,1,1,0,0,1,0,0,1,0,1,1,1,1,0,0,1,0,1,1,0,0,1,0,1,1,1,0,0,0,0,0,0,0,1,1,0,0,1,0,1,1,1,0,0,1,0,1,0,1,1,1,0,0,1],[1,0,1,1,1,0,0,1,0,0,1,0,1,1,1,1,0,0,1,0,1,1,0,0,1,0,1,1,0,0,0,0,0,0,0,1,1,1,0,0,1,0,1,1,1,0,0,1,0,1,0,1,1,1,0,0],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,1,1,1,1,1],[1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]>; 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<K, 56 | [1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0],[0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0],[0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1],[0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,1,1,1,0,1,0,0,0,1,1,1,0,1,1,0,0,1,1,1,0],[0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,1,1,1,0,1,1,0,0,1,1,1,0,0,1,0,0,1,1,1],[0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,1,0,0,1,1,1,0,0,1,0,0,1,1,1,1,0,1,0,0,1,1],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,1,0,0,1,1,1,1,1,0,1,0,0,1,0,0,1,1,1,0,1],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,1,0,1,0,0,1,1,1,1,1,0,1,0,0,1,0,0,1,1,1,0],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,1,1,0,1,1,1,0,1,0,0,1,0,1,1,1,0,1,0,0,1,0,0,1,1,1],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,0,1,0,0,1,1,0,1,0,0,1,1,1,0,1,1,1,0,1,0],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,1,1,0,1,0,0,1,1,0,1,0,0,1,1,0,0,1,1,1,0,1],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,1,1,1,0,1,0,0,1,1,0,1,0,0,1,1,0,0,1,1,1,0],[1,1,1,0,0,1,0,1,1,1,0,0,1,0,1,1,0,0,1,0,1,0,0,1,0,1,1,1,1,0,0,1,0,1,1,0,0,0,0,0,0,0,1,0,0,1,0,1,1,1,1,1,0,0,1,0],[0,1,1,1,0,0,1,0,1,1,1,0,0,1,1,1,1,0,0,1,0,1,0,0,1,0,1,1,1,1,0,0,1,0,1,0,0,0,0,0,0,0,1,1,0,0,1,0,1,0,1,1,1,0,0,1],[1,0,1,1,1,0,0,1,0,1,1,1,0,0,0,1,1,1,0,0,1,1,1,0,0,1,0,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0,1,1,1,0,0,1,0,1,0,1,1,1,0,0],[1,1,1,0,0,1,0,0,1,0,1,1,1,0,0,0,1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0,0,0,0,1,0,0,1,0,1,1,1,0,0,1,0,1,1,1,1,1,0,0,1,0],[0,1,1,1,0,0,1,0,0,1,0,1,1,1,1,0,0,1,0,1,1,0,0,1,0,1,1,1,0,0,0,0,0,0,0,1,1,0,0,1,0,1,1,1,0,0,1,0,1,0,1,1,1,0,0,1],[1,0,1,1,1,0,0,1,0,0,1,0,1,1,1,1,0,0,1,0,1,1,0,0,1,0,1,1,0,0,0,0,0,0,0,1,1,1,0,0,1,0,1,1,1,0,0,1,0,1,0,1,1,1,0,0],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,1,1,1,1,1],[1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]>; 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 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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 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<K, 56 | [1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1 ,1,1,0,1,0,0],[0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0 ,1,1,1,0,1,0,0,1,1,1,0,1,0],[0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0 ,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1],[0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1 ,1,1,0,1,0,0,0,1,1,1,0,1,0,0,0,1,1,1,0,1,1,0,0,1,1,1,0],[0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0 ,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,1,1,1,0,1,1,0,0,1,1,1,0,0,1,0,0,1,1,1],[0,0,0,0,0,0,0,0,0,1,1,1,0,1,0 ,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,1,0,0,1,1,1,0,0,1,0,0,1,1,1,1,0,1,0,0,1,1],[0,0,0,0,0,0,0,0 ,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,1,0,0,1,1,1,1,1,0,1,0,0,1,0,0,1,1,1,0,1],[0 ,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,1,0,1,0,0,1,1,1,1,1,0,1,0,0,1,0 ,0,1,1,1,0],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,1,1,0,1,1,1,0,1,0,0,1,0,1 ,1,1,0,1,0,0,1,0,0,1,1,1],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,0 ,1,0,0,1,1,0,1,0,0,1,1,1,0,1,1,1,0,1,0],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1 ,1,1,0,1,0,1,1,0,1,0,0,1,1,0,1,0,0,1,1,0,0,1,1,1,0,1],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0 ,1,1,1,0,1,0,0,1,1,1,0,1,1,1,1,0,1,0,0,1,1,0,1,0,0,1,1,0,0,1,1,1,0],[1,1,1,0,0,1,0,1,1,1,0,0,1,0,1,1 ,0,0,1,0,1,0,0,1,0,1,1,1,1,0,0,1,0,1,1,0,0,0,0,0,0,0,1,0,0,1,0,1,1,1,1,1,0,0,1,0],[0,1,1,1,0,0,1,0,1 ,1,1,0,0,1,1,1,1,0,0,1,0,1,0,0,1,0,1,1,1,1,0,0,1,0,1,0,0,0,0,0,0,0,1,1,0,0,1,0,1,0,1,1,1,0,0,1],[1,0 ,1,1,1,0,0,1,0,1,1,1,0,0,0,1,1,1,0,0,1,1,1,0,0,1,0,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0,1,1,1,0,0,1,0,1,0,1 ,1,1,0,0],[1,1,1,0,0,1,0,0,1,0,1,1,1,0,0,0,1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0,0,0,0,1,0,0,1,0,1,1,1,0,0 ,1,0,1,1,1,1,1,0,0,1,0],[0,1,1,1,0,0,1,0,0,1,0,1,1,1,1,0,0,1,0,1,1,0,0,1,0,1,1,1,0,0,0,0,0,0,0,1,1,0 ,0,1,0,1,1,1,0,0,1,0,1,0,1,1,1,0,0,1],[1,0,1,1,1,0,0,1,0,0,1,0,1,1,1,1,0,0,1,0,1,1,0,0,1,0,1,1,0,0,0 ,0,0,0,0,1,1,1,0,0,1,0,1,1,1,0,0,1,0,1,0,1,1,1,0,0],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1 ,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1 ,1,1,1,1,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,1,1,1,1,1],[1,1,1,1,1,1,1,1,1,1 ,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]>; 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<K, 8 | > [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<K, 8 | > [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<K, 8 | > [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<K, 8 | > [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<K, 8 | > [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<K, 8 | > [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<K, 8 | > [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<K, 56 | [1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1 ,1,1,0,1,0,0],[0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0 ,1,1,1,0,1,0,0,1,1,1,0,1,0],[0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0 ,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1],[0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1 ,1,1,0,1,0,0,0,1,1,1,0,1,0,0,0,1,1,1,0,1,1,0,0,1,1,1,0],[0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0 ,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,1,1,1,0,1,1,0,0,1,1,1,0,0,1,0,0,1,1,1],[0,0,0,0,0,0,0,0,0,1,1,1,0,1,0 ,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,1,0,0,1,1,1,0,0,1,0,0,1,1,1,1,0,1,0,0,1,1],[0,0,0,0,0,0,0,0 ,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,1,0,0,1,1,1,1,1,0,1,0,0,1,0,0,1,1,1,0,1],[0 ,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,1,0,1,0,0,1,1,1,1,1,0,1,0,0,1,0 ,0,1,1,1,0],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,1,1,0,1,1,1,0,1,0,0,1,0,1 ,1,1,0,1,0,0,1,0,0,1,1,1],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,0 ,1,0,0,1,1,0,1,0,0,1,1,1,0,1,1,1,0,1,0],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1 ,1,1,0,1,0,1,1,0,1,0,0,1,1,0,1,0,0,1,1,0,0,1,1,1,0,1],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0 ,1,1,1,0,1,0,0,1,1,1,0,1,1,1,1,0,1,0,0,1,1,0,1,0,0,1,1,0,0,1,1,1,0],[1,1,1,0,0,1,0,1,1,1,0,0,1,0,1,1 ,0,0,1,0,1,0,0,1,0,1,1,1,1,0,0,1,0,1,1,0,0,0,0,0,0,0,1,0,0,1,0,1,1,1,1,1,0,0,1,0],[0,1,1,1,0,0,1,0,1 ,1,1,0,0,1,1,1,1,0,0,1,0,1,0,0,1,0,1,1,1,1,0,0,1,0,1,0,0,0,0,0,0,0,1,1,0,0,1,0,1,0,1,1,1,0,0,1],[1,0 ,1,1,1,0,0,1,0,1,1,1,0,0,0,1,1,1,0,0,1,1,1,0,0,1,0,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0,1,1,1,0,0,1,0,1,0,1 ,1,1,0,0],[1,1,1,0,0,1,0,0,1,0,1,1,1,0,0,0,1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0,0,0,0,1,0,0,1,0,1,1,1,0,0 ,1,0,1,1,1,1,1,0,0,1,0],[0,1,1,1,0,0,1,0,0,1,0,1,1,1,1,0,0,1,0,1,1,0,0,1,0,1,1,1,0,0,0,0,0,0,0,1,1,0 ,0,1,0,1,1,1,0,0,1,0,1,0,1,1,1,0,0,1],[1,0,1,1,1,0,0,1,0,0,1,0,1,1,1,1,0,0,1,0,1,1,0,0,1,0,1,1,0,0,0 ,0,0,0,0,1,1,1,0,0,1,0,1,1,1,0,0,1,0,1,0,1,1,1,0,0],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1 ,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1 ,1,1,1,1,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,1,1,1,1,1],[1,1,1,1,1,1,1,1,1,1 ,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]>; 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<z> := 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<i> := ExtensionField<Fp, i | i^2 + 1>; mu := i + 1; while IsSquare(mu) do mu +:= 1; end while; "mu =", mu; xi := 1/(lambda^2*mu^3); Fp12<z> := ExtensionField<Fp2, z | z^6 - xi>; 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<z> := 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<i> := ExtensionField<Fp, i | i^2 + 1>; mu := i + 1; while IsSquare(mu) do mu +:= 1; end while; "mu =", mu; xi := 1/(lambda^2*mu^3); Fp12<z> := ExtensionField<Fp2, z | z^6 - xi>; 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<z> := 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<i> := ExtensionField<Fp, i | i^2 + 1>; mu := i + 1; while IsSquare(mu) do mu +:= 1; end while; "mu =", mu; xi := 1/(lambda^2*mu^3); Fp12<z> := ExtensionField<Fp2, z | z^6 - xi>; 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<z> := 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<i> := ExtensionField<Fp, i | i^2 + 1>; mu := i + 1; while IsSquare(mu) do mu +:= 1; end while; "mu =", mu; xi := 1/(lambda^2*mu^3); Fp12<z> := ExtensionField<Fp2, z | z^6 - xi>; 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<z> := 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<i> := ExtensionField<Fp, i | i^2 + 1>; mu := i + 1; while IsSquare(mu) do mu +:= 1; end while; "mu =", mu; xi := 1/(lambda^2*mu^3); Fp12<z> := ExtensionField<Fp2, z | z^6 - xi>; 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<z> := 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<i> := ExtensionField<Fp, i | i^2 + 1>; mu := i + 1; while IsSquare(mu) do mu +:= 1; end while; "mu =", mu; xi := 1/(lambda^2*mu^3); Fp12<z> := ExtensionField<Fp2, z | z^6 - xi>; 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<z> := 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<i> := ExtensionField<Fp, i | i^2 + 1>; mu := i + 1; while IsSquare(mu) do mu +:= 1; end while; "mu =", mu; xi := 1/(lambda^2*mu^3); Fp12<z> := ExtensionField<Fp2, z | z^6 - xi>; 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<z> := 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<i> := ExtensionField<Fp, i | i^2 + 1>; mu := i + 1; while IsSquare(mu) do mu +:= 1; end while; "mu =", mu; xi := 1/(lambda^2*mu^3); Fp12<z> := ExtensionField<Fp2, z | z^6 - xi>; 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<z> := 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<i> := ExtensionField<Fp, i | i^2 + 1>; mu := i + 1; while IsSquare(mu) do mu +:= 1; end while; "mu =", mu; xi := 1/(lambda^2*mu^3); Fp12<z> := ExtensionField<Fp2, z | z^6 - xi>; 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<i> := ExtensionField<Fp, i | i^2 + 1>; lambda := Fp!2; mu := 2 + i;//1 + i; xi := 1/(lambda^2*mu^3); Fp12<z> := ExtensionField<Fp2, z | z^6 - xi>; 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<i> := ExtensionField<Fp, i | i^2 + 1>; lambda := Fp!2; mu := 2 + i;//1 + i; xi := 1/(lambda^2*mu^3); Fp12<z> := ExtensionField<Fp2, z | z^6 - xi>; 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<i> := ExtensionField<Fp, i | i^2 + 1>; lambda := Fp!2; mu := 1 + i; xi := 1/(lambda^2*mu^3); Fp12<z> := ExtensionField<Fp2, z | z^6 - xi>; 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<z> := 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<i> := ExtensionField<K, i | i^2 + 1>; 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<z> := 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<z> := 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<z> := 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<z> := 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<z> := 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<i> := ExtensionField<Fp, i | i^2 + 1>; lambda := 2; mu := 1 + i; xi := 1/(-8 + 8*i); Fp12<z> := ExtensionField<Fp2, z | z^6 - xi>; 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<i> := ExtensionField<Fp, i | i^2 + 1>; lambda := 2; mu := 1 + i; xi := 1/(-8 + 8*i); Fp12<z> := ExtensionField<Fp2, z | z^6 - xi>; 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<i> := ExtensionField<Fp, i | i^2 + 1>; lambda := 2; mu := 1 + i; xi := 1/(-8 + 8*i); Fp12<z> := ExtensionField<Fp2, z | z^6 - xi>; 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<i> := ExtensionField<Fp, i | i^2 + 1>; lambda := 2; mu := 1 + i; xi := 1/(-8 + 8*i); Fp12<z> := ExtensionField<Fp2, z | z^6 - xi>; 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<K, 56 | [1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1 ,1,1,0,1,0,0],[0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0 ,1,1,1,0,1,0,0,1,1,1,0,1,0],[0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0 ,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1],[0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1 ,1,1,0,1,0,0,0,1,1,1,0,1,0,0,0,1,1,1,0,1,1,0,0,1,1,1,0],[0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0 ,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,1,1,1,0,1,1,0,0,1,1,1,0,0,1,0,0,1,1,1],[0,0,0,0,0,0,0,0,0,1,1,1,0,1,0 ,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,1,0,0,1,1,1,0,0,1,0,0,1,1,1,1,0,1,0,0,1,1],[0,0,0,0,0,0,0,0 ,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,1,0,0,1,1,1,1,1,0,1,0,0,1,0,0,1,1,1,0,1],[0 ,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,1,0,1,0,0,1,1,1,1,1,0,1,0,0,1,0 ,0,1,1,1,0],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,1,1,0,1,1,1,0,1,0,0,1,0,1 ,1,1,0,1,0,0,1,0,0,1,1,1],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,0 ,1,0,0,1,1,0,1,0,0,1,1,1,0,1,1,1,0,1,0],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1 ,1,1,0,1,0,1,1,0,1,0,0,1,1,0,1,0,0,1,1,0,0,1,1,1,0,1],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0 ,1,1,1,0,1,0,0,1,1,1,0,1,1,1,1,0,1,0,0,1,1,0,1,0,0,1,1,0,0,1,1,1,0],[1,1,1,0,0,1,0,1,1,1,0,0,1,0,1,1 ,0,0,1,0,1,0,0,1,0,1,1,1,1,0,0,1,0,1,1,0,0,0,0,0,0,0,1,0,0,1,0,1,1,1,1,1,0,0,1,0],[0,1,1,1,0,0,1,0,1 ,1,1,0,0,1,1,1,1,0,0,1,0,1,0,0,1,0,1,1,1,1,0,0,1,0,1,0,0,0,0,0,0,0,1,1,0,0,1,0,1,0,1,1,1,0,0,1],[1,0 ,1,1,1,0,0,1,0,1,1,1,0,0,0,1,1,1,0,0,1,1,1,0,0,1,0,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0,1,1,1,0,0,1,0,1,0,1 ,1,1,0,0],[1,1,1,0,0,1,0,0,1,0,1,1,1,0,0,0,1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0,0,0,0,1,0,0,1,0,1,1,1,0,0 ,1,0,1,1,1,1,1,0,0,1,0],[0,1,1,1,0,0,1,0,0,1,0,1,1,1,1,0,0,1,0,1,1,0,0,1,0,1,1,1,0,0,0,0,0,0,0,1,1,0 ,0,1,0,1,1,1,0,0,1,0,1,0,1,1,1,0,0,1],[1,0,1,1,1,0,0,1,0,0,1,0,1,1,1,1,0,0,1,0,1,1,0,0,1,0,1,1,0,0,0 ,0,0,0,0,1,1,1,0,0,1,0,1,1,1,0,0,1,0,1,0,1,1,1,0,0],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1 ,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1 ,1,1,1,1,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,1,1,1,1,1],[1,1,1,1,1,1,1,1,1,1 ,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]>; 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<K, 56 | [1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1 ,1,1,0,1,0,0],[0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0 ,1,1,1,0,1,0,0,1,1,1,0,1,0],[0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0 ,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1],[0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1 ,1,1,0,1,0,0,0,1,1,1,0,1,0,0,0,1,1,1,0,1,1,0,0,1,1,1,0],[0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0 ,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,1,1,1,0,1,1,0,0,1,1,1,0,0,1,0,0,1,1,1],[0,0,0,0,0,0,0,0,0,1,1,1,0,1,0 ,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,1,0,0,1,1,1,0,0,1,0,0,1,1,1,1,0,1,0,0,1,1],[0,0,0,0,0,0,0,0 ,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,1,0,0,1,1,1,1,1,0,1,0,0,1,0,0,1,1,1,0,1],[0 ,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,1,0,1,0,0,1,1,1,1,1,0,1,0,0,1,0 ,0,1,1,1,0],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,1,1,0,1,1,1,0,1,0,0,1,0,1 ,1,1,0,1,0,0,1,0,0,1,1,1],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,0 ,1,0,0,1,1,0,1,0,0,1,1,1,0,1,1,1,0,1,0],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1 ,1,1,0,1,0,1,1,0,1,0,0,1,1,0,1,0,0,1,1,0,0,1,1,1,0,1],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0 ,1,1,1,0,1,0,0,1,1,1,0,1,1,1,1,0,1,0,0,1,1,0,1,0,0,1,1,0,0,1,1,1,0],[1,1,1,0,0,1,0,1,1,1,0,0,1,0,1,1 ,0,0,1,0,1,0,0,1,0,1,1,1,1,0,0,1,0,1,1,0,0,0,0,0,0,0,1,0,0,1,0,1,1,1,1,1,0,0,1,0],[0,1,1,1,0,0,1,0,1 ,1,1,0,0,1,1,1,1,0,0,1,0,1,0,0,1,0,1,1,1,1,0,0,1,0,1,0,0,0,0,0,0,0,1,1,0,0,1,0,1,0,1,1,1,0,0,1],[1,0 ,1,1,1,0,0,1,0,1,1,1,0,0,0,1,1,1,0,0,1,1,1,0,0,1,0,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0,1,1,1,0,0,1,0,1,0,1 ,1,1,0,0],[1,1,1,0,0,1,0,0,1,0,1,1,1,0,0,0,1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0,0,0,0,1,0,0,1,0,1,1,1,0,0 ,1,0,1,1,1,1,1,0,0,1,0],[0,1,1,1,0,0,1,0,0,1,0,1,1,1,1,0,0,1,0,1,1,0,0,1,0,1,1,1,0,0,0,0,0,0,0,1,1,0 ,0,1,0,1,1,1,0,0,1,0,1,0,1,1,1,0,0,1],[1,0,1,1,1,0,0,1,0,0,1,0,1,1,1,1,0,0,1,0,1,1,0,0,1,0,1,1,0,0,0 ,0,0,0,0,1,1,1,0,0,1,0,1,1,1,0,0,1,0,1,0,1,1,1,0,0],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1 ,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1 ,1,1,1,1,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,1,1,1,1,1],[1,1,1,1,1,1,1,1,1,1 ,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]>; 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<K, 56 | [1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1 ,1,1,0,1,0,0],[0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0 ,1,1,1,0,1,0,0,1,1,1,0,1,0],[0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0 ,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1],[0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1 ,1,1,0,1,0,0,0,1,1,1,0,1,0,0,0,1,1,1,0,1,1,0,0,1,1,1,0],[0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0 ,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,1,1,1,0,1,1,0,0,1,1,1,0,0,1,0,0,1,1,1],[0,0,0,0,0,0,0,0,0,1,1,1,0,1,0 ,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,1,0,0,1,1,1,0,0,1,0,0,1,1,1,1,0,1,0,0,1,1],[0,0,0,0,0,0,0,0 ,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,1,0,0,1,1,1,1,1,0,1,0,0,1,0,0,1,1,1,0,1],[0 ,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,1,0,1,0,0,1,1,1,1,1,0,1,0,0,1,0 ,0,1,1,1,0],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,1,1,0,1,1,1,0,1,0,0,1,0,1 ,1,1,0,1,0,0,1,0,0,1,1,1],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,0 ,1,0,0,1,1,0,1,0,0,1,1,1,0,1,1,1,0,1,0],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1 ,1,1,0,1,0,1,1,0,1,0,0,1,1,0,1,0,0,1,1,0,0,1,1,1,0,1],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0 ,1,1,1,0,1,0,0,1,1,1,0,1,1,1,1,0,1,0,0,1,1,0,1,0,0,1,1,0,0,1,1,1,0],[1,1,1,0,0,1,0,1,1,1,0,0,1,0,1,1 ,0,0,1,0,1,0,0,1,0,1,1,1,1,0,0,1,0,1,1,0,0,0,0,0,0,0,1,0,0,1,0,1,1,1,1,1,0,0,1,0],[0,1,1,1,0,0,1,0,1 ,1,1,0,0,1,1,1,1,0,0,1,0,1,0,0,1,0,1,1,1,1,0,0,1,0,1,0,0,0,0,0,0,0,1,1,0,0,1,0,1,0,1,1,1,0,0,1],[1,0 ,1,1,1,0,0,1,0,1,1,1,0,0,0,1,1,1,0,0,1,1,1,0,0,1,0,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0,1,1,1,0,0,1,0,1,0,1 ,1,1,0,0],[1,1,1,0,0,1,0,0,1,0,1,1,1,0,0,0,1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0,0,0,0,1,0,0,1,0,1,1,1,0,0 ,1,0,1,1,1,1,1,0,0,1,0],[0,1,1,1,0,0,1,0,0,1,0,1,1,1,1,0,0,1,0,1,1,0,0,1,0,1,1,1,0,0,0,0,0,0,0,1,1,0 ,0,1,0,1,1,1,0,0,1,0,1,0,1,1,1,0,0,1],[1,0,1,1,1,0,0,1,0,0,1,0,1,1,1,1,0,0,1,0,1,1,0,0,1,0,1,1,0,0,0 ,0,0,0,0,1,1,1,0,0,1,0,1,1,1,0,0,1,0,1,0,1,1,1,0,0],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1 ,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1 ,1,1,1,1,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,1,1,1,1,1],[1,1,1,1,1,1,1,1,1,1 ,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]>; 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<K, 56 | [1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1 ,1,1,0,1,0,0],[0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0 ,1,1,1,0,1,0,0,1,1,1,0,1,0],[0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0 ,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1],[0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1 ,1,1,0,1,0,0,0,1,1,1,0,1,0,0,0,1,1,1,0,1,1,0,0,1,1,1,0],[0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0 ,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,1,1,1,0,1,1,0,0,1,1,1,0,0,1,0,0,1,1,1],[0,0,0,0,0,0,0,0,0,1,1,1,0,1,0 ,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,1,0,0,1,1,1,0,0,1,0,0,1,1,1,1,0,1,0,0,1,1],[0,0,0,0,0,0,0,0 ,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,1,0,0,1,1,1,1,1,0,1,0,0,1,0,0,1,1,1,0,1],[0 ,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,1,0,1,0,0,1,1,1,1,1,0,1,0,0,1,0 ,0,1,1,1,0],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,1,1,0,1,1,1,0,1,0,0,1,0,1 ,1,1,0,1,0,0,1,0,0,1,1,1],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,0 ,1,0,0,1,1,0,1,0,0,1,1,1,0,1,1,1,0,1,0],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1 ,1,1,0,1,0,1,1,0,1,0,0,1,1,0,1,0,0,1,1,0,0,1,1,1,0,1],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0 ,1,1,1,0,1,0,0,1,1,1,0,1,1,1,1,0,1,0,0,1,1,0,1,0,0,1,1,0,0,1,1,1,0],[1,1,1,0,0,1,0,1,1,1,0,0,1,0,1,1 ,0,0,1,0,1,0,0,1,0,1,1,1,1,0,0,1,0,1,1,0,0,0,0,0,0,0,1,0,0,1,0,1,1,1,1,1,0,0,1,0],[0,1,1,1,0,0,1,0,1 ,1,1,0,0,1,1,1,1,0,0,1,0,1,0,0,1,0,1,1,1,1,0,0,1,0,1,0,0,0,0,0,0,0,1,1,0,0,1,0,1,0,1,1,1,0,0,1],[1,0 ,1,1,1,0,0,1,0,1,1,1,0,0,0,1,1,1,0,0,1,1,1,0,0,1,0,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0,1,1,1,0,0,1,0,1,0,1 ,1,1,0,0],[1,1,1,0,0,1,0,0,1,0,1,1,1,0,0,0,1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0,0,0,0,1,0,0,1,0,1,1,1,0,0 ,1,0,1,1,1,1,1,0,0,1,0],[0,1,1,1,0,0,1,0,0,1,0,1,1,1,1,0,0,1,0,1,1,0,0,1,0,1,1,1,0,0,0,0,0,0,0,1,1,0 ,0,1,0,1,1,1,0,0,1,0,1,0,1,1,1,0,0,1],[1,0,1,1,1,0,0,1,0,0,1,0,1,1,1,1,0,0,1,0,1,1,0,0,1,0,1,1,0,0,0 ,0,0,0,0,1,1,1,0,0,1,0,1,1,1,0,0,1,0,1,0,1,1,1,0,0],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1 ,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1 ,1,1,1,1,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,1,1,1,1,1],[1,1,1,1,1,1,1,1,1,1 ,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]>; 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:={@[email protected]}; for i:=1 to q-1 do dd:=F1.1^i; if Trace(dd) eq 0 then Trace0:=Trace0 join {@[email protected]}; 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<x>:=PolynomialRing(F1); C:=HyperellipticCurve(a*x^5+b*x^3+x,x); N1:=#C; //over Fq R<z>:=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:={@[email protected]}; for i:=1 to q-1 do dd:=F1.1^i; if Trace(dd) eq 0 then Trace0:=Trace0 join {@[email protected]}; 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<x>:=PolynomialRing(F1); C:=HyperellipticCurve(a*x^5+b*x^3+x,x); N1:=#C; //over Fq R<z>:=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:={@[email protected]}; for i:=1 to q-1 do dd:=F1.1^i; if Trace(dd) eq 0 then Trace0:=Trace0 join {@[email protected]}; 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<x>:=PolynomialRing(F1); C:=HyperellipticCurve(a*x^5+b*x^3+x,x); N1:=#C; //over Fq R<z>:=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:={@[email protected]}; for i:=1 to q-1 do dd:=F1.1^i; if Trace(dd) eq 0 then Trace0:=Trace0 join {@[email protected]}; 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<x>:=PolynomialRing(F1); C:=HyperellipticCurve(a*x^5+b*x^3+x,x); N1:=#C; //over Fq R<z>:=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<K, 56 | [1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0],[0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0],[0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1],[0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,1,1,1,0,1,0,0,0,1,1,1,0,1,1,0,0,1,1,1,0], [0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,1,1,1,0,1,1,0,0,1,1,1,0,0,1,0,0,1,1,1], [0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,1,0,0,1,1,1,0,0,1,0,0,1,1,1,1,0,1,0,0,1,1],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,1,0,0,1,1,1,1,1,0,1,0,0,1,0,0,1,1,1,0,1],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,1,0,1,0,0,1,1,1,1,1,0,1,0,0,1,0,0,1,1,1,0],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,1,1,0,1,1,1,0,1,0,0,1,0,1,1,1,0,1,0,0,1,0,0,1,1,1],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,0,1,0,0,1,1,0,1,0,0,1,1,1,0,1,1,1,0,1,0],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,1,1,0,1,0,0,1,1,0,1,0,0,1,1,0,0,1,1,1,0,1],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,1,1,1,0,1,0,0,1,1,0,1,0,0,1,1,0,0,1,1,1,0],[1,1,1,0,0,1,0,1,1,1,0,0,1,0,1,1,0,0,1,0,1,0,0,1,0,1,1,1,1,0,0,1,0,1,1,0,0,0,0,0,0,0,1,0,0,1,0,1,1,1,1,1,0,0,1,0],[0,1,1,1,0,0,1,0,1,1,1,0,0,1,1,1,1,0,0,1,0,1,0,0,1,0,1,1,1,1,0,0,1,0,1,0,0,0,0,0,0,0,1,1,0,0,1,0,1,0,1,1,1,0,0,1],[1,0,1,1,1,0,0,1,0,1,1,1,0,0,0,1,1,1,0,0,1,1,1,0,0,1,0,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0,1,1,1,0,0,1,0,1,0,1,1,1,0,0],[1,1,1,0,0,1,0,0,1,0,1,1,1,0,0,0,1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0,0,0,0,1,0,0,1,0,1,1,1,0,0,1,0,1,1,1,1,1,0,0,1,0],[0,1,1,1,0,0,1,0,0,1,0,1,1,1,1,0,0,1,0,1,1,0,0,1,0,1,1,1,0,0,0,0,0,0,0,1,1,0,0,1,0,1,1,1,0,0,1,0,1,0,1,1,1,0,0,1],[1,0,1,1,1,0,0,1,0,0,1,0,1,1,1,1,0,0,1,0,1,1,0,0,1,0,1,1,0,0,0,0,0,0,0,1,1,1,0,0,1,0,1,1,1,0,0,1,0,1,0,1,1,1,0,0],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,1,1,1,1,1],[1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]>; 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<x>:=PolynomialRing(GF(2)); f:=x^155+x^62+1; K<a>:=ext<GF(2)|f>; I:=ideal<GF(2)|f>; Output: Magma V2.11-10 Tue Dec 13 2005 11:39:59 on modular [Seed = 804427072] ------------------------------------- >> I:=ideal<GF(2)|f>;; ^ 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<x>:=PolynomialRing(GF(2)); f:=x^155+x^62+1; K<a>:=ext<GF(2)|f>; I:=Ideal<GF(2)|f>; Output: Magma V2.11-10 Tue Dec 13 2005 11:38:59 on modular [Seed = 1055097901] ------------------------------------- >> I:=Ideal<GF(2)|f>;; ^ 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<x>:=PolynomialRing(GF(2)); f:=x^155+x^62+1; K<a>:=ext<GF(2)|f>; 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<x>:=PolynomialRing(GF(2)); f:=x^155+x^62+1; K<a>:=ext<GF(2)|f>; 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<x>:=PolynomialRing(GF(2)); f:=x^155+x^62+1; K<a>:=ext<GF(2)|f>; 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<x>:=PolynomialRing(GF(2)); f:=x^155+x^62+1; K<a>:=ext<GF(2)|f>; 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<x>:=PolynomialRing(GF(2)); f:=x^155+x^62+1; K<a>:=ext<GF(2)|definingPolynomial>; Decomposition(f); Output: Magma V2.11-10 Tue Dec 13 2005 11:31:11 on modular [Seed = 3223155902] ------------------------------------- >> K<a>:=ext<GF(2)|definingPolynomial>; ^ 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<x>:=PolynomialRing(GF(2)); definingPolynomial:=x^155+x^62+1; K<a>:=ext<GF(2)|definingPolynomial>; 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<K, 16 | [1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0], [0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0], [0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0], [0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0], [0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0], [0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0], [0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0], [0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0], [0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0], [0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0], [0,0,0,0,0,0,0,0,0,0,1,0,0,0,1,0], [0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0], [0,0,0,0,0,0,0,0,0,0,0,0,1,0,1,0], [0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,0]>; > 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<K, 56 | [1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0], [0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0], [0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1], [0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,1,1,1,0,1,0,0,0,1,1,1,0,1,1,0,0,1,1,1,0], [0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,1,1,1,0,1,1,0,0,1,1,1,0,0,1,0,0,1,1,1], [0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,1,0,0,1,1,1,0,0,1,0,0,1,1,1,1,0,1,0,0,1,1], [0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,1,0,0,1,1,1,1,1,0,1,0,0,1,0,0,1,1,1,0,1], [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,1,0,1,0,0,1,1,1,1,1,0,1,0,0,1,0,0,1,1,1,0], [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,1,1,0,1,1,1,0,1,0,0,1,0,1,1,1,0,1,0,0,1,0,0,1,1,1], [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,0,1,0,0,1,1,0,1,0,0,1,1,1,0,1,1,1,0,1,0], [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,1,1,0,1,0,0,1,1,0,1,0,0,1,1,0,0,1,1,1,0,1], [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,1,1,1,0,1,0,0,1,1,0,1,0,0,1,1,0,0,1,1,1,0], [1,1,1,0,0,1,0,1,1,1,0,0,1,0,1,1,0,0,1,0,1,0,0,1,0,1,1,1,1,0,0,1,0,1,1,0,0,0,0,0,0,0,1,0,0,1,0,1,1,1,1,1,0,0,1,0], [0,1,1,1,0,0,1,0,1,1,1,0,0,1,1,1,1,0,0,1,0,1,0,0,1,0,1,1,1,1,0,0,1,0,1,0,0,0,0,0,0,0,1,1,0,0,1,0,1,0,1,1,1,0,0,1], [1,0,1,1,1,0,0,1,0,1,1,1,0,0,0,1,1,1,0,0,1,1,1,0,0,1,0,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0,1,1,1,0,0,1,0,1,0,1,1,1,0,0], [1,1,1,0,0,1,0,0,1,0,1,1,1,0,0,0,1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0,0,0,0,1,0,0,1,0,1,1,1,0,0,1,0,1,1,1,1,1,0,0,1,0], [0,1,1,1,0,0,1,0,0,1,0,1,1,1,1,0,0,1,0,1,1,0,0,1,0,1,1,1,0,0,0,0,0,0,0,1,1,0,0,1,0,1,1,1,0,0,1,0,1,0,1,1,1,0,0,1], [1,0,1,1,1,0,0,1,0,0,1,0,1,1,1,1,0,0,1,0,1,1,0,0,1,0,1,1,0,0,0,0,0,0,0,1,1,1,0,0,1,0,1,1,1,0,0,1,0,1,0,1,1,1,0,0], [0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0], [0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,1,1,1,1,1], [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]>; > 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<K, 56 | [1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0], [0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0], [0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1], [0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,1,1,1,0,1,0,0,0,1,1,1,0,1,1,0,0,1,1,1,0], [0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,1,1,1,0,1,1,0,0,1,1,1,0,0,1,0,0,1,1,1], [0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,1,0,0,1,1,1,0,0,1,0,0,1,1,1,1,0,1,0,0,1,1], [0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,1,0,0,1,1,1,1,1,0,1,0,0,1,0,0,1,1,1,0,1], [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,1,0,1,0,0,1,1,1,1,1,0,1,0,0,1,0,0,1,1,1,0], [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,1,1,0,1,1,1,0,1,0,0,1,0,1,1,1,0,1,0,0,1,0,0,1,1,1], [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,0,1,0,0,1,1,0,1,0,0,1,1,1,0,1,1,1,0,1,0], [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,1,1,0,1,0,0,1,1,0,1,0,0,1,1,0,0,1,1,1,0,1], [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,1,1,1,0,1,0,0,1,1,0,1,0,0,1,1,0,0,1,1,1,0], [1,1,1,0,0,1,0,1,1,1,0,0,1,0,1,1,0,0,1,0,1,0,0,1,0,1,1,1,1,0,0,1,0,1,1,0,0,0,0,0,0,0,1,0,0,1,0,1,1,1,1,1,0,0,1,0], [0,1,1,1,0,0,1,0,1,1,1,0,0,1,1,1,1,0,0,1,0,1,0,0,1,0,1,1,1,1,0,0,1,0,1,0,0,0,0,0,0,0,1,1,0,0,1,0,1,0,1,1,1,0,0,1], [1,0,1,1,1,0,0,1,0,1,1,1,0,0,0,1,1,1,0,0,1,1,1,0,0,1,0,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0,1,1,1,0,0,1,0,1,0,1,1,1,0,0], [1,1,1,0,0,1,0,0,1,0,1,1,1,0,0,0,1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0,0,0,0,1,0,0,1,0,1,1,1,0,0,1,0,1,1,1,1,1,0,0,1,0], [0,1,1,1,0,0,1,0,0,1,0,1,1,1,1,0,0,1,0,1,1,0,0,1,0,1,1,1,0,0,0,0,0,0,0,1,1,0,0,1,0,1,1,1,0,0,1,0,1,0,1,1,1,0,0,1], [1,0,1,1,1,0,0,1,0,0,1,0,1,1,1,1,0,0,1,0,1,1,0,0,1,0,1,1,0,0,0,0,0,0,0,1,1,1,0,0,1,0,1,1,1,0,0,1,0,1,0,1,1,1,0,0], [0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0], [0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,1,1,1,1,1], [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]>; > 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<K56 | [1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0], [0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0], [0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1], [0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,1,1,1,0,1,0,0,0,1,1,1,0,1,1,0,0,1,1,1,0], [0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,1,1,1,0,1,1,0,0,1,1,1,0,0,1,0,0,1,1,1], [0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,1,0,0,1,1,1,0,0,1,0,0,1,1,1,1,0,1,0,0,1,1], [0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,1,0,0,1,1,1,1,1,0,1,0,0,1,0,0,1,1,1,0,1], [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,1,0,1,0,0,1,1,1,1,1,0,1,0,0,1,0,0,1,1,1,0], [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,1,1,0,1,1,1,0,1,0,0,1,0,1,1,1,0,1,0,0,1,0,0,1,1,1], [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,0,1,0,0,1,1,0,1,0,0,1,1,1,0,1,1,1,0,1,0], [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,1,1,0,1,0,0,1,1,0,1,0,0,1,1,0,0,1,1,1,0,1], [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,1,1,1,0,1,0,0,1,1,0,1,0,0,1,1,0,0,1,1,1,0], [1,1,1,0,0,1,0,1,1,1,0,0,1,0,1,1,0,0,1,0,1,0,0,1,0,1,1,1,1,0,0,1,0,1,1,0,0,0,0,0,0,0,1,0,0,1,0,1,1,1,1,1,0,0,1,0], [0,1,1,1,0,0,1,0,1,1,1,0,0,1,1,1,1,0,0,1,0,1,0,0,1,0,1,1,1,1,0,0,1,0,1,0,0,0,0,0,0,0,1,1,0,0,1,0,1,0,1,1,1,0,0,1], [1,0,1,1,1,0,0,1,0,1,1,1,0,0,0,1,1,1,0,0,1,1,1,0,0,1,0,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0,1,1,1,0,0,1,0,1,0,1,1,1,0,0], [1,1,1,0,0,1,0,0,1,0,1,1,1,0,0,0,1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0,0,0,0,1,0,0,1,0,1,1,1,0,0,1,0,1,1,1,1,1,0,0,1,0], [0,1,1,1,0,0,1,0,0,1,0,1,1,1,1,0,0,1,0,1,1,0,0,1,0,1,1,1,0,0,0,0,0,0,0,1,1,0,0,1,0,1,1,1,0,0,1,0,1,0,1,1,1,0,0,1], [1,0,1,1,1,0,0,1,0,0,1,0,1,1,1,1,0,0,1,0,1,1,0,0,1,0,1,1,0,0,0,0,0,0,0,1,1,1,0,0,1,0,1,1,1,0,0,1,0,1,0,1,1,1,0,0], [0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0], [0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,1,1,1,1,1], [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]>); > 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<K56 | [1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0], [0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0], [0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1], [0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,1,1,1,0,1,0,0,0,1,1,1,0,1,1,0,0,1,1,1,0], [0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,1,1,1,0,1,1,0,0,1,1,1,0,0,1,0,0,1,1,1], [0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,1,0,0,1,1,1,0,0,1,0,0,1,1,1,1,0,1,0,0,1,1], [0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,1,0,0,1,1,1,1,1,0,1,0,0,1,0,0,1,1,1,0,1], [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,1,0,1,0,0,1,1,1,1,1,0,1,0,0,1,0,0,1,1,1,0], [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,1,1,0,1,1,1,0,1,0,0,1,0,1,1,1,0,1,0,0,1,0,0,1,1,1], [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,0,1,0,0,1,1,0,1,0,0,1,1,1,0,1,1,1,0,1,0], [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,1,1,0,1,0,0,1,1,0,1,0,0,1,1,0,0,1,1,1,0,1], [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,1,1,1,0,1,0,0,1,1,0,1,0,0,1,1,0,0,1,1,1,0], [1,1,1,0,0,1,0,1,1,1,0,0,1,0,1,1,0,0,1,0,1,0,0,1,0,1,1,1,1,0,0,1,0,1,1,0,0,0,0,0,0,0,1,0,0,1,0,1,1,1,1,1,0,0,1,0], [0,1,1,1,0,0,1,0,1,1,1,0,0,1,1,1,1,0,0,1,0,1,0,0,1,0,1,1,1,1,0,0,1,0,1,0,0,0,0,0,0,0,1,1,0,0,1,0,1,0,1,1,1,0,0,1], [1,0,1,1,1,0,0,1,0,1,1,1,0,0,0,1,1,1,0,0,1,1,1,0,0,1,0,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0,1,1,1,0,0,1,0,1,0,1,1,1,0,0], [1,1,1,0,0,1,0,0,1,0,1,1,1,0,0,0,1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0,0,0,0,1,0,0,1,0,1,1,1,0,0,1,0,1,1,1,1,1,0,0,1,0], [0,1,1,1,0,0,1,0,0,1,0,1,1,1,1,0,0,1,0,1,1,0,0,1,0,1,1,1,0,0,0,0,0,0,0,1,1,0,0,1,0,1,1,1,0,0,1,0,1,0,1,1,1,0,0,1], [1,0,1,1,1,0,0,1,0,0,1,0,1,1,1,1,0,0,1,0,1,1,0,0,1,0,1,1,0,0,0,0,0,0,0,1,1,1,0,0,1,0,1,1,1,0,0,1,0,1,0,1,1,1,0,0], [0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0], [0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,1,1,1,1,1], [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]>; > 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<K, 56 | [1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0], [0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0], [0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1], [0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,1,1,1,0,1,0,0,0,1,1,1,0,1,1,0,0,1,1,1,0], [0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,1,1,1,0,1,1,0,0,1,1,1,0,0,1,0,0,1,1,1], [0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,1,0,0,1,1,1,0,0,1,0,0,1,1,1,1,0,1,0,0,1,1], [0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,1,0,0,1,1,1,1,1,0,1,0,0,1,0,0,1,1,1,0,1], [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,1,0,1,0,0,1,1,1,1,1,0,1,0,0,1,0,0,1,1,1,0], [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,1,1,0,1,1,1,0,1,0,0,1,0,1,1,1,0,1,0,0,1,0,0,1,1,1], [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,0,1,0,0,1,1,0,1,0,0,1,1,1,0,1,1,1,0,1,0], [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,1,1,0,1,0,0,1,1,0,1,0,0,1,1,0,0,1,1,1,0,1], [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,1,1,1,0,1,0,0,1,1,0,1,0,0,1,1,0,0,1,1,1,0], [1,1,1,0,0,1,0,1,1,1,0,0,1,0,1,1,0,0,1,0,1,0,0,1,0,1,1,1,1,0,0,1,0,1,1,0,0,0,0,0,0,0,1,0,0,1,0,1,1,1,1,1,0,0,1,0], [0,1,1,1,0,0,1,0,1,1,1,0,0,1,1,1,1,0,0,1,0,1,0,0,1,0,1,1,1,1,0,0,1,0,1,0,0,0,0,0,0,0,1,1,0,0,1,0,1,0,1,1,1,0,0,1], [1,0,1,1,1,0,0,1,0,1,1,1,0,0,0,1,1,1,0,0,1,1,1,0,0,1,0,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0,1,1,1,0,0,1,0,1,0,1,1,1,0,0], [1,1,1,0,0,1,0,0,1,0,1,1,1,0,0,0,1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0,0,0,0,1,0,0,1,0,1,1,1,0,0,1,0,1,1,1,1,1,0,0,1,0], [0,1,1,1,0,0,1,0,0,1,0,1,1,1,1,0,0,1,0,1,1,0,0,1,0,1,1,1,0,0,0,0,0,0,0,1,1,0,0,1,0,1,1,1,0,0,1,0,1,0,1,1,1,0,0,1], [1,0,1,1,1,0,0,1,0,0,1,0,1,1,1,1,0,0,1,0,1,1,0,0,1,0,1,1,0,0,0,0,0,0,0,1,1,1,0,0,1,0,1,1,1,0,0,1,0,1,0,1,1,1,0,0], [0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0], [0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,1,1,1,1,1], [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]>; > 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<K, 56 | [1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0], [0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0], [0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1], [0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,1,1,1,0,1,0,0,0,1,1,1,0,1,1,0,0,1,1,1,0], [0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,1,1,1,0,1,1,0,0,1,1,1,0,0,1,0,0,1,1,1], [0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,1,0,0,1,1,1,0,0,1,0,0,1,1,1,1,0,1,0,0,1,1], [0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,1,0,0,1,1,1,1,1,0,1,0,0,1,0,0,1,1,1,0,1], [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,1,0,1,0,0,1,1,1,1,1,0,1,0,0,1,0,0,1,1,1,0], [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,1,1,0,1,1,1,0,1,0,0,1,0,1,1,1,0,1,0,0,1,0,0,1,1,1], [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,0,1,0,0,1,1,0,1,0,0,1,1,1,0,1,1,1,0,1,0], [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,1,1,0,1,0,0,1,1,0,1,0,0,1,1,0,0,1,1,1,0,1], [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,1,1,1,0,1,0,0,1,1,0,1,0,0,1,1,0,0,1,1,1,0], [1,1,1,0,0,1,0,1,1,1,0,0,1,0,1,1,0,0,1,0,1,0,0,1,0,1,1,1,1,0,0,1,0,1,1,0,0,0,0,0,0,0,1,0,0,1,0,1,1,1,1,1,0,0,1,0], [0,1,1,1,0,0,1,0,1,1,1,0,0,1,1,1,1,0,0,1,0,1,0,0,1,0,1,1,1,1,0,0,1,0,1,0,0,0,0,0,0,0,1,1,0,0,1,0,1,0,1,1,1,0,0,1], [1,0,1,1,1,0,0,1,0,1,1,1,0,0,0,1,1,1,0,0,1,1,1,0,0,1,0,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0,1,1,1,0,0,1,0,1,0,1,1,1,0,0], [1,1,1,0,0,1,0,0,1,0,1,1,1,0,0,0,1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0,0,0,0,1,0,0,1,0,1,1,1,0,0,1,0,1,1,1,1,1,0,0,1,0], [0,1,1,1,0,0,1,0,0,1,0,1,1,1,1,0,0,1,0,1,1,0,0,1,0,1,1,1,0,0,0,0,0,0,0,1,1,0,0,1,0,1,1,1,0,0,1,0,1,0,1,1,1,0,0,1], [1,0,1,1,1,0,0,1,0,0,1,0,1,1,1,1,0,0,1,0,1,1,0,0,1,0,1,1,0,0,0,0,0,0,0,1,1,1,0,0,1,0,1,1,1,0,0,1,0,1,0,1,1,1,0,0], [0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0], [0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,1,1,1,1,1], [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]>; 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<K, 56 | [1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0], [0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0], [0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1], [0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,1,1,1,0,1,0,0,0,1,1,1,0,1,1,0,0,1,1,1,0], [0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,1,1,1,0,1,1,0,0,1,1,1,0,0,1,0,0,1,1,1], [0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,1,0,0,1,1,1,0,0,1,0,0,1,1,1,1,0,1,0,0,1,1], [0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,1,0,0,1,1,1,1,1,0,1,0,0,1,0,0,1,1,1,0,1], [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,1,0,1,0,0,1,1,1,1,1,0,1,0,0,1,0,0,1,1,1,0], [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,1,1,0,1,1,1,0,1,0,0,1,0,1,1,1,0,1,0,0,1,0,0,1,1,1], [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,0,1,0,0,1,1,0,1,0,0,1,1,1,0,1,1,1,0,1,0], [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,1,1,0,1,0,0,1,1,0,1,0,0,1,1,0,0,1,1,1,0,1], [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,1,1,1,0,1,0,0,1,1,0,1,0,0,1,1,0,0,1,1,1,0], [1,1,1,0,0,1,0,1,1,1,0,0,1,0,1,1,0,0,1,0,1,0,0,1,0,1,1,1,1,0,0,1,0,1,1,0,0,0,0,0,0,0,1,0,0,1,0,1,1,1,1,1,0,0,1,0], [0,1,1,1,0,0,1,0,1,1,1,0,0,1,1,1,1,0,0,1,0,1,0,0,1,0,1,1,1,1,0,0,1,0,1,0,0,0,0,0,0,0,1,1,0,0,1,0,1,0,1,1,1,0,0,1], [1,0,1,1,1,0,0,1,0,1,1,1,0,0,0,1,1,1,0,0,1,1,1,0,0,1,0,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0,1,1,1,0,0,1,0,1,0,1,1,1,0,0], [1,1,1,0,0,1,0,0,1,0,1,1,1,0,0,0,1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0,0,0,0,1,0,0,1,0,1,1,1,0,0,1,0,1,1,1,1,1,0,0,1,0], [0,1,1,1,0,0,1,0,0,1,0,1,1,1,1,0,0,1,0,1,1,0,0,1,0,1,1,1,0,0,0,0,0,0,0,1,1,0,0,1,0,1,1,1,0,0,1,0,1,0,1,1,1,0,0,1], [1,0,1,1,1,0,0,1,0,0,1,0,1,1,1,1,0,0,1,0,1,1,0,0,1,0,1,1,0,0,0,0,0,0,0,1,1,1,0,0,1,0,1,1,1,0,0,1,0,1,0,1,1,1,0,0], [0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0], [0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,1,1,1,1,1], [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]>; 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<K, 56 | [1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0], [0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0], [0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1], [0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,1,1,1,0,1,0,0,0,1,1,1,0,1,1,0,0,1,1,1,0], [0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,1,1,1,0,1,1,0,0,1,1,1,0,0,1,0,0,1,1,1], [0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,1,0,0,1,1,1,0,0,1,0,0,1,1,1,1,0,1,0,0,1,1], [0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,1,0,0,1,1,1,1,1,0,1,0,0,1,0,0,1,1,1,0,1], [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,1,0,1,0,0,1,1,1,1,1,0,1,0,0,1,0,0,1,1,1,0], [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,1,1,0,1,1,1,0,1,0,0,1,0,1,1,1,0,1,0,0,1,0,0,1,1,1], [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,0,1,0,0,1,1,0,1,0,0,1,1,1,0,1,1,1,0,1,0], [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,1,1,0,1,0,0,1,1,0,1,0,0,1,1,0,0,1,1,1,0,1], [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,1,1,1,0,1,0,0,1,1,0,1,0,0,1,1,0,0,1,1,1,0], [1,1,1,0,0,1,0,1,1,1,0,0,1,0,1,1,0,0,1,0,1,0,0,1,0,1,1,1,1,0,0,1,0,1,1,0,0,0,0,0,0,0,1,0,0,1,0,1,1,1,1,1,0,0,1,0], [0,1,1,1,0,0,1,0,1,1,1,0,0,1,1,1,1,0,0,1,0,1,0,0,1,0,1,1,1,1,0,0,1,0,1,0,0,0,0,0,0,0,1,1,0,0,1,0,1,0,1,1,1,0,0,1], [1,0,1,1,1,0,0,1,0,1,1,1,0,0,0,1,1,1,0,0,1,1,1,0,0,1,0,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0,1,1,1,0,0,1,0,1,0,1,1,1,0,0], [1,1,1,0,0,1,0,0,1,0,1,1,1,0,0,0,1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0,0,0,0,1,0,0,1,0,1,1,1,0,0,1,0,1,1,1,1,1,0,0,1,0], [0,1,1,1,0,0,1,0,0,1,0,1,1,1,1,0,0,1,0,1,1,0,0,1,0,1,1,1,0,0,0,0,0,0,0,1,1,0,0,1,0,1,1,1,0,0,1,0,1,0,1,1,1,0,0,1], [1,0,1,1,1,0,0,1,0,0,1,0,1,1,1,1,0,0,1,0,1,1,0,0,1,0,1,1,0,0,0,0,0,0,0,1,1,1,0,0,1,0,1,1,1,0,0,1,0,1,0,1,1,1,0,0], [0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0], [0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,1,1,1,1,1], [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]>; 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<K, 56 | [1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0], [0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0], [0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1], [0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,1,1,1,0,1,0,0,0,1,1,1,0,1,1,0,0,1,1,1,0], [0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,1,1,1,0,1,1,0,0,1,1,1,0,0,1,0,0,1,1,1], [0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,1,0,0,1,1,1,0,0,1,0,0,1,1,1,1,0,1,0,0,1,1], [0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,1,0,0,1,1,1,1,1,0,1,0,0,1,0,0,1,1,1,0,1], [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,1,0,1,0,0,1,1,1,1,1,0,1,0,0,1,0,0,1,1,1,0], [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,1,1,0,1,1,1,0,1,0,0,1,0,1,1,1,0,1,0,0,1,0,0,1,1,1], [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,0,1,0,0,1,1,0,1,0,0,1,1,1,0,1,1,1,0,1,0], [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,1,1,0,1,0,0,1,1,0,1,0,0,1,1,0,0,1,1,1,0,1], [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,1,1,1,0,1,0,0,1,1,0,1,0,0,1,1,0,0,1,1,1,0], [1,1,1,0,0,1,0,1,1,1,0,0,1,0,1,1,0,0,1,0,1,0,0,1,0,1,1,1,1,0,0,1,0,1,1,0,0,0,0,0,0,0,1,0,0,1,0,1,1,1,1,1,0,0,1,0], [0,1,1,1,0,0,1,0,1,1,1,0,0,1,1,1,1,0,0,1,0,1,0,0,1,0,1,1,1,1,0,0,1,0,1,0,0,0,0,0,0,0,1,1,0,0,1,0,1,0,1,1,1,0,0,1], [1,0,1,1,1,0,0,1,0,1,1,1,0,0,0,1,1,1,0,0,1,1,1,0,0,1,0,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0,1,1,1,0,0,1,0,1,0,1,1,1,0,0], [1,1,1,0,0,1,0,0,1,0,1,1,1,0,0,0,1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0,0,0,0,1,0,0,1,0,1,1,1,0,0,1,0,1,1,1,1,1,0,0,1,0], [0,1,1,1,0,0,1,0,0,1,0,1,1,1,1,0,0,1,0,1,1,0,0,1,0,1,1,1,0,0,0,0,0,0,0,1,1,0,0,1,0,1,1,1,0,0,1,0,1,0,1,1,1,0,0,1], [1,0,1,1,1,0,0,1,0,0,1,0,1,1,1,1,0,0,1,0,1,1,0,0,1,0,1,1,0,0,0,0,0,0,0,1,1,1,0,0,1,0,1,1,1,0,0,1,0,1,0,1,1,1,0,0], [0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0], [0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,1,1,1,1,1], [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]>; 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<i> := ExtensionField<Fp, i | i^2 + 1>; lambda := 2; mu := 1 + i; xi := 1/(-8 + 8*i); Fp12<z> := ExtensionField<Fp2, z | z^6 - xi>; 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<K, 56 | [1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0], [0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0], [0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1], [0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,1,1,1,0,1,0,0,0,1,1,1,0,1,1,0,0,1,1,1,0], [0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,1,1,1,0,1,1,0,0,1,1,1,0,0,1,0,0,1,1,1], [0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,1,0,0,1,1,1,0,0,1,0,0,1,1,1,1,0,1,0,0,1,1], [0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,1,0,0,1,1,1,1,1,0,1,0,0,1,0,0,1,1,1,0,1], [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,1,0,1,0,0,1,1,1,1,1,0,1,0,0,1,0,0,1,1,1,0], [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,1,1,0,1,1,1,0,1,0,0,1,0,1,1,1,0,1,0,0,1,0,0,1,1,1], [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,0,1,0,0,1,1,0,1,0,0,1,1,1,0,1,1,1,0,1,0], [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,1,1,0,1,0,0,1,1,0,1,0,0,1,1,0,0,1,1,1,0,1], [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,1,1,1,0,1,0,0,1,1,0,1,0,0,1,1,0,0,1,1,1,0], [1,1,1,0,0,1,0,1,1,1,0,0,1,0,1,1,0,0,1,0,1,0,0,1,0,1,1,1,1,0,0,1,0,1,1,0,0,0,0,0,0,0,1,0,0,1,0,1,1,1,1,1,0,0,1,0], [0,1,1,1,0,0,1,0,1,1,1,0,0,1,1,1,1,0,0,1,0,1,0,0,1,0,1,1,1,1,0,0,1,0,1,0,0,0,0,0,0,0,1,1,0,0,1,0,1,0,1,1,1,0,0,1], [1,0,1,1,1,0,0,1,0,1,1,1,0,0,0,1,1,1,0,0,1,1,1,0,0,1,0,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0,1,1,1,0,0,1,0,1,0,1,1,1,0,0], [1,1,1,0,0,1,0,0,1,0,1,1,1,0,0,0,1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0,0,0,0,1,0,0,1,0,1,1,1,0,0,1,0,1,1,1,1,1,0,0,1,0], [0,1,1,1,0,0,1,0,0,1,0,1,1,1,1,0,0,1,0,1,1,0,0,1,0,1,1,1,0,0,0,0,0,0,0,1,1,0,0,1,0,1,1,1,0,0,1,0,1,0,1,1,1,0,0,1], [1,0,1,1,1,0,0,1,0,0,1,0,1,1,1,1,0,0,1,0,1,1,0,0,1,0,1,1,0,0,0,0,0,0,0,1,1,1,0,0,1,0,1,1,1,0,0,1,0,1,0,1,1,1,0,0], [0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0], [0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,1,1,1,1,1], [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]>; > 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<K, 56 | [1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0], [0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0], [0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1], [0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,1,1,1,0,1,0,0,0,1,1,1,0,1,1,0,0,1,1,1,0], [0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,1,1,1,0,1,1,0,0,1,1,1,0,0,1,0,0,1,1,1], [0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,1,0,0,1,1,1,0,0,1,0,0,1,1,1,1,0,1,0,0,1,1], [0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,1,0,0,1,1,1,1,1,0,1,0,0,1,0,0,1,1,1,0,1], [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,1,0,1,0,0,1,1,1,1,1,0,1,0,0,1,0,0,1,1,1,0], [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,1,1,0,1,1,1,0,1,0,0,1,0,1,1,1,0,1,0,0,1,0,0,1,1,1], [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,0,1,0,0,1,1,0,1,0,0,1,1,1,0,1,1,1,0,1,0], [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,1,1,0,1,0,0,1,1,0,1,0,0,1,1,0,0,1,1,1,0,1], [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,1,1,1,0,1,0,0,1,1,0,1,0,0,1,1,0,0,1,1,1,0], [1,1,1,0,0,1,0,1,1,1,0,0,1,0,1,1,0,0,1,0,1,0,0,1,0,1,1,1,1,0,0,1,0,1,1,0,0,0,0,0,0,0,1,0,0,1,0,1,1,1,1,1,0,0,1,0], [0,1,1,1,0,0,1,0,1,1,1,0,0,1,1,1,1,0,0,1,0,1,0,0,1,0,1,1,1,1,0,0,1,0,1,0,0,0,0,0,0,0,1,1,0,0,1,0,1,0,1,1,1,0,0,1], [1,0,1,1,1,0,0,1,0,1,1,1,0,0,0,1,1,1,0,0,1,1,1,0,0,1,0,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0,1,1,1,0,0,1,0,1,0,1,1,1,0,0], [1,1,1,0,0,1,0,0,1,0,1,1,1,0,0,0,1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0,0,0,0,1,0,0,1,0,1,1,1,0,0,1,0,1,1,1,1,1,0,0,1,0], [0,1,1,1,0,0,1,0,0,1,0,1,1,1,1,0,0,1,0,1,1,0,0,1,0,1,1,1,0,0,0,0,0,0,0,1,1,0,0,1,0,1,1,1,0,0,1,0,1,0,1,1,1,0,0,1], [1,0,1,1,1,0,0,1,0,0,1,0,1,1,1,1,0,0,1,0,1,1,0,0,1,0,1,1,0,0,0,0,0,0,0,1,1,1,0,0,1,0,1,1,1,0,0,1,0,1,0,1,1,1,0,0], [0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0], [0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,1,1,1,1,1], [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]>; > 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<K, 8 | > [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<i> := ExtensionField<Fp, i | i^2 + 1>; lambda := 2; mu := 1 + i; xi := 1/(-8 + 8*i); Fp12<z> := ExtensionField<Fp2, z | z^6 - xi>; 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<i> := ExtensionField<Fp, i | i^2 + 1>; lambda := 2; mu := 1 + i; xi := 1/(-8 + 8*i); Fp12<z> := ExtensionField<Fp2, z | z^6 - xi>; 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<i> := ExtensionField<Fp, i | i^2 + 1>; lambda := 2; mu := 1 + i; xi := 1/(-8 + 8*i); Fp12<z> := ExtensionField<Fp2, z | z^6 - xi>; 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<i> := ExtensionField<Fp, i | i^2 + 1>; lambda := 2; mu := 1 + i; xi := 1/(-8 + 8*i); Fp12<z> := ExtensionField<Fp2, z | z^6 - xi>; 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<i> := ExtensionField<Fp, i | i^2 + 1>; lambda := 2; mu := 1 + i; xi := 1/(-8 + 8*i); Fp12<z> := ExtensionField<Fp2, z | z^6 - xi>; 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<x>:=PolynomialRing(GF(2)); definingPolynomial:=x^155+x^62+1; K<a>:=ext<GF(2)|definingPolynomial>; 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<x>:=PolynomialRing(GF(2)); definingPolynomial:=x^155+x^62+1; K<a>:=ext<GF(2)|definingPolynomial>; 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<x>:=PolynomialRing(GF(2)); definingPolynomial:=x^155+x^62+1; K<a>:=ext<GF(2)|definingPolynomial>; 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<i> := ExtensionField<Fp, i | i^2 + 1>; lambda := 2; mu := 1 + i; xi := 1/(-8 + 8*i); Fp12<z> := ExtensionField<Fp2, z | z^6 - xi>; 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<i> := ExtensionField<Fp, i | i^2 + 1>; lambda := 2; mu := 1 + i; xi := 1/(-8 + 8*i); Fp12<z> := ExtensionField<Fp2, z | z^6 - xi>; 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<i> := ExtensionField<Fp, i | i^2 + 1>; lambda := 2; mu := 1 + i; xi := 1/(-8 + 8*i); Fp12<z> := ExtensionField<Fp2, z | z^6 - xi>; 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<x>:=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<i> := ExtensionField<Fp, i | i^2 + 1>; lambda := 2; mu := 1 + i; xi := 1/(-8 + 8*i); Fp12<z> := ExtensionField<Fp2, z | z^6 - xi>; 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<i> := ExtensionField<Fp, i | i^2 + 1>; lambda := 2; mu := 1 + i; xi := 1/(-8 + 8*i); Fp12<z> := ExtensionField<Fp2, z | z^6 - xi>; 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<i> := ExtensionField<Fp, i | i^2 + 1>; lambda := 2; mu := 1 + i; xi := 1/(-8 + 8*i); Fp12<z> := ExtensionField<Fp2, z | z^6 - xi>; 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<i> := ExtensionField<Fp, i | i^2 + 1>; lambda := 2; mu := 1 + i; xi := 1/(-8 + 8*i); Fp12<z> := ExtensionField<Fp2, z | z^6 - xi>; 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<i> := ExtensionField<Fp, i | i^2 + 1>; lambda := 2; mu := 1 + i; xi := 1/(-8 + 8*i); Fp12<z> := ExtensionField<Fp2, z | z^6 - xi>; 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<i> := ExtensionField<Fp, i | i^2 + 1>; lambda := 2; mu := 1 + i; xi := 1/(-8 + 8*i); Fp12<z> := ExtensionField<Fp2, z | z^6 - xi>; 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<i> := ExtensionField<Fp, i | i^2 + 1>; lambda := 2; mu := 1 + i; xi := 1/(-8 + 8*i); Fp12<z> := ExtensionField<Fp2, z | z^6 - xi>; 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<i> := ExtensionField<Fp, i | i^2 + 1>; lambda := 2; mu := 1 + i; xi := 1/(-8 + 8*i); Fp12<z> := ExtensionField<Fp2, z | z^6 - xi>; 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<i> := ExtensionField<Fp, i | i^2 + 1>; lambda := 2; mu := 1 + i; xi := 1/(-8 + 8*i); Fp12<z> := ExtensionField<Fp2, z | z^6 - xi>; 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<i> := ExtensionField<Fp, i | i^2 + 1>; lambda := 2; mu := 1 + i; xi := 1/(-8 + 8*i); Fp12<z> := ExtensionField<Fp2, z | z^6 - xi>; 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<i> := ExtensionField<Fp, i | i^2 + 1>; lambda := 2; mu := 1 + i; xi := 1/(-8 + 8*i); Fp12<z> := ExtensionField<Fp2, z | z^6 - xi>; 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<i> := ExtensionField<Fp, i | i^2 + 1>; lambda := 2; mu := 1 + i; xi := 1/(-8 + 8*i); Fp12<z> := ExtensionField<Fp2, z | z^6 - xi>; 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<i> := ExtensionField<Fp, i | i^2 + 1>; lambda := 2; mu := 1 + i; xi := 1/(-8 + 8*i); Fp12<z> := ExtensionField<Fp2, z | z^6 - xi>; 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<i> := ExtensionField<Fp, i | i^2 + 1>; lambda := 2; mu := 1 + i; xi := 1/(-8 + 8*i); Fp12<z> := ExtensionField<Fp2, z | z^6 - xi>; 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<i> := ExtensionField<Fp, i | i^2 + 1>; lambda := 2; mu := 1 + i; xi := 1/(-8 + 8*i); Fp12<z> := ExtensionField<Fp2, z | z^6 - xi>; 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<i> := ExtensionField<Fp, i | i^2 + 1>; lambda := 2; mu := 1 + i; xi := 1/(-8 + 8*i); Fp12<z> := ExtensionField<Fp2, z | z^6 - xi>; 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<i> := ExtensionField<Fp, i | i^2 + 1>; lambda := 2; mu := 1 + i; xi := 1/(-8 + 8*i); Fp12<z> := ExtensionField<Fp2, z | z^6 - xi>; 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<i> := ExtensionField<Fp, i | i^2 + 1>; lambda := 2; mu := 1 + i; xi := 1/(-8 + 8*i); Fp12<z> := ExtensionField<Fp2, z | z^6 - xi>; 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<i> := ExtensionField<Fp, i | i^2 + 1>; lambda := 2; mu := 1 + i; xi := 1/(-8 + 8*i); Fp12<z> := ExtensionField<Fp2, z | z^6 - xi>; 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 ï¿½ 1 to 0 by ï¿½1 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 ^ï¿½ 1 to 0 by ^ï¿½1 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<x> := PolynomialRing(F2); p := x^230 + x^50 + x^33 + x^32 + 1; F<z> := 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<i> := ExtensionField<Fp, i | i^2 + 1>; lambda := 2; mu := 1 + i; xi := 1/(-8 + 8*i); Fp12<z> := ExtensionField<Fp2, z | z^6 - xi>; 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<i> := ExtensionField<Fp, i | i^2 + 1>; lambda := 2; mu := 1 + i; xi := 1/(-8 + 8*i); Fp12<z> := ExtensionField<Fp2, z | z^6 - xi>; 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<i> := ExtensionField<Fp, i | i^2 + 1>; lambda := 2; mu := 1 + i; xi := 1/(-8 + 8*i); Fp12<z> := ExtensionField<Fp2, z | z^6 - xi>; 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<i> := ext<Fp, i | i^2 + 1>; lambda := 2; mu := 1 + i; xi := 1/(-8 + 8*i); Fp12<z> := ext<Fp2, z | z^6 - xi>; Output: Magma V2.11-10 Mon Dec 12 2005 14:04:05 on modular [Seed = 14584475] ------------------------------------- >> Fp2<i> := ext<Fp, i | i^2 + 1>; ^ 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<z> := ext<Fp2, z | z^6 - xi>; ^ 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<x> := PolynomialRing(Integers()); > > f :=x^4-2*x^2+2; > > K<y> := 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<x> := PolynomialRing(Integers()); > > f :=x^4-2*x^2+2; > > K<y> := 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<x> := PolynomialRing(Integers()); > > f :=x^4-2*x^2+2; > > K<y> := 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<x> := PolynomialRing(Integers()); > > f :=x^4-2*x^2+2; > > K<y> := 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<x> := PolynomialRing(Integers()); > > f :=x^8 - 20*x^6 + 98*x^4 - 76*x^2 + 1; > > K<y> := 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<x> := PolynomialRing(F2); p := x^120 + x^41 + x^35 + x^32 + 1; F<z> := 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<x> := 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 [email protected] 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<x> := 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<x> := 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<x> := 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<x> := 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<x,y>:=PolynomialRing(Q,2);
I:=ideal<P| y + (1+x^5+x^10), x^34 +1 >; 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);
^

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<K, 56 |   [1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0],[0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0],[0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1],[0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,1,1,1,0,1,0,0,0,1,1,1,0,1,1,0,0,1,1,1,0],[0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,1,1,1,0,1,1,0,0,1,1,1,0,0,1,0,0,1,1,1],[0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,1,0,0,1,1,1,0,0,1,0,0,1,1,1,1,0,1,0,0,1,1],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,1,0,0,1,1,1,1,1,0,1,0,0,1,0,0,1,1,1,0,1],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,1,0,1,0,0,1,1,1,1,1,0,1,0,0,1,0,0,1,1,1,0],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,1,1,0,1,1,1,0,1,0,0,1,0,1,1,1,0,1,0,0,1,0,0,1,1,1],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,0,1,0,0,1,1,0,1,0,0,1,1,1,0,1,1,1,0,1,0],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,1,1,0,1,0,0,1,1,0,1,0,0,1,1,0,0,1,1,1,0,1],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,1,1,1,0,1,0,0,1,1,0,1,0,0,1,1,0,0,1,1,1,0],[1,1,1,0,0,1,0,1,1,1,0,0,1,0,1,1,0,0,1,0,1,0,0,1,0,1,1,1,1,0,0,1,0,1,1,0,0,0,0,0,0,0,1,0,0,1,0,1,1,1,1,1,0,0,1,0],[0,1,1,1,0,0,1,0,1,1,1,0,0,1,1,1,1,0,0,1,0,1,0,0,1,0,1,1,1,1,0,0,1,0,1,0,0,0,0,0,0,0,1,1,0,0,1,0,1,0,1,1,1,0,0,1],[1,0,1,1,1,0,0,1,0,1,1,1,0,0,0,1,1,1,0,0,1,1,1,0,0,1,0,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0,1,1,1,0,0,1,0,1,0,1,1,1,0,0],[1,1,1,0,0,1,0,0,1,0,1,1,1,0,0,0,1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0,0,0,0,1,0,0,1,0,1,1,1,0,0,1,0,1,1,1,1,1,0,0,1,0],[0,1,1,1,0,0,1,0,0,1,0,1,1,1,1,0,0,1,0,1,1,0,0,1,0,1,1,1,0,0,0,0,0,0,0,1,1,0,0,1,0,1,1,1,0,0,1,0,1,0,1,1,1,0,0,1],[1,0,1,1,1,0,0,1,0,0,1,0,1,1,1,1,0,0,1,0,1,1,0,0,1,0,1,1,0,0,0,0,0,0,0,1,1,1,0,0,1,0,1,1,1,0,0,1,0,1,0,1,1,1,0,0],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0],[0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,1,1,1,1,1],[1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]>;
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<K, 8 |
[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 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<K, 8 |
[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 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<K, 56 |   [1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0],
[0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0],
[0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1],
[0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,1,1,1,0,1,0,0,0,1,1,1,0,1,1,0,0,1,1,1,0],
[0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,1,1,1,0,1,1,0,0,1,1,1,0,0,1,0,0,1,1,1],
[0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,1,0,0,1,1,1,0,0,1,0,0,1,1,1,1,0,1,0,0,1,1],
[0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,1,0,0,1,1,1,1,1,0,1,0,0,1,0,0,1,1,1,0,1],
[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,1,0,1,0,0,1,1,1,1,1,0,1,0,0,1,0,0,1,1,1,0],
[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,1,1,0,1,1,1,0,1,0,0,1,0,1,1,1,0,1,0,0,1,0,0,1,1,1],
[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,0,1,0,0,1,1,0,1,0,0,1,1,1,0,1,1,1,0,1,0],
[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,1,1,0,1,0,0,1,1,0,1,0,0,1,1,0,0,1,1,1,0,1],
[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,1,1,1,0,1,0,0,1,1,0,1,0,0,1,1,0,0,1,1,1,0],
[1,1,1,0,0,1,0,1,1,1,0,0,1,0,1,1,0,0,1,0,1,0,0,1,0,1,1,1,1,0,0,1,0,1,1,0,0,0,0,0,0,0,1,0,0,1,0,1,1,1,1,1,0,0,1,0],
[0,1,1,1,0,0,1,0,1,1,1,0,0,1,1,1,1,0,0,1,0,1,0,0,1,0,1,1,1,1,0,0,1,0,1,0,0,0,0,0,0,0,1,1,0,0,1,0,1,0,1,1,1,0,0,1],
[1,0,1,1,1,0,0,1,0,1,1,1,0,0,0,1,1,1,0,0,1,1,1,0,0,1,0,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0,1,1,1,0,0,1,0,1,0,1,1,1,0,0],
[1,1,1,0,0,1,0,0,1,0,1,1,1,0,0,0,1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0,0,0,0,1,0,0,1,0,1,1,1,0,0,1,0,1,1,1,1,1,0,0,1,0],
[0,1,1,1,0,0,1,0,0,1,0,1,1,1,1,0,0,1,0,1,1,0,0,1,0,1,1,1,0,0,0,0,0,0,0,1,1,0,0,1,0,1,1,1,0,0,1,0,1,0,1,1,1,0,0,1],
[1,0,1,1,1,0,0,1,0,0,1,0,1,1,1,1,0,0,1,0,1,1,0,0,1,0,1,1,0,0,0,0,0,0,0,1,1,1,0,0,1,0,1,1,1,0,0,1,0,1,0,1,1,1,0,0],
[0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0],
[0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,1,1,1,1,1],
[1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]>;
> 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<K, 56 |   [1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0],
[0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0],
[0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1],
[0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,1,1,1,0,1,0,0,0,1,1,1,0,1,1,0,0,1,1,1,0],
[0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,1,1,1,0,1,1,0,0,1,1,1,0,0,1,0,0,1,1,1],
[0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,1,0,0,1,1,1,0,0,1,0,0,1,1,1,1,0,1,0,0,1,1],
[0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,1,0,0,1,1,1,1,1,0,1,0,0,1,0,0,1,1,1,0,1],
[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,1,0,1,0,0,1,1,1,1,1,0,1,0,0,1,0,0,1,1,1,0],
[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,1,1,0,1,1,1,0,1,0,0,1,0,1,1,1,0,1,0,0,1,0,0,1,1,1],
[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,0,1,0,0,1,1,0,1,0,0,1,1,1,0,1,1,1,0,1,0],
[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,1,1,0,1,0,0,1,1,0,1,0,0,1,1,0,0,1,1,1,0,1],
[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,1,1,1,0,1,0,0,1,1,0,1,0,0,1,1,0,0,1,1,1,0],
[1,1,1,0,0,1,0,1,1,1,0,0,1,0,1,1,0,0,1,0,1,0,0,1,0,1,1,1,1,0,0,1,0,1,1,0,0,0,0,0,0,0,1,0,0,1,0,1,1,1,1,1,0,0,1,0],
[0,1,1,1,0,0,1,0,1,1,1,0,0,1,1,1,1,0,0,1,0,1,0,0,1,0,1,1,1,1,0,0,1,0,1,0,0,0,0,0,0,0,1,1,0,0,1,0,1,0,1,1,1,0,0,1],
[1,0,1,1,1,0,0,1,0,1,1,1,0,0,0,1,1,1,0,0,1,1,1,0,0,1,0,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0,1,1,1,0,0,1,0,1,0,1,1,1,0,0],
[1,1,1,0,0,1,0,0,1,0,1,1,1,0,0,0,1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0,0,0,0,1,0,0,1,0,1,1,1,0,0,1,0,1,1,1,1,1,0,0,1,0],
[0,1,1,1,0,0,1,0,0,1,0,1,1,1,1,0,0,1,0,1,1,0,0,1,0,1,1,1,0,0,0,0,0,0,0,1,1,0,0,1,0,1,1,1,0,0,1,0,1,0,1,1,1,0,0,1],
[1,0,1,1,1,0,0,1,0,0,1,0,1,1,1,1,0,0,1,0,1,1,0,0,1,0,1,1,0,0,0,0,0,0,0,1,1,1,0,0,1,0,1,1,1,0,0,1,0,1,0,1,1,1,0,0],
[0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0],
[0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,1,1,1,1,1],
[1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]>;
> 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<K, 56 |   [1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0],
[0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0],
[0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,1,1,0,1],
[0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,1,1,1,0,1,0,0,0,1,1,1,0,1,1,0,0,1,1,1,0],
[0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,1,1,1,0,1,1,0,0,1,1,1,0,0,1,0,0,1,1,1],
[0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,1,0,0,1,1,1,0,0,1,0,0,1,1,1,1,0,1,0,0,1,1],
[0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,1,0,0,1,1,1,1,1,0,1,0,0,1,0,0,1,1,1,0,1],
[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,1,0,1,0,0,1,1,1,1,1,0,1,0,0,1,0,0,1,1,1,0],
[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,1,1,0,1,1,1,0,1,0,0,1,0,1,1,1,0,1,0,0,1,0,0,1,1,1],
[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,0,1,0,1,0,0,1,1,0,1,0,0,1,1,1,0,1,1,1,0,1,0],
[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,0,1,1,0,1,0,0,1,1,0,1,0,0,1,1,0,0,1,1,1,0,1],
[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,0,1,0,0,1,1,1,0,1,1,1,1,0,1,0,0,1,1,0,1,0,0,1,1,0,0,1,1,1,0],
[1,1,1,0,0,1,0,1,1,1,0,0,1,0,1,1,0,0,1,0,1,0,0,1,0,1,1,1,1,0,0,1,0,1,1,0,0,0,0,0,0,0,1,0,0,1,0,1,1,1,1,1,0,0,1,0],
[0,1,1,1,0,0,1,0,1,1,1,0,0,1,1,1,1,0,0,1,0,1,0,0,1,0,1,1,1,1,0,0,1,0,1,0,0,0,0,0,0,0,1,1,0,0,1,0,1,0,1,1,1,0,0,1],
[1,0,1,1,1,0,0,1,0,1,1,1,0,0,0,1,1,1,0,0,1,1,1,0,0,1,0,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0,1,1,1,0,0,1,0,1,0,1,1,1,0,0],
[1,1,1,0,0,1,0,0,1,0,1,1,1,0,0,0,1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0,0,0,0,1,0,0,1,0,1,1,1,0,0,1,0,1,1,1,1,1,0,0,1,0],
[0,1,1,1,0,0,1,0,0,1,0,1,1,1,1,0,0,1,0,1,1,0,0,1,0,1,1,1,0,0,0,0,0,0,0,1,1,0,0,1,0,1,1,1,0,0,1,0,1,0,1,1,1,0,0,1],
[1,0,1,1,1,0,0,1,0,0,1,0,1,1,1,1,0,0,1,0,1,1,0,0,1,0,1,1,0,0,0,0,0,0,0,1,1,1,0,0,1,0,1,1,1,0,0,1,0,1,0,1,1,1,0,0],
[0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0],
[0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,1,1,1,1,1],
[1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]>;
> 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<K, 8 |
>    [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<K, 8 |
>    [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]
^

>>      [0 0 1 0 1 1 0 1] [0 0 0 1 1 1 1 0]>;
^

>>   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<K, 8 |
>    [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<a,b,c>:=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]
-------------------------------------

[
<b, 1>,
<b + c - 1, 1>,
<a + b - 1, 1>,
<a + b + c - 1, 1>
]
[
<b - 1/2, 2>,
<a + b + c - 1, 1>
]

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<a,b,c>:=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]
-------------------------------------

[
<b, 1>,
<b + c - 1, 1>,
<a + b - 1, 1>,
<a + b + c - 1, 1>
]
[
<a + b + c - 1, 1>,
<a - 1/2, 2>
]

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<a,b,c>:=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]
-------------------------------------

[
<a, 1>,
<a + c - 1, 1>,
<a + b - 1, 1>,
<a + b + c - 1, 1>
]
[
<a + b + c - 1, 1>,
<a - 1/2, 2>
]

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<a,b,c>:=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]
-------------------------------------

[
<a, 1>,
<a + c - 1, 1>,
<a + b - 1, 1>,
<a + b + c - 1, 1>
]
[
<a^4 + 2*a^3*b + 2*a^3*c - 3*a^3 + a^2*b^2 + 3*a^2*b*c - 4*a^2*b + a^2*c^2 -
4*a^2*c + 3*a^2 + a*b^2*c - a*b^2 + a*b*c^2 - 4*a*b*c + 2*a*b - a*c^2 +
2*a*c - 3/4*a + 1/4*b + 1/4*c - 1/4, 1>
]

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<a,b,c>:=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]
-------------------------------------

[
<a, 1>,
<a + c - 1, 1>,
<a + b - 1, 1>,
<a + b + c - 1, 1>
]
[
<a + b + c - 1, 2>,
<a - 1/2, 2>
]

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<a,b,c>:=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]
-------------------------------------

[
<a, 1>,
<a + c - 1, 1>,
<a + b - 1, 1>,
<a + b + c - 1, 1>
]
[
<a + b + c - 1, 1>,
<a^3 + a^2*b + a^2*c - 2*a^2 - 3*a*b*c - a*b - a*c + 2*a + b + c - 1, 1>
]

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<a,b,c>:=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]
-------------------------------------

[
<a, 1>,
<a + c - 1, 1>,
<a + b - 1, 1>,
<a + b + c - 1, 1>
]

>> print Factorization(a*(a+c-1)*(a*(a+b-1)-(4*a*b*c-a-b-c+1));;
^

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<a,b,c>:=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]
-------------------------------------

[
<a^4 + 2*a^3*b + 2*a^3*c - 3*a^3 + a^2*b^2 + 3*a^2*b*c - 4*a^2*b + a^2*c^2 -
4*a^2*c + 3*a^2 + a*b^2*c - a*b^2 + a*b*c^2 - 7*a*b*c + 2*a*b - a*c^2 +
2*a*c + b + c - 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: Fri Dec  9 07:55:08 2005

Input: K<a,b,c>:=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]
-------------------------------------

[
<a^3 + a^2*b + a^2*c - 2*a^2 - 3*a*b*c - a*b - a*c + 2*a + b + c - 1, 1>
]

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<a,b,c>:=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]
-------------------------------------

[
<a^3*b^2*c^3 + a^2*b^3*c^3 - a^2*b^2*c^3 - 4*a*b*c + a + b + c - 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: Fri Dec  9 07:54:23 2005

Input: K<a,b,c>:=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]
-------------------------------------

[
<a^4*b^3*c^3 + a^3*b^4*c^3 - a^3*b^3*c^3 - 4*a*b*c + a + b + c - 1, 1>
]

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<a,b,c>:=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]
-------------------------------------

[
<a^2*b + a*b^2 - 4*a*b*c - a*b + a + b + c - 1, 1>
]

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<a,b,c>:=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]
-------------------------------------

[
<a^2*b*c + a*b^2*c - 5*a*b*c + a + b + c - 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: Fri Dec  9 07:49:19 2005

Input: K<a,b,c>:=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]
-------------------------------------

[
<a + b + c - 1, 1>,
<a^3*b^2*c + a^2*b^3*c - a^2*b^2*c - 2*a*b*c + 1/2*a + 1/2*b + 1/2*c - 1/2,
1>
]

Total time: 0.200 seconds, Total memory usage: 3.34MB

'128.139'
************** MAGMA *****************
Host 128.139.226.37 (128.139.226.37)
Time: Fri Dec  9 07:48:58 2005

Input: K<a,b,c>:=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]
-------------------------------------

[
<a + b + c - 1, 1>,
<a^3*b^2*c + a^2*b^3*c - a^2*b^2*c - 4*a*b*c + a + b + c - 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: Fri Dec  9 07:48:40 2005

Input: K<a,b,c>:=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]
-------------------------------------

[
<a + b + c - 1, 1>,
<a^2*b + a*b^2 - 8*a*b*c - a*b + 2*a + 2*b + 2*c - 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: Fri Dec  9 07:48:33 2005

Input: K<a,b,c>:=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]
-------------------------------------

[
<a + b + c - 1, 1>,
<a^2*b + a*b^2 + 8*a*b*c - a*b - 2*a - 2*b - 2*c + 2, 1>
]

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<a,b,c>:=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]
-------------------------------------

[
<a + b + c - 1, 1>,
<a^2*b + a*b^2 + 16*a*b*c - a*b - 4*a - 4*b - 4*c + 4, 1>
]

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<a,b,c>:=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]
-------------------------------------

[
<a + b + c - 1, 1>,
<a^2*b + a*b^2 - 16*a*b*c - a*b + 4*a + 4*b + 4*c - 4, 1>
]

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<a,b,c>:=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]
-------------------------------------

[
<a + b + c - 1, 1>,
<a^2*b*c + a*b^2*c - 17*a*b*c + 4*a + 4*b + 4*c - 4, 1>
]

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<a,b,c>:=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]
-------------------------------------

[
<a + b + c - 1, 1>,
<a^2*b*c + a*b^2*c + 15*a*b*c - 4*a - 4*b - 4*c + 4, 1>
]

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<a,b,c>:=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]
-------------------------------------

[
<a + b + c - 1, 1>,
<a^2*b*c + a*b^2*c + 7*a*b*c - 2*a - 2*b - 2*c + 2, 1>
]

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<a,b,c>:=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]
-------------------------------------

[
<a + b + c - 1, 1>,
<a^2*b*c + a*b^2*c - 9*a*b*c + 2*a + 2*b + 2*c - 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: Fri Dec  9 07:47:36 2005

Input: K<a,b,c>:=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]
-------------------------------------

[
<a + b + c - 1, 1>,
<a^3*b^2*c^2 + a^2*b^3*c^2 - a^2*b^2*c^2 - 1/4*a*b*c + 1/16*a + 1/16*b +
1/16*c - 1/16, 1>
]

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<a,b,c>:=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]
-------------------------------------

[
<a + b + c - 1, 1>,
<a^3*b^2*c^2 + a^2*b^3*c^2 - a^2*b^2*c^2 - 1/2*a*b*c + 1/8*a + 1/8*b + 1/8*c
- 1/8, 1>
]

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<a,b,c>:=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]
-------------------------------------

[
<a + b + c - 1, 1>,
<a^3*b^2*c^2 + a^2*b^3*c^2 - a^2*b^2*c^2 - 2/3*a*b*c + 1/6*a + 1/6*b + 1/6*c
- 1/6, 1>
]

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<a,b,c>:=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]
-------------------------------------

[
<a + b + c - 1, 1>,
<a^2*b*c + a*b^2*c - 5/3*a*b*c + 1/6*a + 1/6*b + 1/6*c - 1/6, 1>
]

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<a,b,c>:=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]
-------------------------------------

[
<a + b + c - 1, 1>,
<a^2*b*c + a*b^2*c - 3/2*a*b*c + 1/8*a + 1/8*b + 1/8*c - 1/8, 1>
]

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<a,b,c>:=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]
-------------------------------------

[
<a + b + c - 1, 1>,
<a^2*b*c + a*b^2*c + a*b*c - 1/2*a - 1/2*b - 1/2*c + 1/2, 1>
]

Total time: 0.190 seconds, Total memory usage: 3.34MB

'128.139'
************** MAGMA *****************
Host 128.139.226.37 (128.139.226.37)
Time: Fri Dec  9 07:46:34 2005

Input: K<a,b,c>:=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]
-------------------------------------

[
<a + b + c - 1, 1>,
<a*b*c - 1/6*a + 1/3*b^2*c - 1/3*b*c - 1/6*b - 1/6*c + 1/6, 1>
]

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<a,b,c>:=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]
-------------------------------------

[
<a + b + c - 1, 1>,
<a*b*c - 1/5*a + 1/5*b^2*c - 1/5*b*c - 1/5*b - 1/5*c + 1/5, 1>
]

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<a,b,c>:=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]
-------------------------------------

[
<a + b + c - 1, 1>,
<a^2*b + a*b^2 + 4*a*b*c - a*b - a - b - c + 1, 1>
]

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<a,b,c>:=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]
-------------------------------------

[
<a + b + c - 1, 1>,
<a^2*c + 5*a*b*c - a*c - a - b - c + 1, 1>
]

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<a,b,c>:=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]
-------------------------------------

[
<a + b + c - 1, 1>,
<a^2*b*c + a*b^2*c + 3*a*b*c - a - b - c + 1, 1>
]

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<a,b,c>:=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]
-------------------------------------

[
<a + b + c - 1, 1>,
<a^2*b*c + a*b^2*c - 1/4*a - 1/4*b - 1/4*c + 1/4, 1>
]

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<a,b,c>:=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]
-------------------------------------

[
<a + b + c - 1, 1>,
<a^3*b^2*c^2 + a^2*b^3*c^2 - a^2*b^2*c^2 + a*b*c - 1/4*a - 1/4*b - 1/4*c +
1/4, 1>
]

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<a,b,c>:=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]
-------------------------------------

[
<a + b + c - 1, 1>,
<a^3*b^2*c^2 + a^2*b^3*c^2 - a^2*b^2*c^2 + 2*a*b*c - 1/2*a - 1/2*b - 1/2*c +
1/2, 1>
]

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<a,b,c>:=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]
-------------------------------------

[
<a + b + c - 1, 1>,
<a^3*b^2*c^2 + a^2*b^3*c^2 - a^2*b^2*c^2 - 2*a*b*c + 1/2*a + 1/2*b + 1/2*c -
1/2, 1>
]

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<a,b,c>:=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]
-------------------------------------

[
<a + b + c - 1, 1>,
<a^3*b^2*c^2 + a^2*b^3*c^2 - a^2*b^2*c^2 - a*b*c + 1/4*a + 1/4*b + 1/4*c -
1/4, 1>
]

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<a,b,c>:=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]
-------------------------------------

[
<a + b + c - 1, 1>,
<a^3*b^2 + a^2*b^3 - a^2*b^2 - a*b*c + 1/4*a + 1/4*b + 1/4*c - 1/4, 1>
]

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<a,b,c>:=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]
-------------------------------------

[
<a + b + c - 1, 1>,
<a^3*b^2 + a^2*b^3 - a^2*b^2 - 2*a*b*c + 1/2*a + 1/2*b + 1/2*c - 1/2, 1>
]

Total time: 0.200 seconds, Total memory usage: 3.34MB

'128.139'
************** MAGMA *****************
Host 128.139.226.36 (128.139.226.36)
Time: Fri Dec  9 07:44:45 2005

Input: K<a,b,c>:=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]
-------------------------------------

[
<a + b + c - 1, 1>,
<a^3*b^2 + a^2*b^3 - a^2*b^2 - 4*a*b*c + a + b + c - 1, 1>
]

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<a,b,c>:=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]
-------------------------------------

[
<a + b + c - 1, 1>,
<a^3*b + a^2*b^2 - a^2*b - 4*a*b*c + a + b + c - 1, 1>
]

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<a,b,c>:=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]
-------------------------------------

[
<a + b + c - 1, 1>,
<a^2*b + a*b^2 - 4*a*b*c - a*b + a + b + c - 1, 1>
]

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<a,b,c>:=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]
-------------------------------------

[
<a + b + c - 1, 1>,
<a^2*c - 3*a*b*c - a*c + a + b + c - 1, 1>
]

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<a,b,c>:=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]
-------------------------------------

[
<a + b + c - 1, 1>,
<a*b*c - 1/3*a - 1/3*b^2*c + 1/3*b*c - 1/3*b - 1/3*c + 1/3, 1>
]

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<a,b,c>:=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]
-------------------------------------

[
<a + b + c - 1, 1>,
<a*b*c - 1/2*a - b^2*c + b*c - 1/2*b - 1/2*c + 1/2, 1>
]

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<a,b,c>:=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]
-------------------------------------

[
<a + b + c - 1, 1>,
<a^2*c - a*b*c - a*c + 1/2*a + 1/2*b + 1/2*c - 1/2, 1>
]

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<a,b,c>:=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]
-------------------------------------

[
<a + b + c - 1, 1>,
<a^2*b + a*b^2 - 2*a*b*c - a*b + 1/2*a + 1/2*b + 1/2*c - 1/2, 1>
]

Total time: 0.200 seconds, Total memory usage: 3.34MB

'128.139'
************** MAGMA *****************
Host 128.139.226.36 (128.139.226.36)
Time: Fri Dec  9 07:43:18 2005

Input: K<a,b,c>:=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]
-------------------------------------

[
<a + b + c - 1, 1>,
<a^2*b*c + a*b^2*c - 2*a*b*c + 1/4*a + 1/4*b + 1/4*c - 1/4, 1>
]

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<a,b,c>:=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]
-------------------------------------

[
<a + b + c - 1, 1>,
<a^2*b*c + a*b^2*c - 3*a*b*c + 1/2*a + 1/2*b + 1/2*c - 1/2, 1>
]

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<a,b,c>:=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]
-------------------------------------

[
<a + b + c - 1, 1>,
<a^2*b*c + a*b^2*c - 5*a*b*c + a + b + c - 1, 1>
]

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<a,b,c>:=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]
-------------------------------------

[
<a, 1>,
<a + c - 1, 1>,
<a + b - 1, 1>,
<a + b + c - 1, 1>
]

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<a,b,c>:=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]
-------------------------------------

[
<a + b + c - 1, 1>,
<a^3 + a^2*b + a^2*c - 2*a^2 - a*b*c - a*b - a*c + 3/2*a + 1/2*b + 1/2*c -
1/2, 1>
]

Total time: 0.190 seconds, Total memory usage: 3.34MB

'128.139'
************** MAGMA *****************
Host 128.139.226.36 (128.139.226.36)
Time: Fri Dec  9 07:39:22 2005

Input: K<a,b,c>:=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]
-------------------------------------

[
<a + b + c - 1, 1>,
<a^3 + a^2*b + a^2*c - 2*a^2 - a*b*c - a*b - a*c + 3/2*a + 1/2*b + 1/2*c -
1/2, 1>
]

Total time: 0.190 seconds, Total memory usage: 3.34MB

'128.139'
************** MAGMA *****************
Host 128.139.226.36 (128.139.226.36)
Time: Fri Dec  9 07:39:13 2005

Input: K<a,b,c>:=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]
-------------------------------------

[
<a + b + c - 1, 1>,
<a^3 - a^2*b + a^2*c - a^2 - a*b^2 - 6*a*b*c + 2*a*b + a + b^3 + b^2*c - b^2
+ b + c - 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: Fri Dec  9 07:38:58 2005

Input: K<a,b,c>:=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]
-------------------------------------

[
<a^2*b*c - 1/2*a^2 + a*b^2*c + a*b*c^2 - a*b*c - 1/2*a*c + 1/2*a - 1/2*b^2 -
1/2*b*c + 1/2*b - 1/4*c^2 + 1/2*c - 1/4, 1>
]

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<K, 8 |
>    [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<K, 8 |
>    [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<K, 8 |
>    [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
^

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<K, 8 |
>    [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<K, 8 |
>    [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<K, 8 |
>    [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<K, 8 |
>    [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
^

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<K, 8 |
>    [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
^

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<K, 8 |
>    [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;
^

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<K, 8 |
>    [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<K, 8 |
>    [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];
^

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<a,b,c>:=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<t>:=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]
[
<a + c - 1, 1>,
<a + b - 1, 1>,
<a + b + c - 1, 1>
]
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<a,b,c>:=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<t>:=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 + c - 1, 1>,
<a + b - 1, 1>,
<a + b + c - 1, 1>
]
(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<a,b,c>:=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<t>:=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]
[
<a + c - 1, 1>,
<a + b - 1, 1>,
<a + b + c - 1, 1>
]
(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 + b + c - 1, 1>,
<a^3*b + a^3*c - a^3 + a^2*b*c - 2*a^2*b + a^2*c^2 - 3*a^2*c + 2*a^2 - a*b^3
- a*b^2*c + 2*a*b^2 - 2*a*b*c - a*c^2 + 2*a*c - 1/2*a - b^3*c + b^3 -
b^2*c^2 + 3*b^2*c - 2*b^2 + b*c^2 - 2*b*c + 3/2*b + 1/2*c - 1/2, 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<a,b,c>:=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<t>:=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]
[
<a + c - 1, 1>,
<a + b - 1, 1>,
<a + b + c - 1, 1>
]
(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 + b + c - 1, 1>,
<a^3*b^2 + a^3*b*c - a^3*b + a^2*b^3 + 2*a^2*b^2*c - 3*a^2*b^2 + a^2*b*c^2 -
3*a^2*b*c + 2*a^2*b + a*b^3*c - a*b^3 + a*b^2*c^2 - 3*a*b^2*c + 2*a*b^2
- a*b*c^2 - a*b + 1/2*a + 1/2*b + 1/2*c - 1/2, 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<a,b,c>:=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<t>:=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]
[
<a + c - 1, 1>,
<a + b - 1, 1>,
<a + b + c - 1, 1>
]
(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 + b + c - 1, 1>,
<a^3*b^2*c + a^3*b*c^2 - a^3*b*c + a^2*b^3*c + 2*a^2*b^2*c^2 - 3*a^2*b^2*c +
a^2*b*c^3 - 3*a^2*b*c^2 + 2*a^2*b*c + a*b^3*c^2 - a*b^3*c + a*b^2*c^3 -
3*a*b^2*c^2 + 2*a*b^2*c - a*b*c^3 + 2*a*b*c^2 - 3*a*b*c + 1/2*a + 1/2*b
+ 1/2*c - 1/2, 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<a,b,c>:=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<t>:=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]
[
<a + c - 1, 1>,
<a + b - 1, 1>,
<a + b + c - 1, 1>
]
(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 + b + c - 1, 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, 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<a,b,c>:=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<t>:=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]
[
<a + c - 1, 1>,
<a + b - 1, 1>,
<a + b + c - 1, 1>
]
(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
[
<a^2*b + a^2*c - a^2 + a*b^2 - 3*a*b + a*c^2 - 3*a*c + 5/2*a + b^2*c - b^2 +
b*c^2 - 3*b*c + 5/2*b - c^2 + 5/2*c - 3/2, 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 18:39:09 2005

Input: K<a,b,c>:=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<t>:=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]
[
<a + c - 1, 1>,
<a + b - 1, 1>,
<a + b + c - 1, 1>
]
(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<a,b,c>:=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<t>:=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]
[
<a + c - 1, 1>,
<a + b - 1, 1>,
<a + b + c - 1, 1>
]
(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<a,b,c>:=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<t>:=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]
[
<a + c - 1, 1>,
<a + b - 1, 1>,
<a + b + c - 1, 1>
]
(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
[
<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, 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 18:36:58 2005

Input: K<a,b,c>:=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<t>:=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]
[
<a + c - 1, 1>,
<a + b - 1, 1>,
<a + b + c - 1, 1>
]
(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<a,b,c>:=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<t>:=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]
[
<a + c - 1, 1>,
<a + b - 1, 1>,
<a + b + c - 1, 1>
]
(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<a,b,c>:=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<t>:=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]
[
<c, 2>,
<b, 2>,
<a, 1>
]
(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<a,b,c>:=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<t>:=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]
[
<c, 2>,
<b, 2>,
<a, 2>
]
(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<a,b,c>:=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<t>:=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]
[
<c, 2>,
<b, 2>,
<a, 2>
]
(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<a,b,c>:=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<t>:=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<a,b,c>:=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<t>:=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]
[
<a*b*c + 3/4*a + 3/4*b + 3/4*c - 3/4, 1>
]
(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<a,b,c>:=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<t>:=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]
[
<a*b*c + 3/4*a + 3/4*b + 3/4*c - 3/4, 1>
]
(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<a,b,c>:=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<t>:=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]
[
<c, 1>,
<b, 1>,
<a, 1>
]
(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<a,b,c>:=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<t>:=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]
[
<c, 1>,
<b, 1>,
<a, 1>
]
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<a,b,c>:=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<t>:=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]
[
<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, 1>
]
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<a,b,c>:=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<t>:=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)));
^
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<a,b,c>:=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<t>:=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<a,b,c>:=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<t>:=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<a,b,c>:=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<t>:=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<a,b,c>:=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<t>:=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)));
^

>>   end for;
^

>> end for;
^

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<a,b,c>:=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<t>:=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<a,b,c>:=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<t>:=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<a,b,c>:=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<t>:=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<a,b,c>:=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<t>:=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<a,b,c>:=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<t>:=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)
[
<b + c - 1, 1>,
<a + b - 1, 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 18:14:46 2005

Input: K<a,b,c>:=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<t>:=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)
[
<a + b - 1, 1>,
<a + b + c - 1, 1>,
<a^2 - 2*a*b + a*c + b^2 + b*c + c - 1, 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 18:12:48 2005

Input: K<a,b,c>:=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<t>:=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)
[
<a + b - 1, 1>,
<a + b + c - 1, 1>,
<a^2 - 2*a*b + a*c + b^2 + b*c + c - 1, 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 18:12:25 2005

Input: K<a,b,c>:=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<t>:=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)
[
<a^2 - 2*a*b + a*c + b^2 + b*c + 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 18:11:51 2005

Input: K<a,b,c>:=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<t>:=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)
[
<a^2 - 2*a*b + a*c + b^2 + b*c + 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 18:11:06 2005

Input: K<a,b,c>:=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<t>:=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)
[
<a^2 - 2*a*b + a*c + b^2 + b*c + c - 1, 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<a,b,c>:=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<t>:=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)
[
<a^2 - 2*a*b + a*c + b^2 + b*c + 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 18:09:23 2005

Input: K<a,b,c>:=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<t>:=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<a,b,c>:=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<t>:=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));
^

>>   end for;
^

>> end for;
^

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<a,b,c>:=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<t>:=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)
[
<a + b - 1, 1>,
<a*b*c + 1/4*a*c^2 - 1/4*a + 1/4*b*c^2 - 1/4*b + 1/4*c^2 - 1/2*c + 1/4, 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 18:04:49 2005

Input: K<a,b,c>:=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<t>:=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<a,b,c>:=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<t>:=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<a,b,c>:=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<t>:=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<a,b,c>:=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<t>:=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<a,b,c>:=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<t>:=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<a,b,c>:=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<t>:=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<a,b,c>:=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<t>:=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<a,b,c>:=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<t>:=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
[
<a + b + c - 1, 1>,
<a*b*c - 1/4*a - 1/4*b - 1/4*c + 1/4, 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<a,b,c>:=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<t>:=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<a,b,c>:=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<t>:=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<a,b,c>:=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<t>:=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<a,b,c>:=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<t>:=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<a,b,c>:=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<t>:=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
[
<a + b + c - 1, 1>,
<a*b*c - 1/4*a - 1/4*b - 1/4*c + 1/4, 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<a,b,c>:=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<t>:=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<a,b,c>:=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<t>:=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<a,b,c>:=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<t>:=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<a,b,c>:=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<t>:=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);
^

>>   end for;
^

>> end for;
^

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<a,b,c>:=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<t>:=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]
[
<c, 2>,
<b, 2>,
<a, 2>,
<a + b - 1, 1>,
<a + b + c - 1, 1>
]
[
<c, 2>,
<b, 2>,
<a, 1>,
<a + b - 1, 1>,
<a + b + c - 1, 1>
]
[
<c, 2>,
<b, 1>,
<a, 1>,
<a + b + c - 1, 1>
]
[
<c, 2>,
<b, 2>,
<a, 2>,
<a + b - 1, 1>,
<a + b + c - 1, 1>
]
[
<c, 2>,
<b, 2>,
<a, 1>,
<a + b - 1, 1>,
<a + b + c - 1, 1>
]
[
<c, 2>,
<b, 1>,
<a, 1>,
<a + b + c - 1, 1>
]
[
<c, 1>,
<b, 2>,
<a, 2>,
<a + b + c - 1, 1>
]
[
<c, 1>,
<b, 2>,
<a, 1>,
<a + b + c - 1, 1>
]
[
<c, 1>,
<b, 1>,
<a, 1>,
<a + b + c - 1, 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 16:57:34 2005

Input: K<a,b,c>:=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<t>:=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<a,b,c>:=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<t>:=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<a,b,c>:=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<t>:=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<a,b,c>:=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<t>:=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<a,b,c>:=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<t>:=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