[was@modular anti-cyclotomic_height_pairing]$ [was@modular anti-cyclotomic_height_pairing]$ Magma V2.7-1 Tue Apr 24 2001 02:46:37 on modular [Seed = 3240039148] Type ? for help. Type -D to quit. Loading startup file "/home/was/modsym/init-magma.m" C IndexGamma0 R ellap idxG0 CS MS S factormod modcharpoly DC ND Tn factorpadic padiccharpoly ES NS Z fcp qexp F Q charpoly fn x > K := QuadraticField(-389); > K; Quadratic Field with defining polynomial $.1^2 + 389 over the Rational Field > ClassNumber(K); 22 > O := MaximalOrder(K); > ClassGroup; Intrinsic 'ClassGroup' Signatures: ( O) -> GrpAb, Map [ Proof: "Bound" | "Current" | "Full" | "GRH" | "Subgroup", Bound: RngIntElt ] The class group of the ring of integers O ( K) -> GrpAb, Map [ Proof: "Bound" | "Current" | "Full" | "GRH" | "Subgroup", Bound: RngIntElt ] The class group of the ring of integers O of K ( Z) -> GrpAb, Map [ Proof: "Bound" | "Current" | "Full" | "GRH" | "Subgroup", Bound: RngIntElt ] The class group of the ring of integers Z (which is trivial) ( Q) -> GrpAb, Map [ Proof: "Bound" | "Current" | "Full" | "GRH" | "Subgroup", Bound: RngIntElt ] The class group of the ring of integers Z of Q (which is trivial) ( Q) -> GrpAb, Map The abelian class group of Q, followed by the isomorphism to Q. ( K) -> GrpAb, Map ( R) -> GrpAb, Map > A,f := ClassGroup(O); > A; Abelian Group isomorphic to Z/22 Defined on 1 generator Relations: 22*A.1 = 0 > f(A.1); <3,2,130> > Type($1); QuadBinElt > a := f(A.1); > ideal; >> ideal; ^ Runtime error in ideal< ... >: Elements must be coercible into the quadratic order > a; <3,2,130> > Type(a); QuadBinElt > a[1]; 3 > a[2]; 2 > a[3]; 130 > Ideal(a); >> Ideal(a); ^ Runtime error in 'Ideal': Bad argument types Argument types given: QuadBinElt > QuadraticForms; Intrinsic 'QuadraticForms' Signatures: ( D) -> QuadBin Create the magma of binary quadratic forms with discriminant D > Lattice(a); Standard Lattice of rank 2 and degree 2 Inner Product Matrix: [ 6 2] [ 2 260] > K := QuadraticField(-389); > I := ideal; > I; Ideal of Maximal Order of Quadratic Field with defining polynomial $.1^2 + 389 over the Rational Field Two element generators : 1, $.1 > IsPrincipal(I); >> IsPrincipal(I); ^ Runtime error in 'IsPrincipal': Bad argument types Argument types given: RngQuadIdl > L := NumberField(x^2+389); > L; Number Field with defining polynomial x^2 + 389 over the Rational Field > O := MaximalOrder(L); > O; Maximal Equation Order of L > I := ideal; > I; Ideal of O Two element generators: [3, 0] [0, 1] > A,f := ClassGroup(L); > A; Abelian Group isomorphic to Z/22 Defined on 1 generator Relations: 22*A.1 = 0 > f(A.1); Ideal of O Two element generators: [118098, 0] [23737, -1] > IsPrincipal(I); true > t,gen := IsPrincipal(I); > gen; 1 > I@@f; 0 > Type(I); RngOrdIdl > Type(w); FldNumElt > Modulus(pAdicRing(3)); >> Modulus(pAdicRing(3)); ^ Runtime error in 'Modulus': Bad argument types Argument types given: RngLoc > Prime(pAdicRing(3)); 3 > I; Principal Ideal of O Generator: [1, 0] > Parent(I); Set of ideals of Maximal Equation Order of L > Order(I); Maximal Equation Order of L > MaximalOrder(I); >> MaximalOrder(I); ^ Runtime error in 'MaximalOrder': Bad argument types Argument types given: RngOrdIdl > UnitGroup(O); Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*$.1 = 0 Mapping from: Abelian Group isomorphic to Z/2 Defined on 1 generator Relations: 2*$.1 = 0 to RngOrd: O > ClassNumber(O); 22 > _,gen := IsPrincipal(I); true > _,gen := IsPrincipal(I); > gen; 1 > Eltseq(gen); [ 1, 0 ] > Eltseq(w); [ 0, 1 ] > Type(L); FldNum > Degree(L); 2 > Discriminant(L); -1556 > L := NumberField(x^2+3); > > O; Maximal Order of L > MinimalPolynomial(O.1); >> MinimalPolynomial(O.1); ^ Runtime error in '.': Bad argument types Argument types given: RngOrd, RngIntElt > DefiningPolynomial(O); $.1^2 + 3 > L := NumberField(x^2+5);O := MaximalOrder(L);DefiningPolynomial(O); > L := NumberField(x^2+5);O := MaximalOrder(L);DefiningPolynomial(O); $.1^2 + 5 > L := NumberField(x^2+7);O := MaximalOrder(L);DefiningPolynomial(O); $.1^2 + 7 > O.1^2+7; >> O.1^2+7; ^ Runtime error in '.': Bad argument types Argument types given: RngOrd, RngIntElt > O; Maximal Order of L > O.1; >> O.1; ^ Runtime error in '.': Bad argument types Argument types given: RngOrd, RngIntElt > Basis(O)[2]; 1/2*$.1 + 1/2 > O![0,1]; [0, 1] > L!(O![0,1]); 1/2*$.1 + 1/2 > MinimalPolynomial(Basis(O)[2]); x^2 - x + 2 > ; > ; > Attach("anti-cyclotomic.m"); > ; > E := EC("389A"); > FindAcceptableK(E,-30,-1); >> FindAcceptableK(E,-30,-1); ^ Runtime error in 'FindAcceptableK': Bad argument types Argument types given: CrvEll, RngIntElt, RngIntElt > FindAcceptableK(E,3,-30,-1); In file "/home/was/papers/anti-cyclotomic_height_pairing/anti-cyclotomic.m", line 119, column 41: >> require Type(K) eq FldNum and Degree(L) eq 2 and Discriminant(L) lt 0 : ^ Runtime error: Undefined reference 'L' in package "/home/was/papers/anti-cyclotomic_height_pairing/anti-cyclotomic.m" > ; > FindAcceptableK(E,3,-30,-1); FindAcceptableK( E: E, p: 3, Dmin: -30, Dmax: -1 ) In file "/home/was/papers/anti-cyclotomic_height_pairing/anti-cyclotomic.m", line 38, column 27: >> if #Factorization(O!p) eq 1 then ^ Runtime error in 'Factorization': factorization not available in this ring > ; > FindAcceptableK(E,3,-30,-1); FindAcceptableK( E: E, p: 3, Dmin: -30, Dmax: -1 ) In file "/home/was/papers/anti-cyclotomic_height_pairing/anti-cyclotomic.m", line 38, column 27: >> if #Factorization(ideal) eq 1 then ^ Runtime error in 'Factorization': Bad argument types Argument types given: RngQuadIdl > ; > FindAcceptableK(E,3,-30,-1); In file "/home/was/papers/anti-cyclotomic_height_pairing/anti-cyclotomic.m", line 36, column 27: >> K := NumberField(x^2-D); ^ Runtime error: Undefined reference 'x' in package "/home/was/papers/anti-cyclotomic_height_pairing/anti-cyclotomic.m" > ; > FindAcceptableK(E,3,-30,-1); FindAcceptableK( E: E, p: 3, Dmin: -30, Dmax: -1 ) In file "/home/was/papers/anti-cyclotomic_height_pairing/anti-cyclotomic.m", line 46, column 48: >> if IsOdd(fac[2]) and #Factorization(O!fac[1]) eq 1 then ^ Runtime error in 'Factorization': Bad argument types Argument types given: RngOrdElt > ; > FindAcceptableK(E,3,-30,-1); [ -20, -11 ] > E11 := QuadraticTwist(E,-11); > E11; Elliptic Curve defined by y^2 = x^3 - 365904*x - 61524144 over Rational Field > QuadraticTwist; Intrinsic 'QuadraticTwist' Signatures: ( E) -> CrvEll The quadratic twist of the elliptic curve E defined over a finite field ( E, d) -> CrvEll The quadratic twist of E by d. ( C, d) -> CrvHyp The quadratic twist of the curve C by d. ( C) -> CrvHyp The quadratic twist of the curve C defined over a finite field. > E; Elliptic Curve defined by y^2 + y = x^3 + x^2 - 2*x over Rational Field > Weierstrass(E); Elliptic Curve defined by y^2 + y = x^3 + x^2 - 2*x over Rational Field > SimplifiedModel(E); >> SimplifiedModel(E); ^ Runtime error in 'SimplifiedModel': Curve must be defined over a finite field > Ew, phi := WeierstrassModel(E); > phi; Elliptic curve isomorphism from: CrvEll: E to CrvEll: Ew Taking (x : y : 1) to (36*x + 12 : 216*y + 108 : 1) > ; In file "/home/was/papers/anti-cyclotomic_height_pairing/anti-cyclotomic.m", line 150, column 1: >> end intrinsic; ^ User error: bad syntax > F := EllipticCurve([1,2]); > aInvariants(F); [ 0, 0, 0, 1, 2 ] > G,f := MordellWeilGroup(E); In file "/home/was/papers/anti-cyclotomic_height_pairing/anti-cyclotomic.m", line 147, column 13: >> Ew, phi, ? := WeierstrassModel(E); ^ User error: bad syntax > G; Abelian Group isomorphic to Z + Z Defined on 2 generators (free) > Rank(G); >> Rank(G); ^ Runtime error in 'Rank': Bad argument types Argument types given: GrpAb > Invariants(G); [ 0, 0 ] > A := AbelianGroup([5,0]); In file "/home/was/papers/anti-cyclotomic_height_pairing/anti-cyclotomic.m", line 147, column 13: >> Ew, phi, ? := WeierstrassModel(E); ^ User error: bad syntax > A; Abelian Group isomorphic to Z/5 + Z Defined on 2 generators Relations: 5*A.1 = 0 > Invariants(A); [ 5, 0 ] > A := AbelianGroup([0,5]); > Invariants(A); [ 5, 0 ] > A.1; A.1 > A.2; A.2 > Ngens(A); 2 > K := NumberField(x^2+4); > s^2; -4 > ; > ; > E; Elliptic Curve defined by y^2 + y = x^3 + x^2 - 2*x over Rational Field > TwistPoint(E,-11); TwistPoint( E: E, D: -11 ) In file "/home/was/papers/anti-cyclotomic_height_pairing/anti-cyclotomic.m", line 154, column 10: >> x := P[1]/P[3]; ^ Runtime error in '[]': Bad argument types > TwistPoint(E,-11); TwistPoint( E: E, D: -11 ) WeierstrassModel( E: E ) In file "/usr/local/Magma2.7/package/EC/models.m", line 23, column 24: >> _, f := IsIsomorphic(E,F); ^ Runtime error in 'IsIsomorphic': Curves must be defined over the rational field or a finite field > ; > ; > TwistPoint(E,-11); TwistPoint( E: E, D: -11 ) In file "/home/was/papers/anti-cyclotomic_height_pairing/anti-cyclotomic.m", line 162, column 11: >> Q := Ew![x/D, (y/D^2) * sqrtD]; ^ Runtime error in '!': Point is not on curve > TwistPoint(E,-11); Elliptic Curve defined by y^2 = x^3 - 3024*x + 46224 over K [ 21, 27*sqrtD ] TwistPoint( E: E, D: -11 ) In file "/home/was/papers/anti-cyclotomic_height_pairing/anti-cyclotomic.m", line 164, column 11: >> Q := Ew![x/D, (y/D^2) * sqrtD]; ^ Runtime error in '!': Point is not on curve > 21^3-3024*21+46224; -8019 > 27^2; 729 > factor(8019); [ <3, 6>, <11, 1> ] 1 > ; In file "/home/was/papers/anti-cyclotomic_height_pairing/anti-cyclotomic.m", line 164, column 33: >> Q := Ew![x/D, (y/(D^2* sqrtD)]; ^ User error: bad syntax > ; > TwistPoint(E,-11); Elliptic Curve defined by y^2 = x^3 - 3024*x + 46224 over K [ 21, 27*sqrtD ] TwistPoint( E: E, D: -11 ) In file "/home/was/papers/anti-cyclotomic_height_pairing/anti-cyclotomic.m", line 164, column 11: >> Q := Ew![x/D, (y/(D^2*sqrtD))]; ^ Runtime error in '!': Point is not on curve > TwistPoint(E,-11); Elliptic Curve defined by y^2 = x^3 - 3024*x + 46224 over K [ 21, 27/11*sqrtD ] TwistPoint( E: E, D: -11 ) In file "/home/was/papers/anti-cyclotomic_height_pairing/anti-cyclotomic.m", line 164, column 11: >> Q := Ew![x/D, (y/(D^2*sqrtD))]; ^ Runtime error in '!': Point is not on curve > ; > TwistPoint(E,-11); Elliptic Curve defined by y^2 = x^3 - 3024*x + 46224 over K [ 21, 27/11*sqrtD ] TwistPoint( E: E, D: -11 ) In file "/home/was/papers/anti-cyclotomic_height_pairing/anti-cyclotomic.m", line 164, column 11: >> Q := Ew![x/D, (y/D^2)*sqrtD]; ^ Runtime error in '!': Point is not on curve > TwistPoint(E,-11); In file "/home/was/papers/anti-cyclotomic_height_pairing/anti-cyclotomic.m", line 164, column 4: >> Q := Ew!try; ^ User error: bad syntax >> TwistPoint(E,-11); ^ User error: Identifier 'TwistPoint' has not been declared or assigned > TwistPoint(E,-11); Elliptic Curve defined by y^2 = x^3 - 3024*x + 46224 over K TwistPoint( E: E, D: -11 ) In file "/home/was/papers/anti-cyclotomic_height_pairing/anti-cyclotomic.m", line 164, column 11: >> Q := Ew!try; ^ Runtime error in '!': Point is not on curve > TwistPoint(E,-11); Elliptic Curve defined by y^2 = x^3 - 3024*x + 46224 over K [ 21, 27*sqrtD ] TwistPoint( E: E, D: -11 ) In file "/home/was/papers/anti-cyclotomic_height_pairing/anti-cyclotomic.m", line 165, column 11: >> Q := Ew!try; ^ Runtime error in '!': Point is not on curve > TwistPoint(E,-11); -3024 46224 Elliptic Curve defined by y^2 = x^3 - 3024*x + 46224 over K [ 21, 27*sqrtD ] TwistPoint( E: E, D: -11 ) In file "/home/was/papers/anti-cyclotomic_height_pairing/anti-cyclotomic.m", line 166, column 11: >> Q := Ew!try; ^ Runtime error in '!': Point is not on curve > TwistPoint(E,-11); Elliptic Curve defined by y^2 = x^3 - 3024*x + 46224 over K [ 21, 27*sqrtD ] rhs = -8019 lhs = 8019 TwistPoint( E: E, D: -11 ) In file "/home/was/papers/anti-cyclotomic_height_pairing/anti-cyclotomic.m", line 167, column 11: >> Q := Ew!try; ^ Runtime error in '!': Point is not on curve > TwistPoint(E,-11); Elliptic Curve defined by y^2 = x^3 - 3024*x + 46224 over K [ 21, 27*sqrtD ] rhs = -8019 lhs = -8019 TwistPoint( E: E, D: -11 ) In file "/home/was/papers/anti-cyclotomic_height_pairing/anti-cyclotomic.m", line 168, column 14: >> return psi(Q); ^ Runtime error in map application: Cannot coerce element into ring > phi; Elliptic curve isomorphism from: CrvEll: E to CrvEll: Ew Taking (x : y : 1) to (36*x + 12 : 216*y + 108 : 1) > Type(phi); Map > BaseExtend(phi,RationalField()); >> BaseExtend(phi,RationalField()); ^ Runtime error in 'BaseExtend': Bad argument types Argument types given: Map, FldRat > Weierstrass(E); [ -3024, 46224 ] > EE := BaseExtend(E,K); > K; Number Field with defining polynomial z^2 + 4 over the Rational Field > Weierstrass(EE); [ -3024, 46224 ] > WeierstrassModel(EE); WeierstrassModel( E: E ) In file "/usr/local/Magma2.7/package/EC/models.m", line 23, column 24: >> _, f := IsIsomorphic(E,F); ^ Runtime error in 'IsIsomorphic': Curves must be defined over the rational field or a finite field > ; > TwistPoint(E,-11); TwistPoint( E: E, D: -11 ) In file "/home/was/papers/anti-cyclotomic_height_pairing/anti-cyclotomic.m", line 160, column 13: >> Ew,_,psi := Weierstrass(E); ^ Runtime error in :=: Expected to assign 3 value(s) but only computed 1 value(s) > Weierstrass; Intrinsic 'Weierstrass' Signatures: ( E) -> SeqEnum Returns the sequence [a, b] of elements defining the Weierstrass B form of the elliptic curve. Note that 2 and 3 must not be 0-divisors > psi; >> psi; ^ User error: Identifier 'psi' has not been declared or assigned > phi; Elliptic curve isomorphism from: CrvEll: E to CrvEll: Ew Taking (x : y : 1) to (36*x + 12 : 216*y + 108 : 1) > Eltseq(phi); >> Eltseq(phi); ^ Runtime error in 'Eltseq': Bad argument types Argument types given: Map > Rule(phi); >> Rule(phi); ^ User error: Identifier 'Rule' has not been declared or assigned > Domain(phi); Elliptic Curve defined by y^2 + y = x^3 + x^2 - 2*x over Rational Field > phi; Elliptic curve isomorphism from: CrvEll: E to CrvEll: Ew Taking (x : y : 1) to (36*x + 12 : 216*y + 108 : 1) > IsomorphismData(phi); [ 12, 0, 108, 6 ] > Ew, phi, psi := WeierstrassModel(E); > IsomorphismData(psi); [ -1/3, 0, -1/2, 1/6 ] > ; > TwistPoint(E,-11); In file "/home/was/papers/anti-cyclotomic_height_pairing/anti-cyclotomic.m", line 162, column 12: >> Q := Pw!try; ^ Runtime error: Undefined reference 'try' in package "/home/was/papers/anti-cyclotomic_height_pairing/anti-cyclotomic.m" > ; > TwistPoint(E,-11); (1/4 : 1/8*sqrtD - 1/2 : 1) > TwistPoint(E,-20); (-61/20 : 319/400*sqrtD - 1/2 : 1) > E; Elliptic Curve defined by y^2 + y = x^3 + x^2 - 2*x over Rational Field > P11 := TwistPoint(E,-11); >> P11 := TwistPoint(E,-11); ^ User error: Identifier 'TwistPoint' has not been declared or assigned > P11 := PointFromTwist(E,-11); > P11; (1/4 : 1/8*sqrtD - 1/2 : 1) > Parent(P11); Elliptic Curve defined by y^2 + y = x^3 + x^2 - 2*x over Number Field with defining polynomial z^2 + 11 over the Rational Field > H_2(P,3); >> H_2(P,3); ^ User error: Identifier 'P' has not been declared or assigned > H_2(P11,3); H_2( P: P, p: 3 ) In file "/home/was/papers/anti-cyclotomic_height_pairing/anti-cyclotomic.m", line 128, column 22: >> pibar := -Evaluate(fac[2][2],0); ^ Runtime error in 'Evaluate': Bad argument types Argument types given: RngIntElt, RngIntElt > H_2(P11,3); 0 > SetVerbose("ac_height",2); > H_2(P11,3); H_2( (1/4 : 1/8*sqrtD - 1/2 : 1) , 3 ) NumeratorIdeal( 1/4 ) rhotilde( Ideal of Maximal Order of Equation Order with defining polynomial x^2 + 11 over its ground order Two element generators: [1, 0] [1, 0] , 1586753292 + O(3^20) , -1586753291 + O(3^20) ) 0 > H_2(3*P11,3)/3^2; H_2( (1859969/2244004 : 394204143/3361517992*sqrtD - 1/2 : 1) , 3 ) NumeratorIdeal( 1859969/2244004 ) rhotilde( Ideal of Maximal Order of Equation Order with defining polynomial x^2 + 11 over its ground order Two element generators: [1859969, 0] [1859969, 0] , 1586753292 + O(3^20) , -1586753291 + O(3^20) ) O(3^18) > n := 2; H_2(3^n*P11,3)/3^(2*n); H_2( (15149121201043150712077534649287181358659069263220165310209/36997655900725382379967627461892952671282761517597531922500 : 1237472676951754508036409981037158464683197464035772760418709017813761949272201574578081/7116415768783413451489917678950449784975851252781995109773021001240894476312667294125000*sqrtD - 1/2 : 1) , 3 ) NumeratorIdeal( 15149121201043150712077534649287181358659069263220165310209/36997655900725382379967627461892952671282761517597531922500 ) rhotilde( Ideal of Maximal Order of Equation Order with defining polynomial x^2 + 11 over its ground order Two element generators: [15149121201043150712077534649287181358659069263220165310209, 0] [15149121201043150712077534649287181358659069263220165310209, 0] , 1586753292 + O(3^20) , -1586753291 + O(3^20) ) O(3^16) > n := 3; H_2(3^n*P11,3)/3^(2*n); H_2( (1120444272546109332262524444478427942598682927385924581795430857449017039718746426114668000026662354576872582676133468309982896406972551181954804075515675740599843804995521215881118990099412964395136805718336604016309469022725383311029015313663491185812718659237032288821803001913358587264714060007604157016368354336059435182679857355320980249653798292346811869964464529911907396139365749613411016039868980109516079411100021334812665655328559622721904920199387572269085940174370850656397731638660048134204872504202655311301468289/1523301494400140365268159660880222613216698418436659001085993984631820607393159695433924203054664879396531375391066734637678262140387046548992906992308010256466093167584212557273018029248063864230594080312199836644612037923343633616475757119397863837966577549937178665392932079280375896456241897006999076280406487766101970330400501948550993710298363195598701590774002190184206125202052412953850962785035504993632355270515007640673961540108329025827851638781864828594065993095965460088832359080280929068361838003256580305806102500 : -301093860348130235898520039145053131152632045110484336653595572588585038053140820781812489044829155524032608528784972810492463804537802696849986895286339387851805991560410573650224042174527111761411121183018291497700401887873316890323922490432024469263906639821907138606938985522205469953810819094832224851770019545483410578641873276240143559176807637072682398334684359389618679621836715922975377826228008037968926187065409025049244187224806452645127127972759574684731024543523130224674152364264639054383043605910508509886937465579694549245646070345023702641529994181377951571658141883834987061115506383968268419859904648440418208516815139342191928091824124262435564075518402938541045747213732232133774675675772213535108319640884159432637380850034273491490582758534228466304864200516150992561/1880090704555965625822008665536679733723229265345996504452646364558307577533932788586752127467761960355813720918610097076980490748471486635343712038729178739328539443766850377471953668035335362520866697003366512827601837075510376627445030860879979315317125854195591873189856059968005078243793850297530079609108659550765778821482740897902875009188480471755491902444390002887555427830141380896987474740992273069394167432791559880333576717612696753516090936565158703519032476187152669658951780482944658722244514589788580363892241066204284898208099889803814853155431095756963950597033448396978241434673672521863830296854037997388781801729442839648234783591722956944518133977511190775892910957590602667466926654289489999842762228479564711737741559639024085347542338424595786262472220037406748625000*sqrtD - 1/2 : 1) , 3 ) NumeratorIdeal( 1120444272546109332262524444478427942598682927385924581795430857449017039718746426114668000026662354576872582676133468309982896406972551181954804075515675740599843804995521215881118990099412964395136805718336604016309469022725383311029015313663491185812718659237032288821803001913358587264714060007604157016368354336059435182679857355320980249653798292346811869964464529911907396139365749613411016039868980109516079411100021334812665655328559622721904920199387572269085940174370850656397731638660048134204872504202655311301468289/1523301494400140365268159660880222613216698418436659001085993984631820607393159695433924203054664879396531375391066734637678262140387046548992906992308010256466093167584212557273018029248063864230594080312199836644612037923343633616475757119397863837966577549937178665392932079280375896456241897006999076280406487766101970330400501948550993710298363195598701590774002190184206125202052412953850962785035504993632355270515007640673961540108329025827851638781864828594065993095965460088832359080280929068361838003256580305806102500 ) rhotilde( Ideal of Maximal Order of Equation Order with defining polynomial x^2 + 11 over its ground order Two element generators: [1120444272546109332262524444478427942598682927385924581795430857449017039718746426114668000026662354576872582676133468309982896406972551181954804075515675740599843804995521215881118990099412964395136805718336604016309469022725383311029015313663491185812718659237032288821803001913358587264714060007604157016368354336059435182679857355320980249653798292346811869964464529911907396139365749613411016039868980109516079411100021334812665655328559622721904920199387572269085940174370850656397731638660048134204872504202655311301468289, 0] [1120444272546109332262524444478427942598682927385924581795430857449017039718746426114668000026662354576872582676133468309982896406972551181954804075515675740599843804995521215881118990099412964395136805718336604016309469022725383311029015313663491185812718659237032288821803001913358587264714060007604157016368354336059435182679857355320980249653798292346811869964464529911907396139365749613411016039868980109516079411100021334812665655328559622721904920199387572269085940174370850656397731638660048134204872504202655311301468289, 0] , 1586753292 + O(3^20) , -1586753291 + O(3^20) ) O(3^14) > H_2(P11,3); H_2( (1/4 : 1/8*sqrtD - 1/2 : 1) , 3 ) NumeratorIdeal( 1/4 ) rhotilde( Ideal of Maximal Order of Equation Order with defining polynomial x^2 + 11 over its ground order Two element generators: [1, 0] [1, 0] , 1586753292 + O(3^20) , -1586753291 + O(3^20) ) 0 > H_2(P11,3); H_2( (1/4 : 1/8*sqrtD - 1/2 : 1) , 3 ) NumeratorIdeal( 1/4 ) rhotilde( 1 , 1586753292 + O(3^20) , -1586753291 + O(3^20) ) 0 > H_2(P11,3); H_2( (1/4 : 1/8*sqrtD - 1/2 : 1) , 3 ) NumeratorIdeal( 1/4 ) rhotilde(I of norm 1 , 1586753292 + O(3^20) , -1586753291 + O(3^20) ) 0 > H_2(P11,3); H_2( (1/4 : 1/8*sqrtD - 1/2 : 1) , 3 ) NumeratorIdeal( 1/4 ) rhotilde(I of norm 1 , 1586753292 + O(3^20) , -1586753291 + O(3^20) ) pi_alpha_v = 1 pibar_alpha_v = 1 lambda(pi_alpha_v) = 0 lambda(pibar_alpha_v) = 0 0 > ; > H_2(P11,3); H_2( (1/4 : 1/8*sqrtD - 1/2 : 1) , 3 ) NumeratorIdeal( 1/4 ) rhotilde(I of norm 1 , 1586753292 + O(3^20) , -1586753291 + O(3^20) ) I^v is principal, generated by 1 pi_alpha_v = 1 pibar_alpha_v = 1 lambda(pi_alpha_v) = 0 lambda(pibar_alpha_v) = 0 0 > P11; (1/4 : 1/8*sqrtD - 1/2 : 1) > ; > K; Number Field with defining polynomial z^2 + 4 over the Rational Field > K := K; > Num_and_Denom_Ideals(K!(1/5)); NumeratorIdeal( 1/5 ) Ideal of Maximal Order of K Two element generators: [1, 0] [1, 0] Ideal of Maximal Order of K Two element generators: [5, 0] [5, 0] > Num_and_Denom_Ideals(K!(5/1)); NumeratorIdeal( 5 ) Ideal of Maximal Order of K Two element generators: [5, 0] [5, 0] Ideal of Maximal Order of K Two element generators: [1, 0] [1, 0] > ab := Num_and_Denom_Ideals(K!(z/1)); NumeratorIdeal( z ) Num_and_Denom_Ideals( x: z ) In file "/home/was/papers/anti-cyclotomic_height_pairing/anti-cyclotomic.m", line 108, column 8: >> a,b := Explode(Generators(ab)); ^ Runtime error in :=: Expected to assign 2 value(s) but only computed 1 value(s) > Num_and_Denom_Ideals(K!(z/1)); NumeratorIdeal( z ) > ab; Principal Ideal of Maximal Order of K Generator: [0, 2] > Basis(ab); Basis(ab); [ [0, 2], [-2, 0] ] > a := $1[1]; > a; [0, 2] > Parent(a); Maximal Order of K > ; > Num_and_Denom_Ideals(K!(z/1)); In file "/home/was/papers/anti-cyclotomic_height_pairing/anti-cyclotomic.m", line 144, column 26: >> return 1/2 * rhotilde(NumeratorIdeal(x), pi, pibar); ^ Runtime error: Undefined reference 'NumeratorIdeal' in package "/home/was/papers/anti-cyclotomic_height_pairing/anti-cyclotomic.m" > ; > Num_and_Denom_Ideals(K!(z/1)); NumeratorIdeal( z ) Ideal of Maximal Order of K Two element generators: [-2, 0] [0, 2] Ideal of Maximal Order of K Two element generators: [1, 0] [0, 1] > ; > H_2(P11,3); H_2( (1/4 : 1/8*sqrtD - 1/2 : 1) , 3 ) NumeratorIdeal( 1/4 ) rhotilde(I of norm 16 , 1586753292 + O(3^20) , -1586753291 + O(3^20) ) I^v is principal, generated by 256 pi_alpha_v = 256 pibar_alpha_v = 256 lambda(pi_alpha_v) = -220851539*3 + O(3^20) lambda(pibar_alpha_v) = -220851539*3 + O(3^20) O(3^20) > n := 1; H_2(3^n*P11,3)/3^(2*n); H_2( (1859969/2244004 : 394204143/3361517992*sqrtD - 1/2 : 1) , 3 ) NumeratorIdeal( 1859969/2244004 ) rhotilde(I of norm 17420421757151974462767376 , 1586753292 + O(3^20) , -1586753291 + O(3^20) ) I^v is principal, generated by 303471094197053885524425462837732696554092289925376 pi_alpha_v = 303471094197053885524425462837732696554092289925376 pibar_alpha_v = 303471094197053885524425462837732696554092289925376 lambda(pi_alpha_v) = 565189054*3 + O(3^20) lambda(pibar_alpha_v) = 565189054*3 + O(3^20) O(3^18) > n := 2; H_2(3^n*P11,3)/3^(2*n); H_2( (15149121201043150712077534649287181358659069263220165310209/36997655900725382379967627461892952671282761517597531922500 : 1237472676951754508036409981037158464683197464035772760418709017813761949272201574578081/7116415768783413451489917678950449784975851252781995109773021001240894476312667294125000*sqrtD - 1/2 : 1) , 3 ) NumeratorIdeal( 15149121201043150712077534649287181358659069263220165310209/36997655900725382379967627461892952671282761517597531922500 ) rhotilde(I of norm 314140042500280571844098481446267818261098901802110443483136046652179413978179426931761393318260077851771748899005563001998374123816127287408926232453060783466792888363910622380708092620317399803437838811655616008750219513474006250000 , 1586753292 + O(3^20) , -1586753291 + O(3^20) ) I^v is principal, generated by 98683966302078083952058879391991768612425738213606608396837705735052635900675055288287437175353714268260131884271122385011112899638600537429683902600064333147575674629235335531836248991510015026838592583928440266211841100404576210519633987512378967162352883381958509316997637333333136277194824385553290437767702738724417592992665271118784559019656609039760852972184568694159069351716101024014788575795552663401916625208020161471027197181274645292594425039062500000000 pi_alpha_v = 98683966302078083952058879391991768612425738213606608396837705735052635900675055288287437175353714268260131884271122385011112899638600537429683902600064333147575674629235335531836248991510015026838592583928440266211841100404576210519633987512378967162352883381958509316997637333333136277194824385553290437767702738724417592992665271118784559019656609039760852972184568694159069351716101024014788575795552663401916625208020161471027197181274645292594425039062500000000 pibar_alpha_v = 98683966302078083952058879391991768612425738213606608396837705735052635900675055288287437175353714268260131884271122385011112899638600537429683902600064333147575674629235335531836248991510015026838592583928440266211841100404576210519633987512378967162352883381958509316997637333333136277194824385553290437767702738724417592992665271118784559019656609039760852972184568694159069351716101024014788575795552663401916625208020161471027197181274645292594425039062500000000 lambda(pi_alpha_v) = 170157796*3^2 + O(3^20) lambda(pibar_alpha_v) = 170157796*3^2 + O(3^20) O(3^16) > P20 := PointFromTwist(E,-20); > n := 0; H_2(3^n*P20,3)/3^(2*n); H_2( (-61/20 : 319/400*sqrtD - 1/2 : 1) , 3 ) NumeratorIdeal( -61/20 ) rhotilde(I of norm 1488400 , 1219441412 + O(3^20) , -1219441412 + O(3^20) ) I^v is principal, generated by 4907707212730393600000000 pi_alpha_v = 4907707212730393600000000 pibar_alpha_v = 4907707212730393600000000 lambda(pi_alpha_v) = -295357981*3 + O(3^20) lambda(pibar_alpha_v) = -295357981*3 + O(3^20) O(3^20) > P20; (-61/20 : 319/400*sqrtD - 1/2 : 1) > FindAcceptableK(E,3,-100,-1); Finding acceptable K for E of conductor 389 Trying D = -95 [Interrupt twice in half a second; exiting] Total time: 12.160 seconds You have new mail in /var/spool/mail/was [was@modular anti-cyclotomic_height_pairing]$ exit exit Process magma finished [was@modular anti-cyclotomic_height_pairing]$ [was@modular anti-cyclotomic_height_pairing]$ Magma V2.7-1 Tue Apr 24 2001 04:21:31 on modular [Seed = 2057617168] Type ? for help. Type -D to quit. Loading startup file "/home/was/modsym/init-magma.m" C IndexGamma0 R ellap idxG0 CS MS S factormod modcharpoly DC ND Tn factorpadic padiccharpoly ES NS Z fcp qexp F Q charpoly fn x > Attach("anti-cyclotomic.m"); > E := EC("389A"); > FindAcceptableK(E,3,-50,-30); [ -35 ] > > > > > SetVerbose("ac_height",2); > P35 := PointFromTwist(E,-35); > P35; (-421/140 : 5711/9800*sqrtD - 1/2 : 1) > E; Elliptic Curve defined by y^2 + y = x^3 + x^2 - 2*x over Rational Field > E := WeierstrassModel(E); > P35 := PointFromTwist(E,-35); > P35; (-3369/35 : 154197/1225*sqrtD : 1) > ; > H_2(P11,3); >> H_2(P11,3); ^ User error: Identifier 'P11' has not been declared or assigned > P11 := PointFromTwist(E,-11); > H_2(P11,3); H_2( (21 : 27*sqrtD : 1) , 3 ) NumeratorIdeal( 21 ) rhotilde(I of norm 441 , 1586753292 + O(3^20) , -1586753291 + O(3^20) ) I^v is principal, generated by 194481 pi_alpha_v = 2401*3^4 pibar_alpha_v = 2401*3^4 H_2( P: (21 : 27*sqrtD : 1), p: 3 ) rhotilde( I: I, pi: 1586753292 + O(3^20), pibar: -1586753291 + O(3^20) ) In file "/home/was/papers/anti-cyclotomic_height_pairing/anti-cyclotomic.m", line 86, column 50: >> vprint ac_height : "lambda(pi_alpha_v) =", Log(pi_alpha_v); ^ Runtime error in 'Log': Series expansion will not converge > E; Elliptic Curve defined by y^2 = x^3 - 3024*x + 46224 over Rational Field > P11; (21 : 27*sqrtD : 1) > ; > H_2(P11,3); H_2( (21 : 27*sqrtD : 1) , 3 ) NumeratorIdeal( 21 ) rhotilde(I of norm 441 , 1586753292 + O(3^20) , -1586753291 + O(3^20) ) I^v is principal, generated by 194481 pi_alpha_v = 5764801*3^8 pibar_alpha_v = 5764801*3^8 H_2( P: (21 : 27*sqrtD : 1), p: 3 ) rhotilde( I: I, pi: 1586753292 + O(3^20), pibar: -1586753291 + O(3^20) ) In file "/home/was/papers/anti-cyclotomic_height_pairing/anti-cyclotomic.m", line 87, column 50: >> vprint ac_height : "lambda(pi_alpha_v) =", Log(pi_alpha_v); ^ Runtime error in 'Log': Series expansion will not converge > P11; (21 : 27*sqrtD : 1) > 2*P11; (-5817/44 : 398547/968*sqrtD : 1) > H_2(2*P11,3); H_2( (-5817/44 : 398547/968*sqrtD : 1) , 3 ) NumeratorIdeal( -5817/44 ) rhotilde(I of norm 33837489 , 1586753292 + O(3^20) , -1586753291 + O(3^20) ) I^v is principal, generated by 1144975661825121 pi_alpha_v = 14135501997841*3^4 pibar_alpha_v = 14135501997841*3^4 H_2( P: (-5817/44 : 398547/968*sqrtD : 1), p: 3 ) rhotilde( I: I, pi: 1586753292 + O(3^20), pibar: -1586753291 + O(3^20) ) In file "/home/was/papers/anti-cyclotomic_height_pairing/anti-cyclotomic.m", line 86, column 50: >> vprint ac_height : "lambda(pi_alpha_v) =", Log(pi_alpha_v); ^ Runtime error in 'Log': Series expansion will not converge > H_2(3*P11,3); H_2( (23471733/561001 : 10643511861/420189749*sqrtD : 1) , 3 ) NumeratorIdeal( 23471733/561001 ) rhotilde(I of norm 550922250023289 , 1586753292 + O(3^20) , -1586753291 + O(3^20) ) I^v is principal, generated by 303515325570723356561042377521 pi_alpha_v = 3747102784823745142728918241*3^4 pibar_alpha_v = 3747102784823745142728918241*3^4 H_2( P: (23471733/561001 : 10643511861/420189749*sqrtD : 1), p: 3 ) rhotilde( I: I, pi: 1586753292 + O(3^20), pibar: -1586753291 + O(3^20) ) In file "/home/was/papers/anti-cyclotomic_height_pairing/anti-cyclotomic.m", line 86, column 50: >> vprint ac_height : "lambda(pi_alpha_v) =", Log(pi_alpha_v); ^ Runtime error in 'Log': Series expansion will not converge > P20 := PointFromTwist(E,-20); > H_2(P20,3); H_2( (-489/5 : 8613/50*sqrtD : 1) , 3 ) NumeratorIdeal( -489/5 ) rhotilde(I of norm 239121 , 1219441412 + O(3^20) , -1219441412 + O(3^20) ) I^v is principal, generated by 3269421189341192674881 pi_alpha_v = 498311414318121121*3^8 pibar_alpha_v = 498311414318121121*3^8 H_2( P: (-489/5 : 8613/50*sqrtD : 1), p: 3 ) rhotilde( I: I, pi: 1219441412 + O(3^20), pibar: -1219441412 + O(3^20) ) In file "/home/was/papers/anti-cyclotomic_height_pairing/anti-cyclotomic.m", line 86, column 50: >> vprint ac_height : "lambda(pi_alpha_v) =", Log(pi_alpha_v); ^ Runtime error in 'Log': Series expansion will not converge > P20; (-489/5 : 8613/50*sqrtD : 1) > P11; (21 : 27*sqrtD : 1) > EE := Parent(P11); > EE; Elliptic Curve defined by y^2 = x^3 - 3024*x + 46224 over Number Field with defining polynomial z^2 + 11 over the Rational Field > G,f := MordellWeilGroup(E); > P0 := EE!Eltseq(f(G.1)); > P1 := EE!Eltseq(f(G.2)); > P0; (12 : 108 : 1) > P1; (-60 : 108 : 1) > P0; (12 : 108 : 1) > P1; (-60 : 108 : 1) > P0+P1; (48 : -108 : 1) > P11+P0; (-72*sqrtD + 12 : -864*sqrtD - 2484 : 1) > H_2(P11+P0,3); H_2( (-72*sqrtD + 12 : -864*sqrtD - 2484 : 1) , 3 ) NumeratorIdeal( -72*sqrtD + 12 ) rhotilde(I of norm 57168 , 1586753292 + O(3^20) , -1586753291 + O(3^20) ) I^v is principal, generated by -196577280*sqrtD - 3202488576 pi_alpha_v = 4098973361*3^4 + O(3^25) pibar_alpha_v = -4180474033*3^4 + O(3^25) H_2( P: (-72*sqrtD + 12 : -864*sqrtD - 2484 : 1), p: 3 ) rhotilde( I: I, pi: 1586753292 + O(3^20), pibar: -1586753291 + O(3^20) ) In file "/home/was/papers/anti-cyclotomic_height_pairing/anti-cyclotomic.m", line 86, column 50: >> vprint ac_height : "lambda(pi_alpha_v) =", Log(pi_alpha_v); ^ Runtime error in 'Log': Series expansion will not converge > D2 := FindAcceptableK(E,2,-50,-1); Finding acceptable K for E of conductor 389 Trying D = -47 Trying D = -43 Trying D = -40 Trying D = -39 Trying D = -35 Trying D = -31 Trying D = -24 Trying D = -23 Trying D = -20 Trying D = -19 Trying D = -15 Trying D = -11 Trying D = -8 Trying D = -7 -7 is acceptable. Trying D = -4 Trying D = -3 > D2; [ -7 ] > p := 2; > D := -7; > P7 := PointFromTwist(E,-7); > P7; (-492/7 : 5508/49*sqrtD : 1) > H_2(P7,2); H_2( (-492/7 : 5508/49*sqrtD : 1) , 2 ) NumeratorIdeal( -492/7 ) rhotilde(I of norm 242064 , 237478 + O(2^20) , -237477 + O(2^20) ) I^v is principal, generated by 242064 pi_alpha_v = 15129*2^4 pibar_alpha_v = 15129*2^4 H_2( P: (-492/7 : 5508/49*sqrtD : 1), p: 2 ) rhotilde( I: I, pi: 237478 + O(2^20), pibar: -237477 + O(2^20) ) In file "/home/was/papers/anti-cyclotomic_height_pairing/anti-cyclotomic.m", line 86, column 50: >> vprint ac_height : "lambda(pi_alpha_v) =", Log(pi_alpha_v); ^ Runtime error in 'Log': Series expansion will not converge > D2 := FindAcceptableK(E,2,-50,-1); Finding acceptable K for E of conductor 389 Trying D = -47 chi_D(-1) = -1 Trying D = -43 Trying D = -40 Trying D = -39 chi_D(-1) = -1 Trying D = -35 Trying D = -31 chi_D(-1) = -1 Trying D = -24 Trying D = -23 chi_D(-1) = -1 Trying D = -20 Trying D = -19 Trying D = -15 chi_D(-1) = -1 Trying D = -11 Trying D = -8 Trying D = -7 chi_D(-1) = -1 -7 is acceptable. Trying D = -4 Trying D = -3 > D2 := FindAcceptableK(E,2,-50,-1); Finding acceptable K for E of conductor 389 Trying D = -47 chi_D(-1) = -1 r = 2 Trying D = -43 Trying D = -40 Trying D = -39 chi_D(-1) = -1 r = 0 Trying D = -35 Trying D = -31 chi_D(-1) = -1 r = 0 Trying D = -24 Trying D = -23 chi_D(-1) = -1 r = 0 Trying D = -20 Trying D = -19 Trying D = -15 chi_D(-1) = -1 r = 0 Trying D = -11 Trying D = -8 Trying D = -7 chi_D(-1) = -1 r = 1 -7 is acceptable. Trying D = -4 Trying D = -3 > D2 := FindAcceptableK(E,2,-50,-1); Finding acceptable K for E of conductor 389 Trying D = -47 chi_D(-1) = 1 r = 2 Trying D = -43 Trying D = -40 Trying D = -39 chi_D(-1) = 1 r = 0 Trying D = -35 Trying D = -31 chi_D(-1) = 1 r = 0 Trying D = -24 Trying D = -23 chi_D(-1) = 1 r = 0 Trying D = -20 Trying D = -19 Trying D = -15 chi_D(-1) = 1 r = 0 Trying D = -11 Trying D = -8 Trying D = -7 chi_D(-1) = -1 r = 1 -7 is acceptable. Trying D = -4 Trying D = -3 > D2 := FindAcceptableK(E,2,-50,-1); In file "/home/was/papers/anti-cyclotomic_height_pairing/anti-cyclotomic.m", line 65, column 11: >> end if; ^ User error: bad syntax >> D2 := FindAcceptableK(E,2,-50,-1); ^ User error: Identifier 'FindAcceptableK' has not been declared or assigned > D2 := FindAcceptableK(E,2,-50,-1); Finding acceptable K for E of conductor 389 Trying D = -47 Trying D = -43 Trying D = -40 Trying D = -39 Trying D = -35 Trying D = -31 Trying D = -24 Trying D = -23 Trying D = -20 Trying D = -19 Trying D = -15 Trying D = -11 Trying D = -8 Trying D = -7 r = 1 -7 is acceptable. Trying D = -4 Trying D = -3 > D2 := FindAcceptableK(E,2,-150,-1); Finding acceptable K for E of conductor 389 Trying D = -148 Trying D = -143 [Interrupt twice in half a second; exiting] Total time: 24.119 seconds [was@modular anti-cyclotomic_height_pairing]$ exit exit Process magma finished