was@form:~/talks/magma_ihp/bsd$ was@form:~/talks/magma_ihp/bsd$ Magma V2.11-8 Wed Oct 13 2004 13:18:57 on form [Seed = 3656776467] Type ? for help. Type -D to quit. Loading startup file "/home/was/magma/local/emacs.m" Loading "/home/was/magma/local/init.m" > > > > > > J := J0(37); > J; Modular abelian variety JZero(37) of dimension 2 and level 37 over Q > Decomposition(J); [ Modular abelian variety 37A of dimension 1, level 37 and conductor 37 over Q, Modular abelian variety 37B of dimension 1, level 37 and conductor 37 over Q ] > S := CuspForms(37); > N := Newforms(S); > N; [* [* q - 2*q^2 - 3*q^3 + 2*q^4 - 2*q^5 + 6*q^6 - q^7 + O(q^8) *], [* q + q^3 - 2*q^4 - q^7 + O(q^8) *] *] > f := N[1][1]; > Af := ModularAbelianVariety(f); > Af; Modular abelian variety Af of dimension 1 and level 37 over Q > J := J0(389); > Decomposition(J); [ Modular abelian variety 389A of dimension 1, level 389 and conductor 389 over Q, Modular abelian variety 389B of dimension 2, level 389 and conductor 389^2 over Q, Modular abelian variety 389C of dimension 3, level 389 and conductor 389^3 over Q, Modular abelian variety 389D of dimension 6, level 389 and conductor 389^6 over Q, Modular abelian variety 389E of dimension 20, level 389 and conductor 389^20 over Q ] > D := $1; > EllipticCurve(D[1]); Elliptic Curve defined by y^2 + y = x^3 + x^2 - 2*x over Rational Field > EllipticCurve(D[2]); >> EllipticCurve(D[2]); ^ Runtime error in 'EllipticCurve': Argument 1 must have dimension 1. > D[2]; Modular abelian variety 389B of dimension 2, level 389 and conductor 389^2 over Q > E := D[1]; > E; Modular abelian variety 389A of dimension 1, level 389 and conductor 389 over Q > L := LSeries(E); > L; L(389A,s): L-series of Modular abelian variety 389A of dimension 1, level 389 and conductor 389 over Q > alpha, r := LeadingCoefficient(L,1, 300); > alpha; 0.75931650029224679065762600319 > r; 2 > EE := EllipticCurve(E); > r, alpha := LeadingCoefficient(L,1, 300); > r, alpha := AnalyticRank(EE); > r; 2 > alpha; 0.7593000000 > J := J0(125); > D := Decomposition(J); > D; [ Modular abelian variety 125A of dimension 2, level 5^3 and conductor 5^6 over Q, Modular abelian variety 125B of dimension 2, level 5^3 and conductor 5^6 over Q, Modular abelian variety 125C of dimension 4, level 5^3 and conductor 5^12 over Q ] > TorsionMultiple(D[3], 2); 0 > TorsionMultiple(D[3], 3); 155 > TorsionMultiple(D[3], 13); 5 > TorsionMultiple(D[3], 37); 5 > [ : A in D]; [ <2, 1>, <2, 5>, <4, 5> ] > RationalCuspidalSubgroup(D[1]); Finitely generated subgroup of abelian variety with invariants [] > RationalCuspidalSubgroup(D[2]); Finitely generated subgroup of abelian variety with invariants [ 5 ] > RationalCuspidalSubgroup(D[3]); Finitely generated subgroup of abelian variety with invariants [ 5 ] > RationalCuspidalSubgroup(J); Finitely generated subgroup of abelian variety with invariants [ 25 ] > D[2] meet D[3]; Finitely generated subgroup of abelian variety with invariants [ 5, 5, 5, 5 ] Modular abelian variety ZERO of dimension 0 and level 5^3 over Q Homomorphism from modular abelian variety of dimension 0 to 125B x 125C given on integral homology by: Matrix with 0 rows and 12 columns Homomorphism from 125B x 125C to JZero(125) (not printing 12x16 matrix) > A := D[2]/RationalCuspidalSubgroup(D[2]); > A; Modular abelian variety of dimension 2 and level 5^3 over Q > IsIsomorphic(A,D[2]); false >