CoCalc Public Filespapers / done / agashe-ribet-stein-congruence / remark-3.7.sagews
Author: William A. Stein
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Compute Environment: Ubuntu 18.04 (Deprecated)

Remark 3.7 Let $f \in S_2(\Gamma,\CC)$ be a newform. When $A_f$ is an elliptic curve, Theorem~3.6 implies that the modular degree divides the congruence number (since for an elliptic curve, the modular degree and modular exponent are the same), and that $n_{A_f} \mid r^2_{A_f}$ (since for an elliptic curve, the modular number is the square of the modular exponent). In general, for a higher dimensional newform quotient, the divisibility $n_{A_f}\mid r^2_{A_f}$ need not hold. For example, there is a newform of degree $24$ in $S_2(\Gamma_0(431))$ such that $n_{A_f} = (2^{11}\cdot 6947)^2 \,\,\nmid\,\, r^2_{A_f} = (2^{10}\cdot 6947)^2.$
Note that $431$ is prime and mod~$2$ multiplicity one fails for $J_0(431)$.

M = ModularSymbols(431,sign=0); M

Modular Symbols space of dimension 73 for Gamma_0(431) of weight 2 with sign 0 over Rational Field
D = M.decomposition(); D

[ Modular Symbols subspace of dimension 1 of Modular Symbols space of dimension 73 for Gamma_0(431) of weight 2 with sign 0 over Rational Field, Modular Symbols subspace of dimension 2 of Modular Symbols space of dimension 73 for Gamma_0(431) of weight 2 with sign 0 over Rational Field, Modular Symbols subspace of dimension 2 of Modular Symbols space of dimension 73 for Gamma_0(431) of weight 2 with sign 0 over Rational Field, Modular Symbols subspace of dimension 6 of Modular Symbols space of dimension 73 for Gamma_0(431) of weight 2 with sign 0 over Rational Field, Modular Symbols subspace of dimension 6 of Modular Symbols space of dimension 73 for Gamma_0(431) of weight 2 with sign 0 over Rational Field, Modular Symbols subspace of dimension 8 of Modular Symbols space of dimension 73 for Gamma_0(431) of weight 2 with sign 0 over Rational Field, Modular Symbols subspace of dimension 48 of Modular Symbols space of dimension 73 for Gamma_0(431) of weight 2 with sign 0 over Rational Field ]
A = D[-1]; A

Modular Symbols subspace of dimension 48 of Modular Symbols space of dimension 73 for Gamma_0(431) of weight 2 with sign 0 over Rational Field
%time factor(A.congruence_number(A.complement().cuspidal_submodule()))

2^10 * 6947 CPU time: 0.81 s, Wall time: 0.81 s
J = J0(431)

D = J.decomposition(); D

[ Simple abelian subvariety 431a(1,431) of dimension 1 of J0(431), Simple abelian subvariety 431b(1,431) of dimension 1 of J0(431), Simple abelian subvariety 431c(1,431) of dimension 3 of J0(431), Simple abelian subvariety 431d(1,431) of dimension 3 of J0(431), Simple abelian subvariety 431e(1,431) of dimension 4 of J0(431), Simple abelian subvariety 431f(1,431) of dimension 24 of J0(431) ]
A = D[-1]; A

Simple abelian subvariety 431f(1,431) of dimension 24 of J0(431)
factor(A.modular_degree())

2^11 * 6947