Families of Elliptic Curves with Prescribed \(n\)-Torsion

In [Rubin and Silverberg, 1995], Rubin and Silverberg explicitly construct families of elliptic curves over \({\mathbb{Q}}\) which have the same Galois representation on the \(n\)-torsion subgroups as a given elliptic curve.

RubinSilverbergPolynomials(n, J : parameters): RngIntElt, RngElt -> RngElt, RngElt

Parameter: RngElt                    Default: 

Suppose that \(n = 2\), 3, 4, or 5 and let \(E : y^2 = x^3 + ax + b\) be an elliptic curve over the rationals with \(j\)-invariant \(1728 J\). This function returns polynomials \(\alpha(t)\) and \(\beta(t)\) that determine a family of elliptic curves with fixed \(n\)-torsion, in the following sense: Every nonsingular member \(F_t\) of the family \(F : y^2 = x^3 + a\alpha(t) x + b\beta(t)\) has \(F_t[n]\) isomorphic to \(E[n]\) as \({\mathbb{Z}}[G]\)-modules, where \(G\) is the absolute Galois group of \({\mathbb{Q}}\), and furthermore the isomorphisms between \(F_t[n]\) and \(E[n]\) preserve the Weil pairing. When \(n\) is 3, 4, or 5, all such “\(n\)-congruent” curves belong to the same family.