Families of Elliptic Curves with Prescribed n-Torsion

In [RS95], Rubin and Silverberg explicitly construct families of elliptic curve over 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 : y² = x³ + ax + b be an elliptic curve over the rationals with j-invariant 1728 J. This function returns polynomials α(t) and β(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 : y² = x³ + aα(t) x + bβ(t) has F_t[n] isomorphic to E[n] as Z[G]-modules, where G is the absolute Galois group of 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.