On families of 9-congruent elliptic curves

Abstract

We compute equations for the families of elliptic curves 9-congruent to a given elliptic curve. We use these to find infinitely many non-trivial pairs of 9-congruent elliptic curves over Q, i.e. pairs of non-isogenous elliptic curves over Q whose 9-torsion subgroups are isomorphic as Galois modules.Comment: 18 pages. This article supersedes parts of arXiv:1105.170