On families of 9-congruent elliptic curves
Volume 171 / 2015
Acta Arithmetica 171 (2015), 371-387
MSC: 11G05, 11F80.
DOI: 10.4064/aa171-4-5
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 ${\mathbb Q}$, i.e. pairs of non-isogenous elliptic curves over ${\mathbb Q}$ whose $9$-torsion subgroups are isomorphic as Galois modules.
