On families of 7- and 11-congruent elliptic curves

We use an invariant-theoretic method to compute certain twists of the modular curves X(n) for n = 7, 11. Searching for rational points on these twists enables us to find non-trivial pairs of n-congruent elliptic curves over Q, that is, pairs of non-isogenous elliptic curves over Q whose n-torsion subgroups are isomorphic as Galois modules. We also find a non-trivial pair of 11-congruent elliptic curves over Q(T ), and hence give an explicit infinite family of non-trivial pairs of 11-congruent elliptic curves over Q. Supplementary materials are available with this article.