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We use an invariant-theoretic method to compute certain twists of the modular curves X(n) for n = 7,9,11. Searching for rational points on these twists enables us to find non-trivial pairs of n-congruent elliptic curves over Q, i.e. pairs of non-isogenous elliptic curves over Q whose n-torsion subgroups are isomorphic as Galois modules. We also show by giving explicit non-trivial examples over Q(T) that there are infinitely many examples over Q in the cases n = 9 and n = 11.

On families of n-congruent elliptic curves   (57 pages)          

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We have used our formulae to produce tables of pairs of n-congruent elliptic curves for n = 9 and n = 11. A Magma file checking all the formulae in the paper is available here, and the quintic forms in Section 6.5 are available here.