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On families of n-congruent elliptic curves

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)     dvi   ps   ps.gz   pdf

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. This paper has been superseded by versions covering the cases n = 7, 11 and n = 9.