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We compute twists of the modular curve X(13) that parametrise the elliptic curves 13-congruent to a given elliptic curve. Searching for rational points on these twists enables us to find non-trivial pairs of 13-congruent elliptic curves over Q, i.e. pairs of non-isogenous elliptic curves over Q whose 13-torsion subgroups are isomorphic as Galois modules. We also find equations for the surfaces parametrising pairs of 13-congruent elliptic curves. There are two such surfaces, corresponding to 13-congruences that do, or do not, respect the Weil pairing. We write each as a double cover of the projective plane ramified over a highly singular model for Baran's modular curve of level 13. By finding suitable rational curves on these surfaces, we show that there are infinitely many non-trivial pairs of 13-congruent elliptic curves over Q.

On families of 13-congruent elliptic curves   (45 pages)          

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The Magma files accompanying this paper are

• congr13-formulae.m A file listing some of the formulae in the paper.
• congr13-check-curves.m Checks the calculations relating to X_E(13,1) and X_E(13,2), and includes Tables 2 and 3.
• congr13-check-surfaces.m Checks the calculations relating to Z(13,1) and Z(13,2), and includes Tables 1, 4, 5 and 6.

Our older table of 13-congruent elliptic curves (available here) is now superseded by Table 6.