
We compute twists of the modular curve X(13) that parametrise the elliptic curves 13congruent to a given elliptic curve. Searching for rational points on these twists enables us to find nontrivial pairs of 13congruent elliptic curves over Q, i.e. pairs of nonisogenous elliptic curves over Q whose 13torsion subgroups are isomorphic as Galois modules. We also find equations for the surfaces parametrising pairs of 13congruent elliptic curves. There are two such surfaces, corresponding to 13congruences 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 nontrivial pairs of 13congruent elliptic curves over Q.