## On *12*-congruences of elliptic curves

We construct infinite families of pairs of (geometrically non-isogenous) elliptic curves defined over ℚ with *12*-torsion subgroups that are isomorphic as Galois modules. This extends previous work of Chen and Fisher where it is assumed that the underlying isomorphism of *12*-torsion subgroups respects the Weil pairing. Our approach is to compute explicit birational models for the modular diagonal quotient surfaces which parametrise such pairs of elliptic curves.

A key ingredient in the proof is to construct simple (algebraic) conditions for the *2*, *3*, or *4*-torsion subgroups of a pair of elliptic curves to be isomorphic as Galois modules. These conditions are given in terms of the *j*-invariants of the pair of elliptic curves.

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