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For a given elliptic curve $E_0$ defined over a number field $k$, weconstruct two families of elliptic curves whose mod 3 representations areisomorphic to that of $E_0$. The isomorphisms in the first family aresymplectic, and those in the second family are anti-symplectic. Ourconstruction is based on the notion of Hessian and Cayleyan curves in classicalgeometry.

Constructing families of elliptic curves with prescribed mod 3 representation via Hessian and Cayleyan curves