  For a given elliptic curve $E_0$ defined over a number field $k$, we
construct two families of elliptic curves whose mod 3 representations are
isomorphic to that of $E_0$. The isomorphisms in the first family are
symplectic, and those in the second family are anti-symplectic. Our
construction is based on the notion of Hessian and Cayleyan curves in classical
geometry.
