  Given an elliptic curve $E$ over a local field $K$ with residue
characteristic $3$, we investigate the action of the absolute Galois group of
$K$ in the case of potentially good reduction. In particular the only not
completely known case is that of the $\ell$-adic Galois representation attached
to an elliptic curve such that the image of inertia is non-cyclic, and
isomorphic to $C_3 \rtimes C_4$. In this work we describe such a representation
explicitly.

Number Theory