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. The hardest case is when the $\ell$-adic Galois representation attached to $E$ has non-cyclic inertia image, isomorphic to $C_3 \rtimes C_4$. In this work we describe such a representation explicitly.
- Elliptic curves over local fields
- Galois representations
- Non-Abelian inertia action