On inertial types of elliptic curves

Abstract

We classify the inertial Weil-Deligne types arising from elliptic curves over all finite extensions $F/\mathbb Q_p$. Based on this classification, we give a fully explicit description of the types and implement an algorithm that computes all inertial types of elliptic curves defined over a given $F$. As an application, we determine all inertial types arising from elliptic curves over any extension $F/\mathbb Q_p$ of degree at most 3.