Wild Galois representations: Elliptic curves over a 2 -adic field with non-abelian inertia action

In this paper, we present a description of the a "-adic Galois representation attached to an elliptic curve defined over a 2-adic field K, in the case where the image of inertia is non-abelian. There are two possibilities for the image of inertia, namely Q8 and SL2(3), and in each case, we need to distinguish whether the inertia degree of K over a 2 is even or odd. The results presented here are being implemented in an algorithm to compute explicitly the Galois representation in these four cases.