According to Grothendieck monodromy theorem, l-adic galois representations of a local field corresponds to Weil-Deligne representations. However, given a galois representation, it is usually difficult to find the Weil-Deligne representations.
My questions are:
(1) Is it possible to describe explicitly the Weil-Deligne representation associated to the Tate module of an elliptic curve over a local field? How about Tate curve (corresponding to a prime element) for example?
(2) In the situation as in (1), is it possible to calculate the L function and epsilon factor associated to the Weil-Deligne representation?
(3) If these are possible, how can one do for Tate curves?
Please give me any advice!