TITLE: (phi, Gamma) module of ordinary elliptic curve
QUESTION [7 upvotes]: Suppose $E$ is an elliptic curve over $\mathbb{Q}_p$ with good ordinary reduction.  Can someone please tell me how to compute the associated $(\phi,\Gamma)$-module of the Tate module of $E$, or give me a reference to where it is computed?

REPLY [4 votes]: As Laurent has already pointed out, the representation is reducible and hence so is the phi-Gamma module, and writing down the two composition factors is easy; describing the extension class is harder. But it can be done: the relevant ext group is one-dimensional, so up to isomophism there is only one non-split extension, and you just need to write down any old non-split extension and you're done. Sarah Zerbes and I describe a way of doing this in our paper "Wach modules and critical slope p-adic L-functions"; see section 4 of http://arxiv.org/abs/1012.0175. There is another approach in one of Pierre Colmez's papers ("La serie principale unitaire" if I remember rightly).