Tate module of an elliptic curve is Hodge-Tate

Question

Let $E$ be an elliptic curve over $\mathbb{Q}_p$, and let $T_p(E)$ be its $p$-adic Tate module. Is there a simple way of seeing that $V_p(E):= T_p(E)\otimes_{\mathbb{Z}_p}\mathbb{Q}_p$ is a Hodge-Tate representation,i.e., $B_{HT}$ admissible? Or maybe equivalently, is there a direct way of computing the Sen's operator for $V_p(E)\otimes \mathbb{C}_p$ when considered as a $\mathbb{C}_p$ representation of $G_{\mathbb{Q}_p}$. To clarify what I mean by simple, I wanted to avoid using Tate's general result on p-divisible groups or the complete machinary of hodge decomposition for etale cohomology of abelian varieties.

Answer 1

You could read this article by Berger , especially pages 17-18-19.