Let $p$ be a prime number, and $E/\mathbf{Q}_p$ and elliptic curve with good reduction $\bar E_p$. Can the Galois representation of $\mathbf{Q}_p$ given by the $p$-adic Tate module of $E$ be recovered from the Dieudonn\'e module associated to the $p$-divisible group $\bar E_p[p^\infty]$ of the reduction $\bar E_p$?
In case of ordinary reduction, can the [EDIT: possible] splitting of the $p$-torsion $E[p]$, as Galois module, be understood in terms of the Dieudonn\'e module of the $p$-torsion $\bar E_p[p]$?
Thanks!