Tate module of an elliptic curve (reviewed)

Let $p \in \mathbb{N}$ be a prime and $E$ an elliptic curve defined over a field $K$. The $p$-adic Tate module of $E$ is the inverse limit \[ T_p(E) = \lim_{\xleftarrow[n \in \mathbb{N}]{}} E[p^n]. \] It carries an action of the absolute Galois group of $K$, and is a free $\Z_p$-module of rank $2$ if $K$ has characteristic not equal to $p$.

Here for $m\in\mathbb{N}$, $E[m]$ denotes the $m$-torsion subgroup of $E$, which is the kernel of the multiplication-by-$m$ endomorphism of $E$.