  The goal of this article is to give an explicit classification of the
possible $p$-adic Galois representations that are attached to elliptic curves
$E$ with CM defined over $\mathbb{Q}(j(E))$. More precisely, let $K$ be an
imaginary quadratic field, and let $\mathcal{O}_{K,f}$ be an order in $K$ of
conductor $f\geq 1$. Let $E$ be an elliptic curve with CM by
$\mathcal{O}_{K,f}$, such that $E$ is defined by a model over
$\mathbb{Q}(j(E))$. Let $p\geq 2$ be a prime, let $G_{\mathbb{Q}(j(E))}$ be the
absolute Galois group of $\mathbb{Q}(j(E))$, and let $\rho_{E,p^\infty}\colon
G_{\mathbb{Q}(j(E))}\to \operatorname{GL}(2,\mathbb{Z}_p)$ be the Galois
representation associated to the Galois action on the Tate module $T_p(E)$. The
goal is then to describe, explicitly, the groups of
$\operatorname{GL}(2,\mathbb{Z}_p)$ that can occur as images of
$\rho_{E,p^\infty}$, up to conjugation, for an arbitrary order
$\mathcal{O}_{K,f}$.
