Abelian varieties isogenous to a power of an elliptic curve

Let $E$ be an elliptic curve over a field $k$ . Let $R:=\operatorname{End}E$ . There is a functor $\mathscr{H}\!\mathit{om}_{R}(-,E)$ from the category of finitely presented torsion-free left $R$ -modules to the category of abelian varieties isogenous to a power of $E$ , and a functor $\operatorname{Hom}(-,E)$ in the opposite direction. We prove necessary and sufficient conditions on $E$ for these functors to be equivalences of categories. We also prove a partial generalization in which $E$ is replaced by a suitable higher-dimensional abelian variety over $\mathbb{F}_{p}$ .