Abelian varieties isogenous to a power of an elliptic curve

Let E be an elliptic curve over a field k. Let R:=End E. There is a functor Hom_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 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 F_p.