Let $E$ and $F$ be two abelian varieties of dimension 1 over $\mathbb{C}$. Let $f : E \to F$ be a surjective homomorphism of abelian varieties ($f(0) = 0$). If $\ker (f) \cong \mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z}$, does this imply that $E$ and $F$ are isomorphic?



isogeny of elliptic curves
Yes. In zero characteristic the image of an isogeny of elliptic curves
is determined up to isomorphism by its kernel. Your isogeny has the same kernel
as the doubling map from $E$ to itself.


