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?
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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. 

