Definition (Isogeny). An isogeny $\varphi :E_1\to E_2$ is a nonconstant morphism with $\varphi (0_{E_1})=0_{E_2}$ (by Theorem 2.8, a morphism is nonconstant if and only if surjective on $\overline{K}$-points).

We say $E_1$ and $E_2$ are isogenous in this case.