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I think that the answer is yes, at least if you are working over an algebraically closed field of char $0$. Let me try to give an argument, I hope that there are no mistakes. You have two flat families $E_1 \to C_1$ and $E_2 \to C_2$ of elliptic curves over some unknown bases $C_1, C_2$. The first assumption says that the map $E_1 \to X$ induces a bijection ...