I can't seem to find the reference for the following fact
Suppose that $E$ is an elliptic curve defined over $k$ and $K, L$ are algebraically closed fields containing $k$. Then $\text{End}_K(E)$ is isomorphic to $\text{End}_L(E)$.
Where $\text{End}_K(E)$ is the ring of endomorphisms of $E(K)$. Of course, it suffices to show the result for $L$ being the algebraic closure of the ground field $k$.