Endomorphism of an elliptic curve (reviewed)

An endomorphism of an elliptic curve defined over a field $K$ is an isogeny $\varphi:E\to E$ defined over the algebraic closure of $K$.

The set of all endomorphisms of $E$ forms a ring called the endomorphism ring of $E$, denoted $\End(E)$. The subring consisting of those endomorphisms defined over $K$ itself is denoted $\End_K(E)$.

$\End(E)$ always contains a subring isomorphic to $\Z$, since for $m\in\Z$ there is the multiplication-by-$m$ map $[m]:E\to E$.