Endomorphism ring of an elliptic curve (reviewed)

The endomorphism ring $\End(E)$ of an elliptic curve \(E\) is the ring of all endomorphisms of \(E\) defined over $K$. For endomorphisms defined over extensions, we speak of the geometric endomorphism ring of $E$.

For elliptic curves defined over $\Q$, this ring is always isomorphic to \(\Z\) consisting of the multiplication-by-$m$ maps $[m] \colon E\to E$ for $m \in \Z$.

This is a special case of the endomorphism ring of an abelian variety.