Complex multiplication (reviewed)

An elliptic curve whose endomorphism ring is larger than \(\Z\) is said to have complex multiplication (often abbreviated to CM). In this case, for curves defined over fields of characteristic zero, the endomorphism ring is isomorphic to an order in an imaginary quadratic field. The discriminant of this order is the CM discriminant.

An elliptic curve whose geometric endomorphism ring is larger than \(\Z\) is said to have potential complex multiplication (potential CM). In the literature, these too are often called CM elliptic curves.

The property of having potential CM depends only on the $j$-invariant of the curve. In characteristic $0$, CM $j$-invariants are algebraic integers, and there are only finitely many in any given number field. There are precisely 13 CM $j$-invariants in $\Q$ (all integers), associated to the 13 imaginary quadratic orders of class number $1$:

$$ \begin{array}{c|ccccccccccccc} j & -12288000 & 54000 & 0 & 287496 & 1728 & 16581375 & -3375 & 8000 & -32768 & -884736 & -884736000 & -147197952000 & -262537412640768000\\ \text{CM discriminant} &-27 & -12 & -3 & -16 & -4 & -28 & -7 & -8 & -11 & -19 & -43 & -67 & -163 \end{array} $$

CM elliptic curves are examples of CM abelian varieties.