CM points (awaiting review)

Rational points on a modular curve $X_H$ that correspond to elliptic curves with complex multiplication are called CM points.

Over $\Q$ these correspond to rational points in the fiber of the map $j\colon X_H\to X(1)$ above one of the $13$ $j$-invariants of CM elliptic curves over $\Q$ corresponding to one of the CM discriminants \[ -3,-4,-7,-8,-11,-12,-16,-19,-27, -27, -43,-67,-163, \] of imaginary quadratic orders of class number one. These $13$ orders lie in $9$ distinct imaginary quadratic fields.