We prove several results on torsion points and Galois representations for complex
multiplication (CM) elliptic curves over a number field containing the CM field. One
result computes the degree in which such an elliptic curve has a rational point of order
,
refining results of Silverberg (Compositio Math.68:3 (1988), 241–249;
Contemp. Math.133 (1992)). Another result bounds the size of the torsion subgroup of an elliptic curve
with CM by a nonmaximal order in terms of the torsion subgroup of an elliptic curve
with CM by the maximal order. Our techniques also yield a complete classification
of both the possible torsion subgroups and the rational cyclic isogenies of a
-CM elliptic
curve
defined over
.
