  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 $N$, refining results of Silverberg. 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
$K$-CM elliptic curve $E$ defined over $K(j(E))$.
