For any elliptic curve E defined over the rationals with complex multiplication (CM) and for every prime p, we describe the image of the mod p Galois representation attached to E. We deduce information about the field of definition of torsion points of these curves; in particular, we classify all cases
where there are torsion points over Galois number fields not containing the field of definition of the CM.
Dieulefait, L.V.; Gonzalez, E.; Jimenez, J. On fields of definition of torsion points of elliptic curves with complex multiplication. "Proceedings of the American Mathematical Society", Juny 2011, vol. 139, núm. 6, p. 1961-1969.