Let A/Q be an abelian variety of dimension g = 1 that is isogenous over Q to Eg, where E is an elliptic curve. If E does not have complex multiplication (CM), by results of Ribet and Elkies concerning fields of definition of elliptic Q-curves E is isogenous to a curve defined over a polyquadratic extension of Q. We show that one can adapt Ribet’s methods to study the field of definition of E up to isogeny also in the CM case. We find two applications of this analysis to the theory of Sato–Tate groups: First, we show that 18 of the 34 possible Sato–Tate groups of abelian surfaces over Q occur among at most 51 Q-isogeny classes of abelian surfaces over Q; Second, we give a positive answer to a question of Serre concerning the existence of a number field over which abelian surfaces can be found realizing each of the 52 possible Sato–Tate groups of abelian surfaces