Elliptic curves with complex multiplication and abelian division fields

Abstract Let be an imaginary quadratic field, and let be the order in of conductor . Let be an elliptic curve with complex multiplication by , such that is defined by a model over , where . In this article, we classify the values of and the elliptic curves such that (i) the division field is an abelian extension of , and (ii) the ‐division field coincides with the th cyclotomic extension of the base field.