Elliptic curves with abelian division fields

Let E be an elliptic curve over , and let . The central object of study of this article is the division field that results by adjoining to the coordinates of all n-torsion points on . In particular, we classify all curves such that is as small as possible, that is, when , and we prove that this is only possible for , or 5. More generally, we classify all curves such that is contained in a cyclotomic extension of or, equivalently (by the Kronecker-Weber theorem), when is an abelian extension. In particular, we prove that this only happens for , or 8, and we classify the possible Galois groups that occur for each value of n.