Entanglement in the family of division fields of elliptic curves with complex multiplication

For every elliptic curve E which has complex multiplication (CM) and is defined over a number field F containing the CM field K, we prove that the family of p(infinity)-division fields of E, with p is an element of N prime, becomes linearly disjoint over F after removing an explicit finite subfamily of fields. We then give a necessary condition for this finite subfamily to be entangled over F, which is always met when F = K. In this case, and under the further assumption that the elliptic curve E is obtained as a base-change from Q, we describe in detail the entanglement in the family of division fields of E.