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