  Consider the elliptic curve $E$ given by the Weierstrass equation $y^2 = x^3
- 11x - 14$, which has complex multiplication by the order of conductor $2$
inside $\mathbb{Z}[i]$. It was recently observed in a paper of Daniels and
Lozano-Robledo that, for each $n \geq 2$, $\mathbb{Q}(\mu_{2^{n+1}}) \subseteq
\mathbb{Q}(E[2^n])$. In this note, we prove that this (a priori surprising)
``tower of vertical entanglements'' is actually more a feature than a bug: it
holds for any elliptic curve $E$ over $\mathbb{Q}$ with complex multiplication
by any order of even discriminant.
