Let $E$ be an elliptic curve with CM by an order $\mathcal O$ in an imaginary quadratic field $K$. Choose a basis for $E[N]$ to get an isomorphism $\operatorname{Aut}(E[N])\cong \operatorname{GL}_2(\mathbb Z/N\mathbb Z) $. Complex multiplication on $E$ induces a homomorphism $(\mathcal O/ N\mathcal O  )^\times\rightarrow \operatorname{GL}_2(\mathbb Z/N\mathbb Z)$. Let $C_N$ be the image. 

Is $C_N$ equal to its centralizer in $\operatorname{GL}_2(\mathbb Z/N\mathbb Z)$?

This should occur in some cases, and should give rise to a rational point on certain modular curves. See [Burcu Baran: Normalizers of non-split Cartan subgroups, modular curves,
and the class number one problem][1], especially Proposition 4.1.


  [1]: https://www.sciencedirect.com/science/article/pii/S0022314X10001782