  For positive integers $M \mid N$ and an order of discriminant $\Delta$ in an
imaginary quadratic field $K$ with discriminant $\Delta_K < -4$, we determine
the fiber of the morphism $X_0(M,N) \rightarrow X(1)$ over the closed point
$J_{\Delta}$ corresponding to $\Delta$. We also show that the fiber of the
natural map $X_1(M,N) \rightarrow X_0(M,N)$ over $J_{\Delta}$ is connected.
Putting this together we deduce the number of points in the fiber of $X_1(M,N)
\rightarrow X(1)$ over $J_{\Delta}$ and their residual degrees. In the
continuation of this work with F. Saia, these results will be extended to
$\Delta_K \in \{-4,3\}$. These works provide all the information needed to
compute, for each positive integer $d$, all subgroups of
$E(F)[\operatorname{tors}]$, where $F$ is a number field of degree $d$ and
$E_{/F}$ is an elliptic curve with complex multiplication (CM).
