Consider an elliptic curve $E/H$ with CM by the entire ring of integers $\mathcal{O}_K$ of $K=\mathbb{Q}[\sqrt{-d}]$ (and such that $j(E)=j(\mathcal{O}_K)$) such that $H$ is the Hilbert class field of $K$. Consider a prime $\mathfrak{B}\nmid 2$ of $H$ such that $E$ has good reduction at $\mathfrak{B}$.
A paper by Rubin and Silverberg and a paper by Stark provide a method to count the number of points on $E(\mathcal{O}_H/\mathfrak{B})$ by finding the Hecke character $\Psi(\mathfrak{B})$, provided that $d\neq 1,3$ and $d$ is squarefree (and $d>0$).
The squarefree assumption seems trivial, but the assumption $d\neq 1,3$ ensures that there are no non-trivial units in $\mathcal{O}_K$ so that $\Psi(\mathfrak{B})=\pm \operatorname{Norm}_{H/K} (\mathfrak{B})$. That is, we only have to consider $\pm$ and no other units.
Question: Are there any references/methods to count the number of points on $E(\mathcal{O}_H/\mathfrak{B})$ for $E/H$ with CM by $\mathbb{Q}[\sqrt{-1}]$ and $\mathbb{Q}[\sqrt{-3}]$? That is, for CM by fields whose rings of integers have non-trivial units.