Let $K$ be an imaginary quadratic field and $E$ be an elliptic curve with CM by $\mathcal{O}_K$. We know that the maximal unramified extension (Hilbert class field) $H/K$ is $K(j(E))$. Can we explicitly write down which are the primes that ramify in the ray class field ie $K(j(E), h(E[\mathfrak{p}]))$ for some $\mathfrak{p}$ where $h$ is the Weber function?
The only sort of information I have for now is the following: a prime $\mathfrak{q}$ splits completely in $K(j(E), h(E[\mathfrak{p}]))$ iff $\mathfrak{q}=(\alpha)$ with $\alpha\in \mathcal{O}_K$ and $\alpha\equiv 1 \mod \mathfrak{p}$.