I am beginning to study the theory of complex multiplication, and I haven't yet got my hands on a copy of Silverman's advanced topics. For now I'm making do with chapter 6 in Silverman-Tate, Rational Points on Elliptic Curves, and C.11 in Silverman's Arithmetic of Elliptic Curves.
If $K$ is an imaginary quadratic field with class group $Cl(K)$. If $\Lambda$ is a fractional ideal of $K$, we get an Elliptic curve $\mathbb{C}/\Lambda$. Now let $H$ be the Hilbert class field for $K$, and for a prime ideal $\mathfrak{p}$, we have the Artin map $\mathfrak{p} \mapsto \sigma_{\mathfrak{p}} \in Gal(H/K)$. If $j(\Lambda) \in \mathbb{C}$ is the $j$-invariant for $\mathbb{C}/\Lambda$, then we have $H = K(j(\Lambda))$ and $\sigma_{\mathfrak{p}}(j(\Lambda)) = j(\Lambda \mathfrak{p}^{-1})$.
Here is my question: This time I'll assume that $Cl(K) = 1$ and $E$ is the Elliptic curve $\mathbb{C}/\mathcal{O}_K$. If $\mathfrak{m}$ is a fractional ideal of $K$, and $H_{\mathfrak{m}}$ is the Ray class field modulo $\mathfrak{m}$, then $H_{\mathfrak{m}} = K(\Phi(E[\mathfrak{m}]))$, where $\Phi$ is the Weber function, and $E[\mathfrak{m}] = ker([\mathfrak{m}]: E \to E)$ are the $\mathfrak{m}$-torsion points of $E/\mathbb{C}$ (note $[\mathfrak{m}] \in End(E)$ since $Cl(K) = 1$).
For $\mathfrak{p}$ co-prime to $\mathfrak{m}$, we again have that the Artin map gives us $\sigma_{\mathfrak{p}} \in Gal(H_{\mathfrak{m}}/K)$, but how does this automorphism act on the $\mathfrak{m}$-torsion points?