I am currently working on a problem and I have no idea how to start. Following the proof of theorem 2.3 (pg.112-113) in Silverman's Book "Advanced Topics in the Arithmetic of Elliptic Curves".
Let $E/\mathbb{C}$ be an elliptic curves with complex multiplication by an order $R_K$ of an imaginary quadratic field $K$. Let $%N\geq 2$ be an integer. There is a representation $\rho: Gal(\bar{K}/K(j(E))) \to \mathrm{Aut}(E[N])$. Since $E$ is defined over $H$, we have that the elements of $Gal(K(E[N],j(E))/K(j(E)))$ commute with the elements of $R_K$ in their action on $E[N]$. So $\rho$ induces a $R_K/NR_K$-module homomorphism. $\phi: Gal(K(E[N],j(E))/K(j(E)))\hookrightarrow \mathrm{Aut}_{R_K/NR_K}(E[N])$ But $E[N]$ is a free $R_K/NR_K$-module of rank 1. Thus $\mathrm{Aut}_{R_K/NR_K}(E[N]) \cong (R_K/NR_K)^*$.
My question is how would one figure out the index of $\phi$? I've thought about this but I don't even know where to start. I'm aware that for an integral ideal $\mathfrak{a}$, $(R_K/\mathfrak{a}R_K)^*$ has order $\varphi(\mathfrak{a})$ (as defined in Lang's Algebraic Number Theory)