Binary quadratic forms and ray class groups

Let $K$ be an imaginary quadratic field different from $\mathbb{Q}(\sqrt{-1})$ and $\mathbb{Q}(\sqrt{-3})$. For a positive integer $N$, let $K_{\mathfrak{n}}$ be the ray class field of $K$ modulo $\mathfrak{n}=N{\mathcal{O}}_K$. By using the congruence subgroup $\pm {\mathrm{\Gamma }}_1(N)$, we construct an extended form class group whose operation is basically the Dirichlet composition, and explicitly show that this group is isomorphic to the Galois group $\mathrm{G}\mathrm{a}\mathrm{l}(K_{\mathfrak{n}}\mathrm{/}K)$. We also present algorithms to find all form classes and show how to multiply two form classes. As an application, we describe $\mathrm{G}\mathrm{a}\mathrm{l}(K_{\mathfrak{n}}^{\mathrm{a}\mathrm{b}}\mathrm{/}K)$ in terms of these extended form class groups for which $K_{\mathfrak{n}}^{\mathrm{a}\mathrm{b}}$ is the maximal abelian extension of $K$ unramified outside prime ideals dividing $\mathfrak{n}$.