  We introduce an algorithm that computes explicit class fields of an imaginary
quadratic field $K$ for a given modulus $\mathfrak{f}\subset\mathcal{O}_K$ more
efficiently than the use of their classical counterparts. Therein, we prove the
fact that certain values of a simple quotient of Siegel $\phi$-function are
elements in the ray class field $K_{\mathfrak{f}}$ of $K$.
