Computing the Hilbert Class Fields of Quartic CM Fields Using Complex Multiplication

Let $K$ be a quartic CM field, that is, a totally imaginary quadratic extension of a real quadratic number field. In a 1962 article titled On the classfields obtained by complex multiplication of abelian varieties, Shimura considered a particular family $\{F_K(m) : m \in \mathbb{Z} >0 \}$ of abelian extensions of $K$, and showed that the Hilbert class field $H_K$ of $K$ is contained in $F_K(m)$ for some positive integer m. We make this m explicit. We then give an algorithm that computes a set of defining polynomials for the Hilbert class field using the field $F_K(m)$. Our proof-of-concept implementation of this algorithm computes a set of defining polynomials much faster than current implementations of the generic Kummer algorithm for certain examples of quartic CM fields.