  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.
