Hilbert class field of Q of square root minus six ($H_{\mathbb{Q}(\sqrt{-6})}=\mathbb{Q}(\sqrt{2},\sqrt{-3})$)

The ideal class group of $K=\mathbb{Q}(\sqrt{-6})$ has order two, andThis quadratic extension of $K$ is unramified at every finite prime and therefore has the degree required of the Hilbert class field.