Computation of the Ideal Class Group of Certain Complex Quartic Fields. II

For quartic fields $K = {F_3}(\sqrt \pi )$, where ${F_3} = Q(\rho )$ and $\pi \equiv 1 \bmod 4$ is a prime of ${F_3}$, the ideal class group is calculated by the same method used previously for quadratic extensions of ${F_1} = Q(i)$, but using Hurwitz’ complex continued fraction over $Q(\rho )$. The class number was found for 10000 such fields, and the previous computation over ${F_1}$ was extended to 10000 cases. The distribution of class numbers is the same for 10000 fields of each type: real quadratic, quadratic over ${F_1}$, quadratic over ${F_3}$. Many fields were found with non-cyclic class group, including the first known real quadratics with groups $5 \times 5$ and $7 \times 7$. Further properties of the continued fractions are also discussed.