Fields with small class group in the family $\mathbb{Q}(\sqrt{9m^2+2m})$

Abstract: Very recently, Issa and Darrag [Arch. Math. (Basel) 123 (2024), no. 4, 379-383] determined partial Dedekind zeta values for certain ideal classes in the real quadratic fields of the form $\mathbb{Q}(\sqrt{9m^2+2m})$, where $9m^2+2m$ is square-free and $m\equiv 2\pmod 3$ is an odd positive integer. We use these partial Dedekind zeta values to investigate the small class numbers of such fields. More precisely, we prove that the class numbers of the fields in the above mentioned family are at least $4$. Further, we provide a sufficient condition permitting to specify the structure of the class groups of order $4$ in this family of fields.