The second moment of the size of the $2$-class group of monogenized cubic fields

We prove that when totally real (resp., complex) monogenized cubic number fields are ordered by height, the second moment of the size of the $2$-class group is at most $3$ (resp., at most $6$). In the totally real case, we further prove that the second moment of the size of the narrow $2$-class group is at most $9$. This result gives further evidence in support of the general observation, first made in work of Bhargava–Hanke–Shankar and recently formalized into a set of heuristics in work of Siad–Venkatesh, that monogenicity has an altering effect on class group distributions. All of the upper bounds we obtain are tight, conditional on tail estimates.