Ideal class group of Q of square root of minus thirty-three ($\mathrm{Cl}(\mathbb{Q}(\sqrt{-33}))$)

The field has ring of integers $\mathbb{Z}[\sqrt{-33}]$, discriminant $-132$, and Minkowski bound below eight. The nonprincipal ramified ideals above two and three generate four classes, and the split primes above seven introduce no new class. Hence