 Ideal class group of Q of square root of minus twenty-two

The ring of integers is $\mathbb{Z}[\sqrt{-22}]$, its discriminant is $-88$, and its Minkowski bound for ideal classes is $4\sqrt{22}/\pi <6$. The primes three and five are inert, whileThe ideal ${\mathfrak{p}}_2$ is nonprincipal because $a^2+22b^2=2$ has no integer solution. Every class is represented by an ideal of norm at most five, so $[{\mathfrak{p}}_2]$ is the only possible nontrivial class and the class group is $C_2$.