Ideal class group of Q of square root of minus twenty-two ($\mathrm{Cl}(\mathbb{Q}(\sqrt{-22}))\cong C_2$)

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 $C_2$.

The equation $y^3=x^2+22$ has no integer solutions. Factoring it in $\mathbb{Z}[\sqrt{-22}]$, the two conjugate factors are coprime. Unique factorization of ideals in a number field and the absence of three-torsion in the class group force $x+\sqrt{-22}$ to be a cube up to a unit. Writing it as $(a+b\sqrt{-22}{)}^3$ makes its $\sqrt{-22}$ coefficientwhich has no integer solution.