 Ideal class group of Q of square root thirty-five

The ring of integers is $\mathbb{Z}[\sqrt{35}]$ and the field discriminant is $140$. The Minkowski bound for ideal classes is $\sqrt{35}<6$. At three the defining polynomial is irreducible, so the only prime ideals needed to generate the ideal class group are $P_2=(2,\sqrt{35}-1)$ and $P_5=(5,\sqrt{35})$, of ideal norms two and five. They satisfy $P_2^2=(2)$, $P_5^2=(5)$ and $P_2{P}_5=(5+\sqrt{35})$, so one class of order of a group element at most two generates. The field norm equation $a^2-35b^2=\pm 2$ is impossible modulo five, proving $P_2$ is not a principal ideal. Hence the ideal class group has exactly two elements.