TITLE: noncyclic class group example of a cubic number field
QUESTION [9 upvotes]: I'm trying to compute ideal class groups of various number fields, and now I'm a little familiar to a class group of a quadratic number field. However, I can't find any non-cyclic example of a class group of a cubic number field. In Marcus' "Number Fields" book, there are some exercises that deal with $\mathbb{Q}(\sqrt[3]{m})$ for integer $m$, but every such exercise has a cyclic class group. Is there any good example of a cubic number field that has a class group isomorphic to the Klein 4-group? How about biquadratic or quartic fields? (I think I can't do with quintic things...) Thanks in advance. Until now, I computed class groups of $\mathbb{Q}(\sqrt{223}), \mathbb{Q}(\sqrt{226}), \mathbb{Q}(\sqrt{-30}), \mathbb{Q}(\sqrt{-89})$.

REPLY [2 votes]: Using a math engine, like PARI/GP one can check the class group of the polynomials of the form $x^3-a$. For $a>0$ we have that the first polynomial with root $\alpha$ for which $\mathbb{Q}(\alpha)$ has non-cyclic class group is $65$. Indeed we have that the class group of $x^3-65$ is $\mathbb{Z}/3\mathbb{Z} \times \mathbb{Z}/6\mathbb{Z}$.
On the other side $a=113$ is the smallest $a$ s.t. $\mathbb{Q}(a)$ has the Klein 4-group as a class group, where $\alpha=\sqrt[3]{113}$. In between the $a'$s producing non-cyclic class groups are $70, 86, 91, 110$ and for all o them $\mathbb{Q}(\sqrt[3]{a})$ has $\mathbb{Z}/3\mathbb{Z} \times \mathbb{Z}/3\mathbb{Z}$ as a class group. 
However considering the size of the discriminant and having to compute at least $20$ some ideals (let alone combine them later) by using Minkowski's Bound it will be too tedious to do it by hand.