(Reference request) List of (isomorphism classes of) ideal class groups of $\mathbb{Q}(\sqrt{d})$.

Question

I'm learning about the ideal class group of a number field, and am trying a few exercises where I calculate $\mathbb{Q}(\sqrt{d})$ for $d \in \mathbb{Z}$ for various $d$.

I'd like to check my work. Unfortunately, I can't seem to find a list anywhere of ideal class groups, nor a calculator to do it. I can only find lists of the class numbers of these number fields.

May someone provide a link to some document listing the ideal class groups for small $d$, hopefully up to $|d|=100$, say? For example, next to $\mathbb{Q}(\sqrt{-30})$ it should list that the ideal class group is $C_2 \times C_2$.

Answer 1

The LMFDB has a database of number fields that includes all quadratic number fields of absolute discriminant up to 2 million, and each number field's entry in the database includes its ideal class group. You can also check the ideal class group of a number field using computer algebra software such as Pari/GP, SageMath, or Magma; entries in the LMFDB have a button near the top to display the relevant commands for these computations.