Given a number field $K$ and an order (not necessarily maximal) $\mathcal{O}$ on it, it's a difficult problem to count the number $h(\mathcal{O})$ of $\mathcal{O}$-fractional ideals in $K$ (I think).
If $\alpha$ is an algebraic integer in $K$, then $h(\mathbb{Z}[\alpha])$ is the number of conjugacy classes over $\mathbb{Z}$ of matrices whose minimal polynomial is equal to the minimal polynomial of $\alpha$?
Suppose now that we have a finite family of number fields (CM-fields) $K_i=\mathbb{Q}(\alpha_i)$. Fix a polynomial $P(X)\in\mathbb{Q}[X]$ and set $\mathcal{O}_i=\mathbb{Z}[P(\alpha_i)]$ (suppose it is an order, that is, $P(\alpha_i)$ is an algebraic integer in $K_i$), do you know (or do you have an idea) a field $K$ and an order $\mathcal{O}$ on it, where
$$ h(\mathcal{O})=\sum_i h(\mathcal{O}_i) $$
