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Let $K$ be a quadratic number field. Let $R$ be an order of $K$, i.e. the subring of $K$ which is a free $\mathbb{Z}$-module of rank $2$.Let $D$ be its discriminant. We use the notation and the result of this question. There exists a bijection between $Cl^+(R)$ and the set of classes of primitive binary quadratic forms of discriminant $D$. So it is natural to ask the following question.

My question How can we compute the ideal class group $G = Cl^+(R)$? Namely how can we compute the following tasks?

  1. Find a set of full representatives of the group $G$.

  2. Compute the multiplication table of the group $G$ in terms of the above representatives.

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Makoto Kato
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  • It is a natural, but obviously hard. We can't even answer this easily, I mean it's really hard, even for a maximal order. Why should it be easier for a general order? – Alex Youcis Nov 14 '13 at 10:29

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