Let $ K = \mathbb{Q}(\sqrt{d})$ be a quadratic number field, and let $ \mathcal{O}_K$ be its ring of integers, and let $ D$ be the discriminant of $ \mathcal{O}_K$ . It is a well-known result (see for example these notes of Andrew Granville: http://www.dms.umontreal.ca/~andrew/Courses/Chapter4.pdf) that an integer $ m$ is the norm of an ideal in $ \mathcal{O}_K$ if and only if

(1) $ $ \displaystyle \sum_{n | m} \left(\frac{D}{n}\right) \geq 1.$ $

In fact, the left hand side of the above is precisely the number of representation of $ m$ by integral binary quadratic forms of discriminant $ D$ .

Is there an analogue of this result for cubic norms? Here, the appropriate substitute for $ D$ ought to he a $ \operatorname{GL}_2(\mathbb{Z})$ -equivalence class of integral binary cubic forms; this is because discriminants $ D$ parametrize quadratic orders, while cubic orders are parametrized by $ \operatorname{GL}_2(\mathbb{Z})$ -classes of binary cubic forms. My question is thus as follows:

Let $ K$ be a cubic field, and let $ \mathcal{O}_K$ be its ring of integers. Let $ F$ be a representative of the $ \operatorname{GL}_2(\mathbb{Z})$ -class of integral binary cubic forms which represents $ \mathcal{O}_K$ . Is there a criterion that depends solely on $ F$ , analogous to (1), which determines whether a given integer $ m$ is the norm of an ideal in $ \mathcal{O}_K$ ?