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In his well-known paper, Nakagawa generalized a construction due to Birch and Merriman to arbitrary binary forms and orders. In particular, his construction gives a canonical algebraic order $\mathcal{O}_F$ which is associated to a binary form $F \in \mathbb{Z}[x,y]$. Nakagawa's construction remains the best way to produce many distinct algebraic orders having bounded discriminant.

A natural question is to ask how to compute the index of these (in general not maximal) orders inside the maximal order of their etale algebras. In particular, in the case when $F$ is irreducible over $\mathbb{Q}$ the order $\mathcal{O}_F$ is a suborder of the ring of integers $\mathcal{O}_K$, where $K = \mathbb{Q}(\theta)$ with $\theta$ a root of $F(x,1)$. In this case the index is simply $[\mathcal{O}_K : \mathcal{O}_F]$.

The answer is essentially the classic theory in the cubic case. This is because by Delone-Faddeev/Levi, every cubic ring comes from a binary cubic form. On the other hand one can reduce the quartic case to the cubic case by using the Bhargava-Wood correspondence between quartic rings and pairs of ternary quadratic forms: this gives a map from a quartic ring to a resolvent cubic ring, and the index of the quartic ring is necessarily equal to the index of its resolvent cubic ring because the map is discriminant preserving.

How does one set this problem up in general? Ideally, there should be some analogue of an index form in this setting: that is, some way of writing down a decomposable form which takes as input a binary $n$-ic form and the output is the index of $\mathcal{O}_F$.

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