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It is known that the set of binary cubic forms

$$\displaystyle T_3 = \{F_{a,b}(x,y) = ax^3 + bx^2 y + (b - 3a)xy^2 - ay^3 : a,b \in \mathbb{Z}\}$$

parametrize the set of cyclic cubic fields, in the sense that cyclic cubic fields correspond precisely to those elements of $T_3$ which are irreducible and maximal. Here we say a binary cubic form $F \in \mathbb{Z}[x,y]$ is maximal if the corresponding cubic ring (via Delone-Faddeev correspondnece) is a maximal cubic ring. See the following paper of Bhargava and Shnidman for details.

For a given field $K$, the index of an element $s$ in $\mathcal{O}_K$ is equal to $g_K(s) = \sqrt{\frac{\text{disc}(\mathbb{Z}[s])}{\text{disc}(\mathcal{O}_K)}}$. In the cubic case the function $g_K(s)$ is a binary cubic form, and precisely equal to the form given by Delone-Faddeev correspondence corresponding to $\mathcal{O}_K$.

For an element $F = F_{a,b} \in T_3$, define $$g(a,b) = \min \{m \in \mathbb{N} : \exists u,v \in \mathbb{Z} \text{ s.t. } F_{a,b}(u,v) = m\}.$$

I ask the following question: as $a,b$ range over some bounded set, say $|a|, |b| \leq X$ (or more naturally, $b^2 - 3ab + 9a^2 \leq X$ which corresponds to considering elements in $T_3$ with bounded discriminant), what is the average value of $g(a,b)$?

Observe that $g(a,b) \leq a$, and since $a \ll \text{disc}(F_{a,b})^{1/4}$, it follows that $g(a,b) \ll \text{disc}(F_{a,b})^{1/4}$. A rough guess is that this should be expected on average.

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