By an elliptic curve over $\mathbb{Q}$, we mean a genus 1 curve with a $\mathbb{Q}$-point. By a monogenic cubic order, we mean a unital cubic ring $R$ isomorphic to $\mathbb{Z}^3$ as a $\mathbb{Z}$-module such that there exists $\theta \in R$ with the property that $\{1, \theta, \theta^2\}$ is a $\mathbb{Z}$-basis of $R$ as a $\mathbb{Z}$-module.

Of course, we know that both objects are naturally associated with monic cubic polynomials: an elliptic curve over $\mathbb{Q}$ has an integral Weierstrass model of the form

$$\displaystyle E/\mathbb{Q} : y^2 = x^3 + Ax + B, A,B \in \mathbb{Z}$$

and by Delone-Faddeev correspondence, a monogenic cubic ring is in unique correspondence with the $\text{GL}_2(\mathbb{Z})$-equivalence class of a monic binary cubic form, say given by $x^3 + rx^2 y + sxy^2 + ty^3$ with $r, s, t \in \mathbb{Z}$ and $r \in \{-1,0,1\}$.

In their seminal paper proving the boundedness of the average rank of elliptic curves (with respect to the height $\max\{4|A|^3, 27B^2\}$), Bhargava and Shankar made use of this connection. In particular, they used the fact that elliptic curves with partial 2-torsion corresponds to reducible monogenic rings, and used results of Nakagawa and Bhargava (in an earlier work) to control the number of binary quartic forms that are potential 2-Selmer elements of such curves.

Are there any deeper connections between monogenic cubic rings and elliptic curves over $\mathbb{Q}$? For example, how would one interpret important invariants of elliptic curves, say the $j$-invariant, on the side of cubic rings?


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