# Monogenic cubic rings and elliptic curves

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?