Let $F(x,y) = a_4 x^4 + a_3 x^3 y + a_2 x^2 y^2 + a_1 xy^3 + a_0 y^4 \in \mathbb{R}[x,y]$ be a binary quartic form, and let $V(\mathbb{R})$ be the 5-dimensional $\mathbb{R}$-vector space of such forms. Let $V(\mathbb{Z})$ be the lattice of binary quartic forms with integer coefficients.
The group $\text{GL}_2(\mathbb{R})$ acts on $V(\mathbb{R})$ via the usual substitution action (i.e., $F_M(x,y) = F(m_1 x + m_2 y, m_3 x + m_4 y)$), with relative invariants given by $I, J$ which are polynomials in $a_4, \cdots, a_0$. Explicitly, they are given by the formulas
$$\displaystyle I(F) = 12 a_4 a_0 - 3 a_3 a_1 + a_2,$$ $$\displaystyle J(F) = 72 a_4 a_2 a_0 + 9 a_3 a_2 a_1 - 27 a_4 a_1^2 - 27 a_0 a_3^2 - 2 a_2^3.$$
In fact the space $\text{GL}_2(\mathbb{R}) \setminus V(\mathbb{R})$ is a co-regular space, so that its ring of polynomial invariants is freely generated; indeed, $I,J$ is a pair of generators. The discriminant $\Delta(F)$ of $F$ is given by
$$\displaystyle \Delta(F) = \frac{4I(F)^3 - J(F)^2}{27}.$$
Bhargava and Shankar showed that one can define a discriminant preserving map from $V(\mathbb{Z})$ to elliptic curves given by short Weierstrass model, via the correspondance
$$\displaystyle F \mapsto E: y^2 = x^3 - \frac{I(F)}{3}x + \frac{J(F)}{27}.$$
With respect to this correspondence, we see that the $\text{GL}_2$-invariant subvarieties of $V(\mathbb{R})$ defined by the vanishing of $I$ and $J$ respectively naturally correspond to elliptic curves with $j$-invariant equal to zero and $1728$ respectively.
M. Matchett-Wood, in her thesis, gave a correspondence between elements of $V(\mathbb{Z})$ and a special subset of quartic rings: in particular, the $\text{GL}_2(\mathbb{Z})$-equivalence classes of integral binary quartic forms correspond exactly (in a discriminant preserving manner) to quartic rings with monogenic cubic resolvent: that is, those quartic rings whose cubic resolvent rings are monogenic rings.
Therefore, binary quartic forms with vanishing $I$ or $J$-invariants should correspond to certain special classes of quartic rings. Have such rings been classified? Are they easy to understand?
