Let $F(x,y) = ax^4 + bx^3y + cx^2y^2 + dxy^3 + ey^4$ be a binary quartic form with integer coefficients. It is well-known that $F$ has two algebraically independent invariants under the action of $\text{GL}_2(\mathbb{Z})$ acting on $F$ via substitution, namely

$$\displaystyle I(F) = 12ae - 3bd + c^2$$

and

$$\displaystyle J(F) = 72ace + 9bcd - 27ad^2 - 27eb^2 - 2c^3.$$

In this paper, Bhargava and Shankar proved (see Theorem 1.7) that if a pair of integers $(A,B)$ satisfy certain congruence conditions modulo 9 and 27 respectively, then there exists a binary quartic form $F$ with integer coefficients such that $I(F) =A$ and $J(F) = B$. Specifically, they determined when $(A,B)$ can appear as $\mathcal{F}(\mathbb{Z},1)$ invariants of monic binary cubic forms, where $\mathcal{F}(\mathbb{Z}, 1)$ is the subgroup of $\text{GL}_2(\mathbb{Z})$ consisting of upper triangular matrices with 1's on the diagonal, and show that each binary cubic form appears as the cubic resolvent of some binary quartic form (under some discriminant preserving map). Indeed, they showed that the binary cubic form

$$\displaystyle x^3 + rx^2y + sxy^2 + ty^3$$

is the cubic resolvent of

$$\displaystyle x^3 y + rx^2y^2 + sxy^3 + ty^4.$$

The problem with this proof is that the quartic form they constructed is always reducible over $\mathbb{Z}$. Certainly, it is possible that this is the only integral quartic form with invariants $A,B$ respectively, but in general there should be other $\text{GL}_2(\mathbb{Z})$ distinct examples since they proved in the same paper that on average, there are $5 \zeta(2)/2$ distinct $\text{GL}_2(\mathbb{Z})$ classes of binary quartic forms with invariants $(A,B)$.

Can anyone give, at least for an infinite family of pairs of $(A,B)$, examples of an irreducible quartic form $F$ whose $I$ and $J$ invariants are equal to $A$ and $B$ respectively?