The special symplectic structure of binary cubics

Let $k$ be a field of characteristic not 2 or 3. Let $V$ be the $k$-space of binary cubic polynomials. The natural symplectic structure on $k^2$ promotes to a symplectic structure $\omega$ on $V$ and from the natural symplectic action of $\textrm{Sl}(2,k)$ one obtains the symplectic module $(V,\omega)$. We give a complete analysis of this symplectic module from the point of view of the associated moment map, its norm square $Q$ (essentially the classical discriminant) and the symplectic gradient of $Q$. Among the results are a symplectic derivation of the Cardano-Tartaglia formulas for the roots of a cubic, detailed parameters for all $\textrm{Sl}(2,k)$ and $\textrm{Gl}(2,k)$-orbits, in particular identifying a group structure on the set of $\textrm{Sl}(2,k)$-orbits of fixed nonzero discriminant, and a purely symplectic generalization of the classical Eisenstein syzygy for the covariants of a binary cubic. Such fine symplectic analysis is due to the special symplectic nature inherited from the ambient exceptional Lie algebra $\mathfrak G_2$.