By a cubic surface $X_F$ we mean the zero locus of a homogeneous cubic polynomial $F(x,y,z,w)$. The group $\text{GL}_4$ acts on $X_F$ via substitution. The ring of polynomial invariants induced by this action is generated by 6 elements, typically denoted $I_8, I_{16}, I_{24}, I_{32}, I_{40}, I_{100}$, where for each $I_j$ the subscript denotes the degree of the invariant in the coefficients of $F$. The first five invariants are algebraically independent, where as $I_{100}^2$ is a polynomial of the other five invariants.
A typical cubic form in 4 variables has 20 coefficients, so these invariants are extremely unwieldy to compute. However, Sylvester showed that any cubic form in 4-variables may be put into what is now called Sylvester standard form. That is, there exist 5 complex numbers $a_0, \cdots, a_4$ and an element $T \in \text{GL}_4(\mathbb{C})$ such that
$$\displaystyle F_T(x,y,z,w) = a_4 x^3 + a_3 y^3 + a_2 z^3 + a_1 w^3 - a_0(x + y + z + w)^3.$$
This dramatic simplification makes it reasonable to ask for an explicit formula for the generators of the ring of invariants. Is this explicitly computed and known?
