The invariants of the third symmetric power representation of \(\mathrm{SL}_2(\mathbb F_p)\) (Q651241)

It is known that the special linear group \(\mathrm{SL}_2(\mathbb F_p)\) over the finite field \(\mathbb F_p\) of characteristic \(p>3\) has a natural linear action on binary forms. The authors compute a finite generating set (not necessarily minimal) for the ring of invariants of the induced action on the finite-dimensional vector space spanned by the cubic monomials.