The special linear group $\text{SL}_2(\mathbb{Z})$ of $2\times 2$ invertible matrices in $\mathbb{Z}$ acts on binary cubic forms $\{ax^3 + bx^2y + cxy^2 + dy^3\}$ by acting on the vector $(x,y)^T$. This action descends to an action of the group $\Gamma_\infty := \left\{ \begin{pmatrix} 1 & n \\ 0 & 1 \end{pmatrix} : n \in \mathbb{Z}\right\}$ on binary cubic forms. It happens to be that $\Gamma_\infty$ is generated by $\begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}$. So understanding the action of this one element allows us to understand the whole $\Gamma_\infty$ action. (This is the action that sends $(x,y) \to (x+y, y)$)
I am interested in the invariants of binary forms under this action. In this (the binary cubic forms) case, the invariants form a 3-dimensional algebra, and I know the invariants. This action makes sense on other binary forms as well. It happens to be that the invariants of the binary quadratic case form a 2-dimensional algebra, which I also know. But I found these through the "brute force" method.
I am not really familiar with invariant theory at all, so I don't know if this is an easy or a hard question. But I'm wondering if there's some nice form for the invariants of binary n-degree forms under $\Gamma_\infty$? Alternately, perhaps there's a nice form for the invariants under $\text{SL}_2(\mathbb{Z})$ that one might perhaps be able to modify?