Let $G$ be a reductive group, and let $V$ be a finite dimensional vector space on which $G$ acts linearly. Let $\mathcal{P}$ be the ring of polynomial invariants of the action of $G$ on $V$. We say that the pair $(G,V)$ is a coregular space if the ring $\mathcal{P}$ is a polynomial ring, that is, it is freely generated over the field of definition of $V$.
Bhargava and Shankar have idenitifed that coregular spaces are of particular interest to arithmetic. In particular, they showed that the coregular spaces consisting of binary quartic forms with $\operatorname{GL}_2$ action, ternary cubic forms with $\operatorname{GL}_3$ action, etc. can be used to bound the average rank of elliptic curves.
My question is, is there a coregular space where $\mathcal{P}$ is generated by two invariants $U,V$ such that $\deg U = 10, \deg V = 15$? For example, when $(G,V) = (\operatorname{GL}_2(\mathbb{C}), V_4(\mathbb{C})$ where $V_n(\mathbb{C})$ is the space of binary $n$-ic forms, the ring $\mathcal{P}$ is generated by two invariants $I$ and $J$ of degrees 2 and 3, respectively.
More generally, is there a classification result known for coregular spaces, like there is for prehomogeneous spaces? Bhargava and Ho wrote extensively on coregular spaces associated to genus one curves, but did not appear to classify all coregular spaces.
