Coregular spaces and genus one curves

A coregular space is a representation of an algebraic group for which the ring of polynomial invariants is freely generated. We show that the orbits of many coregular irreducible representations of algebraic groups where the number of generating invariants is at least two, over a (not necessarily algebraically closed) field k, correspond to genus one curves over k together with line bundles, vector bundles, and/or points on their Jacobian curves. In particular, we give explicit descriptions of certain moduli spaces of genus one curves with extra structure as quotients by algebraic groups of open subsets of affine spaces.