  A coregular space is a representation of an algebraic group for which the
ring of polynomial invariants is free. In this paper, we show that the orbits
of many coregular irreducible representations where the number of 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 Jacobians.
  In forthcoming work, we use these orbit parametrizations to determine the
average sizes of Selmer groups for various families of elliptic curves.
