Other than the trivial case of an ellipse with equivalent major and minor axes (a circle) or the degenerate case where one axis is $0$, is there any known ellipse that has rational length major and minor axes and whose perimeter can be represented in closed form as either an algebraic number, or else an algebraic multiple of $\pi$?*
If, by some chance, the answer is yes and such an ellipse is possible I would very much like to know what proportions one such ellipse has.
EDIT: *Or more generally, an algebraic multiple of an algebraic power of $\pi$.