Algebraicity of the division points of the trifolium and related topics

Gauss and Abel proved that the points dividing the unit circle and the lemniscate of Bernoulli in parts of equal length have algebraic coordinates. In this note we generalise these results to the Erd\H{o}s lemniscate with three leaves. We also study further questions related to the algebraicity of division points and transcendence of length of a class of curves including polynomial lemniscates. To do this we analyse the structure and periods of the Jacobian of certain hyperelliptic curves.