  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.
