About new examples of Serre curves

Abstract

Abel's theorem claims that the Lemniscate can be divided into n equal arcs by ruler and compass iff $n=2^kp_1\ldots p_m$, where $p_j$ are pairwise distinct Fermat primes. The proof is based on the fact that the lemniscate can be parametrised by rational functions and the arc length is a first type elliptic integral of the parameter. Joseph Alfred Serret proposed a method to describe all such curves in [1]. In papers [1, 2, 3] he found series of such curves and described its important properties. Since then no new examples of curves with rational parametrisation such that arc length is a first type elliptic integral of the parameter are known. In this note we describe new example of such a curve.