Geodesic
About new examples of Serre curves
Čebyševskij sbornik, Tome 21 (2020) no. 2, pp. 266-274

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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.
Keywords: Serret curve, elliptic integral, algebraic curve. 
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     author = {A. T. Lipkovski and F. Yu. Popelensky},
     title = {About new examples of {Serre} curves},
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A. T. Lipkovski; F. Yu. Popelensky. About new examples of Serre curves. Čebyševskij sbornik, Tome 21 (2020) no. 2, pp. 266-274. http://geodesic.mathdoc.fr/item/CHEB_2020_21_2_a18/