Geodesic
About new examples of Serre curves
Čebyševskij sbornik, Tome 21 (2020) no. 2, pp. 266-274 Cet article a éte moissonné depuis la source Math-Net.Ru

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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.
Mots-clés : Serret curve
Keywords: elliptic integral, algebraic curve. 
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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/

[1] J. A. Serret, “Mémoire sur la représentation géométrique des fonctions elliptiques and ultra-elliptiques”, Journal de mathématiques pures et appliquées, 10 (1845), 257–285

[2] J. A. Serret, “Développements sur une classe d'équations relatives a la représentation géométrique des fonctions elliptiques”, Journal de mathématiques pures et appliquées, 10 (1845), 351–363

[3] J. A. Serret, “Note sur les courbes elliptiques de la première espèce”, Journal de mathématiques pures et appliquées, 10 (1845), 421–429

[4] J. Liouville, “Sur un Mémoire de M. Serret, relatif à la representation des fonctions elliptiques”, Journal de mathématiques pures et appliquées, 10 (1845), 456–465

[5] Prasolov V. V., Soloviev Y. P., Elliptic Functions and Algebraic Equations, M., 2020