Counting points on the Fricke–Macbeath curve over finite fields
Journal de Théorie des Nombres de Bordeaux, Volume 30 (2018) no. 1, pp. 117-129.

The Fricke-Macbeath curve is a smooth projective algebraic curve of genus 7 with automorphism group PSL 2 (𝔽 8 ). We recall two models of it (introduced, respectively, by Maxim Hendriks and by Bradley Brock) defined over , and we establish an explicit isomorphism defined over (-7) between these models. Moreover, we decompose up to isogeny over the jacobian of one of these models. As a consequence we obtain a simple formula for the number of points over 𝔽 q on (the reduction of) this model, in terms of the elliptic curve with equation y 2 =x 3 +x 2 -114x-127. Moreover, twists by elements of PSL 2 (𝔽 8 ) of the curve over finite fields are described. The curve leads to a number of new records as maintained on manYPoints of curves of genus 7 with many rational points over finite fields.

La courbe de Fricke-Macbeath est une courbe projective lisse de genre 7 avec groupe d’automorphismes PSL 2 (𝔽 8 ). Nous rappelons deux modèles de cette courbe (introduits respectivement par Maxim Hendriks et Bradley Brock) définis sur , et nous établissons un isomorphisme explicite, défini sur (-7), entre ces deux modèles. De plus, nous décomposons à isogénie sur près la jacobienne de l’un des modèles. Comme une conséquence nous obtenons une formule simple pour le nombre de points sur 𝔽 q de (la réduction de) ce modèle, en termes de la courbe elliptique d’équation y 2 =x 3 +x 2 -114x-127. Enfin, des tordus de cette courbe par des éléments de PSL 2 (𝔽 8 ) sur des corps finis sont décrits. La courbe donne un certain nombre de nouveaux records maintenus par manYPoints de courbes de genre 7 avec beaucoup de points rationnels sur des corps finis.

Published online:
DOI: 10.5802/jtnb.1019
Classification: 11N56,  14G42
Keywords: Hurwitz curve, automorphism group, jacobian, point counting
     author = {Jaap Top and Carlo Verschoor},
     title = {Counting points on the {Fricke{\textendash}Macbeath} curve over finite fields},
     journal = {Journal de Th\'eorie des Nombres de Bordeaux},
     pages = {117--129},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
     volume = {30},
     number = {1},
     year = {2018},
     doi = {10.5802/jtnb.1019},
     language = {en},
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Jaap Top; Carlo Verschoor. Counting points on the Fricke–Macbeath curve over finite fields. Journal de Théorie des Nombres de Bordeaux, Volume 30 (2018) no. 1, pp. 117-129. doi : 10.5802/jtnb.1019.

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