Entry details for q = 711 = 71, g = 3
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Lower bound Nmin = 120

Submitted by C. Ritzenthaler
Date 06/01/2009
Reference Jean-Pierre Serre
Rational points on curves over finite fields. With contributions by Everett Howe, Joseph Oesterlé and Christophe Ritzenthaler. Edited by Alp Bassa, Elisa Lorenzo García, Christophe Ritzenthaler and René Schoof
Documents Mathématiques 18, Société Mathématique de France, Paris, 2020
Comments
This number of points is reached by the curve C: x^4+y^4+z^4+2*a*(x^2*y^2+y^2*z^2+z^2*x^2)=0 with a=36. This curve was found by M. Nitzberg (see loc. cit. p.72). See also Top : Curves of genus 3 over small finite fields.

Its Jacobian is isogenous to E^3 with trace(E)=-16 and the automorphism group of C is S_4.
Tags Explicit curves

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Upper bound Nmax = 120

Submitted by Everett Howe
Date 06/10/2010
Reference Jean-Pierre Serre
Sur le nombre de points rationnels d'une courbe algébrique sur un corps fini
C. R. Acad. Sci. Paris Sér. I Math. 296 (1983), 397–402. (= Œuvres III, No. 128, 658–663).
Comments
The Hasse-Weil-Serre bound
Tags Hasse-Weil-Serre bound

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