Entry details for q = 471 = 47, g = 3
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Lower bound Nmin = 87

Submitted by C. Ritzenthaler
Date 06/25/2009
Reference Jaap Top
Curves of genus 3 over small finite fields
Indag. Math. (N.S.) 14 (2003), no. 2, 275–283
Comments
This number of points is reached by the curve C : x^4+(1/9)* y^4+(2/3)*x^2*y^2-190*y^2*z^2-570*x^2*z^2+(152/9)*y^3*z-152* x^2*y*z-1083*z^4=0.

Its Jacobian is isomorphic to E^3 with trace(E)=-13. The automorphism group of C contains S_3.

This curve is actually missing in loc. cit. but can be found at http://arxiv.org/abs/0901.2920.
Tags Explicit curves

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Upper bound Nmax = 87

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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