Entry details for q = 411 = 41, g = 3
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Lower bound Nmin = 78

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^2*y^2 + z^2*(x^2+y^2)+a*(x^3*y+y^3*z+z^3*x)+b*(x^3*z+y^3*x+z^3*y)=0 with a=b=22.

Its Jacobian is isogenous to E^3 with trace(E)=-12 and S_3 is contained in the automorphism group of C.

Note that another curve is given in loc. cit. but this curve is not irreducible. The present one comes from a personal correspondence with Top.
Tags Explicit curves

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Upper bound Nmax = 78

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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