Entry details for q = 34 = 81, g = 5
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Lower bound Nmin = 167

Submitted by Everett Howe
Date 04/16/2010
Reference Everett W. Howe and Kristin E. Lauter
New methods for bounding the number of points on curves over finite fields
Geometry and Arithmetic (C. Faber, G. Farkas, and R. de Jong, eds.), European Mathematical Society, 2012, pp. 173–212
Comments
Let s in F_9 satisfy s^2 = s + 1, and let E be the elliptic curve y^2 = x^3 + s*x^2 + s^5. Let C be the degree-5 Kummer cover of E defined by z^5 = ((s^3*x + s^5)*y + (s^5*x^3 + s^3*x - 1))/(x + s^3). Then C is a genus-5 curve with real Weil polynomial equal to (x - 1)^2 * (x + 1)^3, and C has 167 points over F_81.

This curve was found by a computer search of degree-5 Kummer extensions of E of genus 5.
Tags Explicit curves

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Upper bound Nmax = 172

Submitted by Everett Howe
Date 04/14/2010
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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