Entry details for q = 26 = 64, g = 5
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Lower bound Nmin = 140

Submitted by Everett Howe
Date 05/21/2010
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 be an element of F_8 such that s^3 + s + 1 = 0. Let D be the plane quartic defined by
(s*x^2 + s*y^2 + s^3*z^2 + s^3*x*y + s^3*y*z + s^3*x*z)^2 + x*y*z*(x + y + z) = 0,
and let C be the double cover of D defined by
w^2 + w*z = s*x*z + s^2*y*z.
Then the real Weil polynomial of C over F_8 is (t + 1)^5, and C has 140 points over F_64.

The Magma routines in the file IsogenyClasses.magma (available at the URL above) show that there are no genus-5 curves over F_64 that have 141, 142, 143, or 144 points. Thus, N_64(5) is equal to either 140 or 145.
Tags Explicit curves

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Upper bound Nmax = 145

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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