Entry details for q = 194 = 130321, g = 3
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Lower bound Nmin = 132488

Submitted by C. Ritzenthaler
Date 11-15-2009
Reference Tomoyoshi Ibukiyama
On rational points of curves of genus 3 over finite fields
Tohoku Math. J. (2) 45 (1993) 311–329
This number is reached by a supersingular hyperelliptic curve of genus 3. This is proved using the class number formula of ternary quaternion Hermitian forms by Hashimoto.
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Explicit example     
Everett Howe
05-21-2010 00:08
If n is a nonsquare in F_{19^4}, then the hyperelliptic curve n*y^2 = x^7 - x has 132488 points.
Upper bound Nmax = 132488

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).
The Hasse-Weil-Serre bound
Tags Hasse-Weil-Serre bound

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