manYPoints – Table of Curves with Many Points
Entry details for q =
11
4
= 14641
, g =
3
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Lower bound
N
min
= 15368
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
Comments
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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User comments
Explicit example
Everett Howe
05-20-2010 23:50
One such hyperelliptic curve is n*y^2 = x^7 + 5*x^5 + 5*x^3 + x, where n is any element of F_{11^4} that is not a square.
Upper bound
N
max
= 15368
Earlier entry
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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