manYPoints – Table of Curves with Many Points
Entry details for q =
19
5
= 2476099
, g =
2
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Lower bound
N
min
= 2482394
Submitted by
M. Afzal Soomro
Date
07/07/2010
Reference
Howe, Everett W.; Leprévost, Franck; Poonen, Bjorn
Large torsion subgroups of split Jacobians of curves of genus two or three
Forum Math. 12 (2000), no. 3, 315–364
Comments
This number of point is reached by the curve
y^2=(5*a^4 + 3*a^3 + 5*a^2 + 10*a + 9)*x^6 + (14*a^4 + 7*a^3 + 6*a^2 + 8*a + 8)*x^4 + (9*a^4 + 8*a^3 + 14*a^2 + a + 3)*x^2 + 15*a^4 + 9*a^3 + 5*a^2 + 14*a + 7,
where a satisfies the minimal polynomial X^5+5*X+17 in F_19[X].
By Corollary 6 and Proposition 3 and 4, we can get explicit equation of maximal curve of genus 2.
Tags
Explicit curves, Formulas for N_q(1) and N_q(2)
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Upper bound
N
max
= 2482394
Submitted by
Everett Howe
Date
05/03/2010
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, Formulas for N_q(1) and N_q(2)
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