Entry details for q = 175 = 1419857, g = 2
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Lower bound Nmin = 1424624

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 points is reached by the curve

y^2=y^2= (15*a^4 + 5*a^3 + 4*a^2 + 12*a + 5)*x^6 + (12*a^4 + 13*a^3 +14*a^2 + 16*a)*x^4 + (7*a^4 + 7*a^3 + 2*a^2 + 12*a + 9)*x^2 + 13*a^4 + 10*a^3 + 13*a^2 + 13*a + 11

where a satisfies the minimal polynomial X^5+X+14 in F_17[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 Nmax = 1424624

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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