Entry details for q = 21 = 2, g = 5
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Lower bound Nmin = 9

Submitted by Gerrit Oomens
Date 01/01/1900
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
Tags Methods from general class field theory

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Explicit Curve     
S.E.Fischer
02/27/2015 15:07
We can assume such a curve C as an extension E of degree 2 of a curve F of genus 2 as follows:
F:= y^3 * (x^3 + x) + y^2 * (x^3 + x^2 + x) + 1 ;
E/F := z^2 + z + x^3 + x.
Defining equation     
Isabel Pirsic
06/18/2012 11:21
E.g.,

y4([0],[2,1,0],[2,1],[7,3]) =

y^4 + (x^2+x+1)*y^2 + (x^2+x)*y + x^7+ x^3
Upper bound Nmax = 9

Submitted by Everett Howe
Date 04/14/2010
Reference Jean-Pierre Serre
Rational points on curves over finite fields
Notes by Fernando Q. Gouvêa of lectures at Harvard University, 1985.
Comments
The Oesterlé bound. Here is a real Weil polynomial that we don't know how to eliminate: x * (x + 2)^2 * (x^2 + 2*x - 2)
Tags Oesterlé bound

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