Entry details for q = 21 = 2, g = 7
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Lower bound Nmin = 10

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

User comments

Explicit Curve     
S.E.Fischer
11/27/2014 23:10
We can construct such a curve C as an deg2 extension of a curve C' of genus 2 as follows:

C': y^3 *( x^3 + x ) + y * (x^3 + x^2 + x) + 1
C/C': z^2 + z + x^3 + x

This information, appropriately provided for MAGMA, reads as:

> R<x> := FunctionField(GF(2));
> P<y> := PolynomialRing(R);
> S<z> := PolynomialRing(P);
FF := FunctionField( y^3 *( x^3 + x ) + y * (x^3 + x^2 + x) + 1 );
> FF;
F := ( ext < FF | z^2 + z + x^3 + x >);
F;
print "Genus =", Genus(F);
print "NoP =", #Places(F,1);
print Places(F,1);
Upper bound Nmax = 10

Submitted by Gerrit Oomens
Date 01/01/1900
Reference Jean-Pierre Serre
Rational points on curves over finite fields
Notes by Fernando Q. Gouvêa of lectures at Harvard University, 1985.
Comments
See the table at the beginning of the lecture notes, and the discussion on pages SeTh 38–40.
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