Submitted by 
Gerrit Oomens 
Date 
01/01/1900 
Reference 
JP. Serre Nombre de points des courbes algébriques sur F_q Sém. de Théorie des Nombres de Bordeaux, 1982/83, exp. no. 22. (= Oeuvres III, No. 129, p. 664668).

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Tags 
Explicit curves 
User comments
Explicit Curve

S.E.Fischer
12/26/2014 00:31

We can assume such a curve C as an extension E of degree 3 of a curve F of genus 1 with eight rational points as follows:
F:= y^2 +y +x + 1/x ;
E/F := z^3*x^2 +y +1 .

Explicit example

Everett Howe
05/20/2010 21:51

In his Harvard notes, Serre notes that a model of the Klein curve gives an example of a genus3 curve with 14 points:
(x + y + z)^4 + (x*y + x*z + y*z)^2 + x*y*z*(x + y + z) = 0


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