manYPoints – Table of Curves with Many Points
Entry details for q = 35 = 243, g = 3
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Lower bound Nmin = 334

Submitted by C. Ritzenthaler
Date 11-25-2009
Reference Not available
Comments
This number of point is reached by the curve

a^49*t1^4 + a*t1*t3 + t2^2=0

where

t1:=X+Y+Z;t2:=X*Y+Y*Z+Z*X;t3:=X*Y*Z;

and

a is a root of u^5-u^3+u^2+1=0 in F_{3^5}.

This curve has been found thanks to Mestre's preprint

"courbes de genre 3 avec S_3 comme groupe d'automorphismes".
Tags Explicit curves

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Upper bound Nmax = 334

Submitted by C. Ritzenthaler
Date 11-27-2009
Reference Kristin Lauter
The Maximum or Minimum Number of Rational Points on Genus Three Curves over Finite Fields
Compositio Mathematica 134 87-111 (2002)
Comments
This is Th.1 p.89 of the paper, since 15^2+15+3=3^5.
Tags None

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