manYPoints – Table of Curves with Many Points
Entry details for q = 311 = 31, g = 3
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Lower bound Nmin = 56

Submitted by C. Ritzenthaler
Date 03-20-2009
Reference Roland Auer, Jaap Top
Some genus 3 curves with many points
Algorithmic number theory (Sydney, 2002), 163–171, Lecture Notes in Comput. Sci., 2369, Springer, Berlin, 2002
Comments
This was found explicitly as the curve x^4+y^4+z^4= 5*(x^2*y^2+x^2*z^2+y^2*z^2). This curve has 24 automorphisms and its Jacobian is isogenous to the cube of the elliptic curve y^2=x*(x-1)*(x-4) which has trace -8.
Tags Explicit curves

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Upper bound Nmax = 62

Submitted by Everett Howe
Date 06-11-2010
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 upper bound follows from Theorem 1 (p. 89) of the cited reference, because 31 = 5^2 + 5 + 1.
Tags None

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