manYPoints – Table of Curves with Many Points
Entry details for q = 591 = 59, g = 3
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Lower bound Nmin = 102

Submitted by C. Ritzenthaler
Date 06-01-2009
Reference Jaap Top
Curves of genus 3 over small finite fields
Indag. Math. (N.S.) 14 (2003), no. 2, 275–283
Comments
This number of points is reached by the curve C : (3*x^2+y^2)^2+a*x*(x^2-y^2)*z+b*z^4=0 with a=4 and b=6.

Its Jacobian is isogenous to E^3 with trace(E)=-14. Its automorphism group is isomorphic to S_3.
Tags Explicit curves

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Upper bound Nmax = 102

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 59 = 7^2 + 7 + 3.
Tags None

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