manYPoints – Table of Curves with Many Points
Entry details for q = 731 = 73, g = 3
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Lower bound Nmin = 122

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 point is reached by the curve C: x^3*z+y^3*z+x^2*y^2+a*x*y*z^2+b*z^4=0 with a=2 and b=48.

The Jacobian of C is isogenous to E^3 with trace(E)=-16 and its automorphism group is S_3 (over F_q and geometric).
Tags Explicit curves

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Upper bound Nmax = 122

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 73 = 8^2 + 8 + 1.
Tags None

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