manYPoints – Table of Curves with Many Points
Entry details for q = 311 = 31, g = 3
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Lower bound Nmin = 62

Submitted by C. Ritzenthaler
Date 03-20-2009
Reference Jaap Top
Curves of genus 3 over small finite fields
Indag. Math. (N.S.) 14 (2003), no. 2, 275–283
Comments
This value is reached by the curve
x^3*z+y^3*z+x^2*y^2+4*x*y*z^2+2*z^4=0 which has 6 automorphisms. Its Jacobian is isogenous to the cube of an elliptic curve with trace -10.
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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