manYPoints – Table of Curves with Many Points
Entry details for q = 371 = 37, g = 3
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Lower bound Nmin = 72

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:
(s-1)*(s*x^4+b*y^4+b*z^4+s*a*x^2*y^2+s*a*x^2*z^2)-(s*a^2-2*b*(s+1))*y^2*z^2=0 with s=7,a=0 and b=2.

It was constructed using Prop.15 of Howe, E.W., F. Leprevost and B. Poonen : Large torsion subgroups of split Jacobians of
curves of genus two or three. Forum Math. 12,315-364 (2000).

The Jacobian of C is isogenous to E1*E2^2 with trace(E1)=-10 and trace(E2)=-12 and its automorphism group has order 16.
Tags Explicit curves

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Upper bound Nmax = 72

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 2 (p. 89) of the cited reference, because 37 = 6^2 + 1.
Tags None

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