manYPoints

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: x^4+y^4+z^4+2a(x^2y^2+y^2z^2+z^2*x^2)=0 with a=27. This curve was found by M. Nitzberg (see loc. cit. p.72) but is not given there. See Top : Curves of genus 3 over small finite fields for the equation. Its Jacobian is isogenous to E^3 with trace(E)=-12 and the automorphism group of C is S_4.
Tags	Explicit curves

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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 43 = 6^2 + 6 + 1.
Tags	None

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