manYPoints

Submitted by	C. Ritzenthaler
Date	02/23/2009
Reference	Howe, Everett W.; Leprévost, Franck; Poonen, Bjorn Large torsion subgroups of split Jacobians of curves of genus two or three Forum Math. 12 (2000), no. 3, 315–364
Comments	The curve C : (6a+2a^2)(X^4+Y^4+Z^4)+(3+4a^2)(X^2Y^2 + X^2Z^2+ Y^2Z^2)=0 where a is a root of x^3+6x^2+4=0 over F_7 reaches this number of rational points. It was constructed using the loc. cit. article with the elliptic curve E: y^2 + xy = x^3 + a^213x^2 + a^79x + a^90. Hence the Jacobian of C is isomorphic to a quotient of E^3 by a (2-2-2) rational subgroup. The automorphism group of C is S_4.
Tags	Explicit curves

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Submitted by	C. Ritzenthaler
Date	02/23/2009
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 is Th.1 p.89 of the paper, since 18^2+18+1=7^3. Actually, defect 1 and 2 are eliminated through standard techniques when for defect 0, this comes from the deeper fact that there is no unimodular definite positive hermitian matrix of rank 3 over Z[(1+Sqrt(-3))/2].
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