manYPoints

Submitted by	C. Ritzenthaler
Date	02-23-2009
Reference	Everett W. Howe, Franck Leprévost, Bjorn Poonen 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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