Entry details for q = 75 = 16807, g = 3
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Lower bound Nmin = 17582

Submitted by C. Ritzenthaler
Date 11/15/2009
Reference Not available
This number is reached bu the following curve




where a is a root of u^5+u-3=0 in F_{7^5}.

This curve has been found by J.-F. Mestre using his preprint

"courbes de genre 3 avec S_3 comme groupe d'automorphismes".
Tags Explicit curves

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Upper bound Nmax = 17582

Submitted by C. Ritzenthaler
Date 11/28/2009
Reference Kristin Lauter
Geometric methods for improving the upper bounds on the number of rational points on algebraic curves over finite fields
J. Algebraic Geom. 10 (2001) 19–36
Since m:=Floor(2*Sqrt(q)) is divisible by p=7 there does not exist an elliptic curve with trace -m over F_q. One can also check that (x^2+m*x+q)^3 cannot be the Weil polynomial of a simple abelian variety because this implies that q is a cube (see Prop. 2.5 in Maisner, Nart : Abelian surfaces over finite fi elds as Jacobians. With an appendix by Everett W. Howe, Exp. Math. 11 (2002), 321–337). So defect 0 is impossible.

Defect 1 is impossible as well due to the "resultant 1" method.

Now one can check that all the possibilities of Table 1 in the article for defect 2 are excluded (some because p divides m, some because of the "resultant 1" method and the last one because of the value of the fractional part of 2*Sqrt(q)).
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