manYPoints – Table of Curves with Many Points
Entry details for q = 21 = 2, g = 3
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Lower bound Nmin = 7

Submitted by Everett Howe
Date 05-20-2010
Reference L. E. Dickson
Quartic curves modulo 2
Trans. Amer. Math. Soc. 16 (1915) 111–120.
Comments
Dickson gives an explicit quartic curve with 7 points:
x^3*y + x^2*y^2 + x*z^3 + x^2*z^2 + y^3*z + y*z^3 = 0
Tags Explicit curves

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Upper bound Nmax = 7

Submitted by Everett Howe
Date 05-20-2010
Reference L. E. Dickson
Quartic curves modulo 2
Trans. Amer. Math. Soc. 16 (1915) 111–120.
Comments
Dickson uses without comment the fact that the number of points on a plane quartic over F_2 is at most the number of points in the projective plane over F_2, which is 7. A hyperelliptic curve over F_2 can have at most 6 points, or else its quadratic twists would have a negative number of points.
Tags Oesterlé bound

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