manYPoints – Table of Curves with Many Points
Entry details for q = 21 = 2, g = 10
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Lower bound Nmin = 13

Submitted by Gerrit Oomens
Date 01-01-1900
Reference Jean-Pierre Serre
Letter to G. van der Geer
September 1, 1997.
Comments
Tags Methods from general class field theory

User comments

Correction     
S.E.Fischer
12-20-2014 03:39
The extension E/F leads to a curve of genus 4 with 8 points. Extending once again by H of degree 2 leads to the desired result, as we write
H/(E/F):= v^2 + v*x + x^3 + x .
Explicit Curve     
S.E.Fischer
12-18-2014 15:27
We can assume such a curve C as an extension E of degree 2 of a curve F of genus 1 as follows:
F:= (x + y + x*y) * x*y + x + 1
E/F := z^2 * x + z + x^2 * y^2
My reference corrected     
Isabel Pirsic
06-19-2012 16:18
G. van der Geer
Hunting for curves over finite fields with many points
arXiv:0902.3882
Defining equation     
Isabel Pirsic
06-19-2012 14:17
y^8 +
(x^4 + x + 1)*y^4 +
(x^2 + x)*y^2 +
(x^4 + x^2)*y +
(x^12 + x^4)

Ref: G. van der Geer, M. van der Vlugt
How to construct curves over finite fields with many points
In: Arithmetic Geometry, (Cortona 1994), F. Catanese Ed.,
Cambridge Univ. Press, Cambridge, 1997, p. 169-189.
Upper bound Nmax = 13

Submitted by Everett Howe
Date 04-14-2010
Reference Jean-Pierre Serre
Rational points on curves over finite fields. With contributions by Everett Howe, Joseph Oesterlé and Christophe Ritzenthaler. Edited by Alp Bassa, Elisa Lorenzo García, Christophe Ritzenthaler and René Schoof
Documents Mathématiques 18, Société Mathématique de France, Paris, 2020
Comments
The Oesterlé bound. Here is a real Weil polynomial that we don't know how to eliminate: (x + 1) * (x + 2) * (x^2 + 2*x - 1) * (x^6 + 5*x^5 + 2*x^4 - 18*x^3 - 14*x^2 + 11*x + 1)
Tags Oesterlé bound

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