Entry details for q = 21 = 2, g = 12
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Lower bound Nmin = 14

Submitted by Gerrit Oomens
Date 01/01/1900
Reference J-P. Serre
Nombre de points des courbes algébriques sur F_q
Sém. de Théorie des Nombres de Bordeaux, 1982/83, exp. no. 22. (= Oeuvres III, No. 129, p. 664-668).
Comments
Tags Methods from general class field theory

User comments

Explicit equation     
Marc Masdeu
07/27/2021 08:56
Let L be the Carlitz extension over F2(t) attached to M = (t^3 + t + 1) * (t^3 + t^2 + 1). It has genus 78. The subfield of L fixed by H=<t> has genus 12, and sequence (d, a_d) of numbers of places

(1, 14), (2, 0), (3, 2), (4, 0), (5, 0), (6, 0), (7, 29), (8, 28), (9, 42), (10, 119), (11, 196), (12, 315), ...

A defining equation for the curve attached to the subfield cut out by H is

x^7 + (t^6 + t^5 + t^4 + t^3 + 1)*x^6 + (t^10 + t^9 + t^7 + t^6 + t^5 + t^2 + t)*x^5 + (t^13 + t^12 + t^10 + t^9 + t^7 + t^6 + t^5 + t^3 + t)*x^4 + (t^14 + t^13 + t^12 + t^10 + t^9)*x^3 + (t^12 + t^9 + t^7 + t^5 + t^4 + t^2 + t)*x^2 + (t^8 + t^7 + t^4 + t^2 + t)*x + 1,

and its Weil polynomial is

(x^6 + 4*x^5 + 9*x^4 + 15*x^3 + 18*x^2 + 16*x + 8)^2 * (x^12 + 3*x^11 + 7*x^10 + 15*x^9 + 31*x^8 + 49*x^7 + 71*x^6 + 98*x^5 + 124*x^4 + 120*x^3 + 112*x^2 + 96*x + 64).
Upper bound Nmax = 15

Submitted by Everett Howe
Date 04/14/2010
Reference Jean-Pierre Serre
Rational points on curves over finite fields
Notes by Fernando Q. Gouvêa of lectures at Harvard University, 1985.
Comments
The Oesterlé bound. Here is a real Weil polynomial that we don't know how to eliminate: (x + 1)^2 * (x + 2)^2 * (x^2 - 2) * (x^2 + 2*x - 2)^3
Tags Oesterlé bound

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