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| Submitted by |
Gerrit Oomens |
| Date |
01-01-1900 |
| Reference |
Jean-Pierre 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).
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| Tags |
Methods from general class field theory |
User comments
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Explicit equation
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Marc Masdeu
07-27-2021 08:56
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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).
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