manYPoints – Table of Curves with Many Points
Entry details for q =
2
1
= 2
, g =
48
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Lower bound
N
min
= 35
Earlier entry
Submitted by
Karl Rökaeus
Date
04-25-2012
Reference
Not available
Comments
Let F be the hyperelliptic genus 2 field given by
y^2 + x*y + x^5 + x.
Using the notation of Magma, let D be the divisor
(x^4 + x + 1, y + x^3 + x^2) +
(x^4 + x + 1, y + x^3 + x^2 + x) + 2*(1/x, y/x^3).
(The first two places in the support of D has degree 4, the last is rational.)
Let S be the set { (x, y), (x + 1, y + x + 1),
(x + 1, y + 1), (x^2 + x + 1)}.
(The first three places in S are rational, the last has degree 4.)
Then F^D_S, the largest abelian extension of F with conductor <=D such that all places in S split completely,
has genus 48 and 35 rational places. This can easily be verified using Magma.
Tags
Methods from general class field theory
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Upper bound
N
max
= 39
Earlier entry
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
Tags
Oesterlé bound
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