Entry details for q = 21 = 2, g = 48
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Lower bound Nmin = 35

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 Nmax = 39

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
Tags Oesterlé bound

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