manYPoints – Table of Curves with Many Points
Entry details for q =
5
1
= 5
, g =
12
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Lower bound
N
min
= 33
Earlier entry
Later entry
Submitted by
Karl Rökaeus
Date
02/22/2011
Reference
Not available
Comments
The hyperelliptic curve of genus 2 given by
y^2=x^5+x^4+3*x^2+1
has class group isomorphic to Z/55, in which \infty, (0,1) and (0,-1) map to 44, 0 and 33. By class field theory it has an unramified cover of degree 11 in which these points split completely. This cover therefore has genus 1+(2-1)*11=12 and at least 3*11=33 rational points.
Tags
Methods from general class field theory
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Upper bound
N
max
= 38
Earlier entry
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 + 3)^4 * (x^2 + 2*x - 1) * (x^2 + 6*x + 7)^3
Tags
Oesterlé bound
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