manYPoints – Table of Curves with Many Points
Entry details for q =
5
1
= 5
, g =
9
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Lower bound
N
min
= 32
Earlier entry
Submitted by
Karl Rökaeus
Date
02/22/2011
Reference
Not available
Comments
The hyperelliptic curve C of genus 2 given by
y^2=x^5-x^3+x
has class group isomorphic to Z/8*Z/8. The points \infty, (0,0), (4,3) and (4,2) on C map to (0,4), 0, (0,5) and (0,3) in the class group, hence to a subgroup of index 8. By class field theory, C has an unramified cover of degree 8 in which these points split completely. This cover therefore has genus 1+8*(2-1)=9 and at least 8*4=32 rational points (in fact exactly 32 since this was already known to be an upper bound).
Tags
Methods from general class field theory
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Upper bound
N
max
= 32
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 + 2)^5 * (x + 4)^4
Tags
Oesterlé bound
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