manYPoints – Table of Curves with Many Points
Entry details for q =
3
3
= 27
, g =
11
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Lower bound
N
min
= 104
Earlier entry
Submitted by
Everett Howe
Date
06/12/2023
Reference
Not available
Comments
Let r be an element of GF(27) with r^3 - r + 1 = 0 and let
f := (x^2 + r*x + 2) * (x^2 + r^9*x + r^3) * (x^2 + r^25*x + 2);
g1 := x^4 + r^9*x^3 + r^17*x^2 + r^16*x + r^17;
g2 := x^4 + r^12*x^3 + r^10*x^2 + r*x + r^23;
Then the curve defined by
y^2 = f*g1
z^2 = f*g2
has genus 11 and has 104 points.
This example was found by searching through curves of the given form.
Tags
Analysis and enumeration, Towers of curves with many points
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Upper bound
N
max
= 133
Earlier entry
Submitted by
Everett Howe
Date
04/14/2010
Reference
E. W. Howe, K. E. Lauter
Improved upper bounds for the number of points on curves over finite fields
Ann. Inst. Fourier (Grenoble) 53 (2003) 1677–1737.
Comments
A real Weil polynomial we don't know how to eliminate: (x + 5) * (x + 10)^10
Tags
None
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