Entry details for q = 51 = 5, g = 7
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Lower bound Nmin = 24

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^2-x
has class group isomorphic to Z/60. The points \infty, (0,0), (4,4) and (4,1) on C map to 30, 0, 18 and 42 in the class group, hence to a subgroup of index 6. By class field theory, C has an unramified cover of degree 6 in which these points split completely. This cover therefore has genus 1+6*(2-1)=7 and at least 6*4=24 rational points.
Tags Methods from general class field theory

User comments

Alternate curve     
Everett Howe
08/24/2016 20:25
Another example is the curve defined by
y^2 = x^6 + 4*x^4 + 1
z^2 = x^6 + 4*x^2 + 4,
which also has 24 points.
Upper bound Nmax = 26

Submitted by Everett Howe
Date 04/14/2010
Reference Everett W. Howe and Kristin E. Lauter
New methods for bounding the number of points on curves over finite fields
Geometry and Arithmetic (C. Faber, G. Farkas, and R. de Jong, eds.), European Mathematical Society, 2012, pp. 173–212
Comments
A real Weil polynomial we don't know how to eliminate: (x + 1) * (x + 3) * (x + 4) * (x^2 + 6*x + 7)^2
Tags None

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