manYPoints

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 64=24 rational points.
Tags	Methods from general class field theory

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

Submitted by	Everett Howe
Date	04-14-2010
Reference	Everett W. Howe, 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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