Entry details for q = 51 = 5, g = 9
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Lower bound Nmin = 32

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 Nmax = 32

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
Tags Oesterlé bound

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