Entry details for q = 51 = 5, g = 12
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Lower bound Nmin = 33

Submitted by Karl Rökaeus
Date 02/22/2011
Reference Not available
Comments
The hyperelliptic curve of genus 2 given by
y^2=x^5+x^4+3*x^2+1
has class group isomorphic to Z/55, in which \infty, (0,1) and (0,-1) map to 44, 0 and 33. By class field theory it has an unramified cover of degree 11 in which these points split completely. This cover therefore has genus 1+(2-1)*11=12 and at least 3*11=33 rational points.
Tags Methods from general class field theory

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Upper bound Nmax = 38

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 + 3)^4 * (x^2 + 2*x - 1) * (x^2 + 6*x + 7)^3
Tags Oesterlé bound

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