Entry details for q = 31 = 3, g = 23
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Lower bound Nmin = 33

Submitted by Markus Grassl
Date 04/18/2013
Reference Markus Grassl
calculations for a curve over GF(3) with genus 23 and 33 rational points using Magma
Comments
The curve has been found the algorithms for ray class fields in Magma, together with an explicit equation.
Using various randomised techniques, the following somewhat sparser defining equation has been found:

y^22 + (x+2)*y^21 + (2*x^6+x^5+2*x^4+x^3+2*x+1)*y^19 + (2*x^7+2*x^6+x^5+2*x^4+2*x^3+2*x^2+2*x+2)*y^18
+ (2*x^10+2*x^5+x^4+2*x^3+2*x)*y^17 + (x^12+x^8+x^7+x^5+x^3+2)*y^16
+ (x^12+2*x^11+2*x^10+2*x^8+2*x^5+x^4+x^3+x^2+2*x+1)*y^15+ (2*x^12+2*x^10+x^9+x^8+x^6+x^4+x^2)*y^14
+ (2*x^17+2*x^16+2*x^15+2*x^10+2*x^5+x^3+x^2+x+2)*y^13
+ (2*x^20+x^19+x^17+2*x^15+2*x^14+x^12+2*x^8+x^4+2*x^3+2*x^2+x+1)*y^12
+ (x^22+x^21+x^19+x^18+2*x^17+2*x^12+x^11+x^10+x^9+x^8+x^6+2*x^5+x^3+x^2+x)*y^11
+ (2*x^24+2*x^22+2*x^21+2*x^20+x^19+2*x^18+2*x^17+x^16+x^14+x^13+x^10+2*x^8+2*x^5+2*x^4+x^3+x^2)*y^10
+ (x^25+x^21+x^20+2*x^18+x^17+2*x^15+x^14+x^11+2*x^10+2*x^9+2*x^8+x^7+2*x^5+2*x^4)*y^9
+ (x^28+2*x^23+x^21+2*x^20+2*x^19+x^17+x^14+x^6+x^5)*y^8
+ (2*x^28+2*x^26+2*x^25+x^24+2*x^23+2*x^22+x^21+x^19+x^16+x^15+x^14+x^13+x^11)*y^7
+ (x^32+x^30+2*x^27+x^26+2*x^24+2*x^23+x^22+x^21+x^19+x^16+x^15+x^13+x^11+2*x^10+x^8+2*x^6)*y^6
+ (2*x^34+x^32+x^31+x^30+2*x^29+x^27+x^26+x^25+2*x^24+x^23+x^22+2*x^19+2*x^17+2*x^16+x^15+x^14+2*x^13+2*x^10+x^9+x^8+2*x^7)*y^5
+ (2*x^36+x^35+2*x^33+2*x^32+2*x^31+2*x^30+x^29+2*x^26+x^24+x^23+2*x^22+x^21+x^20+2*x^19+x^17+2*x^16+2*x^15+2*x^14+x^13+2*x^12+2*x^11+2*x^8)*y^4
+ (2*x^38+x^34+x^32+2*x^30+x^27+x^26+x^24+2*x^23+x^21+2*x^20+x^19+2*x^17+2*x^15+x^14+x^13+2*x^12+x^11)*y^3
+ (x^39+x^38+x^34+x^33+x^31+x^29+x^27+2*x^26+2*x^25+x^23+x^22+2*x^21+x^19+2*x^18+x^17+2*x^15)*y^2
+ (x^42+2*x^40+x^38+x^37+2*x^36+x^35+2*x^34+2*x^33+x^32+x^31+2*x^30+2*x^29+2*x^28+2*x^26+x^25+x^23+x^21+2*x^20+x^18+2*x^17)*y
+ 2*x^44+x^43+2*x^42+2*x^41+2*x^38+x^36+x^35+2*x^33+x^31+x^29+x^28+x^26+2*x^23+x^21+x^20

The added reference mainly serves for verification of the data.
Tags Explicit curves, Methods from general class field theory

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Upper bound Nmax = 37

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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