manYPoints – Table of Curves with Many Points
Entry details for q =
3
1
= 3
, g =
23
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Lower bound
N
min
= 33
Earlier entry
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
User comments
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Upper bound
N
max
= 37
Earlier entry
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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