Entry details for q = 21 = 2, g = 39
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Lower bound Nmin = 33

Submitted by Gerrit Oomens
Date 01/01/1900
Reference Jean-Pierre Serre
Sur le nombre de points rationnels d'une courbe algébrique sur un corps fini
C. R. Acad. Sci. Paris Sér. I Math. 296 (1983), 397–402. (= Œuvres III, No. 128, 658–663).
Comments
Tags Methods from general class field theory

User comments

alternative defining equation     
Markus Grassl
04/14/2013 17:17
y^16
+ (x^18 + x^16 + x^12 + x^8 + x^6 + x^4 + 1)*y^8
+ (x^24 + x^23 + x^22 + x^20 + x^19 + x^16 + x^15 + x^14 + x^11 + x^10 + x^6 + x^4)*y^4
+ (x^26 + x^23 + x^19 + x^16 + x^15 + x^11 + x^10 + x^8)*y^2
+ (x^26 + x^24 + x^22 + x^20 + x^18 + x^16 + x^14 + x^12)*y
+ x^48 + x^44 + x^40 + x^36 + x^35 + x^34 + x^30 + x^26 + x^22 + x^19


optimal curve with sparser defining equation     
Isabel Pirsic
06/19/2012 16:32
y^16+
(x^32+x^28+x^24+x^22+x^21+x^20+x^18+x^14+x^13+x^12+
x^6+x^4+1)*y^8+
(x^40+x^38+x^37+x^35+x^33+x^32+x^31+x^30+
x^27+x^26+x^25+x^23+x^22+x^20+x^19+x^16+x^14+x^13+x^11+x^8+x^6+x^4)*y^4+
(x^42+x^41+x^38+x^37+x^35+x^33+x^32+x^31+
x^30+x^29+x^28+x^24+x^21+x^20+x^18+x^16+x^15+x^13+x^11+x^8)*y^2+
(x^42+x^41+x^40+x^32+x^29+x^27+x^26+x^25+x^23+x^20
+x^19+x^15+x^13+x^12)*y+
x^76+x^72+x^68+x^64+x^59+x^58+x^57+x^56+x^55+x^53+x^52+x^51+x^49+x^48+x^44+x^40+x^38+x^36+x^35+x^33+x^32+x^30+x^28+x^27+x^25+x^23+x^21+x^18

Ref:
G. van der Geer
Hunting for curves over finite fields with many points
arXiv:0902.3882
Defining equation     
Isabel Pirsic
06/18/2012 13:13
x^36*y^16 + (x^30 + x^29)*y^15 + (x^31 + x^29 + x^28 + x^25)*y^14 + (x^30 + x^28 + x^26 + x^25 + x^24 + x^23 +
x^22 + x^19)*y^13 + (x^30 + x^29 + x^27 + x^26 + x^23 + x^19 + x^17 + x^16)*y^12 + (x^29 + x^27 + x^26 +
x^24 + x^21 + x^17 + x^15 + x^12)*y^11 + (x^31 + x^30 + x^28 + x^27 + x^26 + x^21 + x^18 + x^17 + x^15 +
x^13 + x^12 + x^10 + x^9 + x^8)*y^10 + (x^27 + x^26 + x^22 + x^21 + x^16 + x^14 + x^13 + x^12 + x^10 + x^8
+ x^6 + x^5 + x^4 + x^3)*y^9 + (x^34 + x^33 + x^32 + x^30 + x^27 + x^25 + x^20 + x^19 + x^17 + x^15 + x^14
+ x^12 + x^10 + x^8 + x^7 + x^4 + x^3 + x)*y^8 + (x^29 + x^27 + x^21 + x^20 + x^19 + x^15 + x^11 + x^10 +
x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + 1)*y^7 + (x^29 + x^25 + x^24 + x^22 + x^21 + x^20 + x^18 + x^17
+ x^15 + x^14 + x^9 + x^5 + x^2 + 1)*y^6 + (x^28 + x^26 + x^22 + x^20 + x^18 + x^17 + x^15 + x^13 + x^12 +
x^10 + x^8 + x^6 + x^5 + x^3)*y^5 + (x^32 + x^29 + x^27 + x^25 + x^24 + x^23 + x^20 + x^19 + x^16 + x^15 +
x^14 + x^12 + x^11 + x^7 + x^3 + x^2)*y^4 + (x^27 + x^21 + x^18 + x^17 + x^16 + x^14 + x^13 + x^12 + x^7 +
x^6)*y^3 + (x^30 + x^29 + x^27 + x^26 + x^24 + x^22 + x^21 + x^20 + x^18 + x^16 + x^14 + x^12 + x^10 +
x^8)*y^2 + (x^27 + x^26 + x^25 + x^24 + x^23 + x^22 + x^21 + x^19 + x^18 + x^17 + x^16 + x^15)*y + x^35
Upper bound Nmax = 33

Submitted by Gerrit Oomens
Date 01/01/1900
Reference Jean-Pierre Serre
Rational points on curves over finite fields
Notes by Fernando Q. Gouvêa of lectures at Harvard University, 1985.
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