manYPoints – Table of Curves with Many Points
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).
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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. With contributions by Everett Howe, Joseph Oesterlé and Christophe Ritzenthaler. Edited by Alp Bassa, Elisa Lorenzo García, Christophe Ritzenthaler and René Schoof
Documents Mathématiques 18, Société Mathématique de France, Paris, 2020
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