manYPoints – Table of Curves with Many Points
Entry details for q = 31 = 3, g = 11
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Lower bound Nmin = 21

Submitted by Virgile Ducet
Date 05/21/2012
Reference V. Ducet, C. Fieker
Computing Equations of Curves with Many Points
pp. 317–334 in: ANTS X—Proceedings of the Tenth Algorithmic Number Theory Symposium, Open Book Ser., 1, Math. Sci. Publ., Berkeley, CA, 2013
Comments
The curve has equation:

y^8 + (2*x^10 + x^9 + 2*x^7 + x^6 + 2*x^5 + 2*x^4 + x^3 + x^2)*y^6 + (2*x^20 + 2*x^19 + 2*x^18 + x^17 + x^16 + x^15 + 2*x^14 + 2*x^12 + 2*x^11 + x^10 + 2*x^9 + 2*x^8 + x^7 + 2*x^6 + x^4)*y^4 + (x^30 + 2*x^28 + x^24 + x^23 + 2*x^22 + x^21 + x^20 + 2*x^19 + x^18 + x^17 + 2*x^16 + x^15 + x^14 + 2*x^13 + x^11 + x^9 + x^8 + 2*x^7)*y^2 + x^40 + x^39 + 2*x^37 + x^35 + 2*x^32 + x^31 + x^30 + x^29 + 2*x^28 + 2*x^22 + 2*x^21 + x^19 + 2*x^17 + x^14 + 2*x^13 + 2*x^12 + 2*x^11 + x^10
Tags Methods from general class field theory

User comments

sparser defining equation     
Isabel Pirsic
06/01/2012 12:37
Just wanted to note that y/x has a slightly sparser minimal polynomial,
85 monomials instead of 105:

y^8 + (2*x^8 + x^7 + 2*x^5 + x^4 + 2*x^3 + 2*x^2 + x + 1)*y^6 + (2*x^16 + 2*x^15 + 2*x^14 + x^13 + x^12 + x^11 + 2*x^10 + 2*x^8 + 2*x^7 + x^6 + 2*x^5 + 2*x^4 + x^3 + 2*x^2 + 1)*y^4 + (x^24 + 2*x^22 + x^18 + x^17 + 2*x^16 + x^15 + x^14 + 2*x^13 + x^12 + x^11 + 2*x^10 + x^9 + x^8 + 2*x^7 + x^5 + x^3 + x^2 + 2*x)*y^2 + x^32 + x^31 + 2*x^29 + x^27 + 2*x^24 + x^23 + x^22 + x^21 + 2*x^20 + 2*x^14 + 2*x^13 + x^11 + 2*x^9 + x^6 + 2*x^5 + 2*x^4 + 2*x^3 + x^2
Upper bound Nmax = 22

Submitted by Everett Howe
Date 04/14/2010
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
Comments
The Oesterlé bound. Here is a real Weil polynomial that we don't know how to eliminate: (x^2 + 4*x + 2)^2 * (x^7 + 10*x^6 + 35*x^5 + 42*x^4 - 26*x^3 - 99*x^2 - 67*x - 9)
Tags Oesterlé bound

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