manYPoints

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

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

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. Here is a real Weil polynomial that we don't know how to eliminate: (x^2 + 4x + 2)^2 (x^7 + 10x^6 + 35x^5 + 42x^4 - 26x^3 - 99x^2 - 67x - 9)
Tags	Oesterlé bound

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