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
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
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 + 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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