Entry details for q = 21 = 2, g = 24
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Lower bound Nmin = 23

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^16 + (x^18 + x^17 + x^15 + x^14 + x^10 + x^8)*y^12 + (x^28 + x^27 + x^26 + x^22 + x^21 + x^14)*y^10 + (x^31 + x^30 + x^29 + x^27 + x^26 + x^25 + x^23 + x^19)*y^9 + (x^48 + x^43 + x^39 + x^38 + x^37 + x^34 + x^33 + x^32 + x^31 + x^30 + x^29 + x^28 + x^27 + x^25 + x^23 + x^22 + x^21 + x^20 + x^18)*y^8 + (x^48 + x^47 + x^46 + x^40 + x^34 + x^33 + x^32 + x^26)*y^6 + (x^51 + x^50 + x^49 + x^43 + x^39 + x^38 + x^37 + x^31)*y^5 + (x^72 + x^71 + x^70 + x^69 + x^68 + x^67 + x^65 + x^59 + x^58 + x^56 + x^51 + x^49 + x^47 + x^45 + x^42 + x^41 + x^40 + x^38 + x^36 + x^34 + x^33 + x^31 + x^30 + x^28)*y^4 + (x^57 + x^56 + x^55 + x^48 + x^47 + x^41)*y^3 + (x^82 + x^81 + x^77 + x^76 + x^75 + x^73 + x^72 + x^69 + x^68 + x^67 + x^64 + x^61 + x^58 + x^57 + x^55 + x^54 + x^53 + x^50 + x^49 + x^48 + x^45 + x^44 + x^40 + x^39 + x^38 + x^37 + x^36 + x^34)*y^2 + (x^85 + x^84 + x^81 + x^77 + x^75 + x^74 + x^72 + x^71 + x^70 + x^69 + x^68 + x^65 + x^63 + x^62 + x^60 + x^59 + x^58 + x^57 + x^55 + x^54 + x^53 + x^50 + x^48 + x^46 + x^44 + x^42 + x^41 + x^39)*y + x^108 + x^97 + x^96 + x^95 + x^91 + x^90 + x^88 + x^87 + x^86 + x^83 + x^82 + x^81 + x^78 + x^76 + x^75 + x^72 + x^70 + x^69 + x^68 + x^64 + x^63 + x^58 + x^55 + x^53 + x^52 + x^51 + x^50 + x^46
Tags Methods from general class field theory

User comments

sparser defining equation     
Isabel Pirsic
06/01/2012 12:40
Just wanted to note that y/x^2 has a slightly sparser minimal polynomial, 408 instead of 640 monomials (might be relevant for some applications):

y^16 + (x^10 + x^9 + x^7 + x^6 + x^2 + 1)*y^12 + (x^16 + x^15 + x^14 + x^10 + x^9 + x^2)*y^10 + (x^17 + x^16 + x^15 + x^13 + x^12 + x^11 + x^9 + x^5)*y^9 + (x^32 + x^27 + x^23 + x^22 + x^21 + x^18 + x^17 + x^16 + x^15 + x^14 + x^13 +x^12 + x^11 + x^9 + x^7 + x^6 + x^5 + x^4 + x^2)*y^8 + (x^28 + x^27 + x^26 +
x^20 + x^14 + x^13 + x^12 + x^6)*y^6 + (x^29 + x^28 + x^27 + x^21 + x^17 + x^16+ x^15 + x^9)*y^5 + (x^48 + x^47 + x^46 + x^45 + x^44 + x^43 + x^41 + x^35 + x^34 + x^32 + x^27 + x^25 + x^23 + x^21 + x^18 + x^17 + x^16 + x^14 + x^12 + x^10 + x^9 + x^7 + x^6 + x^4)*y^4 + (x^31 + x^30 + x^29 + x^22 + x^21 +
x^15)*y^3 + (x^54 + x^53 + x^49 + x^48 + x^47 + x^45 + x^44 + x^41 + x^40 + x^39 + x^36 + x^33 + x^30 + x^29 + x^27 + x^26 + x^25 + x^22 + x^21 + x^20 + x^17 + x^16 + x^12 + x^11 + x^10 +
x^47 + x^45 + x^44 + x^42 + x^41 + x^40 + x^39 + x^38 + x^35 + x^33 + x^32 + x^30 + x^29 + x^28 + x^27 + x^25 + x^24 + x^23 + x^20 + x^18 + x^16 + x^14 + x^12 + x^11 + x^9)*y + x^76 + x^65 + x^64 + x^63 + x^59 + x^58 + x^56 + x^55 + x^54 + x^51 + x^50 + x^49 + x^46 + x^44 + x^43 + x^40 + x^38 + x^37 + x^36 +
x^32 + x^31 + x^26 + x^23 + x^21 + x^20 + x^19 + x^18 + x^14
Upper bound Nmax = 23

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
Tags Oesterlé bound

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