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 Gerrit Oomens
Date 01-01-1900
Reference H. Niederreiter, C. P. Xing
Cyclotomic function fields, Hilbert class fields and global function fields with many rational places
Acta Arithm. 79 (1997), p. 59-76.
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