Entry details for q = 21 = 2, g = 10
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Lower bound Nmin = 13

 Submitted by Gerrit Oomens Date 01/01/1900 Reference J-P. SerreLetter to G. van der GeerSeptember 1, 1997. Comments Tags Methods from general class field theory

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 Correction S.E.Fischer 12/20/2014 03:39 The extension E/F leads to a curve of genus 4 with 8 points. Extending once again by H of degree 2 leads to the desired result, as we write H/(E/F):= v^2 + v*x + x^3 + x . Explicit Curve S.E.Fischer 12/18/2014 15:27 We can assume such a curve C as an extension E of degree 2 of a curve F of genus 1 as follows: F:= (x + y + x*y) * x*y + x + 1 E/F := z^2 * x + z + x^2 * y^2 My reference corrected Isabel Pirsic 06/19/2012 16:18 G. van der Geer Hunting for curves over finite fields with many points arXiv:0902.3882 Defining equation Isabel Pirsic 06/19/2012 14:17 y^8 + (x^4 + x + 1)*y^4 + (x^2 + x)*y^2 + (x^4 + x^2)*y + (x^12 + x^4) Ref: G. van der Geer, M. van der Vlugt How to construct curves over finite fields with many points In: Arithmetic Geometry, (Cortona 1994), F. Catanese Ed., Cambridge Univ. Press, Cambridge, 1997, p. 169-189.
Upper bound Nmax = 13

 Submitted by Everett Howe Date 04/14/2010 Reference Jean-Pierre SerreRational points on curves over finite fieldsNotes 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 + 1) * (x + 2) * (x^2 + 2*x - 1) * (x^6 + 5*x^5 + 2*x^4 - 18*x^3 - 14*x^2 + 11*x + 1) Tags Oesterlé bound

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