Entry details for q = 21 = 2, g = 5
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Lower bound Nmin = 9

 Submitted by Gerrit Oomens Date 01-01-1900 Reference Jean-Pierre SerreSur le nombre de points rationnels d'une courbe algébrique sur un corps finiC. R. Acad. Sci. Paris Sér. I Math. 296 (1983), 397–402. (= Œuvres III, No. 128, 658–663). Comments Tags Methods from general class field theory

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 Explicit Curve S.E.Fischer 02-27-2015 15:07 We can assume such a curve C as an extension E of degree 2 of a curve F of genus 2 as follows: F:= y^3 * (x^3 + x) + y^2 * (x^3 + x^2 + x) + 1 ; E/F := z^2 + z + x^3 + x. Defining equation Isabel Pirsic 06-18-2012 11:21 E.g., y4([0],[2,1,0],[2,1],[7,3]) = y^4 + (x^2+x+1)*y^2 + (x^2+x)*y + x^7+ x^3
Upper bound Nmax = 9

 Submitted by Everett Howe Date 04-14-2010 Reference Jean-Pierre SerreRational points on curves over finite fields. With contributions by Everett Howe, Joseph Oesterlé and Christophe Ritzenthaler. Edited by Alp Bassa, Elisa Lorenzo García, Christophe Ritzenthaler and René SchoofDocuments Mathématiques 18, Société Mathématique de France, Paris, 2020 Comments The Oesterlé bound. Here is a real Weil polynomial that we don't know how to eliminate: x * (x + 2)^2 * (x^2 + 2*x - 2) Tags Oesterlé bound

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