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Certain sextics with many rational points

This research was partially supported by JSPS Grant-in-Aid for Young Scientists (B) 25800090.

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  • We construct a family of sextics from the Wiman and Edge sextics. We find a curve over $\mathbb{F}_{5^7}$ attaining the Serre bound, and update $9$ entries of genus $6$ in manYPoints.org by computer search on these sextics.

    Mathematics Subject Classification: Primary: 11G20, 14G05; Secondary: 14G50.

    Citation:

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  • Table 1.  Maximal curves over $\mathbb{F}_{p^2}$ of genus $6$

    5 7 11 13 17 19 23 29
    W W W W W
    31 37 41 43 47 53 59 61
    W S W S W S W
    67 71 73 79 83 89 97 101
    S W W W W S W
    103 107 109 113 127 131 137 139
    W W E S W W S W
    149 151 157 163 167 173 179 181
    E S S W S W S
    191 193 197 199
    W W S W
     | Show Table
    DownLoad: CSV

    Table 2.  $S$ with many points over $\mathbb{F}_p$

    $\mathbb{F}_p$ $(a,b,c)$ $\#S(\mathbb{F}_p)$ old entry
    $19 $ $(13,6,16) $ $56$ $[50 -68]$
    $37 $ $(29,28,14) $ $98 $ $[86 -104]$
    $43 $ $(2,4,2)$ $104 $ $[100-116]$
    $53 $ $(51,36,1) $ $ 132 $ $[120-138 ]$
    $61 $ $(42,54,17)$ $140 $ $[134-152] $
    $67 $ $(65,2,45) $ $152$ $[140-164] $
    $71 $ $(29,65,70)$ $ 156 $ $[150-168] $
     | Show Table
    DownLoad: CSV

    Table 3.  $S$ with many points over $\mathbb{F}_q$

    $\mathbb{F}_q$ $(a,b,c)$ $\#S(\mathbb{F}_q)$ old entry
    $5^3 $ $(\beta^4,\beta^{56},\beta^{38})$ $ 240 $ $[210-255] $
    $u^3+3u+3=0$
    $7^3 $ $(\beta^{22},\beta^{94},\beta^{8})$ $ 542 $ $[512-564] $
    $u^3-u^2+4=0$
     | Show Table
    DownLoad: CSV
  •   W. Bosma  and  J. Cannon andC. Playoust , The Magma algebra system. I. The user language, J. Symb. Comput., 24 (1997) , 235-265.  doi: 10.1006/jsco.1996.0125.
      W. L. Edge , A pencil of four-nodal plane sextics, Math. Proc. Cambridge Philos. Soc., 89 (1981) , 413-421.  doi: 10.1017/S0305004100058321.
      A. Garcia , G. Güneri  and  H. Stichtenoth , A generalization of the Giulietti--Korchmáros maximal curve, Adv. Geom., 10 (2010) , 427-434.  doi: 10.1515/ADVGEOM.2010.020.
      G. van der Geer, E. Howe, K. Lauter and C. Ritzenthaler, Table of curves with many points available at http://www.manypoints.org
      M. Giulietti , M. Montanucci  and  G. Zini , On maximal curves that are not quotients of the Hermitian curve, Finite Fields Appl, 41 (2016) , 72-88.  doi: 10.1016/j.ffa.2016.05.005.
      E. Kani  and  M. Rosen , Idempotent relations and factors of Jacobians, Math. Ann., 284 (1989) , 307-327.  doi: 10.1007/BF01442878.
      M. Q. Kawakita, Wiman's and Edge's sextic attaining Serre's bound preprint, available at http://www.manypoints.org/upload/kawakita.pdf
      M. Q. Kawakita, Wiman's and Edge's sextic attaining Serre's bound Ⅱ, in Algorithmic Arithmetic, Geometry, and Coding Theory, Amer. Math. Soc., 2015, 191–203. doi: 10.1090/conm/637/12758.
      K. Lauter , Geometric methods for improving the upper bounds on the number of rational points on algebraic curves over finite fields, J. Algebraic Geom., 10 (2001) , 19-36. 
      C. Moreno, Algebraic Curves over Finite Fields Cambridge Univ. Press, 1991. doi: 10.1017/CBO9780511608766.
      A. Wiman , Ueber eine einfache Gruppe von 360 ebenen Collineationen, Math. Ann., 47 (1896) , 531-556.  doi: 10.1007/BF01445800.
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