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A005526
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Maximal number of rational points that a (smooth, geometrically irreducible) curve of genus 3 over the finite field GF(q) can have, where q is the n-th prime power >= 2.
(Formerly M4338)
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1
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7, 10, 14, 16, 20, 24, 28, 28, 32, 38, 40, 44, 48, 56, 56, 60, 62, 64, 72, 78, 80, 87, 92, 96, 102, 107, 113, 116, 120, 122, 131, 136, 136, 144, 155
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,1
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REFERENCES
| R. Auer and J. Top, Some genus 3 curves with many points, pp. 163-171 of ANTS 2002, Lect. Notes Computer Sci. 2369 (2002).
J. W. P. Hirschfeld, Linear codes and algebraic codes, pp. 35-53 of F. C. Holroyd and R. J. Wilson, editors, Geometrical Combinatorics. Pitman, Boston, 1984.
J.-P. Serre, Sur le nombre des points rationnels d'une courbe algebrique sur un corps fini, Compt. Rend. Acad. Sci. Paris, 296 (1983), 397-402; Oeuvres, vol. 3, pp. 658-663.
J.-P. Serre, Nombres de points des courbe algebriques sur F_q, Semin. Theorie Nombres Bordeaux, 1982/83, No. 22; Oeuvres, vol. 3, pp. 664-669.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
W. C. Waterhouse, Abelian varieties over finite fields. Ann. Sci. Ecole Norm. Sup. (4) 2 1969, 521-560.
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LINKS
| K. Lauter and J.-P. Serre, The maximum or minimum number of rational points ... , Compos. Math., accepted, 2002.
Jaap Top, Curves of genus 3 over small finite fields
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EXAMPLE
| For q=23 the value is 48: this maximum is attained by the following curve (due to Serre): x^4+y^4+z^4-5(x^2y^2 +y^2z^2 + z^2x^2)=0, over the field with 23 elements.
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CROSSREFS
| Cf. A080205.
Sequence in context: A059752 A080205 A108980 * A192292 A030123 A191833
Adjacent sequences: A005523 A005524 A005525 * A005527 A005528 A005529
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KEYWORD
| nonn,more
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| More terms from A. E. Brouwer, Sep 15 1997.
Edited by Dean Hickerson (dean.hickerson(AT)yahoo.com), Feb 05 2003 and Feb 23 2003, adding more terms from the paper by Jaap Top.
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