A005526 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)

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

Offset

1,1

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. See N_q(3) on page 51.

J.-P. Serre, Sur le nombre des points rationnels d'une courbe algébrique 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 courbes algébriques 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.

Links

Table of n, a(n) for n=1..35.

K. Lauter and J.-P. Serre, The maximum or minimum number of rational points on curves of genus three over finite fields, arXiv:math/0104086 [math.AG], 2001; Compos. Math.134 (p. 87-111) 2002.

Jean-Pierre Serre, Sur le nombre des points rationnels d'une courbe algébrique sur un corps fini, C. R. Acad. Sci. Paris Ser. I Math. 296 (1983), no. 9, 397-402.

Jaap Top, Curves of genus 3 over small finite fields, arXiv:math/0301264 [math.NT], 2003.

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.

Crossrefs

Cf. A080205.

Sequence in context: A059752 A080205 A108980 * A192292 A030123 A191833

Adjacent sequences: A005523 A005524 A005525 * A005527 A005528 A005529

Keyword

nonn,more

Author

N. J. A. Sloane.

Extensions

More terms from A. E. Brouwer, Sep 15 1997.

Edited by Dean Hickerson, Feb 05 2003 and Feb 23 2003, adding more terms from the paper by Jaap Top.

Status

approved