A169878 Maximum number of rational points on a smooth absolutely irreducible projective curve of genus 2 over the field F_3^n.

8, 20, 48, 118, 306, 838, 2372, 6886, 20244, 60022, 178830, 534358, 1599374, 4791718, 14364057, 43072966, 129185618, 387499222, 1162397834, 3487020598, 10460762306, 31381768198, 94144406138, 282431662246, 847292291373, 2541872205622, 7625608530780, 22876811586838, 68630410502264

Offset

1,1

References

J. W. P. Hirschfeld, Linear codes and algebraic curves, 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 algébrique sur un corps fini. C. R. Acad. Sci. Paris Ser. I Math. 296 (1983), no. 9, 397-402.

Links

Robin Visser, Table of n, a(n) for n = 1..2000

Gerard van der Geer et al., Tables of curves with many points

Gerard van der Geer and Marcel van der Vlugt, Tables of curves with many points, Math. Comp. 69 (2000) 797-810.

Prog

(Sage)
def a(n):
    if n==2: return 20
    elif (n%2 == 0): return 3^n + 1 + 4*3^(n/2)
    elif ((floor(2*3^(n/2))%3 == 0) or (3^n-1).is_square()
        or (4*3^n-3).is_square() or (4*3^n-7).is_square()):
        if (frac(2*3^(n/2)) > ((sqrt(5)-1)/2)): return 3^n + 2*floor(2*3^(n/2))
        else: return 3^n + 2*floor(2*3^(n/2)) - 1
    else: return 3^n + 1 + 2*floor(2*3^(n/2))  # Robin Visser, Oct 01 2023

Crossrefs

Cf. A005525, A169869-A169883.

Sequence in context: A273295 A273266 A273304 * A107816 A361073 A361215

Adjacent sequences: A169875 A169876 A169877 * A169879 A169880 A169881

Keyword

nonn

Author

N. J. A. Sloane, Jul 05 2010

Extensions

More terms from Robin Visser, Oct 01 2023

Status

approved