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

Author

N. J. A. Sloane, Jul 05 2010

Keywords

• nonn

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.

Links

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

Crossrefs

Cf. A005525, A169869-A169883.

Programs

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

Extensions

More terms from Robin Visser, Oct 01 2023