A169881 Maximum number of rational points on a smooth absolutely irreducible projective curve of genus 2 over the field F_5^n.

12, 46, 170, 726, 3348, 16126, 79244, 393126, 1958714, 9778126, 48856074, 244203126, 1220842880, 6103828126, 30518276895, 152589453126, 762942946982, 3814705078126, 19073503797404, 95367470703126, 476837245549530, 2384185986328126, 11920929391810152, 59604645751953126, 298023226060613260

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..1400

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

Crossrefs

Cf. A005525, A169869-A169883.

Sequence in context: A340231 A100183 A050490 * A200669 A197471 A159013

Adjacent sequences: A169878 A169879 A169880 * A169882 A169883 A169884

Keyword

nonn

Author

N. J. A. Sloane, Jul 05 2010

Extensions

More terms from Robin Visser, Oct 01 2023

Status

approved