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

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..1400
• 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 == 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

Extensions

More terms from Robin Visser, Oct 01 2023