cp's OEIS Frontend

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

%I A169881 #15 May 02 2024 09:47:20
%S A169881 12,46,170,726,3348,16126,79244,393126,1958714,9778126,48856074,
%T A169881 244203126,1220842880,6103828126,30518276895,152589453126,
%U A169881 762942946982,3814705078126,19073503797404,95367470703126,476837245549530,2384185986328126,11920929391810152,59604645751953126,298023226060613260
%N A169881 Maximum number of rational points on a smooth absolutely irreducible projective curve of genus 2 over the field F_5^n.
%D A169881 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.
%H A169881 Robin Visser, <a href="/A169881/b169881.txt">Table of n, a(n) for n = 1..1400</a>
%H A169881 Jean-Pierre Serre, <a href="https://gallica.bnf.fr/ark:/12148/bpt6k55351747/f35.item">Sur le nombre des points rationnels d'une courbe algébrique sur un corps fini</a>, C. R. Acad. Sci. Paris Ser. I Math. 296 (1983), no. 9, 397-402.
%H A169881 Gerard van der Geer et al., <a href="http://www.manypoints.org">Tables of curves with many points</a>
%H A169881 Gerard van der Geer and Marcel van der Vlugt, <a href="https://doi.org/10.1090/S0025-5718-99-01143-6">Tables of curves with many points</a>, Math. Comp. 69 (2000) 797-810.
%o A169881 (Sage)
%o A169881 def a(n):
%o A169881     if (n%2 == 0): return 5^n + 1 + 4*5^(n/2)
%o A169881     elif ((floor(2*5^(n/2))%5 == 0) or (5^n-1).is_square()
%o A169881         or (4*5^n-3).is_square() or (4*5^n-7).is_square()):
%o A169881         if (frac(2*5^(n/2)) > ((sqrt(5)-1)/2)): return 5^n + 2*floor(2*5^(n/2))
%o A169881         else: return 5^n + 2*floor(2*5^(n/2)) - 1
%o A169881     else: return 5^n + 1 + 2*floor(2*5^(n/2))  # _Robin Visser_, Oct 01 2023
%Y A169881 Cf. A005525, A169869-A169883.
%K A169881 nonn
%O A169881 1,1
%A A169881 _N. J. A. Sloane_, Jul 05 2010
%E A169881 More terms from _Robin Visser_, Oct 01 2023