A169883 Maximum number of rational points on a smooth absolutely irreducible projective curve of genus 1 over the field F_7^n.

13, 64, 381, 2500, 17066, 118336, 825358, 5769604, 40366312, 282508864, 1977415678, 13841522500, 96889632947, 678224719936, 4747565867723, 33232942099204, 232630544491667, 1628413678617664, 11398895398904361, 79792266862562500, 558545865578002528, 3909821052537641536

Offset: 1

Author

N. J. A. Sloane, Jul 05 2010

Keywords

• nonn

Links

• Robin Visser, Table of n, a(n) for n = 1..1000
• Max Deuring, Die Typen der Multiplikatorenringe elliptischer Funktionenkörper, Abh. Math. Sem. Hansischen Univ. 14 (1941), 197-272.
• 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.
• W. C. Waterhouse, Abelian varieties over finite fields, Ann Sci. E.N.S., (4) 2 (1969), 521-560.

Crossrefs

Cf. A005523, A169869-A169883.

Programs

Sage

def a(n):
    if (n==1) or (n%2 == 0) or (floor(2*7^(n/2))%7 != 0):
        return 7^n + 1 + floor(2*7^(n/2))
    else:
        return 7^n + floor(2*7^(n/2))  # Robin Visser, Aug 17 2023

Formula

a(n) = 7^n + 1 + floor(2*7^(n/2)) if 7 does not divide floor(2*7^(n/2)), n is even, or n = 1. Otherwise a(n) = 7^n + floor(2*7^(n/2)) [Deuring-Waterhouse]. - Robin Visser, Aug 17 2023

Extensions

More terms from Robin Visser, Aug 17 2023