A169872 Maximum number of rational points on a smooth absolutely irreducible projective curve of genus 1 over the field F_2^n.

5, 9, 14, 25, 44, 81, 150, 289, 558, 1089, 2138, 4225, 8374, 16641, 33130, 66049, 131796, 263169, 525736, 1050625, 2100048, 4198401, 8394400, 16785409, 33566018, 67125249, 134240898, 268468225, 536917252, 1073807361, 2147576330, 4295098369, 8590119956, 17180131329, 34360109096

Offset

1,1

Links

Robin Visser, Table of n, a(n) for n = 1..3000

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.

Formula

a(n) = 2^n + 1 + floor(2^(n/2 + 1)) if floor(2^(n/2 + 1)) is odd, n is even, or n = 1. Otherwise a(n) = 2^n + floor(2^(n/2 + 1)) [Deuring-Waterhouse]. - Robin Visser, Aug 17 2023

Prog

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

Crossrefs

Cf. A005523, A169869-A169883.

Sequence in context: A314992 A022810 A109329 * A115380 A107980 A163161

Adjacent sequences: A169869 A169870 A169871 * A169873 A169874 A169875

Keyword

nonn

Author

N. J. A. Sloane, Jul 05 2010

Extensions

More terms from Robin Visser, Aug 17 2023

Status

approved