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

Author

N. J. A. Sloane, Jul 05 2010

Keywords

• nonn

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.

Crossrefs

Cf. A005523, A169869-A169883.

Programs

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

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

Extensions

More terms from Robin Visser, Aug 17 2023