A169877 Maximum number of rational points on a smooth absolutely irreducible projective curve of genus 1 over the field F_3^n.

7, 16, 38, 100, 275, 784, 2280, 6724, 19964, 59536, 177989, 532900, 1596849, 4787344, 14356482, 43059844, 129162891, 387459856, 1162329651, 3486902500, 10460557755, 31381413904, 94143792483, 282430599364, 847290450408, 2541869016976, 7625603007884, 22876802020900, 68630393933574

Offset

1,1

Links

Robin Visser, Table of n, a(n) for n = 1..2000

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) = 3^n + 1 + floor(2*3^(n/2)) if 3 does not divide floor(2*3^(n/2)), n is even, or n = 1. Otherwise a(n) = 3^n + floor(2*3^(n/2)) [Deuring-Waterhouse]. - Robin Visser, Aug 17 2023

Prog

(Sage)
def a(n):
    if (n==1) or (n%2 == 0) or (floor(2*3^(n/2))%3 != 0):
        return 3^n + 1 + floor(2*3^(n/2))
    else:
        return 3^n + floor(2*3^(n/2))  # Robin Visser, Aug 17 2023

Crossrefs

Cf. A005523, A169869-A169883.

Sequence in context: A176449 A278945 A327628 * A286710 A036834 A020941

Adjacent sequences: A169874 A169875 A169876 * A169878 A169879 A169880

Keyword

nonn

Author

N. J. A. Sloane, Jul 05 2010

Extensions

More terms from Robin Visser, Aug 17 2023

Status

approved