A169880 Maximum number of rational points on a smooth absolutely irreducible projective curve of genus 1 over the field F_5^n.

10, 36, 148, 676, 3237, 15876, 78685, 391876, 1955920, 9771876, 48842100, 244171876, 1220773003, 6103671876, 30517927510, 152588671876, 762941200054, 3814701171876, 19073495062765, 95367451171876, 476837201876328, 2384185888671876, 11920929173444139, 59604645263671876

Offset: 1

Author

N. J. A. Sloane, Jul 05 2010

Keywords

• nonn

Links

• Robin Visser, Table of n, a(n) for n = 1..1400
• 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*5^(n/2))%5 != 0):
        return 5^n + 1 + floor(2*5^(n/2))
    else:
        return 5^n + floor(2*5^(n/2))  # Robin Visser, Aug 17 2023

Formula

a(n) = 5^n + 1 + floor(2*5^(n/2)) if 5 does not divide floor(2*5^(n/2)), n is even, or n = 1. Otherwise a(n) = 5^n + floor(2*5^(n/2)) [Deuring-Waterhouse]. - Robin Visser, Aug 17 2023

Extensions

More terms from Robin Visser, Aug 17 2023