A169869 -id:A169869 - cp's OEIS frontend

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

N. J. A. Sloane, Jul 05 2010

nonn

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.

Cf. A005523, A169869-A169883.

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

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

More terms from Robin Visser, Aug 17 2023

A096336 Spin(2n+1) and Spin(2n+2) have torsion index 2^a(n).

Original entry on oeis.org

0, 0, 0, 1, 1, 1, 2, 3, 4, 4, 5, 5, 6, 7, 8, 9, 10, 10, 11, 12, 13, 14, 15, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 55, 56, 57

Offset: 0

Richard Borcherds (reb(AT)math.berkeley.edu), Jun 28 2004

First several terms agree with A169869 but the two sequences are distinct as can be seen where the values are 19 and 20. - Skip Garibaldi, Mar 05 2017

Michael De Vlieger, Table of n, a(n) for n = 0..10000
Burt Totaro, The torsion index of the spin groups, Duke Math. J. 129 (2005), no. 2, 249-290, doi:10.1215/S0012-7094-05-12923-4.

a[0] = 0; a[n_] := a[n] = Module[{e = Floor[Log2@n], b}, b = n - 2^e; n - Floor[Log2[(n + 1) n/2 + 1]] + Boole[2 b - a[b] <= e - 3]]; Table[a@ n, {n, 0, 120}] (* Michael De Vlieger, Mar 06 2017 *)

import numpy as np
def a_typical(n):
    '''
    For most n, this is the value of a(n)
    '''
    return int(n - np.floor(np.log2( n*(n+1)/2 + 1)))
def a(n):
    '''
    The torsion index of Spin_{2n+1} and Spin_{2n+2} is 2^a(n)
    Totaro denotes it by u(ell)
    '''
    if n >= 0 and n <= 18:   # Table 1 in Totaro's paper
        return [0,0,0,1,1,1,2,3,4,4,5,5,6,7,8,9,10,10,11][n];
    maxe = int(np.floor(np.log2(n)))
    for e in range(maxe+1):
        b = n - 2**e
        if 2*b - a(b) <= e - 3: # occurs for n = 8, 16, 32, 33, ...
            return a_typical(n)+1
    return a_typical(n)
# Skip Garibaldi, Mar 05 2017

a(n) is usually n-floor(log_2((n+1)n/2 + 1)), but is this number plus 1 if n = 2^e+b for nonnegative integers e, b such that 2b-a(b) <= e-3.

Edited and a(19)-a(49) added by Skip Garibaldi, Mar 05 2017

More terms from Michael De Vlieger, Mar 06 2017

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

Original entry on oeis.org

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

N. J. A. Sloane, Jul 05 2010

nonn

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.

Cf. A005523, A169869-A169883.

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

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

More terms from Robin Visser, Aug 17 2023

A169873 Maximum number of rational points on a smooth absolutely irreducible projective curve of genus 2 over the field F_2^n.

Original entry on oeis.org

6, 10, 18, 33, 53, 97, 172, 321, 603, 1153, 2227, 4353, 8553, 16897, 33491, 66561, 132519, 264193, 527183, 1052673, 2102943, 4202497, 8400192, 16793601, 33577603, 67141633, 134264067, 268500993, 536963592, 1073872897, 2147669011, 4295229441, 8590305319, 17180393473, 34360479823

Offset: 1

N. J. A. Sloane, Jul 05 2010

nonn

J. W. P. Hirschfeld, Linear codes and algebraic curves, pp. 35-53 of F. C. Holroyd and R. J. Wilson, editors, Geometrical Combinatorics. Pitman, Boston, 1984.

Robin Visser, Table of n, a(n) for n = 1..3000
Jean-Pierre Serre, Sur le nombre des points rationnels d'une courbe algébrique sur un corps fini, C. R. Acad. Sci. Paris Ser. I Math. 296 (1983), no. 9, 397-402.
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.

Cf. A005525, A169869-A169883.

def a(n):
    if n==2: return 10
    elif (n%2 == 0): return 2^n + 1 + 2^(n/2+2)
    elif ((floor(2^(n/2+1))%2 == 0) or (2^n-1).is_square()
        or (4*2^n-3).is_square() or (4*2^n-7).is_square()):
        if (frac(2^(n/2+1)) > ((sqrt(5)-1)/2)): return 2^n + 2*floor(2^(n/2+1))
        else: return 2^n + 2*floor(2^(n/2+1)) - 1
    else: return 2^n + 1 + 2*floor(2^(n/2+1))  # Robin Visser, Oct 01 2023

More terms from Robin Visser, Oct 01 2023

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

Original entry on oeis.org

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

N. J. A. Sloane, Jul 05 2010

nonn

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.

Cf. A005523, A169869-A169883.

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

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

More terms from Robin Visser, Aug 17 2023

A169878 Maximum number of rational points on a smooth absolutely irreducible projective curve of genus 2 over the field F_3^n.

Original entry on oeis.org

8, 20, 48, 118, 306, 838, 2372, 6886, 20244, 60022, 178830, 534358, 1599374, 4791718, 14364057, 43072966, 129185618, 387499222, 1162397834, 3487020598, 10460762306, 31381768198, 94144406138, 282431662246, 847292291373, 2541872205622, 7625608530780, 22876811586838, 68630410502264

Offset: 1

N. J. A. Sloane, Jul 05 2010

nonn

J. W. P. Hirschfeld, Linear codes and algebraic curves, pp. 35-53 of F. C. Holroyd and R. J. Wilson, editors, Geometrical Combinatorics. Pitman, Boston, 1984.

Robin Visser, Table of n, a(n) for n = 1..2000
Jean-Pierre Serre, Sur le nombre des points rationnels d'une courbe algébrique sur un corps fini, C. R. Acad. Sci. Paris Ser. I Math. 296 (1983), no. 9, 397-402.
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.

Cf. A005525, A169869-A169883.

def a(n):
    if n==2: return 20
    elif (n%2 == 0): return 3^n + 1 + 4*3^(n/2)
    elif ((floor(2*3^(n/2))%3 == 0) or (3^n-1).is_square()
        or (4*3^n-3).is_square() or (4*3^n-7).is_square()):
        if (frac(2*3^(n/2)) > ((sqrt(5)-1)/2)): return 3^n + 2*floor(2*3^(n/2))
        else: return 3^n + 2*floor(2*3^(n/2)) - 1
    else: return 3^n + 1 + 2*floor(2*3^(n/2))  # Robin Visser, Oct 01 2023

More terms from Robin Visser, Oct 01 2023

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

Original entry on oeis.org

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

N. J. A. Sloane, Jul 05 2010

nonn

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.

Cf. A005523, A169869-A169883.

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

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

More terms from Robin Visser, Aug 17 2023

A169881 Maximum number of rational points on a smooth absolutely irreducible projective curve of genus 2 over the field F_5^n.

Original entry on oeis.org

12, 46, 170, 726, 3348, 16126, 79244, 393126, 1958714, 9778126, 48856074, 244203126, 1220842880, 6103828126, 30518276895, 152589453126, 762942946982, 3814705078126, 19073503797404, 95367470703126, 476837245549530, 2384185986328126, 11920929391810152, 59604645751953126, 298023226060613260

Offset: 1

N. J. A. Sloane, Jul 05 2010

nonn

J. W. P. Hirschfeld, Linear codes and algebraic curves, pp. 35-53 of F. C. Holroyd and R. J. Wilson, editors, Geometrical Combinatorics. Pitman, Boston, 1984.

Robin Visser, Table of n, a(n) for n = 1..1400
Jean-Pierre Serre, Sur le nombre des points rationnels d'une courbe algébrique sur un corps fini, C. R. Acad. Sci. Paris Ser. I Math. 296 (1983), no. 9, 397-402.
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.

Cf. A005525, A169869-A169883.

def a(n):
    if (n%2 == 0): return 5^n + 1 + 4*5^(n/2)
    elif ((floor(2*5^(n/2))%5 == 0) or (5^n-1).is_square()
        or (4*5^n-3).is_square() or (4*5^n-7).is_square()):
        if (frac(2*5^(n/2)) > ((sqrt(5)-1)/2)): return 5^n + 2*floor(2*5^(n/2))
        else: return 5^n + 2*floor(2*5^(n/2)) - 1
    else: return 5^n + 1 + 2*floor(2*5^(n/2))  # Robin Visser, Oct 01 2023

More terms from Robin Visser, Oct 01 2023

A169870 Maximum number of rational points on a smooth absolutely irreducible projective curve of genus n over the field F_4.

Original entry on oeis.org

9, 10, 14, 15, 17, 20, 21

Offset: 1

N. J. A. Sloane, Jul 05 2010

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.

Cf. A169869-A169883.

A169871 Maximum number of rational points on a smooth absolutely irreducible projective curve of genus n over the field F_8.

Original entry on oeis.org

14, 18, 24, 25, 29

Offset: 1

N. J. A. Sloane, Jul 05 2010

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.

Cf. A169869-A169883.

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

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A096336 Spin(2n+1) and Spin(2n+2) have torsion index 2^a(n).

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

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A169873 Maximum number of rational points on a smooth absolutely irreducible projective curve of genus 2 over the field F_2^n.

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

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A169878 Maximum number of rational points on a smooth absolutely irreducible projective curve of genus 2 over the field F_3^n.

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

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A169881 Maximum number of rational points on a smooth absolutely irreducible projective curve of genus 2 over the field F_5^n.

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