A005525 -id:A005525 - cp's OEIS frontend

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

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

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

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

A262587 "Special" prime powers in Serre's sense.

Original entry on oeis.org

2, 3, 5, 7, 8, 13, 17, 31, 32, 37, 43, 73, 101, 128, 157, 197, 211, 241, 257, 307, 343, 401, 421, 463, 577, 601, 677, 757, 1123, 1297, 1483, 1601, 1723, 2048, 2187, 2551, 2917, 2971, 3137, 3307, 3541, 3907, 4357, 4423, 4831, 5113, 5477, 5701, 6007, 6163, 6481, 7057, 8011, 8101, 8191

Offset: 1

N. J. A. Sloane, Oct 21 2015

nonn

See Hirschfeld, pp. 49-50 for precise definition.

By a theorem of Hasse-Weil and Serre, every (absolutely irreducible, smooth) genus 2 curve over GF(q) has cardinality at most q + 1 + 2*floor(2*sqrt(q)). This sequence consists exactly of the prime powers q (excluding 4 and 9) for which there does not exist any genus 2 curve over GF(q) with cardinality equal to q + 1 + 2*floor(2*sqrt(q)). - Robin Visser, Aug 26 2023

J. W. P. Hirschfeld, Linear codes and algebraic codes, pp. 35-53 of F. C. Holroyd and R. J. Wilson, editors, Geometrical Combinatorics. Pitman, Boston, 1984.
J.-P. Serre, Oeuvres, vol. 3, pp. 658-663 and 664-669.

Robin Visser, Table of n, a(n) for n = 1..10000
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.
Jean-Pierre Serre, Nombres de points des courbe algebriques sur F_q, Sémin. Théorie Nombres Bordeaux, 1982/83, No. 22; Oeuvres, vol. 3, pp. 664-669.

Cf. A005525, A364690.

Subsequence of A246655.

for q in range(1, 1000):
    if Integer(q).is_prime_power():
        p = Integer(q).prime_factors()[0]
        if (not Integer(q).is_square()):
            if ((floor(2*sqrt(q))%p == 0) or (q-1).is_square() or
                (4*q-3).is_square() or (4*q-7).is_square()): print(q) # Robin Visser, Aug 26 2023

More terms from Robin Visser, Aug 26 2023

cp's OEIS Frontend

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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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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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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References

Links

Crossrefs

Programs

Sage

Extensions

A262587 "Special" prime powers in Serre's sense.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Sage

Extensions