A005523 -id:A005523 - 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

A005525 Maximal number of rational points on a curve of genus 2 over GF(q), where q = A246655(n) is the n-th prime power > 1.

Original entry on oeis.org

6, 8, 10, 12, 16, 18, 20, 24, 26, 33, 32, 36, 42, 46, 48, 50, 52, 53, 60, 66, 68, 74, 78, 82, 90, 92, 97, 100, 104, 106, 114, 118, 120, 126, 136, 140, 144, 148, 150, 156, 166, 170, 172, 172, 176, 184, 186, 198, 200, 206, 214, 218, 222, 226, 232, 234, 246, 248, 252, 256, 268, 282

Offset: 1

N. J. A. Sloane

The successive values of q are 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 23, ... (see A246655).

			a(2) = 8 because 8 is the maximal number of rational points on a genus 2 curve over GF(3). One example of such a maximal curve is the genus 2 curve y^2 = x^6 + 2*x^2 + 1 which consists of the rational points (x,y) = (0, 1), (0, 2), (1, 1), (1, 2), (1, 1), (1, 2), and two points at infinity. - _Robin Visser_, Aug 03 2023

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. See N_q(2) on page 51.
J.-P. Serre, Oeuvres, vol. 3, pp. 658-663 and 664-669.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

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.
W. C. Waterhouse, Abelian varieties over finite fields, Ann Sci. E.N.S., (4) 2 (1969), 521-560.

Cf. A005523.

for q in range(1, 1000):
    if Integer(q).is_prime_power():
        p = Integer(q).prime_factors()[0]
        if q==4: print(10)
        elif q==9: print(20)
        elif (Integer(q).is_square()): print(q + 1 + 4*sqrt(q))
        elif ((floor(2*sqrt(q))%p == 0) or (q-1).is_square()
            or (4*q-3).is_square() or (4*q-7).is_square()):
            if (frac(2*sqrt(q)) > ((sqrt(5)-1)/2)):  print(q + 2*floor(2*sqrt(q)))
            else:  print(q + 2*floor(2*sqrt(q)) - 1)
        else:  print(q + 1 + 2*floor(2*sqrt(q)))  # Robin Visser, Aug 03 2023

a(n) <= q + 1 + 4*sqrt(q) where q = A246655(n) [Hasse-Weil theorem]. - Robin Visser, Aug 03 2023

a(n) >= q - 1 + 2*floor(2*sqrt(q)) for all n except for 3 and 7, where q = A246655(n) [Serre]. - Robin Visser, Aug 03 2023

More terms from Robin Visser, Aug 03 2023

A005524 Values k arising from a construction of Hirschfeld of k-arcs on elliptic curves over GF(q), where q = A246655(n) is the n-th prime power > 1.

Original entry on oeis.org

2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 14, 16, 18, 19, 20, 21, 22, 25, 27, 28, 30, 32, 34, 37, 38, 40, 42, 44, 45, 48, 50, 51, 54, 58, 61, 62, 64, 65, 67, 72, 74, 75, 75, 77, 80, 81, 87, 88, 91, 94, 96, 98, 100, 103, 104, 109, 110, 113, 114, 120, 126, 129, 130, 132, 135, 136, 137, 141

Offset: 1

N. J. A. Sloane

nonn

Let E be an elliptic curve over GF(q). A k-arc on E is a set of k points in E(GF(q)) such that no three are collinear (in the projective plane over GF(q)). Hirschfeld showed that if the number #E(GF(q)) of GF(q)-rational points on E is even, then there exists a k-arc on E for k = #E(GF(q))/2. Here, a(n) denotes the largest possible k arising from this construction, hence a(n) = floor(A005523(n)/2). Note that a(n) is not necessarily the maximal k such that there exists a k-arc on an elliptic curve over GF(q); e.g. the elliptic curve y^2 = x^3 + x + 1 over GF(5) contains a 6-arc consisting of the points {(0,1), (3,1), (4,2), (4,3), (0,4), (3,4)}. - Robin Visser, Aug 26 2023

			For n = 4, the elliptic curve E : y^2 = x^3 + 3*x over GF(5) has 10 rational points.  As this is the maximal number of rational points an elliptic curve over GF(5) can have, this implies a(4) = 10/2 = 5. - _Robin Visser_, Aug 26 2023

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. See M_q(1) on page 51.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Robin Visser, Table of n, a(n) for n = 1..10000
J. W. P. Hirschfeld, G. Korchmáros, and F. Torres, Algebraic curves over a finite field, Princeton Ser. Appl. Math., Princeton University Press, Princeton, NJ, 2008.
Mathematica Information Center, Item 5175, for full code.
Ed Pegg Jr, Integer Complexity.

Cf. A000961 (values of q), A005523, A365216.

for q in range(1, 1000):
    if Integer(q).is_prime_power():
        p = Integer(q).prime_factors()[0]
        if (floor(2*sqrt(q))%p != 0) or (Integer(q).is_square()) or (q==p):
            print(floor((q + 1 + floor(2*sqrt(q)))/2))
        else:
            print(floor((q + floor(2*sqrt(q)))/2))  # Robin Visser, Aug 26 2023

a(n) = floor(A005523(n)/2) [Hirschfeld]. - Robin Visser, Aug 26 2023

New name and more terms from Robin Visser, Aug 26 2023

A364681 a(n) is the number of isogeny classes of elliptic curves over GF(q), where q = A246655(n) is the n-th prime power > 1.

Original entry on oeis.org

5, 7, 9, 9, 11, 9, 13, 13, 15, 13, 17, 17, 19, 20, 17, 21, 23, 15, 25, 25, 27, 27, 27, 29, 31, 31, 21, 33, 33, 35, 35, 29, 37, 37, 39, 41, 41, 41, 41, 43, 45, 37, 45, 25, 45, 47, 47, 49, 49, 51, 51, 51, 50, 53, 53, 53, 55, 55, 57, 57, 59, 59, 61, 61, 61, 61, 63, 45, 63, 37, 65, 65

Offset: 1

Robin Visser, Aug 02 2023

nonn

Two elliptic curves over a finite field F_q are isogenous if and only if they have the same trace of Frobenius, or equivalently, have the same number of points over F_q.

Thus a(n) is the number of integers k such that there exists an elliptic curve over GF(q) with trace k, where q = A246655(n).

			For n = 1, the a(1) = 5 isogeny classes of elliptic curves over GF(2) are parametrized by the 5 possible values for the trace of Frobenius: -2, -1, 0, 1, 2.
For n = 2, the a(2) = 7 isogeny classes of elliptic curves over GF(3) are parametrized by the 7 possible values for the trace of Frobenius: -3, -2, -1, 0, 1, 2, 3.

Robin Visser, Table of n, a(n) for n = 1..10000
Max Deuring, Die Typen der Multiplikatorenringe elliptischer Funktionenkörper, Abh. Math. Sem. Hansischen Univ. 14 (1941), 197-272.
W. C. Waterhouse, Abelian varieties over finite fields, Ann Sci. E.N.S., (4) 2 (1969), 521-560.

Cf. A005523, A362570.

for q in range(1, 1000):
    if Integer(q).is_prime_power():
        p, ans = Integer(q).prime_factors()[0], 0
        for a in range(-floor(2*sqrt(q)), floor(2*sqrt(q))+1):
            if (a%p != 0) or (Integer(q).is_square() and ((abs(a) == 2*sqrt(q))
                  or ((p%3 != 1) and (abs(a) == sqrt(q))) or ((p%4 != 1) and
                  (a==0)))) or ((not Integer(q).is_square()) and
                  (((p in [2,3]) and (abs(a) == sqrt(p*q))) or (a==0))):
                ans += 1
        print(ans)

a(n) = 2*floor(2*sqrt(q)) + 1 if q is prime, where q = A246655(n).

A364690 Prime powers q such that there does not exist an elliptic curve E over GF(q) with cardinality q + 1 + floor(2*sqrt(q)).

Original entry on oeis.org

128, 2048, 2187, 16807, 32768, 131072, 524288, 1953125, 2097152, 8388608, 14348907, 48828125, 134217728, 536870912, 30517578125, 549755813888, 847288609443, 2199023255552, 19073486328125, 140737488355328, 562949953421312, 36028797018963968, 144115188075855872, 450283905890997363

Offset: 1

Robin Visser, Aug 02 2023

nonn

By Hasse's theorem, every elliptic curve E over GF(q) has cardinality at most q + 1 + floor(2*sqrt(q)). Moreover, for every prime power q, there exists an elliptic curve E over GF(q) with cardinality at least q + floor(2*sqrt(q)). Thus these are the prime powers q for which A005523(n) = q + floor(2*sqrt(q)), where q = A246655(n).

By a theorem of Deuring and Waterhouse, these are exactly the prime powers q = p^k such that q is not prime, q is not a square, and p divides floor(2*sqrt(q)).

			The first few values of the sequence (factorized) are 2^7, 2^11, 3^7, 7^5, 2^15, 2^17, 2^19, 5^9, 2^21, 2^23, 3^15, 5^11, 2^27, 2^29, ...

Katie Ahrens and Jon Grantham, Table of n, a(n) for n = 1..146
Max Deuring, Die Typen der Multiplikatorenringe elliptischer Funktionenkörper, Abh. Math. Sem. Hansischen Univ. 14 (1941), 197-272.
W. C. Waterhouse, Abelian varieties over finite fields, Ann Sci. E.N.S., (4) 2 (1969), 521-560.

Cf. A005523, A246655.

Subsequence of A246547.

for q in range(1, 100000):
    if Integer(q).is_prime_power():
        p = Integer(q).prime_factors()[0]
        if (floor(2*sqrt(q))%p == 0) and (not Integer(q).is_square()) and (q != p):
            print(q)

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

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

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

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.

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A005525 Maximal number of rational points on a curve of genus 2 over GF(q), where q = A246655(n) is the n-th prime power > 1.

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A005524 Values k arising from a construction of Hirschfeld of k-arcs on elliptic curves over GF(q), where q = A246655(n) is the n-th prime power > 1.

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A364681 a(n) is the number of isogeny classes of elliptic curves over GF(q), where q = A246655(n) is the n-th prime power > 1.

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A364690 Prime powers q such that there does not exist an elliptic curve E over GF(q) with cardinality q + 1 + floor(2*sqrt(q)).

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Sage

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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Sage

Formula

Extensions

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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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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