A002151 -id:A002151 - cp's OEIS frontend

A060950 Rank of elliptic curve y^2 = x^3 + n.

0, 1, 1, 0, 1, 0, 0, 1, 1, 1, 1, 1, 0, 0, 2, 0, 2, 1, 1, 0, 0, 1, 0, 2, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 1, 2, 1, 1, 1, 1, 0, 2, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 2, 1, 0, 0, 1, 1, 2, 0, 2, 1, 1, 1, 1, 0, 1, 1, 2, 1, 0, 1, 1, 0, 2, 1, 0, 1, 1, 0, 0, 0, 0, 0, 2, 0, 1, 1, 0, 1, 0, 0, 1, 1, 1

Offset: 1

N. J. A. Sloane, May 10 2001

The curves for n and -27*n are isogenous (as Noam Elkies points out--see Womack), so they have the same rank. - Jonathan Sondow, Sep 10 2013

			a(1) = A060951(27) = a(729) = 0. - _Jonathan Sondow_, Sep 10 2013

T. D. Noe, Table of n, a(n) for n = 1..10000 (from Gebel)
J. Gebel, Integer points on Mordell curves [Cached copy, after the original web site tnt.math.se.tmu.ac.jp was shut down in 2017]
H. Mishima, Tables of Elliptic Curves
T. Womack, Minimal-known positive and negative k for Mordell curves of given rank

Cf. A031507, A002151, A002153, A002155, A102833, A179124.
Cf. A060748, A060838, A060951, A060952, A060953.
Cf. A081119 (number of integral solutions to Mordell's equation y^2 = x^3 + n).

a(n) = ellanalyticrank(ellinit([0, 0, 0, 0, n]))[1] \\ Jianing Song, Aug 24 2022

apply( {A060950(n)=ellrank(ellinit([0, n]))[1]}, [1..99]) \\ For PARI version  < 2.14, use ellanalyticrank(...). - M. F. Hasler, Jul 01 2024

a(n) = A060951(27*n) and A060951(n) = a(27*n), so a(n) = a(729*n). - Jonathan Sondow, Sep 10 2013

Corrected by James R. Buddenhagen, Feb 18 2005

A002155 Numbers k for which the rank of the elliptic curve y^2 = x^3 + k is 2.

Original entry on oeis.org

15, 17, 24, 37, 43, 57, 63, 65, 73, 79, 89, 101, 106, 122, 129, 131, 142, 145, 148, 151, 161, 164, 168, 171, 186, 195, 197, 198, 204, 217, 222, 223, 225, 229, 232, 233, 248, 252, 260, 265, 268, 269, 281, 294, 295, 297, 303, 322, 331, 337, 347, 350, 353, 360, 366, 369, 373, 377, 381, 388, 389, 392, 404, 409, 412, 414, 433, 449, 464, 469, 481, 483, 485, 492

Offset: 1

N. J. A. Sloane

nonn

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 1..1724 (using Gebel)
B. J. Birch and H. P. F. Swinnerton-Dyer, Notes on elliptic curves, I, J. Reine Angew. Math., 212 (1963), 7-25.
J. Gebel, Integer points on Mordell curves, web.archive.org copy of the "MORDELL+" file on the SIMATH web site shut down in 2017. [Locally cached copy].
L. Lehman, Elliptic Curves of Rank Two. [broken link]
H. Mishima, Tables of Elliptic Curves.

Cf. A060950, A002151, A002153, A102833, A060748, A060838, A060951-A060953.

for k in[1..500] do if Rank(EllipticCurve([0,0,0,0,k])) eq 2 then print k; end if; end for; // Vaclav Kotesovec, Jul 07 2019

More terms from James R. Buddenhagen, Feb 18 2005

A002153 Numbers k for which the rank of the elliptic curve y^2 = x^3 + k is 1.

Original entry on oeis.org

2, 3, 5, 8, 9, 10, 11, 12, 18, 19, 22, 26, 28, 30, 31, 33, 35, 36, 38, 39, 40, 41, 44, 46, 47, 48, 50, 52, 54, 55, 56, 58, 61, 62, 66, 67, 68, 69, 71, 72, 74, 76, 77, 80, 82, 83, 91, 92, 94, 97, 98, 99, 100, 102, 103, 105, 107, 108, 110, 111, 112, 117, 118, 119

Offset: 1

N. J. A. Sloane

nonn

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 1..5111 (using Gebel).
B. J. Birch and H. P. F. Swinnerton-Dyer, Notes on elliptic curves, I, J. Reine Angew. Math., 212 (1963), 7-25.
J. Gebel, Integer points on Mordell curves, web.archive.org copy of the "MORDELL+" file on the SIMATH web site shut down in 2017. [Locally cached copy].
H. Mishima, Tables of Elliptic Curves.

Cf. A060950, A002151, A002155, A102833, A060748, A060838, A060951-A060953.

for k in[1..200] do if Rank(EllipticCurve([0,0,0,0,k])) eq 1 then print k; end if; end for; // Vaclav Kotesovec, Jul 07 2019

Corrected and extended by James R. Buddenhagen, Feb 18 2005

A102833 Numbers n for which rank of the elliptic curve y^2=x^3+n is 3.

Original entry on oeis.org

113, 141, 316, 346, 359, 427, 443, 506, 537, 568, 659, 681, 730, 745, 873, 892, 899, 940, 997, 1016, 1025, 1090, 1149, 1157, 1171, 1213, 1304, 1305, 1342, 1367, 1373, 1478, 1522, 1639, 1646, 1737, 1753, 1772, 1811, 1841, 1897, 1907, 1954, 2024, 2143

Offset: 1

James R. Buddenhagen, Feb 18 2005. Entry revised by N. J. A. Sloane, Jun 10 2012

nonn

T. D. Noe, Table of n, a(n) for n=1..250 (using Gebel).
B. J. Birch and H. P. F. Swinnerton-Dyer, Notes on elliptic curves, I, J. Reine Angew. Math., 212 (1963), 7-25.
J. Gebel, Integer points on Mordell curves [Cached copy, after the original web site tnt.math.se.tmu.ac.jp was shut down in 2017]
H. Mishima, Tables of Elliptic Curves

Cf. A060950, A002151, A002153, A002155, A060748, A060838, A060951-A060953.

for k in[1..2000] do if Rank(EllipticCurve([0,0,0,0,k])) eq 3 then print k; end if; end for; // Vaclav Kotesovec, Jul 07 2019

More terms from T. D. Noe, Jul 24 2007

A179124 Parameters n for which the elliptic curve y^2=x^3+n has rank 4.

Original entry on oeis.org

2089, 3391, 4481, 4910, 6856, 7057, 7954, 9052, 10333, 10636, 10942, 11321, 11665, 12092, 12742, 13191, 13897, 14129, 14668, 15193, 15501, 15641, 15661, 15689, 16306, 16376, 16649

Offset: 1

Artur Jasinski, Jun 30 2010

nonn

See A031507 for the smallest n such that rank of the elliptic curve y^2=x^3+n is some given k.

J. Gebel, Integer points on Mordell curves [Cached copy, after the original web site tnt.math.se.tmu.ac.jp was shut down in 2017]

Cf. A002151 (rank 0), A002153 (rank 1), A002155 (rank 2), A102833 (rank 3), A031507.

a(9)-a(27) from Seiichi Manyama, Jul 07 2019

A179127 Numbers n for which the order of Tate-Shafarevich group Ш (Sha) of the elliptic curve y^2=x^3+n is 4.

Original entry on oeis.org

123, 174, 214, 231, 286, 362, 383, 445, 487, 510, 527, 546, 566, 571, 608, 627, 669, 706, 718, 734, 741, 762, 805, 914, 942, 965, 970, 1019, 1042, 1059, 1075, 1131, 1155, 1166, 1189, 1203, 1210, 1230, 1236, 1245, 1287, 1320, 1355, 1392, 1397, 1410, 1411

Offset: 1

Artur Jasinski, Jun 30 2010

nonn

For n<123 the order of the Tate-Shafarevich group Ш of the elliptic curve y^2=x^3+n is 1.

J. Gebel, Integer points on Mordell curves [Cached copy, after the original web site tnt.math.se.tmu.ac.jp was shut down in 2017]

Cf. A002151, A002153, A002155, A002833, A031507.

Edited by N. J. A. Sloane, Jul 05 2010

A179125 a(n) = A000037(n)^3.

Original entry on oeis.org

8, 27, 125, 216, 343, 512, 1000, 1331, 1728, 2197, 2744, 3375, 4913, 5832, 6859, 8000, 9261, 10648, 12167, 13824, 17576, 19683, 21952, 24389, 27000, 29791, 32768, 35937, 39304, 42875, 50653, 54872, 59319, 64000, 68921, 74088, 79507, 85184, 91125

Offset: 1

Artur Jasinski, Jun 30 2010

nonn

Parameters n for which the torsion subgroup of the elliptic curve y^2=x^3+n has order 2.

Numbers which are perfect cubes (A000578) but not perfect squares (A000290).

Amiram Eldar, Table of n, a(n) for n = 1..10000
Josef Gebel, Integer points on Mordell curves. [Cached copy, after the original web site tnt.math.se.tmu.ac.jp was shut down in 2017]
Josef Gebel, Attila Pethö and Horst G. Zimmer, Computing integral points on Mordell's elliptic curves, Collectanea Mathematica, Vol. 48, No. 1-2 (1997), pp. 115-136; alternative link.
Index to sequences related to elliptic curves

Cf. A002117, A002151, A002153, A002155, A013664, A102833, A031507.

[(n+Floor(1/2+Sqrt(n)))^3: n in [1..60]]; // Vincenzo Librandi, Apr 11 2020

a[n_]:=(n + Floor[1/2 + Sqrt[n]])^3; Array[a,50] (* Vincenzo Librandi, Apr 11 2020 *)

isok(n) = !issquare(n) && ispower(n, 3); \\ Michel Marcus, Nov 02 2013

a(n) = (n + (1+sqrtint(4*n))\2)^3; \\ Michel Marcus, Nov 02 2013

from math import isqrt
def A179125(n): return (n+(k:=isqrt(n))+int(n>k*(k+1)))**3 # Chai Wah Wu, Jun 05 2025

Sum_{n>=1} 1/a(n) = zeta(3) - zeta(6) = A002117 - A013664 = 0.1847138411... - Amiram Eldar, Nov 21 2020

Exponent in the definition corrected by R. J. Mathar, Jul 20 2010

A179126 Positive integers m for which the torsion subgroup of the elliptic curve y^2 = x^3 + m has order 3.

Original entry on oeis.org

4, 9, 16, 25, 36, 49, 81, 100, 121, 144, 169, 196, 225, 256, 289, 324, 361, 400, 441, 484, 529, 576, 625, 676, 784, 841, 900, 961, 1024, 1089, 1156, 1225, 1296, 1369, 1444, 1521, 1600, 1681, 1764, 1849, 1936, 2025, 2116, 2209, 2304, 2401, 2500, 2601, 2704

Offset: 1

Artur Jasinski, Jun 30 2010

Apparently equal to the set of integers (A004709(k))^2, k>=2. [This is incorrect, as shown by the terms 256, 576, 1024, 1600, and 2304. - Jianing Song, Aug 25 2022]
From Jianing Song, Aug 25 2022: (Start)
Numbers which are perfect squares (A000290) but not perfect cubes (A000578). This follows from the complete description of the torsion group of y^2 = x^3 + n, using O to denote the point at infinity (see Exercise 10.19 of Chapter X of Silverman's Arithmetic of elliptic curves):
- If n = t^6 is a sixth power, then the torsion group consists of O, (2*t^2,+-3*t^3), (0,+-t^3), and (-t^2, 0).
- If n = t^2 is not a sixth power, then the torsion group consists of O and (0,+-t).
- If n = t^3 is not a sixth power, then the torsion group consists of O and (-t,0).
- If n is of the form -432*t^6, then the torsion group consists of O and (12*t^2,+-36*t^3).
- In all the other cases, the torsion group is trivial. (End)

Jianing Song, Table of n, a(n) for n = 1..10000
J. Gebel, Integer points on Mordell curves [Cached copy, after the original web site tnt.math.se.tmu.ac.jp was shut down in 2017]
Joseph H. Silverman, The Arithmetic of Elliptic Curves

Cf. A179125, A002151, A002153, A002155, A002833, A031507, A228532.

isA179126(n) = my(k=ispower(n)); !(k%2) && (k%6) \\ Jianing Song, Aug 25 2022

is_A179126 = lambda n: EllipticCurve([0,n]).torsion_order() == 3  # D. S. McNeil, Jan 04 2011

A179128 Numbers n for which order of Tate-Shafarevich group Ш of the elliptic curve y^2=x^3+n is 5.

Original entry on oeis.org

8798, 9834

Offset: 1

Artur Jasinski, Jun 30 2010

For n<123 order of Tate-Shafarevich group Ш of the elliptic curve y^2=x^3+n is 1.

For #Ш=4 see A179127.

J. Gebel, Integer points on Mordell curves [Cached copy, after the original web site tnt.math.se.tmu.ac.jp was shut down in 2017]

Cf. A002151, A002153, A002155, A002833, A031507.

A179130 Parameters k for which the Tate-Shafarevich group Ш of the elliptic curve y^2=x^3+k has order 16.

Original entry on oeis.org

3686, 4010, 4631, 4694, 5730, 6395, 6467, 6493, 7221, 7683, 8222, 8726, 8950, 9237, 9762, 9951, 9965

Offset: 1

Artur Jasinski, Jun 30 2010

For parameters k<123, the order of the Tate-Shafarevich group Ш of the elliptic curve y^2=x^3+k is 1.

J. Gebel, Integer points on Mordell curves [Cached copy, after the original web site tnt.math.se.tmu.ac.jp was shut down in 2017]

Cf. A002151, A002153, A002155, A031507, A179127 (order 4), A179128 (order 5), A179129 (order 9), A179138-A179144.

cp's OEIS Frontend

A060950 Rank of elliptic curve y^2 = x^3 + n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

PARI

Formula

Extensions

A002155 Numbers k for which the rank of the elliptic curve y^2 = x^3 + k is 2.

Views

Author

Keywords

References

Links

Crossrefs

Programs

Magma

Extensions

A002153 Numbers k for which the rank of the elliptic curve y^2 = x^3 + k is 1.

Views

Author

Keywords

References

Links

Crossrefs

Programs

Magma

Extensions

A102833 Numbers n for which rank of the elliptic curve y^2=x^3+n is 3.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Extensions

A179124 Parameters n for which the elliptic curve y^2=x^3+n has rank 4.

Views

Author

Keywords

Comments

Links

Crossrefs

Extensions

A179127 Numbers n for which the order of Tate-Shafarevich group Ш (Sha) of the elliptic curve y^2=x^3+n is 4.

Views

Author

Keywords

Comments

Links

Crossrefs

Extensions

A179125 a(n) = A000037(n)^3.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

PARI

Python

Formula

Extensions

A179126 Positive integers m for which the torsion subgroup of the elliptic curve y^2 = x^3 + m has order 3.

Views

Author

Keywords

Comments

Links