A031507 -id:A031507 - 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

A031508 a(n) = smallest k > 0 such that the elliptic curve y^2 = x^3 - k has rank n, or -1 if no such k exists.

Original entry on oeis.org

1, 2, 11, 174, 2351, 28279, 975379

Offset: 0

Noam D. Elkies

See A031507 for the smallest k>0 such that the elliptic curve y^2 = x^3 + k has rank n. - Jonathan Sondow, Sep 06 2013
See A060951 for the rank of y^2 = x^3 - n. - Jonathan Sondow, Sep 10 2013
Gebel, Pethö, & Zimmer: "One experimental observation derived from the tables is that the rank r of Mordell's curves grows according to r = O(log |k|/|log log |k||^(2/3))." Hence this fit suggests a(n) >> exp(n (log n)^(1/3)) where >> is the Vinogradov symbol. - Charles R Greathouse IV, Sep 10 2013
a(7) <= 56877643. a(8) <= 2520963512. a(9) <= 463066403167. a(10) <= 56736325657288. a(11) <= 46111487743732324. a(12) <= 6533891544658786928. See Table 3.3 in [Womack 2003]. - Jose Aranda, Jun 30 2024
The three questions for arbitrary k, positive k, and negative k are not very far from each other because the curves for k and -27k are related by a 3-isogeny and therefore have the same rank. It would be most natural to ask for the minimal |k| for k of either sign [see A373795].  - Noam D. Elkies,  Jul 02 2024
a(16) <= 1160221354461565256631205207888 (Elkies, ANTS-XVI, 2024). The same article also establishes the existence of a value of k which has rank >= 17. - N. J. A. Sloane, Jul 05 2024

			From _M. F. Hasler_, Jul 01 2024: (Start)
Sequence A060951 = (0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 2, 0, 1, 0, 1, 0, 0, 1, ...) gives the analytic rank of the elliptic curve y^2 = x^3 - k for k = 1, 2, 3, ...
We can see that:
  - the smallest k that gives rank 0 is k = 1 = a(0);
  - the smallest k that gives rank 1 is k = 2 = a(1);
  - the smallest k that gives rank 2 is k = 11 = a(2); etc. (End)

Noam D. Elkies, Rank of an elliptic curve and 3-rank of a quadratic field via the Burgess bounds, 2024 Algorithmic Number Theory Symposium, ANTS-XVI, MIT, July 2024.

J. E. Cremona, Elliptic Curve Data
Noam D. Elkies and Zev Klagsbrun, New rank records for elliptic curves having rational torsion, ANTS XIV—Proceedings of the Fourteenth Algorithmic Number Theory Symposium, 233-250. Mathematical Sciences Publishers, Berkeley, CA, 2020.
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]
J. Gebel, A. Pethö, and H. G. Zimmer, On Mordell's equation, Compositio Mathematica 110 (1998), 335-367. MR1602064.
Tom Womack, Explicit Descent on Elliptic Curves, PhD thesis, University of Nottingham, July 2003
Tom Womack, Minimal-known positive and negative k for Mordell curves of given rank (personal web page, latest available snapshot on web.archive.org from Jan. 2017), last modified 10/2002.

Cf. A031507, A373795.

{a(n) = my(k=1); while(ellanalyticrank(ellinit([0, 0, 0, 0, -k]))[1]<>n, k++); k} \\ Seiichi Manyama, Aug 24 2019

{A031508(n)=for(k=1,oo, ellrank(ellinit([0, -k]))[1]==n && return(k))} \\ M. F. Hasler, Jul 01 2024

a(n) = min { k >= 1 | A060951(k) == n }. - M. F. Hasler, Jul 01 2024

Definition clarified by Jonathan Sondow, Oct 26 2013.

Escape clause added to definition by N. J. A. Sloane, Jun 29 2024, because, as John Cremona reminds me, it is not known if k always exists.

A309028 Smallest k>0 such that the elliptic curve y^2 = x^3 + k*x has rank n, if k exists.

Original entry on oeis.org

1, 3, 14, 323, 1918, 195843

Offset: 0

Seiichi Manyama, Jul 08 2019

See A309029 for the smallest negative k.

B. J. Birch and H. P. F. Swinnerton-Dyer, Notes on elliptic curves, I, J. Reine Angew. Math., 212 (1963), 7-25.

Cf. A031507, A060953, A194687, A309029.

Cf. A002158, A002159, A076329, A309030, A309031, A309190.

a(5) from Vaclav Kotesovec, Jul 14 2019

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

A309069 Least k such that the rank of the elliptic curve y^2 = x^3 + k^2 is n.

Original entry on oeis.org

1, 3, 15, 427, 17353

Offset: 0

Seiichi Manyama, Jul 10 2019

Cf. A031507, A060950, A309060, A309068.

{a(n) = my(k=1); while(ellanalyticrank(ellinit([0, 0, 0, 0, k^2]))[1]<>n, k++); k}

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.

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A031508 a(n) = smallest k > 0 such that the elliptic curve y^2 = x^3 - k has rank n, or -1 if no such k exists.

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A309028 Smallest k>0 such that the elliptic curve y^2 = x^3 + k*x has rank n, if k exists.

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

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

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A309069 Least k such that the rank of the elliptic curve y^2 = x^3 + k^2 is n.

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

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

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