A060953 -id:A060953 - 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

A002151 Numbers k for which rank of the elliptic curve y^2 = x^3 + k is 0.

Original entry on oeis.org

1, 4, 6, 7, 13, 14, 16, 20, 21, 23, 25, 27, 29, 32, 34, 42, 45, 49, 51, 53, 59, 60, 64, 70, 75, 78, 81, 84, 85, 86, 87, 88, 90, 93, 95, 96, 104, 109, 114, 115, 116, 123, 124, 125, 135, 137, 140, 144, 153, 157, 158, 159, 160, 162, 165, 167, 173, 175, 176, 178

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..2907 (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]
L. Lehman, Elliptic Curves of Rank Zero.
H. Mishima, Tables of Elliptic Curves.

Cf. A060950, A002153, A002155, A102833, A060748, A060838, A060951, A060952, A060953.

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

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

The missing entry 123 was added by T. D. Noe, Jul 24 2007

A060951 Rank of elliptic curve y^2 = x^3 - n.

Original entry on oeis.org

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

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) = A060950(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. A031508, A002150, A002152, A002154, A179136, A179137.
Cf. A060748, A060838, A060950, A060952, A060953.
Cf. A081120 (number of integral solutions to Mordell's equation y^2 = x^3 - n).

{a(n) = if( n<1, 0, length( ellgenerators( ellinit( [ 0, 0, 0, 0, -n], 1))))} /* Michael Somos, Mar 17 2011 */

apply( {A060951(n)=ellrank(ellinit([0,-n]))[1]}, [1..99]) \\ For version < 2.14, use ellanalyticrank(...). - M. F. Hasler, Jul 01 2024

a(n) = A060950(27*n) and A060950(n) = a(27*n), so a(n) = a(729*n). - Jonathan Sondow, Sep 10 2013

Corrected Apr 08 2005 at the suggestion of James R. Buddenhagen. There were errors caused by the fact that Mishima lists each curve of rank two twice, once for each generator.

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

A060952 Rank of elliptic curve y^2 = x^3 - n*x.

Original entry on oeis.org

0, 1, 0, 0, 1, 1, 1, 0, 0, 1, 0, 1, 0, 1, 1, 0, 2, 0, 0, 1, 1, 1, 1, 0, 1, 1, 0, 0, 0, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 0, 1, 1, 0, 0, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 1, 2, 1, 1, 0, 1, 0, 1, 0, 0, 2, 1, 0, 0, 1, 1, 1, 1, 1, 1, 0, 1, 2, 1, 1, 1, 0, 3, 0, 1, 1, 1, 1, 0, 1, 2, 0, 0

Offset: 1

N. J. A. Sloane, May 10 2001

Seiichi Manyama, Table of n, a(n) for n = 1..1000
H. Mishima, Tables of Elliptic Curves

Cf. A060748, A060838, A060950-A060953.

{a(n) = ellanalyticrank(ellinit([0, 0, 0, -n, 0]))[1]} \\ Seiichi Manyama, Sep 16 2018

Corrected Apr 08 2005 at the suggestion of James R. Buddenhagen. There were errors caused by the fact that Mishima lists each curve of rank two twice, once for each generator and each curve of rank three thrice.

A002159 Numbers k for which the rank of the elliptic curve y^2 = x^3 + k*x is 1.

Original entry on oeis.org

3, 5, 8, 9, 13, 15, 18, 19, 20, 21, 24, 28, 29, 31, 35, 37, 40, 47, 48, 49, 51, 53, 56, 60, 61, 67, 69, 77, 79, 80, 83, 84, 85, 88, 90, 92, 93, 95, 98, 100, 101, 104, 109, 111, 115, 120, 121, 124, 125, 126, 127, 128, 131, 133, 136, 141, 143, 144, 148, 149, 152, 153, 156

Offset: 1

N. J. A. Sloane

nonn

Terms 80 and 128 are missing in the article by Birch and Swinnerton-Dyer, page 25, table 4b. - Vaclav Kotesovec, Jul 07 2019

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).

Vincenzo Librandi, Table of n, a(n) for n = 1..2040
B. J. Birch and H. P. F. Swinnerton-Dyer, Notes on elliptic curves, I, J. Reine Angew. Math., 212 (1963), 7-25.

Cf. A060953, A076329.

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

for(k=1, 200, if(ellanalyticrank(ellinit([0, 0, 0, k, 0]))[1]==1, print1(k", "))) \\ Seiichi Manyama, Jul 07 2019

More terms added by Seiichi Manyama, Jul 07 2019

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

A002158 Numbers k for which the rank of the elliptic curve y^2 = x^3 + k*x is 0.

Original entry on oeis.org

1, 2, 4, 6, 7, 10, 11, 12, 16, 17, 22, 23, 25, 26, 27, 30, 32, 36, 38, 41, 42, 43, 44, 45, 50, 52, 54, 57, 58, 59, 62, 64, 70, 71, 72, 74, 75, 76, 78, 81, 82, 86, 87, 91, 96, 97, 102, 103, 106, 107, 108, 110, 112, 116, 117, 118, 119, 122, 123, 130, 132, 134, 135, 137, 139, 140, 142, 146, 147, 151, 160, 161, 162, 166, 167, 169, 170, 172, 174, 176, 177, 182, 186, 187, 190, 192, 193, 194, 199

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).

Vincenzo Librandi, Table of n, a(n) for n = 1..2000
B. J. Birch and H. P. F. Swinnerton-Dyer, Notes on elliptic curves, I, J. Reine Angew. Math., 212 (1963), 7-25.

Cf. A002159 (rank 1), A076329 (rank 2).

Cf. A060953.

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

for(k=1, 200, if(ellanalyticrank(ellinit([0, 0, 0, k, 0]))[1]==0, print1(k", "))) \\ Seiichi Manyama, Jul 07 2019

Corrected and extended by Vaclav Kotesovec, Jul 07 2019

New name by Vaclav Kotesovec, Jul 07 2019

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

cp's OEIS Frontend

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

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

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A002151 Numbers k for which rank of the elliptic curve y^2 = x^3 + k is 0.

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A060951 Rank of elliptic curve y^2 = x^3 - n.

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

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A060952 Rank of elliptic curve y^2 = x^3 - n*x.

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A002159 Numbers k for which the rank of the elliptic curve y^2 = x^3 + k*x is 1.

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

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