A060952 -id:A060952 - cp's OEIS frontend

A060953 Rank of elliptic curve y^2 = x^3 + n*x.

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

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
F. Richman, Elliptic curves
K. Rubin and A. Silverberg, Ranks of elliptic curves

Cf. A060748, A060838, A060950-A060953.

{ A060953(n) = ellanalyticrank( ellinit([0,0,0,n,0]) )[1]; }

a(-n) = A060952(n). - Michael Somos, Dec 15 2011

Lambert Klasen (Lambert.Klasen(AT)gmx.net), Mar 31 2005, kindly rechecked this sequence against the Mishima web site and found no errors.
Corrected Apr 10 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.
Extended by Max Alekseyev, Mar 09 2009

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

Original entry on oeis.org

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

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.

A002156 Numbers k for which the rank of the elliptic curve y^2 = x^3 - k*x is 0.

Original entry on oeis.org

1, 3, 4, 8, 9, 11, 13, 16, 18, 19, 24, 27, 28, 29, 33, 35, 40, 43, 44, 48, 51, 59, 61, 63, 64, 67, 68, 75, 81, 83, 88, 91, 92, 93, 98, 100, 104, 107, 108, 109, 113, 115, 120, 121, 123, 125, 126, 128, 129, 131, 139, 144, 152, 153, 157, 163, 164, 168, 172, 173, 176, 177, 179, 180, 187, 189, 193, 195, 198, 200

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..1730
B. J. Birch and H. P. F. Swinnerton-Dyer, Notes on elliptic curves, I, J. Reine Angew. Math., 212 (1963), 7-25.
B. J. Birch and H. P. F. Swinnerton-Dyer, Notes on elliptic curves, I, J. Reine Angew. Math., 212 (1963), 7-25 (open access).

Cf. A060952.

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

A002157 Numbers k for which the rank of the elliptic curve y^2 = x^3 - k*x is 1.

Original entry on oeis.org

2, 5, 6, 7, 10, 12, 14, 15, 20, 21, 22, 23, 25, 26, 30, 31, 32, 34, 36, 37, 38, 39, 41, 42, 45, 46, 47, 49, 50, 52, 53, 54, 55, 57, 58, 60, 62, 66, 69, 70, 71, 72, 73, 74, 76, 78, 79, 80, 84, 85, 86, 87, 89, 94, 95, 96, 99, 101, 102, 103, 105, 106, 110, 111, 112, 114, 116

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).

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

Cf. A060952.

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

A309034 Numbers k for which rank of the elliptic curve y^2=x^3-k*x is 4.

Original entry on oeis.org

5037, 5795, 6497, 7585, 7672, 8701, 10001, 10081, 10605, 14547, 14637, 15805, 20091, 20737, 20760, 21177, 21571, 22321, 23137, 24492, 27812, 30877, 31595, 33026, 34241, 36737, 38412, 38497, 41021, 41907, 41922, 42347, 43036

Offset: 1

Seiichi Manyama, Jul 08 2019

nonn

Cf. A002156 (rank 0), A002157 (rank 1). A309032 (rank 2), A309033 (rank 3), this sequence (rank 4), A309100 (rank 5).

Cf. A060952, A309031.

for k in[1..10000] do if Rank(EllipticCurve([0,0,0,-k,0])) eq 4 then print k; end if; end for; // Vaclav Kotesovec, Jul 08 2019

for(k=1, 1e4, if(ellanalyticrank(ellinit([0, 0, 0, -k, 0]))[1]==4, print1(k", ")))

a(29)-a(33) from Seiichi Manyama, Jul 09 2019

A319510 Rank of elliptic curve y^2 = x^3 - n^2 * x.

Original entry on oeis.org

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

Offset: 1

Seiichi Manyama, Sep 24 2018

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..5000

Cf. A003273, A060952, A062693, A062695, A194687, A273929.

{a(n) = ellanalyticrank(ellinit([0, 0, 0, -n^2, 0]))[1]}

a(n) = A060952(n^2).
a(A003273(n)) > 0.
a(A194687(n)) = n.
Empirical: a(n) = a(4*n). - Jose Aranda, Jul 02 2024

A309029 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, 2, 17, 82, 5037, 49042

Offset: 0

Seiichi Manyama, Jul 08 2019

See A309028 for the smallest positive k.

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

Cf. A031508, A060952, A194687, A309028.

Cf. A002156, A002157, A309032, A309033, A309034, A309100.

a(5) from Vaclav Kotesovec, Jul 09 2019

A385881 Algebraic rank of elliptic curve y^2 = x^3 - n*x - n.

Original entry on oeis.org

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

Offset: 1

Shreyansh Jaiswal, Aug 20 2025

Terms from n = 43 onward are the analytic ranks (see PARI code) of the corresponding elliptic curves. By the BSD conjecture, these are expected to equal the algebraic ranks. Thus, the validity of these terms is conditional on BSD.

			a(1) = 0 because y^2 = x^3 - x - 1 has rank 0.

LMFDB, y^2 = x^3 - x - 1.

Cf. A060951, A060952.

a(n) = ellanalyticrank( ellinit([0, 0, 0, -n, -n]) )[1]; \\ Michel Marcus, Aug 20 2025

for k in range(1, 43):
    E = EllipticCurve([-k, -k])
    print(E.rank(), end=", ")

More terms from Michel Marcus, Aug 20 2025

cp's OEIS Frontend

A060953 Rank of elliptic curve y^2 = x^3 + n*x.

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

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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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A002156 Numbers k for which the rank of the elliptic curve y^2 = x^3 - k*x is 0.

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Magma

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A002157 Numbers k for which the rank of the elliptic curve y^2 = x^3 - k*x is 1.

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References

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Magma

PARI

Extensions

A309034 Numbers k for which rank of the elliptic curve y^2=x^3-k*x is 4.

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A319510 Rank of elliptic curve y^2 = x^3 - n^2 * x.

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