A060838 -id:A060838 - 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

A060748 a(n) is the smallest m such that the elliptic curve x^3 + y^3 = m has rank n, or -1 if no such m exists.

Original entry on oeis.org

1, 6, 19, 657, 21691, 489489, 9902523, 1144421889, 1683200989470, 349043376293530, 137006962414679910, 13293998056584952174157235

Offset: 0

N. J. A. Sloane, Apr 23 2001

From Nick Rogers (rogers(AT)fas.harvard.edu), Jul 03 2003: (Start)
I have verified that the first 5 entries are correct; the first two are basically trivial and the third is due to Selmer. I'm not sure who first discovered entries 4 and 5 and I expect that they had been previously proved to be the smallest values.
But I have rechecked that they are minimal for their respective rank using a combination of 3-descent, Magma and John Cremona's program mwrank.
There are new smaller values for ranks 6 and 7, namely k = 9902523 has rank 6 and k = 1144421889 has rank 7. 3-descent combined with Ian Connell's package apecs for Maple verifies that these are minimal subject to the Birch and Swinnerton-Dyer conjecture and the Generalized Riemann Hypothesis for L-functions associated to elliptic curves.
Finally, there are new entries for ranks 8 and 9: k = 1683200989470 has rank 8 and k = 148975046052222390 has rank 9. It seems somewhat likely that the rank 8 example is minimal. (End)
The sequence might be finite, even if it is redefined as smallest m such that x^3 + y^3 = m has rank >= n. - Jonathan Sondow, Oct 27 2013

Noam D. Elkies, Yet more rank records for x^3+y^3=k, Posting to Number Theory List, Oct 19 2003, for a(9)
Noam D. Elkies and Nicholas F. Rogers, Rank records for x^3+y^3=k, cont'd, Posting to Number Theory List, Jul 18 2003, for a(8) and a(9).
Noam D. Elkies and Nicholas F. Rogers, Elliptic curves x^3 + y^3 = k of high rank, Algorithmic Number Theory, 6th International Symposium, ANTS-VI, Burlington, VT, USA, June 13-18, 2004, Proceedings, Springer, Berlin, Heidelberg, 2004, pp. 184-193. See also the arXiv versionarXiv:math/0403116 [math.NT], 2004.
Troy Kessler, 3 descent on elliptic curve, Posting to Number Theory List, Apr 22, 2001.
Nick Rogers, Rank computations for the congruent number elliptic curves. Experimental Mathematics 9 (2000), no. 4, 591-594.

Positions of records in A060838.

Cf. A230564.

{a(n) = my(k=1); while(ellanalyticrank(ellinit([0, 0, 0, 0, -432*k^2]))[1]<>n, k++); k} \\ Seiichi Manyama, Aug 24 2019

Definition clarified by Jonathan Sondow, Oct 27 2013
a(10)-a(11) from Amiram Eldar were taken from the paper by Elkies & Rogers, Jul 27 2017.
Escape clause added by N. J. A. Sloane, Oct 26 2017

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

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.

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

A309960 Numbers k for which rank of the elliptic curve y^2 = x^3 - 432*k^2 is 0.

Original entry on oeis.org

1, 2, 3, 4, 5, 8, 10, 11, 14, 16, 18, 21, 23, 24, 25, 27, 29, 32, 36, 38, 39, 40, 41, 44, 45, 46, 47, 52, 54, 55, 57, 59, 60, 64, 66, 73, 74, 76, 77, 80, 81, 82, 83, 88, 93, 95, 99, 100, 101, 102, 108, 109, 111, 112, 113, 116, 118, 119, 121, 122, 125, 128, 129, 131, 135, 137

Offset: 1

Seiichi Manyama, Aug 25 2019

nonn

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Complement of A159843 \ A000578.

Cf. A060748, A060838, A309961 (rank 1), A309962 (rank 2), A309963 (rank 3), A309964 (rank 4).

for(k=1, 200, if(ellanalyticrank(ellinit([0, 0, 0, 0, -432*k^2]))[1]==0, print1(k", ")))

is(n, f=factor(n))=my(c=prod(i=1, #f~, f[i, 1]^(f[i, 2]\3)), r=n/c^3, E, eri, mwr, ar); if(r<6, return(1)); E=ellinit([0, 16*r^2]); eri=ellrankinit(E); mwr=ellrank(eri); if(mwr[1], return(0)); ar=ellanalyticrank(E)[1]; if(ar<2, return(!ar)); for(effort=1, 99, mwr=ellrank(eri, effort); if(mwr[1]>0, return(0), mwr[2]<1, return(1))); "unknown (0 under BSD conjecture)" \\ Charles R Greathouse IV, Jan 24 2023

A060838(a(n)) = 0.

cp's OEIS Frontend

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

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A060748 a(n) is the smallest m such that the elliptic curve x^3 + y^3 = m has rank n, or -1 if no such m exists.

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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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Magma

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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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Magma

Extensions

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

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Author

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Examples

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Crossrefs

Programs

PARI

PARI

Formula

Extensions

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

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Keywords

References

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Programs

Magma

Extensions

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

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