A002150 -id:A002150 - cp's OEIS frontend

A031507 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.

1, 2, 15, 113, 2089, 66265, 1358556

Offset: 0

See A031508 for the smallest negative k. - Artur Jasinski, Nov 21 2011
See A060950 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
The curves for k and -27*k are isogenous (as Noam Elkies points out---see Womack), so they have the same rank. - Jonathan Sondow, Sep 10 2013
Womack (2003) gives further upper bounds: a(7) <= 47550317, a(8) <= 1632201497, a(9) <= 185418133372, a(10) <=  68513487607153. - M. F. Hasler, Jul 01 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

			a(12) <= 27*A031508(12) <= 27*6533891544658786928 = 176415071705787247056 (from Quer 1987 and Womack). - _Jonathan Sondow_, Sep 10 2013

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, web.archive.org copy of the "MORDELL+" file on the SIMATH web site shut down in 2017. [Locally cached copy].
J. Gebel, A. Pethö and H. G. Zimmer, On Mordell's equation, Compositio Math. 110 (1998), 335-367. (doi:10.1023/A:1000281602647 not working as of July 2024.)
J. Quer, Corps quadratiques de 3-rang 6 et courbes elliptiques de rang 12, C. R. Acad. Sc. Paris I, 305 (1987), 215-218.
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 Oct. 2002.

Cf. A031508, A373795.

{A031507(n)=for(k=1, oo, ellrank(ellinit([0, k]))[1]==n && return(k))} \\ Use ellanalyticrank() for PARI version < 2.14. - M. F. Hasler, Jul 01 2024

a(n) <= 27*A031508(n) and A031508(n) <= 27*a(n). - Jonathan Sondow, Sep 10 2013

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.

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.

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.

A076329 Numbers k for which rank of the elliptic curve y^2=x^3+k*x is 2.

Original entry on oeis.org

14, 33, 34, 39, 46, 55, 63, 65, 66, 68, 73, 89, 94, 99, 105, 113, 114, 129, 138, 145, 150, 154, 155, 158, 178, 183, 185, 201, 203, 206, 209, 219, 224, 226, 233, 238, 254, 258, 260, 273, 274, 281, 289, 295, 299, 308, 310, 333, 334, 337, 345, 353, 354, 360, 385, 386, 388

Offset: 1

N. J. A. Sloane, Nov 06 2002

nonn

D. S. Jandu, Elliptic Curve, Infinite Bandwidth Publishing, N. Hollywood CA 2007.
D. S. Jandu, Birch And Swinnerton Dyer Conjecture, Infinite Bandwidth Publishing, N. Hollywood CA 2007.
D. Zagier & G. Harder, "La conjecture de Birch et Swinnerton-Dyer" in Les Dossiers de La Recherche, pp. 48-53 No. 20 August-October 2005 Paris.

B. J. Birch and H. P. F. Swinnerton-Dyer, Notes on elliptic curves, I, J. Reine Angew. Math., 212 (1963), 7-25.
A. L. Robledo, PlanetMath.org, Birch and Swinnerton-Dyer conjecture
W. A. Stein, The Birch and Swinnerton-Dyer Conjecture, Part 1
W. A. Stein, The Birch and Swinnerton-Dyer Conjecture, Part 2
W. A. Stein, The Birch and Swinnerton-Dyer Conjecture, Part 3
Wikipedia, Birch and Swinnerton-Dyer conjecture

Cf. A002150-A002159.

for(k=1, 1e3, if(ellanalyticrank(ellinit([0, 0, 0, k, 0]))[1]==2, print1(k", "))) \\ Seiichi Manyama, Jul 07 2019

More terms added by Seiichi Manyama, Jul 07 2019

A179136 Parameters n for which the elliptic curve y^2=x^3-n has rank 3.

Original entry on oeis.org

174, 307, 362, 431, 503, 516, 706, 713, 741, 755, 804, 984, 1048, 1075, 1173, 1187, 1192, 1208, 1236, 1259, 1315, 1356, 1439, 1478, 1588, 1607, 1668, 1712, 1724, 1727, 1763, 1777, 1812, 1902, 1951, 1966, 1999, 2001, 2036, 2071, 2181, 2188, 2198, 2219

Offset: 1

Artur Jasinski, Jun 30 2010

nonn

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. A002150 (rank 0), A002152 (rank 1), A002154 (rank 2), A031508, A179137 (rank 4).

A179137 Parameters n for which the elliptic curve y^2=x^3-n has rank 4.

Original entry on oeis.org

2351, 3896, 4799, 4827, 5417, 5835, 6691, 6843, 9748, 9967, 10723, 11559, 12163, 12394, 12891, 13971, 14188, 14907, 15049, 15544

Offset: 1

Artur Jasinski, Jun 30 2010

nonn

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. A002150 (rank 0), A002152 (rank 1), A002154 (rank 2), A179136 (rank 3).

Cf. A031508.

for(k=1, 1e4, if(ellanalyticrank(ellinit([0, 0, 0, 0, -k]))[1]==4, print1(k", "))) \\ Seiichi Manyama, Jul 07 2019

a(11)-a(20) from Seiichi Manyama, Jul 07 2019

A076330 Elliptic curves (see reference for precise definition).

Original entry on oeis.org

17, 41, 57, 62, 82, 97, 117, 137, 142, 146, 161, 177, 193, 194, 392, 452, 792

Offset: 1

N. J. A. Sloane, Nov 06 2002

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

Cf. A002150-A002159. Subsequence of A002158.

A076331 Elliptic curves (see reference for precise definition).

Original entry on oeis.org

136, 584, 712, 776

Offset: 1

N. J. A. Sloane, Nov 06 2002

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

Cf. A002150-A002159. Subsequence of A002159.

cp's OEIS Frontend

A031507 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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A060951 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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PARI

PARI

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A076329 Numbers k for which rank of the elliptic curve y^2=x^3+k*x is 2.

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Programs

PARI

Extensions

A179136 Parameters n for which the elliptic curve y^2=x^3-n has rank 3.

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A179137 Parameters n for which the elliptic curve y^2=x^3-n has rank 4.

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Author

Keywords

Links

Crossrefs

Programs

PARI

Extensions

A076330 Elliptic curves (see reference for precise definition).

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Author

Keywords

Links

Crossrefs

A076331 Elliptic curves (see reference for precise definition).

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Author

Keywords

Links

Crossrefs