A060951 -id:A060951 - cp's OEIS frontend

A081120 Number of integral solutions to Mordell's equation y^2 = x^3 - n.

1, 2, 0, 4, 0, 0, 4, 1, 0, 0, 4, 0, 2, 0, 2, 0, 0, 2, 2, 2, 0, 0, 2, 0, 2, 4, 1, 6, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 6, 2, 0, 0, 0, 2, 2, 0, 6, 4, 2, 0, 0, 0, 4, 2, 4, 2, 0, 0, 0, 4, 2, 0, 4, 1, 0, 0, 2, 0, 0, 0, 2, 2, 0, 2, 0, 4, 0, 0, 2, 0, 2, 0, 2, 0, 0, 0, 2, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 6

Offset: 1

T. D. Noe, Mar 06 2003

Mordell's equation has a finite number of integral solutions for all nonzero n.
Gebel, Pethö, and Zimmer (1998) computed the solutions for |n| <= 10^4. Bennett and Ghadermarzi (2015) extended this bound to |n| <= 10^7.
Sequence A081121 gives n for which there are no integral solutions. See A081119 for the number of integral solutions to y^2 = x^3 + n.
From Jianing Song, Aug 24 2022: (Start)
If A060951(n) = 0 (namely the elliptic curve y^2 = x^3 - n has rank 0), then:
- a(n) = 2 if n is of the form 432*t^6;
- a(n) = 1 if n is a cube;
- a(n) = 0 otherwise.
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.
So a torsion point on y^2 = x^3 + n other than O is an integral point. If y^2 = x^3 + n has rank 0, then all the integral points on y^2 = x^3 + n are exactly the torsion points other than O.
Note that this result implies particularly that a(n) = a(n*t^6) for all t if A060951(n) = 0: the elliptic curve y^2 = x^3 - n*t^6 can be written as (y/t^3)^2 = (x/t^2)^3 - n, so it has the same Mordell-Weil group (hence the same rank and isomorphic torsion group) as y^2 = x^3 - n. (End)

			a(4)=4 refers to (x,y) = (2,+-2) and (5,+-11).

T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, page 191.

Jean-François Alcover, Table of n, a(n) for n = 1..10000 [There were errors in the previous b-file, which had 10000 terms contributed by T. D. Noe and based on the work of J. Gebel.]
M. A. Bennett and A. Ghadermarzi, Mordell's equation: a classical approach. LMS J. Compute. Math. 18 (2015): 633-646. doi:10.1112/S1461157015000182 arXiv:1311.7077
J. Gebel, A. Pethö, and H. G. Zimmer, On Mordell's equation, Compositio Mathematica. 110:3 (1998): 335-367.
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.
Eric Weisstein's World of Mathematics, Mordell Curve.
Wikipedia, Mordell curve.

Cf. A081119, A081121. See A134109 for another version.

(* This naive approach gives correct results up to n=1000 *) xmax[] = 10^4; Do[ xmax[n] = 10^5, {n, {366, 775, 999}}]; Do[ xmax[n] = 10^6, {n, {207, 307, 847}}]; f[n] := (x = Floor[n^(1/3)] - 1; s = {}; While[ x <= xmax[n], x++; y2 = x^3 - n; If[y2 >= 0, y = Sqrt[y2]; If[ IntegerQ[y], AppendTo[s, y]]]]; s); a[n_] := (fn = f[n]; If[fn == {}, 0, 2 Length[fn] - If[ First[fn] == 0, 1, 0]]); Table[ an = a[n]; Print["a[", n, "] = ", an]; an, {n, 1, 100}] (* Jean-François Alcover, Mar 06 2012 *)

Edited by Max Alekseyev, Feb 06 2021

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

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.

Original entry on oeis.org

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

Offset: 0

Noam D. Elkies

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.

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

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

A002150 Numbers k for which the rank of the elliptic curve y^2 = x^3 - k is 0.

Original entry on oeis.org

1, 3, 5, 6, 8, 9, 10, 12, 14, 16, 17, 24, 27, 31, 32, 33, 34, 36, 37, 41, 42, 46, 52, 62, 64, 68, 69, 70, 73, 77, 78, 80, 82, 86, 88, 90, 92, 96, 97, 98, 99, 103, 105, 108, 111, 113, 114, 117, 119, 122, 125, 132, 133, 134, 136, 141, 142, 144, 145, 149, 154, 156, 158, 160

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. A002152 (rank 1), A002154 (rank 2), A179136 (rank 3), A179137 (rank 4).

Cf. A060951.

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

for(k=1, 200, if(ellanalyticrank(ellinit([0, 0, 0, 0, -k]))[1]==0, print1(k", "))) \\ Seiichi Manyama, Jul 06 2019

Better definition from Artur Jasinski, Jun 30 2010

More terms added by Seiichi Manyama, Jul 06 2019

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.

A309068 Least k such that the rank of the elliptic curve y^2 = x^3 - k^2 is n.

Original entry on oeis.org

1, 2, 11, 362, 7954

Offset: 0

Seiichi Manyama, Jul 10 2019

Cf. A031508, A060951, A194687, A309069.

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

cp's OEIS Frontend

A081120 Number of integral solutions to Mordell's equation y^2 = x^3 - n.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

Extensions

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

PARI

Formula

Extensions

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

Views

Author

Keywords

References

Links

Crossrefs

Programs

Magma

Extensions

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

Views

Author

Keywords

References

Links

Crossrefs

Programs

Magma

Extensions

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.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

PARI

Formula

Extensions

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

Views

Author

Keywords

References

Links

Crossrefs

Programs

Magma

Extensions

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

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Extensions

A002150 Numbers k for which the rank of the elliptic curve y^2 = x^3 - k is 0.

Views