A060950 -id:A060950 - cp's OEIS frontend

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.

1, 2, 11, 174, 2351, 28279, 975379

Offset: 0

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.

A309069 Least k such that the rank of the elliptic curve y^2 = x^3 + k^2 is n.

Original entry on oeis.org

1, 3, 15, 427, 17353

Offset: 0

Seiichi Manyama, Jul 10 2019

Cf. A031507, A060950, A309060, A309068.

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

A356730 Conductor of the elliptic curve y^2 = x^3 + n.

Original entry on oeis.org

36, 1728, 3888, 108, 2700, 15552, 21168, 576, 972, 14400, 52272, 3888, 18252, 84672, 97200, 27, 10404, 15552, 51984, 2700, 47628, 209088, 228528, 15552, 2700, 97344, 144, 7056, 90828, 388800, 415152, 1728, 117612, 499392, 176400, 972, 49284, 623808, 657072, 43200, 181548

Offset: 1

Jianing Song, Aug 24 2022

nonn

The discriminant of the elliptic curve y^2 = x^3 - n is -432*n^2 and the rank is A060950(n).
a(n*t^6) = a(n) for all t since the elliptic curve y^2 = x^3 + n*t^6 can be written as (y/t^3)^2 = (x/t^2)^3 + n, and the conductor is an invariant of elliptic curves.
Conjectures: (Start)
(i) a(27*n) = A356731(n) for all n.
(ii) a(n) is divisible by 36, and a(n) = 36 <=> n is a sixth power, a(n) = 108 <=> n is 4 times a sixth power, a(n) = 144 <=> n is 27 times a sixth power; moreover, it seems that a(n) is divisible by 36*n^2 if n is squarefree. (End)

Jianing Song, Table of n, a(n) for n = 1..10000
LMFDB, Elliptic Curves Over Q

Cf. A356731, A060950.

a(n) = ellglobalred(ellinit([0,0,0,0,n]))[1]

A386928 Algebraic rank of elliptic curve y^2 = x^3 + n*x + n.

Original entry on oeis.org

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

Offset: 1

Shreyansh Jaiswal, Aug 08 2025

Terms from n = 29 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) = 1 because y^2 = x^3 + x + 1 has rank 1.

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

Cf. A060950, A060953.

a(n) = ellanalyticrank(ellinit([n, n]))[1]; \\ Jinyuan Wang, Aug 08 2025

for k in range(1,29):
    E = EllipticCurve([k,k])
    print(E.rank(),end=", ")

More terms from Jinyuan Wang, Aug 08 2025

cp's OEIS Frontend

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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A309069 Least k such that the rank of the elliptic curve y^2 = x^3 + k^2 is n.

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A356730 Conductor of the elliptic curve y^2 = x^3 + n.

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A386928 Algebraic rank of elliptic curve y^2 = x^3 + n*x + n.

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