A031507 -id:A031507 - cp's OEIS frontend

A373795 a(n) = smallest |k| such that the elliptic curve y^2 = x^3 + k has rank n, or -1 if no such k exists.

1, 2, 11, 113, 2089, 28279, 975379

Offset: 0

N. J. A. Sloane, Jul 04 2024

a(n) = min{ A031507(n), A031508(n) }.
See A031507 and A031508 for further information.
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

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.

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.

Cf. A031507, A031508, A081119.

A179129 Numbers k for which order of Tate-Shafarevich group Ш of the elliptic curve y^2=x^3+k is 9.

Original entry on oeis.org

410, 790, 851, 1294, 1383, 1546, 1635, 1735, 1866, 2139, 2167, 2230, 2363, 2419, 2685, 2743, 2757, 2867, 2958, 3021, 3028, 3119, 3355, 3422, 3490, 3630, 3719, 3903, 3962, 4199, 4365, 4421, 4498, 4722, 4731, 4765, 4927, 4954, 4974, 5011, 5018, 5109

Offset: 1

Artur Jasinski, Jun 30 2010

For k<123 order of Tate-Shafarevich group Ш of the elliptic curve y^2=x^3+k is 1.

For #Ш=4 see A179127. For #Ш=5 see A179128.

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. A002151, A002153, A002155, A002833, A031507, A179127, A179128, A179130.

cp's OEIS Frontend

A373795 a(n) = smallest |k| such that the elliptic curve y^2 = x^3 + k has rank n, or -1 if no such k exists.

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