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

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

Author

N. J. A. Sloane, May 10 2001

Keywords

• nonn
• nice

Comments

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

Examples

			a(1) = A060951(27) = a(729) = 0. - _Jonathan Sondow_, Sep 10 2013
		

Links

• 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

Crossrefs

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).

Programs

PARI

a(n) = ellanalyticrank(ellinit([0, 0, 0, 0, n]))[1] \\ Jianing Song, Aug 24 2022

PARI

apply( {A060950(n)=ellrank(ellinit([0, n]))[1]}, [1..99]) \\ For PARI version  < 2.14, use ellanalyticrank(...). - M. F. Hasler, Jul 01 2024

Formula

a(n) = A060951(27*n) and A060951(n) = a(27*n), so a(n) = a(729*n). - Jonathan Sondow, Sep 10 2013

Extensions

Corrected by James R. Buddenhagen, Feb 18 2005