A060951 Rank of elliptic curve y^2 = x^3 - n.

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

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) = A060950(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. 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).

Programs

PARI

{a(n) = if( n<1, 0, length( ellgenerators( ellinit( [ 0, 0, 0, 0, -n], 1))))} /* Michael Somos, Mar 17 2011 */

PARI

apply( {A060951(n)=ellrank(ellinit([0,-n]))[1]}, [1..99]) \\ For version < 2.14, use ellanalyticrank(...). - M. F. Hasler, Jul 01 2024

Formula

a(n) = A060950(27*n) and A060950(n) = a(27*n), so a(n) = a(729*n). - Jonathan Sondow, Sep 10 2013

Extensions

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.