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

1,11

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

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

Formula

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

Example

a(1) = A060950(27) = a(729) = 0. - Jonathan Sondow, Sep 10 2013

Prog

(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

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).

Sequence in context: A230001 A070100 A070095 * A115525 A241910 A065717

Adjacent sequences: A060948 A060949 A060950 * A060952 A060953 A060954

Keyword

nonn,nice

Author

N. J. A. Sloane, May 10 2001

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.

Status

approved