cp's OEIS Frontend

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

%I A060951 #38 Jul 02 2024 02:16:33
%S A060951 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,
%T A060951 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,
%U A060951 0,0,1,1,0,1,1,2,0,0,1,0,1,0,2,1,1,0,1,0,2
%N A060951 Rank of elliptic curve y^2 = x^3 - n.
%C A060951 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
%H A060951 T. D. Noe, <a href="/A060951/b060951.txt">Table of n, a(n) for n = 1..10000</a> (from Gebel)
%H A060951 J. Gebel, <a href="/A001014/a001014.txt">Integer points on Mordell curves</a> [Cached copy, after the original web site tnt.math.se.tmu.ac.jp was shut down in 2017]
%H A060951 H. Mishima, <a href="http://www.asahi-net.or.jp/~KC2H-MSM/ec/eca1/ec01rm.txt">Tables of Elliptic Curves</a>
%H A060951 T. Womack, <a href="http://www.tom.womack.net/maths/mordellc.htm">Minimal-known positive and negative k for Mordell curves of given rank</a>
%F A060951 a(n) = A060950(27*n) and A060950(n) = a(27*n), so a(n) = a(729*n). - _Jonathan Sondow_, Sep 10 2013
%e A060951 a(1) = A060950(27) = a(729) = 0. - _Jonathan Sondow_, Sep 10 2013
%o A060951 (PARI) {a(n) = if( n<1, 0, length( ellgenerators( ellinit( [ 0, 0, 0, 0, -n], 1))))} /* _Michael Somos_, Mar 17 2011 */
%o A060951 (PARI) apply( {A060951(n)=ellrank(ellinit([0,-n]))[1]}, [1..99]) \\ For version < 2.14, use ellanalyticrank(...). - _M. F. Hasler_, Jul 01 2024
%Y A060951 Cf. A031508, A002150, A002152, A002154, A179136, A179137.
%Y A060951 Cf. A060748, A060838, A060950, A060952, A060953.
%Y A060951 Cf. A081120 (number of integral solutions to Mordell's equation y^2 = x^3 - n).
%K A060951 nonn,nice
%O A060951 1,11
%A A060951 _N. J. A. Sloane_, May 10 2001
%E A060951 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.