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Rank of elliptic curve y^2 = x^3 + n.
14

%I #28 Jul 02 2024 02:11:38

%S 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,

%T 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,

%U 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

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

%C 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 T. D. Noe, <a href="/A060950/b060950.txt">Table of n, a(n) for n = 1..10000</a> (from Gebel)

%H 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 H. Mishima, <a href="http://www.asahi-net.or.jp/~KC2H-MSM/ec/eca1/ec01rp.txt">Tables of Elliptic Curves</a>

%H 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 a(n) = A060951(27*n) and A060951(n) = a(27*n), so a(n) = a(729*n). - _Jonathan Sondow_, Sep 10 2013

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

%o (PARI) a(n) = ellanalyticrank(ellinit([0, 0, 0, 0, n]))[1] \\ _Jianing Song_, Aug 24 2022

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

%Y Cf. A031507, A002151, A002153, A002155, A102833, A179124.

%Y Cf. A060748, A060838, A060951, A060952, A060953.

%Y Cf. A081119 (number of integral solutions to Mordell's equation y^2 = x^3 + n).

%K nonn,nice

%O 1,15

%A _N. J. A. Sloane_, May 10 2001

%E Corrected by _James R. Buddenhagen_, Feb 18 2005