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A060838 Rank of elliptic curve x^3 + y^3 = n. 18

%I #71 Feb 17 2022 00:09:57

%S 0,0,0,0,0,1,1,0,1,0,0,1,1,0,1,0,1,0,2,1,0,1,0,0,0,1,0,1,0,2,1,0,1,1,

%T 1,0,2,0,0,0,0,1,1,0,0,0,0,1,1,1,1,0,1,0,0,1,0,1,0,0,1,1,1,0,2,0,1,1,

%U 1,1,1,1,0,0,1,0,0,1,1,0,0,0,0,1,1

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

%C The elliptic curve X^3 + Y^3 = D*Z^3 where D is a rational integer has a birationally equivalent form y^2*z = x^3 - 2^4*3^3*D^2*z^3 where x = 2^2*3*D*Z, y = 2^2*3^3*D*(Y - X), z = X + Y (see p. 123 of Stephens). Taking z = 1 and 2^2*3^3 = 432 yields y^2 = x^3 - 432*D^2, which is the Weierstrass form of the elliptic curve used by John Voight in the Magma program below. - _Ralf Steiner_, Nov 11 2017

%C Zagier and Kramarz studied the analytic rank of the curve E: x^3 + y^3 = m, where m is cubefree. They computed L(E,1) for 0 < m <= 70000 and also L'(E,1) if the sign of the functional equation for L(E,1) was negative. In the second case the range was only 0 < m <= 20000. - Attila Pethő, Posting to the Number Theory List, Nov 11 2017

%H John Voight and Joseph L. Wetherell, <a href="/A060838/b060838.txt">Table of n, a(n) for n = 1..10000</a>

%H Nakao Hisayasu, <a href="http://www.kaynet.or.jp/~kay/misc/nna2.html">Tables of the rank and rational points for the elliptic curve x^3 + y^3 = n for n cubefree, 1<=n<=200</a> (text in Japanese)

%H Nakao Hisayasu, <a href="http://www.kaynet.or.jp/~kay/misc/nna3.html">Tables of the rank and rational points for the elliptic curve x^3 + y^3 = n for n cubefree, 201<=n<=500</a> (text in Japanese)

%H Nakao Hisayasu, <a href="http://www.kaynet.or.jp/~kay/misc/nna4.html">Tables of the rank and rational points for the elliptic curve x^3 + y^3 = n for n cubefree, 501<=n<=1000</a> (text in Japanese)

%H Nakao Hisayasu, <a href="http://www.kaynet.or.jp/~kay/misc/nna5.html">Tables of the rank and rational points for the elliptic curve x^3 + y^3 = n for n cubefree, 1001<=n<=1500</a> (text in Japanese)

%H Nakao Hisayasu, <a href="http://www.kaynet.or.jp/~kay/misc/nna6.html">Tables of the rank and rational points for the elliptic curve x^3 + y^3 = n for n cubefree, 1501<=n<=2000</a> (text in Japanese)

%H Nakao Hisayasu, <a href="http://www.kaynet.or.jp/~kay/misc/nna7.html">Tables of the rank and rational points for the elliptic curve x^3 + y^3 = n for n cubefree, 2001<=n<=2500</a> (text in Japanese)

%H ...

%H Nakao Hisayasu, <a href="http://www.kaynet.or.jp/~kay/misc/nna20.html">Tables of the rank and rational points for the elliptic curve x^3 + y^3 = n for n cubefree, 8501<=n<=9000</a> (text in Japanese)

%H Nakao Hisayasu, <a href="http://www.kaynet.or.jp/~kay/misc/nna21.html">Tables of the rank and rational points for the elliptic curve x^3 + y^3 = n for n cubefree, 9001<=n<=9500</a> (text in Japanese)

%H Nakao Hisayasu, <a href="http://www.kaynet.or.jp/~kay/misc/nna22.html">Tables of the rank and rational points for the elliptic curve x^3 + y^3 = n for n cubefree, 9501<=n<=10000</a> (text in Japanese)

%H N. M. Stephens, <a href="https://doi.org/10.1515/crll.1968.231.121">The Diophantine equation X^3 + Y^3 = D Z^3 and the conjectures of Birch and Swinnerton-Dyer</a>, J. Reine Angew. Math. 231 (1968), 121-162.

%H D. Zagier and G. Kramarz, <a href="http://www.informaticsjournals.com/index.php/jims/article/view/21978">Numerical investigations related to the L-series of certain elliptic curves</a>, J. Indian Math. Soc. 52 (1987), 51-60 (the Ramanujan Centenary volume).

%o (Magma)

%o seq := [];

%o M := 10000;

%o for m := 1 to M do

%o E := EllipticCurve([0,-432*m^2]);

%o Append(~seq, Rank(E));

%o end for;

%o seq;

%o // John Voight, Nov 02 2017

%o (PARI) {a(n) = ellanalyticrank(ellinit([0, 0, 0, 0, -432*n^2]))[1]} \\ _Seiichi Manyama_, Aug 25 2019

%Y Cf. A060748 (positions of records in this sequence), A060950.

%K nonn,nice

%O 1,19

%A Noam Katz (noamkj(AT)hotmail.com), May 02 2001

%E Many thanks to _Andrew V. Sutherland_, John Voight, and _Joseph L. Wetherell_, who all responded to my request for additional terms for this sequence. - _N. J. A. Sloane_, Nov 01 2017

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Last modified March 28 18:04 EDT 2024. Contains 371254 sequences. (Running on oeis4.)