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

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, 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, 1, 1, 1, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1

Offset

1,19

Comments

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

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

Links

John Voight and Joseph L. Wetherell, Table of n, a(n) for n = 1..10000

Nakao Hisayasu, Tables of the rank and rational points for the elliptic curve x^3 + y^3 = n for n cubefree, 1<=n<=200 (text in Japanese)

Nakao Hisayasu, Tables of the rank and rational points for the elliptic curve x^3 + y^3 = n for n cubefree, 201<=n<=500 (text in Japanese)

Nakao Hisayasu, Tables of the rank and rational points for the elliptic curve x^3 + y^3 = n for n cubefree, 501<=n<=1000 (text in Japanese)

Nakao Hisayasu, Tables of the rank and rational points for the elliptic curve x^3 + y^3 = n for n cubefree, 1001<=n<=1500 (text in Japanese)

Nakao Hisayasu, Tables of the rank and rational points for the elliptic curve x^3 + y^3 = n for n cubefree, 1501<=n<=2000 (text in Japanese)

Nakao Hisayasu, Tables of the rank and rational points for the elliptic curve x^3 + y^3 = n for n cubefree, 2001<=n<=2500 (text in Japanese)

...

Nakao Hisayasu, Tables of the rank and rational points for the elliptic curve x^3 + y^3 = n for n cubefree, 8501<=n<=9000 (text in Japanese)

Nakao Hisayasu, Tables of the rank and rational points for the elliptic curve x^3 + y^3 = n for n cubefree, 9001<=n<=9500 (text in Japanese)

Nakao Hisayasu, Tables of the rank and rational points for the elliptic curve x^3 + y^3 = n for n cubefree, 9501<=n<=10000 (text in Japanese)

N. M. Stephens, The Diophantine equation X^3 + Y^3 = D Z^3 and the conjectures of Birch and Swinnerton-Dyer, J. Reine Angew. Math. 231 (1968), 121-162.

D. Zagier and G. Kramarz, Numerical investigations related to the L-series of certain elliptic curves, J. Indian Math. Soc. 52 (1987), 51-60 (the Ramanujan Centenary volume).

Prog

(Magma)
seq := [];
M := 10000;
for m := 1 to M do
E := EllipticCurve([0, -432*m^2]);
Append(~seq, Rank(E));
end for;
seq;
// John Voight, Nov 02 2017
(PARI) {a(n) = ellanalyticrank(ellinit([0, 0, 0, 0, -432*n^2]))[1]} \\ Seiichi Manyama, Aug 25 2019

Crossrefs

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

Sequence in context: A321886 A060154 A061007 * A206567 A362422 A085252

Adjacent sequences: A060835 A060836 A060837 * A060839 A060840 A060841

Keyword

nonn,nice

Author

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

Extensions

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

Status

approved