A052432 Smallest conductor of elliptic curve with rank n.

11, 37, 389, 5077, 234446

Offset

0,1

Comments

The smallest known conductors for ranks 5, 6, 7, and 11 are 19047851, 5187563742, 382623908456, 18031737725935636520843, respectively. These are just upper bounds on a(n).

Links

Table of n, a(n) for n=0..4.

J. E. Cremona, Elliptic Curve Data

J. E. Cremona, Best known conductors for elliptic curves of given rank

Noam D. Elkies and Mark Watkins, Elliptic curves of large rank and small conductor, arXiv:math/0403374 [math.NT], 2004.

M. O. Rubenstein, Elliptic curves of high rank and the Riemann zeta function, arXiv:1307.0420 [math.NT], 2013.

Example

From Michael Somos, Apr 12 2012: (Start)
The curve "11a3": y^2 + y = x^3 - x^2 has rank 0.
The curve "37a1": y^2 + y = x^3 - x has rank 1 with generator [0, 0].
The curve "389a1": y^2 + y = x^3 + x^2 - 2 * x has rank 2 with generators [0, 0], [-1, 1].
The curve "5077a1": y^2 + y = x^3 - 7 * x + 6 has rank 3 with generators [0, 2], [-1, 3], [-2, 3]. (End)

Crossrefs

Sequence in context: A120833 A261420 A034969 * A104269 A084014 A232976

Adjacent sequences: A052429 A052430 A052431 * A052433 A052434 A052435

Keyword

nonn,nice,hard,more

Author

Jesper Petersen (u943254(AT)daimi.au.dk), Mar 14 2000

Extensions

Added value for rank 4 from Cremona's extended tables, by John Cremona, Apr 02 2012

Upper bounds for a(5)-a(7) and a(11) from Elkies & Watkins added by Jonathan Sondow, Oct 29 2013.

Unproved values for a(5)-a(7) and a(11) removed by N. J. A. Sloane, Jan 25 2016 at the suggestion of John Cremona.

Status

approved