A374926 Least k such that the rank of the elliptic curve y^2 = x^3 - x + k^2 is n, or -1 if no such k exists.

1, 2, 5, 24, 113, 337, 6310, 78560, 423515, 765617

Offset: 1

Author

Jose Aranda, Jul 24 2024

Keywords

• nonn
• hard
• more

Comments

This family of curves quickly reaches a moderate value of rank with a relatively low "k" parameter. And is fully analyzed in Tadik's work (see link). Tadik finds 11 terms, a rank lower bound and shows the torsion group is always trivial. The evolution of the rank is shown in detail, finding that a(11) <= 1118245045.

I have sequentially checked the first 10 terms, thus proving that they are the least k for each rank.

Examples

			The curve C[1] = [-1,1^2] has rank one, with generator [1,-1].The rank of C[2] = [-1,2^2] is 2 because it has two generators:PARI> e=ellinit([-1,2^2] );ellgenerators(e) = [[-1, 2], [0, 2]].If k>1, the curve C[k] always has at least two generators: [0,k], [-1,k], then its minimum rank is two.
		

Links

• Ezra Brown and Bruce T. Myers, Elliptic Curves from Mordell to Diophantus and Back, Amer. Math. Monthly 109 (2002), 639-649.
• Edward Vincent Eikenberg, Rational points on some families of Elliptic Curves, PhD thesis, University of Maryland, 2004.
• Petra Tadik, The rank of certain subfamilies of the elliptic curve y^2 = x^3 -x +t^2, Ann. Math. Inform. 40 (2012), 145-153.

Crossrefs

Cf. A194687, A309060, A309068, A309069.