A309060 -id:A309060 - cp's OEIS frontend

A309069 Least k such that the rank of the elliptic curve y^2 = x^3 + k^2 is n.

1, 3, 15, 427, 17353

Offset: 0

Seiichi Manyama, Jul 10 2019

{a(n) = my(k=1); while(ellanalyticrank(ellinit([0, 0, 0, 0, k^2]))[1]<>n, k++); k}

A309061 Rank of elliptic curve y^2 = x^3 + n^2 * x.

Original entry on oeis.org

0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 1, 1, 0, 1, 1, 0, 2, 0, 1, 0, 0, 0, 1, 0, 0, 1, 1, 1, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 0, 1, 0, 0, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0, 0, 0, 1, 2, 2, 0, 1, 0, 0, 1, 1, 1, 2, 1, 1, 0, 0, 2, 1, 0, 0, 0, 1, 0, 0, 1, 1, 1, 0, 1, 1, 0, 2

Offset: 1

Seiichi Manyama, Jul 09 2019

nonn

Cf. A060953, A309060, A319510.

{a(n) = ellanalyticrank(ellinit([0, 0, 0, n^2, 0]))[1]}

a(n) = A060953(n^2).

A372543 Least k such that the rank of the elliptic curve y^2 = x^3 - k^2*x + 1 is n, or -1 if no such k exists.

Original entry on oeis.org

0, 1, 2, 4, 8, 17, 61, 347, 3778, 11416

Offset: 0

Jose Aranda, Jul 04 2024

This family of curves quickly reaches a moderate value of rank with a relatively small parameter k.

By heuristic search (see links), a(10) <= 216493 and a(11) <= 1448203.

Anna Antoniewicz, On a family of elliptic curves, (2005) Iagellonicae Acta Mathematica, XLIII.
Jose Aranda, Non sequential search for upper bounds (PARI-GP Script)
Cecylia Bocovich, Elliptic Curves of High Rank, (2012) Macalester College, Science Honors Projects.

Cf. A194687, A309060, A309068, A309069, A309146, A309147, A309164, A309165.

a(n,startAt=0)=for(k=startAt, oo, my(t=ellrank(ellinit([-k^2, +1]))); if(t[2]n, warning("k=",k," has rank in ",t[1..2]); next); if(t[1]n, error("Cannot determine if a(",n,") is ",k," or larger; rank is in ",t[1..2])); return(k)) \\ Charles R Greathouse IV, Jul 08 2024

PARI
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\\ See Aranda link.
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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.

Original entry on oeis.org

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

Offset: 1

Jose Aranda, Jul 24 2024

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.

			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.

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.

Cf. A194687, A309060, A309068, A309069.

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A309069 Least k such that the rank of the elliptic curve y^2 = x^3 + k^2 is n.

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A309061 Rank of elliptic curve y^2 = x^3 + n^2 * x.

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A372543 Least k such that the rank of the elliptic curve y^2 = x^3 - k^2*x + 1 is n, or -1 if no such k exists.

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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.

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