A194687 -id:A194687 - cp's OEIS frontend

A319510 Rank of elliptic curve y^2 = x^3 - n^2 * x.

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

Offset: 1

Seiichi Manyama, Sep 24 2018

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..5000

Cf. A003273, A060952, A062693, A062695, A194687, A273929.

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

a(n) = A060952(n^2).
a(A003273(n)) > 0.
a(A194687(n)) = n.
Empirical: a(n) = a(4*n). - Jose Aranda, Jul 02 2024

A309028 Smallest k>0 such that the elliptic curve y^2 = x^3 + k*x has rank n, if k exists.

Original entry on oeis.org

1, 3, 14, 323, 1918, 195843

Offset: 0

Seiichi Manyama, Jul 08 2019

See A309029 for the smallest negative k.

B. J. Birch and H. P. F. Swinnerton-Dyer, Notes on elliptic curves, I, J. Reine Angew. Math., 212 (1963), 7-25.

Cf. A031507, A060953, A194687, A309029.

Cf. A002158, A002159, A076329, A309030, A309031, A309190.

a(5) from Vaclav Kotesovec, Jul 14 2019

A309060 Least k such that the rank of the elliptic curve y^2 = x^3 + k^2*x is n.

Original entry on oeis.org

1, 3, 17, 627, 14637

Offset: 0

Seiichi Manyama, Jul 09 2019

From Jose Aranda, Jun 30 2024: (Start)
A319510(n even) = A309061(n/2), A319510(n odd) = A309061(2*n) (Empirical).
A194687(5) =      48272239 which implies a(5) <=      96544478 (Checked).
A194687(6) =    6611719866 which implies a(6) <=    3305859933 (Checked).
A194687(7) <= 797507543735 which implies a(7) <= 1595015087470 (Checked). (End)

			A309061(1) = 0.
A309061(3) = 1.
A309061(17) = 2.

Cf. A194687, A309028, A309061, A319510.

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

A309061(a(n)) = n.

A309029 Smallest k>0 such that the elliptic curve y^2 = x^3 - k*x has rank n, if k exists.

Original entry on oeis.org

1, 2, 17, 82, 5037, 49042

Offset: 0

Seiichi Manyama, Jul 08 2019

See A309028 for the smallest positive k.

B. J. Birch and H. P. F. Swinnerton-Dyer, Notes on elliptic curves, I, J. Reine Angew. Math., 212 (1963), 7-25.

Cf. A031508, A060952, A194687, A309028.

Cf. A002156, A002157, A309032, A309033, A309034, A309100.

a(5) from Vaclav Kotesovec, Jul 09 2019

A309068 Least k such that the rank of the elliptic curve y^2 = x^3 - k^2 is n.

Original entry on oeis.org

1, 2, 11, 362, 7954

Offset: 0

Seiichi Manyama, Jul 10 2019

Cf. A031508, A060951, A194687, A309069.

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

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.

cp's OEIS Frontend

A319510 Rank of elliptic curve y^2 = x^3 - n^2 * x.

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A309028 Smallest k>0 such that the elliptic curve y^2 = x^3 + k*x has rank n, if k exists.

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A309060 Least k such that the rank of the elliptic curve y^2 = x^3 + k^2*x is n.

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A309029 Smallest k>0 such that the elliptic curve y^2 = x^3 - k*x has rank n, if k exists.

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Extensions

A309068 Least k such that the rank of the elliptic curve y^2 = x^3 - k^2 is n.

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PARI

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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PARI

PARI

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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Author

Keywords

Comments

Examples

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Crossrefs