A002159 -id:A002159 - cp's OEIS frontend

A076329 Numbers k for which rank of the elliptic curve y^2=x^3+k*x is 2.

14, 33, 34, 39, 46, 55, 63, 65, 66, 68, 73, 89, 94, 99, 105, 113, 114, 129, 138, 145, 150, 154, 155, 158, 178, 183, 185, 201, 203, 206, 209, 219, 224, 226, 233, 238, 254, 258, 260, 273, 274, 281, 289, 295, 299, 308, 310, 333, 334, 337, 345, 353, 354, 360, 385, 386, 388

Offset: 1

N. J. A. Sloane, Nov 06 2002

nonn

D. S. Jandu, Elliptic Curve, Infinite Bandwidth Publishing, N. Hollywood CA 2007.
D. S. Jandu, Birch And Swinnerton Dyer Conjecture, Infinite Bandwidth Publishing, N. Hollywood CA 2007.
D. Zagier & G. Harder, "La conjecture de Birch et Swinnerton-Dyer" in Les Dossiers de La Recherche, pp. 48-53 No. 20 August-October 2005 Paris.

B. J. Birch and H. P. F. Swinnerton-Dyer, Notes on elliptic curves, I, J. Reine Angew. Math., 212 (1963), 7-25.
A. L. Robledo, PlanetMath.org, Birch and Swinnerton-Dyer conjecture
W. A. Stein, The Birch and Swinnerton-Dyer Conjecture, Part 1
W. A. Stein, The Birch and Swinnerton-Dyer Conjecture, Part 2
W. A. Stein, The Birch and Swinnerton-Dyer Conjecture, Part 3
Wikipedia, Birch and Swinnerton-Dyer conjecture

Cf. A002150-A002159.

for(k=1, 1e3, if(ellanalyticrank(ellinit([0, 0, 0, k, 0]))[1]==2, print1(k", "))) \\ Seiichi Manyama, Jul 07 2019

More terms added by Seiichi Manyama, Jul 07 2019

A002158 Numbers k for which the rank of the elliptic curve y^2 = x^3 + k*x is 0.

Original entry on oeis.org

1, 2, 4, 6, 7, 10, 11, 12, 16, 17, 22, 23, 25, 26, 27, 30, 32, 36, 38, 41, 42, 43, 44, 45, 50, 52, 54, 57, 58, 59, 62, 64, 70, 71, 72, 74, 75, 76, 78, 81, 82, 86, 87, 91, 96, 97, 102, 103, 106, 107, 108, 110, 112, 116, 117, 118, 119, 122, 123, 130, 132, 134, 135, 137, 139, 140, 142, 146, 147, 151, 160, 161, 162, 166, 167, 169, 170, 172, 174, 176, 177, 182, 186, 187, 190, 192, 193, 194, 199

Offset: 1

N. J. A. Sloane

nonn

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 1..2000
B. J. Birch and H. P. F. Swinnerton-Dyer, Notes on elliptic curves, I, J. Reine Angew. Math., 212 (1963), 7-25.

Cf. A002159 (rank 1), A076329 (rank 2).

Cf. A060953.

for k in[1..200] do if Rank(EllipticCurve([0,0,0,k,0])) eq 0 then print k; end if; end for; // Vaclav Kotesovec, Jul 07 2019

for(k=1, 200, if(ellanalyticrank(ellinit([0, 0, 0, k, 0]))[1]==0, print1(k", "))) \\ Seiichi Manyama, Jul 07 2019

Corrected and extended by Vaclav Kotesovec, Jul 07 2019

New name by Vaclav Kotesovec, Jul 07 2019

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

A309030 Numbers k for which rank of the elliptic curve y^2=x^3+k*x is 3.

Original entry on oeis.org

323, 328, 399, 445, 579, 723, 904, 943, 1023, 1139, 1288, 1314, 1443, 1508, 1679, 1743, 1763, 1768, 1953, 2005, 2035, 2159, 2275, 2328, 2419, 2451, 2504, 2533, 2725, 2739, 2790, 2793, 2824, 2915, 2980, 3029, 3038, 3043, 3108, 3196, 3199, 3245, 3341, 3363, 3443, 3459, 3465

Offset: 1

Seiichi Manyama, Jul 08 2019

nonn

Cf. A002158 (rank 0), A002159 (rank 1), A076329 (rank 2), this sequence (rank 3), A309031 (rank 4).

Cf. A309033.

for k in[1..4000] do if Rank(EllipticCurve([0,0,0,k,0])) eq 3 then print k; end if; end for; // Vaclav Kotesovec, Jul 08 2019

for(k=1, 3e3, if(ellanalyticrank(ellinit([0, 0, 0, k, 0]))[1]==3, print1(k", ")))

A309031 Numbers k for which rank of the elliptic curve y^2=x^3+k*x is 4.

Original entry on oeis.org

1918, 5190, 6123, 6953, 9603, 10759, 12483, 13398, 14673, 14795, 15910, 15934, 16238, 17753, 18278, 18705, 18814, 20148, 20398, 20658, 23180, 23953, 24475, 25988, 26809, 28633, 29274, 30340, 30688, 31073, 31098, 31174, 32118, 33218, 33278, 34804, 36955, 37214, 37298

Offset: 1

Seiichi Manyama, Jul 08 2019

nonn

Cf. A002158 (rank 0), A002159 (rank 1), A076329 (rank 2), A309030 (rank 3), this sequence (rank 4).

Cf. A060953, A309034.

for k in[1..10000] do if Rank(EllipticCurve([0,0,0,k,0])) eq 4 then print k; end if; end for; // Vaclav Kotesovec, Jul 08 2019

for(k=1, 1e4, if(ellanalyticrank(ellinit([0, 0, 0, k, 0]))[1]==4, print1(k", ")))

A309190 Numbers k for which rank of the elliptic curve y^2 = x^3 + k*x is 5.

Original entry on oeis.org

195843, 196168, 233864

Offset: 1

Vaclav Kotesovec, Jul 16 2019

Cf. A309028, A309100.

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

A076330 Elliptic curves (see reference for precise definition).

Original entry on oeis.org

17, 41, 57, 62, 82, 97, 117, 137, 142, 146, 161, 177, 193, 194, 392, 452, 792

Offset: 1

N. J. A. Sloane, Nov 06 2002

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

Cf. A002150-A002159. Subsequence of A002158.

A076331 Elliptic curves (see reference for precise definition).

Original entry on oeis.org

136, 584, 712, 776

Offset: 1

N. J. A. Sloane, Nov 06 2002

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

Cf. A002150-A002159. Subsequence of A002159.

cp's OEIS Frontend

A076329 Numbers k for which rank of the elliptic curve y^2=x^3+k*x is 2.

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A002158 Numbers k for which the rank of the elliptic curve y^2 = x^3 + k*x is 0.

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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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A309030 Numbers k for which rank of the elliptic curve y^2=x^3+k*x is 3.

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A309031 Numbers k for which rank of the elliptic curve y^2=x^3+k*x is 4.

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Magma

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A309190 Numbers k for which rank of the elliptic curve y^2 = x^3 + k*x is 5.

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A076330 Elliptic curves (see reference for precise definition).

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A076331 Elliptic curves (see reference for precise definition).

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