A076329 -id:A076329 - cp's OEIS frontend

A002159 Numbers k for which the rank of the elliptic curve y^2 = x^3 + k*x is 1.

3, 5, 8, 9, 13, 15, 18, 19, 20, 21, 24, 28, 29, 31, 35, 37, 40, 47, 48, 49, 51, 53, 56, 60, 61, 67, 69, 77, 79, 80, 83, 84, 85, 88, 90, 92, 93, 95, 98, 100, 101, 104, 109, 111, 115, 120, 121, 124, 125, 126, 127, 128, 131, 133, 136, 141, 143, 144, 148, 149, 152, 153, 156

Offset: 1

N. J. A. Sloane

nonn

Terms 80 and 128 are missing in the article by Birch and Swinnerton-Dyer, page 25, table 4b. - Vaclav Kotesovec, Jul 07 2019

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..2040
B. J. Birch and H. P. F. Swinnerton-Dyer, Notes on elliptic curves, I, J. Reine Angew. Math., 212 (1963), 7-25.

Cf. A060953, A076329.

for k in[1..200] do if Rank(EllipticCurve([0,0,0,k,0])) eq 1 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]==1, 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

A309032 Numbers k for which rank of the elliptic curve y^2=x^3-k*x is 2.

Original entry on oeis.org

17, 56, 65, 77, 90, 97, 117, 132, 136, 141, 145, 155, 156, 161, 184, 205, 207, 219, 220, 221, 241, 252, 257, 259, 260, 264, 272, 275, 285, 291, 292, 301, 305, 306, 337, 342, 355, 356, 371, 376, 395, 396, 401, 420, 429, 433, 445, 449, 452, 456, 465, 481, 497, 507, 516

Offset: 1

Seiichi Manyama, Jul 08 2019

nonn

Cf. A002156 (rank 0), A002157 (rank 1). this sequence (rank 2), A309033 (rank 3), A309034(rank 4).

Cf. A076329.

for k in[1..1000] do if Rank(EllipticCurve([0,0,0,-k,0])) eq 2 then print k; end if; end for; // Vaclav Kotesovec, Jul 08 2019

for(k=1, 1e3, if(ellanalyticrank(ellinit([0, 0, 0, -k, 0]))[1]==2, print1(k", ")))

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.

A107484 Some numbers k such that the elliptic curve y^2 = x^3 + k*x has rank 2.

Original entry on oeis.org

14, 33, 34, 39, 46, -17, -56, -65, -77

Offset: 1

Sam Alexander, May 28 2005

Dale Husemoller, "Elliptic Curves", Springer-Verlag: New York, 1987, p. 35.

Cf. A002151.

This is the first few terms of A076329 followed by the first few negated terms of A309032.

cp's OEIS Frontend

A002159 Numbers k for which the rank of the elliptic curve y^2 = x^3 + k*x is 1.

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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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A309032 Numbers k for which rank of the elliptic curve y^2=x^3-k*x is 2.

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

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A107484 Some numbers k such that the elliptic curve y^2 = x^3 + k*x has rank 2.

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