A060953 -id:A060953 - cp's OEIS frontend

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

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", ")))

A228227 Primes congruent to {7, 11} mod 16.

Original entry on oeis.org

7, 11, 23, 43, 59, 71, 103, 107, 139, 151, 167, 199, 251, 263, 283, 311, 331, 347, 359, 379, 439, 443, 487, 491, 503, 523, 571, 587, 599, 619, 631, 647, 683, 727, 743, 811, 823, 827, 839, 859, 887, 907, 919, 967, 971, 983, 1019, 1031, 1051, 1063, 1163, 1223

Offset: 1

Arkadiusz Wesolowski, Aug 16 2013

nonn

Union of A141194 and A141195.

Let p be a prime number and let E(p) denote the elliptic curve y^2 = x^3 + p*x. If p is in the sequence, then the rank of E(p) is 0. Therefore A060953(a(n)) = 0.

J. H. Silverman, The arithmetic of elliptic curves, Springer, NY, 1986, p. 311.

Arkadiusz Wesolowski, Table of n, a(n) for n = 1..1000

Cf. A007519, A060953, A141194, A141195, A228228.

[p: p in PrimesUpTo(1223) | p mod 16 in {7, 11}];

Select[Prime@Range[200], MemberQ[{7, 11}, Mod[#, 16]] &]

A228228 Primes congruent to {3, 5, 13, 15} mod 16.

Original entry on oeis.org

3, 5, 13, 19, 29, 31, 37, 47, 53, 61, 67, 79, 83, 101, 109, 127, 131, 149, 157, 163, 173, 179, 181, 191, 197, 211, 223, 227, 229, 239, 269, 271, 277, 293, 307, 317, 349, 367, 373, 383, 389, 397, 419, 421, 431, 461, 463, 467, 479, 499, 509, 541, 547, 557, 563

Offset: 1

Arkadiusz Wesolowski, Aug 16 2013

nonn

Union of A091968, A127589, A141196, and A127576.

Let p be a prime number and let E(p) denote the elliptic curve y^2 = x^3 + p*x. If p is in the sequence, then the rank of E(p) is 0 or 1. Therefore, A060953(a(n)) must be one of only two values: 0 or 1.

J. H. Silverman, The arithmetic of elliptic curves, Springer, NY, 1986, p. 311.

Arkadiusz Wesolowski, Table of n, a(n) for n = 1..1000

Cf. A007519, A060953, A091968, A127576, A127589, A141196, A228227.

[p: p in PrimesUpTo(563) | p mod 16 in {3, 5, 13, 15}];

Select[Prime@Range[103], MemberQ[{3, 5, 13, 15}, Mod[#, 16]] &]

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).

A386928 Algebraic rank of elliptic curve y^2 = x^3 + n*x + n.

Original entry on oeis.org

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

Offset: 1

Shreyansh Jaiswal, Aug 08 2025

Terms from n = 29 onward are the analytic ranks (see PARI code) of the corresponding elliptic curves. By the BSD conjecture, these are expected to equal the algebraic ranks. Thus, the validity of these terms is conditional on BSD.

			a(1) = 1 because y^2 = x^3 + x + 1 has rank 1.

LMFDB, y^2 = x^3 + x + 1.

Cf. A060950, A060953.

a(n) = ellanalyticrank(ellinit([n, n]))[1]; \\ Jinyuan Wang, Aug 08 2025

for k in range(1,29):
    E = EllipticCurve([k,k])
    print(E.rank(),end=", ")

More terms from Jinyuan Wang, Aug 08 2025

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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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A228227 Primes congruent to {7, 11} mod 16.

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A228228 Primes congruent to {3, 5, 13, 15} mod 16.

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

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A386928 Algebraic rank of elliptic curve y^2 = x^3 + n*x + n.

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