A309960 Numbers k for which rank of the elliptic curve y^2 = x^3 - 432*k^2 is 0.

1, 2, 3, 4, 5, 8, 10, 11, 14, 16, 18, 21, 23, 24, 25, 27, 29, 32, 36, 38, 39, 40, 41, 44, 45, 46, 47, 52, 54, 55, 57, 59, 60, 64, 66, 73, 74, 76, 77, 80, 81, 82, 83, 88, 93, 95, 99, 100, 101, 102, 108, 109, 111, 112, 113, 116, 118, 119, 121, 122, 125, 128, 129, 131, 135, 137

Offset

1,2

Links

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Formula

A060838(a(n)) = 0.

Prog

(PARI) for(k=1, 200, if(ellanalyticrank(ellinit([0, 0, 0, 0, -432*k^2]))[1]==0, print1(k", ")))
(PARI) is(n, f=factor(n))=my(c=prod(i=1, #f~, f[i, 1]^(f[i, 2]\3)), r=n/c^3, E, eri, mwr, ar); if(r<6, return(1)); E=ellinit([0, 16*r^2]); eri=ellrankinit(E); mwr=ellrank(eri); if(mwr[1], return(0)); ar=ellanalyticrank(E)[1]; if(ar<2, return(!ar)); for(effort=1, 99, mwr=ellrank(eri, effort); if(mwr[1]>0, return(0), mwr[2]<1, return(1))); "unknown (0 under BSD conjecture)" \\ Charles R Greathouse IV, Jan 24 2023

Crossrefs

Complement of A159843 \ A000578.

Cf. A060748, A060838, A309961 (rank 1), A309962 (rank 2), A309963 (rank 3), A309964 (rank 4).

Sequence in context: A305399 A101547 A047597 * A247935 A378983 A005233

Adjacent sequences: A309957 A309958 A309959 * A309961 A309962 A309963

Keyword

nonn

Author

Seiichi Manyama, Aug 25 2019

Status

approved