A356730 Conductor of the elliptic curve y^2 = x^3 + n.

36, 1728, 3888, 108, 2700, 15552, 21168, 576, 972, 14400, 52272, 3888, 18252, 84672, 97200, 27, 10404, 15552, 51984, 2700, 47628, 209088, 228528, 15552, 2700, 97344, 144, 7056, 90828, 388800, 415152, 1728, 117612, 499392, 176400, 972, 49284, 623808, 657072, 43200, 181548

Offset: 1

Author

Jianing Song, Aug 24 2022

Keywords

• nonn

Comments

The discriminant of the elliptic curve y^2 = x^3 - n is -432*n^2 and the rank is A060950(n).

a(n*t^6) = a(n) for all t since the elliptic curve y^2 = x^3 + n*t^6 can be written as (y/t^3)^2 = (x/t^2)^3 + n, and the conductor is an invariant of elliptic curves.

Conjectures: (Start)

(i) a(27*n) = A356731(n) for all n.

(ii) a(n) is divisible by 36, and a(n) = 36 <=> n is a sixth power, a(n) = 108 <=> n is 4 times a sixth power, a(n) = 144 <=> n is 27 times a sixth power; moreover, it seems that a(n) is divisible by 36*n^2 if n is squarefree. (End)

Links

• Jianing Song, Table of n, a(n) for n = 1..10000
• LMFDB, Elliptic Curves Over Q

Crossrefs

Cf. A356731, A060950.

Programs

PARI

a(n) = ellglobalred(ellinit([0,0,0,0,n]))[1]