A356731 Conductor of the elliptic curve y^2 = x^3 - n.

144, 1728, 972, 432, 10800, 15552, 5292, 576, 3888, 14400, 13068, 972, 73008, 84672, 24300, 432, 41616, 15552, 12996, 10800, 190512, 209088, 57132, 15552, 10800, 97344, 36, 1764, 363312, 388800, 103788, 1728, 470448, 499392, 44100, 3888, 197136, 623808, 164268, 43200, 726192

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 A060951(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) = A356730(n) for all n.

(ii) a(n) is divisible by 36, and a(n) = 36 <=> n is 27 times a sixth power, a(n) = 108 <=> n is 108 times a sixth power, a(n) = 144 <=> n is 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. A356730, A060951.

Programs

PARI

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