A356731 - OEIS

%I #11 Aug 25 2022 09:54:39

%S 144,1728,972,432,10800,15552,5292,576,3888,14400,13068,972,73008,

%T 84672,24300,432,41616,15552,12996,10800,190512,209088,57132,15552,

%U 10800,97344,36,1764,363312,388800,103788,1728,470448,499392,44100,3888,197136,623808,164268,43200,726192

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

%C The discriminant of the elliptic curve y^2 = x^3 - n is -432*n^2 and the rank is A060951(n).

%C 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.

%C Conjectures: (Start)

%C (i) a(27*n) = A356730(n) for all n.

%C (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)

%H Jianing Song, <a href="/A356731/b356731.txt">Table of n, a(n) for n = 1..10000</a>

%H LMFDB, <a href="https://www.lmfdb.org/EllipticCurve/Q/">Elliptic Curves Over Q</a>

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

%Y Cf. A356730, A060951.

%K nonn

%O 1,1

%A _Jianing Song_, Aug 24 2022