A060951 -id:A060951 - cp's OEIS frontend

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

Jianing Song, Aug 24 2022

nonn

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)

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

Cf. A356730, A060951.

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

A385881 Algebraic rank of elliptic curve y^2 = x^3 - n*x - n.

Original entry on oeis.org

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

Offset: 1

Shreyansh Jaiswal, Aug 20 2025

Terms from n = 43 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) = 0 because y^2 = x^3 - x - 1 has rank 0.

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

Cf. A060951, A060952.

a(n) = ellanalyticrank( ellinit([0, 0, 0, -n, -n]) )[1]; \\ Michel Marcus, Aug 20 2025

for k in range(1, 43):
    E = EllipticCurve([-k, -k])
    print(E.rank(), end=", ")

More terms from Michel Marcus, Aug 20 2025

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

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