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

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

Author

Shreyansh Jaiswal, Aug 20 2025

Keywords

• nonn
• new

Comments

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.

Examples

			a(1) = 0 because y^2 = x^3 - x - 1 has rank 0.
		

Links

• LMFDB, y^2 = x^3 - x - 1.

Crossrefs

Cf. A060951, A060952.

Programs

PARI

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

SageMath

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

Extensions

More terms from Michel Marcus, Aug 20 2025