A060564 Number of elliptic curves (up to isogeny) of conductor n.

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 1, 0, 1, 1, 1, 0, 0, 1, 0, 2, 1, 0, 0, 1, 0, 1, 1, 1, 1, 1, 2, 2, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 2, 1, 1, 1, 2, 1, 2, 3, 2, 0, 0, 1, 1, 1, 1, 1, 3, 1, 0, 1, 1, 0, 1, 1, 0, 3, 1, 3, 1, 1, 2, 0, 1, 1, 2, 1, 0, 0, 1, 2, 3, 2, 2, 0, 1, 0, 2, 0, 1, 4

Offset

1,26

Comments

By the modularity of elliptic curves over Q (proved by Breuil-Conrad-Diamond-Taylor), a(n) is equivalently the number of integral normalized weight 2 newforms for Gamma_0(n). - Robin Visser, Nov 04 2024

Links

J. E. Cremona, Table of n, a(n) for n = 1..10000

J. E. Cremona, Elliptic Curve Data

A. Dujella, History of elliptic rank records

LMFDB, Elliptic curves over Q

F. Richman, Elliptic curves [broken link]

A. L. Robledo, PlanetMath.org, The Arithmetic of Elliptic Curves [broken link]

E. Savaş, T. A. Schmidt, and Ç. K. Koç, Generating Elliptic Curves of Prime Order. In: Koç, Ç.K., Naccache, D., Paar, C. (eds) Cryptographic Hardware and Embedded Systems — CHES 2001. CHES 2001. Lecture Notes in Computer Science, vol 2162. Springer, Berlin, Heidelberg.

J. S. Silverman, An Introduction to the Theory of Elliptic Curves

Eric Weisstein's World of Mathematics, Elliptic Curve

Example

a(11) = 1, as there is exactly one isogeny class of elliptic curves over Q of conductor 11, represented by E : y^2 + y = x^3 - x^2. - Robin Visser, Nov 04 2024

Prog

(Sage)  # Uses Cremona's database of elliptic curves (works for all n < 500000)
def a(n):
    return CremonaDatabase().number_of_isogeny_classes(n)  # Robin Visser, Nov 04 2024

Crossrefs

Cf. A005788, A110620, A127788.

Sequence in context: A345236 A321223 A082926 * A031267 A247763 A065360

Adjacent sequences: A060561 A060562 A060563 * A060565 A060566 A060567

Keyword

nonn,nice

Author

N. J. A. Sloane, Apr 12 2001

Status

approved