A127788 -id:A127788 - cp's OEIS frontend

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

N. J. A. Sloane, Apr 12 2001

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

			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

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

Cf. A005788, A110620, A127788.

# 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

A159046 Dimension of the space of newforms of weight 2 on the subgroup Gamma_1(n).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 2, 1, 1, 2, 5, 2, 7, 3, 5, 4, 12, 5, 12, 6, 13, 8, 22, 7, 26, 13, 19, 11, 25, 13, 40, 14, 29, 19, 51, 13, 57, 25, 39, 21, 70, 23, 69, 24, 55, 37, 92, 22, 79, 42, 71, 34, 117, 34, 126, 39, 87, 61, 117, 31, 155, 68, 109, 45, 176, 55, 187, 56, 119, 87

Offset: 1

Steven Finch, Apr 03 2009

nonn

			a(p) = A029937(p) = (p-5)*(p-7)/24 for any prime p>3.
G.f. = x^11 + 2*x^13 + x^14 + x^15 + 2*x^16 + 5*x^17 + 2*x^18 + 7*x^19 + ...

G. C. Greubel, Table of n, a(n) for n = 1..10000
G. Martin, Dimensions of the spaces of cusp forms and newforms on Gamma_0(N) and Gamma_1(N), J. Numb. Theory 112 (2005) 298-331.

Cf. A007427, A029937, A029938, A127788.

a[ n_] := If[ n < 1, 0, Sum[ DivisorSum[ n/j, MoebiusMu[#] MoebiusMu[n/j/#] &] If[ j < 5, 0, 1 + DivisorSum[ j, #^2 MoebiusMu[ j/#] / 24 - EulerPhi [#] EulerPhi[j/#] / 4 &]], {j, Divisors@n}]]; (* Michael Somos, May 10 2015 *)

{a(n) = if( n<1, 0, sumdiv(n, j, sumdiv(n/j, k, moebius(k) * moebius(n/j/k)) * if( j<5, 0, 1 + sumdiv(j, k, k^2 * moebius(j/k) / 24 - eulerphi(k) * eulerphi(j/k) / 4))))}; /* Michael Somos, May 10 2015 */

a(n) = A029937(n) - sum a(m)*d(n/m), where the summation is over all divisors 1 < m < n of n and d is the divisor function.

Dirichlet convolution of A007247 and A029937. - Michael Somos, May 10 2015

A260088 Dimension of the space of newforms of weight 4 and level n.

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 2, 1, 3, 2, 2, 1, 4, 1, 4, 1, 4, 3, 5, 1, 3, 3, 4, 2, 7, 2, 7, 3, 6, 4, 6, 1, 9, 5, 6, 3, 10, 2, 10, 3, 5, 6, 11, 3, 8, 5, 8, 3, 13, 4, 10, 4, 10, 7, 14, 2, 15, 8, 7, 5, 12, 4, 16, 4, 12, 6, 17, 4, 18, 9, 10, 5, 16, 6, 19, 6

Offset: 1

R. J. Mathar, Jul 15 2015

Cf. A127788, A063195 - A063247.

seq(g0star(4,N),N=1..80); # using functions coded in A063195

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

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A159046 Dimension of the space of newforms of weight 2 on the subgroup Gamma_1(n).

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A260088 Dimension of the space of newforms of weight 4 and level n.

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