A060564 -id:A060564 - cp's OEIS frontend

A005788 Conductors of elliptic curves.

11, 14, 15, 17, 19, 20, 21, 24, 26, 27, 30, 32, 33, 34, 35, 36, 37, 38, 39, 40, 42, 43, 44, 45, 46, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 61, 62, 63, 64, 65, 66, 67, 69, 70, 72, 73, 75, 76, 77, 78

Offset: 1

N. J. A. Sloane

nonn

By the modularity of elliptic curves over Q (proved by Breuil-Conrad-Diamond-Taylor), these are equivalently the positive integers k such that there exists a rational weight 2 newform for Gamma_0(k). - Robin Visser, Nov 04 2024

			a(1) = 11, as there are no elliptic curves over Q of conductor less than 11, but there are exactly three elliptic curves over Q of conductor equal to 11, for example E : y^2 + y = x^3 - x^2. - _Robin Visser_, Nov 04 2024

B. J. Birch and W. Kuyk, eds., Modular Functions of One Variable IV (Antwerp, 1972), Lect. Notes Math. 476 (1975), see pp. 82ff.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

J. E. Cremona, Table of n, a(n) for n = 1..10000
J. E. Cremona, Elliptic Curve Data.
Sean Howe and Kirti Joshi, Asymptotics of conductors of elliptic curves over Q, arXiv:1201.4566 [math.NT], 2012.
LMFDB, Elliptic curves over Q.
Eric Weisstein's World of Mathematics, Elliptic Curve.

Cf. A060564, A110563, A110620, A217055.

# Uses Cremona's database of elliptic curves (works for all k < 500000)
def is_A005788(k):
    return CremonaDatabase().number_of_curves(k) > 0
print([k for k in range(1, 1000) if is_A005788(k)])  # Robin Visser, Nov 04 2024

A110620 Number of elliptic curves (up to isomorphism) of conductor n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 6, 8, 0, 4, 0, 3, 4, 6, 0, 0, 6, 0, 5, 4, 0, 0, 8, 0, 4, 4, 4, 3, 4, 4, 5, 4, 4, 0, 6, 1, 2, 8, 2, 0, 6, 4, 8, 2, 2, 1, 6, 4, 6, 7, 3, 0, 0, 1, 4, 6, 4, 2, 12, 1, 0, 2, 4, 0, 6, 2, 0, 12, 1, 6, 4, 1, 8, 0, 2, 1, 6, 2, 0, 0, 1, 3, 16, 4, 3, 0, 2, 0, 8, 0, 6, 11, 4, 1, 12, 0

Offset: 1

Steven Finch, Sep 14 2005

nonn

			a(11)=3 since there are three non-isomorphic elliptic curves of conductor eleven, represented by the minimal models y^2+y=x^3-x^2-10*x-20, y^2+y=x^3-x^2-7820*x-263580 and y^2+y=x^3-x^2.

J. E. Cremona, Table of n, a(n) for n = 1..10000
A. Brumer and J. H. Silverman, The number of elliptic curves over Q with conductor N, Manuscripta Math. 91 (1996), no. 1, 95-102.
J. E. Cremona, Elliptic Curve Data.
LMFDB, Elliptic curves over Q.

Cf. A005788, A060564.

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

A217055 Prime numbers which are conductors of elliptic curves.

Original entry on oeis.org

11, 17, 19, 37, 43, 53, 61, 67, 73, 79, 83, 89, 101, 109, 113, 131, 139, 163, 179, 197, 229, 233, 269, 277, 307, 331, 347, 353, 359, 373, 389, 431, 433, 443, 467, 503, 557, 563, 571, 593, 643, 659, 677, 701, 709, 733, 739, 797, 811, 827, 829, 997, 1019, 1051

Offset: 1

Gene Ward Smith, Sep 25 2012

nonn

Taken from the data by Armand Brumer and Oisin McGuinness listing 310716 elliptic curves with prime conductor. Note that for some primes, there is more than one elliptic curve with that conductor.

All primes p of the form p = u^2 + 64 for some integer u are in this sequence, as Setzer (1975) proved that for such primes p that there are exactly two elliptic curves E/Q of conductor p. - Robin Visser, Sep 04 2024

			a(1) = 11, as there are no elliptic curves over Q of conductor less than 11, but there are exactly three elliptic curves over Q of conductor equal to 11, for example E : y^2 + y = x^3 - x^2. - _Robin Visser_, Sep 04 2024

Robin Visser, Table of n, a(n) for n = 1..10000 (taken from the Bennett-Gherga-Rechnitzer database).
M. A. Bennett, A. Gherga, and A. Rechnitzer, Computing elliptic curves over Q, Math. Comp., 88 (2019), no. 317, 1341-1390.
Armand Brumer and Oisin McGuiness, 310716 Elliptic Curves of Prime Conductor
J. E. Cremona, Elliptic Curve Data
B. Setzer, Elliptic Curves of Prime Conductor, J. London Math. Soc. (2) 10 (1975), 367-378.

Cf. A005788, A060564, A363793.

# Uses Cremona's database of elliptic curves (works for all p < 500000)
def is_A217055(p):
    if not Integer(p).is_prime(): return False
    return CremonaDatabase().number_of_curves(p) > 0
print([p for p in range(1, 1000) if is_A217055(p)])  # Robin Visser, Sep 04 2024

A381352 Conductors admitting a normalized weight-2 eta-quotient newform.

Original entry on oeis.org

11, 14, 15, 20, 24, 27, 32, 36, 48, 64, 80, 144

Offset: 1

Dimitris Cardaris, May 11 2025

By the modularity theorem, each Q-isogeny class of elliptic curves of conductor N corresponds to a unique normalized rational weight-2 newform on Gamma_0(N). This sequence (classified by Martin & Ono) lists exactly those 12 conductors N for which that newform at level N can be written as a single eta-quotient.

			a(1) = 11 since f(z) = q-2*q^2-q^3+2*q^4+ ... = eta^2(z)*eta^2(11*z) on Gamma_0(11)

Yves Martin and Ken Ono, Eta-Quotients and Elliptic Curves, Proc. Amer. Math. Soc. 125, No 11 (1997), 3169-3176.

Cf. A005788, A060564, A110620.

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A005788 Conductors of elliptic curves.

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A110620 Number of elliptic curves (up to isomorphism) of conductor n.

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A217055 Prime numbers which are conductors of elliptic curves.

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A381352 Conductors admitting a normalized weight-2 eta-quotient newform.

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