A005788 -id:A005788 - 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

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

A110563 Conductors of positive-rank elliptic curves.

Original entry on oeis.org

37, 43, 53, 57, 58, 61, 65, 77, 79, 82, 83, 88, 89, 91, 92, 99, 101, 102, 106, 112, 117, 118, 121, 122, 123, 124, 128, 129, 130, 131, 135, 136, 138, 141, 142, 143, 145, 148, 152, 153, 154, 155, 156, 158, 160, 162, 163, 166, 170, 171, 172, 175, 176, 184, 185, 189, 190, 192, 196, 197, 198

Offset: 1

Steven Finch, Sep 12 2005

nonn

			a(1) = 37, as there are no positive rank elliptic curves over Q of conductor less than 37, but there is an elliptic curve of rank 1 over Q of conductor equal to 37, given by E : y^2 + y = x^3 - x. - _Robin Visser_, Nov 07 2024

J. E. Cremona, Table of n, a(n) for n = 1..10000
J. E. Cremona, Elliptic Curve Data
Steven R. Finch, Elliptic curves over Q [Broken link]
Steven R. Finch, Elliptic curves over Q
LMFDB, Elliptic curves over Q.
Heinz M. Tschoepe and Horst G. Zimmer, Computation of the Néron-Tate height on elliptic curves, Math. Comp. 48 (1987) 351-370.

Cf. A005788.

# Uses Cremona's database of elliptic curves (works for all k < 500000)
def is_A110563(k):
    curves = [EllipticCurve(i[0]) for i in CremonaDatabase().allcurves(k).values()]
    return any([(E.rank() > 0) for E in curves])
print([k for k in range(1, 100) if is_A110563(k)])  # Robin Visser, Nov 07 2024

More terms added by Robin Visser, Nov 07 2024, taken from J. E. Cremona's database of elliptic curves.

A375818 Primes p such that there exists an elliptic cuve E/Q with good reduction away from p.

Original entry on oeis.org

2, 3, 7, 11, 17, 19, 37, 43, 47, 53, 61, 67, 73, 79, 83, 89, 101, 109, 113, 131, 139, 157, 163, 179, 191, 197, 229, 233, 269, 277, 307, 331, 347, 353, 359, 373, 389, 397, 431, 433, 443, 467, 503, 557, 563, 571, 593, 643, 659, 673, 677, 701, 709, 733, 739, 797, 811, 827, 829, 863, 877

Offset: 1

Robin Visser, Aug 30 2024

nonn

Equivalently, primes p such that there exists an elliptic curve E/Q whose conductor is a power of p.

			a(1) = 2, as there exists an elliptic curve over Q with good reduction away from 2, e.g. E : y^2 = x^3 + x.
a(2) = 3, as there exists an elliptic curve over Q with good reduction away from 3, e.g. E : y^2 + y = x^3.
a(3) = 7, as there exists an elliptic curve over Q with good reduction away from 7, e.g. E : y^2 + xy = x^3 - x^2 - 2x - 1, but there does not exist an elliptic curve over Q with good reduction away from 5.

M. A. Bennett, A. Gherga, and A. Rechnitzer, Computing elliptic curves over Q, Math. Comp., 88 (2019), no. 317, 1341-1390.
J. E. Cremona, Elliptic Curve Data
B. Edixhoven, A. de Groot, and J. Top, Elliptic curves over the rationals with bad reduction at only one prime, Math. Comp. 54 (1990), no. 189, 413-419.
R. von Känel and B. Matschke, Number of rational elliptic curves up to Q-isomorphisms with good reduction outside S, 2016.

Cf. A005788, A217055, A363793.

def is_A375818(p):
    if not Integer(p).is_prime(): return False
    EC = EllipticCurves_with_good_reduction_outside_S([p])
    return len(EC) > 0
print([p for p in range(1, 1000) if is_A375818(p)])

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.

cp's OEIS Frontend

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

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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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A110563 Conductors of positive-rank elliptic curves.

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A375818 Primes p such that there exists an elliptic cuve E/Q with good reduction away from p.

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Sage

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

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