A110620 -id:A110620 - 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

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

Original entry on oeis.org

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

A363793 Number of Q-isomorphism classes of elliptic curves E/Q with good reduction away from prime(n).

Original entry on oeis.org

24, 8, 0, 4, 12, 0, 8, 8, 0, 0, 0, 16, 0, 6, 2, 2, 0, 2, 4, 0, 4, 4, 2, 6, 0, 2, 0, 0, 2, 4, 0, 2, 0, 2, 0, 0, 2, 4, 0, 0, 4, 0, 2, 0, 2, 0, 0, 0, 0, 2, 4, 0, 0, 0, 0, 0, 2, 0, 4, 0, 0, 0, 10, 0, 0, 0, 2, 0, 2, 0, 4, 6, 0, 2, 0, 0, 2, 4, 0, 0, 0, 0, 6, 4, 0, 8, 0, 0, 0, 0, 2, 0, 0, 0, 0, 8

Offset: 1

Robin Visser, Jun 22 2023

nonn

R. von Känel and B. Matschke conjecture that a(n) <= 24 for all n.

			For n = 1, there are a(1) = 24 elliptic curves over Q with good reduction outside 2, classified by Ogg (1966), with j-invariants given in A332545.
For n = 2, there are a(2) = 8 elliptic curves over Q with good reduction outside 3. A set of 8 Weierstrass equations for these curves can be given as: y^2 + y = x^3 - 270x - 1708, y^2 + y = x^3 - 30x + 63, y^2 + y = x^3 - 7, y^2 + y = x^3, y^2 + y = x^3 - 1, y^2 + y = x^3 + 20, y^2 + y = x^3 - 61, and y^2 + y = x^3 + 2.
For n = 3, Edixhoven-Groot-Top proved there are no elliptic curves over Q with good reduction away from 5, so a(3) = 0.

M. A. Bennett and A. Rechnitzer, Computing elliptic curves over Q: bad reduction at one prime, In: Melnik, R., Makarov, R., Belair, J. (eds) Recent Progress and Modern Challenges in Applied Mathematics, Modeling and Computational Science. Fields Institute Communications, vol 79. Springer, New York, NY.
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.
A. P. Ogg, Abelian curves of 2-power conductor, Proc. Cambridge Philos. Soc. 62 (1966), 143-148.
B. Setzer, Elliptic curves of prime conductor, J. London Math. Soc. (2)10(1975), 367-378.
R. von Känel and B. Matschke, List of certain finite sets S of rational primes together with the number of rational elliptic curves up to rational isomorphisms with good reduction outside S, 2016.

Cf. A000040, A110620, A332545, A359480, A361661.

def a(n):
    EC = EllipticCurves_with_good_reduction_outside_S([Primes()[n-1]])
    return len(EC)

a(n) = A110620(prime(n)) + A110620(prime(n)^2) for all n > 2.

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

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A363793 Number of Q-isomorphism classes of elliptic curves E/Q with good reduction away from prime(n).

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

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