A217055 -id:A217055 - 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

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)])

cp's OEIS Frontend

A005788 Conductors of elliptic curves.

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