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

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

Author

Robin Visser, Aug 30 2024

Keywords

• nonn

Comments

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

Examples

			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.
		

Links

• 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.

Crossrefs

Cf. A005788, A217055, A363793.

Programs

Sage

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)])