A217055 Prime numbers which are conductors of elliptic curves.

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

Author

Gene Ward Smith, Sep 25 2012

Keywords

• nonn

Comments

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

Examples

			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
		

Links

• 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.

Crossrefs

Cf. A005788, A060564, A363793.

Programs

Sage

# 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