cp's OEIS Frontend

A217055 Prime numbers which are conductors of elliptic curves.

%I A217055 #19 Sep 13 2024 08:17:23
%S A217055 11,17,19,37,43,53,61,67,73,79,83,89,101,109,113,131,139,163,179,197,
%T A217055 229,233,269,277,307,331,347,353,359,373,389,431,433,443,467,503,557,
%U A217055 563,571,593,643,659,677,701,709,733,739,797,811,827,829,997,1019,1051
%N A217055 Prime numbers which are conductors of elliptic curves.
%C A217055 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.
%C A217055 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
%H A217055 Robin Visser, <a href="/A217055/b217055.txt">Table of n, a(n) for n = 1..10000</a> (taken from the Bennett-Gherga-Rechnitzer database).
%H A217055 M. A. Bennett, A. Gherga, and A. Rechnitzer, <a href="https://doi.org/10.1090/mcom/3370">Computing elliptic curves over Q</a>, Math. Comp., 88 (2019), no. 317, 1341-1390.
%H A217055 Armand Brumer and Oisin McGuiness, <a href="http://wstein.org/oisin/">310716 Elliptic Curves of Prime Conductor</a>
%H A217055 J. E. Cremona, <a href="https://johncremona.github.io/ecdata/">Elliptic Curve Data</a>
%H A217055 B. Setzer, <a href="https://doi.org/10.1112/jlms/s2-10.3.367">Elliptic Curves of Prime Conductor</a>, J. London Math. Soc. (2) 10 (1975), 367-378.
%e A217055 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
%o A217055 (Sage)  # Uses Cremona's database of elliptic curves (works for all p < 500000)
%o A217055 def is_A217055(p):
%o A217055     if not Integer(p).is_prime(): return False
%o A217055     return CremonaDatabase().number_of_curves(p) > 0
%o A217055 print([p for p in range(1, 1000) if is_A217055(p)])  # _Robin Visser_, Sep 04 2024
%Y A217055 Cf. A005788, A060564, A363793.
%K A217055 nonn
%O A217055 1,1
%A A217055 _Gene Ward Smith_, Sep 25 2012