cp's OEIS Frontend

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

%I A375818 #7 Sep 13 2024 08:17:39
%S A375818 2,3,7,11,17,19,37,43,47,53,61,67,73,79,83,89,101,109,113,131,139,157,
%T A375818 163,179,191,197,229,233,269,277,307,331,347,353,359,373,389,397,431,
%U A375818 433,443,467,503,557,563,571,593,643,659,673,677,701,709,733,739,797,811,827,829,863,877
%N A375818 Primes p such that there exists an elliptic cuve E/Q with good reduction away from p.
%C A375818 Equivalently, primes p such that there exists an elliptic curve E/Q whose conductor is a power of p.
%H A375818 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 A375818 J. E. Cremona, <a href="https://johncremona.github.io/ecdata/">Elliptic Curve Data</a>
%H A375818 B. Edixhoven, A. de Groot, and J. Top, <a href="https://doi.org/10.1090/S0025-5718-1990-0995209-4">Elliptic curves over the rationals with bad reduction at only one prime</a>, Math. Comp. 54 (1990), no. 189, 413-419.
%H A375818 R. von Känel and B. Matschke, <a href="https://github.com/bmatschke/solving-classical-diophantine-equations/blob/master/elliptic-curve-database/numberOfCurves.txt">Number of rational elliptic curves up to Q-isomorphisms with good reduction outside S</a>, 2016.
%e A375818 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.
%e A375818 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.
%e A375818 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.
%o A375818 (Sage)
%o A375818 def is_A375818(p):
%o A375818     if not Integer(p).is_prime(): return False
%o A375818     EC = EllipticCurves_with_good_reduction_outside_S([p])
%o A375818     return len(EC) > 0
%o A375818 print([p for p in range(1, 1000) if is_A375818(p)])
%Y A375818 Cf. A005788, A217055, A363793.
%K A375818 nonn
%O A375818 1,1
%A A375818 _Robin Visser_, Aug 30 2024