OFFSET
1,1
COMMENTS
R. von Känel and B. Matschke conjecture that a(n) <= a(2) = 752 for all n.
REFERENCES
N. M. Stephens, The Birch Swinnerton-Dyer Conjecture for Selmer curves of positive rank, Ph.D. Thesis (1965), The University of Manchester.
LINKS
F. B. Coghlan, Elliptic Curves with Conductor N = 2^m 3^n, Ph.D. Thesis (1967), The University of Manchester.
J. E. Cremona and M. P. Lingham, Finding all elliptic curves with good reduction outside a given set of primes, Experiment. Math. 16 (2007), no. 3, 303-312.
A. P. Ogg, Abelian curves of 2-power conductor, Proc. Cambridge Philos. Soc. 62 (1966), 143-148.
R. von Känel and B. Matschke, List of all rational elliptic curves with good reduction outside {2, p} up to Q-isomorphisms, 2015.
EXAMPLE
For n = 1, there are a(1) = 24 elliptic curves over Q with good reduction outside 2, classified by Ogg (1966), with j-invariants given in A332545.
For n = 2, there are a(2) = 752 elliptic curves over Q with good reduction outside {2,3}, classified independently by Coghlan (1967) and Stephens (1965).
PROG
(Sage) # This is very slow!
def a(n):
S = list(set([2, Primes()[n-1]]))
EC = EllipticCurves_with_good_reduction_outside_S(S)
return len(EC)
CROSSREFS
KEYWORD
nonn
AUTHOR
Robin Visser, Mar 31 2023
STATUS
approved