A368081 - OEIS

%I #6 Dec 12 2023 08:19:51

%S 5,83,32,33,26,39,29,39,29,34,32,27,19,18,14,34,35,19,11,33,14,35,19,

%T 21,16,24,10,27,17,15,32,17,16,18,11,13,14,26,15,20,13,16,7,8,11,11,

%U 11,32,11,33,17,12,18

%N Number of Qbar-isomorphism classes of elliptic curves E/Q with good reduction away from 2 and prime(n).

%C Two elliptic curves are isomorphic over Qbar (the algebraic numbers) if and only if they share the same j-invariant, thus a(n) is also the number of distinct j-invariants of elliptic curves E/Q with good reduction outside 2 and prime(n).

%D N. M. Stephens, The Birch Swinnerton-Dyer Conjecture for Selmer curves of positive rank, Ph.D. Thesis (1965), The University of Manchester.

%H F. B. Coghlan, <a href="https://www.proquest.com/openview/451c146777dad63e754d1a3d0f9eb5ee">Elliptic Curves with Conductor N = 2^m 3^n</a>, Ph.D. Thesis (1967), The University of Manchester.

%H J. E. Cremona and M. P. Lingham, <a href="https://projecteuclid.org/journals/experimental-mathematics/volume-16/issue-3/Finding-All-Elliptic-Curves-with-Good-Reduction-Outside-a-Given/em/1204928531.full">Finding all elliptic curves with good reduction outside a given set of primes</a>, Experiment. Math. 16 (2007), no. 3, 303-312.

%H A. P. Ogg, <a href="https://doi.org/10.1017/S0305004100039670">Abelian curves of 2-power conductor</a>, Proc. Cambridge Philos. Soc. 62 (1966), 143-148.

%H R. von Känel and B. Matschke, <a href="https://github.com/bmatschke/solving-classical-diophantine-equations/blob/master/elliptic-curve-database/curves__S_2_p_pMax250.txt">List of all rational elliptic curves with good reduction outside {2, p} up to Q-isomorphisms</a>, 2015.

%e For n = 1, there are 24 elliptic curves over Q with good reduction outside 2, classified by Ogg (1966). These are divided into a(1) = 5 Qbar-isomorphism classes, where the 5 corresponding j-invariants are given by 128, 1728, 8000, 10976, and 287496 (sequence A332545).

%o (Sage) # This is very slow for n > 4

%o def a(n):

%o     S = list(set([2, Primes()[n-1]]))

%o     EC = EllipticCurves_with_good_reduction_outside_S(S)

%o     return len(set(E.j_invariant() for E in EC))

%Y Cf. A332545, A359480.

%K nonn,more

%O 1,1

%A _Robin Visser_, Dec 10 2023