A368080 - OEIS

%I #6 Dec 12 2023 08:20:15

%S 0,5,83,442,2140,8980,34960,124124,418816

%N Number of Qbar-isomorphism classes of elliptic curves E/Q with good reduction outside the first n prime numbers.

%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 the first n prime numbers.

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

%H 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(317):1341-1390, 2019.

%H A. J. Best and B. Matschke, <a href="https://github.com/elliptic-curve-data/ec-data-S6">Elliptic curves with good reduction outside {2, 3, 5, 7, 11, 13}</a>.

%H A. J. Best and B. Matschke, <a href="https://arxiv.org/abs/2007.10535">Elliptic curves with good reduction outside of the first six primes</a>, arXiv:2007.10535 [math.NT], 2020.

%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 B. Matschke, <a href="https://github.com/bmatschke/s-unit-equations/tree/main/elliptic-curve-tables">Elliptic curve tables</a>.

%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://arxiv.org/abs/1605.06079">Solving S-unit, Mordell, Thue, Thue-Mahler and generalized Ramanujan-Nagell equations via Shimura-Taniyama conjecture</a>, arXiv:1605.06079 [math.NT], 2016.

%e For n = 0, Tate proved there are no elliptic curves over Q with good reduction everywhere, so a(0) = 0.

%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 > 2

%o def a(n):

%o     S = Primes()[:n]

%o     EC = EllipticCurves_with_good_reduction_outside_S(S)

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

%Y Cf. A332545, A361661.

%K nonn,more

%O 0,2

%A _Robin Visser_, Dec 10 2023