OFFSET
0,2
COMMENTS
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.
REFERENCES
N. M. Stephens, The Birch Swinnerton-Dyer Conjecture for Selmer curves of positive rank, Ph.D. Thesis (1965), The University of Manchester.
LINKS
M. A. Bennett, A. Gherga, and A. Rechnitzer, Computing elliptic curves over Q, Math. Comp., 88(317):1341-1390, 2019.
A. J. Best and B. Matschke, Elliptic curves with good reduction outside {2, 3, 5, 7, 11, 13}.
A. J. Best and B. Matschke, Elliptic curves with good reduction outside of the first six primes, arXiv:2007.10535 [math.NT], 2020.
F. B. Coghlan, Elliptic Curves with Conductor N = 2^m 3^n, Ph.D. Thesis (1967), The University of Manchester.
B. Matschke, Elliptic curve tables.
A. P. Ogg, Abelian curves of 2-power conductor, Proc. Cambridge Philos. Soc. 62 (1966), 143-148.
R. von Känel and B. Matschke, Solving S-unit, Mordell, Thue, Thue-Mahler and generalized Ramanujan-Nagell equations via Shimura-Taniyama conjecture, arXiv:1605.06079 [math.NT], 2016.
EXAMPLE
For n = 0, Tate proved there are no elliptic curves over Q with good reduction everywhere, so a(0) = 0.
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).
PROG
(Sage) # This is very slow for n > 2
def a(n):
S = Primes()[:n]
EC = EllipticCurves_with_good_reduction_outside_S(S)
return len(set(E.j_invariant() for E in EC))
CROSSREFS
KEYWORD
nonn,more
AUTHOR
Robin Visser, Dec 10 2023
STATUS
approved