OFFSET
0,2
COMMENTS
A. Best and B. Matschke performed a heuristic computation which suggests a(6) = 4576128.
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.
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 a(1) = 24 elliptic curves over Q with good reduction outside 2, classified by Ogg (1966), with j-invariants given in A332545. E.g., a set of 24 Weierstrass equations for these curves can be given as: y^2 = x^3 - 11*x - 14, y^2 = x^3 - 11*x + 14, y^2 = x^3 - x, y^2 = x^3 + 4*x, y^2 = x^3 - 44*x - 112, y^2 = x^3 - 44*x + 112, y^2 = x^3 - 4*x, y^2 = x^3 + x, y^2 = x^3 + x^2 - 9*x + 7, y^2 = x^3 + x^2 + x + 1, y^2 = x^3 + x^2 - 2*x - 2, y^2 = x^3 + x^2 + 3*x - 5, y^2 = x^3 - x^2 - 9*x - 7, y^2 = x^3 - x^2 + x - 1, y^2 = x^3 - x^2 - 2*x + 2, y^2 = x^3 - x^2 + 3*x + 5, y^2 = x^3 + x^2 - 13*x - 21, y^2 = x^3 + x^2 - 3*x + 1, y^2 = x^3 - 2*x, y^2 = x^3 + 8*x, y^2 = x^3 - 8*x, y^2 = x^3 + 2*x, y^2 = x^3 - x^2 - 13*x + 21, y^2 = x^3 - x^2 - 3*x - 1.
PROG
(Sage)
def a(n):
S = Primes()[:n]
EC = EllipticCurves_with_good_reduction_outside_S(S)
return len(EC)
CROSSREFS
KEYWORD
nonn,more
AUTHOR
Robin Visser, Mar 21 2023
STATUS
approved