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A361661
Number of Q-isomorphism classes of elliptic curves E/Q with good reduction outside the first n prime numbers.
5
0, 24, 752, 7600, 71520, 592192
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 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.
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
Sequence in context: A175604 A283873 A359480 * A269147 A269209 A270252
KEYWORD
nonn,more
AUTHOR
Robin Visser, Mar 21 2023
STATUS
approved