A361661 Number of Q-isomorphism classes of elliptic curves E/Q with good reduction outside the first n prime numbers.

0, 24, 752, 7600, 71520, 592192

Offset: 0

Author

Robin Visser, Mar 21 2023

Keywords

• nonn
• more

Comments

A. Best and B. Matschke performed a heuristic computation which suggests a(6) = 4576128.

Examples

			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.
		

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.

Crossrefs

Cf. A332545, A359480.

Programs

Sage

def a(n):
    S = Primes()[:n]
    EC = EllipticCurves_with_good_reduction_outside_S(S)
    return len(EC)