A368080 Number of Qbar-isomorphism classes of elliptic curves E/Q with good reduction outside the first n prime numbers.

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

Offset: 0

Author

Robin Visser, Dec 10 2023

Keywords

• nonn
• more

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.

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 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).
		

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.

Crossrefs

Cf. A332545, A361661.

Programs

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))