A368081 Number of Qbar-isomorphism classes of elliptic curves E/Q with good reduction away from 2 and prime(n).

5, 83, 32, 33, 26, 39, 29, 39, 29, 34, 32, 27, 19, 18, 14, 34, 35, 19, 11, 33, 14, 35, 19, 21, 16, 24, 10, 27, 17, 15, 32, 17, 16, 18, 11, 13, 14, 26, 15, 20, 13, 16, 7, 8, 11, 11, 11, 32, 11, 33, 17, 12, 18

Offset: 1

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 2 and prime(n).

Examples

			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

• F. B. Coghlan, Elliptic Curves with Conductor N = 2^m 3^n, Ph.D. Thesis (1967), The University of Manchester.
• J. E. Cremona and M. P. Lingham, Finding all elliptic curves with good reduction outside a given set of primes, Experiment. Math. 16 (2007), no. 3, 303-312.
• A. P. Ogg, Abelian curves of 2-power conductor, Proc. Cambridge Philos. Soc. 62 (1966), 143-148.
• R. von Känel and B. Matschke, List of all rational elliptic curves with good reduction outside {2, p} up to Q-isomorphisms, 2015.

Crossrefs

Cf. A332545, A359480.

Programs

Sage

# This is very slow for n > 4
def a(n):
    S = list(set([2, Primes()[n-1]]))
    EC = EllipticCurves_with_good_reduction_outside_S(S)
    return len(set(E.j_invariant() for E in EC))