A363793 Number of Q-isomorphism classes of elliptic curves E/Q with good reduction away from prime(n).

24, 8, 0, 4, 12, 0, 8, 8, 0, 0, 0, 16, 0, 6, 2, 2, 0, 2, 4, 0, 4, 4, 2, 6, 0, 2, 0, 0, 2, 4, 0, 2, 0, 2, 0, 0, 2, 4, 0, 0, 4, 0, 2, 0, 2, 0, 0, 0, 0, 2, 4, 0, 0, 0, 0, 0, 2, 0, 4, 0, 0, 0, 10, 0, 0, 0, 2, 0, 2, 0, 4, 6, 0, 2, 0, 0, 2, 4, 0, 0, 0, 0, 6, 4, 0, 8, 0, 0, 0, 0, 2, 0, 0, 0, 0, 8

Offset

1,1

Comments

R. von Känel and B. Matschke conjecture that a(n) <= 24 for all n.

Links

Table of n, a(n) for n=1..96.

M. A. Bennett and A. Rechnitzer, Computing elliptic curves over Q: bad reduction at one prime, In: Melnik, R., Makarov, R., Belair, J. (eds) Recent Progress and Modern Challenges in Applied Mathematics, Modeling and Computational Science. Fields Institute Communications, vol 79. Springer, New York, NY.

B. Edixhoven, A. de Groot, and J. Top, Elliptic curves over the rationals with bad reduction at only one prime, Math. Comp. 54 (1990), no.189, 413-419.

A. P. Ogg, Abelian curves of 2-power conductor, Proc. Cambridge Philos. Soc. 62 (1966), 143-148.

B. Setzer, Elliptic curves of prime conductor, J. London Math. Soc. (2)10(1975), 367-378.

R. von Känel and B. Matschke, List of certain finite sets S of rational primes together with the number of rational elliptic curves up to rational isomorphisms with good reduction outside S, 2016.

Formula

a(n) = A110620(prime(n)) + A110620(prime(n)^2) for all n > 2.

Example

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.
For n = 2, there are a(2) = 8 elliptic curves over Q with good reduction outside 3. A set of 8 Weierstrass equations for these curves can be given as: y^2 + y = x^3 - 270x - 1708, y^2 + y = x^3 - 30x + 63, y^2 + y = x^3 - 7, y^2 + y = x^3, y^2 + y = x^3 - 1, y^2 + y = x^3 + 20, y^2 + y = x^3 - 61, and y^2 + y = x^3 + 2.
For n = 3, Edixhoven-Groot-Top proved there are no elliptic curves over Q with good reduction away from 5, so a(3) = 0.

Prog

(Sage)
def a(n):
    EC = EllipticCurves_with_good_reduction_outside_S([Primes()[n-1]])
    return len(EC)

Crossrefs

Cf. A000040, A110620, A332545, A359480, A361661.

Sequence in context: A297983 A298632 A241362 * A033823 A040557 A040556

Adjacent sequences: A363790 A363791 A363792 * A363794 A363795 A363796

Keyword

nonn

Author

Robin Visser, Jun 22 2023

Status

approved