

A363793


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


0



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
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OFFSET

1,1


COMMENTS

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


LINKS

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.


FORMULA



EXAMPLE

For n = 1, there are a(1) = 24 elliptic curves over Q with good reduction outside 2, classified by Ogg (1966), with jinvariants 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, EdixhovenGrootTop 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()[n1]])
return len(EC)


CROSSREFS



KEYWORD

nonn


AUTHOR



STATUS

approved



