A359480 -id:A359480 - cp's OEIS frontend

A332545 The j-invariants of the elliptic curves defined over Q with good reduction away from 2.

128, 1728, 8000, 10976, 287496

Offset: 1

N. J. A. Sloane, Feb 19 2020

			From _Robin Visser_, Nov 26 2023: (Start)
There are exactly 24 isomorphism classes of elliptic curves defined over Q with good reduction away from 2, classified by Ogg (1966).
There are 4 curves with j-invariant 128 given by y^2 = x^3 + x^2 + x + 1, y^2 = x^3 + x^2 + 3x - 5, y^2 = x^3 - x^2 + x - 1, and y^2 = x^3 - x^2 + 3x + 5.
There are 8 curves with j-invariant 1728 given by y^2 = x^3 - x, y^2 = x^3 + 4x, y^2 = x^3 - 4x, y^2 = x^3 + x, y^2 = x^3 - 2x, y^2 = x^3 + 8x, y^2 = x^3 -8x, and y^2 = x^3 + 2x.
There are 4 curves with j-invariant 8000 given by y^2 = x^3 + x^2 - 13x - 21, y^2 = x^3 + x^2 - 3x + 1, y^2 = x^3 - x^2 - 13x + 21, and y^2 = x^3 - x^2 - 3x - 1.
There are 4 curves with j-invariant 10976 given by y^2 = x^3 + x^2 - 9x + 7, y^2 = x^3 + x^2 - 2x - 2, y^2 = x^3 - x^2 - 9x - 7, and y^2 = x^3 - x^2 - 2x + 2.
There are 4 curves with j-invariant 287496 given by y^2 = x^3 - 11x - 14, y^2 = x^3 - 11x + 14, y^2 = x^3 - 44x - 112, and y^2 = x^3 - 44x + 112. (End)

Pinch, R. G. (1984, July). Elliptic curves with good reduction away from 2. Mathematical Proceedings of the Cambridge Philosophical Society (Vol. 96, No. 1, pp. 25-38). Cambridge University Press. See Prop. 4.1.

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

Cf. A359480.

set(E.j_invariant() for E in EllipticCurves_with_good_reduction_outside_S([2]))  # Robin Visser, Nov 26 2023

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

Original entry on oeis.org

0, 24, 752, 7600, 71520, 592192

Offset: 0

Robin Visser, Mar 21 2023

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

			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.

N. M. Stephens, The Birch Swinnerton-Dyer Conjecture for Selmer curves of positive rank, Ph.D. Thesis (1965), The University of Manchester.

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.

Cf. A332545, A359480.

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

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

Original entry on oeis.org

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

Robin Visser, Jun 22 2023

nonn

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

			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.

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.

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

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

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

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

Original entry on oeis.org

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

Robin Visser, Dec 10 2023

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).

			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).

N. M. Stephens, The Birch Swinnerton-Dyer Conjecture for Selmer curves of positive rank, Ph.D. Thesis (1965), The University of Manchester.

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.

Cf. A332545, A359480.

# 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))

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A332545 The j-invariants of the elliptic curves defined over Q with good reduction away from 2.

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

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

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

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