Generator a, with minimal polynomial
x2−6; class number 1.
sage:R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-6, 0, 1]))
gp:K = nfinit(Polrev([-6, 0, 1]));
magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-6, 0, 1]);
y2+(a+1)xy+y=x3+ax2+(−34a−69)x+112a+223
sage:E = EllipticCurve([K([1,1]),K([0,1]),K([1,0]),K([-69,-34]),K([223,112])])
gp:E = ellinit([Polrev([1,1]),Polrev([0,1]),Polrev([1,0]),Polrev([-69,-34]),Polrev([223,112])], K);
magma:E := EllipticCurve([K![1,1],K![0,1],K![1,0],K![-69,-34],K![223,112]]);
This is a global minimal model.
sage:E.is_global_minimal_model()
Z⊕Z/2Z
P | h^(P) | Order |
(5:−4a+3:1) | 0.52031257234450943068603338021579827619 | ∞ |
(21a+417:−819a−833:1) | 0 | 2 |
Conductor: |
N |
= |
(−10a−15) |
= |
(a+3)⋅(−a−1)⋅(−a+1)2 |
sage:E.conductor()
gp:ellglobalred(E)[1]
magma:Conductor(E);
|
Conductor norm: |
N(N) |
= |
375 |
= |
3⋅5⋅52 |
sage:E.conductor().norm()
gp:idealnorm(ellglobalred(E)[1])
magma:Norm(Conductor(E));
|
Discriminant: |
Δ |
= |
43125a+13125 |
Discriminant ideal:
|
Dmin=(Δ)
|
= |
(43125a+13125) |
= |
(a+3)2⋅(−a−1)4⋅(−a+1)9 |
sage:E.discriminant()
gp:E.disc
magma:Discriminant(E);
|
Discriminant norm:
|
N(Dmin)=N(Δ)
|
= |
−10986328125 |
= |
−32⋅54⋅59 |
sage:E.discriminant().norm()
gp:norm(E.disc)
magma:Norm(Discriminant(E));
|
j-invariant: |
j |
= |
187514550280309a+187535641186181 |
sage:E.j_invariant()
gp:E.j
magma:jInvariant(E);
|
Endomorphism ring: |
End(E) |
= |
Z
|
Geometric endomorphism ring: |
End(EQ) |
= |
Z
(no potential complex multiplication)
|
sage:E.has_cm(), E.cm_discriminant()
magma:HasComplexMultiplication(E);
|
Sato-Tate group: |
ST(E) |
= |
SU(2) |
Analytic rank: |
ran | = |
1
|
sage:E.rank()
magma:Rank(E);
|
Mordell-Weil rank: |
r |
= |
1 |
Regulator:
|
Reg(E/K) |
≈ |
0.52031257234450943068603338021579827619
|
Néron-Tate Regulator:
|
RegNT(E/K) |
≈ |
1.04062514468901886137206676043159655238
|
Global period: |
Ω(E/K) | ≈ |
5.2662521191952088406195224235229681575 |
Tamagawa product: |
∏pcp | = |
16
= 2⋅22⋅2
|
Torsion order: |
#E(K)tor | = |
2 |
Special value: |
L(r)(E/K,1)/r! |
≈ | 4.4745599687873097608045487742521184675 |
Analytic order of Ш:
|
Шan | = |
1 (rounded) |
4.474559969≈L′(E/K,1)=?#E(K)tor2⋅∣dK∣1/2#Ш(E/K)⋅Ω(E/K)⋅RegNT(E/K)⋅∏pcp≈22⋅4.8989791⋅5.266252⋅1.040625⋅16≈4.474559969
sage:E.local_data()
magma:LocalInformation(E);
This elliptic curve is not semistable.
There
are 3 primes p
of bad reduction.
This curve has non-trivial cyclic isogenies of degree d for d=
2.
Its isogeny class
375.2-a
consists of curves linked by isogenies of
degree 2.
This elliptic curve is not a Q-curve.
It is not the base change of an elliptic curve defined over any subfield.