Generator a, with minimal polynomial
x2−7; class number 1.
sage:R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-7, 0, 1]))
gp:K = nfinit(Polrev([-7, 0, 1]));
magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-7, 0, 1]);
y2+(a+1)xy+ay=x3+(a+1)x2+(148a−380)x+1439a−3796
sage:E = EllipticCurve([K([1,1]),K([1,1]),K([0,1]),K([-380,148]),K([-3796,1439])])
gp:E = ellinit([Polrev([1,1]),Polrev([1,1]),Polrev([0,1]),Polrev([-380,148]),Polrev([-3796,1439])], K);
magma:E := EllipticCurve([K![1,1],K![1,1],K![0,1],K![-380,148],K![-3796,1439]]);
This is a global minimal model.
sage:E.is_global_minimal_model()
Z⊕Z/2Z
P | h^(P) | Order |
(3a−10:a:1) | 0.87881968258904483276103482454452875940 | ∞ |
(−213a+15:−419a+461:1) | 0 | 2 |
Conductor: |
N |
= |
(9a) |
= |
(−a+2)2⋅(−a−2)2⋅(a) |
sage:E.conductor()
gp:ellglobalred(E)[1]
magma:Conductor(E);
|
Conductor norm: |
N(N) |
= |
567 |
= |
32⋅32⋅7 |
sage:E.conductor().norm()
gp:idealnorm(ellglobalred(E)[1])
magma:Norm(Conductor(E));
|
Discriminant: |
Δ |
= |
−18522a−9261 |
Discriminant ideal:
|
Dmin=(Δ)
|
= |
(−18522a−9261) |
= |
(−a+2)3⋅(−a−2)6⋅(a)6 |
sage:E.discriminant()
gp:E.disc
magma:Discriminant(E);
|
Discriminant norm:
|
N(Dmin)=N(Δ)
|
= |
−2315685267 |
= |
−33⋅36⋅76 |
sage:E.discriminant().norm()
gp:norm(E.disc)
magma:Norm(Discriminant(E));
|
j-invariant: |
j |
= |
−3431204736a+3433558976 |
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) |
≈ |
0.87881968258904483276103482454452875940
|
Néron-Tate Regulator:
|
RegNT(E/K) |
≈ |
1.75763936517808966552206964908905751880
|
Global period: |
Ω(E/K) | ≈ |
4.4101128779332199173848050291868327107 |
Tamagawa product: |
∏pcp | = |
8
= 2⋅2⋅2
|
Torsion order: |
#E(K)tor | = |
2 |
Special value: |
L(r)(E/K,1)/r! |
≈ | 2.9297492801828298741848710359144752013 |
Analytic order of Ш:
|
Шan | = |
1 (rounded) |
2.929749280≈L′(E/K,1)=?#E(K)tor2⋅∣dK∣1/2#Ш(E/K)⋅Ω(E/K)⋅RegNT(E/K)⋅∏pcp≈22⋅5.2915031⋅4.410113⋅1.757639⋅8≈2.929749280
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, 3 and 6.
Its isogeny class
567.1-m
consists of curves linked by isogenies of
degrees dividing 6.
This elliptic curve is not a Q-curve.
It is not the base change of an elliptic curve defined over any subfield.