Generator i, with minimal polynomial
x2+1; class number 1.
sage:R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([1, 0, 1]))
gp:K = nfinit(Polrev([1, 0, 1]));
magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 0, 1]);
y2=x3+(−18i+99)x
sage:E = EllipticCurve([K([0,0]),K([0,0]),K([0,0]),K([99,-18]),K([0,0])])
gp:E = ellinit([Polrev([0,0]),Polrev([0,0]),Polrev([0,0]),Polrev([99,-18]),Polrev([0,0])], K);
magma:E := EllipticCurve([K![0,0],K![0,0],K![0,0],K![99,-18],K![0,0]]);
This is a global minimal model.
sage:E.is_global_minimal_model()
Z/2Z
Conductor: |
N |
= |
(144i+108) |
= |
(i+1)4⋅(−i−2)2⋅(3)2 |
sage:E.conductor()
gp:ellglobalred(E)[1]
magma:Conductor(E);
|
Conductor norm: |
N(N) |
= |
32400 |
= |
24⋅52⋅92 |
sage:E.conductor().norm()
gp:idealnorm(ellglobalred(E)[1])
magma:Norm(Conductor(E));
|
Discriminant: |
Δ |
= |
33499008i−55940544 |
Discriminant ideal:
|
Dmin=(Δ)
|
= |
(33499008i−55940544) |
= |
(i+1)12⋅(−i−2)9⋅(3)6 |
sage:E.discriminant()
gp:E.disc
magma:Discriminant(E);
|
Discriminant norm:
|
N(Dmin)=N(Δ)
|
= |
4251528000000000 |
= |
212⋅59⋅96 |
sage:E.discriminant().norm()
gp:norm(E.disc)
magma:Norm(Discriminant(E));
|
j-invariant: |
j |
= |
1728 |
sage:E.j_invariant()
gp:E.j
magma:jInvariant(E);
|
Endomorphism ring: |
End(E) |
= |
Z[−1]
(complex multiplication) |
Geometric endomorphism ring: |
End(EQ) |
= |
Z[−1]
|
sage:E.has_cm(), E.cm_discriminant()
magma:HasComplexMultiplication(E);
|
Sato-Tate group: |
ST(E) |
= |
U(1) |
Analytic rank: |
ran | = |
0
|
sage:E.rank()
magma:Rank(E);
|
Mordell-Weil rank: |
r |
= |
0 |
Regulator:
|
Reg(E/K) |
= |
1
|
Néron-Tate Regulator:
|
RegNT(E/K) |
= |
1
|
Global period: |
Ω(E/K) | ≈ |
1.37077343115282462308804099624222274768 |
Tamagawa product: |
∏pcp | = |
4
= 1⋅2⋅2
|
Torsion order: |
#E(K)tor | = |
2 |
Special value: |
L(r)(E/K,1)/r! |
≈ | 0.68538671557641231154402049812111137384 |
Analytic order of Ш:
|
Шan | = |
1 (rounded) |
0.685386716≈L(E/K,1)=?#E(K)tor2⋅∣dK∣1/2#Ш(E/K)⋅Ω(E/K)⋅RegNT(E/K)⋅∏pcp≈22⋅2.0000001⋅1.370773⋅1⋅4≈0.685386716
sage:E.local_data()
magma:LocalInformation(E);
This elliptic curve is not semistable.
There
are 3 primes p
of bad reduction.
This curve has no rational isogenies other than
endomorphisms.
Its isogeny class 32400.1-CMb consists of this curve only.
This elliptic curve is a Q-curve.
It is not the base change of an elliptic curve defined over any subfield.