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+xy+iy=x3+(i−1)x2+(−i+166)x−927i−367
sage:E = EllipticCurve([K([1,0]),K([-1,1]),K([0,1]),K([166,-1]),K([-367,-927])])
gp:E = ellinit([Polrev([1,0]),Polrev([-1,1]),Polrev([0,1]),Polrev([166,-1]),Polrev([-367,-927])], K);
magma:E := EllipticCurve([K![1,0],K![-1,1],K![0,1],K![166,-1],K![-367,-927]]);
This is a global minimal model.
sage:E.is_global_minimal_model()
Z
P | h^(P) | Order |
(−21i−19:−58i+146:1) | 0.12945349623647181623622699107400263408 | ∞ |
Conductor: |
N |
= |
(25i+275) |
= |
(i+1)⋅(−i−2)2⋅(2i+1)2⋅(−6i−5) |
sage:E.conductor()
gp:ellglobalred(E)[1]
magma:Conductor(E);
|
Conductor norm: |
N(N) |
= |
76250 |
= |
2⋅52⋅52⋅61 |
sage:E.conductor().norm()
gp:idealnorm(ellglobalred(E)[1])
magma:Norm(Conductor(E));
|
Discriminant: |
Δ |
= |
−276523125i+77556875 |
Discriminant ideal:
|
Dmin=(Δ)
|
= |
(−276523125i+77556875) |
= |
(i+1)⋅(−i−2)7⋅(2i+1)4⋅(−6i−5)5 |
sage:E.discriminant()
gp:E.disc
magma:Discriminant(E);
|
Discriminant norm:
|
N(Dmin)=N(Δ)
|
= |
82480107519531250 |
= |
2⋅57⋅54⋅615 |
sage:E.discriminant().norm()
gp:norm(E.disc)
magma:Norm(Discriminant(E));
|
j-invariant: |
j |
= |
−844596301014362283266683i−84459630104028212136069 |
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) |
≈ |
0.12945349623647181623622699107400263408
|
Néron-Tate Regulator:
|
RegNT(E/K) |
≈ |
0.258906992472943632472453982148005268160
|
Global period: |
Ω(E/K) | ≈ |
1.04647644809975611835187191133278244792 |
Tamagawa product: |
∏pcp | = |
20
= 1⋅22⋅1⋅5
|
Torsion order: |
#E(K)tor | = |
1 |
Special value: |
L(r)(E/K,1)/r! |
≈ | 2.7094006987127634519724927618884770900 |
Analytic order of Ш:
|
Шan | = |
1 (rounded) |
2.709400699≈L′(E/K,1)=?#E(K)tor2⋅∣dK∣1/2#Ш(E/K)⋅Ω(E/K)⋅RegNT(E/K)⋅∏pcp≈12⋅2.0000001⋅1.046476⋅0.258907⋅20≈2.709400699
sage:E.local_data()
magma:LocalInformation(E);
This elliptic curve is not semistable.
There
are 4 primes p
of bad reduction.
This elliptic curve is not a Q-curve.
It is not the base change of an elliptic curve defined over any subfield.