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+x2+(532i−534)x+7530i−3252
sage:E = EllipticCurve([K([0,0]),K([1,0]),K([0,0]),K([-534,532]),K([-3252,7530])])
gp:E = ellinit([Polrev([0,0]),Polrev([1,0]),Polrev([0,0]),Polrev([-534,532]),Polrev([-3252,7530])], K);
magma:E := EllipticCurve([K![0,0],K![1,0],K![0,0],K![-534,532],K![-3252,7530]]);
This is a global minimal model.
sage:E.is_global_minimal_model()
Z⊕Z/2Z
P | h^(P) | Order |
(2i−18:−25i+25:1) | 0.81253488526093829540591544489796009346 | ∞ |
(6i−15:0:1) | 0 | 2 |
Conductor: |
N |
= |
(−80i−160) |
= |
(i+1)8⋅(−i−2)2⋅(2i+1) |
sage:E.conductor()
gp:ellglobalred(E)[1]
magma:Conductor(E);
|
Conductor norm: |
N(N) |
= |
32000 |
= |
28⋅52⋅5 |
sage:E.conductor().norm()
gp:idealnorm(ellglobalred(E)[1])
magma:Norm(Conductor(E));
|
Discriminant: |
Δ |
= |
793600i+435200 |
Discriminant ideal:
|
Dmin=(Δ)
|
= |
(793600i+435200) |
= |
(i+1)21⋅(−i−2)6⋅(2i+1)2 |
sage:E.discriminant()
gp:E.disc
magma:Discriminant(E);
|
Discriminant norm:
|
N(Dmin)=N(Δ)
|
= |
819200000000 |
= |
221⋅56⋅52 |
sage:E.discriminant().norm()
gp:norm(E.disc)
magma:Norm(Discriminant(E));
|
j-invariant: |
j |
= |
25358400014i−251259500802 |
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) |
≈ |
0.81253488526093829540591544489796009346
|
Néron-Tate Regulator:
|
RegNT(E/K) |
≈ |
1.62506977052187659081183088979592018692
|
Global period: |
Ω(E/K) | ≈ |
1.17404614738686914652730130763337763422 |
Tamagawa product: |
∏pcp | = |
16
= 22⋅2⋅2
|
Torsion order: |
#E(K)tor | = |
2 |
Special value: |
L(r)(E/K,1)/r! |
≈ | 3.8158138066321454916424828883538092399 |
Analytic order of Ш:
|
Шan | = |
1 (rounded) |
3.815813807≈L′(E/K,1)=?#E(K)tor2⋅∣dK∣1/2#Ш(E/K)⋅Ω(E/K)⋅RegNT(E/K)⋅∏pcp≈22⋅2.0000001⋅1.174046⋅1.625070⋅16≈3.815813807
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, 4 and 8.
Its isogeny class
32000.2-f
consists of curves linked by isogenies of
degrees dividing 8.
This elliptic curve is not a Q-curve.
It is not the base change of an elliptic curve defined over any subfield.