Generator a, with minimal polynomial
x2−x+2; class number 1.
sage:R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([2, -1, 1]))
gp:K = nfinit(Polrev([2, -1, 1]));
magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2, -1, 1]);
y2+xy+y=x3−x2−95x−697
sage:E = EllipticCurve([K([1,0]),K([-1,0]),K([1,0]),K([-95,0]),K([-697,0])])
gp:E = ellinit([Polrev([1,0]),Polrev([-1,0]),Polrev([1,0]),Polrev([-95,0]),Polrev([-697,0])], K);
magma:E := EllipticCurve([K![1,0],K![-1,0],K![1,0],K![-95,0],K![-697,0]]);
This is a global minimal model.
sage:E.is_global_minimal_model()
Z⊕Z/3Z
P | h^(P) | Order |
(−4a−1:−12a−6:1) | 0.23990086462566904313136453079413008001 | ∞ |
(19:−74:1) | 0 | 3 |
Conductor: |
N |
= |
(162) |
= |
(a)⋅(−a+1)⋅(3)4 |
sage:E.conductor()
gp:ellglobalred(E)[1]
magma:Conductor(E);
|
Conductor norm: |
N(N) |
= |
26244 |
= |
2⋅2⋅94 |
sage:E.conductor().norm()
gp:idealnorm(ellglobalred(E)[1])
magma:Norm(Conductor(E));
|
Discriminant: |
Δ |
= |
−169869312 |
Discriminant ideal:
|
Dmin=(Δ)
|
= |
(−169869312) |
= |
(a)21⋅(−a+1)21⋅(3)4 |
sage:E.discriminant()
gp:E.disc
magma:Discriminant(E);
|
Discriminant norm:
|
N(Dmin)=N(Δ)
|
= |
28855583159353344 |
= |
221⋅221⋅94 |
sage:E.discriminant().norm()
gp:norm(E.disc)
magma:Norm(Discriminant(E));
|
j-invariant: |
j |
= |
−20971521159088625 |
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) |
≈ |
0.23990086462566904313136453079413008001
|
Néron-Tate Regulator:
|
RegNT(E/K) |
≈ |
0.479801729251338086262729061588260160020
|
Global period: |
Ω(E/K) | ≈ |
1.16697043968117750273764330319746246122 |
Tamagawa product: |
∏pcp | = |
441
= (3⋅7)⋅(3⋅7)⋅1
|
Torsion order: |
#E(K)tor | = |
3 |
Special value: |
L(r)(E/K,1)/r! |
≈ | 10.369760452363677327351334332217861112 |
Analytic order of Ш:
|
Шan | = |
1 (rounded) |
10.369760452≈L′(E/K,1)=?#E(K)tor2⋅∣dK∣1/2#Ш(E/K)⋅Ω(E/K)⋅RegNT(E/K)⋅∏pcp≈32⋅2.6457511⋅1.166970⋅0.479802⋅441≈10.369760452
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=
3, 7 and 21.
Its isogeny class
26244.2-l
consists of curves linked by isogenies of
degrees dividing 21.