Generator a, with minimal polynomial
x2−14; class number 1.
sage:R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-14, 0, 1]))
gp:K = nfinit(Polrev([-14, 0, 1]));
magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-14, 0, 1]);
y2+(a+1)xy=x3+(1285a−4941)x−48650a+182431
sage:E = EllipticCurve([K([1,1]),K([0,0]),K([0,0]),K([-4941,1285]),K([182431,-48650])])
gp:E = ellinit([Polrev([1,1]),Polrev([0,0]),Polrev([0,0]),Polrev([-4941,1285]),Polrev([182431,-48650])], K);
magma:E := EllipticCurve([K![1,1],K![0,0],K![0,0],K![-4941,1285],K![182431,-48650]]);
This is a global minimal model.
sage:E.is_global_minimal_model()
Z
P | h^(P) | Order |
(−4a+20:−43a+149:1) | 0.27200247337390478371354217345331712666 | ∞ |
Conductor: |
N |
= |
(a+2) |
= |
(−a+4)⋅(−a+3) |
sage:E.conductor()
gp:ellglobalred(E)[1]
magma:Conductor(E);
|
Conductor norm: |
N(N) |
= |
10 |
= |
2⋅5 |
sage:E.conductor().norm()
gp:idealnorm(ellglobalred(E)[1])
magma:Norm(Conductor(E));
|
Discriminant: |
Δ |
= |
−2a+6 |
Discriminant ideal:
|
Dmin=(Δ)
|
= |
(−2a+6) |
= |
(−a+4)2⋅(−a+3) |
sage:E.discriminant()
gp:E.disc
magma:Discriminant(E);
|
Discriminant norm:
|
N(Dmin)=N(Δ)
|
= |
−20 |
= |
−22⋅5 |
sage:E.discriminant().norm()
gp:norm(E.disc)
magma:Norm(Discriminant(E));
|
j-invariant: |
j |
= |
107114676554418062503a+1026620682081199569989 |
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) |
≈ |
0.27200247337390478371354217345331712666
|
Néron-Tate Regulator:
|
RegNT(E/K) |
≈ |
0.544004946747809567427084346906634253320
|
Global period: |
Ω(E/K) | ≈ |
14.163134738548721329178436662472233718 |
Tamagawa product: |
∏pcp | = |
2
= 2⋅1
|
Torsion order: |
#E(K)tor | = |
1 |
Special value: |
L(r)(E/K,1)/r! |
≈ | 2.0591985216127295013337256651752767828 |
Analytic order of Ш:
|
Шan | = |
1 (rounded) |
2.059198522≈L′(E/K,1)=?#E(K)tor2⋅∣dK∣1/2#Ш(E/K)⋅Ω(E/K)⋅RegNT(E/K)⋅∏pcp≈12⋅7.4833151⋅14.163135⋅0.544005⋅2≈2.059198522
sage:E.local_data()
magma:LocalInformation(E);
This elliptic curve is semistable.
There
are 2 primes p
of bad reduction.
This curve has non-trivial cyclic isogenies of degree d for d=
5.
Its isogeny class
10.1-c
consists of curves linked by isogenies of
degree 5.
This elliptic curve is not a Q-curve.
It is not the base change of an elliptic curve defined over any subfield.