Generator a, with minimal polynomial
x2−x+6; class number 3.
sage:R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([6, -1, 1]))
gp:K = nfinit(Polrev([6, -1, 1]));
magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![6, -1, 1]);
y2+xy+y=x3+(−a−1)x2+(−3a+10)x+a+7
sage:E = EllipticCurve([K([1,0]),K([-1,-1]),K([1,0]),K([10,-3]),K([7,1])])
gp:E = ellinit([Polrev([1,0]),Polrev([-1,-1]),Polrev([1,0]),Polrev([10,-3]),Polrev([7,1])], K);
magma:E := EllipticCurve([K![1,0],K![-1,-1],K![1,0],K![10,-3],K![7,1]]);
This is a global minimal model.
sage:E.is_global_minimal_model()
Z⊕Z/2Z⊕Z/2Z
P | h^(P) | Order |
(−1:a+1:1) | 0.27386528121531218143334392311940850638 | ∞ |
(−43a+41:83a−85:1) | 0 | 2 |
(−41a−21:81a−41:1) | 0 | 2 |
Conductor: |
N |
= |
(18,6a) |
= |
(2,a)⋅(2,a+1)⋅(3,a)2⋅(3,a+2) |
sage:E.conductor()
gp:ellglobalred(E)[1]
magma:Conductor(E);
|
Conductor norm: |
N(N) |
= |
108 |
= |
2⋅2⋅32⋅3 |
sage:E.conductor().norm()
gp:idealnorm(ellglobalred(E)[1])
magma:Norm(Conductor(E));
|
Discriminant: |
Δ |
= |
13104a−41040 |
Discriminant ideal:
|
Dmin=(Δ)
|
= |
(13104a−41040) |
= |
(2,a)4⋅(2,a+1)8⋅(3,a)10⋅(3,a+2)2 |
sage:E.discriminant()
gp:E.disc
magma:Discriminant(E);
|
Discriminant norm:
|
N(Dmin)=N(Δ)
|
= |
2176782336 |
= |
24⋅28⋅310⋅32 |
sage:E.discriminant().norm()
gp:norm(E.disc)
magma:Norm(Discriminant(E));
|
j-invariant: |
j |
= |
−2073654256267a+1036822576079 |
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) |
≈ |
0.27386528121531218143334392311940850638
|
Néron-Tate Regulator:
|
RegNT(E/K) |
≈ |
0.547730562430624362866687846238817012760
|
Global period: |
Ω(E/K) | ≈ |
4.2631400829787424910520982906442720882 |
Tamagawa product: |
∏pcp | = |
64
= 22⋅2⋅22⋅2
|
Torsion order: |
#E(K)tor | = |
4 |
Special value: |
L(r)(E/K,1)/r! |
≈ | 1.9475680945168386727622166971056384619 |
Analytic order of Ш:
|
Шan | = |
1 (rounded) |
1.947568095≈L′(E/K,1)=?#E(K)tor2⋅∣dK∣1/2#Ш(E/K)⋅Ω(E/K)⋅RegNT(E/K)⋅∏pcp≈42⋅4.7958321⋅4.263140⋅0.547731⋅64≈1.947568095
sage:E.local_data()
magma:LocalInformation(E);
This elliptic curve is not semistable.
There
are 4 primes p
of bad reduction.
p |
N(p) |
Tamagawa number |
Kodaira symbol |
Reduction type |
Root number |
ordp(N) |
ordp(Dmin) |
ordp(den(j)) |
(2,a)
|
2
|
4
|
I4
|
Split multiplicative
|
−1 |
1 |
4 |
4 |
(2,a+1)
|
2
|
2
|
I8
|
Non-split multiplicative
|
1 |
1 |
8 |
8 |
(3,a)
|
3
|
4
|
I4∗
|
Additive
|
−1 |
2 |
10 |
4 |
(3,a+2)
|
3
|
2
|
I2
|
Non-split multiplicative
|
1 |
1 |
2 |
2 |
This curve has non-trivial cyclic isogenies of degree d for d=
2 and 4.
Its isogeny class
108.6-b
consists of curves linked by isogenies of
degrees dividing 8.
This elliptic curve is a Q-curve.
It is not the base change of an elliptic curve defined over any subfield.