Base field \(\Q(\sqrt{-2}) \)
Generator \(a\), with minimal polynomial \( x^{2} + 2 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([2, 0, 1]))
gp: K = nfinit(Polrev([2, 0, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2, 0, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([0,0]),K([1,-1]),K([0,0]),K([-2,-4]),K([-4,2])])
gp: E = ellinit([Polrev([0,0]),Polrev([1,-1]),Polrev([0,0]),Polrev([-2,-4]),Polrev([-4,2])], K);
magma: E := EllipticCurve([K![0,0],K![1,-1],K![0,0],K![-2,-4],K![-4,2]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((-8a-4)\) | = | \((a)^{4}\cdot(a-1)^{2}\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 144 \) | = | \(2^{4}\cdot3^{2}\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((-640a-1472)\) | = | \((a)^{12}\cdot(a-1)^{6}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 2985984 \) | = | \(2^{12}\cdot3^{6}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( 8000 \) | ||
sage: E.j_invariant()
gp: E.j
magma: jInvariant(E);
| |||
| Endomorphism ring: | \(\Z[\sqrt{-2}]\) | (complex multiplication) | |
| Geometric endomorphism ring: | \(\Z[\sqrt{-2}]\) | ||
sage: E.has_cm(), E.cm_discriminant()
magma: HasComplexMultiplication(E);
| |||
| Sato-Tate group: | $\mathrm{U}(1)$ | ||
Mordell-Weil group
| Rank: | \(0\) |
| Torsion structure: | \(\Z/2\Z\) |
| sage: T = E.torsion_subgroup(); T.invariants()
gp: T = elltors(E); T[2]
magma: T,piT := TorsionSubgroup(E); Invariants(T);
| |
| Torsion generator: | $\left(a + 1 : 0 : 1\right)$ |
| sage: T.gens()
gp: T[3]
magma: [piT(P) : P in Generators(T)];
| |
BSD invariants
| Analytic rank: | \( 0 \) | ||
|
sage: E.rank()
magma: Rank(E);
|
|||
| Mordell-Weil rank: | \(0\) | ||
| Regulator: | \( 1 \) | ||
| Period: | \( 3.6632836862650086779156541051659805576 \) | ||
| Tamagawa product: | \( 2 \) = \(1\cdot2\) | ||
| Torsion order: | \(2\) | ||
| Leading coefficient: | \( 1.2951663679840203510143703764735140393 \) | ||
| Analytic order of Ш: | \( 1 \) (rounded) | ||
Local data at primes of bad reduction
sage: E.local_data()
magma: LocalInformation(E);
| prime | Norm | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord(\(\mathfrak{N}\)) | ord(\(\mathfrak{D}\)) | ord\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((a)\) | \(2\) | \(1\) | \(II^{*}\) | Additive | \(1\) | \(4\) | \(12\) | \(0\) |
| \((a-1)\) | \(3\) | \(2\) | \(I_0^{*}\) | Additive | \(-1\) | \(2\) | \(6\) | \(0\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) .
The image is a Borel subgroup if \(p=2\), a split Cartan subgroup if \(\left(\frac{ -2 }{p}\right)=+1\) or a nonsplit Cartan subgroup if \(\left(\frac{ -2 }{p}\right)=-1\).
Isogenies and isogeny class
This curve has no rational isogenies other than endomorphisms. Its isogeny class 144.3-CMa consists of this curve only.
Base change
This elliptic curve is a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.