Base field \(\Q(\sqrt{-3}) \)
Generator \(a\), with minimal polynomial \( x^{2} - x + 1 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([1, -1, 1]))
gp: K = nfinit(Polrev([1, -1, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -1, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([0,0]),K([0,-1]),K([0,0]),K([-4,4]),K([4,-4])])
gp: E = ellinit([Polrev([0,0]),Polrev([0,-1]),Polrev([0,0]),Polrev([-4,4]),Polrev([4,-4])], K);
magma: E := EllipticCurve([K![0,0],K![0,-1],K![0,0],K![-4,4],K![4,-4]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((192a-160)\) | = | \((2)^{5}\cdot(6a-5)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 31744 \) | = | \(4^{5}\cdot31\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((2560a+512)\) | = | \((2)^{9}\cdot(6a-5)\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 8126464 \) | = | \(4^{9}\cdot31\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( \frac{53240}{31} a - \frac{63888}{31} \) | ||
sage: E.j_invariant()
gp: E.j
magma: jInvariant(E);
| |||
| Endomorphism ring: | \(\Z\) | ||
| Geometric endomorphism ring: | \(\Z\) | (no potential complex multiplication) | |
sage: E.has_cm(), E.cm_discriminant()
magma: HasComplexMultiplication(E);
| |||
| Sato-Tate group: | $\mathrm{SU}(2)$ | ||
Mordell-Weil group
| Rank: | \(1\) |
| Generator | $\left(2 : 2 : 1\right)$ |
| Height | \(0.21902608408926328664792264626033995039\) |
| Torsion structure: | trivial |
| sage: T = E.torsion_subgroup(); T.invariants()
gp: T = elltors(E); T[2]
magma: T,piT := TorsionSubgroup(E); Invariants(T);
| |
BSD invariants
| Analytic rank: | \( 1 \) | ||
|
sage: E.rank()
magma: Rank(E);
|
|||
| Mordell-Weil rank: | \(1\) | ||
| Regulator: | \( 0.21902608408926328664792264626033995039 \) | ||
| Period: | \( 3.5521509974230378981181366874571409086 \) | ||
| Tamagawa product: | \( 2 \) = \(2\cdot1\) | ||
| Torsion order: | \(1\) | ||
| Leading coefficient: | \( 3.5934914595322574411363463853804307168 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((2)\) | \(4\) | \(2\) | \(I_0^{*}\) | Additive | \(1\) | \(5\) | \(9\) | \(0\) |
| \((6a-5)\) | \(31\) | \(1\) | \(I_{1}\) | Non-split multiplicative | \(1\) | \(1\) | \(1\) | \(1\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) .
Isogenies and isogeny class
This curve has no rational isogenies. Its isogeny class 31744.2-c consists of this curve only.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.