Base field 3.3.148.1
Generator \(a\), with minimal polynomial \( x^{3} - x^{2} - 3 x + 1 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([1, -3, -1, 1]))
gp: K = nfinit(Polrev([1, -3, -1, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -3, -1, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([-1,0,1]),K([1,1,-1]),K([0,0,0]),K([56622738702290433353,-143775828060934351842,-122876240093597390564]),K([-617397594907508652287429548778,1567689103089350829411708766103,1339806177584198660985008156380])])
gp: E = ellinit([Polrev([-1,0,1]),Polrev([1,1,-1]),Polrev([0,0,0]),Polrev([56622738702290433353,-143775828060934351842,-122876240093597390564]),Polrev([-617397594907508652287429548778,1567689103089350829411708766103,1339806177584198660985008156380])], K);
magma: E := EllipticCurve([K![-1,0,1],K![1,1,-1],K![0,0,0],K![56622738702290433353,-143775828060934351842,-122876240093597390564],K![-617397594907508652287429548778,1567689103089350829411708766103,1339806177584198660985008156380]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((2)\) | = | \((a^2-a-2)^{3}\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 8 \) | = | \(2^{3}\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((4a^2-12)\) | = | \((a^2-a-2)^{8}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 256 \) | = | \(2^{8}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( 5088 a^{2} + 8224 a + 3904 \) | ||
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: | \(0\) | |
| Torsion structure: | \(\Z/2\Z\oplus\Z/4\Z\) | |
| sage: T = E.torsion_subgroup(); T.invariants()
gp: T = elltors(E); T[2]
magma: T,piT := TorsionSubgroup(E); Invariants(T);
| ||
| Torsion generators: | $\left(\frac{4347879635}{2} a^{2} + \frac{5087395205}{2} a - 1001775658 : -\frac{8063741397}{2} a^{2} - 4717637420 a + 1857930881 : 1\right)$ | $\left(4785876661 a^{2} + 5599889607 a - 2205385219 : 594441845638255 a^{2} + 695548370666361 a - 273925416936940 : 1\right)$ |
| 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: | \( 223.18826420022044179173853209997750472 \) | ||
| Tamagawa product: | \( 2 \) | ||
| Torsion order: | \(8\) | ||
| Leading coefficient: | \( 0.57331132207744951681123408265720299983 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((a^2-a-2)\) | \(2\) | \(2\) | \(I_{1}^{*}\) | Additive | \(-1\) | \(3\) | \(8\) | \(0\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.
| prime | Image of Galois Representation |
|---|---|
| \(2\) | 2Cs |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2 and 4.
Its isogeny class
8.1-a
consists of curves linked by isogenies of
degrees dividing 8.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.