Base field 4.4.19821.1
Generator \(a\), with minimal polynomial \( x^{4} - x^{3} - 8 x^{2} + 6 x + 3 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([3, 6, -8, -1, 1]))
gp: K = nfinit(Polrev([3, 6, -8, -1, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3, 6, -8, -1, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([-3,3,2/3,-1/3]),K([3,1,-1,0]),K([-2,3,2/3,-1/3]),K([-18,14,11/3,-7/3]),K([-96,79,41/3,-31/3])])
gp: E = ellinit([Polrev([-3,3,2/3,-1/3]),Polrev([3,1,-1,0]),Polrev([-2,3,2/3,-1/3]),Polrev([-18,14,11/3,-7/3]),Polrev([-96,79,41/3,-31/3])], K);
magma: E := EllipticCurve([K![-3,3,2/3,-1/3],K![3,1,-1,0],K![-2,3,2/3,-1/3],K![-18,14,11/3,-7/3],K![-96,79,41/3,-31/3]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((-2/3a^3-2/3a^2+6a+4)\) | = | \((-1/3a^3-1/3a^2+3a+2)\cdot(2)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 48 \) | = | \(3\cdot16\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((2/3a^3+2/3a^2-4a-4)\) | = | \((-1/3a^3-1/3a^2+3a+2)^{2}\cdot(2)\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 144 \) | = | \(3^{2}\cdot16\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( \frac{185538109}{6} a^{3} - \frac{83396763}{2} a^{2} - \frac{465693059}{2} a + \frac{1599654271}{6} \) | ||
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 | $\left(\frac{7}{3} a^{3} - \frac{8}{3} a^{2} - 18 a + 18 : 16 a^{3} - 22 a^{2} - 121 a + 139 : 1\right)$ |
| Height | \(0.35256003684125163928178834270172678052\) |
| 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: | \( 0.35256003684125163928178834270172678052 \) | ||
| Period: | \( 351.64059273226286747282399401659089367 \) | ||
| Tamagawa product: | \( 2 \) = \(2\cdot1\) | ||
| Torsion order: | \(1\) | ||
| Leading coefficient: | \( 7.04464791824163 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((-1/3a^3-1/3a^2+3a+2)\) | \(3\) | \(2\) | \(I_{2}\) | Split multiplicative | \(-1\) | \(1\) | \(2\) | \(2\) |
| \((2)\) | \(16\) | \(1\) | \(I_{1}\) | 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 48.1-g 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.