Base field 3.3.229.1
Generator \(a\), with minimal polynomial \( x^{3} - 4 x - 1 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-1, -4, 0, 1]))
gp: K = nfinit(Polrev([-1, -4, 0, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, -4, 0, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([-2,0,1]),K([-2,-1,1]),K([1,0,0]),K([0,-4,-1]),K([0,1,1])])
gp: E = ellinit([Polrev([-2,0,1]),Polrev([-2,-1,1]),Polrev([1,0,0]),Polrev([0,-4,-1]),Polrev([0,1,1])], K);
magma: E := EllipticCurve([K![-2,0,1],K![-2,-1,1],K![1,0,0],K![0,-4,-1],K![0,1,1]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((a-3)\) | = | \((a+1)\cdot(-a^2+2)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 14 \) | = | \(2\cdot7\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((2a^2-3a+5)\) | = | \((a+1)^{4}\cdot(-a^2+2)^{2}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 784 \) | = | \(2^{4}\cdot7^{2}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( \frac{2334119}{784} a^{2} - \frac{3847349}{784} a + \frac{1948435}{784} \) | ||
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: | \(\Z/2\Z\oplus\Z/6\Z\) | |
| sage: T = E.torsion_subgroup(); T.invariants()
gp: T = elltors(E); T[2]
magma: T,piT := TorsionSubgroup(E); Invariants(T);
| ||
| Torsion generators: | $\left(-\frac{1}{4} a^{2} + \frac{1}{2} a + \frac{3}{4} : -\frac{1}{8} a^{2} - \frac{3}{8} a : 1\right)$ | $\left(a^{2} + 3 a + 1 : -6 a^{2} - 13 a - 3 : 1\right)$ |
| 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: | \( 342.33371535775197455865191864295447637 \) | ||
| Tamagawa product: | \( 4 \) = \(2\cdot2\) | ||
| Torsion order: | \(12\) | ||
| Leading coefficient: | \( 0.62839024075361314906521287939860880115 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((a+1)\) | \(2\) | \(2\) | \(I_{4}\) | Non-split multiplicative | \(1\) | \(1\) | \(4\) | \(4\) |
| \((-a^2+2)\) | \(7\) | \(2\) | \(I_{2}\) | Split multiplicative | \(-1\) | \(1\) | \(2\) | \(2\) |
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 |
| \(3\) | 3B.1.1 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2, 3, 4, 6 and 12.
Its isogeny class
14.1-b
consists of curves linked by isogenies of
degrees dividing 24.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.