Base field 4.4.4752.1
Generator \(a\), with minimal polynomial \( x^{4} - 2 x^{3} - 3 x^{2} + 4 x + 1 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([1, 4, -3, -2, 1]))
gp: K = nfinit(Polrev([1, 4, -3, -2, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 4, -3, -2, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([-1,-4,0,1]),K([3,-3,-2,1]),K([-1,0,1,0]),K([-8,9,4,-3]),K([5,1,-2,0])])
gp: E = ellinit([Polrev([-1,-4,0,1]),Polrev([3,-3,-2,1]),Polrev([-1,0,1,0]),Polrev([-8,9,4,-3]),Polrev([5,1,-2,0])], K);
magma: E := EllipticCurve([K![-1,-4,0,1],K![3,-3,-2,1],K![-1,0,1,0],K![-8,9,4,-3],K![5,1,-2,0]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((a^2-4)\) | = | \((a^3-a^2-4a+1)\cdot(a^2+a-2)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 39 \) | = | \(3\cdot13\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((27a^2-108a+27)\) | = | \((a^3-a^2-4a+1)^{14}\cdot(a^2+a-2)\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 62178597 \) | = | \(3^{14}\cdot13\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( \frac{15263744}{1053} a^{3} - \frac{34884416}{1053} a^{2} - \frac{39642112}{1053} a + \frac{24189824}{351} \) | ||
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 | $\left(-a^{3} + 4 a^{2} - 4 a - 1 : 7 a^{3} - 22 a^{2} + 11 a + 2 : 1\right)$ |
| Height | \(0.74264786874714864685746146636994417285\) |
| Torsion structure: | \(\Z/14\Z\) |
| sage: T = E.torsion_subgroup(); T.invariants()
gp: T = elltors(E); T[2]
magma: T,piT := TorsionSubgroup(E); Invariants(T);
| |
| Torsion generator: | $\left(-a^{3} + 2 a^{2} + 3 a - 4 : -2 a^{3} + 3 a^{2} + 6 a - 4 : 1\right)$ |
| sage: T.gens()
gp: T[3]
magma: [piT(P) : P in Generators(T)];
| |
BSD invariants
| Analytic rank: | \( 1 \) | ||
|
sage: E.rank()
magma: Rank(E);
|
|||
| Mordell-Weil rank: | \(1\) | ||
| Regulator: | \( 0.74264786874714864685746146636994417285 \) | ||
| Period: | \( 751.20686456935556936470852532490741900 \) | ||
| Tamagawa product: | \( 14 \) = \(( 2 \cdot 7 )\cdot1\) | ||
| Torsion order: | \(14\) | ||
| Leading coefficient: | \( 2.31225765868508 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((a^3-a^2-4a+1)\) | \(3\) | \(14\) | \(I_{14}\) | Split multiplicative | \(-1\) | \(1\) | \(14\) | \(14\) |
| \((a^2+a-2)\) | \(13\) | \(1\) | \(I_{1}\) | Non-split multiplicative | \(1\) | \(1\) | \(1\) | \(1\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.
| prime | Image of Galois Representation |
|---|---|
| \(2\) | 2B |
| \(7\) | 7B.1.1 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2, 7 and 14.
Its isogeny class
39.2-e
consists of curves linked by isogenies of
degrees dividing 14.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.