Base field 4.4.15952.1
Generator \(a\), with minimal polynomial \( x^{4} - 6 x^{2} - 2 x + 1 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([1, -2, -6, 0, 1]))
gp: K = nfinit(Polrev([1, -2, -6, 0, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -2, -6, 0, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([-1,-6,0,1]),K([0,0,0,0]),K([-2,5,1,-1]),K([8,-32,-1,7]),K([-24,0,281,125])])
gp: E = ellinit([Polrev([-1,-6,0,1]),Polrev([0,0,0,0]),Polrev([-2,5,1,-1]),Polrev([8,-32,-1,7]),Polrev([-24,0,281,125])], K);
magma: E := EllipticCurve([K![-1,-6,0,1],K![0,0,0,0],K![-2,5,1,-1],K![8,-32,-1,7],K![-24,0,281,125]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((-a^3+6a+4)\) | = | \((a^2+2a)^{2}\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 9 \) | = | \(3^{2}\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((2a^3-2a^2-12a-3)\) | = | \((a^2+2a)^{7}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 2187 \) | = | \(3^{7}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( -\frac{40153079888}{3} a^{3} - \frac{11056865288}{3} a^{2} + \frac{237872824648}{3} a + \frac{145810516136}{3} \) | ||
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 | $\left(\frac{53}{2} a^{3} - \frac{69}{4} a^{2} - \frac{1191}{8} a + \frac{167}{4} : -\frac{10525}{32} a^{3} + \frac{3445}{16} a^{2} + \frac{59229}{32} a - \frac{4151}{8} : 1\right)$ |
| Height | \(2.6461419687788688587386558132584028034\) |
| Torsion structure: | \(\Z/6\Z\) |
| sage: T = E.torsion_subgroup(); T.invariants()
gp: T = elltors(E); T[2]
magma: T,piT := TorsionSubgroup(E); Invariants(T);
| |
| Torsion generator: | $\left(3 a^{3} + a^{2} - 11 a + 3 : -16 a^{3} - 9 a^{2} + 53 a - 13 : 1\right)$ |
| 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: | \( 2.6461419687788688587386558132584028034 \) | ||
| Period: | \( 545.67965286233041576339077855959251426 \) | ||
| Tamagawa product: | \( 2 \) | ||
| Torsion order: | \(6\) | ||
| Leading coefficient: | \( 2.54056796087507 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((a^2+2a)\) | \(3\) | \(2\) | \(I_{1}^{*}\) | Additive | \(-1\) | \(2\) | \(7\) | \(1\) |
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 |
| \(3\) | 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
9.1-b
consists of curves linked by isogenies of
degrees dividing 12.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.