Base field 4.4.10512.1
Generator \(a\), with minimal polynomial \( x^{4} - 7 x^{2} - 6 x + 1 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([1, -6, -7, 0, 1]))
gp: K = nfinit(Polrev([1, -6, -7, 0, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -6, -7, 0, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([-4,-6,0,1]),K([-6,-6,0,1]),K([-1,-5,-1,1]),K([-4830,-4324,153,589]),K([137439,131800,-3626,-18547])])
gp: E = ellinit([Polrev([-4,-6,0,1]),Polrev([-6,-6,0,1]),Polrev([-1,-5,-1,1]),Polrev([-4830,-4324,153,589]),Polrev([137439,131800,-3626,-18547])], K);
magma: E := EllipticCurve([K![-4,-6,0,1],K![-6,-6,0,1],K![-1,-5,-1,1],K![-4830,-4324,153,589],K![137439,131800,-3626,-18547]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((a^3-a^2-5a)\) | = | \((a^3-a^2-5a)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 4 \) | = | \(4\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((-4a^3+4a^2+20a)\) | = | \((a^3-a^2-5a)^{5}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 1024 \) | = | \(4^{5}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( -\frac{46866366404774471142406706382678508925}{8} a^{3} - 837204824113363505263545929366924088 a^{2} + \frac{81776852534182441962754365040569479719}{2} a + \frac{327944882292203164723438266897992960759}{8} \) | ||
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 | $\left(-\frac{1018865163}{175256642} a^{3} + \frac{28337535}{87628321} a^{2} + \frac{7520340789}{175256642} a + \frac{3222895534}{87628321} : \frac{19209079236899}{3281154851524} a^{3} - \frac{94348329286071}{3281154851524} a^{2} - \frac{53561053813909}{3281154851524} a + \frac{177020755105829}{1640577425762} : 1\right)$ |
| Height | \(3.2274654342991159095460600047808694235\) |
| Torsion structure: | \(\Z/3\Z\) |
| sage: T = E.torsion_subgroup(); T.invariants()
gp: T = elltors(E); T[2]
magma: T,piT := TorsionSubgroup(E); Invariants(T);
| |
| Torsion generator: | $\left(-\frac{20}{3} a^{3} + 3 a^{2} + 41 a + 37 : \frac{52}{9} a^{3} - \frac{89}{3} a^{2} + \frac{37}{9} a + 105 : 1\right)$ |
| 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: | \( 3.2274654342991159095460600047808694235 \) | ||
| Period: | \( 28.659471545381856813935013773342846714 \) | ||
| Tamagawa product: | \( 5 \) | ||
| Torsion order: | \(3\) | ||
| Leading coefficient: | \( 2.00481631535149 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((a^3-a^2-5a)\) | \(4\) | \(5\) | \(I_{5}\) | Split multiplicative | \(-1\) | \(1\) | \(5\) | \(5\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.
| prime | Image of Galois Representation |
|---|---|
| \(3\) | 3B.1.1 |
| \(5\) | 5B.4.2 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
3, 5, 9, 15 and 45.
Its isogeny class
4.1-b
consists of curves linked by isogenies of
degrees dividing 45.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.