Base field 6.6.1312625.1
Generator \(a\), with minimal polynomial \( x^{6} - x^{5} - 7 x^{4} + 7 x^{3} + 12 x^{2} - 12 x - 1 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-1, -12, 12, 7, -7, -1, 1]))
gp: K = nfinit(Polrev([-1, -12, 12, 7, -7, -1, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, -12, 12, 7, -7, -1, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([1,1,0,0,0,0]),K([-4,-29,6,20,-1,-3]),K([1,19,-4,-13,1,2]),K([-8,-61,21,47,-5,-8]),K([-4,-37,10,26,-2,-4])])
gp: E = ellinit([Polrev([1,1,0,0,0,0]),Polrev([-4,-29,6,20,-1,-3]),Polrev([1,19,-4,-13,1,2]),Polrev([-8,-61,21,47,-5,-8]),Polrev([-4,-37,10,26,-2,-4])], K);
magma: E := EllipticCurve([K![1,1,0,0,0,0],K![-4,-29,6,20,-1,-3],K![1,19,-4,-13,1,2],K![-8,-61,21,47,-5,-8],K![-4,-37,10,26,-2,-4]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((-a^5+6a^3+a^2-8a-2)\) | = | \((-a^5+6a^3+a^2-8a-2)\) |
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: | \((12a^5+6a^4-76a^3-35a^2+117a+17)\) | = | \((-a^5+6a^3+a^2-8a-2)^{12}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 16777216 \) | = | \(4^{12}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( -\frac{976778627465}{4096} a^{5} - \frac{483215871649}{4096} a^{4} + \frac{1528794165519}{1024} a^{3} + \frac{2302932872201}{4096} a^{2} - \frac{4139545058523}{2048} a - \frac{653446315795}{4096} \) | ||
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: | \(\Z/2\Z\oplus\Z/2\Z\) | |
| sage: T = E.torsion_subgroup(); T.invariants()
gp: T = elltors(E); T[2]
magma: T,piT := TorsionSubgroup(E); Invariants(T);
| ||
| Torsion generators: | $\left(-a^{3} - 2 a^{2} + 2 a + 3 : -a^{5} + 8 a^{3} + 2 a^{2} - 12 a - 2 : 1\right)$ | $\left(\frac{7}{4} a^{5} + \frac{1}{2} a^{4} - 11 a^{3} - 2 a^{2} + \frac{61}{4} a + \frac{1}{4} : -3 a^{5} - \frac{11}{8} a^{4} + \frac{153}{8} a^{3} + \frac{47}{8} a^{2} - \frac{111}{4} a - \frac{3}{2} : 1\right)$ |
| 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: | \( 11197.348885915551916931370938467701497 \) | ||
| Tamagawa product: | \( 2 \) | ||
| Torsion order: | \(4\) | ||
| Leading coefficient: | \( 1.22167 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((-a^5+6a^3+a^2-8a-2)\) | \(4\) | \(2\) | \(I_{12}\) | Non-split multiplicative | \(1\) | \(1\) | \(12\) | \(12\) |
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 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2 and 4.
Its isogeny class
4.1-a
consists of curves linked by isogenies of
degrees dividing 8.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.