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Properties

Label 6.6.434581.1-27.2-b1
Base field 6.6.434581.1
Conductor norm \( 27 \)
CM no
Base change no
Q-curve no
Torsion order \( 1 \)
Rank \( 0 \)

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Base field 6.6.434581.1

Generator \(a\), with minimal polynomial \( x^{6} - 2 x^{5} - 4 x^{4} + 5 x^{3} + 4 x^{2} - 2 x - 1 \); class number \(1\).

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-1, -2, 4, 5, -4, -2, 1]))
 
gp: K = nfinit(Polrev([-1, -2, 4, 5, -4, -2, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, -2, 4, 5, -4, -2, 1]);
 

Weierstrass equation

\({y}^2+\left(a^{5}-2a^{4}-3a^{3}+3a^{2}+2a+1\right){x}{y}+\left(3a^{5}-6a^{4}-10a^{3}+12a^{2}+5a-2\right){y}={x}^{3}+\left(-3a^{5}+8a^{4}+6a^{3}-18a^{2}+a+6\right){x}^{2}+\left(11a^{5}-13a^{4}-58a^{3}+16a^{2}+61a+21\right){x}-560a^{5}+744a^{4}+2735a^{3}-952a^{2}-2871a-825\)
sage: E = EllipticCurve([K([1,2,3,-3,-2,1]),K([6,1,-18,6,8,-3]),K([-2,5,12,-10,-6,3]),K([21,61,16,-58,-13,11]),K([-825,-2871,-952,2735,744,-560])])
 
gp: E = ellinit([Polrev([1,2,3,-3,-2,1]),Polrev([6,1,-18,6,8,-3]),Polrev([-2,5,12,-10,-6,3]),Polrev([21,61,16,-58,-13,11]),Polrev([-825,-2871,-952,2735,744,-560])], K);
 
magma: E := EllipticCurve([K![1,2,3,-3,-2,1],K![6,1,-18,6,8,-3],K![-2,5,12,-10,-6,3],K![21,61,16,-58,-13,11],K![-825,-2871,-952,2735,744,-560]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((-2a^5+4a^4+7a^3-9a^2-3a+2)\) = \((-2a^5+4a^4+7a^3-9a^2-3a+2)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 27 \) = \(27\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((-21a^5+35a^4+81a^3-59a^2-38a+8)\) = \((-2a^5+4a^4+7a^3-9a^2-3a+2)^{5}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 14348907 \) = \(27^{5}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{1612895}{243} a^{5} - \frac{688339}{27} a^{4} + \frac{4362505}{243} a^{3} + \frac{2436914}{243} a^{2} - \frac{72826}{27} a - \frac{317266}{81} \)
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: trivial
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 

BSD invariants

Analytic rank: \( 0 \)
sage: E.rank()
 
magma: Rank(E);
 
Mordell-Weil rank: \(0\)
Regulator: \( 1 \)
Period: \( 858.10163488632067458395645830559475695 \)
Tamagawa product: \( 1 \)
Torsion order: \(1\)
Leading coefficient: \( 1.30168 \)
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\((j)_{-}\)
\((-2a^5+4a^4+7a^3-9a^2-3a+2)\) \(27\) \(1\) \(I_{5}\) Non-split multiplicative \(1\) \(1\) \(5\) \(5\)

Galois Representations

The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) .

Isogenies and isogeny class

This curve has no rational isogenies. Its isogeny class 27.2-b consists of this curve only.

Base change

This elliptic curve is not a \(\Q\)-curve.

It is not the base change of an elliptic curve defined over any subfield.