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Properties

Label 6.6.1541581.1-27.1-a1
Base field 6.6.1541581.1
Conductor norm \( 27 \)
CM no
Base change no
Q-curve no
Torsion order \( 1 \)
Rank \( 0 \)

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Base field 6.6.1541581.1

Generator \(a\), with minimal polynomial \( x^{6} - x^{5} - 6 x^{4} + 2 x^{3} + 9 x^{2} + x - 1 \); class number \(1\).

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-1, 1, 9, 2, -6, -1, 1]))
 
gp: K = nfinit(Polrev([-1, 1, 9, 2, -6, -1, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 1, 9, 2, -6, -1, 1]);
 

Weierstrass equation

\({y}^2+\left(2a^{5}-4a^{4}-7a^{3}+9a^{2}+5a+1\right){x}{y}+\left(2a^{5}-4a^{4}-7a^{3}+9a^{2}+5a\right){y}={x}^{3}+\left(-a^{5}+2a^{4}+4a^{3}-5a^{2}-4a-1\right){x}^{2}+\left(2a^{5}-3a^{4}-10a^{3}+9a^{2}+13a-7\right){x}+4a^{5}-7a^{4}-18a^{3}+20a^{2}+21a-8\)
sage: E = EllipticCurve([K([1,5,9,-7,-4,2]),K([-1,-4,-5,4,2,-1]),K([0,5,9,-7,-4,2]),K([-7,13,9,-10,-3,2]),K([-8,21,20,-18,-7,4])])
 
gp: E = ellinit([Polrev([1,5,9,-7,-4,2]),Polrev([-1,-4,-5,4,2,-1]),Polrev([0,5,9,-7,-4,2]),Polrev([-7,13,9,-10,-3,2]),Polrev([-8,21,20,-18,-7,4])], K);
 
magma: E := EllipticCurve([K![1,5,9,-7,-4,2],K![-1,-4,-5,4,2,-1],K![0,5,9,-7,-4,2],K![-7,13,9,-10,-3,2],K![-8,21,20,-18,-7,4]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((-a^5+a^4+6a^3-3a^2-7a)\) = \((-a^5+a^4+6a^3-3a^2-7a)\)
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: \((2a^5-3a^4-12a^3+8a^2+21a+1)\) = \((-a^5+a^4+6a^3-3a^2-7a)^{3}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 19683 \) = \(27^{3}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{54670142}{9} a^{5} + 3282314 a^{4} - \frac{845766476}{27} a^{3} - \frac{974660561}{27} a^{2} - \frac{11484862}{9} a + \frac{104745092}{27} \)
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: \( 1729.5158174036084871855359859605764774 \)
Tamagawa product: \( 1 \)
Torsion order: \(1\)
Leading coefficient: \( 1.39297 \)
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\((j)_{-}\)
\((-a^5+a^4+6a^3-3a^2-7a)\) \(27\) \(1\) \(I_{3}\) Non-split multiplicative \(1\) \(1\) \(3\) \(3\)

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 27.1-a 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.