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Properties

Label 6.6.434581.1-29.2-c1
Base field 6.6.434581.1
Conductor norm \( 29 \)
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

\({y}^2+\left(a^{5}-a^{4}-5a^{3}+a^{2}+3a+1\right){x}{y}+\left(2a^{5}-4a^{4}-7a^{3}+9a^{2}+3a-3\right){y}={x}^{3}+\left(-2a^{5}+5a^{4}+4a^{3}-10a^{2}+a+1\right){x}^{2}+\left(2a^{5}-2a^{4}-11a^{3}+2a^{2}+11a+4\right){x}+6a^{5}-8a^{4}-29a^{3}+11a^{2}+28a+6\)
sage: E = EllipticCurve([K([1,3,1,-5,-1,1]),K([1,1,-10,4,5,-2]),K([-3,3,9,-7,-4,2]),K([4,11,2,-11,-2,2]),K([6,28,11,-29,-8,6])])
 
gp: E = ellinit([Polrev([1,3,1,-5,-1,1]),Polrev([1,1,-10,4,5,-2]),Polrev([-3,3,9,-7,-4,2]),Polrev([4,11,2,-11,-2,2]),Polrev([6,28,11,-29,-8,6])], K);
 
magma: E := EllipticCurve([K![1,3,1,-5,-1,1],K![1,1,-10,4,5,-2],K![-3,3,9,-7,-4,2],K![4,11,2,-11,-2,2],K![6,28,11,-29,-8,6]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((3a^5-7a^4-9a^3+17a^2+3a-5)\) = \((3a^5-7a^4-9a^3+17a^2+3a-5)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 29 \) = \(29\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((a^5-3a^4-2a^3+9a^2-a-3)\) = \((3a^5-7a^4-9a^3+17a^2+3a-5)\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 29 \) = \(29\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( -\frac{328404104530776}{29} a^{5} + \frac{906495307821765}{29} a^{4} + \frac{624404013794482}{29} a^{3} - \frac{2116757679534948}{29} a^{2} + \frac{295763857235945}{29} a + \frac{431937635205700}{29} \)
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: \( 790.58986778460347096675407034132779872 \)
Tamagawa product: \( 1 \)
Torsion order: \(1\)
Leading coefficient: \( 1.19927 \)
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\((j)_{-}\)
\((3a^5-7a^4-9a^3+17a^2+3a-5)\) \(29\) \(1\) \(I_{1}\) Non-split multiplicative \(1\) \(1\) \(1\) \(1\)

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 29.2-c 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.