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Properties

Label 6.6.434581.1-29.1-b1
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

\({y}^2+\left(2a^{5}-5a^{4}-5a^{3}+12a^{2}+a-3\right){x}{y}+\left(2a^{5}-5a^{4}-5a^{3}+12a^{2}+a-3\right){y}={x}^{3}+\left(-2a^{5}+5a^{4}+4a^{3}-10a^{2}+3a+3\right){x}^{2}+\left(3a^{5}-9a^{4}-8a^{3}+29a^{2}+7a-15\right){x}-16a^{5}+35a^{4}+53a^{3}-84a^{2}-28a+32\)
sage: E = EllipticCurve([K([-3,1,12,-5,-5,2]),K([3,3,-10,4,5,-2]),K([-3,1,12,-5,-5,2]),K([-15,7,29,-8,-9,3]),K([32,-28,-84,53,35,-16])])
 
gp: E = ellinit([Polrev([-3,1,12,-5,-5,2]),Polrev([3,3,-10,4,5,-2]),Polrev([-3,1,12,-5,-5,2]),Polrev([-15,7,29,-8,-9,3]),Polrev([32,-28,-84,53,35,-16])], K);
 
magma: E := EllipticCurve([K![-3,1,12,-5,-5,2],K![3,3,-10,4,5,-2],K![-3,1,12,-5,-5,2],K![-15,7,29,-8,-9,3],K![32,-28,-84,53,35,-16]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((a^4-a^3-4a^2+a+3)\) = \((a^4-a^3-4a^2+a+3)\)
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: \((23a^5-84a^4-32a^3+236a^2+28a-48)\) = \((a^4-a^3-4a^2+a+3)^{7}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( -17249876309 \) = \(-29^{7}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( -\frac{489143918834519418}{17249876309} a^{5} + \frac{1350726974773319371}{17249876309} a^{4} + \frac{928906959941769094}{17249876309} a^{3} - \frac{3154651692992057986}{17249876309} a^{2} + \frac{442831889196161377}{17249876309} a + \frac{644411244050480300}{17249876309} \)
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: \( 644.76010496030301995022496233308184557 \)
Tamagawa product: \( 1 \)
Torsion order: \(1\)
Leading coefficient: \( 0.978054 \)
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\((j)_{-}\)
\((a^4-a^3-4a^2+a+3)\) \(29\) \(1\) \(I_{7}\) Non-split multiplicative \(1\) \(1\) \(7\) \(7\)

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 29.1-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.