↑

Properties

Label 6.6.434581.1-29.2-b1
Base field 6.6.434581.1
Conductor norm \( 29 \)
CM no
Base change no
Q-curve no
Torsion order \( 1 \)
Rank \( 0 \)

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

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

\({y}^2+\left(2a^{5}-4a^{4}-7a^{3}+9a^{2}+4a-2\right){x}{y}+\left(2a^{5}-5a^{4}-5a^{3}+12a^{2}-4\right){y}={x}^{3}+\left(a^{3}-2a^{2}-a+1\right){x}^{2}+\left(-a^{5}+3a^{4}+a^{3}-6a^{2}+a+1\right){x}+2a^{5}-5a^{4}-4a^{3}+12a^{2}-a-4\)
sage: E = EllipticCurve([K([-2,4,9,-7,-4,2]),K([1,-1,-2,1,0,0]),K([-4,0,12,-5,-5,2]),K([1,1,-6,1,3,-1]),K([-4,-1,12,-4,-5,2])])
 
gp: E = ellinit([Polrev([-2,4,9,-7,-4,2]),Polrev([1,-1,-2,1,0,0]),Polrev([-4,0,12,-5,-5,2]),Polrev([1,1,-6,1,3,-1]),Polrev([-4,-1,12,-4,-5,2])], K);
 
magma: E := EllipticCurve([K![-2,4,9,-7,-4,2],K![1,-1,-2,1,0,0],K![-4,0,12,-5,-5,2],K![1,1,-6,1,3,-1],K![-4,-1,12,-4,-5,2]]);
 

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: \((-87a^5+222a^4+227a^3-538a^2-76a+166)\) = \((3a^5-7a^4-9a^3+17a^2+3a-5)^{7}\)
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: \( \frac{163462751293821202}{17249876309} a^{5} - \frac{111681935365476176}{17249876309} a^{4} - \frac{801547698431869756}{17249876309} a^{3} - \frac{236802257690773383}{17249876309} a^{2} + \frac{343172721161084264}{17249876309} a + \frac{124367830204079372}{17249876309} \)
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\((j)_{-}\)
\((3a^5-7a^4-9a^3+17a^2+3a-5)\) \(29\) \(1\) \(I_{7}\) 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 29.2-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.