Properties

Label 6.6.703493.1-41.3-c1
Base field 6.6.703493.1
Conductor norm \( 41 \)
CM no
Base change no
Q-curve no
Torsion order \( 1 \)
Rank \( 0 \)

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Base field 6.6.703493.1

Generator \(a\), with minimal polynomial \( x^{6} - 2 x^{5} - 5 x^{4} + 11 x^{3} + 2 x^{2} - 9 x + 1 \); class number \(1\).

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([1, -9, 2, 11, -5, -2, 1]))
 
gp: K = nfinit(Polrev([1, -9, 2, 11, -5, -2, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -9, 2, 11, -5, -2, 1]);
 

Weierstrass equation

\({y}^2+\left(a^{5}-6a^{3}+a^{2}+7a-1\right){x}{y}+\left(2a^{5}-2a^{4}-11a^{3}+11a^{2}+10a-6\right){y}={x}^{3}+\left(-a^{4}+4a^{2}-2a-2\right){x}^{2}+\left(-a^{5}-a^{4}+6a^{3}+4a^{2}-9a-2\right){x}+4a^{5}-18a^{3}+8a^{2}+13a-8\)
sage: E = EllipticCurve([K([-1,7,1,-6,0,1]),K([-2,-2,4,0,-1,0]),K([-6,10,11,-11,-2,2]),K([-2,-9,4,6,-1,-1]),K([-8,13,8,-18,0,4])])
 
gp: E = ellinit([Polrev([-1,7,1,-6,0,1]),Polrev([-2,-2,4,0,-1,0]),Polrev([-6,10,11,-11,-2,2]),Polrev([-2,-9,4,6,-1,-1]),Polrev([-8,13,8,-18,0,4])], K);
 
magma: E := EllipticCurve([K![-1,7,1,-6,0,1],K![-2,-2,4,0,-1,0],K![-6,10,11,-11,-2,2],K![-2,-9,4,6,-1,-1],K![-8,13,8,-18,0,4]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((a^3-a^2-4a+3)\) = \((a^3-a^2-4a+3)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 41 \) = \(41\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((-2a^5+a^4+12a^3-4a^2-14a+1)\) = \((a^3-a^2-4a+3)\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 41 \) = \(41\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( -\frac{8584162099}{41} a^{5} + \frac{25376641760}{41} a^{4} + \frac{18654917222}{41} a^{3} - 2738133595 a^{2} + \frac{90181405656}{41} a - \frac{8977485322}{41} \)
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: \( 1351.4372366033521367077827131602868696 \)
Tamagawa product: \( 1 \)
Torsion order: \(1\)
Leading coefficient: \( 1.61126 \)
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\((j)_{-}\)
\((a^3-a^2-4a+3)\) \(41\) \(1\) \(I_{1}\) 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 41.3-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.