Properties

Label 6.6.453789.1-41.1-b2
Base field \(\Q(\zeta_{21})^+\)
Conductor norm \( 41 \)
CM no
Base change no
Q-curve no
Torsion order \( 1 \)
Rank \( 0 \)

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Base field \(\Q(\zeta_{21})^+\)

Generator \(a\), with minimal polynomial \( x^{6} - x^{5} - 6 x^{4} + 6 x^{3} + 8 x^{2} - 8 x + 1 \); class number \(1\).

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([1, -8, 8, 6, -6, -1, 1]))
 
gp: K = nfinit(Polrev([1, -8, 8, 6, -6, -1, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -8, 8, 6, -6, -1, 1]);
 

Weierstrass equation

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

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((a^5-6a^3+a^2+7a-2)\) = \((a^5-6a^3+a^2+7a-2)\)
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: \((5a^4+2a^3-19a^2-6a+6)\) = \((a^5-6a^3+a^2+7a-2)^{3}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( -68921 \) = \(-41^{3}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{1692444184785}{68921} a^{5} + \frac{1104206748561}{68921} a^{4} - \frac{8329796043201}{68921} a^{3} - \frac{3610036693062}{68921} a^{2} + \frac{7573650019722}{68921} a - \frac{1024189061643}{68921} \)
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: \( 798.28831229797171681793933071028557800 \)
Tamagawa product: \( 1 \)
Torsion order: \(1\)
Leading coefficient: \( 1.18504 \)
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\((j)_{-}\)
\((a^5-6a^3+a^2+7a-2)\) \(41\) \(1\) \(I_{3}\) Non-split multiplicative \(1\) \(1\) \(3\) \(3\)

Galois Representations

The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.

prime Image of Galois Representation
\(3\) 3B

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 3.
Its isogeny class 41.1-b consists of curves linked by isogenies of degree 3.

Base change

This elliptic curve is not a \(\Q\)-curve.

It is not the base change of an elliptic curve defined over any subfield.