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Properties

Label 6.6.453789.1-41.5-d1
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

\({y}^2+\left(a^{5}-4a^{3}+2a^{2}+2a-4\right){x}{y}+\left(a^{5}-5a^{3}+a^{2}+6a-2\right){y}={x}^{3}+\left(a^{5}-a^{4}-4a^{3}+6a^{2}+a-7\right){x}^{2}+\left(6a^{5}-6a^{4}-32a^{3}+31a^{2}+35a-26\right){x}+4a^{5}-5a^{4}-20a^{3}+25a^{2}+18a-20\)
sage: E = EllipticCurve([K([-4,2,2,-4,0,1]),K([-7,1,6,-4,-1,1]),K([-2,6,1,-5,0,1]),K([-26,35,31,-32,-6,6]),K([-20,18,25,-20,-5,4])])
 
gp: E = ellinit([Polrev([-4,2,2,-4,0,1]),Polrev([-7,1,6,-4,-1,1]),Polrev([-2,6,1,-5,0,1]),Polrev([-26,35,31,-32,-6,6]),Polrev([-20,18,25,-20,-5,4])], K);
 
magma: E := EllipticCurve([K![-4,2,2,-4,0,1],K![-7,1,6,-4,-1,1],K![-2,6,1,-5,0,1],K![-26,35,31,-32,-6,6],K![-20,18,25,-20,-5,4]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((a^4+2a^3-4a^2-6a+3)\) = \((a^4+2a^3-4a^2-6a+3)\)
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: \((a^5+a^4-5a^3-4a^2+5a+3)\) = \((a^4+2a^3-4a^2-6a+3)\)
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: \( -\frac{1959802}{41} a^{5} + \frac{7305801}{41} a^{4} - \frac{1686142}{41} a^{3} - \frac{25057762}{41} a^{2} + \frac{36500036}{41} a - \frac{14311424}{41} \)
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: \( 603.94057741679419555345677352454947071 \)
Tamagawa product: \( 1 \)
Torsion order: \(1\)
Leading coefficient: \( 0.896535 \)
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\((j)_{-}\)
\((a^4+2a^3-4a^2-6a+3)\) \(41\) \(1\) \(I_{1}\) 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 41.5-d 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.