Properties

Label 6.6.371293.1-1.1-a1
Base field \(\Q(\zeta_{13})^+\)
Conductor norm \( 1 \)
CM no
Base change yes
Q-curve yes
Torsion order \( 1 \)
Rank \( 0 \)

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Base field \(\Q(\zeta_{13})^+\)

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

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-1, -3, 6, 4, -5, -1, 1]))
 
gp: K = nfinit(Polrev([-1, -3, 6, 4, -5, -1, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, -3, 6, 4, -5, -1, 1]);
 

Weierstrass equation

\({y}^2+\left(a^{5}-4a^{3}+a^{2}+2a-2\right){x}{y}+\left(a^{3}+a^{2}-2a-1\right){y}={x}^{3}+\left(-a^{5}+6a^{3}-8a+1\right){x}^{2}+\left(-6649a^{5}+3526a^{4}+36471a^{3}-7581a^{2}-48724a-11069\right){x}-549202a^{5}+227195a^{4}+2939749a^{3}-400010a^{2}-3734843a-839159\)
sage: E = EllipticCurve([K([-2,2,1,-4,0,1]),K([1,-8,0,6,0,-1]),K([-1,-2,1,1,0,0]),K([-11069,-48724,-7581,36471,3526,-6649]),K([-839159,-3734843,-400010,2939749,227195,-549202])])
 
gp: E = ellinit([Polrev([-2,2,1,-4,0,1]),Polrev([1,-8,0,6,0,-1]),Polrev([-1,-2,1,1,0,0]),Polrev([-11069,-48724,-7581,36471,3526,-6649]),Polrev([-839159,-3734843,-400010,2939749,227195,-549202])], K);
 
magma: E := EllipticCurve([K![-2,2,1,-4,0,1],K![1,-8,0,6,0,-1],K![-1,-2,1,1,0,0],K![-11069,-48724,-7581,36471,3526,-6649],K![-839159,-3734843,-400010,2939749,227195,-549202]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((1)\) = \((1)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 1 \) = 1
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((1)\) = \((1)\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 1 \) = 1
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( -3387888351672962316333 a^{4} + 3387888351672962316333 a^{3} + 13551553406691849265332 a^{2} - 6775776703345924632666 a - 14577323462934449612494 \)
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: \( 0.0026716469568848618763422560119573014187 \)
Tamagawa product: \( 1 \)
Torsion order: \(1\)
Leading coefficient: \( 0.571393 \)
Analytic order of Ш: \( 130321 \) (rounded)

Local data at primes of bad reduction

sage: E.local_data()
 
magma: LocalInformation(E);
 
No primes of bad reduction.

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
\(19\) 19B.1.2

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 3, 19 and 57.
Its isogeny class 1.1-a consists of curves linked by isogenies of degrees dividing 57.

Base change

This elliptic curve is a \(\Q\)-curve. It is the base change of the following elliptic curve:

Base field Curve
\(\Q(\sqrt{13}) \) 2.2.13.1-169.1-a1