Properties

Label 6.6.453789.1-27.1-b2
Base field \(\Q(\zeta_{21})^+\)
Conductor \((a^3+a^2-2a-1)\)
Conductor norm \( 27 \)
CM no
Base change yes: 147.c2,441.a2
Q-curve yes
Torsion order \( 13 \)
Rank \( 1 \)

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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(Pol(Vecrev([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

\({y}^2+\left(a^{5}-5a^{3}+a^{2}+5a-2\right){y}={x}^{3}+\left(-a^{5}+5a^{3}-a^{2}-5a+2\right){x}^{2}+\left(-a^{5}+7a^{3}-a^{2}-11a+2\right){x}+a^{5}-7a^{3}+a^{2}+11a-2\)
sage: E = EllipticCurve([K([0,0,0,0,0,0]),K([2,-5,-1,5,0,-1]),K([-2,5,1,-5,0,1]),K([2,-11,-1,7,0,-1]),K([-2,11,1,-7,0,1])])
 
gp: E = ellinit([Pol(Vecrev([0,0,0,0,0,0])),Pol(Vecrev([2,-5,-1,5,0,-1])),Pol(Vecrev([-2,5,1,-5,0,1])),Pol(Vecrev([2,-11,-1,7,0,-1])),Pol(Vecrev([-2,11,1,-7,0,1]))], K);
 
magma: E := EllipticCurve([K![0,0,0,0,0,0],K![2,-5,-1,5,0,-1],K![-2,5,1,-5,0,1],K![2,-11,-1,7,0,-1],K![-2,11,1,-7,0,1]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((a^3+a^2-2a-1)\) = \((a^3+a^2-2a-1)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 27 \) = \(27\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((-3)\) = \((a^3+a^2-2a-1)^{2}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 729 \) = \(27^{2}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( -\frac{28672}{3} \)
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: \(1\)
Generator $\left(-a^{3} + a^{2} + 3 a - 1 : -2 a^{5} + 9 a^{3} - 8 a + 2 : 1\right)$
Height \(0.0762700343945709\)
Torsion structure: \(\Z/13\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Torsion generator: $\left(1 : -1 : 1\right)$
sage: T.gens()
 
gp: T[3]
 
magma: [piT(P) : P in Generators(T)];
 

BSD invariants

Analytic rank: \( 1 \)
sage: E.rank()
 
magma: Rank(E);
 
Mordell-Weil rank: \(1\)
Regulator: \( 0.0762700343945709 \)
Period: not available
Tamagawa product: \( 2 \)
Torsion order: \(13\)
Leading coefficient: not available
Analytic order of Ш: not available

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\((j)_{-}\)
\((a^3+a^2-2a-1)\) \(27\) \(2\) \(I_{2}\) Split multiplicative \(-1\) \(1\) \(2\) \(2\)

Galois Representations

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

prime Image of Galois Representation
\(13\) 13B.1.1

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 13.
Its isogeny class 27.1-b consists of curves linked by isogenies of degree 13.

Base change

This curve is the base change of 147.c2, 441.a2, defined over \(\Q\), so it is also a \(\Q\)-curve.