Properties

Label 6.6.453789.1-27.1-a2
Base field \(\Q(\zeta_{21})^+\)
Conductor norm \( 27 \)
CM no
Base change yes
Q-curve yes
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

\({y}^2+\left(a^{4}-3a^{2}+1\right){y}={x}^{3}+\left(-a^{5}+4a^{3}-2a^{2}-2a+5\right){x}^{2}+\left(-2a^{5}+7a^{4}+4a^{3}-28a^{2}+17a+4\right){x}-6a^{5}+13a^{4}+15a^{3}-52a^{2}+33a-3\)
sage: E = EllipticCurve([K([0,0,0,0,0,0]),K([5,-2,-2,4,0,-1]),K([1,0,-3,0,1,0]),K([4,17,-28,4,7,-2]),K([-3,33,-52,15,13,-6])])
 
gp: E = ellinit([Polrev([0,0,0,0,0,0]),Polrev([5,-2,-2,4,0,-1]),Polrev([1,0,-3,0,1,0]),Polrev([4,17,-28,4,7,-2]),Polrev([-3,33,-52,15,13,-6])], K);
 
magma: E := EllipticCurve([K![0,0,0,0,0,0],K![5,-2,-2,4,0,-1],K![1,0,-3,0,1,0],K![4,17,-28,4,7,-2],K![-3,33,-52,15,13,-6]]);
 

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: \(0\)
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: \( 350.73439133252966262407759738984136477 \)
Tamagawa product: \( 2 \)
Torsion order: \(1\)
Leading coefficient: \( 1.04131 \)
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\((j)_{-}\)
\((a^3+a^2-2a-1)\) \(27\) \(2\) \(I_{2}\) Non-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.12.1

Isogenies and isogeny class

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

Base change

This elliptic curve is a \(\Q\)-curve. It is the base change of the following 5 elliptic curves:

Base field Curve
\(\Q\) 147.b2
\(\Q\) 441.b2
\(\Q(\sqrt{21}) \) 2.2.21.1-147.1-b2
\(\Q(\zeta_{7})^+\) 3.3.49.1-1323.1-b2
\(\Q(\zeta_{7})^+\) 3.3.49.1-729.1-a2