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Properties

Label 6.6.371293.1-27.2-b2
Base field \(\Q(\zeta_{13})^+\)
Conductor norm \( 27 \)
CM no
Base change yes
Q-curve no
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

\({y}^2+\left(a^{4}+a^{3}-4a^{2}-3a+3\right){x}{y}+\left(a^{5}-5a^{3}+a^{2}+5a-1\right){y}={x}^{3}+\left(-a^{5}+5a^{3}-a^{2}-6a+1\right){x}^{2}+\left(-a^{5}+a^{4}+6a^{3}-6a^{2}-9a+9\right){x}+a^{5}-3a^{4}-4a^{3}+10a^{2}+3a-6\)
sage: E = EllipticCurve([K([3,-3,-4,1,1,0]),K([1,-6,-1,5,0,-1]),K([-1,5,1,-5,0,1]),K([9,-9,-6,6,1,-1]),K([-6,3,10,-4,-3,1])])
 
gp: E = ellinit([Polrev([3,-3,-4,1,1,0]),Polrev([1,-6,-1,5,0,-1]),Polrev([-1,5,1,-5,0,1]),Polrev([9,-9,-6,6,1,-1]),Polrev([-6,3,10,-4,-3,1])], K);
 
magma: E := EllipticCurve([K![3,-3,-4,1,1,0],K![1,-6,-1,5,0,-1],K![-1,5,1,-5,0,1],K![9,-9,-6,6,1,-1],K![-6,3,10,-4,-3,1]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((a^4-a^3-4a^2+2a+3)\) = \((a^4-a^3-4a^2+2a+3)\)
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: \((-79a^4+79a^3+316a^2-158a-222)\) = \((a^4-a^3-4a^2+2a+3)^{9}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 7625597484987 \) = \(27^{9}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( -\frac{9799933}{19683} a^{4} + \frac{9799933}{19683} a^{3} + \frac{39199732}{19683} a^{2} - \frac{19599866}{19683} a - \frac{8637512}{19683} \)
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: \( 604.60496476542178552784047956000625778 \)
Tamagawa product: \( 1 \)
Torsion order: \(1\)
Leading coefficient: \( 0.992232 \)
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\((j)_{-}\)
\((a^4-a^3-4a^2+2a+3)\) \(27\) \(1\) \(I_{9}\) Non-split multiplicative \(1\) \(1\) \(9\) \(9\)

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 27.2-b consists of curves linked by isogenies of degree 3.

Base change

This elliptic curve is not a \(\Q\)-curve. It is the base change of the following elliptic curve:

Base field Curve
\(\Q(\sqrt{13}) \) 2.2.13.1-507.1-d2