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Properties

Label 6.6.1387029.1-27.1-d1
Base field 6.6.1387029.1
Conductor norm \( 27 \)
CM no
Base change no
Q-curve no
Torsion order \( 3 \)
Rank \( 0 \)

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Base field 6.6.1387029.1

Generator \(a\), with minimal polynomial \( x^{6} - 3 x^{5} - 2 x^{4} + 9 x^{3} - x^{2} - 4 x + 1 \); class number \(1\).

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([1, -4, -1, 9, -2, -3, 1]))
 
gp: K = nfinit(Polrev([1, -4, -1, 9, -2, -3, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -4, -1, 9, -2, -3, 1]);
 

Weierstrass equation

\({y}^2+\left(a^{2}-a-1\right){x}{y}+a^{2}{y}={x}^{3}+\left(a^{2}-a\right){x}^{2}+\left(5a^{5}-13a^{4}-15a^{3}+41a^{2}+6a-19\right){x}-8a^{5}+22a^{4}+21a^{3}-65a^{2}-10a+28\)
sage: E = EllipticCurve([K([-1,-1,1,0,0,0]),K([0,-1,1,0,0,0]),K([0,0,1,0,0,0]),K([-19,6,41,-15,-13,5]),K([28,-10,-65,21,22,-8])])
 
gp: E = ellinit([Polrev([-1,-1,1,0,0,0]),Polrev([0,-1,1,0,0,0]),Polrev([0,0,1,0,0,0]),Polrev([-19,6,41,-15,-13,5]),Polrev([28,-10,-65,21,22,-8])], K);
 
magma: E := EllipticCurve([K![-1,-1,1,0,0,0],K![0,-1,1,0,0,0],K![0,0,1,0,0,0],K![-19,6,41,-15,-13,5],K![28,-10,-65,21,22,-8]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((a^5-3a^4-3a^3+8a^2+2a-2)\) = \((a^5-3a^4-2a^3+8a^2+a-2)^{3}\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 27 \) = \(3^{3}\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((-3a^5+7a^4+12a^3-22a^2-10a+2)\) = \((a^5-3a^4-2a^3+8a^2+a-2)^{9}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 19683 \) = \(3^{9}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( -132954280 a^{5} + 362404713 a^{4} + 365155892 a^{3} - 1096319488 a^{2} - 167101878 a + 485890469 \)
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: \(\Z/3\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Torsion generator: $\left(-a^{2} + a + 2 : a^{5} - 2 a^{4} - 4 a^{3} + 5 a^{2} + 4 a : 1\right)$
sage: T.gens()
 
gp: T[3]
 
magma: [piT(P) : P in Generators(T)];
 

BSD invariants

Analytic rank: \( 0 \)
sage: E.rank()
 
magma: Rank(E);
 
Mordell-Weil rank: \(0\)
Regulator: \( 1 \)
Period: \( 12888.622955572952108445390960199837682 \)
Tamagawa product: \( 1 \)
Torsion order: \(3\)
Leading coefficient: \( 1.21597 \)
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\((j)_{-}\)
\((a^5-3a^4-2a^3+8a^2+a-2)\) \(3\) \(1\) \(IV^{*}\) Additive \(1\) \(3\) \(9\) \(0\)

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.1.1

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 3.
Its isogeny class 27.1-d consists of curves linked by isogenies of degree 3.

Base change

This elliptic curve is not a \(\Q\)-curve.

It is not the base change of an elliptic curve defined over any subfield.