Properties

Label 6.6.1397493.1-27.1-h1
Base field 6.6.1397493.1
Conductor norm \( 27 \)
CM no
Base change no
Q-curve no
Torsion order \( 3 \)
Rank \( 0 \)

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Base field 6.6.1397493.1

Generator \(a\), with minimal polynomial \( x^{6} - 3 x^{5} - 3 x^{4} + 10 x^{3} + 3 x^{2} - 6 x + 1 \); class number \(1\).

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([1, -6, 3, 10, -3, -3, 1]))
 
gp: K = nfinit(Polrev([1, -6, 3, 10, -3, -3, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -6, 3, 10, -3, -3, 1]);
 

Weierstrass equation

\({y}^2+\left(a^{3}-2a^{2}-2a+2\right){x}{y}+\left(a^{2}-1\right){y}={x}^{3}+\left(a^{5}-2a^{4}-6a^{3}+7a^{2}+10a-2\right){x}^{2}+\left(4a^{5}-12a^{4}-11a^{3}+35a^{2}+14a-9\right){x}-a^{5}+3a^{4}+2a^{3}-10a^{2}-a+7\)
sage: E = EllipticCurve([K([2,-2,-2,1,0,0]),K([-2,10,7,-6,-2,1]),K([-1,0,1,0,0,0]),K([-9,14,35,-11,-12,4]),K([7,-1,-10,2,3,-1])])
 
gp: E = ellinit([Polrev([2,-2,-2,1,0,0]),Polrev([-2,10,7,-6,-2,1]),Polrev([-1,0,1,0,0,0]),Polrev([-9,14,35,-11,-12,4]),Polrev([7,-1,-10,2,3,-1])], K);
 
magma: E := EllipticCurve([K![2,-2,-2,1,0,0],K![-2,10,7,-6,-2,1],K![-1,0,1,0,0,0],K![-9,14,35,-11,-12,4],K![7,-1,-10,2,3,-1]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((-a^5+3a^4+3a^3-8a^2-6a)\) = \((a^3-a^2-3a)^{3}\)
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: \((-a^3+3a+1)\) = \((a^3-a^2-3a)^{3}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 27 \) = \(3^{3}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( -607008006 a^{5} + 193715388 a^{4} + 2340294876 a^{3} + 203925438 a^{2} - 1274124573 a + 226434609 \)
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: $\left(-\frac{1}{3} a^{4} + \frac{4}{3} a^{3} - \frac{8}{3} a + \frac{2}{3} : \frac{1}{3} a^{4} - 2 a^{3} + a^{2} + \frac{14}{3} a - 3 : 1\right)$
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: \( 17180.226605878382826046126270597151993 \)
Tamagawa product: \( 1 \)
Torsion order: \(3\)
Leading coefficient: \( 1.61477 \)
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\((j)_{-}\)
\((a^3-a^2-3a)\) \(3\) \(1\) \(II\) Additive \(1\) \(3\) \(3\) \(0\)

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-h 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.