Properties

Label 6.6.1229312.1-392.1-k2
Base field 6.6.1229312.1
Conductor \((1/4a^5-5/2a^3+4a)\)
Conductor norm \( 392 \)
CM no
Base change yes: 14.a5,448.g5
Q-curve yes
Torsion order \( 18 \)
Rank \( 0 \)

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Base field 6.6.1229312.1

Generator \(a\), with minimal polynomial \( x^{6} - 10 x^{4} + 24 x^{2} - 8 \); class number \(1\).

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-8, 0, 24, 0, -10, 0, 1]))
 
gp: K = nfinit(Pol(Vecrev([-8, 0, 24, 0, -10, 0, 1])));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-8, 0, 24, 0, -10, 0, 1]);
 

Weierstrass equation

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

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((1/4a^5-5/2a^3+4a)\) = \((-1/4a^5+2a^3-1/2a^2-3a+1)\cdot(-1/2a^3+3a+1)\cdot(1/4a^5-2a^3+3a)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 392 \) = \(7\cdot7\cdot8\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((-28)\) = \((-1/4a^5+2a^3-1/2a^2-3a+1)^{3}\cdot(-1/2a^3+3a+1)^{3}\cdot(1/4a^5-2a^3+3a)^{4}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 481890304 \) = \(7^{3}\cdot7^{3}\cdot8^{4}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( -\frac{15625}{28} \)
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: \(\Z/18\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Torsion generator: $\left(\frac{1}{2} a^{4} - 5 a^{2} + 11 : -2 a^{4} + 19 a^{2} - 40 : 1\right)$
sage: T.gens()
 
gp: T[3]
 
magma: [piT(P) : P in Generators(T)];
 

BSD invariants

Analytic rank: not available
sage: E.rank()
 
magma: Rank(E);
 
Mordell-Weil rank: \(0\)
Regulator: \( 1 \)
Period: not available
Tamagawa product: \( 18 \)  =  \(3\cdot3\cdot2\)
Torsion order: \(18\)
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\((j)_{-}\)
\((-1/4a^5+2a^3-1/2a^2-3a+1)\) \(7\) \(3\) \(I_{3}\) Split multiplicative \(-1\) \(1\) \(3\) \(3\)
\((-1/2a^3+3a+1)\) \(7\) \(3\) \(I_{3}\) Split multiplicative \(-1\) \(1\) \(3\) \(3\)
\((1/4a^5-2a^3+3a)\) \(8\) \(2\) \(I_{4}\) Non-split multiplicative \(1\) \(1\) \(4\) \(4\)

Galois Representations

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

prime Image of Galois Representation
\(2\) 2B
\(3\) 3B.1.1

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2, 3, 4, 6, 9, 12, 18 and 36.
Its isogeny class 392.1-k consists of curves linked by isogenies of degrees dividing 36.

Base change

This curve is the base change of 14.a5, 448.g5, defined over \(\Q\), so it is also a \(\Q\)-curve.