Properties

Label 4.4.5125.1-55.1-d1
Base field 4.4.5125.1
Conductor \((a^3-4a)\)
Conductor norm \( 55 \)
CM no
Base change no
Q-curve no
Torsion order \( 6 \)
Rank \( 0 \)

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Base field 4.4.5125.1

Generator \(a\), with minimal polynomial \( x^{4} - 2 x^{3} - 6 x^{2} + 7 x + 11 \); class number \(1\).

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([11, 7, -6, -2, 1]))
 
gp: K = nfinit(Pol(Vecrev([11, 7, -6, -2, 1])));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![11, 7, -6, -2, 1]);
 

Weierstrass equation

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

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((a^3-4a)\) = \((-a^3+a^2+4a+1)\cdot(-a)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 55 \) = \(5\cdot11\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((a^3-7a-11)\) = \((-a^3+a^2+4a+1)^{2}\cdot(-a)^{2}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 3025 \) = \(5^{2}\cdot11^{2}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{131300318}{605} a^{3} + \frac{14981064}{605} a^{2} - \frac{151139983}{121} a - \frac{681395002}{605} \)
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: \(\Z/6\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Torsion generator: $\left(a^{3} - 4 a - 1 : -7 a^{3} - 3 a^{2} + 28 a + 22 : 1\right)$
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: \( 669.043009920361 \)
Tamagawa product: \( 4 \)  =  \(2\cdot2\)
Torsion order: \(6\)
Leading coefficient: \( 1.03839979400277 \)
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\((j)_{-}\)
\((-a^3+a^2+4a+1)\) \(5\) \(2\) \(I_{2}\) Non-split multiplicative \(1\) \(1\) \(2\) \(2\)
\((-a)\) \(11\) \(2\) \(I_{2}\) 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
\(2\) 2B
\(3\) 3B.1.1

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2, 3 and 6.
Its isogeny class 55.1-d consists of curves linked by isogenies of degrees dividing 6.

Base change

This curve is not the base change of an elliptic curve defined over \(\Q\). It is not a \(\Q\)-curve.