Properties

Label 4.4.19429.1-3.1-a1
Base field 4.4.19429.1
Conductor norm \( 3 \)
CM no
Base change no
Q-curve no
Torsion order \( 1 \)
Rank \( 0 \)

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Base field 4.4.19429.1

Generator \(a\), with minimal polynomial \( x^{4} - x^{3} - 7 x^{2} - x + 5 \); class number \(1\).

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([5, -1, -7, -1, 1]))
 
gp: K = nfinit(Polrev([5, -1, -7, -1, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![5, -1, -7, -1, 1]);
 

Weierstrass equation

\({y}^2+\left(a^{2}-a-4\right){x}{y}={x}^{3}+\left(-a^{3}+a^{2}+4a+3\right){x}^{2}+\left(-2a^{3}-2a^{2}+20a+24\right){x}+2a^{3}-15a^{2}+14a+41\)
sage: E = EllipticCurve([K([-4,-1,1,0]),K([3,4,1,-1]),K([0,0,0,0]),K([24,20,-2,-2]),K([41,14,-15,2])])
 
gp: E = ellinit([Polrev([-4,-1,1,0]),Polrev([3,4,1,-1]),Polrev([0,0,0,0]),Polrev([24,20,-2,-2]),Polrev([41,14,-15,2])], K);
 
magma: E := EllipticCurve([K![-4,-1,1,0],K![3,4,1,-1],K![0,0,0,0],K![24,20,-2,-2],K![41,14,-15,2]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((a^3-a^2-6a-3)\) = \((a^3-a^2-6a-3)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 3 \) = \(3\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((-2a^3+5a^2+8a-11)\) = \((a^3-a^2-6a-3)^{3}\)
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: \( -\frac{46448}{27} a^{3} - \frac{6578}{9} a^{2} + \frac{120947}{27} a - \frac{52073}{27} \)
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: \( 233.08328296804033248101138795674134903 \)
Tamagawa product: \( 1 \)
Torsion order: \(1\)
Leading coefficient: \( 1.67219107963240 \)
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\((j)_{-}\)
\((a^3-a^2-6a-3)\) \(3\) \(1\) \(I_{3}\) Non-split multiplicative \(1\) \(1\) \(3\) \(3\)

Galois Representations

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

Isogenies and isogeny class

This curve has no rational isogenies. Its isogeny class 3.1-a consists of this curve only.

Base change

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

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