Properties

Label 4.4.5125.1-41.1-e1
Base field 4.4.5125.1
Conductor \((a^3-a^2-4a-3)\)
Conductor norm \( 41 \)
CM no
Base change no
Q-curve no
Torsion order \( 1 \)
Rank \( 1 \)

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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

\({y}^2+\left(a^{2}-3\right){y}={x}^{3}+\left(-a^{2}+4\right){x}^{2}+\left(6a^{3}-20a^{2}-3a+39\right){x}-9a^{3}+32a^{2}-5a-48\)
sage: E = EllipticCurve([K([0,0,0,0]),K([4,0,-1,0]),K([-3,0,1,0]),K([39,-3,-20,6]),K([-48,-5,32,-9])])
 
gp: E = ellinit([Pol(Vecrev([0,0,0,0])),Pol(Vecrev([4,0,-1,0])),Pol(Vecrev([-3,0,1,0])),Pol(Vecrev([39,-3,-20,6])),Pol(Vecrev([-48,-5,32,-9]))], K);
 
magma: E := EllipticCurve([K![0,0,0,0],K![4,0,-1,0],K![-3,0,1,0],K![39,-3,-20,6],K![-48,-5,32,-9]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((a^3-a^2-4a-3)\) = \((a^3-a^2-4a-3)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 41 \) = \(41\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((-37a^2+37a+124)\) = \((a^3-a^2-4a-3)^{4}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 2825761 \) = \(41^{4}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{71073792}{1681} a^{3} - \frac{64847872}{1681} a^{2} - \frac{296812544}{1681} a - \frac{149258240}{1681} \)
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: \(1\)
Generator $\left(a^{3} - 2 a^{2} - a + 2 : -2 a^{3} + 4 a^{2} + 3 a - 6 : 1\right)$
Height \(0.00931209044957870\)
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: \( 1 \)
sage: E.rank()
 
magma: Rank(E);
 
Mordell-Weil rank: \(1\)
Regulator: \( 0.00931209044957870 \)
Period: \( 1120.90385756440 \)
Tamagawa product: \( 4 \)
Torsion order: \(1\)
Leading coefficient: \( 2.33285957371122 \)
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\((j)_{-}\)
\((a^3-a^2-4a-3)\) \(41\) \(4\) \(I_{4}\) Split multiplicative \(-1\) \(1\) \(4\) \(4\)

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 41.1-e consists of this curve only.

Base change

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