Properties

Label 4.4.725.1-109.1-b1
Base field 4.4.725.1
Conductor norm \( 109 \)
CM no
Base change no
Q-curve no
Torsion order \( 1 \)
Rank \( 0 \)

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Base field 4.4.725.1

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

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

Weierstrass equation

\({y}^2+\left(a^{3}-a^{2}-2a+2\right){x}{y}+\left(a^{2}-a-1\right){y}={x}^{3}+\left(-a^{2}+2a+1\right){x}^{2}+\left(4a^{3}-15a-8\right){x}-8a^{3}+2a^{2}+25a+8\)
sage: E = EllipticCurve([K([2,-2,-1,1]),K([1,2,-1,0]),K([-1,-1,1,0]),K([-8,-15,0,4]),K([8,25,2,-8])])
 
gp: E = ellinit([Polrev([2,-2,-1,1]),Polrev([1,2,-1,0]),Polrev([-1,-1,1,0]),Polrev([-8,-15,0,4]),Polrev([8,25,2,-8])], K);
 
magma: E := EllipticCurve([K![2,-2,-1,1],K![1,2,-1,0],K![-1,-1,1,0],K![-8,-15,0,4],K![8,25,2,-8]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((-a^3+a^2+5a)\) = \((-a^3+a^2+5a)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 109 \) = \(109\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((a^2-a-5)\) = \((-a^3+a^2+5a)\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 109 \) = \(109\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( -\frac{5067738978911}{109} a^{3} + \frac{7485983125541}{109} a^{2} + \frac{11630034812778}{109} a - \frac{10618018451270}{109} \)
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: \( 22.907881739628105984929846954239419613 \)
Tamagawa product: \( 1 \)
Torsion order: \(1\)
Leading coefficient: \( 0.850777369312031 \)
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\((j)_{-}\)
\((-a^3+a^2+5a)\) \(109\) \(1\) \(I_{1}\) Non-split multiplicative \(1\) \(1\) \(1\) \(1\)

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 109.1-b 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.