Properties

Label 4.4.725.1-109.1-a1
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

\({y}^2+\left(a^{3}-3a\right){x}{y}+\left(a^{2}-a\right){y}={x}^{3}+\left(-a^{3}+2a^{2}+a-3\right){x}^{2}+\left(540a^{3}-71a^{2}-1805a-989\right){x}+10516a^{3}-2406a^{2}-33972a-15480\)
sage: E = EllipticCurve([K([0,-3,0,1]),K([-3,1,2,-1]),K([0,-1,1,0]),K([-989,-1805,-71,540]),K([-15480,-33972,-2406,10516])])
 
gp: E = ellinit([Polrev([0,-3,0,1]),Polrev([-3,1,2,-1]),Polrev([0,-1,1,0]),Polrev([-989,-1805,-71,540]),Polrev([-15480,-33972,-2406,10516])], K);
 
magma: E := EllipticCurve([K![0,-3,0,1],K![-3,1,2,-1],K![0,-1,1,0],K![-989,-1805,-71,540],K![-15480,-33972,-2406,10516]]);
 

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: \((-399760a^3+396997a^2+867558a-254938)\) = \((-a^3+a^2+5a)^{11}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 25804264053054077850709 \) = \(109^{11}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{16235216636009362328759362787732}{25804264053054077850709} a^{3} - \frac{4260010497864756660724731056719}{25804264053054077850709} a^{2} - \frac{51855684642834574989082645270777}{25804264053054077850709} a - \frac{22019771839812368650729782793234}{25804264053054077850709} \)
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: \( 0.18823653156954479714313859813894756693 \)
Tamagawa product: \( 1 \)
Torsion order: \(1\)
Leading coefficient: \( 0.845902442567382 \)
Analytic order of Ш: \( 121 \) (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\((j)_{-}\)
\((-a^3+a^2+5a)\) \(109\) \(1\) \(I_{11}\) Non-split multiplicative \(1\) \(1\) \(11\) \(11\)

Galois Representations

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

prime Image of Galois Representation
\(11\) 11B.1.2

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 11.
Its isogeny class 109.1-a consists of curves linked by isogenies of degree 11.

Base change

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

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