Properties

Label 5.5.81509.1-9.1-a2
Base field 5.5.81509.1
Conductor norm \( 9 \)
CM no
Base change no
Q-curve no
Torsion order \( 1 \)
Rank \( 1 \)

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Base field 5.5.81509.1

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

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-2, 5, 3, -5, -1, 1]))
 
gp: K = nfinit(Polrev([-2, 5, 3, -5, -1, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-2, 5, 3, -5, -1, 1]);
 

Weierstrass equation

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

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((a^2-a-1)\) = \((a^2-a-1)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 9 \) = \(9\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((2a^4-3a^3-4a^2+7a-3)\) = \((a^2-a-1)^{3}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( -729 \) = \(-9^{3}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{286946845}{27} a^{4} + \frac{61919467}{27} a^{3} - \frac{1359497734}{27} a^{2} - \frac{791995172}{27} a + \frac{472024472}{27} \)
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: \(1\)
Generator $\left(0 : -a^{4} + 4 a^{2} + a - 1 : 1\right)$
Height \(0.016060513871737468040706215511518590902\)
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: \( 0.016060513871737468040706215511518590902 \)
Period: \( 6296.0619081737570036912786411028007017 \)
Tamagawa product: \( 1 \)
Torsion order: \(1\)
Leading coefficient: \( 1.77090657 \)
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\((j)_{-}\)
\((a^2-a-1)\) \(9\) \(1\) \(I_{3}\) Non-split multiplicative \(1\) \(1\) \(3\) \(3\)

Galois Representations

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

prime Image of Galois Representation
\(3\) 3B

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 3.
Its isogeny class 9.1-a consists of curves linked by isogenies of degree 3.

Base change

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

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