Properties

Label 5.5.160801.1-9.3-b1
Base field 5.5.160801.1
Conductor norm \( 9 \)
CM no
Base change no
Q-curve no
Torsion order \( 1 \)
Rank \( 1 \)

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Base field 5.5.160801.1

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

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-1, 3, 4, -5, -1, 1]))
 
gp: K = nfinit(Polrev([-1, 3, 4, -5, -1, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 3, 4, -5, -1, 1]);
 

Weierstrass equation

\({y}^2+\left(a^{4}-a^{3}-5a^{2}+5a+3\right){x}{y}+\left(2a^{4}-a^{3}-9a^{2}+5a+2\right){y}={x}^{3}+\left(a^{2}+a-1\right){x}^{2}+\left(-a^{3}+2a^{2}+8a-4\right){x}+3a^{4}-2a^{3}-12a^{2}+9a-2\)
sage: E = EllipticCurve([K([3,5,-5,-1,1]),K([-1,1,1,0,0]),K([2,5,-9,-1,2]),K([-4,8,2,-1,0]),K([-2,9,-12,-2,3])])
 
gp: E = ellinit([Polrev([3,5,-5,-1,1]),Polrev([-1,1,1,0,0]),Polrev([2,5,-9,-1,2]),Polrev([-4,8,2,-1,0]),Polrev([-2,9,-12,-2,3])], K);
 
magma: E := EllipticCurve([K![3,5,-5,-1,1],K![-1,1,1,0,0],K![2,5,-9,-1,2],K![-4,8,2,-1,0],K![-2,9,-12,-2,3]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((a^4-4a^2+a+1)\) = \((-a^4+5a^2+a-3)^{2}\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 9 \) = \(3^{2}\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((a^4-a^3-5a^2+4a+1)\) = \((-a^4+5a^2+a-3)^{3}\)
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: \( -10331 a^{4} - 1485 a^{3} + 14206 a^{2} + 5063 a - 1551 \)
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 $\left(-a^{4} + a^{3} + 4 a^{2} - 4 a : 0 : 1\right)$
Height \(0.026884599041736376433805165300361462984\)
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: \( 0.026884599041736376433805165300361462984 \)
Period: \( 4295.4716666641474930090802859562762546 \)
Tamagawa product: \( 2 \)
Torsion order: \(1\)
Leading coefficient: \( 2.87985121 \)
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\((j)_{-}\)
\((-a^4+5a^2+a-3)\) \(3\) \(2\) \(III\) Additive \(1\) \(2\) \(3\) \(0\)

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 9.3-b 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.