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Properties

Label 6.6.592661.1-83.1-d1
Base field 6.6.592661.1
Conductor norm \( 83 \)
CM no
Base change no
Q-curve no
Torsion order \( 1 \)
Rank \( 1 \)

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Base field 6.6.592661.1

Generator \(a\), with minimal polynomial \( x^{6} - x^{5} - 5 x^{4} + 4 x^{3} + 5 x^{2} - 2 x - 1 \); class number \(1\).

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-1, -2, 5, 4, -5, -1, 1]))
 
gp: K = nfinit(Polrev([-1, -2, 5, 4, -5, -1, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, -2, 5, 4, -5, -1, 1]);
 

Weierstrass equation

\({y}^2+\left(2a^{5}-2a^{4}-9a^{3}+9a^{2}+5a-6\right){x}{y}+\left(a^{4}-4a^{2}+a+1\right){y}={x}^{3}+\left(a^{5}-5a^{3}+a^{2}+3a-4\right){x}^{2}+\left(-11a^{5}+14a^{4}+48a^{3}-56a^{2}-32a+32\right){x}-23a^{5}+31a^{4}+105a^{3}-128a^{2}-76a+72\)
sage: E = EllipticCurve([K([-6,5,9,-9,-2,2]),K([-4,3,1,-5,0,1]),K([1,1,-4,0,1,0]),K([32,-32,-56,48,14,-11]),K([72,-76,-128,105,31,-23])])
 
gp: E = ellinit([Polrev([-6,5,9,-9,-2,2]),Polrev([-4,3,1,-5,0,1]),Polrev([1,1,-4,0,1,0]),Polrev([32,-32,-56,48,14,-11]),Polrev([72,-76,-128,105,31,-23])], K);
 
magma: E := EllipticCurve([K![-6,5,9,-9,-2,2],K![-4,3,1,-5,0,1],K![1,1,-4,0,1,0],K![32,-32,-56,48,14,-11],K![72,-76,-128,105,31,-23]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((-2a^5+a^4+9a^3-4a^2-6a+1)\) = \((-2a^5+a^4+9a^3-4a^2-6a+1)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 83 \) = \(83\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((a^5-a^4-3a^3+4a^2-2a-2)\) = \((-2a^5+a^4+9a^3-4a^2-6a+1)\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( -83 \) = \(-83\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{46638196}{83} a^{5} + \frac{119264171}{83} a^{4} - \frac{90683712}{83} a^{3} - \frac{273899140}{83} a^{2} + \frac{36697985}{83} a + \frac{37204710}{83} \)
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 $\left(1 : -4 a^{5} + 4 a^{4} + 19 a^{3} - 18 a^{2} - 14 a + 12 : 1\right)$
Height \(0.0056130727761519343794235617972804318303\)
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: \( 0.0056130727761519343794235617972804318303 \)
Period: \( 70481.431168469981956049921990849354023 \)
Tamagawa product: \( 1 \)
Torsion order: \(1\)
Leading coefficient: \( 3.08335 \)
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\((j)_{-}\)
\((-2a^5+a^4+9a^3-4a^2-6a+1)\) \(83\) \(1\) \(I_{1}\) 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 83.1-d 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.