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Properties

Label 6.6.485125.1-59.3-b1
Base field 6.6.485125.1
Conductor norm \( 59 \)
CM no
Base change no
Q-curve no
Torsion order \( 1 \)
Rank \( 1 \)

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Base field 6.6.485125.1

Generator \(a\), with minimal polynomial \( x^{6} - 2 x^{5} - 4 x^{4} + 8 x^{3} + 2 x^{2} - 5 x + 1 \); class number \(1\).

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([1, -5, 2, 8, -4, -2, 1]))
 
gp: K = nfinit(Polrev([1, -5, 2, 8, -4, -2, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -5, 2, 8, -4, -2, 1]);
 

Weierstrass equation

\({y}^2+\left(a^{5}-a^{4}-4a^{3}+4a^{2}+2a-2\right){x}{y}+\left(-2a^{5}+3a^{4}+9a^{3}-10a^{2}-7a+4\right){y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-a^{5}+6a^{3}-2a^{2}-6a+3\right){x}+a^{4}-2a^{3}-a^{2}+a-1\)
sage: E = EllipticCurve([K([-2,2,4,-4,-1,1]),K([1,-1,0,0,0,0]),K([4,-7,-10,9,3,-2]),K([3,-6,-2,6,0,-1]),K([-1,1,-1,-2,1,0])])
 
gp: E = ellinit([Polrev([-2,2,4,-4,-1,1]),Polrev([1,-1,0,0,0,0]),Polrev([4,-7,-10,9,3,-2]),Polrev([3,-6,-2,6,0,-1]),Polrev([-1,1,-1,-2,1,0])], K);
 
magma: E := EllipticCurve([K![-2,2,4,-4,-1,1],K![1,-1,0,0,0,0],K![4,-7,-10,9,3,-2],K![3,-6,-2,6,0,-1],K![-1,1,-1,-2,1,0]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((a^5-a^4-4a^3+4a^2+a-2)\) = \((a^5-a^4-4a^3+4a^2+a-2)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 59 \) = \(59\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((a^5-a^4-4a^3+4a^2+a-2)\) = \((a^5-a^4-4a^3+4a^2+a-2)\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 59 \) = \(59\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{1370532}{59} a^{5} + \frac{445702}{59} a^{4} - \frac{5167788}{59} a^{3} - \frac{815081}{59} a^{2} + \frac{3174888}{59} a - \frac{498817}{59} \)
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 $\left(2 a^{5} - 3 a^{4} - 10 a^{3} + 11 a^{2} + 11 a - 4 : 6 a^{5} - 8 a^{4} - 29 a^{3} + 29 a^{2} + 29 a - 10 : 1\right)$
Height \(0.0016309701807232352652592625863679613321\)
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: \( 0.0016309701807232352652592625863679613321 \)
Period: \( 164018.23084338464361015068009646073436 \)
Tamagawa product: \( 1 \)
Torsion order: \(1\)
Leading coefficient: \( 2.30442 \)
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\((j)_{-}\)
\((a^5-a^4-4a^3+4a^2+a-2)\) \(59\) \(1\) \(I_{1}\) 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 59.3-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.