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Properties

Label 6.6.485125.1-64.1-a2
Base field 6.6.485125.1
Conductor norm \( 64 \)
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

\({y}^2+\left(a^{4}-3a^{2}-a+1\right){x}{y}+\left(2a^{5}-2a^{4}-9a^{3}+7a^{2}+8a-2\right){y}={x}^{3}+\left(a^{5}-a^{4}-4a^{3}+2a^{2}+a+1\right){x}^{2}+\left(-4a^{4}+16a^{2}+a-5\right){x}+4a^{5}-5a^{4}-17a^{3}+18a^{2}+9a-12\)
sage: E = EllipticCurve([K([1,-1,-3,0,1,0]),K([1,1,2,-4,-1,1]),K([-2,8,7,-9,-2,2]),K([-5,1,16,0,-4,0]),K([-12,9,18,-17,-5,4])])
 
gp: E = ellinit([Polrev([1,-1,-3,0,1,0]),Polrev([1,1,2,-4,-1,1]),Polrev([-2,8,7,-9,-2,2]),Polrev([-5,1,16,0,-4,0]),Polrev([-12,9,18,-17,-5,4])], K);
 
magma: E := EllipticCurve([K![1,-1,-3,0,1,0],K![1,1,2,-4,-1,1],K![-2,8,7,-9,-2,2],K![-5,1,16,0,-4,0],K![-12,9,18,-17,-5,4]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((2)\) = \((2)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 64 \) = \(64\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((-8)\) = \((2)^{3}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( -262144 \) = \(-64^{3}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( -\frac{372514795}{2} a^{5} + \frac{1962808611}{8} a^{4} + \frac{3650149817}{4} a^{3} - \frac{6936306195}{8} a^{2} - \frac{7715765979}{8} a + \frac{2182489307}{8} \)
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 $\left(-5 a^{5} + 8 a^{4} + 22 a^{3} - 31 a^{2} - 17 a + 18 : 16 a^{5} - 29 a^{4} - 71 a^{3} + 115 a^{2} + 59 a - 69 : 1\right)$
Height \(0.0077837535249815245985547087823022324145\)
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: \( 0.0077837535249815245985547087823022324145 \)
Period: \( 13990.337260106950258133421748028974557 \)
Tamagawa product: \( 3 \)
Torsion order: \(1\)
Leading coefficient: \( 2.81425 \)
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\((j)_{-}\)
\((2)\) \(64\) \(3\) \(I_{3}\) 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 64.1-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.