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Properties

Label 6.6.703493.1-64.1-b1
Base field 6.6.703493.1
Conductor norm \( 64 \)
CM no
Base change no
Q-curve no
Torsion order \( 1 \)
Rank \( 1 \)

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Base field 6.6.703493.1

Generator \(a\), with minimal polynomial \( x^{6} - 2 x^{5} - 5 x^{4} + 11 x^{3} + 2 x^{2} - 9 x + 1 \); class number \(1\).

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([1, -9, 2, 11, -5, -2, 1]))
 
gp: K = nfinit(Polrev([1, -9, 2, 11, -5, -2, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -9, 2, 11, -5, -2, 1]);
 

Weierstrass equation

\({y}^2+\left(a^{5}-a^{4}-6a^{3}+5a^{2}+6a-1\right){x}{y}+\left(a^{5}-a^{4}-5a^{3}+6a^{2}+3a-3\right){y}={x}^{3}+\left(a^{5}-2a^{4}-5a^{3}+10a^{2}+2a-5\right){x}^{2}+\left(a^{5}+2a^{4}-6a^{3}-13a^{2}+7a+18\right){x}-11a^{5}+2a^{4}+64a^{3}-8a^{2}-66a-8\)
sage: E = EllipticCurve([K([-1,6,5,-6,-1,1]),K([-5,2,10,-5,-2,1]),K([-3,3,6,-5,-1,1]),K([18,7,-13,-6,2,1]),K([-8,-66,-8,64,2,-11])])
 
gp: E = ellinit([Polrev([-1,6,5,-6,-1,1]),Polrev([-5,2,10,-5,-2,1]),Polrev([-3,3,6,-5,-1,1]),Polrev([18,7,-13,-6,2,1]),Polrev([-8,-66,-8,64,2,-11])], K);
 
magma: E := EllipticCurve([K![-1,6,5,-6,-1,1],K![-5,2,10,-5,-2,1],K![-3,3,6,-5,-1,1],K![18,7,-13,-6,2,1],K![-8,-66,-8,64,2,-11]]);
 

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: \( -\frac{4155719}{8} a^{5} - \frac{2581671}{8} a^{4} + \frac{16657899}{8} a^{3} + \frac{900641}{4} a^{2} - 1374892 a + \frac{1236175}{8} \)
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 $\left(-5 a^{5} + 12 a^{4} + 25 a^{3} - 66 a^{2} - 12 a + 56 : 42 a^{5} - 84 a^{4} - 218 a^{3} + 465 a^{2} + 128 a - 391 : 1\right)$
Height \(0.0073938860060503256631341928380573164348\)
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: \( 0.0073938860060503256631341928380573164348 \)
Period: \( 19202.856664731547168155101042550707485 \)
Tamagawa product: \( 3 \)
Torsion order: \(1\)
Leading coefficient: \( 3.04706 \)
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 \( p < 1000 \) .

Isogenies and isogeny class

This curve has no rational isogenies. Its isogeny class 64.1-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.