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Properties

Label 6.6.703493.1-71.1-e1
Base field 6.6.703493.1
Conductor norm \( 71 \)
CM no
Base change no
Q-curve no
Torsion order \( 1 \)
Rank \( 0 \)

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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

\({y}^2+\left(3a^{5}-2a^{4}-17a^{3}+11a^{2}+16a-5\right){x}{y}+\left(a^{5}-6a^{3}+a^{2}+7a\right){y}={x}^{3}+\left(-a^{5}+7a^{3}+a^{2}-9a-3\right){x}^{2}+\left(-69a^{5}+47a^{4}+407a^{3}-221a^{2}-427a+54\right){x}-236a^{5}+160a^{4}+1392a^{3}-756a^{2}-1470a+178\)
sage: E = EllipticCurve([K([-5,16,11,-17,-2,3]),K([-3,-9,1,7,0,-1]),K([0,7,1,-6,0,1]),K([54,-427,-221,407,47,-69]),K([178,-1470,-756,1392,160,-236])])
 
gp: E = ellinit([Polrev([-5,16,11,-17,-2,3]),Polrev([-3,-9,1,7,0,-1]),Polrev([0,7,1,-6,0,1]),Polrev([54,-427,-221,407,47,-69]),Polrev([178,-1470,-756,1392,160,-236])], K);
 
magma: E := EllipticCurve([K![-5,16,11,-17,-2,3],K![-3,-9,1,7,0,-1],K![0,7,1,-6,0,1],K![54,-427,-221,407,47,-69],K![178,-1470,-756,1392,160,-236]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((-2a^5+a^4+12a^3-4a^2-12a-1)\) = \((-2a^5+a^4+12a^3-4a^2-12a-1)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 71 \) = \(71\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((-57a^5+27a^4+325a^3-113a^2-318a-18)\) = \((-2a^5+a^4+12a^3-4a^2-12a-1)^{5}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 1804229351 \) = \(71^{5}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( -\frac{124315078720314611694730}{1804229351} a^{5} + \frac{84498863776551023357538}{1804229351} a^{4} + \frac{733137948754350072437846}{1804229351} a^{3} - \frac{399515065412184994079968}{1804229351} a^{2} - \frac{776103776934549949304354}{1804229351} a + \frac{94157783375202703558381}{1804229351} \)
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: \(0\)
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: \( 0 \)
sage: E.rank()
 
magma: Rank(E);
 
Mordell-Weil rank: \(0\)
Regulator: \( 1 \)
Period: \( 866.13008303550372599995575534786055498 \)
Tamagawa product: \( 1 \)
Torsion order: \(1\)
Leading coefficient: \( 1.03265 \)
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\((j)_{-}\)
\((-2a^5+a^4+12a^3-4a^2-12a-1)\) \(71\) \(1\) \(I_{5}\) Non-split multiplicative \(1\) \(1\) \(5\) \(5\)

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 71.1-e 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.