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Properties

Label 6.6.1202933.1-79.1-a1
Base field 6.6.1202933.1
Conductor norm \( 79 \)
CM no
Base change no
Q-curve no
Torsion order \( 1 \)
Rank \( 0 \)

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Base field 6.6.1202933.1

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

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-1, 1, 6, -2, -6, 0, 1]))
 
gp: K = nfinit(Polrev([-1, 1, 6, -2, -6, 0, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 1, 6, -2, -6, 0, 1]);
 

Weierstrass equation

\({y}^2+\left(2a^{5}-11a^{3}-4a^{2}+7a+2\right){x}{y}+\left(a^{5}-a^{4}-5a^{3}+3a^{2}+4a-2\right){y}={x}^{3}+\left(-a^{5}+2a^{4}+6a^{3}-9a^{2}-9a+5\right){x}^{2}+\left(-4a^{5}+2a^{4}+22a^{3}-4a^{2}-16a+10\right){x}-2a^{5}+2a^{4}+12a^{3}-8a^{2}-14a+7\)
sage: E = EllipticCurve([K([2,7,-4,-11,0,2]),K([5,-9,-9,6,2,-1]),K([-2,4,3,-5,-1,1]),K([10,-16,-4,22,2,-4]),K([7,-14,-8,12,2,-2])])
 
gp: E = ellinit([Polrev([2,7,-4,-11,0,2]),Polrev([5,-9,-9,6,2,-1]),Polrev([-2,4,3,-5,-1,1]),Polrev([10,-16,-4,22,2,-4]),Polrev([7,-14,-8,12,2,-2])], K);
 
magma: E := EllipticCurve([K![2,7,-4,-11,0,2],K![5,-9,-9,6,2,-1],K![-2,4,3,-5,-1,1],K![10,-16,-4,22,2,-4],K![7,-14,-8,12,2,-2]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((-2a^5+11a^3+4a^2-6a-1)\) = \((-2a^5+11a^3+4a^2-6a-1)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 79 \) = \(79\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((-2a^5+11a^3+4a^2-6a-1)\) = \((-2a^5+11a^3+4a^2-6a-1)\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 79 \) = \(79\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( -\frac{2248311}{79} a^{5} + \frac{344041}{79} a^{4} + \frac{12225319}{79} a^{3} + \frac{3867560}{79} a^{2} - \frac{8350654}{79} a - \frac{3874311}{79} \)
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: \( 2447.4086598746841222998622933965141359 \)
Tamagawa product: \( 1 \)
Torsion order: \(1\)
Leading coefficient: \( 2.23144 \)
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\((j)_{-}\)
\((-2a^5+11a^3+4a^2-6a-1)\) \(79\) \(1\) \(I_{1}\) Non-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 79.1-a 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.