Properties

Label 6.6.592661.1-47.1-a3
Base field 6.6.592661.1
Conductor norm \( 47 \)
CM no
Base change no
Q-curve no
Torsion order \( 4 \)
Rank \( 0 \)

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Base field 6.6.592661.1

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

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

Weierstrass equation

\({y}^2+\left(a^{2}-2\right){x}{y}+\left(a^{5}-4a^{3}+a^{2}+a-3\right){y}={x}^{3}+\left(-a^{5}+2a^{4}+5a^{3}-9a^{2}-4a+4\right){x}^{2}+\left(-640a^{5}+216a^{4}+3345a^{3}-347a^{2}-3444a-992\right){x}+4631a^{5}-1611a^{4}-24188a^{3}+2739a^{2}+24854a+6981\)
sage: E = EllipticCurve([K([-2,0,1,0,0,0]),K([4,-4,-9,5,2,-1]),K([-3,1,1,-4,0,1]),K([-992,-3444,-347,3345,216,-640]),K([6981,24854,2739,-24188,-1611,4631])])
 
gp: E = ellinit([Polrev([-2,0,1,0,0,0]),Polrev([4,-4,-9,5,2,-1]),Polrev([-3,1,1,-4,0,1]),Polrev([-992,-3444,-347,3345,216,-640]),Polrev([6981,24854,2739,-24188,-1611,4631])], K);
 
magma: E := EllipticCurve([K![-2,0,1,0,0,0],K![4,-4,-9,5,2,-1],K![-3,1,1,-4,0,1],K![-992,-3444,-347,3345,216,-640],K![6981,24854,2739,-24188,-1611,4631]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((a^5-a^4-4a^3+4a^2-2)\) = \((a^5-a^4-4a^3+4a^2-2)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 47 \) = \(47\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((-3a^5+a^4+14a^3-a^2-11a-6)\) = \((a^5-a^4-4a^3+4a^2-2)^{2}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 2209 \) = \(47^{2}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( -\frac{800826235730}{2209} a^{5} + \frac{1672321121383}{2209} a^{4} + \frac{574341648699}{2209} a^{3} - \frac{1394105750193}{2209} a^{2} + \frac{76684984327}{2209} a + \frac{175128065644}{2209} \)
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: \(\Z/2\Z\oplus\Z/2\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Torsion generators: $\left(13 a^{5} - 6 a^{4} - 67 a^{3} + 15 a^{2} + 67 a + 14 : 10 a^{5} - 5 a^{4} - 52 a^{3} + 12 a^{2} + 53 a + 12 : 1\right)$ $\left(3 a^{5} - \frac{9}{4} a^{4} - 15 a^{3} + 8 a^{2} + 14 a - 1 : \frac{17}{8} a^{5} - \frac{17}{8} a^{4} - 11 a^{3} + \frac{55}{8} a^{2} + \frac{45}{4} a + \frac{1}{8} : 1\right)$
sage: T.gens()
 
gp: T[3]
 
magma: [piT(P) : P in Generators(T)];
 

BSD invariants

Analytic rank: \( 0 \)
sage: E.rank()
 
magma: Rank(E);
 
Mordell-Weil rank: \(0\)
Regulator: \( 1 \)
Period: \( 2089.9632873453821689478948175333639018 \)
Tamagawa product: \( 2 \)
Torsion order: \(4\)
Leading coefficient: \( 1.35739 \)
Analytic order of Ш: \( 4 \) (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\((j)_{-}\)
\((a^5-a^4-4a^3+4a^2-2)\) \(47\) \(2\) \(I_{2}\) Non-split multiplicative \(1\) \(1\) \(2\) \(2\)

Galois Representations

The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.

prime Image of Galois Representation
\(2\) 2Cs

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2.
Its isogeny class 47.1-a consists of curves linked by isogenies of degrees dividing 4.

Base change

This elliptic curve is not a \(\Q\)-curve.

It is not the base change of an elliptic curve defined over any subfield.