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Properties

Label 6.6.592661.1-49.1-a1
Base field 6.6.592661.1
Conductor norm \( 49 \)
CM no
Base change no
Q-curve no
Torsion order \( 1 \)
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

\({y}^2+\left(a^{5}-a^{4}-4a^{3}+4a^{2}+2a-1\right){x}{y}+\left(a^{5}-a^{4}-4a^{3}+4a^{2}+a-1\right){y}={x}^{3}+\left(2a^{5}-3a^{4}-9a^{3}+13a^{2}+6a-7\right){x}^{2}+\left(-88a^{5}+34a^{4}+458a^{3}-68a^{2}-465a-120\right){x}+658a^{5}-224a^{4}-3439a^{3}+364a^{2}+3542a+1014\)
sage: E = EllipticCurve([K([-1,2,4,-4,-1,1]),K([-7,6,13,-9,-3,2]),K([-1,1,4,-4,-1,1]),K([-120,-465,-68,458,34,-88]),K([1014,3542,364,-3439,-224,658])])
 
gp: E = ellinit([Polrev([-1,2,4,-4,-1,1]),Polrev([-7,6,13,-9,-3,2]),Polrev([-1,1,4,-4,-1,1]),Polrev([-120,-465,-68,458,34,-88]),Polrev([1014,3542,364,-3439,-224,658])], K);
 
magma: E := EllipticCurve([K![-1,2,4,-4,-1,1],K![-7,6,13,-9,-3,2],K![-1,1,4,-4,-1,1],K![-120,-465,-68,458,34,-88],K![1014,3542,364,-3439,-224,658]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((a^5-2a^4-5a^3+7a^2+4a-2)\) = \((a^5-2a^4-5a^3+7a^2+4a-2)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 49 \) = \(49\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((-62a^5+61a^4+240a^3-269a^2-53a+130)\) = \((a^5-2a^4-5a^3+7a^2+4a-2)^{7}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 678223072849 \) = \(49^{7}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{7276612344201570}{823543} a^{5} + \frac{2816047654779374}{823543} a^{4} - \frac{32477199071902533}{823543} a^{3} - \frac{15939419409793606}{823543} a^{2} + \frac{14275065920439475}{823543} a + \frac{5246284168272092}{823543} \)
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: \( 1384.1480731872470432254362869454666247 \)
Tamagawa product: \( 1 \)
Torsion order: \(1\)
Leading coefficient: \( 1.79796 \)
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\((j)_{-}\)
\((a^5-2a^4-5a^3+7a^2+4a-2)\) \(49\) \(1\) \(I_{7}\) Non-split multiplicative \(1\) \(1\) \(7\) \(7\)

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 49.1-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.