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Properties

Label 6.6.485125.1-49.1-b1
Base field 6.6.485125.1
Conductor norm \( 49 \)
CM no
Base change no
Q-curve no
Torsion order \( 1 \)
Rank \( 0 \)

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Base field 6.6.485125.1

Generator \(a\), with minimal polynomial \( x^{6} - 2 x^{5} - 4 x^{4} + 8 x^{3} + 2 x^{2} - 5 x + 1 \); class number \(1\).

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([1, -5, 2, 8, -4, -2, 1]))
 
gp: K = nfinit(Polrev([1, -5, 2, 8, -4, -2, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -5, 2, 8, -4, -2, 1]);
 

Weierstrass equation

\({y}^2+\left(-a^{5}+2a^{4}+4a^{3}-7a^{2}-2a+3\right){x}{y}+\left(a^{5}-a^{4}-5a^{3}+4a^{2}+6a-1\right){y}={x}^{3}+\left(a^{5}-a^{4}-5a^{3}+4a^{2}+4a-1\right){x}^{2}+\left(a^{5}-4a^{4}-8a^{3}+14a^{2}+9a-10\right){x}+6a^{5}-4a^{4}-24a^{3}+17a^{2}+13a-14\)
sage: E = EllipticCurve([K([3,-2,-7,4,2,-1]),K([-1,4,4,-5,-1,1]),K([-1,6,4,-5,-1,1]),K([-10,9,14,-8,-4,1]),K([-14,13,17,-24,-4,6])])
 
gp: E = ellinit([Polrev([3,-2,-7,4,2,-1]),Polrev([-1,4,4,-5,-1,1]),Polrev([-1,6,4,-5,-1,1]),Polrev([-10,9,14,-8,-4,1]),Polrev([-14,13,17,-24,-4,6])], K);
 
magma: E := EllipticCurve([K![3,-2,-7,4,2,-1],K![-1,4,4,-5,-1,1],K![-1,6,4,-5,-1,1],K![-10,9,14,-8,-4,1],K![-14,13,17,-24,-4,6]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((-2a^5+3a^4+10a^3-12a^2-11a+6)\) = \((-2a^5+3a^4+10a^3-12a^2-11a+6)\)
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: \((-a^5+4a^4+8a^3-19a^2-13a+11)\) = \((-2a^5+3a^4+10a^3-12a^2-11a+6)^{3}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 117649 \) = \(49^{3}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{30176477175473918504}{49} a^{5} - \frac{370959595583589700360}{343} a^{4} - \frac{935402224099909297679}{343} a^{3} + \frac{1461778923930007873681}{343} a^{2} + \frac{778934793586578494395}{343} a - \frac{866228484050474713990}{343} \)
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: \( 787.04812874992592451012497093567924312 \)
Tamagawa product: \( 1 \)
Torsion order: \(1\)
Leading coefficient: \( 1.12999 \)
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\((j)_{-}\)
\((-2a^5+3a^4+10a^3-12a^2-11a+6)\) \(49\) \(1\) \(I_{3}\) Non-split multiplicative \(1\) \(1\) \(3\) \(3\)

Galois Representations

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

prime Image of Galois Representation
\(3\) 3B

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 3.
Its isogeny class 49.1-b consists of curves linked by isogenies of degree 3.

Base change

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

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