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Properties

Label 6.6.434581.1-43.2-b1
Base field 6.6.434581.1
Conductor norm \( 43 \)
CM no
Base change no
Q-curve no
Torsion order \( 1 \)
Rank \( 1 \)

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Base field 6.6.434581.1

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

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

Weierstrass equation

\({y}^2+\left(3a^{5}-6a^{4}-10a^{3}+12a^{2}+4a-3\right){x}{y}+\left(a^{5}-2a^{4}-3a^{3}+4a^{2}-1\right){y}={x}^{3}+\left(a^{5}-3a^{4}-2a^{3}+8a^{2}+a-3\right){x}^{2}+\left(-2a^{5}-2a^{4}+13a^{3}+19a^{2}-4a-9\right){x}-13a^{5}+16a^{4}+56a^{3}-13a^{2}-25a+4\)
sage: E = EllipticCurve([K([-3,4,12,-10,-6,3]),K([-3,1,8,-2,-3,1]),K([-1,0,4,-3,-2,1]),K([-9,-4,19,13,-2,-2]),K([4,-25,-13,56,16,-13])])
 
gp: E = ellinit([Polrev([-3,4,12,-10,-6,3]),Polrev([-3,1,8,-2,-3,1]),Polrev([-1,0,4,-3,-2,1]),Polrev([-9,-4,19,13,-2,-2]),Polrev([4,-25,-13,56,16,-13])], K);
 
magma: E := EllipticCurve([K![-3,4,12,-10,-6,3],K![-3,1,8,-2,-3,1],K![-1,0,4,-3,-2,1],K![-9,-4,19,13,-2,-2],K![4,-25,-13,56,16,-13]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((a^5-3a^4-a^3+7a^2-3a-4)\) = \((a^5-3a^4-a^3+7a^2-3a-4)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 43 \) = \(43\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((-a^5+8a^3-10a)\) = \((a^5-3a^4-a^3+7a^2-3a-4)^{2}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 1849 \) = \(43^{2}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( -\frac{35119198}{1849} a^{5} + \frac{13127784}{1849} a^{4} + \frac{79757853}{1849} a^{3} - \frac{41392973}{1849} a^{2} - \frac{11112889}{1849} a + \frac{3697481}{1849} \)
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: \(1\)
Generator $\left(-26 a^{5} + 63 a^{4} + 79 a^{3} - 165 a^{2} - 42 a + 70 : -232 a^{5} + 554 a^{4} + 716 a^{3} - 1441 a^{2} - 382 a + 613 : 1\right)$
Height \(0.00086005804389502969946120799255837935665\)
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: \( 1 \)
sage: E.rank()
 
magma: Rank(E);
 
Mordell-Weil rank: \(1\)
Regulator: \( 0.00086005804389502969946120799255837935665 \)
Period: \( 143341.29272308864628257012305302180804 \)
Tamagawa product: \( 2 \)
Torsion order: \(1\)
Leading coefficient: \( 2.24411 \)
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\((j)_{-}\)
\((a^5-3a^4-a^3+7a^2-3a-4)\) \(43\) \(2\) \(I_{2}\) Split multiplicative \(-1\) \(1\) \(2\) \(2\)

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 43.2-b 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.