Properties

Label 6.6.703493.1-41.4-d1
Base field 6.6.703493.1
Conductor \((a^5-a^4-6a^3+4a^2+7a+1)\)
Conductor norm \( 41 \)
CM no
Base change no
Q-curve no
Torsion order \( 1 \)
Rank \( 0 \)

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Base field 6.6.703493.1

Generator \(a\), with minimal polynomial \( x^{6} - 2 x^{5} - 5 x^{4} + 11 x^{3} + 2 x^{2} - 9 x + 1 \); class number \(1\).

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([1, -9, 2, 11, -5, -2, 1]))
 
gp: K = nfinit(Pol(Vecrev([1, -9, 2, 11, -5, -2, 1])));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -9, 2, 11, -5, -2, 1]);
 

Weierstrass equation

\({y}^2+\left(2a^{5}-a^{4}-12a^{3}+6a^{2}+13a-2\right){x}{y}+\left(a^{5}-6a^{3}+8a+2\right){y}={x}^{3}+\left(-2a^{5}+2a^{4}+11a^{3}-11a^{2}-9a+5\right){x}^{2}+\left(-2a^{5}+3a^{4}+10a^{3}-16a^{2}-7a+14\right){x}-6a^{5}+16a^{4}+27a^{3}-92a^{2}-3a+86\)
sage: E = EllipticCurve([K([-2,13,6,-12,-1,2]),K([5,-9,-11,11,2,-2]),K([2,8,0,-6,0,1]),K([14,-7,-16,10,3,-2]),K([86,-3,-92,27,16,-6])])
 
gp: E = ellinit([Pol(Vecrev([-2,13,6,-12,-1,2])),Pol(Vecrev([5,-9,-11,11,2,-2])),Pol(Vecrev([2,8,0,-6,0,1])),Pol(Vecrev([14,-7,-16,10,3,-2])),Pol(Vecrev([86,-3,-92,27,16,-6]))], K);
 
magma: E := EllipticCurve([K![-2,13,6,-12,-1,2],K![5,-9,-11,11,2,-2],K![2,8,0,-6,0,1],K![14,-7,-16,10,3,-2],K![86,-3,-92,27,16,-6]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((a^5-a^4-6a^3+4a^2+7a+1)\) = \((a^5-a^4-6a^3+4a^2+7a+1)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 41 \) = \(41\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((-85a^5+78a^4+436a^3-375a^2-283a+87)\) = \((a^5-a^4-6a^3+4a^2+7a+1)^{7}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 194754273881 \) = \(41^{7}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{2193085400250940456}{194754273881} a^{5} - \frac{9040335834589460238}{194754273881} a^{4} + \frac{8220320611440950985}{194754273881} a^{3} + \frac{6678172413832281891}{194754273881} a^{2} - \frac{9786903378511537191}{194754273881} a + \frac{1033336974108669203}{194754273881} \)
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: not available
Tamagawa product: \( 1 \)
Torsion order: \(1\)
Leading coefficient: not available
Analytic order of Ш: not available

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\((j)_{-}\)
\((a^5-a^4-6a^3+4a^2+7a+1)\) \(41\) \(1\) \(I_{7}\) 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 41.4-d consists of this curve only.

Base change

This curve is not the base change of an elliptic curve defined over \(\Q\). It is not a \(\Q\)-curve.