Properties

Label 6.6.703493.1-41.4-e2
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 \( 2 \)
Rank \( 1 \)

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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

\({y}^2+\left(2a^{5}-a^{4}-12a^{3}+5a^{2}+14a-1\right){x}{y}+\left(2a^{5}-2a^{4}-11a^{3}+11a^{2}+9a-5\right){y}={x}^{3}+\left(-a^{5}+a^{4}+6a^{3}-6a^{2}-7a+4\right){x}^{2}+\left(-30a^{5}+3a^{4}+169a^{3}-71a^{2}-156a+28\right){x}-115a^{5}-28a^{4}+635a^{3}-157a^{2}-561a+63\)
sage: E = EllipticCurve([K([-1,14,5,-12,-1,2]),K([4,-7,-6,6,1,-1]),K([-5,9,11,-11,-2,2]),K([28,-156,-71,169,3,-30]),K([63,-561,-157,635,-28,-115])])
 
gp: E = ellinit([Pol(Vecrev([-1,14,5,-12,-1,2])),Pol(Vecrev([4,-7,-6,6,1,-1])),Pol(Vecrev([-5,9,11,-11,-2,2])),Pol(Vecrev([28,-156,-71,169,3,-30])),Pol(Vecrev([63,-561,-157,635,-28,-115]))], K);
 
magma: E := EllipticCurve([K![-1,14,5,-12,-1,2],K![4,-7,-6,6,1,-1],K![-5,9,11,-11,-2,2],K![28,-156,-71,169,3,-30],K![63,-561,-157,635,-28,-115]]);
 

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: \((3a^5-a^4-18a^3+4a^2+20a+1)\) = \((a^5-a^4-6a^3+4a^2+7a+1)\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 41 \) = \(41\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{1661681620471232}{41} a^{5} + \frac{213704000156457}{41} a^{4} - \frac{7841722250015960}{41} a^{3} + \frac{1577558381739890}{41} a^{2} + \frac{6635361185868735}{41} a - \frac{778052671995566}{41} \)
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 $\left(7 a^{5} - 6 a^{4} - 41 a^{3} + 29 a^{2} + 36 a - 4 : -27 a^{5} + 13 a^{4} + 169 a^{3} - 60 a^{2} - 216 a + 25 : 1\right)$
Height \(0.738892988210579\)
Torsion structure: \(\Z/2\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Torsion generator: $\left(2 a^{4} - 5 a^{2} : -a^{4} - 3 a^{3} + a^{2} + 8 a + 1 : 1\right)$
sage: T.gens()
 
gp: T[3]
 
magma: [piT(P) : P in Generators(T)];
 

BSD invariants

Analytic rank: \( 1 \)
sage: E.rank()
 
magma: Rank(E);
 
Mordell-Weil rank: \(1\)
Regulator: \( 0.738892988210579 \)
Period: not available
Tamagawa product: \( 1 \)
Torsion order: \(2\)
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\((j)_{-}\)
\((a^5-a^4-6a^3+4a^2+7a+1)\) \(41\) \(1\) \(I_{1}\) Split multiplicative \(-1\) \(1\) \(1\) \(1\)

Galois Representations

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

prime Image of Galois Representation
\(2\) 2B
\(3\) 3B.1.2

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2, 3 and 6.
Its isogeny class 41.4-e consists of curves linked by isogenies of degrees dividing 6.

Base change

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