Properties

Label 6.6.703493.1-41.4-b2
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 \( 0 \)

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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

\({y}^2+\left(2a^{5}-2a^{4}-11a^{3}+10a^{2}+9a-4\right){x}{y}+\left(3a^{5}-2a^{4}-17a^{3}+10a^{2}+17a-2\right){y}={x}^{3}+\left(-a^{5}+a^{4}+5a^{3}-4a^{2}-3a-1\right){x}^{2}+\left(19a^{5}-10a^{4}-106a^{3}+47a^{2}+92a-16\right){x}+55a^{5}-29a^{4}-312a^{3}+137a^{2}+286a-40\)
sage: E = EllipticCurve([K([-4,9,10,-11,-2,2]),K([-1,-3,-4,5,1,-1]),K([-2,17,10,-17,-2,3]),K([-16,92,47,-106,-10,19]),K([-40,286,137,-312,-29,55])])
 
gp: E = ellinit([Pol(Vecrev([-4,9,10,-11,-2,2])),Pol(Vecrev([-1,-3,-4,5,1,-1])),Pol(Vecrev([-2,17,10,-17,-2,3])),Pol(Vecrev([-16,92,47,-106,-10,19])),Pol(Vecrev([-40,286,137,-312,-29,55]))], K);
 
magma: E := EllipticCurve([K![-4,9,10,-11,-2,2],K![-1,-3,-4,5,1,-1],K![-2,17,10,-17,-2,3],K![-16,92,47,-106,-10,19],K![-40,286,137,-312,-29,55]]);
 

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: \((-a^5+5a^3-5a+2)\) = \((a^5-a^4-6a^3+4a^2+7a+1)^{2}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( -1681 \) = \(-41^{2}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( -\frac{46785115689974520}{1681} a^{5} + \frac{25523470799745310}{1681} a^{4} + \frac{267495305548152369}{1681} a^{3} - \frac{124251905123044313}{1681} a^{2} - \frac{253745891578685595}{1681} a + \frac{46114837230979857}{1681} \)
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: \(\Z/2\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Torsion generator: $\left(\frac{9}{2} a^{5} - \frac{11}{4} a^{4} - \frac{105}{4} a^{3} + \frac{53}{4} a^{2} + \frac{107}{4} a - \frac{19}{4} : -\frac{23}{8} a^{5} + \frac{3}{2} a^{4} + \frac{137}{8} a^{3} - 7 a^{2} - \frac{155}{8} a + \frac{1}{8} : 1\right)$
sage: T.gens()
 
gp: T[3]
 
magma: [piT(P) : P in Generators(T)];
 

BSD invariants

Analytic rank: \( 0 \)
sage: E.rank()
 
magma: Rank(E);
 
Mordell-Weil rank: \(0\)
Regulator: \( 1 \)
Period: not available
Tamagawa product: \( 2 \)
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\((j)_{-}\)
\((a^5-a^4-6a^3+4a^2+7a+1)\) \(41\) \(2\) \(I_{2}\) Non-split multiplicative \(1\) \(1\) \(2\) \(2\)

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

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2.
Its isogeny class 41.4-b consists of curves linked by isogenies of degree 2.

Base change

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