Properties

Label 5.5.14641.1-43.1-a1
Base field \(\Q(\zeta_{11})^+\)
Conductor \((-a^3-a^2+2a+3)\)
Conductor norm \( 43 \)
CM no
Base change no
Q-curve no
Torsion order \( 7 \)
Rank \( 0 \)

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Base field \(\Q(\zeta_{11})^+\)

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

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-1, 3, 3, -4, -1, 1]))
 
gp: K = nfinit(Pol(Vecrev([-1, 3, 3, -4, -1, 1])));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 3, 3, -4, -1, 1]);
 

Weierstrass equation

\({y}^2+\left(a^{3}+a^{2}-3a-1\right){x}{y}+\left(a^{2}-2\right){y}={x}^{3}+\left(-a^{3}+a^{2}+2a-2\right){x}^{2}+\left(-a^{4}+3a^{2}-2a-1\right){x}+a^{4}-3a^{2}-a\)
sage: E = EllipticCurve([K([-1,-3,1,1,0]),K([-2,2,1,-1,0]),K([-2,0,1,0,0]),K([-1,-2,3,0,-1]),K([0,-1,-3,0,1])])
 
gp: E = ellinit([Pol(Vecrev([-1,-3,1,1,0])),Pol(Vecrev([-2,2,1,-1,0])),Pol(Vecrev([-2,0,1,0,0])),Pol(Vecrev([-1,-2,3,0,-1])),Pol(Vecrev([0,-1,-3,0,1]))], K);
 
magma: E := EllipticCurve([K![-1,-3,1,1,0],K![-2,2,1,-1,0],K![-2,0,1,0,0],K![-1,-2,3,0,-1],K![0,-1,-3,0,1]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((-a^3-a^2+2a+3)\) = \((-a^3-a^2+2a+3)\)
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: \((a^3-2a^2-2a+2)\) = \((-a^3-a^2+2a+3)\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( -43 \) = \(-43\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( -\frac{166996603}{43} a^{4} - \frac{153441602}{43} a^{3} + \frac{373478096}{43} a^{2} + \frac{215672423}{43} a - \frac{87008899}{43} \)
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: \(\Z/7\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Torsion generator: $\left(-a^{4} + a^{3} + 3 a^{2} - 2 a : a^{4} - 2 a^{3} - 3 a^{2} + 5 a : 1\right)$
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: \( 5702.69397717770 \)
Tamagawa product: \( 1 \)
Torsion order: \(7\)
Leading coefficient: \( 0.961830658994384 \)
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\((j)_{-}\)
\((-a^3-a^2+2a+3)\) \(43\) \(1\) \(I_{1}\) 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
\(7\) 7B.1.1

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 7.
Its isogeny class 43.1-a consists of curves linked by isogenies of degree 7.

Base change

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