Properties

Label 5.5.14641.1-43.2-b1
Base field \(\Q(\zeta_{11})^+\)
Conductor norm \( 43 \)
CM no
Base change no
Q-curve no
Torsion order \( 5 \)
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(Polrev([-1, 3, 3, -4, -1, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 3, 3, -4, -1, 1]);
 

Weierstrass equation

\({y}^2+\left(a^{4}-3a^{2}+1\right){x}{y}+\left(a^{4}+a^{3}-3a^{2}-2a+1\right){y}={x}^{3}+\left(-a^{4}-a^{3}+3a^{2}+4a+1\right){x}^{2}+\left(-6a^{4}-a^{3}+20a^{2}+4a-8\right){x}-6a^{4}+a^{3}+23a^{2}-3a-15\)
sage: E = EllipticCurve([K([1,0,-3,0,1]),K([1,4,3,-1,-1]),K([1,-2,-3,1,1]),K([-8,4,20,-1,-6]),K([-15,-3,23,1,-6])])
 
gp: E = ellinit([Polrev([1,0,-3,0,1]),Polrev([1,4,3,-1,-1]),Polrev([1,-2,-3,1,1]),Polrev([-8,4,20,-1,-6]),Polrev([-15,-3,23,1,-6])], K);
 
magma: E := EllipticCurve([K![1,0,-3,0,1],K![1,4,3,-1,-1],K![1,-2,-3,1,1],K![-8,4,20,-1,-6],K![-15,-3,23,1,-6]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((-a^4+2a^2+a+1)\) = \((-a^4+2a^2+a+1)\)
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: \((-10a^4-25a^3+43a^2+80a-55)\) = \((-a^4+2a^2+a+1)^{5}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( -147008443 \) = \(-43^{5}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{245417473145}{147008443} a^{4} - \frac{392753447771}{147008443} a^{3} - \frac{61634765294}{3418801} a^{2} - \frac{2240734555627}{147008443} a - \frac{148495909040}{147008443} \)
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: \(\Z/5\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Torsion generator: $\left(-a^{3} + 3 a + 2 : a^{4} + a^{3} - 3 a^{2} - 6 a - 3 : 1\right)$
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: \( 537.84618223957766282372374176009879432 \)
Tamagawa product: \( 5 \)
Torsion order: \(5\)
Leading coefficient: \( 0.889001954 \)
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\((j)_{-}\)
\((-a^4+2a^2+a+1)\) \(43\) \(5\) \(I_{5}\) Split multiplicative \(-1\) \(1\) \(5\) \(5\)

Galois Representations

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

prime Image of Galois Representation
\(5\) 5B.1.1

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 5.
Its isogeny class 43.2-b consists of curves linked by isogenies of degree 5.

Base change

This elliptic curve is not a \(\Q\)-curve.

It is not the base change of an elliptic curve defined over any subfield.