Properties

Base field \(\Q(\zeta_{11})^+\)
Label 5.5.14641.1-43.2-b1
Conductor \((43,a^{4} - 2 a^{2} - a - 1)\)
Conductor norm \( 43 \)
CM no
base-change no
Q-curve no
Torsion order \( 5 \)
Rank not available

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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: x = polygen(QQ); K.<a> = NumberField(x^5 - x^4 - 4*x^3 + 3*x^2 + 3*x - 1)
 
gp: K = nfinit(a^5 - a^4 - 4*a^3 + 3*a^2 + 3*a - 1);
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 3, 3, -4, -1, 1]);
 

Weierstrass equation

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

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

\(\mathfrak{N} \) = \((43,a^{4} - 2 a^{2} - a - 1)\) = \( \left(-a^{4} + 2 a^{2} + a + 1\right) \)
sage: E.conductor()
 
magma: Conductor(E);
 
\(N(\mathfrak{N}) \) = \( 43 \) = \( 43 \)
sage: E.conductor().norm()
 
magma: Norm(Conductor(E));
 
\(\mathfrak{D}\) = \((147008443,a + 137461936,a^{2} + 30218599,a^{3} - 3 a + 77402164,a^{4} - 4 a^{2} + 119504652)\) = \( \left(-a^{4} + 2 a^{2} + a + 1\right)^{5} \)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
\(N(\mathfrak{D})\) = \( 147008443 \) = \( 43^{5} \)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
\(j\) = \( \frac{245417473145}{147008443} a^{4} - \frac{392753447771}{147008443} a^{3} - \frac{61634765294}{3418801} a^{2} - \frac{2240734555627}{147008443} a - \frac{148495909040}{147008443} \)
sage: E.j_invariant()
 
gp: E.j
 
magma: jInvariant(E);
 
\( \text{End} (E) \) = \(\Z\)   (no Complex Multiplication )
sage: E.has_cm(), E.cm_discriminant()
 
magma: HasComplexMultiplication(E);
 
\( \text{ST} (E) \) = $\mathrm{SU}(2)$

Mordell-Weil group

Rank not available.

sage: E.rank()
 
magma: Rank(E);
 

Regulator: not available

sage: gens = E.gens(); gens
 
magma: gens := [P:P in Generators(E)|Order(P) eq 0]; gens;
 
sage: E.regulator_of_points(gens)
 
magma: Regulator(gens);
 

Torsion subgroup

Structure: \(\Z/5\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
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)];
 

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\((j)_{-}\)
\( \left(-a^{4} + 2 a^{2} + a + 1\right) \) \(43\) \(5\) \(I_{5}\) 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 curve is not the base-change of an elliptic curve defined over \(\Q\). It is not a \(\Q\)-curve.