Properties

Base field 5.5.65657.1
Label 5.5.65657.1-27.1-e1
Conductor \((27,-a^{4} + a^{3} + 5 a^{2} - 3 a - 2)\)
Conductor norm \( 27 \)
CM no
base-change no
Q-curve yes
Torsion order \( 1 \)
Rank not available

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Base field 5.5.65657.1

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

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 5, 2, -5, -1, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^5 - x^4 - 5*x^3 + 2*x^2 + 5*x + 1)
gp (2.8): K = nfinit(a^5 - a^4 - 5*a^3 + 2*a^2 + 5*a + 1);

Weierstrass equation

\( y^2 + \left(-a^{4} + a^{3} + 5 a^{2} - 2 a - 3\right) x y + \left(-a^{4} + 2 a^{3} + 5 a^{2} - 6 a - 4\right) y = x^{3} + \left(-a^{4} + 2 a^{3} + 3 a^{2} - 4 a\right) x^{2} + \left(2 a^{4} - 8 a^{3} + 5 a^{2} + 6 a + 1\right) x - 2 a^{4} + 5 a^{3} + a^{2} - 5 a - 1 \)
magma: E := ChangeRing(EllipticCurve([-a^4 + a^3 + 5*a^2 - 2*a - 3, -a^4 + 2*a^3 + 3*a^2 - 4*a, -a^4 + 2*a^3 + 5*a^2 - 6*a - 4, 2*a^4 - 8*a^3 + 5*a^2 + 6*a + 1, -2*a^4 + 5*a^3 + a^2 - 5*a - 1]),K);
sage: E = EllipticCurve(K, [-a^4 + a^3 + 5*a^2 - 2*a - 3, -a^4 + 2*a^3 + 3*a^2 - 4*a, -a^4 + 2*a^3 + 5*a^2 - 6*a - 4, 2*a^4 - 8*a^3 + 5*a^2 + 6*a + 1, -2*a^4 + 5*a^3 + a^2 - 5*a - 1])
gp (2.8): E = ellinit([-a^4 + a^3 + 5*a^2 - 2*a - 3, -a^4 + 2*a^3 + 3*a^2 - 4*a, -a^4 + 2*a^3 + 5*a^2 - 6*a - 4, 2*a^4 - 8*a^3 + 5*a^2 + 6*a + 1, -2*a^4 + 5*a^3 + a^2 - 5*a - 1],K)

This is a global minimal model.

sage: E.is_global_minimal_model()

Invariants

\(\mathfrak{N} \) = \((27,-a^{4} + a^{3} + 5 a^{2} - 3 a - 2)\) = \( \left(-a^{4} + a^{3} + 4 a^{2} - 2 a - 2\right)^{3} \)
magma: Conductor(E);
sage: E.conductor()
\(N(\mathfrak{N}) \) = \( 27 \) = \( 3^{3} \)
magma: Norm(Conductor(E));
sage: E.conductor().norm()
\(\mathfrak{D}\) = \((177147,a + 83450,a^{4} - a^{3} - 4 a^{2} + 2 a + 162743,-a^{4} + a^{3} + 5 a^{2} - 3 a + 31318,-a^{4} + 2 a^{3} + 4 a^{2} - 6 a + 169258)\) = \( \left(-a^{4} + a^{3} + 4 a^{2} - 2 a - 2\right)^{11} \)
magma: Discriminant(E);
sage: E.discriminant()
gp (2.8): E.disc
\(N(\mathfrak{D})\) = \( 177147 \) = \( 3^{11} \)
magma: Norm(Discriminant(E));
sage: E.discriminant().norm()
gp (2.8): norm(E.disc)
\(j\) = \( 124002 a^{4} - 33919 a^{3} - 523999 a^{2} - 180889 a - 13950 \)
magma: jInvariant(E);
sage: E.j_invariant()
gp (2.8): E.j
\( \text{End} (E) \) = \(\Z\)   (no Complex Multiplication )
magma: HasComplexMultiplication(E);
sage: E.has_cm(), E.cm_discriminant()
\( \text{ST} (E) \) = $\mathrm{SU}(2)$

Mordell-Weil group

Rank not available.
magma: Rank(E);
sage: E.rank()
magma: Generators(E); // includes torsion
sage: E.gens()

Regulator: not available

magma: Regulator(Generators(E));
sage: E.regulator_of_points(E.gens())

Torsion subgroup

Structure: Trivial
magma: TorsionSubgroup(E);
sage: E.torsion_subgroup().gens()
gp (2.8): elltors(E)[2]
magma: Order(TorsionSubgroup(E));
sage: E.torsion_order()
gp (2.8): elltors(E)[1]

Local data at primes of bad reduction

magma: LocalInformation(E);
sage: E.local_data()
prime Norm Tamagawa number Kodaira symbol Reduction type ord(\(\mathfrak{N}\)) ord(\(\mathfrak{D}\)) ord\((j)_{-}\)
\( \left(-a^{4} + a^{3} + 4 a^{2} - 2 a - 2\right) \) \(3\) \(1\) \( II^* \) Additive \(3\) \(11\) \(0\)

Galois Representations

The mod \( p \) Galois Representation has maximal image for all primes \( p \) .

Isogenies and isogeny class

This curve has no rational isogenies. Its isogeny class 27.1-e consists of this curve only.

Base change

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