Properties

Base field 6.6.434581.1
Label 6.6.434581.1-27.2-a1
Conductor \((3,-2 a^{5} + 5 a^{4} + 5 a^{3} - 12 a^{2} - a + 5)\)
Conductor norm \( 27 \)
CM no
base-change no
Q-curve no
Torsion order \( 1 \)
Rank not available

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Base field 6.6.434581.1

Generator \(a\), with minimal polynomial \( x^{6} - 2 x^{5} - 4 x^{4} + 5 x^{3} + 4 x^{2} - 2 x - 1 \); class number \(1\).

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, -2, 4, 5, -4, -2, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^6 - 2*x^5 - 4*x^4 + 5*x^3 + 4*x^2 - 2*x - 1)
gp (2.8): K = nfinit(a^6 - 2*a^5 - 4*a^4 + 5*a^3 + 4*a^2 - 2*a - 1);

Weierstrass equation

\( y^2 + \left(a + 1\right) x y + \left(a^{5} - 2 a^{4} - 4 a^{3} + 5 a^{2} + 4 a - 1\right) y = x^{3} + \left(a^{5} - 4 a^{4} + 12 a^{2} - 3 a - 4\right) x^{2} + \left(-2 a^{5} + 3 a^{4} + 8 a^{3} - 6 a - 1\right) x - 3 a^{5} + 6 a^{4} + 10 a^{3} - 10 a^{2} - 3 a \)
magma: E := ChangeRing(EllipticCurve([a + 1, a^5 - 4*a^4 + 12*a^2 - 3*a - 4, a^5 - 2*a^4 - 4*a^3 + 5*a^2 + 4*a - 1, -2*a^5 + 3*a^4 + 8*a^3 - 6*a - 1, -3*a^5 + 6*a^4 + 10*a^3 - 10*a^2 - 3*a]),K);
sage: E = EllipticCurve(K, [a + 1, a^5 - 4*a^4 + 12*a^2 - 3*a - 4, a^5 - 2*a^4 - 4*a^3 + 5*a^2 + 4*a - 1, -2*a^5 + 3*a^4 + 8*a^3 - 6*a - 1, -3*a^5 + 6*a^4 + 10*a^3 - 10*a^2 - 3*a])
gp (2.8): E = ellinit([a + 1, a^5 - 4*a^4 + 12*a^2 - 3*a - 4, a^5 - 2*a^4 - 4*a^3 + 5*a^2 + 4*a - 1, -2*a^5 + 3*a^4 + 8*a^3 - 6*a - 1, -3*a^5 + 6*a^4 + 10*a^3 - 10*a^2 - 3*a],K)

This is a global minimal model.

sage: E.is_global_minimal_model()

Invariants

\(\mathfrak{N} \) = \((3,-2 a^{5} + 5 a^{4} + 5 a^{3} - 12 a^{2} - a + 5)\) = \( \left(-2 a^{5} + 4 a^{4} + 7 a^{3} - 9 a^{2} - 3 a + 2\right) \)
magma: Conductor(E);
sage: E.conductor()
\(N(\mathfrak{N}) \) = \( 27 \) = \( 27 \)
magma: Norm(Conductor(E));
sage: E.conductor().norm()
\(\mathfrak{D}\) = \((27,27 a^{4} - 54 a^{3} - 81 a^{2} + 81 a + 27,27 a^{5} - 81 a^{4} - 54 a^{3} + 216 a^{2} - 81,12 a^{5} - 10 a^{4} - 76 a^{3} + 18 a^{2} + 79 a,15 a^{5} - 34 a^{4} - 51 a^{3} + 85 a^{2} + 31 a - 19,17 a^{5} - 37 a^{4} - 61 a^{3} + 93 a^{2} + 39 a - 32)\) = \( \left(-2 a^{5} + 4 a^{4} + 7 a^{3} - 9 a^{2} - 3 a + 2\right)^{3} \)
magma: Discriminant(E);
sage: E.discriminant()
gp (2.8): E.disc
\(N(\mathfrak{D})\) = \( 19683 \) = \( 27^{3} \)
magma: Norm(Discriminant(E));
sage: E.discriminant().norm()
gp (2.8): norm(E.disc)
\(j\) = \( \frac{1286151805}{27} a^{5} - \frac{97850411}{3} a^{4} - \frac{6302966594}{27} a^{3} - \frac{1859319866}{27} a^{2} + \frac{2699241622}{27} a + \frac{977886878}{27} \)
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 Root number ord(\(\mathfrak{N}\)) ord(\(\mathfrak{D}\)) ord\((j)_{-}\)
\( \left(-2 a^{5} + 4 a^{4} + 7 a^{3} - 9 a^{2} - 3 a + 2\right) \) \(27\) \(1\) \(I_{3}\) Non-split multiplicative \(1\) \(1\) \(3\) \(3\)

Galois Representations

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

prime Image of Galois Representation
\(3\) 3B

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 3.
Its isogeny class 27.2-a consists of curves linked by isogenies of degree3.

Base change

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