Properties

Base field 5.5.36497.1
Label 5.5.36497.1-9.1-a3
Conductor \((9,a^{2} - a - 2)\)
Conductor norm \( 9 \)
CM no
base-change no
Q-curve not determined
Torsion order \( 2 \)
Rank not available

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Base field 5.5.36497.1

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

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

Weierstrass equation

\( y^2 + \left(a^{2} - a - 2\right) x y + \left(a^{2} - a - 2\right) y = x^{3} + \left(a^{3} - a^{2} - 4 a + 1\right) x^{2} + \left(-a^{4} - 2 a^{3} + 6 a^{2} + 8 a - 11\right) x + 36 a^{4} - 88 a^{3} - 60 a^{2} + 206 a - 75 \)
magma: E := ChangeRing(EllipticCurve([a^2 - a - 2, a^3 - a^2 - 4*a + 1, a^2 - a - 2, -a^4 - 2*a^3 + 6*a^2 + 8*a - 11, 36*a^4 - 88*a^3 - 60*a^2 + 206*a - 75]),K);
sage: E = EllipticCurve(K, [a^2 - a - 2, a^3 - a^2 - 4*a + 1, a^2 - a - 2, -a^4 - 2*a^3 + 6*a^2 + 8*a - 11, 36*a^4 - 88*a^3 - 60*a^2 + 206*a - 75])
gp (2.8): E = ellinit([a^2 - a - 2, a^3 - a^2 - 4*a + 1, a^2 - a - 2, -a^4 - 2*a^3 + 6*a^2 + 8*a - 11, 36*a^4 - 88*a^3 - 60*a^2 + 206*a - 75],K)

This is a global minimal model.

sage: E.is_global_minimal_model()

Invariants

\(\mathfrak{N} \) = \((9,a^{2} - a - 2)\) = \( \left(a^{2} - 1\right)^{2} \)
magma: Conductor(E);
sage: E.conductor()
\(N(\mathfrak{N}) \) = \( 9 \) = \( 3^{2} \)
magma: Norm(Conductor(E));
sage: E.conductor().norm()
\(\mathfrak{D}\) = \((387420489,a + 306972940,a^{4} - 2 a^{3} - 3 a^{2} + 5 a + 323254835,a^{4} - a^{3} - 4 a^{2} + 2 a + 145780798,a^{2} - a + 69580600)\) = \( \left(a^{2} - 1\right)^{18} \)
magma: Discriminant(E);
sage: E.discriminant()
gp (2.8): E.disc
\(N(\mathfrak{D})\) = \( 387420489 \) = \( 3^{18} \)
magma: Norm(Discriminant(E));
sage: E.discriminant().norm()
gp (2.8): norm(E.disc)
\(j\) = \( \frac{679068463130598740}{531441} a^{4} - \frac{359821343885971159}{177147} a^{3} - \frac{826730295123875959}{177147} a^{2} + \frac{2377533227165763589}{531441} a + \frac{1654749596218451752}{531441} \)
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 rank and generators

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: \(\Z/2\Z\)
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]
Generator: $\left(-2 a^{4} + 4 a^{3} + 4 a^{2} - 8 a + 1 : -a^{4} + 2 a^{3} + 3 a^{2} - 5 a + 1 : 1\right)$
magma: [f(P): P in Generators(T)] where T,f:=TorsionSubgroup(E);
sage: E.torsion_subgroup().gens()
gp (2.8): elltors(E)[3]

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\((j)_{-}\)
\( \left(a^{2} - 1\right) \) 3 \(2\) \( I_{12}^* \) Additive 2 18 12

Galois Representations

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

prime Image of Galois Representation
\(2\) 2B
\(3\) 3B

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2, 3 and 6.
Its isogeny class 9.1-a consists of curves linked by isogenies of degrees dividing 6.

Base change

This curve is not the base-change of an elliptic curve defined over \(\Q\). It has not yet been determined whether or not it is a \(\Q\)-curve.