Properties

Base field 5.5.65657.1
Label 5.5.65657.1-9.1-a2
Conductor \((9,a^{4} - 2 a^{3} - 4 a^{2} + 5 a + 3)\)
Conductor norm \( 9 \)
CM no
base-change no
Q-curve no
Torsion order \( 3 \)
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

\( y^2 + \left(a^{4} - a^{3} - 4 a^{2} + 3 a + 3\right) x y + \left(-2 a^{4} + 3 a^{3} + 9 a^{2} - 8 a - 6\right) y = x^{3} + \left(-a^{4} + 5 a^{2} + 2 a - 4\right) x^{2} + \left(2 a^{4} - 2 a^{3} - 13 a^{2} + 9 a + 17\right) x - 58 a^{4} + 72 a^{3} + 271 a^{2} - 176 a - 247 \)
magma: E := ChangeRing(EllipticCurve([a^4 - a^3 - 4*a^2 + 3*a + 3, -a^4 + 5*a^2 + 2*a - 4, -2*a^4 + 3*a^3 + 9*a^2 - 8*a - 6, 2*a^4 - 2*a^3 - 13*a^2 + 9*a + 17, -58*a^4 + 72*a^3 + 271*a^2 - 176*a - 247]),K);
sage: E = EllipticCurve(K, [a^4 - a^3 - 4*a^2 + 3*a + 3, -a^4 + 5*a^2 + 2*a - 4, -2*a^4 + 3*a^3 + 9*a^2 - 8*a - 6, 2*a^4 - 2*a^3 - 13*a^2 + 9*a + 17, -58*a^4 + 72*a^3 + 271*a^2 - 176*a - 247])
gp (2.8): E = ellinit([a^4 - a^3 - 4*a^2 + 3*a + 3, -a^4 + 5*a^2 + 2*a - 4, -2*a^4 + 3*a^3 + 9*a^2 - 8*a - 6, 2*a^4 - 2*a^3 - 13*a^2 + 9*a + 17, -58*a^4 + 72*a^3 + 271*a^2 - 176*a - 247],K)

This is a global minimal model.

sage: E.is_global_minimal_model()

Invariants

\(\mathfrak{N} \) = \((9,a^{4} - 2 a^{3} - 4 a^{2} + 5 a + 3)\) = \( \left(-a^{4} + a^{3} + 4 a^{2} - 2 a - 2\right)^{2} \)
magma: Conductor(E);
sage: E.conductor()
\(N(\mathfrak{N}) \) = \( 9 \) = \( 3^{2} \)
magma: Norm(Conductor(E));
sage: E.conductor().norm()
\(\mathfrak{D}\) = \((6561,a + 4718,a^{4} - a^{3} - 4 a^{2} + 2 a + 5279,-a^{4} + a^{3} + 5 a^{2} - 3 a + 5074,-a^{4} + 2 a^{3} + 4 a^{2} - 6 a + 5233)\) = \( \left(-a^{4} + a^{3} + 4 a^{2} - 2 a - 2\right)^{8} \)
magma: Discriminant(E);
sage: E.discriminant()
gp (2.8): E.disc
\(N(\mathfrak{D})\) = \( 6561 \) = \( 3^{8} \)
magma: Norm(Discriminant(E));
sage: E.discriminant().norm()
gp (2.8): norm(E.disc)
\(j\) = \( \frac{463637}{9} a^{4} - \frac{90008}{3} a^{3} - \frac{1905559}{9} a^{2} - 8065 a + \frac{82504}{9} \)
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: \(\Z/3\Z\)
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: $\left(3 a^{4} - 4 a^{3} - 13 a^{2} + 8 a + 12 : 4 a^{4} - 4 a^{3} - 21 a^{2} + 13 a + 18 : 1\right)$
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 Root number ord(\(\mathfrak{N}\)) ord(\(\mathfrak{D}\)) ord\((j)_{-}\)
\( \left(-a^{4} + a^{3} + 4 a^{2} - 2 a - 2\right) \) \(3\) \(2\) \(I_{2}^*\) Additive \(-1\) \(2\) \(8\) \(2\)

Galois Representations

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

prime Image of Galois Representation
\(3\) 3B.1.1

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 3.
Its isogeny class 9.1-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.