Properties

Base field 5.5.65657.1
Label 5.5.65657.1-9.1-a1
Conductor \((9,a^{4} - 2 a^{3} - 4 a^{2} + 5 a + 3)\)
Conductor norm \( 9 \)
CM no
base-change no
Q-curve not determined
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

\( y^2 + \left(-a^{4} + 2 a^{3} + 4 a^{2} - 6 a - 2\right) x y + \left(-a^{4} + 2 a^{3} + 4 a^{2} - 5 a - 3\right) y = x^{3} + \left(-2 a^{4} + 2 a^{3} + 9 a^{2} - 6 a - 6\right) x^{2} + \left(-45 a^{4} + 79 a^{3} + 158 a^{2} - 211 a - 53\right) x - 242 a^{4} + 430 a^{3} + 867 a^{2} - 1149 a - 312 \)
magma: E := ChangeRing(EllipticCurve([-a^4 + 2*a^3 + 4*a^2 - 6*a - 2, -2*a^4 + 2*a^3 + 9*a^2 - 6*a - 6, -a^4 + 2*a^3 + 4*a^2 - 5*a - 3, -45*a^4 + 79*a^3 + 158*a^2 - 211*a - 53, -242*a^4 + 430*a^3 + 867*a^2 - 1149*a - 312]),K);
sage: E = EllipticCurve(K, [-a^4 + 2*a^3 + 4*a^2 - 6*a - 2, -2*a^4 + 2*a^3 + 9*a^2 - 6*a - 6, -a^4 + 2*a^3 + 4*a^2 - 5*a - 3, -45*a^4 + 79*a^3 + 158*a^2 - 211*a - 53, -242*a^4 + 430*a^3 + 867*a^2 - 1149*a - 312])
gp (2.8): E = ellinit([-a^4 + 2*a^3 + 4*a^2 - 6*a - 2, -2*a^4 + 2*a^3 + 9*a^2 - 6*a - 6, -a^4 + 2*a^3 + 4*a^2 - 5*a - 3, -45*a^4 + 79*a^3 + 158*a^2 - 211*a - 53, -242*a^4 + 430*a^3 + 867*a^2 - 1149*a - 312],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()
\(\mathfrak{D}\) = \((531441,a + 260597,a^{4} - a^{3} - 4 a^{2} + 2 a + 339890,-a^{4} + a^{3} + 5 a^{2} - 3 a + 385612,-a^{4} + 2 a^{3} + 4 a^{2} - 6 a + 346405)\) = \( \left(-a^{4} + a^{3} + 4 a^{2} - 2 a - 2\right)^{12} \)
magma: Discriminant(E);
sage: E.discriminant()
gp (2.8): E.disc
\(N(\mathfrak{D})\) = \( 531441 \) = \( 3^{12} \)
magma: Norm(Discriminant(E));
sage: E.discriminant().norm()
gp (2.8): norm(E.disc)
\(j\) = \( -\frac{346538649913}{729} a^{4} + \frac{304213159354}{243} a^{3} + \frac{131211798509}{729} a^{2} - \frac{95505911702}{81} a - \frac{199860865145}{729} \)
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: 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 \(2\) \( I_{6}^* \) Additive 2 12 6

Galois Representations

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

prime Image of Galois Representation
\(3\) 3B.1.2

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 has not yet been determined whether or not it is a \(\Q\)-curve.