Properties

Base field 5.5.65657.1
Label 5.5.65657.1-45.1-a2
Conductor \((45,-a^{3} + 4 a)\)
Conductor norm \( 45 \)
CM no
base-change no
Q-curve yes
Torsion order \( 2 \)
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

\( y^2 + \left(-2 a^{4} + 3 a^{3} + 9 a^{2} - 9 a - 6\right) x y + \left(-a^{4} + 2 a^{3} + 5 a^{2} - 6 a - 5\right) y = x^{3} + \left(a^{4} - a^{3} - 5 a^{2} + 4 a + 3\right) x^{2} + \left(16 a^{4} - 20 a^{3} - 76 a^{2} + 58 a + 56\right) x - 11 a^{4} + 10 a^{3} + 56 a^{2} - 13 a - 77 \)
magma: E := ChangeRing(EllipticCurve([-2*a^4 + 3*a^3 + 9*a^2 - 9*a - 6, a^4 - a^3 - 5*a^2 + 4*a + 3, -a^4 + 2*a^3 + 5*a^2 - 6*a - 5, 16*a^4 - 20*a^3 - 76*a^2 + 58*a + 56, -11*a^4 + 10*a^3 + 56*a^2 - 13*a - 77]),K);
sage: E = EllipticCurve(K, [-2*a^4 + 3*a^3 + 9*a^2 - 9*a - 6, a^4 - a^3 - 5*a^2 + 4*a + 3, -a^4 + 2*a^3 + 5*a^2 - 6*a - 5, 16*a^4 - 20*a^3 - 76*a^2 + 58*a + 56, -11*a^4 + 10*a^3 + 56*a^2 - 13*a - 77])
gp (2.8): E = ellinit([-2*a^4 + 3*a^3 + 9*a^2 - 9*a - 6, a^4 - a^3 - 5*a^2 + 4*a + 3, -a^4 + 2*a^3 + 5*a^2 - 6*a - 5, 16*a^4 - 20*a^3 - 76*a^2 + 58*a + 56, -11*a^4 + 10*a^3 + 56*a^2 - 13*a - 77],K)

This is a global minimal model.

sage: E.is_global_minimal_model()

Invariants

\(\mathfrak{N} \) = \((45,-a^{3} + 4 a)\) = \( \left(-a^{4} + a^{3} + 4 a^{2} - 2 a - 2\right)^{2} \cdot \left(-a^{2} + a + 2\right) \)
magma: Conductor(E);
sage: E.conductor()
\(N(\mathfrak{N}) \) = \( 45 \) = \( 3^{2} \cdot 5 \)
magma: Norm(Conductor(E));
sage: E.conductor().norm()
\(\mathfrak{D}\) = \((675,a + 668,a^{4} - a^{3} - 4 a^{2} + 2 a + 149,-a^{4} + a^{3} + 5 a^{2} - 3 a + 484,-a^{4} + 2 a^{3} + 4 a^{2} - 6 a + 211)\) = \( \left(-a^{4} + a^{3} + 4 a^{2} - 2 a - 2\right)^{3} \cdot \left(-a^{2} + a + 2\right)^{2} \)
magma: Discriminant(E);
sage: E.discriminant()
gp (2.8): E.disc
\(N(\mathfrak{D})\) = \( 675 \) = \( 3^{3} \cdot 5^{2} \)
magma: Norm(Discriminant(E));
sage: E.discriminant().norm()
gp (2.8): norm(E.disc)
\(j\) = \( \frac{11314724189}{25} a^{4} - \frac{30736776391}{25} a^{3} - \frac{3562664832}{25} a^{2} + \frac{29192288229}{25} a + \frac{6597394298}{25} \)
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: \(\Z/2\Z\)
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: $\left(\frac{5}{4} a^{4} - \frac{7}{4} a^{3} - \frac{11}{2} a^{2} + 5 a + \frac{11}{4} : -\frac{1}{2} a^{4} + a^{3} + \frac{9}{8} a^{2} - \frac{15}{4} a + 3 : 1\right)$
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\((j)_{-}\)
\( \left(-a^{4} + a^{3} + 4 a^{2} - 2 a - 2\right) \) \(3\) \(2\) \( III \) Additive \(2\) \(3\) \(0\)
\( \left(-a^{2} + a + 2\right) \) \(5\) \(2\) \( I_{2} \) Non-split multiplicative \(1\) \(2\) \(2\)

Galois Representations

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

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

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2.
Its isogeny class 45.1-a consists of curves linked by isogenies of degree2.

Base change

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