# Properties

 Base field 5.5.65657.1 Label 5.5.65657.1-27.1-e1 Conductor $$(27,-a^{4} + a^{3} + 5 a^{2} - 3 a - 2)$$ Conductor norm $$27$$ CM no base-change no Q-curve no Torsion order $$1$$ Rank not available

# Related objects

Show commands for: Magma / SageMath / Pari/GP

## 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

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

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 $$\mathfrak{N}$$ = $$(27,-a^{4} + a^{3} + 5 a^{2} - 3 a - 2)$$ = $$\left(-a^{4} + a^{3} + 4 a^{2} - 2 a - 2\right)^{3}$$ magma: Conductor(E); sage: E.conductor() $$N(\mathfrak{N})$$ = $$27$$ = $$3^{3}$$ magma: Norm(Conductor(E)); sage: E.conductor().norm() $$\mathfrak{D}$$ = $$(177147,a + 83450,a^{4} - a^{3} - 4 a^{2} + 2 a + 162743,-a^{4} + a^{3} + 5 a^{2} - 3 a + 31318,-a^{4} + 2 a^{3} + 4 a^{2} - 6 a + 169258)$$ = $$\left(-a^{4} + a^{3} + 4 a^{2} - 2 a - 2\right)^{11}$$ magma: Discriminant(E); sage: E.discriminant() gp (2.8): E.disc $$N(\mathfrak{D})$$ = $$177147$$ = $$3^{11}$$ magma: Norm(Discriminant(E)); sage: E.discriminant().norm() gp (2.8): norm(E.disc) $$j$$ = $$124002 a^{4} - 33919 a^{3} - 523999 a^{2} - 180889 a - 13950$$ 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$$(j)_{-}$$
$$\left(-a^{4} + a^{3} + 4 a^{2} - 2 a - 2\right)$$ $$3$$ $$1$$ $$II^*$$ Additive $$-1$$ $$3$$ $$11$$ $$0$$

## Galois Representations

The mod $$p$$ Galois Representation has maximal image for all primes $$p$$ .

## Isogenies and isogeny class

This curve has no rational isogenies. Its isogeny class 27.1-e consists of this curve only.

## Base change

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