# Properties

 Base field 5.5.65657.1 Label 5.5.65657.1-27.1-a1 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 / Pari/GP / SageMath

## 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 $$1$$.

sage: x = polygen(QQ); K.<a> = NumberField(x^5 - x^4 - 5*x^3 + 2*x^2 + 5*x + 1)

gp: K = nfinit(a^5 - a^4 - 5*a^3 + 2*a^2 + 5*a + 1);

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 5, 2, -5, -1, 1]);

## Weierstrass equation

$$y^2 + \left(a^{3} - 4 a\right) x y + \left(-2 a^{4} + 3 a^{3} + 9 a^{2} - 8 a - 7\right) y = x^{3} + \left(2 a^{4} - 3 a^{3} - 9 a^{2} + 10 a + 6\right) x^{2} + \left(2 a^{4} - a^{3} - 15 a^{2} + 10 a + 16\right) x + a^{3} - 4 a^{2} + 4 a + 3$$
sage: E = EllipticCurve(K, [a^3 - 4*a, 2*a^4 - 3*a^3 - 9*a^2 + 10*a + 6, -2*a^4 + 3*a^3 + 9*a^2 - 8*a - 7, 2*a^4 - a^3 - 15*a^2 + 10*a + 16, a^3 - 4*a^2 + 4*a + 3])

gp: E = ellinit([a^3 - 4*a, 2*a^4 - 3*a^3 - 9*a^2 + 10*a + 6, -2*a^4 + 3*a^3 + 9*a^2 - 8*a - 7, 2*a^4 - a^3 - 15*a^2 + 10*a + 16, a^3 - 4*a^2 + 4*a + 3],K)

magma: E := ChangeRing(EllipticCurve([a^3 - 4*a, 2*a^4 - 3*a^3 - 9*a^2 + 10*a + 6, -2*a^4 + 3*a^3 + 9*a^2 - 8*a - 7, 2*a^4 - a^3 - 15*a^2 + 10*a + 16, a^3 - 4*a^2 + 4*a + 3]),K);

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 $$\mathfrak{N}$$ = $$(27,-a^{4} + a^{3} + 5 a^{2} - 3 a - 2)$$ = $$\left(-a^{4} + a^{3} + 4 a^{2} - 2 a - 2\right)^{3}$$ sage: E.conductor()  magma: Conductor(E); $$N(\mathfrak{N})$$ = $$27$$ = $$3^{3}$$ sage: E.conductor().norm()  magma: Norm(Conductor(E)); $$\mathfrak{D}$$ = $$(177147,a + 83450,a^{4} - a^{3} - 4 a^{2} + 2 a + 162743,-a^{4} + a^{3} + 5 a^{2} - 3 a + 31318,-a^{4} + 2 a^{3} + 4 a^{2} - 6 a + 169258)$$ = $$\left(-a^{4} + a^{3} + 4 a^{2} - 2 a - 2\right)^{11}$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); $$N(\mathfrak{D})$$ = $$177147$$ = $$3^{11}$$ sage: E.discriminant().norm()  gp: norm(E.disc)  magma: Norm(Discriminant(E)); $$j$$ = $$-169969542 a^{4} + 461869972 a^{3} + 56647563 a^{2} - 437225155 a - 98969253$$ sage: E.j_invariant()  gp: E.j  magma: jInvariant(E); $$\text{End} (E)$$ = $$\Z$$ (no Complex Multiplication ) sage: E.has_cm(), E.cm_discriminant()  magma: HasComplexMultiplication(E); $$\text{ST} (E)$$ = $\mathrm{SU}(2)$

## Mordell-Weil group

Rank not available.

sage: E.rank()

magma: Rank(E);

Regulator: not available

sage: gens = E.gens(); gens

magma: gens := [P:P in Generators(E)|Order(P) eq 0]; gens;

sage: E.regulator_of_points(gens)

magma: Regulator(gens);

## Torsion subgroup

Structure: Trivial sage: T = E.torsion_subgroup(); T.invariants()  gp: T = elltors(E); T[2]  magma: T,piT := TorsionSubgroup(E); Invariants(T);

## Local data at primes of bad reduction

sage: E.local_data()

magma: LocalInformation(E);

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 $$p < 1000$$ .

## Isogenies and isogeny class

This curve has no rational isogenies. Its isogeny class 27.1-a 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.