# Properties

 Base field 6.6.434581.1 Label 6.6.434581.1-27.2-a1 Conductor $$(3,-2 a^{5} + 5 a^{4} + 5 a^{3} - 12 a^{2} - a + 5)$$ 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 6.6.434581.1

Generator $$a$$, with minimal polynomial $$x^{6} - 2 x^{5} - 4 x^{4} + 5 x^{3} + 4 x^{2} - 2 x - 1$$; class number $$1$$.

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, -2, 4, 5, -4, -2, 1]);

sage: x = polygen(QQ); K.<a> = NumberField(x^6 - 2*x^5 - 4*x^4 + 5*x^3 + 4*x^2 - 2*x - 1)

gp: K = nfinit(a^6 - 2*a^5 - 4*a^4 + 5*a^3 + 4*a^2 - 2*a - 1);

## Weierstrass equation

$$y^2 + \left(a + 1\right) x y + \left(a^{5} - 2 a^{4} - 4 a^{3} + 5 a^{2} + 4 a - 1\right) y = x^{3} + \left(a^{5} - 4 a^{4} + 12 a^{2} - 3 a - 4\right) x^{2} + \left(-2 a^{5} + 3 a^{4} + 8 a^{3} - 6 a - 1\right) x - 3 a^{5} + 6 a^{4} + 10 a^{3} - 10 a^{2} - 3 a$$
magma: E := ChangeRing(EllipticCurve([a + 1, a^5 - 4*a^4 + 12*a^2 - 3*a - 4, a^5 - 2*a^4 - 4*a^3 + 5*a^2 + 4*a - 1, -2*a^5 + 3*a^4 + 8*a^3 - 6*a - 1, -3*a^5 + 6*a^4 + 10*a^3 - 10*a^2 - 3*a]),K);

sage: E = EllipticCurve(K, [a + 1, a^5 - 4*a^4 + 12*a^2 - 3*a - 4, a^5 - 2*a^4 - 4*a^3 + 5*a^2 + 4*a - 1, -2*a^5 + 3*a^4 + 8*a^3 - 6*a - 1, -3*a^5 + 6*a^4 + 10*a^3 - 10*a^2 - 3*a])

gp: E = ellinit([a + 1, a^5 - 4*a^4 + 12*a^2 - 3*a - 4, a^5 - 2*a^4 - 4*a^3 + 5*a^2 + 4*a - 1, -2*a^5 + 3*a^4 + 8*a^3 - 6*a - 1, -3*a^5 + 6*a^4 + 10*a^3 - 10*a^2 - 3*a],K)

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 $$\mathfrak{N}$$ = $$(3,-2 a^{5} + 5 a^{4} + 5 a^{3} - 12 a^{2} - a + 5)$$ = $$\left(-2 a^{5} + 4 a^{4} + 7 a^{3} - 9 a^{2} - 3 a + 2\right)$$ magma: Conductor(E);  sage: E.conductor() $$N(\mathfrak{N})$$ = $$27$$ = $$27$$ magma: Norm(Conductor(E));  sage: E.conductor().norm() $$\mathfrak{D}$$ = $$(27,27 a^{4} - 54 a^{3} - 81 a^{2} + 81 a + 27,27 a^{5} - 81 a^{4} - 54 a^{3} + 216 a^{2} - 81,12 a^{5} - 10 a^{4} - 76 a^{3} + 18 a^{2} + 79 a,15 a^{5} - 34 a^{4} - 51 a^{3} + 85 a^{2} + 31 a - 19,17 a^{5} - 37 a^{4} - 61 a^{3} + 93 a^{2} + 39 a - 32)$$ = $$\left(-2 a^{5} + 4 a^{4} + 7 a^{3} - 9 a^{2} - 3 a + 2\right)^{3}$$ magma: Discriminant(E);  sage: E.discriminant()  gp: E.disc $$N(\mathfrak{D})$$ = $$19683$$ = $$27^{3}$$ magma: Norm(Discriminant(E));  sage: E.discriminant().norm()  gp: norm(E.disc) $$j$$ = $$\frac{1286151805}{27} a^{5} - \frac{97850411}{3} a^{4} - \frac{6302966594}{27} a^{3} - \frac{1859319866}{27} a^{2} + \frac{2699241622}{27} a + \frac{977886878}{27}$$ magma: jInvariant(E);  sage: E.j_invariant()  gp: 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()

Regulator: not available

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

sage: gens = E.gens(); gens

magma: Regulator(gens);

sage: E.regulator_of_points(gens)

## Torsion subgroup

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

## 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$$(j)_{-}$$
$$\left(-2 a^{5} + 4 a^{4} + 7 a^{3} - 9 a^{2} - 3 a + 2\right)$$ $$27$$ $$1$$ $$I_{3}$$ Non-split multiplicative $$1$$ $$1$$ $$3$$ $$3$$

## Galois Representations

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

prime Image of Galois Representation
$$3$$ 3B

## Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 3.
Its isogeny class 27.2-a consists of curves linked by isogenies of degree 3.

## Base change

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