# Properties

 Base field 6.6.434581.1 Label 6.6.434581.1-43.2-a1 Conductor $$(43,a^{4} - a^{3} - 5 a^{2} + 4)$$ Conductor norm $$43$$ 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 (2.8): K = nfinit(a^6 - 2*a^5 - 4*a^4 + 5*a^3 + 4*a^2 - 2*a - 1);

## Weierstrass equation

$$y^2 + \left(a^{5} - a^{4} - 5 a^{3} + a^{2} + 3 a\right) x y + \left(2 a^{5} - 5 a^{4} - 5 a^{3} + 12 a^{2} - 3\right) y = x^{3} + \left(a^{5} - 4 a^{4} + 11 a^{2} - 3 a - 4\right) x^{2} + \left(-3 a^{5} + 2 a^{4} + 16 a^{3} + 2 a^{2} - 9 a - 1\right) x + 9 a^{5} - 24 a^{4} - 27 a^{3} + 71 a^{2} + 19 a - 34$$
magma: E := ChangeRing(EllipticCurve([a^5 - a^4 - 5*a^3 + a^2 + 3*a, a^5 - 4*a^4 + 11*a^2 - 3*a - 4, 2*a^5 - 5*a^4 - 5*a^3 + 12*a^2 - 3, -3*a^5 + 2*a^4 + 16*a^3 + 2*a^2 - 9*a - 1, 9*a^5 - 24*a^4 - 27*a^3 + 71*a^2 + 19*a - 34]),K);
sage: E = EllipticCurve(K, [a^5 - a^4 - 5*a^3 + a^2 + 3*a, a^5 - 4*a^4 + 11*a^2 - 3*a - 4, 2*a^5 - 5*a^4 - 5*a^3 + 12*a^2 - 3, -3*a^5 + 2*a^4 + 16*a^3 + 2*a^2 - 9*a - 1, 9*a^5 - 24*a^4 - 27*a^3 + 71*a^2 + 19*a - 34])
gp (2.8): E = ellinit([a^5 - a^4 - 5*a^3 + a^2 + 3*a, a^5 - 4*a^4 + 11*a^2 - 3*a - 4, 2*a^5 - 5*a^4 - 5*a^3 + 12*a^2 - 3, -3*a^5 + 2*a^4 + 16*a^3 + 2*a^2 - 9*a - 1, 9*a^5 - 24*a^4 - 27*a^3 + 71*a^2 + 19*a - 34],K)

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 $$\mathfrak{N}$$ = $$(43,a^{4} - a^{3} - 5 a^{2} + 4)$$ = $$\left(a^{5} - 3 a^{4} - a^{3} + 7 a^{2} - 3 a - 4\right)$$ magma: Conductor(E); sage: E.conductor() $$N(\mathfrak{N})$$ = $$43$$ = $$43$$ magma: Norm(Conductor(E)); sage: E.conductor().norm() $$\mathfrak{D}$$ = $$(6321363049,a^{4} - 2 a^{3} - 3 a^{2} + 3 a + 643142085,a^{5} - 3 a^{4} - 2 a^{3} + 8 a^{2} + 825524395,a + 5525344834,a^{5} - 2 a^{4} - 3 a^{3} + 3 a^{2} + a + 1459486535,a^{5} - 2 a^{4} - 3 a^{3} + 4 a^{2} + 2977171159)$$ = $$\left(a^{5} - 3 a^{4} - a^{3} + 7 a^{2} - 3 a - 4\right)^{6}$$ magma: Discriminant(E); sage: E.discriminant() gp (2.8): E.disc $$N(\mathfrak{D})$$ = $$6321363049$$ = $$43^{6}$$ magma: Norm(Discriminant(E)); sage: E.discriminant().norm() gp (2.8): norm(E.disc) $$j$$ = $$\frac{25867021722146251}{6321363049} a^{5} - \frac{89822684726406532}{6321363049} a^{4} + \frac{29268467321015368}{6321363049} a^{3} + \frac{85250804762932381}{6321363049} a^{2} - \frac{22977539272511937}{6321363049} a - \frac{17388365353799698}{6321363049}$$ 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$$(j)_{-}$$
$$\left(a^{5} - 3 a^{4} - a^{3} + 7 a^{2} - 3 a - 4\right)$$ $$43$$ $$2$$ $$I_{6}$$ Non-split multiplicative $$1$$ $$1$$ $$6$$ $$6$$

## Galois Representations

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

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

## Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 3.
Its isogeny class 43.2-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 is not a $$\Q$$-curve.