# Properties

 Base field 6.6.434581.1 Label 6.6.434581.1-71.2-a1 Conductor $$(71,2 a^{4} - 4 a^{3} - 6 a^{2} + 7 a + 2)$$ Conductor norm $$71$$ CM no base-change no Q-curve no Torsion order $$2$$ 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

$$y^2 + a x y + \left(3 a^{5} - 7 a^{4} - 8 a^{3} + 15 a^{2} + a - 4\right) y = x^{3} + \left(-a^{5} + 2 a^{4} + 4 a^{3} - 4 a^{2} - 4 a + 2\right) x^{2} + \left(-5 a^{5} + 14 a^{4} + 8 a^{3} - 25 a^{2} + 6\right) x + a^{5} + a^{4} - 11 a^{3} - a^{2} + 18 a - 8$$
magma: E := ChangeRing(EllipticCurve([a, -a^5 + 2*a^4 + 4*a^3 - 4*a^2 - 4*a + 2, 3*a^5 - 7*a^4 - 8*a^3 + 15*a^2 + a - 4, -5*a^5 + 14*a^4 + 8*a^3 - 25*a^2 + 6, a^5 + a^4 - 11*a^3 - a^2 + 18*a - 8]),K);

sage: E = EllipticCurve(K, [a, -a^5 + 2*a^4 + 4*a^3 - 4*a^2 - 4*a + 2, 3*a^5 - 7*a^4 - 8*a^3 + 15*a^2 + a - 4, -5*a^5 + 14*a^4 + 8*a^3 - 25*a^2 + 6, a^5 + a^4 - 11*a^3 - a^2 + 18*a - 8])

gp: E = ellinit([a, -a^5 + 2*a^4 + 4*a^3 - 4*a^2 - 4*a + 2, 3*a^5 - 7*a^4 - 8*a^3 + 15*a^2 + a - 4, -5*a^5 + 14*a^4 + 8*a^3 - 25*a^2 + 6, a^5 + a^4 - 11*a^3 - a^2 + 18*a - 8],K)

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 $$\mathfrak{N}$$ = $$(71,2 a^{4} - 4 a^{3} - 6 a^{2} + 7 a + 2)$$ = $$\left(a^{3} - 3 a\right)$$ magma: Conductor(E);  sage: E.conductor() $$N(\mathfrak{N})$$ = $$71$$ = $$71$$ magma: Norm(Conductor(E));  sage: E.conductor().norm() $$\mathfrak{D}$$ = $$(71,a^{4} - 2 a^{3} - 3 a^{2} + 3 a + 58,a^{5} - 3 a^{4} - 2 a^{3} + 8 a^{2} + 16,a + 28,a^{5} - 2 a^{4} - 3 a^{3} + 3 a^{2} + a + 37,a^{5} - 2 a^{4} - 3 a^{3} + 4 a^{2} + 6)$$ = $$\left(a^{3} - 3 a\right)$$ magma: Discriminant(E);  sage: E.discriminant()  gp: E.disc $$N(\mathfrak{D})$$ = $$71$$ = $$71$$ magma: Norm(Discriminant(E));  sage: E.discriminant().norm()  gp: norm(E.disc) $$j$$ = $$-\frac{771436057367340}{71} a^{5} + \frac{1021204229550191}{71} a^{4} + \frac{3776605848171226}{71} a^{3} - \frac{1302945474751863}{71} a^{2} - \frac{3967265080250394}{71} a - \frac{1140546291203617}{71}$$ 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: $$\Z/2\Z$$ magma: T,piT := TorsionSubgroup(E); Invariants(T);  sage: T = E.torsion_subgroup(); T.invariants()  gp: T = elltors(E); T[2] $\left(-2 a^{5} + 5 a^{4} + 5 a^{3} - 12 a^{2} - a + 3 : -2 a^{5} + 5 a^{4} + 5 a^{3} - 11 a^{2} + 3 : 1\right)$ magma: [piT(P) : P in Generators(T)];  sage: T.gens()  gp: T[3]

## 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$$(j)_{-}$$
$$\left(a^{3} - 3 a\right)$$ $$71$$ $$1$$ $$I_{1}$$ Non-split multiplicative $$1$$ $$1$$ $$1$$ $$1$$

## Galois Representations

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

prime Image of Galois Representation
$$2$$ 2B
$$3$$ 3B

## Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 2, 3 and 6.
Its isogeny class 71.2-a consists of curves linked by isogenies of degrees dividing 6.

## Base change

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