# Properties

 Base field $$\Q(\zeta_{9})^+$$ Label 3.3.81.1-8.1-a2 Conductor $$(2,2)$$ Conductor norm $$8$$ CM no base-change yes: 162.c2 Q-curve yes Torsion order $$3$$ Rank not available

# Related objects

Show commands for: Magma / SageMath / Pari/GP

## Base field $$\Q(\zeta_{9})^+$$

Generator $$a$$, with minimal polynomial $$x^{3} - 3 x - 1$$; class number $$1$$.

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, -3, 0, 1]);

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

gp (2.8): K = nfinit(a^3 - 3*a - 1);

## Weierstrass equation

$$y^2 + a x y + \left(a^{2} - 1\right) y = x^{3} - x^{2} + \left(125040 a^{2} - 43189 a - 360486\right) x + 56242078 a^{2} - 19527401 a - 161952511$$
magma: E := ChangeRing(EllipticCurve([a, -1, a^2 - 1, 125040*a^2 - 43189*a - 360486, 56242078*a^2 - 19527401*a - 161952511]),K);

sage: E = EllipticCurve(K, [a, -1, a^2 - 1, 125040*a^2 - 43189*a - 360486, 56242078*a^2 - 19527401*a - 161952511])

gp (2.8): E = ellinit([a, -1, a^2 - 1, 125040*a^2 - 43189*a - 360486, 56242078*a^2 - 19527401*a - 161952511],K)

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 $$\mathfrak{N}$$ = $$(2,2)$$ = $$\left(2\right)$$ magma: Conductor(E);  sage: E.conductor() $$N(\mathfrak{N})$$ = $$8$$ = $$8$$ magma: Norm(Conductor(E));  sage: E.conductor().norm() $$\mathfrak{D}$$ = $$(2097152,2097152 a,2097152 a^{2} - 4194304)$$ = $$\left(2\right)^{21}$$ magma: Discriminant(E);  sage: E.discriminant()  gp (2.8): E.disc $$N(\mathfrak{D})$$ = $$9223372036854775808$$ = $$8^{21}$$ magma: Norm(Discriminant(E));  sage: E.discriminant().norm()  gp (2.8): norm(E.disc) $$j$$ = $$-\frac{1159088625}{2097152}$$ 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: $$\Z/3\Z$$ 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] $\left(-\frac{1219}{3} a^{2} + 125 a + \frac{3601}{3} : \frac{33091}{3} a^{2} - \frac{11812}{3} a - \frac{94685}{3} : 1\right)$ magma: [f(P): P in Generators(T)] where T,f:=TorsionSubgroup(E);  sage: E.torsion_subgroup().gens()  gp (2.8): elltors(E)[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$$(j)_{-}$$
$$\left(2\right)$$ $$8$$ $$21$$ $$I_{21}$$ Split multiplicative $$-1$$ $$1$$ $$21$$ $$21$$

## Galois Representations

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

prime Image of Galois Representation
$$3$$ 3B.1.1
$$7$$ 7B.1.3

## Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 3, 7 and 21.
Its isogeny class 8.1-a consists of curves linked by isogenies of degrees dividing 21.

## Base change

This curve is the base-change of elliptic curves 162.c2, defined over $$\Q$$, so it is also a $$\Q$$-curve.