# Properties

 Base field $$\Q(\sqrt{-2})$$ Label 2.0.8.1-22050.2-e1 Conductor $$(105 a)$$ Conductor norm $$22050$$ CM no base-change yes: 6720.cc6,210.d6 Q-curve yes Torsion order $$12$$ Rank not available

# Related objects

Show commands for: Magma / SageMath / Pari/GP

## Base field $$\Q(\sqrt{-2})$$

Generator $$a$$, with minimal polynomial $$x^{2} + 2$$; class number $$1$$.

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2, 0, 1]);

sage: x = polygen(QQ); K.<a> = NumberField(x^2 + 2)

gp (2.8): K = nfinit(a^2 + 2);

## Weierstrass equation

$$y^2 + x y = x^{3} - 81 x + 6561$$
magma: E := ChangeRing(EllipticCurve([1, 0, 0, -81, 6561]),K);

sage: E = EllipticCurve(K, [1, 0, 0, -81, 6561])

gp (2.8): E = ellinit([1, 0, 0, -81, 6561],K)

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 $$\mathfrak{N}$$ = $$(105 a)$$ = $$\left(a\right) \cdot \left(-a - 1\right) \cdot \left(a - 1\right) \cdot \left(5\right) \cdot \left(7\right)$$ magma: Conductor(E);  sage: E.conductor() $$N(\mathfrak{N})$$ = $$22050$$ = $$2 \cdot 3^{2} \cdot 25 \cdot 49$$ magma: Norm(Conductor(E));  sage: E.conductor().norm() $$\mathfrak{D}$$ = $$(18600435000)$$ = $$\left(a\right)^{6} \cdot \left(-a - 1\right)^{12} \cdot \left(a - 1\right)^{12} \cdot \left(5\right)^{4} \cdot \left(7\right)$$ magma: Discriminant(E);  sage: E.discriminant()  gp (2.8): E.disc $$N(\mathfrak{D})$$ = $$345976182189225000000$$ = $$2^{6} \cdot 3^{24} \cdot 25^{4} \cdot 49$$ magma: Norm(Discriminant(E));  sage: E.discriminant().norm()  gp (2.8): norm(E.disc) $$j$$ = $$-\frac{58818484369}{18600435000}$$ 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: $$\Z/12\Z$$ 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] $\left(-15 a + 6 : -45 a - 63 : 1\right)$ 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$$(j)_{-}$$
$$\left(a\right)$$ $$2$$ $$6$$ $$I_{6}$$ Split multiplicative $$-1$$ $$1$$ $$6$$ $$6$$
$$\left(-a - 1\right)$$ $$3$$ $$12$$ $$I_{12}$$ Split multiplicative $$-1$$ $$1$$ $$12$$ $$12$$
$$\left(a - 1\right)$$ $$3$$ $$12$$ $$I_{12}$$ Split multiplicative $$-1$$ $$1$$ $$12$$ $$12$$
$$\left(5\right)$$ $$25$$ $$4$$ $$I_{4}$$ Split multiplicative $$-1$$ $$1$$ $$4$$ $$4$$
$$\left(7\right)$$ $$49$$ $$1$$ $$I_{1}$$ 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.1.1

## Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 2, 3, 4, 6 and 12.
Its isogeny class 22050.2-e consists of curves linked by isogenies of degrees dividing 12.

## Base change

This curve is the base-change of elliptic curves 6720.cc6, 210.d6, defined over $$\Q$$, so it is also a $$\Q$$-curve.