# Properties

 Base field $$\Q(\sqrt{6})$$ Label 2.2.24.1-147.1-g4 Conductor $$(7 a + 21)$$ Conductor norm $$147$$ CM no base-change yes: 21.a3,4032.k3 Q-curve yes Torsion order $$8$$ Rank not available

# Related objects

Show commands for: Magma / SageMath / Pari/GP

## Base field $$\Q(\sqrt{6})$$

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

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-6, 0, 1]);

sage: x = polygen(QQ); K.<a> = NumberField(x^2 - 6)

gp (2.8): K = nfinit(a^2 - 6);

## Weierstrass equation

$$y^2 + x y = x^{3} - 39 x + 90$$
magma: E := ChangeRing(EllipticCurve([1, 0, 0, -39, 90]),K);

sage: E = EllipticCurve(K, [1, 0, 0, -39, 90])

gp (2.8): E = ellinit([1, 0, 0, -39, 90],K)

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 $$\mathfrak{N}$$ = $$(7 a + 21)$$ = $$\left(a + 3\right) \cdot \left(7\right)$$ magma: Conductor(E);  sage: E.conductor() $$N(\mathfrak{N})$$ = $$147$$ = $$3 \cdot 49$$ magma: Norm(Conductor(E));  sage: E.conductor().norm() $$\mathfrak{D}$$ = $$(45927)$$ = $$\left(a + 3\right)^{16} \cdot \left(7\right)$$ magma: Discriminant(E);  sage: E.discriminant()  gp (2.8): E.disc $$N(\mathfrak{D})$$ = $$2109289329$$ = $$3^{16} \cdot 49$$ magma: Norm(Discriminant(E));  sage: E.discriminant().norm()  gp (2.8): norm(E.disc) $$j$$ = $$\frac{6570725617}{45927}$$ 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: $$\Z/8\Z$$ 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] $\left(-3 : 15 : 1\right)$ 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$$(j)_{-}$$
$$\left(a + 3\right)$$ $$3$$ $$16$$ $$I_{16}$$ Split multiplicative $$-1$$ $$1$$ $$16$$ $$16$$
$$\left(7\right)$$ $$49$$ $$1$$ $$I_{1}$$ 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

## Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 2, 4 and 8.
Its isogeny class 147.1-g consists of curves linked by isogenies of degrees dividing 8.

## Base change

This curve is the base-change of elliptic curves 21.a3, 4032.k3, defined over $$\Q$$, so it is also a $$\Q$$-curve.