# Properties

 Base field $$\Q(\sqrt{-3})$$ Label 2.0.3.1-1444.2-b5 Conductor $$(38)$$ Conductor norm $$1444$$ CM no base-change yes: 38.a3,342.e3 Q-curve yes Torsion order $$9$$ Rank $$0$$

# Related objects

Show commands for: Magma / SageMath / Pari/GP

## Base field $$\Q(\sqrt{-3})$$

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

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -1, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^2 - x + 1)
gp (2.8): K = nfinit(a^2 - a + 1);

## Weierstrass equation

$$y^2 + a x y + y = x^{3} - 10 a x + 90$$
magma: E := ChangeRing(EllipticCurve([a, 0, 1, -10*a, 90]),K);
sage: E = EllipticCurve(K, [a, 0, 1, -10*a, 90])
gp (2.8): E = ellinit([a, 0, 1, -10*a, 90],K)

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 $$\mathfrak{N}$$ = $$(38)$$ = $$\left(2\right) \cdot \left(-5 a + 3\right) \cdot \left(-5 a + 2\right)$$ magma: Conductor(E); sage: E.conductor() $$N(\mathfrak{N})$$ = $$1444$$ = $$4 \cdot 19^{2}$$ magma: Norm(Conductor(E)); sage: E.conductor().norm() $$\mathfrak{D}$$ = $$(3511808)$$ = $$\left(2\right)^{9} \cdot \left(-5 a + 3\right)^{3} \cdot \left(-5 a + 2\right)^{3}$$ magma: Discriminant(E); sage: E.discriminant() gp (2.8): E.disc $$N(\mathfrak{D})$$ = $$12332795428864$$ = $$4^{9} \cdot 19^{6}$$ magma: Norm(Discriminant(E)); sage: E.discriminant().norm() gp (2.8): norm(E.disc) $$j$$ = $$\frac{94196375}{3511808}$$ 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: $$0$$
magma: Rank(E);
sage: E.rank()
magma: Generators(E); // includes torsion
sage: E.gens()
magma: Regulator(Generators(E));
sage: E.regulator_of_points(E.gens())

## Torsion subgroup

Structure: $$\Z/3\Z\times\Z/3\Z$$ 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] $\left(-a - 7 : 24 a - 13 : 1\right)$,$\left(0 : -10 : 1\right)$ 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$$(j)_{-}$$
$$\left(-5 a + 3\right)$$ $$19$$ $$3$$ $$I_{3}$$ Split multiplicative $$-1$$ $$1$$ $$3$$ $$3$$
$$\left(-5 a + 2\right)$$ $$19$$ $$3$$ $$I_{3}$$ Split multiplicative $$-1$$ $$1$$ $$3$$ $$3$$
$$\left(2\right)$$ $$4$$ $$9$$ $$I_{9}$$ Split multiplicative $$-1$$ $$1$$ $$9$$ $$9$$

## Galois Representations

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

prime Image of Galois Representation
$$3$$ 3Cs.1.1[2]

## Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 3.
Its isogeny class 1444.2-b consists of curves linked by isogenies of degrees dividing 9.

## Base change

This curve is the base-change of elliptic curves 38.a3, 342.e3, defined over $$\Q$$, so it is also a $$\Q$$-curve.