Properties

 Base field $$\Q(\sqrt{3})$$ Label 2.2.12.1-162.1-a1 Conductor $$(9 a + 9)$$ Conductor norm $$162$$ CM no base-change yes: 54.b1,432.g2 Q-curve yes Torsion order $$1$$ Rank not available

Related objects

Show commands for: Magma / SageMath / Pari/GP

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

Generator $$a$$, with minimal polynomial $$x^{2} - 3$$; class number $$1$$.

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-3, 0, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^2 - 3)
gp (2.8): K = nfinit(a^2 - 3);

Weierstrass equation

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

This is a global minimal model.

sage: E.is_global_minimal_model()

Invariants

 $$\mathfrak{N}$$ = $$(9 a + 9)$$ = $$\left(a + 1\right) \cdot \left(a\right)^{4}$$ magma: Conductor(E); sage: E.conductor() $$N(\mathfrak{N})$$ = $$162$$ = $$2 \cdot 3^{4}$$ magma: Norm(Conductor(E)); sage: E.conductor().norm() $$\mathfrak{D}$$ = $$(54)$$ = $$\left(a + 1\right)^{2} \cdot \left(a\right)^{6}$$ magma: Discriminant(E); sage: E.discriminant() gp (2.8): E.disc $$N(\mathfrak{D})$$ = $$2916$$ = $$2^{2} \cdot 3^{6}$$ magma: Norm(Discriminant(E)); sage: E.discriminant().norm() gp (2.8): norm(E.disc) $$j$$ = $$-\frac{132651}{2}$$ 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: Trivial 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]

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$$(j)_{-}$$
$$\left(a + 1\right)$$ $$2$$ $$2$$ $$I_{2}$$ Split multiplicative $$-1$$ $$1$$ $$2$$ $$2$$
$$\left(a\right)$$ $$3$$ $$1$$ $$IV$$ Additive $$-1$$ $$4$$ $$6$$ $$0$$

Galois Representations

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

prime Image of Galois Representation
$$3$$ 3B.1.2

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 3 and 9.
Its isogeny class 162.1-a consists of curves linked by isogenies of degrees dividing 9.

Base change

This curve is the base-change of elliptic curves 54.b1, 432.g2, defined over $$\Q$$, so it is also a $$\Q$$-curve.