Properties

 Base field $$\Q(\sqrt{-2})$$ Label 2.0.8.1-392.1-a3 Conductor $$(14 a)$$ Conductor norm $$392$$ CM no base-change yes: 448.e3,56.a3 Q-curve yes Torsion order $$4$$ Rank $$1$$

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

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

This is a global minimal model.

sage: E.is_global_minimal_model()

Invariants

 $$\mathfrak{N}$$ = $$(14 a)$$ = $$\left(a\right)^{3} \cdot \left(7\right)$$ magma: Conductor(E); sage: E.conductor() $$N(\mathfrak{N})$$ = $$392$$ = $$2^{3} \cdot 49$$ magma: Norm(Conductor(E)); sage: E.conductor().norm() $$\mathfrak{D}$$ = $$(784)$$ = $$\left(a\right)^{8} \cdot \left(7\right)^{2}$$ magma: Discriminant(E); sage: E.discriminant() gp (2.8): E.disc $$N(\mathfrak{D})$$ = $$614656$$ = $$2^{8} \cdot 49^{2}$$ magma: Norm(Discriminant(E)); sage: E.discriminant().norm() gp (2.8): norm(E.disc) $$j$$ = $$\frac{740772}{49}$$ 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: $$1$$
magma: Rank(E);
sage: E.rank()

Generator: $\left(-a - 1 : 2 : 1\right)$

Height: 0.5345542883986127

magma: Generators(E); // includes torsion
sage: E.gens()

Regulator: 0.534554288399

magma: Regulator(Generators(E));
sage: E.regulator_of_points(E.gens())

Torsion subgroup

Structure: $$\Z/2\Z\times\Z/2\Z$$ 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] $\left(-1 : 0 : 1\right)$,$\left(-\frac{1}{2} : -\frac{1}{4} a : 1\right)$ 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$$(j)_{-}$$
$$\left(a\right)$$ $$2$$ $$4$$ $$I_{1}^*$$ Additive $$-1$$ $$3$$ $$8$$ $$0$$
$$\left(7\right)$$ $$49$$ $$2$$ $$I_{2}$$ Split multiplicative $$-1$$ $$1$$ $$2$$ $$2$$

Galois Representations

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

prime Image of Galois Representation
$$2$$ 2Cs

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 2.
Its isogeny class 392.1-a consists of curves linked by isogenies of degrees dividing 4.

Base change

This curve is the base-change of elliptic curves 448.e3, 56.a3, defined over $$\Q$$, so it is also a $$\Q$$-curve.