# Properties

 Label 2.0.3.1-16384.1-g1 Base field $$\Q(\sqrt{-3})$$ Conductor norm $$16384$$ CM no Base change yes Q-curve yes Torsion order $$2$$ Rank $$0$$

# Related objects

Show commands: Magma / PariGP / SageMath

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

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

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([1, -1, 1]))

gp: K = nfinit(Polrev([1, -1, 1]));

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

## Weierstrass equation

$${y}^2={x}^{3}+a{x}^{2}+\left(3a-3\right){x}+5$$
sage: E = EllipticCurve([K([0,0]),K([0,1]),K([0,0]),K([-3,3]),K([5,0])])

gp: E = ellinit([Polrev([0,0]),Polrev([0,1]),Polrev([0,0]),Polrev([-3,3]),Polrev([5,0])], K);

magma: E := EllipticCurve([K![0,0],K![0,1],K![0,0],K![-3,3],K![5,0]]);

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 Conductor: $$(128)$$ = $$(2)^{7}$$ sage: E.conductor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor norm: $$16384$$ = $$4^{7}$$ sage: E.conductor().norm()  gp: idealnorm(ellglobalred(E)[1])  magma: Norm(Conductor(E)); Discriminant: $$(-16384)$$ = $$(2)^{14}$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); Discriminant norm: $$268435456$$ = $$4^{14}$$ sage: E.discriminant().norm()  gp: norm(E.disc)  magma: Norm(Discriminant(E)); j-invariant: $$128$$ sage: E.j_invariant()  gp: E.j  magma: jInvariant(E); Endomorphism ring: $$\Z$$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) sage: E.has_cm(), E.cm_discriminant()  magma: HasComplexMultiplication(E); Sato-Tate group: $\mathrm{SU}(2)$

## Mordell-Weil group

 Rank: $$0$$ Torsion structure: $$\Z/2\Z$$ sage: T = E.torsion_subgroup(); T.invariants()  gp: T = elltors(E); T[2]  magma: T,piT := TorsionSubgroup(E); Invariants(T); Torsion generator: $\left(a : 0 : 1\right)$ sage: T.gens()  gp: T[3]  magma: [piT(P) : P in Generators(T)];

## BSD invariants

 Analytic rank: $$0$$ sage: E.rank()  magma: Rank(E); Mordell-Weil rank: $$0$$ Regulator: $$1$$ Period: $$2.7723970052074913588500971127911723848$$ Tamagawa product: $$2$$ Torsion order: $$2$$ Leading coefficient: $$1.6006441572570574384826241804601277627$$ Analytic order of Ш: $$1$$ (rounded)

## Local data at primes of bad reduction

sage: E.local_data()

magma: LocalInformation(E);

prime Norm Tamagawa number Kodaira symbol Reduction type Root number ord($$\mathfrak{N}$$) ord($$\mathfrak{D}$$) ord$$(j)_{-}$$
$$(2)$$ $$4$$ $$2$$ $$III^{*}$$ Additive $$-1$$ $$7$$ $$14$$ $$0$$

## Galois Representations

The mod $$p$$ Galois Representation has maximal image for all primes $$p < 1000$$ 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.
Its isogeny class 16384.1-g consists of curves linked by isogenies of degree 2.

## Base change

This elliptic curve is a $$\Q$$-curve. It is the base change of the following 2 elliptic curves:

Base field Curve
$$\Q$$ 128.d2
$$\Q$$ 1152.c2