# Properties

 Label 2.0.8.1-16200.3-f2 Base field $$\Q(\sqrt{-2})$$ Conductor $$(90a)$$ Conductor norm $$16200$$ CM no Base change yes: 360.c1,2880.w1 Q-curve yes Torsion order $$2$$ Rank $$1$$

# Related objects

Show commands for: Magma / Pari/GP / SageMath

## Base field$$\Q(\sqrt{-2})$$

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

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([2, 0, 1]))

gp: K = nfinit(Pol(Vecrev([2, 0, 1])));

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2, 0, 1]);

## Weierstrass equation

$${y}^2+a{x}{y}+a{y}={x}^{3}-{x}^{2}-29{x}+81$$
sage: E = EllipticCurve([K([0,1]),K([-1,0]),K([0,1]),K([-29,0]),K([81,0])])

gp: E = ellinit([Pol(Vecrev([0,1])),Pol(Vecrev([-1,0])),Pol(Vecrev([0,1])),Pol(Vecrev([-29,0])),Pol(Vecrev([81,0]))], K);

magma: E := EllipticCurve([K![0,1],K![-1,0],K![0,1],K![-29,0],K![81,0]]);

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 Conductor: $$(90a)$$ = $$(a)^{3}\cdot(-a-1)^{2}\cdot(a-1)^{2}\cdot(5)$$ sage: E.conductor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor norm: $$16200$$ = $$2^{3}\cdot3^{2}\cdot3^{2}\cdot25$$ sage: E.conductor().norm()  gp: idealnorm(ellglobalred(E)[1])  magma: Norm(Conductor(E)); Discriminant: $$(21600)$$ = $$(a)^{10}\cdot(-a-1)^{3}\cdot(a-1)^{3}\cdot(5)^{2}$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); Discriminant norm: $$466560000$$ = $$2^{10}\cdot3^{3}\cdot3^{3}\cdot25^{2}$$ sage: E.discriminant().norm()  gp: norm(E.disc)  magma: Norm(Discriminant(E)); j-invariant: $$\frac{3721734}{25}$$ 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: $$1$$ Generator $\left(3 : -2 a - 2 : 1\right)$ Height $$0.437561463636797$$ 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: $\left(\frac{7}{2} : -\frac{9}{4} a : 1\right)$ sage: T.gens()  gp: T[3]  magma: [piT(P) : P in Generators(T)];

## BSD invariants

 Analytic rank: $$1$$ sage: E.rank()  magma: Rank(E); Mordell-Weil rank: $$1$$ Regulator: $$0.437561463636797$$ Period: $$1.94870920223201$$ Tamagawa product: $$16$$  =  $$2\cdot2\cdot2\cdot2$$ Torsion order: $$2$$ Leading coefficient: $$4.82348676843580$$ 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$$(j)_{-}$$
$$(a)$$ $$2$$ $$2$$ $$III^{*}$$ Additive $$-1$$ $$3$$ $$10$$ $$0$$
$$(-a-1)$$ $$3$$ $$2$$ $$III$$ Additive $$1$$ $$2$$ $$3$$ $$0$$
$$(a-1)$$ $$3$$ $$2$$ $$III$$ Additive $$1$$ $$2$$ $$3$$ $$0$$
$$(5)$$ $$25$$ $$2$$ $$I_{2}$$ Split multiplicative $$-1$$ $$1$$ $$2$$ $$2$$

## 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 16200.3-f consists of curves linked by isogenies of degree 2.

## Base change

This curve is the base change of elliptic curves 360.c1, 2880.w1, defined over $$\Q$$, so it is also a $$\Q$$-curve.