# Properties

 Label 2.2.8.1-144.1-b2 Base field $$\Q(\sqrt{2})$$ Conductor norm $$144$$ CM no Base change yes Q-curve yes Torsion order $$8$$ Rank $$0$$

# Learn more

Show commands: Magma / PariGP / 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(Polrev([-2, 0, 1]));

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

## Weierstrass equation

$${y}^2+a{x}{y}+a{y}={x}^{3}+3{x}+22$$
sage: E = EllipticCurve([K([0,1]),K([0,0]),K([0,1]),K([3,0]),K([22,0])])

gp: E = ellinit([Polrev([0,1]),Polrev([0,0]),Polrev([0,1]),Polrev([3,0]),Polrev([22,0])], K);

magma: E := EllipticCurve([K![0,1],K![0,0],K![0,1],K![3,0],K![22,0]]);

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 Conductor: $$(12)$$ = $$(a)^{4}\cdot(3)$$ sage: E.conductor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor norm: $$144$$ = $$2^{4}\cdot9$$ sage: E.conductor().norm()  gp: idealnorm(ellglobalred(E)[1])  magma: Norm(Conductor(E)); Discriminant: $$(-209952)$$ = $$(a)^{10}\cdot(3)^{8}$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); Discriminant norm: $$44079842304$$ = $$2^{10}\cdot9^{8}$$ sage: E.discriminant().norm()  gp: norm(E.disc)  magma: Norm(Discriminant(E)); j-invariant: $$\frac{207646}{6561}$$ 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: $$\Z/8\Z$$ sage: T = E.torsion_subgroup(); T.invariants()  gp: T = elltors(E); T[2]  magma: T,piT := TorsionSubgroup(E); Invariants(T); Torsion generator: $\left(-1 : 3 a : 1\right)$ 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: $$5.6835085175015869010371476279434336897$$ Tamagawa product: $$32$$  =  $$2^{2}\cdot2^{3}$$ Torsion order: $$8$$ Leading coefficient: $$1.0047118534142184299587942673228949529$$ 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$$(j)_{-}$$
$$(a)$$ $$2$$ $$4$$ $$I_{2}^{*}$$ Additive $$-1$$ $$4$$ $$10$$ $$0$$
$$(3)$$ $$9$$ $$8$$ $$I_{8}$$ Split multiplicative $$-1$$ $$1$$ $$8$$ $$8$$

## 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, 4 and 8.
Its isogeny class 144.1-b consists of curves linked by isogenies of degrees dividing 16.

## Base change

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

Base field Curve
$$\Q$$ 48.a6
$$\Q$$ 192.b6