Properties

 Label 2.0.8.1-41616.5-f2 Base field $$\Q(\sqrt{-2})$$ Conductor $$(204)$$ Conductor norm $$41616$$ CM no Base change yes: 816.g2,3264.a2 Q-curve yes Torsion order $$1$$ Rank $$1$$

Related objects

Show commands: 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

$${y}^2={x}^{3}-{x}^{2}+3{x}-9$$
sage: E = EllipticCurve([K([0,0]),K([-1,0]),K([0,0]),K([3,0]),K([-9,0])])

gp: E = ellinit([Pol(Vecrev([0,0])),Pol(Vecrev([-1,0])),Pol(Vecrev([0,0])),Pol(Vecrev([3,0])),Pol(Vecrev([-9,0]))], K);

magma: E := EllipticCurve([K![0,0],K![-1,0],K![0,0],K![3,0],K![-9,0]]);

This is a global minimal model.

sage: E.is_global_minimal_model()

Invariants

 Conductor: $$(204)$$ = $$(a)^{4}\cdot(-a-1)\cdot(a-1)\cdot(-2a+3)\cdot(2a+3)$$ sage: E.conductor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor norm: $$41616$$ = $$2^{4}\cdot3\cdot3\cdot17\cdot17$$ sage: E.conductor().norm()  gp: idealnorm(ellglobalred(E)[1])  magma: Norm(Conductor(E)); Discriminant: $$(-29376)$$ = $$(a)^{12}\cdot(-a-1)^{3}\cdot(a-1)^{3}\cdot(-2a+3)\cdot(2a+3)$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); Discriminant norm: $$862949376$$ = $$2^{12}\cdot3^{3}\cdot3^{3}\cdot17\cdot17$$ sage: E.discriminant().norm()  gp: norm(E.disc)  magma: Norm(Discriminant(E)); j-invariant: $$\frac{32768}{459}$$ 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 $\left(2 : -1 : 1\right)$ Height $$1.18091958746321$$ Torsion structure: trivial sage: T = E.torsion_subgroup(); T.invariants()  gp: T = elltors(E); T[2]  magma: T,piT := TorsionSubgroup(E); Invariants(T);

BSD invariants

 Analytic rank: $$1$$ sage: E.rank()  magma: Rank(E); Mordell-Weil rank: $$1$$ Regulator: $$1.18091958746321$$ Period: $$2.51912659859620$$ Tamagawa product: $$1$$  =  $$1\cdot1\cdot1\cdot1\cdot1$$ Torsion order: $$1$$ Leading coefficient: $$4.20712404792651$$ 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$$(j)_{-}$$
$$(a)$$ $$2$$ $$1$$ $$II^{*}$$ Additive $$1$$ $$4$$ $$12$$ $$0$$
$$(-a-1)$$ $$3$$ $$1$$ $$I_{3}$$ Non-split multiplicative $$1$$ $$1$$ $$3$$ $$3$$
$$(a-1)$$ $$3$$ $$1$$ $$I_{3}$$ Non-split multiplicative $$1$$ $$1$$ $$3$$ $$3$$
$$(-2a+3)$$ $$17$$ $$1$$ $$I_{1}$$ Non-split multiplicative $$1$$ $$1$$ $$1$$ $$1$$
$$(2a+3)$$ $$17$$ $$1$$ $$I_{1}$$ Non-split multiplicative $$1$$ $$1$$ $$1$$ $$1$$

Galois Representations

The mod $$p$$ Galois Representation has maximal image for all primes $$p < 1000$$ except those listed.

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

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 3.
Its isogeny class 41616.5-f consists of curves linked by isogenies of degree 3.

Base change

This curve is the base change of 816.g2, 3264.a2, defined over $$\Q$$, so it is also a $$\Q$$-curve.