# Properties

 Label 2.0.8.1-41616.5-p2 Base field $$\Q(\sqrt{-2})$$ Conductor $$(204)$$ Conductor norm $$41616$$ CM no Base change no Q-curve no Torsion order $$2$$ 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

$${y}^2+a{x}{y}+a{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-51a+143\right){x}+396a+422$$
sage: E = EllipticCurve([K([0,1]),K([-1,-1]),K([0,1]),K([143,-51]),K([422,396])])

gp: E = ellinit([Pol(Vecrev([0,1])),Pol(Vecrev([-1,-1])),Pol(Vecrev([0,1])),Pol(Vecrev([143,-51])),Pol(Vecrev([422,396]))], K);

magma: E := EllipticCurve([K![0,1],K![-1,-1],K![0,1],K![143,-51],K![422,396]]);

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: $$(-6977616a-4924560)$$ = $$(a)^{8}\cdot(-a-1)\cdot(a-1)^{8}\cdot(-2a+3)^{2}\cdot(2a+3)^{4}$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); Discriminant norm: $$121625541280512$$ = $$2^{8}\cdot3\cdot3^{8}\cdot17^{2}\cdot17^{4}$$ sage: E.discriminant().norm()  gp: norm(E.disc)  magma: Norm(Discriminant(E)); j-invariant: $$-\frac{8951763861508}{547981281} a - \frac{18527853018220}{547981281}$$ 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 $\left(-\frac{338}{81} a - \frac{178}{81} : \frac{2485}{729} a - \frac{2668}{729} : 1\right)$ Height $$1.20689736671273$$ 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: $\left(-5 a - \frac{1}{2} : -\frac{1}{4} a - 5 : 1\right)$ 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: $$1.20689736671273$$ Period: $$0.761049375648576$$ Tamagawa product: $$16$$  =  $$1\cdot1\cdot2\cdot2\cdot2^{2}$$ Torsion order: $$2$$ Leading coefficient: $$5.19586864019237$$ 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$$(j)_{-}$$
$$(a)$$ $$2$$ $$1$$ $$I_0^{*}$$ Additive $$1$$ $$4$$ $$8$$ $$0$$
$$(-a-1)$$ $$3$$ $$1$$ $$I_{1}$$ Split multiplicative $$-1$$ $$1$$ $$1$$ $$1$$
$$(a-1)$$ $$3$$ $$2$$ $$I_{8}$$ Non-split multiplicative $$1$$ $$1$$ $$8$$ $$8$$
$$(-2a+3)$$ $$17$$ $$2$$ $$I_{2}$$ Non-split multiplicative $$1$$ $$1$$ $$2$$ $$2$$
$$(2a+3)$$ $$17$$ $$4$$ $$I_{4}$$ Split multiplicative $$-1$$ $$1$$ $$4$$ $$4$$

## 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 and 4.
Its isogeny class 41616.5-p consists of curves linked by isogenies of degrees dividing 4.

## Base change

This curve is not the base change of an elliptic curve defined over $$\Q$$. It is not a $$\Q$$-curve.