# Properties

 Label 2.0.8.1-41616.5-g3 Base field $$\Q(\sqrt{-2})$$ Conductor norm $$41616$$ CM no Base change no Q-curve yes Torsion order $$4$$ Rank $$1$$

# Related objects

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

$${y}^2+a{x}{y}={x}^{3}+{x}^{2}+\left(-17220a-3696\right){x}+1101912a-1018512$$
sage: E = EllipticCurve([K([0,1]),K([1,0]),K([0,0]),K([-3696,-17220]),K([-1018512,1101912])])

gp: E = ellinit([Polrev([0,1]),Polrev([1,0]),Polrev([0,0]),Polrev([-3696,-17220]),Polrev([-1018512,1101912])], K);

magma: E := EllipticCurve([K![0,1],K![1,0],K![0,0],K![-3696,-17220],K![-1018512,1101912]]);

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: $$(363184549852992a+189318828026880)$$ = $$(a)^{13}\cdot(-a-1)\cdot(a-1)\cdot(-2a+3)^{4}\cdot(2a+3)^{16}$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); Discriminant norm: $$299647653149312226931625238528$$ = $$2^{13}\cdot3\cdot3\cdot17^{4}\cdot17^{16}$$ sage: E.discriminant().norm()  gp: norm(E.disc)  magma: Norm(Discriminant(E)); j-invariant: $$\frac{61437923106397764191713}{291967151254001210886} a - \frac{146204680417750243067160}{48661191875666868481}$$ 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 $\left(-\frac{640}{9} a + \frac{48823}{288} : -\frac{14034967}{6912} a + \frac{46496}{27} : 1\right)$ Height $$3.8269817694884771133715028814573805997$$ Torsion structure: $$\Z/4\Z$$ sage: T = E.torsion_subgroup(); T.invariants()  gp: T = elltors(E); T[2]  magma: T,piT := TorsionSubgroup(E); Invariants(T); Torsion generator: $\left(12 a + 239 : 25 a - 3456 : 1\right)$ 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: $$3.8269817694884771133715028814573805997$$ Period: $$0.045938536802987021535045449156671775130$$ Tamagawa product: $$256$$  =  $$2^{2}\cdot1\cdot1\cdot2^{2}\cdot2^{4}$$ Torsion order: $$4$$ Leading coefficient: $$3.9780343798598310070341526577267025247$$ 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$$(j)_{-}$$
$$(a)$$ $$2$$ $$4$$ $$I_{5}^{*}$$ Additive $$1$$ $$4$$ $$13$$ $$1$$
$$(-a-1)$$ $$3$$ $$1$$ $$I_{1}$$ Non-split multiplicative $$1$$ $$1$$ $$1$$ $$1$$
$$(a-1)$$ $$3$$ $$1$$ $$I_{1}$$ Non-split multiplicative $$1$$ $$1$$ $$1$$ $$1$$
$$(-2a+3)$$ $$17$$ $$4$$ $$I_{4}$$ Split multiplicative $$-1$$ $$1$$ $$4$$ $$4$$
$$(2a+3)$$ $$17$$ $$16$$ $$I_{16}$$ Split multiplicative $$-1$$ $$1$$ $$16$$ $$16$$

## 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, 8 and 16.
Its isogeny class 41616.5-g consists of curves linked by isogenies of degrees dividing 16.

## Base change

This elliptic curve is a $$\Q$$-curve.

It is not the base change of an elliptic curve defined over any subfield.