# Properties

 Label 2.0.8.1-5202.5-i4 Base field $$\Q(\sqrt{-2})$$ Conductor norm $$5202$$ CM no Base change yes Q-curve yes Torsion order $$8$$ Rank $$0$$

# 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

$${y}^2+{x}{y}={x}^{3}+226{x}-2232$$
sage: E = EllipticCurve([K([1,0]),K([0,0]),K([0,0]),K([226,0]),K([-2232,0])])

gp: E = ellinit([Polrev([1,0]),Polrev([0,0]),Polrev([0,0]),Polrev([226,0]),Polrev([-2232,0])], K);

magma: E := EllipticCurve([K![1,0],K![0,0],K![0,0],K![226,0],K![-2232,0]]);

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 Conductor: $$(51a)$$ = $$(a)\cdot(-a-1)\cdot(a-1)\cdot(-2a+3)\cdot(2a+3)$$ sage: E.conductor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor norm: $$5202$$ = $$2\cdot3\cdot3\cdot17\cdot17$$ sage: E.conductor().norm()  gp: idealnorm(ellglobalred(E)[1])  magma: Norm(Conductor(E)); Discriminant: $$(-2927177028)$$ = $$(a)^{4}\cdot(-a-1)^{16}\cdot(a-1)^{16}\cdot(-2a+3)\cdot(2a+3)$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); Discriminant norm: $$8568365353250912784$$ = $$2^{4}\cdot3^{16}\cdot3^{16}\cdot17\cdot17$$ sage: E.discriminant().norm()  gp: norm(E.disc)  magma: Norm(Discriminant(E)); j-invariant: $$\frac{1276229915423}{2927177028}$$ 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: $\left(-9 a - 8 : 36 a - 32 : 1\right)$ 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: $$0.36750829442389617228036359325337420104$$ Tamagawa product: $$1024$$  =  $$2^{2}\cdot2^{4}\cdot2^{4}\cdot1\cdot1$$ Torsion order: $$8$$ Leading coefficient: $$4.1578817140710277500951007224568595386$$ 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$$(j)_{-}$$
$$(a)$$ $$2$$ $$4$$ $$I_{4}$$ Split multiplicative $$-1$$ $$1$$ $$4$$ $$4$$
$$(-a-1)$$ $$3$$ $$16$$ $$I_{16}$$ Split multiplicative $$-1$$ $$1$$ $$16$$ $$16$$
$$(a-1)$$ $$3$$ $$16$$ $$I_{16}$$ Split multiplicative $$-1$$ $$1$$ $$16$$ $$16$$
$$(-2a+3)$$ $$17$$ $$1$$ $$I_{1}$$ Split multiplicative $$-1$$ $$1$$ $$1$$ $$1$$
$$(2a+3)$$ $$17$$ $$1$$ $$I_{1}$$ 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
$$2$$ 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 5202.5-i 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
$$\Q$$ 102.c6
$$\Q$$ 3264.bc6