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## Base field$$\Q(\zeta_{16})^+$$

Generator $$a$$, with minimal polynomial $$x^{4} - 4 x^{2} + 2$$; class number $$1$$.

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([2, 0, -4, 0, 1]))

gp: K = nfinit(Pol(Vecrev([2, 0, -4, 0, 1])));

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2, 0, -4, 0, 1]);

## Weierstrass equation

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

gp: E = ellinit([Pol(Vecrev([0,-2,0,1])),Pol(Vecrev([-1,0,1,0])),Pol(Vecrev([0,-2,0,1])),Pol(Vecrev([-2,0,1,0])),Pol(Vecrev([0,0,0,0]))], K);

magma: E := EllipticCurve([K![0,-2,0,1],K![-1,0,1,0],K![0,-2,0,1],K![-2,0,1,0],K![0,0,0,0]]);

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 Conductor: $$(2a^2-4)$$ = $$(a)^{6}$$ sage: E.conductor()  gp: ellglobalred(E)  magma: Conductor(E); Conductor norm: $$64$$ = $$2^{6}$$ sage: E.conductor().norm()  gp: idealnorm(ellglobalred(E))  magma: Norm(Conductor(E)); Discriminant: $$(8)$$ = $$(a)^{12}$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); Discriminant norm: $$4096$$ = $$2^{12}$$ sage: E.discriminant().norm()  gp: norm(E.disc)  magma: Norm(Discriminant(E)); j-invariant: $$8000$$ sage: E.j_invariant()  gp: E.j  magma: jInvariant(E); Endomorphism ring: $$\Z$$ Geometric endomorphism ring: $$\Z[\sqrt{-2}]$$ (potential complex multiplication) sage: E.has_cm(), E.cm_discriminant()  magma: HasComplexMultiplication(E); Sato-Tate group: $N(\mathrm{U}(1))$

## Mordell-Weil group

 Rank: $$1$$ Generator $\left(a : a : 1\right)$ Height $$0.120031449377074$$ Torsion structure: $$\Z/2\Z\times\Z/4\Z$$ sage: T = E.torsion_subgroup(); T.invariants()  gp: T = elltors(E); T  magma: T,piT := TorsionSubgroup(E); Invariants(T); Torsion generators: $\left(-1 : 0 : 1\right)$ $\left(0 : 0 : 1\right)$ sage: T.gens()  gp: T  magma: [piT(P) : P in Generators(T)];

## BSD invariants

 Analytic rank: $$1$$ sage: E.rank()  magma: Rank(E); Mordell-Weil rank: $$1$$ Regulator: $$0.120031449377074$$ Period: $$2576.57229428263$$ Tamagawa product: $$4$$ Torsion order: $$8$$ Leading coefficient: $$1.70848989820242$$ 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 $$-1$$ $$6$$ $$12$$ $$0$$

## Galois Representations

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

prime Image of Galois Representation
$$2$$ 2Cs

For all other primes $$p$$, the image is the normalizer of a split Cartan subgroup if $$\left(\frac{ -2 }{p}\right)=+1$$ or the normalizer of a nonsplit Cartan subgroup if $$\left(\frac{ -2 }{p}\right)=-1$$.

## Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 2 and 4.
Its isogeny class 64.1-a consists of curves linked by isogenies of degrees dividing 8.

## Base change

This curve is the base change of 256.d1, 256.a2, defined over $$\Q$$, so it is also a $$\Q$$-curve.