# Properties

 Label 4.4.2000.1-320.1-g6 Base field $$\Q(\zeta_{20})^+$$ Conductor $$(-2a^2-2a+10)$$ Conductor norm $$320$$ CM no Base change yes: 40.a3,200.c3 Q-curve yes Torsion order $$16$$ Rank $$0$$

# Related objects

Show commands: Magma / Pari/GP / SageMath

## Base field$$\Q(\zeta_{20})^+$$

Generator $$a$$, with minimal polynomial $$x^{4} - 5 x^{2} + 5$$; class number $$1$$.

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([5, 0, -5, 0, 1]))

gp: K = nfinit(Pol(Vecrev([5, 0, -5, 0, 1])));

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![5, 0, -5, 0, 1]);

## Weierstrass equation

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

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

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

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 Conductor: $$(-2a^2-2a+10)$$ = $$(a^3-a^2-3a+2)^{3}\cdot(a)$$ sage: E.conductor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor norm: $$320$$ = $$4^{3}\cdot5$$ sage: E.conductor().norm()  gp: idealnorm(ellglobalred(E)[1])  magma: Norm(Conductor(E)); Discriminant: $$(80)$$ = $$(a^3-a^2-3a+2)^{8}\cdot(a)^{4}$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); Discriminant norm: $$40960000$$ = $$4^{8}\cdot5^{4}$$ sage: E.discriminant().norm()  gp: norm(E.disc)  magma: Norm(Discriminant(E)); j-invariant: $$\frac{55296}{5}$$ 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: $$\Z/2\Z\times\Z/8\Z$$ sage: T = E.torsion_subgroup(); T.invariants()  gp: T = elltors(E); T[2]  magma: T,piT := TorsionSubgroup(E); Invariants(T); Torsion generators: $\left(a^{2} - 3 : 0 : 1\right)$ $\left(a^{3} + 2 a^{2} - 2 a - 3 : 2 a^{3} + 4 a^{2} - 2 a - 5 : 1\right)$ 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: $$1242.96341366577$$ Tamagawa product: $$16$$  =  $$2^{2}\cdot2^{2}$$ Torsion order: $$16$$ Leading coefficient: $$1.73709417906366$$ 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$$(j)_{-}$$
$$(a^3-a^2-3a+2)$$ $$4$$ $$4$$ $$I_{1}^{*}$$ Additive $$-1$$ $$3$$ $$8$$ $$0$$
$$(a)$$ $$5$$ $$4$$ $$I_{4}$$ 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
$$2$$ 2Cs

## Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 2, 4 and 8.
Its isogeny class 320.1-g consists of curves linked by isogenies of degrees dividing 16.

## Base change

This curve is the base change of 40.a3, 200.c3, defined over $$\Q$$, so it is also a $$\Q$$-curve.