# Properties

 Label 2.2.5.1-605.2-a2 Base field $$\Q(\sqrt{5})$$ Conductor norm $$605$$ CM no Base change no Q-curve no Torsion order $$2$$ Rank $$0$$

# Related objects

Show commands: Magma / Pari/GP / SageMath

## Base field$$\Q(\sqrt{5})$$

Generator $$\phi$$, with minimal polynomial $$x^{2} - x - 1$$; class number $$1$$.

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

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

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

## Weierstrass equation

$${y}^2+\phi{x}{y}={x}^{3}+\phi{x}^{2}+\left(3\phi-195\right){x}+847\phi-500$$
sage: E = EllipticCurve([K([0,1]),K([0,1]),K([0,0]),K([-195,3]),K([-500,847])])

gp: E = ellinit([Pol(Vecrev([0,1])),Pol(Vecrev([0,1])),Pol(Vecrev([0,0])),Pol(Vecrev([-195,3])),Pol(Vecrev([-500,847]))], K);

magma: E := EllipticCurve([K![0,1],K![0,1],K![0,0],K![-195,3],K![-500,847]]);

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 Conductor: $$(-4\phi-23)$$ = $$(-2\phi+1)\cdot(-3\phi+2)^{2}$$ sage: E.conductor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor norm: $$605$$ = $$5\cdot11^{2}$$ sage: E.conductor().norm()  gp: idealnorm(ellglobalred(E)[1])  magma: Norm(Conductor(E)); Discriminant: $$(499750\phi+5845625)$$ = $$(-2\phi+1)^{6}\cdot(-3\phi+2)^{9}$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); Discriminant norm: $$36842932671875$$ = $$5^{6}\cdot11^{9}$$ sage: E.discriminant().norm()  gp: norm(E.disc)  magma: Norm(Discriminant(E)); j-invariant: $$-\frac{393466}{25} \phi + \frac{17856881}{125}$$ 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: $$\Z/2\Z$$ sage: T = E.torsion_subgroup(); T.invariants()  gp: T = elltors(E); T[2]  magma: T,piT := TorsionSubgroup(E); Invariants(T); Torsion generator: $\left(\frac{27}{4} \phi - \frac{17}{4} : -\frac{5}{4} \phi - \frac{27}{8} : 1\right)$ 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: $$3.03830961336023$$ Tamagawa product: $$4$$  =  $$2\cdot2$$ Torsion order: $$2$$ Leading coefficient: $$1.35877336643291$$ 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$$(j)_{-}$$
$$(-2\phi+1)$$ $$5$$ $$2$$ $$I_{6}$$ Non-split multiplicative $$1$$ $$1$$ $$6$$ $$6$$
$$(-3\phi+2)$$ $$11$$ $$2$$ $$III^{*}$$ Additive $$1$$ $$2$$ $$9$$ $$0$$

## Galois Representations

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

prime Image of Galois Representation
$$2$$ 2B
$$3$$ 3Ns

## Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 2.
Its isogeny class 605.2-a consists of curves linked by isogenies of degree 2.

## Base change

This elliptic curve is not a $$\Q$$-curve.

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