# Properties

 Label 2.2.5.1-81.1-a6 Base field $$\Q(\sqrt{5})$$ Conductor norm $$81$$ CM yes ($$-60$$) Base change no Q-curve yes Torsion order $$2$$ Rank $$0$$

# Related objects

Show commands: Magma / PariGP / 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(Polrev([-1, -1, 1]));

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

## Weierstrass equation

$${y}^2+\phi{x}{y}+\phi{y}={x}^{3}+\left(-\phi-1\right){x}^{2}+\left(-113\phi-125\right){x}+867\phi+384$$
sage: E = EllipticCurve([K([0,1]),K([-1,-1]),K([0,1]),K([-125,-113]),K([384,867])])

gp: E = ellinit([Polrev([0,1]),Polrev([-1,-1]),Polrev([0,1]),Polrev([-125,-113]),Polrev([384,867])], K);

magma: E := EllipticCurve([K![0,1],K![-1,-1],K![0,1],K![-125,-113],K![384,867]]);

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 Conductor: $$(9)$$ = $$(3)^{2}$$ sage: E.conductor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor norm: $$81$$ = $$9^{2}$$ sage: E.conductor().norm()  gp: idealnorm(ellglobalred(E)[1])  magma: Norm(Conductor(E)); Discriminant: $$(19683)$$ = $$(3)^{9}$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); Discriminant norm: $$387420489$$ = $$9^{9}$$ sage: E.discriminant().norm()  gp: norm(E.disc)  magma: Norm(Discriminant(E)); j-invariant: $$-16554983445 \phi + 26786530035$$ sage: E.j_invariant()  gp: E.j  magma: jInvariant(E); Endomorphism ring: $$\Z$$ Geometric endomorphism ring: $$\Z[\sqrt{-15}]$$ (potential complex multiplication) sage: E.has_cm(), E.cm_discriminant()  magma: HasComplexMultiplication(E); Sato-Tate group: $N(\mathrm{U}(1))$

## 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: $\left(\frac{19}{4} \phi - \frac{5}{4} : -\frac{9}{4} \phi - \frac{19}{8} : 1\right)$ 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: $$2.6250240279447933084927913747793015088$$ Tamagawa product: $$2$$ Torsion order: $$2$$ Leading coefficient: $$0.58697321690548654029979273929421712342$$ 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$$(j)_{-}$$
$$(3)$$ $$9$$ $$2$$ $$III^{*}$$ 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
$$3$$ 3B.1.2

For all other primes $$p$$, the image is a Borel subgroup if $$p\in \{ 2, 5\}$$, the normalizer of a split Cartan subgroup if $$\left(\frac{ -15 }{p}\right)=+1$$ or the normalizer of a nonsplit Cartan subgroup if $$\left(\frac{ -15 }{p}\right)=-1$$.

## Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 2, 3, 5, 6, 10, 15 and 30.
Its isogeny class 81.1-a consists of curves linked by isogenies of degrees dividing 30.

## Base change

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

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