# Properties

 Base field $$\Q(\sqrt{5})$$ Label 2.2.5.1-2420.1-e1 Conductor $$(-44 \phi + 22)$$ Conductor norm $$2420$$ CM no base-change yes: 110.a1,550.i1 Q-curve yes Torsion order $$3$$ Rank $$1$$

# Related objects

Show commands for: Magma / SageMath / Pari/GP

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

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

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

sage: x = polygen(QQ); K.<phi> = NumberField(x^2 - x - 1)

gp (2.8): K = nfinit(phi^2 - phi - 1);

## Weierstrass equation

$$y^2 + x y + y = x^{3} - 89 x + 316$$
magma: E := ChangeRing(EllipticCurve([1, 0, 1, -89, 316]),K);

sage: E = EllipticCurve(K, [1, 0, 1, -89, 316])

gp (2.8): E = ellinit([1, 0, 1, -89, 316],K)

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 $$\mathfrak{N}$$ = $$(-44 \phi + 22)$$ = $$\left(2\right) \cdot \left(-2 \phi + 1\right) \cdot \left(-3 \phi + 2\right) \cdot \left(-3 \phi + 1\right)$$ magma: Conductor(E);  sage: E.conductor() $$N(\mathfrak{N})$$ = $$2420$$ = $$4 \cdot 5 \cdot 11^{2}$$ magma: Norm(Conductor(E));  sage: E.conductor().norm() $$\mathfrak{D}$$ = $$(851840)$$ = $$\left(2\right)^{7} \cdot \left(-2 \phi + 1\right)^{2} \cdot \left(-3 \phi + 2\right)^{3} \cdot \left(-3 \phi + 1\right)^{3}$$ magma: Discriminant(E);  sage: E.discriminant()  gp (2.8): E.disc $$N(\mathfrak{D})$$ = $$725631385600$$ = $$4^{7} \cdot 5^{2} \cdot 11^{6}$$ magma: Norm(Discriminant(E));  sage: E.discriminant().norm()  gp (2.8): norm(E.disc) $$j$$ = $$-\frac{76711450249}{851840}$$ magma: jInvariant(E);  sage: E.j_invariant()  gp (2.8): E.j $$\text{End} (E)$$ = $$\Z$$ (no Complex Multiplication ) magma: HasComplexMultiplication(E);  sage: E.has_cm(), E.cm_discriminant() $$\text{ST} (E)$$ = $\mathrm{SU}(2)$

## Mordell-Weil group

Rank: $$1$$
magma: Rank(E);

sage: E.rank()

Generator: $\left(2 \phi + 7 : -10 \phi - 12 : 1\right)$

Height: 0.02140014800682888

magma: Generators(E); // includes torsion

sage: E.gens()

Regulator: 0.0214001480068

magma: Regulator(Generators(E));

sage: E.regulator_of_points(E.gens())

## Torsion subgroup

Structure: $$\Z/3\Z$$ magma: TorsionSubgroup(E);  sage: E.torsion_subgroup().gens()  gp (2.8): elltors(E)[2] magma: Order(TorsionSubgroup(E));  sage: E.torsion_order()  gp (2.8): elltors(E)[1] $\left(4 : -8 : 1\right)$ magma: [f(P): P in Generators(T)] where T,f:=TorsionSubgroup(E);  sage: E.torsion_subgroup().gens()  gp (2.8): elltors(E)[3]

## Local data at primes of bad reduction

magma: LocalInformation(E);

sage: E.local_data()

prime Norm Tamagawa number Kodaira symbol Reduction type Root number ord($$\mathfrak{N}$$) ord($$\mathfrak{D}$$) ord$$(j)_{-}$$
$$\left(-2 \phi + 1\right)$$ $$5$$ $$2$$ $$I_{2}$$ Non-split multiplicative $$1$$ $$1$$ $$2$$ $$2$$
$$\left(-3 \phi + 2\right)$$ $$11$$ $$3$$ $$I_{3}$$ Split multiplicative $$-1$$ $$1$$ $$3$$ $$3$$
$$\left(-3 \phi + 1\right)$$ $$11$$ $$3$$ $$I_{3}$$ Split multiplicative $$-1$$ $$1$$ $$3$$ $$3$$
$$\left(2\right)$$ $$4$$ $$7$$ $$I_{7}$$ Split multiplicative $$-1$$ $$1$$ $$7$$ $$7$$

## Galois Representations

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

prime Image of Galois Representation
$$3$$ 3B.1.1

## Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 3.
Its isogeny class 2420.1-e consists of curves linked by isogenies of degree 3.

## Base change

This curve is the base-change of elliptic curves 110.a1, 550.i1, defined over $$\Q$$, so it is also a $$\Q$$-curve.