# Properties

 Base field $$\Q(\sqrt{5})$$ Label 2.2.5.1-2205.1-e3 Conductor $$(-42 \phi + 21)$$ Conductor norm $$2205$$ CM no base-change yes: 105.a3,525.a3 Q-curve yes Torsion order $$4$$ Rank $$1$$

# Learn more about

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: K = nfinit(phi^2 - phi - 1);

## Weierstrass equation

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

sage: E = EllipticCurve(K, [1, 0, 1, -3, 1])

gp: E = ellinit([1, 0, 1, -3, 1],K)

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 $$\mathfrak{N}$$ = $$(-42 \phi + 21)$$ = $$\left(3\right) \cdot \left(-2 \phi + 1\right) \cdot \left(7\right)$$ magma: Conductor(E);  sage: E.conductor() $$N(\mathfrak{N})$$ = $$2205$$ = $$5 \cdot 9 \cdot 49$$ magma: Norm(Conductor(E));  sage: E.conductor().norm() $$\mathfrak{D}$$ = $$(105)$$ = $$\left(3\right) \cdot \left(-2 \phi + 1\right)^{2} \cdot \left(7\right)$$ magma: Discriminant(E);  sage: E.discriminant()  gp: E.disc $$N(\mathfrak{D})$$ = $$11025$$ = $$5^{2} \cdot 9 \cdot 49$$ magma: Norm(Discriminant(E));  sage: E.discriminant().norm()  gp: norm(E.disc) $$j$$ = $$\frac{1771561}{105}$$ magma: jInvariant(E);  sage: E.j_invariant()  gp: 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: $\left(-\phi : \phi : 1\right)$

magma: gens := [P:P in Generators(E)|Order(P) eq 0]; gens;

sage: gens = E.gens(); gens

Height: 0.4578054292188009

magma: [Height(P):P in gens];

sage: [P.height() for P in gens]

Regulator: 0.457805429219

magma: Regulator(gens);

sage: E.regulator_of_points(gens)

## Torsion subgroup

Structure: $$\Z/4\Z$$ magma: T,piT := TorsionSubgroup(E); Invariants(T);  sage: T = E.torsion_subgroup(); T.invariants()  gp: T = elltors(E); T[2] $\left(0 : -\phi : 1\right)$ magma: [piT(P) : P in Generators(T)];  sage: T.gens()  gp: T[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$$(j)_{-}$$
$$\left(-2 \phi + 1\right)$$ $$5$$ $$2$$ $$I_{2}$$ Split multiplicative $$-1$$ $$1$$ $$2$$ $$2$$
$$\left(3\right)$$ $$9$$ $$1$$ $$I_{1}$$ Split multiplicative $$-1$$ $$1$$ $$1$$ $$1$$
$$\left(7\right)$$ $$49$$ $$1$$ $$I_{1}$$ Split multiplicative $$-1$$ $$1$$ $$1$$ $$1$$

## 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

## Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 2 and 4.
Its isogeny class 2205.1-e consists of curves linked by isogenies of degrees dividing 4.

## Base change

This curve is the base-change of elliptic curves 105.a3, 525.a3, defined over $$\Q$$, so it is also a $$\Q$$-curve.