# Properties

 Base field $$\Q(\sqrt{5})$$ Label 2.2.5.1-31.1-a1 Conductor $$(5 \phi - 2)$$ Conductor norm $$31$$ CM no base-change no Q-curve no Torsion order $$8$$ Rank $$0$$

# 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

$$y^2 + x y + \phi y = x^{3} + \left(\phi + 1\right) x^{2} + \phi x$$
magma: E := ChangeRing(EllipticCurve([1, phi + 1, phi, phi, 0]),K);
sage: E = EllipticCurve(K, [1, phi + 1, phi, phi, 0])
gp (2.8): E = ellinit([1, phi + 1, phi, phi, 0],K)

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 $$\mathfrak{N}$$ = $$(5 \phi - 2)$$ = $$\left(5 \phi - 2\right)$$ magma: Conductor(E); sage: E.conductor() $$N(\mathfrak{N})$$ = $$31$$ = $$31$$ magma: Norm(Conductor(E)); sage: E.conductor().norm() $$\mathfrak{D}$$ = $$(5 \phi - 2)$$ = $$\left(5 \phi - 2\right)$$ magma: Discriminant(E); sage: E.discriminant() gp (2.8): E.disc $$N(\mathfrak{D})$$ = $$31$$ = $$31$$ magma: Norm(Discriminant(E)); sage: E.discriminant().norm() gp (2.8): norm(E.disc) $$j$$ = $$-\frac{106208}{31} \phi + \frac{51455}{31}$$ 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: $$0$$
magma: Rank(E);
sage: E.rank()
magma: Generators(E); // includes torsion
sage: E.gens()
magma: Regulator(Generators(E));
sage: E.regulator_of_points(E.gens())

## Torsion subgroup

Structure: $$\Z/8\Z$$ 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] $\left(-1 : 0 : 1\right)$ 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$$(j)_{-}$$
$$\left(5 \phi - 2\right)$$ $$31$$ $$1$$ $$I_{1}$$ Non-split multiplicative $$1$$ $$1$$ $$1$$ $$1$$

## Galois Representations

The mod $$p$$ Galois Representation has maximal image for all primes $$p$$ 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, 4 and 8.
Its isogeny class 31.1-a consists of curves linked by isogenies of degrees dividing 8.

## Base change

This curve is not the base-change of an elliptic curve defined over $$\Q$$. It is not a $$\Q$$-curve.