# Properties

 Base field $\Q(\sqrt{5})$ Label 2.2.5.1-31.1-a1 Conductor $\left(5 \phi - 2\right)$ 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

 $\mathfrak{N}$ = $\left(5 \phi - 2\right)$ = $\left(5 \phi - 2\right)$ magma: Conductor(E); sage: E.conductor() $N(\mathfrak{N})$ = $31$ = $31$ magma: Norm(Conductor(E)); sage: E.conductor().norm() $\mathfrak{D}$ = $\left(5 \phi - 2\right)$ = $\left(5 \phi - 2\right)$ 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 rank and generators

Rank: $0$
magma: Rank(E);
sage: E.rank()
magma: Generators(E); // includes torsion
sage: E.gens()

Regulator: 1

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 ord($\mathfrak{N}$) ord($\mathfrak{D}$) ord$(j)_{-}$
$\left(5 \phi - 2\right)$ 31 $1$ $I_{1}$ Non-split multiplicative 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.