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
Q-curve no
Torsion order \( 8 \)
Rank \( 0 \)

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

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

magma: K<phi> := NumberField(x^2 - x - 1);
sage: 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]
Generator: $\left(-1 : 0 : 1\right)$
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