Properties

 Label 2.2.5.1-31.1-a1 Base field $$\Q(\sqrt{5})$$ 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 / Pari/GP / SageMath

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

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

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

gp: K = nfinit(phi^2 - phi - 1);

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

Weierstrass equation

$$y^2+xy+\phiy=x^{3}+\left(\phi+1\right)x^{2}+\phix$$
sage: E = EllipticCurve(K, [1, phi + 1, phi, phi, 0])

gp: E = ellinit([1, phi + 1, phi, phi, 0],K)

magma: E := ChangeRing(EllipticCurve([1, phi + 1, phi, phi, 0]),K);

This is a global minimal model.

sage: E.is_global_minimal_model()

Invariants

 Conductor: $$(5 \phi - 2)$$ = $$\left(5 \phi - 2\right)$$ sage: E.conductor()  magma: Conductor(E); Conductor norm: $$31$$ = $$31$$ sage: E.conductor().norm()  magma: Norm(Conductor(E)); Discriminant: $$(5 \phi - 2)$$ = $$\left(5 \phi - 2\right)$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); Discriminant norm: $$31$$ = $$31$$ sage: E.discriminant().norm()  gp: norm(E.disc)  magma: Norm(Discriminant(E)); j-invariant: $$-\frac{106208}{31} \phi + \frac{51455}{31}$$ sage: E.j_invariant()  gp: E.j  magma: jInvariant(E); Endomorphism ring: $$\Z$$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) sage: E.has_cm(), E.cm_discriminant()  magma: HasComplexMultiplication(E); Sato-Tate group: $\mathrm{SU}(2)$

Mordell-Weil group

 Rank: $$0$$ Torsion structure: $$\Z/8\Z$$ sage: T = E.torsion_subgroup(); T.invariants()  gp: T = elltors(E); T[2]  magma: T,piT := TorsionSubgroup(E); Invariants(T); Torsion generator: $\left(-1 : 0 : 1\right)$ sage: T.gens()  gp: T[3]  magma: [piT(P) : P in Generators(T)];

BSD invariants

 Analytic rank: $$0$$ sage: E.rank()  magma: Rank(E); Mordell-Weil rank: $$0$$ Regulator: $$1$$ Period: $$51.5088397125366$$ Tamagawa product: $$1$$ Torsion order: $$8$$ Leading coefficient: $$0.359928959498039$$ Analytic order of Ш: $$1$$ (rounded)

Local data at primes of bad reduction

sage: E.local_data()

magma: LocalInformation(E);

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 $$p < 1000$$ 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, 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.