# Properties

 Base field $$\Q(\sqrt{5})$$ Label 2.2.5.1-31.1-a4 Conductor $$(5 \phi - 2)$$ Conductor norm $$31$$ CM no base-change no Q-curve no Torsion order $$2$$ 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

$$y^2 + x y + \phi y = x^{3} + \left(\phi + 1\right) x^{2} + \left(31 \phi - 75\right) x + 141 \phi - 303$$
sage: E = EllipticCurve(K, [1, phi + 1, phi, 31*phi - 75, 141*phi - 303])

gp: E = ellinit([1, phi + 1, phi, 31*phi - 75, 141*phi - 303],K)

magma: E := ChangeRing(EllipticCurve([1, phi + 1, phi, 31*phi - 75, 141*phi - 303]),K);

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 $$\mathfrak{N}$$ = $$(5 \phi - 2)$$ = $$\left(5 \phi - 2\right)$$ sage: E.conductor()  magma: Conductor(E); $$N(\mathfrak{N})$$ = $$31$$ = $$31$$ sage: E.conductor().norm()  magma: Norm(Conductor(E)); $$\mathfrak{D}$$ = $$(204376 \phi - 1053557)$$ = $$\left(5 \phi - 2\right)^{8}$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); $$N(\mathfrak{D})$$ = $$852891037441$$ = $$31^{8}$$ sage: E.discriminant().norm()  gp: norm(E.disc)  magma: Norm(Discriminant(E)); $$j$$ = $$\frac{11889611722383394}{852891037441} \phi - \frac{8629385062119691}{852891037441}$$ sage: E.j_invariant()  gp: E.j  magma: jInvariant(E); $$\text{End} (E)$$ = $$\Z$$ (no Complex Multiplication ) sage: E.has_cm(), E.cm_discriminant()  magma: HasComplexMultiplication(E); $$\text{ST} (E)$$ = $\mathrm{SU}(2)$

## Mordell-Weil group

Rank: $$0$$

sage: E.rank()

magma: Rank(E);

sage: gens = E.gens(); gens

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

sage: E.regulator_of_points(gens)

magma: Regulator(gens);

## Torsion subgroup

Structure: $$\Z/2\Z$$ sage: T = E.torsion_subgroup(); T.invariants()  gp: T = elltors(E); T[2]  magma: T,piT := TorsionSubgroup(E); Invariants(T); $\left(-7 \phi + \frac{27}{4} : 3 \phi - \frac{27}{8} : 1\right)$ sage: T.gens()  gp: T[3]  magma: [piT(P) : P in Generators(T)];

## 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$$(j)_{-}$$
$$\left(5 \phi - 2\right)$$ $$31$$ $$2$$ $$I_{8}$$ Non-split multiplicative $$1$$ $$1$$ $$8$$ $$8$$

## 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.