# Properties

 Base field $$\Q(\sqrt{5})$$ Label 2.2.5.1-3249.1-f3 Conductor $$(57)$$ Conductor norm $$3249$$ CM no base-change yes: 1425.a2,57.c2 Q-curve yes Torsion order $$4$$ Rank $$1$$

# Learn more about

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

$$y^2 + x y + y = x^{3} - 7 x + 5$$
magma: E := ChangeRing(EllipticCurve([1, 0, 1, -7, 5]),K);

sage: E = EllipticCurve(K, [1, 0, 1, -7, 5])

gp (2.8): E = ellinit([1, 0, 1, -7, 5],K)

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 $$\mathfrak{N}$$ = $$(57)$$ = $$\left(3\right) \cdot \left(4 \phi - 3\right) \cdot \left(-4 \phi + 1\right)$$ magma: Conductor(E);  sage: E.conductor() $$N(\mathfrak{N})$$ = $$3249$$ = $$9 \cdot 19^{2}$$ magma: Norm(Conductor(E));  sage: E.conductor().norm() $$\mathfrak{D}$$ = $$(3249)$$ = $$\left(3\right)^{2} \cdot \left(4 \phi - 3\right)^{2} \cdot \left(-4 \phi + 1\right)^{2}$$ magma: Discriminant(E);  sage: E.discriminant()  gp (2.8): E.disc $$N(\mathfrak{D})$$ = $$10556001$$ = $$9^{2} \cdot 19^{4}$$ magma: Norm(Discriminant(E));  sage: E.discriminant().norm()  gp (2.8): norm(E.disc) $$j$$ = $$\frac{30664297}{3249}$$ 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: $$1$$
magma: Rank(E);

sage: E.rank()

Generator: $\left(\phi + 1 : 1 : 1\right)$

Height: 0.1954606462301571

magma: Generators(E); // includes torsion

sage: E.gens()

Regulator: 0.19546064623

magma: Regulator(Generators(E));

sage: E.regulator_of_points(E.gens())

## Torsion subgroup

Structure: $$\Z/2\Z\times\Z/2\Z$$ 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] $\left(-3 : 1 : 1\right)$,$\left(1 : -1 : 1\right)$ 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$$(j)_{-}$$
$$\left(4 \phi - 3\right)$$ $$19$$ $$2$$ $$I_{2}$$ Non-split multiplicative $$1$$ $$1$$ $$2$$ $$2$$
$$\left(-4 \phi + 1\right)$$ $$19$$ $$2$$ $$I_{2}$$ Non-split multiplicative $$1$$ $$1$$ $$2$$ $$2$$
$$\left(3\right)$$ $$9$$ $$2$$ $$I_{2}$$ Split multiplicative $$-1$$ $$1$$ $$2$$ $$2$$

## Galois Representations

The mod $$p$$ Galois Representation has maximal image for all primes $$p$$ except those listed.

prime Image of Galois Representation
$$2$$ 2Cs

## Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 2.
Its isogeny class 3249.1-f consists of curves linked by isogenies of degrees dividing 4.

## Base change

This curve is the base-change of elliptic curves 1425.a2, 57.c2, defined over $$\Q$$, so it is also a $$\Q$$-curve.