# Properties

 Label 2.2.5.1-196.1-a4 Base field $$\Q(\sqrt{5})$$ Conductor $$(14)$$ Conductor norm $$196$$ CM no Base change yes: 350.f3,14.a3 Q-curve yes Torsion order $$6$$ Rank $$0$$

# Related objects

Show commands: Magma / Pari/GP / SageMath

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

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

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-1, -1, 1]))

gp: K = nfinit(Pol(Vecrev([-1, -1, 1])));

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

## Weierstrass equation

$${y}^2+{x}{y}+{y}={x}^{3}-36{x}-70$$
sage: E = EllipticCurve([K([1,0]),K([0,0]),K([1,0]),K([-36,0]),K([-70,0])])

gp: E = ellinit([Pol(Vecrev([1,0])),Pol(Vecrev([0,0])),Pol(Vecrev([1,0])),Pol(Vecrev([-36,0])),Pol(Vecrev([-70,0]))], K);

magma: E := EllipticCurve([K![1,0],K![0,0],K![1,0],K![-36,0],K![-70,0]]);

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 Conductor: $$(14)$$ = $$(2)\cdot(7)$$ sage: E.conductor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor norm: $$196$$ = $$4\cdot49$$ sage: E.conductor().norm()  gp: idealnorm(ellglobalred(E)[1])  magma: Norm(Conductor(E)); Discriminant: $$(941192)$$ = $$(2)^{3}\cdot(7)^{6}$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); Discriminant norm: $$885842380864$$ = $$4^{3}\cdot49^{6}$$ sage: E.discriminant().norm()  gp: norm(E.disc)  magma: Norm(Discriminant(E)); j-invariant: $$\frac{4956477625}{941192}$$ 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: $$\Z/6\Z$$ sage: T = E.torsion_subgroup(); T.invariants()  gp: T = elltors(E); T[2]  magma: T,piT := TorsionSubgroup(E); Invariants(T); Torsion generator: $\left(-4 : -2 : 1\right)$ 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: $$3.92571594687094$$ Tamagawa product: $$18$$  =  $$3\cdot( 2 \cdot 3 )$$ Torsion order: $$6$$ Leading coefficient: $$0.877816771755838$$ 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$$(j)_{-}$$
$$(2)$$ $$4$$ $$3$$ $$I_{3}$$ Split multiplicative $$-1$$ $$1$$ $$3$$ $$3$$
$$(7)$$ $$49$$ $$6$$ $$I_{6}$$ Split multiplicative $$-1$$ $$1$$ $$6$$ $$6$$

## 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
$$3$$ 3Cs.1.1

## Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 2, 3 and 6.
Its isogeny class 196.1-a consists of curves linked by isogenies of degrees dividing 18.

## Base change

This curve is the base change of 350.f3, 14.a3, defined over $$\Q$$, so it is also a $$\Q$$-curve.