# Properties

 Label 2.2.5.1-225.1-c2 Base field $$\Q(\sqrt{5})$$ Conductor norm $$225$$ CM no Base change yes Q-curve yes Torsion order $$5$$ Rank $$0$$

# Related objects

Show commands: Magma / PariGP / 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(Polrev([-1, -1, 1]));

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

## Weierstrass equation

$${y}^2+{y}={x}^{3}+{x}^{2}+2{x}+4$$
sage: E = EllipticCurve([K([0,0]),K([1,0]),K([1,0]),K([2,0]),K([4,0])])

gp: E = ellinit([Polrev([0,0]),Polrev([1,0]),Polrev([1,0]),Polrev([2,0]),Polrev([4,0])], K);

magma: E := EllipticCurve([K![0,0],K![1,0],K![1,0],K![2,0],K![4,0]]);

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 Conductor: $$(15)$$ = $$(-2\phi+1)^{2}\cdot(3)$$ sage: E.conductor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor norm: $$225$$ = $$5^{2}\cdot9$$ sage: E.conductor().norm()  gp: idealnorm(ellglobalred(E)[1])  magma: Norm(Conductor(E)); Discriminant: $$(-6075)$$ = $$(-2\phi+1)^{4}\cdot(3)^{5}$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); Discriminant norm: $$36905625$$ = $$5^{4}\cdot9^{5}$$ sage: E.discriminant().norm()  gp: norm(E.disc)  magma: Norm(Discriminant(E)); j-invariant: $$\frac{20480}{243}$$ 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: $$\Z/5\Z$$ sage: T = E.torsion_subgroup(); T.invariants()  gp: T = elltors(E); T[2]  magma: T,piT := TorsionSubgroup(E); Invariants(T); Torsion generator: $\left(-1 : -2 : 1\right)$ 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: $$9.8355914195086055647680533000922758832$$ Tamagawa product: $$5$$  =  $$1\cdot5$$ Torsion order: $$5$$ Leading coefficient: $$0.87972204051739572915450891547308160320$$ 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$$(j)_{-}$$
$$(-2\phi+1)$$ $$5$$ $$1$$ $$IV$$ Additive $$-1$$ $$2$$ $$4$$ $$0$$
$$(3)$$ $$9$$ $$5$$ $$I_{5}$$ Split multiplicative $$-1$$ $$1$$ $$5$$ $$5$$

## Galois Representations

The mod $$p$$ Galois Representation has maximal image for all primes $$p < 1000$$ except those listed.

prime Image of Galois Representation
$$5$$ 5B.1.1[2]

## Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 5.
Its isogeny class 225.1-c consists of curves linked by isogenies of degree 5.

## Base change

This elliptic curve is a $$\Q$$-curve. It is the base change of the following 2 elliptic curves:

Base field Curve
$$\Q$$ 75.a2
$$\Q$$ 75.c2