# Properties

 Label 2.2.5.1-2025.1-d2 Base field $$\Q(\sqrt{5})$$ Conductor norm $$2025$$ CM yes ($$-3$$) Base change yes Q-curve yes Torsion order $$1$$ Rank $$1$$

# 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

$${y}^2+{y}={x}^{3}-34$$
sage: E = EllipticCurve([K([0,0]),K([0,0]),K([1,0]),K([0,0]),K([-34,0])])

gp: E = ellinit([Pol(Vecrev([0,0])),Pol(Vecrev([0,0])),Pol(Vecrev([1,0])),Pol(Vecrev([0,0])),Pol(Vecrev([-34,0]))], K);

magma: E := EllipticCurve([K![0,0],K![0,0],K![1,0],K![0,0],K![-34,0]]);

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 Conductor: $$(45)$$ = $$(-2\phi+1)^{2}\cdot(3)^{2}$$ sage: E.conductor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor norm: $$2025$$ = $$5^{2}\cdot9^{2}$$ sage: E.conductor().norm()  gp: idealnorm(ellglobalred(E)[1])  magma: Norm(Conductor(E)); Discriminant: $$(-492075)$$ = $$(-2\phi+1)^{4}\cdot(3)^{9}$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); Discriminant norm: $$242137805625$$ = $$5^{4}\cdot9^{9}$$ sage: E.discriminant().norm()  gp: norm(E.disc)  magma: Norm(Discriminant(E)); j-invariant: $$0$$ sage: E.j_invariant()  gp: E.j  magma: jInvariant(E); Endomorphism ring: $$\Z$$ Geometric endomorphism ring: $$\Z[(1+\sqrt{-3})/2]$$ (potential complex multiplication) sage: E.has_cm(), E.cm_discriminant()  magma: HasComplexMultiplication(E); Sato-Tate group: $N(\mathrm{U}(1))$

## Mordell-Weil group

 Rank: $$1$$ Generator $\left(6 : -14 : 1\right)$ Height $$0.460920105039883$$ Torsion structure: trivial sage: T = E.torsion_subgroup(); T.invariants()  gp: T = elltors(E); T[2]  magma: T,piT := TorsionSubgroup(E); Invariants(T);

## BSD invariants

 Analytic rank: $$1$$ sage: E.rank()  magma: Rank(E); Mordell-Weil rank: $$1$$ Regulator: $$0.460920105039883$$ Period: $$1.82517916703785$$ Tamagawa product: $$2$$  =  $$1\cdot2$$ Torsion order: $$1$$ Leading coefficient: $$1.50489480973353$$ 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 $$-1$$ $$2$$ $$4$$ $$0$$
$$(3)$$ $$9$$ $$2$$ $$III^{*}$$ Additive $$1$$ $$2$$ $$9$$ $$0$$

## Galois Representations

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

prime Image of Galois Representation
$$3$$ 3B.1.2
$$5$$ 5Cs[2]

For all other primes $$p$$, the image is the normalizer of a split Cartan subgroup if $$\left(\frac{ -3 }{p}\right)=+1$$ or the normalizer of a nonsplit Cartan subgroup if $$\left(\frac{ -3 }{p}\right)=-1$$.

## Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 3.
Its isogeny class 2025.1-d consists of curves linked by isogenies of degree 3.

## Base change

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

Base field Curve
$$\Q$$ 225.c1
$$\Q$$ 225.d1