Properties

 Label 2.2.5.1-2025.1-e3 Base field $$\Q(\sqrt{5})$$ Conductor norm $$2025$$ CM yes ($$-15$$) Base change no Q-curve yes Torsion order $$2$$ 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

$${y}^2+\left(\phi+1\right){x}{y}+\left(\phi+1\right){y}={x}^{3}+{x}^{2}+\left(6\phi-1\right){x}+13\phi+2$$
sage: E = EllipticCurve([K([1,1]),K([1,0]),K([1,1]),K([-1,6]),K([2,13])])

gp: E = ellinit([Polrev([1,1]),Polrev([1,0]),Polrev([1,1]),Polrev([-1,6]),Polrev([2,13])], K);

magma: E := EllipticCurve([K![1,1],K![1,0],K![1,1],K![-1,6],K![2,13]]);

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: $$(-3375)$$ = $$(-2\phi+1)^{6}\cdot(3)^{3}$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); Discriminant norm: $$11390625$$ = $$5^{6}\cdot9^{3}$$ sage: E.discriminant().norm()  gp: norm(E.disc)  magma: Norm(Discriminant(E)); j-invariant: $$85995 \phi - 138510$$ sage: E.j_invariant()  gp: E.j  magma: jInvariant(E); Endomorphism ring: $$\Z$$ Geometric endomorphism ring: $$\Z[(1+\sqrt{-15})/2]$$ (potential complex multiplication) sage: E.has_cm(), E.cm_discriminant()  magma: HasComplexMultiplication(E); Sato-Tate group: $N(\mathrm{U}(1))$

Mordell-Weil group

 Rank: $$0$$ Torsion structure: $$\Z/2\Z$$ sage: T = E.torsion_subgroup(); T.invariants()  gp: T = elltors(E); T[2]  magma: T,piT := TorsionSubgroup(E); Invariants(T); Torsion generator: $\left(-2 \phi + 1 : \phi : 1\right)$ 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: $$3.5218393014329192417987564357653027406$$ Tamagawa product: $$4$$  =  $$2\cdot2$$ Torsion order: $$2$$ Leading coefficient: $$1.5750144167668759850956748248675809053$$ 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$$(j)_{-}$$
$$(-2\phi+1)$$ $$5$$ $$2$$ $$I_0^{*}$$ Additive $$1$$ $$2$$ $$6$$ $$0$$
$$(3)$$ $$9$$ $$2$$ $$III$$ Additive $$1$$ $$2$$ $$3$$ $$0$$

Galois Representations

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

The image is a Borel subgroup if $$p\in \{ 3, 5\}$$, the normalizer of a split Cartan subgroup if $$\left(\frac{ -15 }{p}\right)=+1$$ or the normalizer of a nonsplit Cartan subgroup if $$\left(\frac{ -15 }{p}\right)=-1$$.

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 2, 3, 5, 6, 10, 15 and 30.
Its isogeny class 2025.1-e consists of curves linked by isogenies of degrees dividing 30.

Base change

This elliptic curve is a $$\Q$$-curve.

It is not the base change of an elliptic curve defined over any subfield.