# Properties

 Label 2.0.3.1-1521.1-CMb1 Base field $$\Q(\sqrt{-3})$$ Conductor norm $$1521$$ CM yes ($$-3$$) Base change no Q-curve yes Torsion order $$1$$ Rank $$0$$

# Related objects

Show commands: Magma / Pari/GP / SageMath

## Base field$$\Q(\sqrt{-3})$$

Generator $$a$$, 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

$${y}^2+a{y}={x}^{3}-136a+165$$
sage: E = EllipticCurve([K([0,0]),K([0,0]),K([0,1]),K([0,0]),K([165,-136])])

gp: E = ellinit([Pol(Vecrev([0,0])),Pol(Vecrev([0,0])),Pol(Vecrev([0,1])),Pol(Vecrev([0,0])),Pol(Vecrev([165,-136]))], K);

magma: E := EllipticCurve([K![0,0],K![0,0],K![0,1],K![0,0],K![165,-136]]);

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 Conductor: $$(-45a+21)$$ = $$(-2a+1)^{2}\cdot(-4a+1)^{2}$$ sage: E.conductor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor norm: $$1521$$ = $$3^{2}\cdot13^{2}$$ sage: E.conductor().norm()  gp: idealnorm(ellglobalred(E)[1])  magma: Norm(Conductor(E)); Discriminant: $$(11362275a-3764664)$$ = $$(-2a+1)^{6}\cdot(-4a+1)^{10}$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); Discriminant norm: $$100498840557921$$ = $$3^{6}\cdot13^{10}$$ 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: $$\Z[(1+\sqrt{-3})/2]$$ (complex multiplication) Geometric endomorphism ring: $$\Z[(1+\sqrt{-3})/2]$$ sage: E.has_cm(), E.cm_discriminant()  magma: HasComplexMultiplication(E); Sato-Tate group: $\mathrm{U}(1)$

## Mordell-Weil group

 Rank: $$0$$ 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: $$0$$ sage: E.rank()  magma: Rank(E); Mordell-Weil rank: $$0$$ Regulator: $$1$$ Period: $$0.956447781058546$$ Tamagawa product: $$1$$  =  $$1\cdot1$$ Torsion order: $$1$$ Leading coefficient: $$1.10441076771994$$ 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$$(j)_{-}$$
$$(-2a+1)$$ $$3$$ $$1$$ $$I_0^{*}$$ Additive $$-1$$ $$2$$ $$6$$ $$0$$
$$(-4a+1)$$ $$13$$ $$1$$ $$II^{*}$$ Additive $$1$$ $$2$$ $$10$$ $$0$$

## Galois Representations

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

prime Image of Galois Representation
$$13$$ 13Cs.5.1

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

## Isogenies and isogeny class

This curve has no rational isogenies other than endomorphisms. Its isogeny class 1521.1-CMb consists of this curve only.

## Base change

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

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