# Properties

 Label 2.0.3.1-21609.3-d4 Base field $$\Q(\sqrt{-3})$$ Conductor norm $$21609$$ CM yes ($$-28$$) Base change no Q-curve yes Torsion order $$2$$ Rank $$0$$

# Related objects

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

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

## Weierstrass equation

$${y}^2+{x}{y}+a{y}={x}^{3}-{x}^{2}+\left(-893a+558\right){x}-5014a+9549$$
sage: E = EllipticCurve([K([1,0]),K([-1,0]),K([0,1]),K([558,-893]),K([9549,-5014])])

gp: E = ellinit([Polrev([1,0]),Polrev([-1,0]),Polrev([0,1]),Polrev([558,-893]),Polrev([9549,-5014])], K);

magma: E := EllipticCurve([K![1,0],K![-1,0],K![0,1],K![558,-893],K![9549,-5014]]);

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 Conductor: $$(147)$$ = $$(-2a+1)^{2}\cdot(-3a+1)^{2}\cdot(3a-2)^{2}$$ sage: E.conductor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor norm: $$21609$$ = $$3^{2}\cdot7^{2}\cdot7^{2}$$ sage: E.conductor().norm()  gp: idealnorm(ellglobalred(E)[1])  magma: Norm(Conductor(E)); Discriminant: $$(-3333960a+2991303)$$ = $$(-2a+1)^{6}\cdot(-3a+1)^{9}\cdot(3a-2)^{3}$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); Discriminant norm: $$10090298369529$$ = $$3^{6}\cdot7^{9}\cdot7^{3}$$ sage: E.discriminant().norm()  gp: norm(E.disc)  magma: Norm(Discriminant(E)); j-invariant: $$16581375$$ sage: E.j_invariant()  gp: E.j  magma: jInvariant(E); Endomorphism ring: $$\Z$$ Geometric endomorphism ring: $$\Z[\sqrt{-7}]$$ (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: $\left(-14 a - \frac{13}{4} : \frac{13}{2} a + \frac{13}{8} : 1\right)$ 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: $$0.53948968106790808373343712452636469645$$ Tamagawa product: $$16$$  =  $$2^{2}\cdot2\cdot2$$ Torsion order: $$2$$ Leading coefficient: $$2.4917961007166566610245869408698200625$$ 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$$(j)_{-}$$
$$(-2a+1)$$ $$3$$ $$4$$ $$I_0^{*}$$ Additive $$-1$$ $$2$$ $$6$$ $$0$$
$$(-3a+1)$$ $$7$$ $$2$$ $$III^{*}$$ Additive $$-1$$ $$2$$ $$9$$ $$0$$
$$(3a-2)$$ $$7$$ $$2$$ $$III$$ 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 $$p\in \{ 2, 7\}$$, the normalizer of a split Cartan subgroup if $$\left(\frac{ -7 }{p}\right)=+1$$ or the normalizer of a nonsplit Cartan subgroup if $$\left(\frac{ -7 }{p}\right)=-1$$.

## Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 2, 7 and 14.
Its isogeny class 21609.3-d consists of curves linked by isogenies of degrees dividing 14.

## Base change

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

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