# Properties

 Label 2.0.3.1-43776.1-o2 Base field $$\Q(\sqrt{-3})$$ Conductor $$(-240a+144)$$ Conductor norm $$43776$$ CM no Base change no Q-curve no Torsion order $$2$$ Rank $$1$$

# Related objects

Show commands for: 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

$${y}^2={x}^{3}+\left(405a-93\right){x}-888a-3478$$
sage: E = EllipticCurve([K([0,0]),K([0,0]),K([0,0]),K([-93,405]),K([-3478,-888])])

gp: E = ellinit([Pol(Vecrev([0,0])),Pol(Vecrev([0,0])),Pol(Vecrev([0,0])),Pol(Vecrev([-93,405])),Pol(Vecrev([-3478,-888]))], K);

magma: E := EllipticCurve([K![0,0],K![0,0],K![0,0],K![-93,405],K![-3478,-888]]);

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 Conductor: $$(-240a+144)$$ = $$(-2a+1)^{2}\cdot(2)^{4}\cdot(-5a+3)$$ sage: E.conductor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor norm: $$43776$$ = $$3^{2}\cdot4^{4}\cdot19$$ sage: E.conductor().norm()  gp: idealnorm(ellglobalred(E)[1])  magma: Norm(Conductor(E)); Discriminant: $$(-752799744a-3510853632)$$ = $$(-2a+1)^{9}\cdot(2)^{12}\cdot(-5a+3)^{6}$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); Discriminant norm: $$15535770395285127168$$ = $$3^{9}\cdot4^{12}\cdot19^{6}$$ sage: E.discriminant().norm()  gp: norm(E.disc)  magma: Norm(Discriminant(E)); j-invariant: $$-\frac{36038181633}{47045881} a - \frac{39546962313}{47045881}$$ 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: $$1$$ Generator $\left(-7 a + 16 : 12 a + 10 : 1\right)$ Height $$2.39012195071157$$ 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: $\left(-14 a + 8 : 0 : 1\right)$ sage: T.gens()  gp: T[3]  magma: [piT(P) : P in Generators(T)];

## BSD invariants

 Analytic rank: $$1$$ sage: E.rank()  magma: Rank(E); Mordell-Weil rank: $$1$$ Regulator: $$2.39012195071157$$ Period: $$0.339142190988430$$ Tamagawa product: $$8$$  =  $$2\cdot2\cdot2$$ Torsion order: $$2$$ Leading coefficient: $$3.74396035752145$$ 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$$(j)_{-}$$
$$(-2a+1)$$ $$3$$ $$2$$ $$III^{*}$$ Additive $$1$$ $$2$$ $$9$$ $$0$$
$$(2)$$ $$4$$ $$2$$ $$I_4^{*}$$ Additive $$1$$ $$4$$ $$12$$ $$0$$
$$(-5a+3)$$ $$19$$ $$2$$ $$I_{6}$$ Non-split multiplicative $$1$$ $$1$$ $$6$$ $$6$$

## Galois Representations

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

prime Image of Galois Representation
$$2$$ 2B
$$3$$ 3B[2]

## Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 2, 3 and 6.
Its isogeny class 43776.1-o consists of curves linked by isogenies of degrees dividing 6.

## Base change

This curve is not the base change of an elliptic curve defined over $$\Q$$. It is not a $$\Q$$-curve.