# Properties

 Label 3.3.361.1-19.1-a3 Base field 3.3.361.1 Conductor norm $$19$$ CM no Base change yes Q-curve yes Torsion order $$9$$ Rank $$0$$

# Related objects

Show commands: Magma / PariGP / SageMath

## Base field3.3.361.1

Generator $$a$$, with minimal polynomial $$x^{3} - x^{2} - 6 x + 7$$; class number $$1$$.

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([7, -6, -1, 1]))

gp: K = nfinit(Polrev([7, -6, -1, 1]));

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![7, -6, -1, 1]);

## Weierstrass equation

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

gp: E = ellinit([Polrev([0,0,0]),Polrev([1,0,0]),Polrev([1,0,0]),Polrev([1,0,0]),Polrev([0,0,0])], K);

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

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 Conductor: $$(a^2-a-4)$$ = $$(a^2-a-4)$$ sage: E.conductor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor norm: $$19$$ = $$19$$ sage: E.conductor().norm()  gp: idealnorm(ellglobalred(E)[1])  magma: Norm(Conductor(E)); Discriminant: $$(-19)$$ = $$(a^2-a-4)^{3}$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); Discriminant norm: $$-6859$$ = $$-19^{3}$$ sage: E.discriminant().norm()  gp: norm(E.disc)  magma: Norm(Discriminant(E)); j-invariant: $$\frac{32768}{19}$$ 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: $$0$$ Torsion structure: $$\Z/9\Z$$ sage: T = E.torsion_subgroup(); T.invariants()  gp: T = elltors(E); T[2]  magma: T,piT := TorsionSubgroup(E); Invariants(T); Torsion generator: $\left(-a + 2 : a^{2} - 2 a - 1 : 1\right)$ 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: $$67.881322206799970684689363509881789017$$ Tamagawa product: $$3$$ Torsion order: $$9$$ Leading coefficient: $$1.1909003895929819418366555001733647196$$ Analytic order of Ш: $$9$$ (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$$(j)_{-}$$
$$(a^2-a-4)$$ $$19$$ $$3$$ $$I_{3}$$ Split multiplicative $$-1$$ $$1$$ $$3$$ $$3$$

## 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.1.1

## Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 3 and 9.
Its isogeny class 19.1-a consists of curves linked by isogenies of degrees dividing 9.

## Base change

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

Base field Curve
$$\Q$$ 19.a3