# Properties

 Label 2.2.76.1-171.1-j1 Base field $$\Q(\sqrt{19})$$ Conductor $$(3 a)$$ Conductor norm $$171$$ CM no Base change yes: 912.g1,1083.e1 Q-curve yes Torsion order $$1$$ Rank $$1$$

# Related objects

Show commands for: Magma / Pari/GP / SageMath

## Base field$$\Q(\sqrt{19})$$

Generator $$a$$, with minimal polynomial $$x^{2} - 19$$; class number $$1$$.

sage: x = polygen(QQ); K.<a> = NumberField(x^2 - 19)

gp: K = nfinit(a^2 - 19);

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-19, 0, 1]);

## Weierstrass equation

$$y^2+ay=x^{3}+x^{2}-2x-7$$
sage: E = EllipticCurve(K, [0, 1, a, -2, -7])

gp: E = ellinit([0, 1, a, -2, -7],K)

magma: E := ChangeRing(EllipticCurve([0, 1, a, -2, -7]),K);

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 Conductor: $$(3 a)$$ = $$\left(-a - 4\right) \cdot \left(-a + 4\right) \cdot \left(a\right)$$ sage: E.conductor()  magma: Conductor(E); Conductor norm: $$171$$ = $$3^{2} \cdot 19$$ sage: E.conductor().norm()  magma: Norm(Conductor(E)); Discriminant: $$(171)$$ = $$\left(-a - 4\right)^{2} \cdot \left(-a + 4\right)^{2} \cdot \left(a\right)^{2}$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); Discriminant norm: $$29241$$ = $$3^{4} \cdot 19^{2}$$ sage: E.discriminant().norm()  gp: norm(E.disc)  magma: Norm(Discriminant(E)); j-invariant: $$-\frac{1404928}{171}$$ 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 $\left(\frac{7}{4} : -\frac{1}{8} a : 1\right)$ Height $$1.19415681026877$$ 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: $$1$$ sage: E.rank()  magma: Rank(E); Mordell-Weil rank: $$1$$ Regulator: $$1.19415681026877$$ Period: $$3.67998808513525$$ Tamagawa product: $$8$$  =  $$2\cdot2\cdot2$$ Torsion order: $$1$$ Leading coefficient: $$8.06530803396437$$ 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$$(j)_{-}$$
$$\left(-a - 4\right)$$ $$3$$ $$2$$ $$I_{2}$$ Split multiplicative $$-1$$ $$1$$ $$2$$ $$2$$
$$\left(-a + 4\right)$$ $$3$$ $$2$$ $$I_{2}$$ Split multiplicative $$-1$$ $$1$$ $$2$$ $$2$$
$$\left(a\right)$$ $$19$$ $$2$$ $$I_{2}$$ Split multiplicative $$-1$$ $$1$$ $$2$$ $$2$$

## Galois Representations

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

## Isogenies and isogeny class

This curve has no rational isogenies. Its isogeny class 171.1-j consists of this curve only.

## Base change

This curve is the base change of elliptic curves 912.g1, 1083.e1, defined over $$\Q$$, so it is also a $$\Q$$-curve.