# Properties

 Label 2.2.76.1-171.1-f6 Base field $$\Q(\sqrt{19})$$ Conductor $$(3 a)$$ Conductor norm $$171$$ CM no Base change no Q-curve yes Torsion order $$2$$ 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

$$y^2+axy+ay=x^{3}+x^{2}+\left(-175a-860\right)x-3178a-14456$$
sage: E = EllipticCurve(K, [a, 1, a, -175*a - 860, -3178*a - 14456])

gp: E = ellinit([a, 1, a, -175*a - 860, -3178*a - 14456],K)

magma: E := ChangeRing(EllipticCurve([a, 1, a, -175*a - 860, -3178*a - 14456]),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: $$(-1125 a - 4788)$$ = $$\left(-a - 4\right)^{2} \cdot \left(-a + 4\right)^{8} \cdot \left(a\right)$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); Discriminant norm: $$1121931$$ = $$3^{10} \cdot 19$$ sage: E.discriminant().norm()  gp: norm(E.disc)  magma: Norm(Discriminant(E)); j-invariant: $$\frac{80836867580206264835}{124659} a + \frac{18545249299658286056}{6561}$$ 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 $\left(-\frac{414}{25} a + \frac{21149}{450} : \frac{577481}{4500} a - \frac{6839941}{13500} : 1\right)$ Height $$5.20378782686344$$ 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: $\left(-4 a - \frac{31}{4} : \frac{27}{8} a + 38 : 1\right)$ 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: $$5.20378782686344$$ Period: $$0.565538563228076$$ Tamagawa product: $$4$$  =  $$2\cdot2\cdot1$$ Torsion order: $$2$$ Leading coefficient: $$0.675157357182630$$ 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$$(j)_{-}$$
$$\left(-a - 4\right)$$ $$3$$ $$2$$ $$I_{2}$$ Non-split multiplicative $$1$$ $$1$$ $$2$$ $$2$$
$$\left(-a + 4\right)$$ $$3$$ $$2$$ $$I_{8}$$ Non-split multiplicative $$1$$ $$1$$ $$8$$ $$8$$
$$\left(a\right)$$ $$19$$ $$1$$ $$I_{1}$$ Split multiplicative $$-1$$ $$1$$ $$1$$ $$1$$

## 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

## Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 2, 4 and 8.
Its isogeny class 171.1-f consists of curves linked by isogenies of degrees dividing 8.

## Base change

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