# Properties

 Label 2.2.76.1-171.1-g1 Base field $$\Q(\sqrt{19})$$ Conductor $$(3 a)$$ Conductor norm $$171$$ CM no Base change no Q-curve no Torsion order $$1$$ Rank $$0$$

# 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

$$y^2+axy+y=x^{3}+\left(659571a-2874998\right)x+609801836a-2658064575$$
sage: E = EllipticCurve(K, [a, 0, 1, 659571*a - 2874998, 609801836*a - 2658064575])

gp: E = ellinit([a, 0, 1, 659571*a - 2874998, 609801836*a - 2658064575],K)

magma: E := ChangeRing(EllipticCurve([a, 0, 1, 659571*a - 2874998, 609801836*a - 2658064575]),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: $$(-1701 a - 9234)$$ = $$\left(-a - 4\right)^{5} \cdot \left(-a + 4\right)^{8} \cdot \left(a\right)$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); Discriminant norm: $$30292137$$ = $$3^{13} \cdot 19$$ sage: E.discriminant().norm()  gp: norm(E.disc)  magma: Norm(Discriminant(E)); j-invariant: $$\frac{12364905437067631}{124659} a - \frac{2836704000281954}{6561}$$ 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: $$0$$ 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: $$0$$ sage: E.rank()  magma: Rank(E); Mordell-Weil rank: $$0$$ Regulator: $$1$$ Period: $$1.31649668297455$$ Tamagawa product: $$40$$  =  $$5\cdot2^{3}\cdot1$$ Torsion order: $$1$$ Leading coefficient: $$6.04050105325535$$ 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$$(j)_{-}$$
$$\left(-a - 4\right)$$ $$3$$ $$5$$ $$I_{5}$$ Split multiplicative $$-1$$ $$1$$ $$5$$ $$5$$
$$\left(-a + 4\right)$$ $$3$$ $$8$$ $$I_{8}$$ Split multiplicative $$-1$$ $$1$$ $$8$$ $$8$$
$$\left(a\right)$$ $$19$$ $$1$$ $$I_{1}$$ Non-split multiplicative $$1$$ $$1$$ $$1$$ $$1$$

## 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-g consists of this curve only.

## Base change

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