# Properties

 Label 2.2.57.1-171.1-i1 Base field $$\Q(\sqrt{57})$$ Conductor norm $$171$$ CM no Base change no Q-curve yes Torsion order $$1$$ Rank $$0$$

# Related objects

Show commands: Magma / PariGP / SageMath

## Base field$$\Q(\sqrt{57})$$

Generator $$a$$, with minimal polynomial $$x^{2} - x - 14$$; class number $$1$$.

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-14, -1, 1]))

gp: K = nfinit(Polrev([-14, -1, 1]));

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-14, -1, 1]);

## Weierstrass equation

$${y}^2+{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-279a-912\right){x}+5943a+19463$$
sage: E = EllipticCurve([K([0,0]),K([-1,-1]),K([1,0]),K([-912,-279]),K([19463,5943])])

gp: E = ellinit([Polrev([0,0]),Polrev([-1,-1]),Polrev([1,0]),Polrev([-912,-279]),Polrev([19463,5943])], K);

magma: E := EllipticCurve([K![0,0],K![-1,-1],K![1,0],K![-912,-279],K![19463,5943]]);

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 Conductor: $$(30a-129)$$ = $$(4a+13)^{2}\cdot(10a-43)$$ sage: E.conductor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor norm: $$171$$ = $$3^{2}\cdot19$$ sage: E.conductor().norm()  gp: idealnorm(ellglobalred(E)[1])  magma: Norm(Conductor(E)); Discriminant: $$(-4617)$$ = $$(4a+13)^{10}\cdot(10a-43)^{2}$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); Discriminant norm: $$21316689$$ = $$3^{10}\cdot19^{2}$$ 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: $$0$$ 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: $$6.1529882294741298010739064002062539590$$ Tamagawa product: $$4$$  =  $$2\cdot2$$ Torsion order: $$1$$ Leading coefficient: $$3.2599328010315359167746304887339166801$$ 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$$(j)_{-}$$
$$(4a+13)$$ $$3$$ $$2$$ $$I_{4}^{*}$$ Additive $$-1$$ $$2$$ $$10$$ $$4$$
$$(10a-43)$$ $$19$$ $$2$$ $$I_{2}$$ 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-i consists of this curve only.

## Base change

This elliptic curve is a $$\Q$$-curve.

It is not the base change of an elliptic curve defined over any subfield.