# Properties

 Label 2.2.57.1-57.1-e1 Base field $$\Q(\sqrt{57})$$ Conductor norm $$57$$ CM no Base change yes Q-curve yes Torsion order $$1$$ Rank $$1$$

# 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

$${y}^2+{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-28187a-92304\right){x}-5536728a-18132329$$
sage: E = EllipticCurve([K([0,0]),K([-1,1]),K([1,0]),K([-92304,-28187]),K([-18132329,-5536728])])

gp: E = ellinit([Polrev([0,0]),Polrev([-1,1]),Polrev([1,0]),Polrev([-92304,-28187]),Polrev([-18132329,-5536728])], K);

magma: E := EllipticCurve([K![0,0],K![-1,1],K![1,0],K![-92304,-28187],K![-18132329,-5536728]]);

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 Conductor: $$(-2a+1)$$ = $$(4a+13)\cdot(10a-43)$$ sage: E.conductor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor norm: $$57$$ = $$3\cdot19$$ sage: E.conductor().norm()  gp: idealnorm(ellglobalred(E)[1])  magma: Norm(Conductor(E)); Discriminant: $$(-171)$$ = $$(4a+13)^{4}\cdot(10a-43)^{2}$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); Discriminant norm: $$29241$$ = $$3^{4}\cdot19^{2}$$ 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: $$1$$ Generator $\left(83 a + \frac{1093}{4} : \frac{4545}{4} a + \frac{29765}{8} : 1\right)$ Height $$1.1941568102687657884912118247284091796$$ 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: $$1.1941568102687657884912118247284091796$$ Period: $$3.6799880851352519841225526402809908993$$ Tamagawa product: $$8$$  =  $$2^{2}\cdot2$$ Torsion order: $$1$$ Leading coefficient: $$9.3130155290131626709094480889472566230$$ 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$$(j)_{-}$$
$$(4a+13)$$ $$3$$ $$4$$ $$I_{4}$$ Split multiplicative $$-1$$ $$1$$ $$4$$ $$4$$
$$(10a-43)$$ $$19$$ $$2$$ $$I_{2}$$ Non-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 57.1-e consists of this curve only.

## Base change

This elliptic curve is a $$\Q$$-curve. It is the base change of the following 2 elliptic curves:

Base field Curve
$$\Q$$ 171.d1
$$\Q$$ 1083.e1