# Properties

 Label 2.2.33.1-539.1-e1 Base field $$\Q(\sqrt{33})$$ Conductor norm $$539$$ CM no Base change yes Q-curve yes Torsion order $$2$$ Rank $$1$$

# Related objects

Show commands: Magma / PariGP / SageMath

## Base field$$\Q(\sqrt{33})$$

Generator $$a$$, with minimal polynomial $$x^{2} - x - 8$$; class number $$1$$.

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-8, -1, 1]))

gp: K = nfinit(Polrev([-8, -1, 1]));

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-8, -1, 1]);

## Weierstrass equation

$${y}^2+{x}{y}+a{y}={x}^{3}-a{x}^{2}+\left(1280a+3040\right){x}-161701a-383601$$
sage: E = EllipticCurve([K([1,0]),K([0,-1]),K([0,1]),K([3040,1280]),K([-383601,-161701])])

gp: E = ellinit([Polrev([1,0]),Polrev([0,-1]),Polrev([0,1]),Polrev([3040,1280]),Polrev([-383601,-161701])], K);

magma: E := EllipticCurve([K![1,0],K![0,-1],K![0,1],K![3040,1280],K![-383601,-161701]]);

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 Conductor: $$(-28a-63)$$ = $$(-4a-9)\cdot(7)$$ sage: E.conductor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor norm: $$539$$ = $$11\cdot49$$ sage: E.conductor().norm()  gp: idealnorm(ellglobalred(E)[1])  magma: Norm(Conductor(E)); Discriminant: $$(-41503)$$ = $$(-4a-9)^{4}\cdot(7)^{3}$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); Discriminant norm: $$1722499009$$ = $$11^{4}\cdot49^{3}$$ sage: E.discriminant().norm()  gp: norm(E.disc)  magma: Norm(Discriminant(E)); j-invariant: $$\frac{4657463}{41503}$$ 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 $\left(\frac{205}{9} a + \frac{479}{9} : -\frac{2417}{9} a - \frac{17158}{27} : 1\right)$ Height $$1.1839287023003136282328748766856106169$$ 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: $\left(13 a + 30 : -7 a - 15 : 1\right)$ 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: $$1.1839287023003136282328748766856106169$$ Period: $$3.2101873408696119145423660433455397356$$ Tamagawa product: $$6$$  =  $$2\cdot3$$ Torsion order: $$2$$ Leading coefficient: $$1.9848158161738105597445225579762314471$$ 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$$(j)_{-}$$
$$(-4a-9)$$ $$11$$ $$2$$ $$I_{4}$$ Non-split multiplicative $$1$$ $$1$$ $$4$$ $$4$$
$$(7)$$ $$49$$ $$3$$ $$I_{3}$$ Split multiplicative $$-1$$ $$1$$ $$3$$ $$3$$

## 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.
Its isogeny class 539.1-e consists of curves linked by isogenies of degree 2.

## Base change

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

Base field Curve
$$\Q$$ 693.a2
$$\Q$$ 847.a2