# Properties

 Base field $$\Q(\sqrt{209})$$ Label 2.2.209.1-2.2-a1 Conductor $$(-11 a + 85)$$ Conductor norm $$2$$ CM no base-change no Q-curve no Torsion order $$1$$ Rank not available

# Related objects

Show commands for: Magma / SageMath / Pari/GP

## Base field $$\Q(\sqrt{209})$$

Generator $$a$$, with minimal polynomial $$x^{2} - x - 52$$; class number $$1$$.

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-52, -1, 1]);

sage: x = polygen(QQ); K.<a> = NumberField(x^2 - x - 52)

gp (2.8): K = nfinit(a^2 - a - 52);

## Weierstrass equation

$$y^2 + a x y + \left(a + 1\right) y = x^{3} + \left(-a - 1\right) x^{2} + \left(-333486 a - 2243755\right) x + 412430147 a + 2775001836$$
magma: E := ChangeRing(EllipticCurve([a, -a - 1, a + 1, -333486*a - 2243755, 412430147*a + 2775001836]),K);

sage: E = EllipticCurve(K, [a, -a - 1, a + 1, -333486*a - 2243755, 412430147*a + 2775001836])

gp (2.8): E = ellinit([a, -a - 1, a + 1, -333486*a - 2243755, 412430147*a + 2775001836],K)

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 $$\mathfrak{N}$$ = $$(-11 a + 85)$$ = $$\left(-11 a + 85\right)$$ magma: Conductor(E);  sage: E.conductor() $$N(\mathfrak{N})$$ = $$2$$ = $$2$$ magma: Norm(Conductor(E));  sage: E.conductor().norm() $$\mathfrak{D}$$ = $$(5 a + 71)$$ = $$\left(-11 a + 85\right)^{12}$$ magma: Discriminant(E);  sage: E.discriminant()  gp (2.8): E.disc $$N(\mathfrak{D})$$ = $$4096$$ = $$2^{12}$$ magma: Norm(Discriminant(E));  sage: E.discriminant().norm()  gp (2.8): norm(E.disc) $$j$$ = $$\frac{1714092597}{4096} a - \frac{3313345635}{1024}$$ magma: jInvariant(E);  sage: E.j_invariant()  gp (2.8): E.j $$\text{End} (E)$$ = $$\Z$$ (no Complex Multiplication ) magma: HasComplexMultiplication(E);  sage: E.has_cm(), E.cm_discriminant() $$\text{ST} (E)$$ = $\mathrm{SU}(2)$

## Mordell-Weil group

Rank not available.
magma: Rank(E);

sage: E.rank()

magma: Generators(E); // includes torsion

sage: E.gens()

Regulator: not available

magma: Regulator(Generators(E));

sage: E.regulator_of_points(E.gens())

## Torsion subgroup

Structure: Trivial magma: TorsionSubgroup(E);  sage: E.torsion_subgroup().gens()  gp (2.8): elltors(E)[2] magma: Order(TorsionSubgroup(E));  sage: E.torsion_order()  gp (2.8): elltors(E)[1]

## Local data at primes of bad reduction

magma: LocalInformation(E);

sage: E.local_data()

prime Norm Tamagawa number Kodaira symbol Reduction type Root number ord($$\mathfrak{N}$$) ord($$\mathfrak{D}$$) ord$$(j)_{-}$$
$$\left(-11 a + 85\right)$$ $$2$$ $$12$$ $$I_{12}$$ Split multiplicative $$-1$$ $$1$$ $$12$$ $$12$$

## Galois Representations

The mod $$p$$ Galois Representation has maximal image for all primes $$p$$ except those listed.

prime Image of Galois Representation
$$3$$ 3Nn

## Isogenies and isogeny class

This curve has no rational isogenies. Its isogeny class 2.2-a 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.