# Properties

 Base field $$\Q(\sqrt{209})$$ Label 2.2.209.1-4.1-d2 Conductor $$(2)$$ Conductor norm $$4$$ CM no base-change no Q-curve yes Torsion order $$2$$ Rank not available

# Related objects

Show commands for: Magma / Pari/GP / SageMath

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

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

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

gp: K = nfinit(a^2 - a - 52);

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

## Weierstrass equation

$$y^2 + x y = x^{3} - x^{2} + \left(950 a + 6392\right) x - 42668 a - 287088$$
sage: E = EllipticCurve(K, [1, -1, 0, 950*a + 6392, -42668*a - 287088])

gp: E = ellinit([1, -1, 0, 950*a + 6392, -42668*a - 287088],K)

magma: E := ChangeRing(EllipticCurve([1, -1, 0, 950*a + 6392, -42668*a - 287088]),K);

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 $$\mathfrak{N}$$ = $$(2)$$ = $$\left(11 a + 74\right) \cdot \left(-11 a + 85\right)$$ sage: E.conductor()  magma: Conductor(E); $$N(\mathfrak{N})$$ = $$4$$ = $$2^{2}$$ sage: E.conductor().norm()  magma: Norm(Conductor(E)); $$\mathfrak{D}$$ = $$(-2624 a - 17600)$$ = $$\left(11 a + 74\right)^{6} \cdot \left(-11 a + 85\right)^{15}$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); $$N(\mathfrak{D})$$ = $$2097152$$ = $$2^{21}$$ sage: E.discriminant().norm()  gp: norm(E.disc)  magma: Norm(Discriminant(E)); $$j$$ = $$-\frac{239886134047017}{32768} a + \frac{463484975124303}{8192}$$ sage: E.j_invariant()  gp: E.j  magma: jInvariant(E); $$\text{End} (E)$$ = $$\Z$$ (no Complex Multiplication ) sage: E.has_cm(), E.cm_discriminant()  magma: HasComplexMultiplication(E); $$\text{ST} (E)$$ = $\mathrm{SU}(2)$

## Mordell-Weil group

Rank not available.

sage: E.rank()

magma: Rank(E);

Regulator: not available

sage: gens = E.gens(); gens

magma: gens := [P:P in Generators(E)|Order(P) eq 0]; gens;

sage: E.regulator_of_points(gens)

magma: Regulator(gens);

## Torsion subgroup

Structure: $$\Z/2\Z$$ sage: T = E.torsion_subgroup(); T.invariants()  gp: T = elltors(E); T[2]  magma: T,piT := TorsionSubgroup(E); Invariants(T); $\left(\frac{11}{4} a + 19 : -\frac{11}{8} a - \frac{19}{2} : 1\right)$ sage: T.gens()  gp: T[3]  magma: [piT(P) : P in Generators(T)];

## 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$$(j)_{-}$$
$$\left(11 a + 74\right)$$ $$2$$ $$2$$ $$I_{6}$$ Non-split multiplicative $$1$$ $$1$$ $$6$$ $$6$$
$$\left(-11 a + 85\right)$$ $$2$$ $$1$$ $$I_{15}$$ Non-split multiplicative $$1$$ $$1$$ $$15$$ $$15$$

## 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
$$3$$ 3Nn
$$5$$ 5B

## Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 2, 5 and 10.
Its isogeny class 4.1-d consists of curves linked by isogenies of degrees dividing 10.

## Base change

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