# Properties

 Base field $$\Q(\sqrt{209})$$ Label 2.2.209.1-4.1-d3 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 / 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: K = nfinit(a^2 - a - 52);

## Weierstrass equation

$$y^2 + x y = x^{3} - x^{2} + \left(1485209680 a - 11478318473\right) x - 83988775088712 a + 649100205595989$$
magma: E := ChangeRing(EllipticCurve([1, -1, 0, 1485209680*a - 11478318473, -83988775088712*a + 649100205595989]),K);

sage: E = EllipticCurve(K, [1, -1, 0, 1485209680*a - 11478318473, -83988775088712*a + 649100205595989])

gp: E = ellinit([1, -1, 0, 1485209680*a - 11478318473, -83988775088712*a + 649100205595989],K)

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 $$\mathfrak{N}$$ = $$(2)$$ = $$\left(11 a + 74\right) \cdot \left(-11 a + 85\right)$$ magma: Conductor(E);  sage: E.conductor() $$N(\mathfrak{N})$$ = $$4$$ = $$2^{2}$$ magma: Norm(Conductor(E));  sage: E.conductor().norm() $$\mathfrak{D}$$ = $$(-34584 a + 249464)$$ = $$\left(11 a + 74\right)^{3} \cdot \left(-11 a + 85\right)^{30}$$ magma: Discriminant(E);  sage: E.discriminant()  gp: E.disc $$N(\mathfrak{D})$$ = $$8589934592$$ = $$2^{33}$$ magma: Norm(Discriminant(E));  sage: E.discriminant().norm()  gp: norm(E.disc) $$j$$ = $$\frac{24345866052441}{1073741824} a - \frac{46176275773215}{268435456}$$ magma: jInvariant(E);  sage: E.j_invariant()  gp: 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()

Regulator: not available

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

sage: gens = E.gens(); gens

magma: Regulator(gens);

sage: E.regulator_of_points(gens)

## Torsion subgroup

Structure: $$\Z/2\Z$$ magma: T,piT := TorsionSubgroup(E); Invariants(T);  sage: T = E.torsion_subgroup(); T.invariants()  gp: T = elltors(E); T[2] $\left(\frac{46829}{4} a - \frac{361913}{4} : -\frac{46829}{8} a + \frac{361913}{8} : 1\right)$ magma: [piT(P) : P in Generators(T)];  sage: T.gens()  gp: T[3]

## 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$$(j)_{-}$$
$$\left(11 a + 74\right)$$ $$2$$ $$1$$ $$I_{3}$$ Non-split multiplicative $$1$$ $$1$$ $$3$$ $$3$$
$$\left(-11 a + 85\right)$$ $$2$$ $$2$$ $$I_{30}$$ Non-split multiplicative $$1$$ $$1$$ $$30$$ $$30$$

## Galois Representations

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