# Properties

 Base field $$\Q(\sqrt{209})$$ Label 2.2.209.1-16.5-a1 Conductor $$(15 a + 101)$$ Conductor norm $$16$$ CM yes ($$-11$$) base-change no Q-curve yes 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: K = nfinit(a^2 - a - 52);

## Weierstrass equation

$$y^2 + \left(a + 1\right) y = x^{3} + \left(a - 1\right) x^{2} + \left(-931337 a - 6266403\right) x - 1351685915 a - 9094705362$$
magma: E := ChangeRing(EllipticCurve([0, a - 1, a + 1, -931337*a - 6266403, -1351685915*a - 9094705362]),K);

sage: E = EllipticCurve(K, [0, a - 1, a + 1, -931337*a - 6266403, -1351685915*a - 9094705362])

gp: E = ellinit([0, a - 1, a + 1, -931337*a - 6266403, -1351685915*a - 9094705362],K)

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 $$\mathfrak{N}$$ = $$(15 a + 101)$$ = $$\left(-11 a + 85\right)^{4}$$ magma: Conductor(E);  sage: E.conductor() $$N(\mathfrak{N})$$ = $$16$$ = $$2^{4}$$ 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: E.disc $$N(\mathfrak{D})$$ = $$4096$$ = $$2^{12}$$ magma: Norm(Discriminant(E));  sage: E.discriminant().norm()  gp: norm(E.disc) $$j$$ = $$-32768$$ magma: jInvariant(E);  sage: E.j_invariant()  gp: E.j $$\text{End} (E)$$ = $$\Z[(1+\sqrt{-11})/2]$$ ( Complex Multiplication ) magma: HasComplexMultiplication(E);  sage: E.has_cm(), E.cm_discriminant() $$\text{ST} (E)$$ = $N(\mathrm{U}(1))$

## 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: Trivial magma: T,piT := TorsionSubgroup(E); Invariants(T);  sage: T = E.torsion_subgroup(); T.invariants()  gp: T = elltors(E); T[2]

## 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$$(j)_{-}$$
$$\left(-11 a + 85\right)$$ $$2$$ $$1$$ $$II^*$$ Additive $$-1$$ $$4$$ $$12$$ $$0$$

## Galois Representations

The mod $$p$$ Galois Representation has maximal image for all primes $$p < 1000$$ except those listed.

prime Image of Galois Representation
$$11$$ 11B.10.3

For all other primes $$p$$, the image is the normalizer of a split Cartan subgroup if $$\left(\frac{ -11 }{p}\right)=+1$$ or the normalizer of a nonsplit Cartan subgroup if $$\left(\frac{ -11 }{p}\right)=-1$$.

## Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 11.
Its isogeny class 16.5-a consists of curves linked by isogenies of degree 11.

## Base change

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