# Properties

 Base field $$\Q(\sqrt{209})$$ Label 2.2.209.1-16.4-a1 Conductor $$(-15 a + 116)$$ 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 (2.8): K = nfinit(a^2 - a - 52);

## Weierstrass equation

$$y^2 + a y = x^{3} + \left(a - 1\right) x^{2} + \left(-317 a - 2113\right) x + 7753 a + 52137$$
magma: E := ChangeRing(EllipticCurve([0, a - 1, a, -317*a - 2113, 7753*a + 52137]),K);

sage: E = EllipticCurve(K, [0, a - 1, a, -317*a - 2113, 7753*a + 52137])

gp (2.8): E = ellinit([0, a - 1, a, -317*a - 2113, 7753*a + 52137],K)

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 $$\mathfrak{N}$$ = $$(-15 a + 116)$$ = $$\left(11 a + 74\right)^{4}$$ magma: Conductor(E);  sage: E.conductor() $$N(\mathfrak{N})$$ = $$16$$ = $$2^{4}$$ magma: Norm(Conductor(E));  sage: E.conductor().norm() $$\mathfrak{D}$$ = $$(-5 a + 76)$$ = $$\left(11 a + 74\right)^{12}$$ 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) $$j$$ = $$-32768$$ magma: jInvariant(E);  sage: E.j_invariant()  gp (2.8): 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()

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$$(j)_{-}$$
$$\left(11 a + 74\right)$$ $$2$$ $$1$$ $$II^*$$ Additive $$-1$$ $$4$$ $$12$$ $$0$$

## Galois Representations

The mod $$p$$ Galois Representation has maximal image for all primes $$p$$ 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 16.4-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.