Properties

Base field \(\Q(\sqrt{209}) \)
Label 2.2.209.1-16.4-a2
Conductor \((-15 a + 116)\)
Conductor norm \( 16 \)
CM yes (\(-11\))
base-change no
Q-curve yes
Torsion order \( 1 \)
Rank not available

Related objects

Downloads

Learn more about

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

\( y^2 + a y = x^{3} - a x^{2} + \left(931337 a - 7197740\right) x + 1351685914 a - 10446391276 \)
magma: E := ChangeRing(EllipticCurve([0, -a, a, 931337*a - 7197740, 1351685914*a - 10446391276]),K);
 
sage: E = EllipticCurve(K, [0, -a, a, 931337*a - 7197740, 1351685914*a - 10446391276])
 
gp (2.8): E = ellinit([0, -a, a, 931337*a - 7197740, 1351685914*a - 10446391276],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.