Properties

Base field \(\Q(\sqrt{209}) \)
Label 2.2.209.1-1.1-a2
Conductor \((1)\)
Conductor norm \( 1 \)
CM yes (\(-19\))
base-change no
Q-curve yes
Torsion order \( 1 \)
Rank not available

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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

\( y^2 + y = x^{3} + \left(-12880 a - 86662\right) x + 2141323 a + 14407712 \)
magma: E := ChangeRing(EllipticCurve([0, 0, 1, -12880*a - 86662, 2141323*a + 14407712]),K);
 
sage: E = EllipticCurve(K, [0, 0, 1, -12880*a - 86662, 2141323*a + 14407712])
 
gp (2.8): E = ellinit([0, 0, 1, -12880*a - 86662, 2141323*a + 14407712],K)
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

\(\mathfrak{N} \) = \((1)\) = \((1)\)
magma: Conductor(E);
 
sage: E.conductor()
 
\(N(\mathfrak{N}) \) = \( 1 \) = 1
magma: Norm(Conductor(E));
 
sage: E.conductor().norm()
 
\(\mathfrak{D}\) = \((1)\) = \((1)\)
magma: Discriminant(E);
 
sage: E.discriminant()
 
gp (2.8): E.disc
 
\(N(\mathfrak{D})\) = \( 1 \) = 1
magma: Norm(Discriminant(E));
 
sage: E.discriminant().norm()
 
gp (2.8): norm(E.disc)
 
\(j\) = \( -884736 \)
magma: jInvariant(E);
 
sage: E.j_invariant()
 
gp (2.8): E.j
 
\( \text{End} (E) \) = \(\Z[(1+\sqrt{-19})/2]\)   ( 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()
 
No primes of bad reduction.

Galois Representations

The mod \( p \) Galois Representation has maximal image for all primes \( p \) except those listed.

prime Image of Galois Representation
\(11\) 11Ns.3.1
\(19\) 19B.18.6

For all other primes \(p\), the image is the normalizer of a split Cartan subgroup if \(\left(\frac{ -19 }{p}\right)=+1\) or the normalizer of a nonsplit Cartan subgroup if \(\left(\frac{ -19 }{p}\right)=-1\).

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 19.
Its isogeny class 1.1-a consists of curves linked by isogenies of degree 19.

Base change

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