Properties

Base field \(\Q(\sqrt{209}) \)
Label 2.2.209.1-4.1-c2
Conductor \((2)\)
Conductor norm \( 4 \)
CM no
base-change no
Q-curve yes
Torsion order \( 2 \)
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

\( y^2 + x y + y = x^{3} - x^{2} + \left(109626655 a - 847240411\right) x + 1679777810708 a - 12982021956321 \)
magma: E := ChangeRing(EllipticCurve([1, -1, 1, 109626655*a - 847240411, 1679777810708*a - 12982021956321]),K);
 
sage: E = EllipticCurve(K, [1, -1, 1, 109626655*a - 847240411, 1679777810708*a - 12982021956321])
 
gp (2.8): E = ellinit([1, -1, 1, 109626655*a - 847240411, 1679777810708*a - 12982021956321],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()
 
\(\mathfrak{D}\) = \((-2624 a - 17600)\) = \( \left(11 a + 74\right)^{6} \cdot \left(-11 a + 85\right)^{15} \)
magma: Discriminant(E);
 
sage: E.discriminant()
 
gp (2.8): E.disc
 
\(N(\mathfrak{D})\) = \( 2097152 \) = \( 2^{21} \)
magma: Norm(Discriminant(E));
 
sage: E.discriminant().norm()
 
gp (2.8): norm(E.disc)
 
\(j\) = \( -\frac{239886134047017}{32768} a + \frac{463484975124303}{8192} \)
magma: jInvariant(E);
 
sage: E.j_invariant()
 
gp (2.8): 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()
 
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: \(\Z/2\Z\)
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]
 
Generator: $\left(\frac{6359}{4} a - 12286 : -\frac{6359}{8} a + \frac{12285}{2} : 1\right)$
magma: [f(P): P in Generators(T)] where T,f:=TorsionSubgroup(E);
 
sage: E.torsion_subgroup().gens()
 
gp (2.8): elltors(E)[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\((j)_{-}\)
\( \left(11 a + 74\right) \) \(2\) \(6\) \(I_{6}\) Split multiplicative \(-1\) \(1\) \(6\) \(6\)
\( \left(-11 a + 85\right) \) \(2\) \(15\) \(I_{15}\) Split multiplicative \(-1\) \(1\) \(15\) \(15\)

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-c 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.