Properties

Base field \(\Q(\sqrt{-11}) \)
Label 2.0.11.1-6400.2-k4
Conductor \((80)\)
Conductor norm \( 6400 \)
CM no
base-change yes: 9680.q1,80.a1
Q-curve yes
Torsion order \( 4 \)
Rank \( 1 \)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

Base field \(\Q(\sqrt{-11}) \)

Generator \(a\), with minimal polynomial \( x^{2} - x + 3 \); class number \(1\).

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^2 - x + 3)
 
gp (2.8): K = nfinit(a^2 - a + 3);
 

Weierstrass equation

\( y^2 = x^{3} - 107 x + 426 \)
magma: E := ChangeRing(EllipticCurve([0, 0, 0, -107, 426]),K);
 
sage: E = EllipticCurve(K, [0, 0, 0, -107, 426])
 
gp (2.8): E = ellinit([0, 0, 0, -107, 426],K)
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

\(\mathfrak{N} \) = \((80)\) = \( \left(2\right)^{4} \cdot \left(-a - 1\right) \cdot \left(a - 2\right) \)
magma: Conductor(E);
 
sage: E.conductor()
 
\(N(\mathfrak{N}) \) = \( 6400 \) = \( 4^{4} \cdot 5^{2} \)
magma: Norm(Conductor(E));
 
sage: E.conductor().norm()
 
\(\mathfrak{D}\) = \((5120)\) = \( \left(2\right)^{10} \cdot \left(-a - 1\right) \cdot \left(a - 2\right) \)
magma: Discriminant(E);
 
sage: E.discriminant()
 
gp (2.8): E.disc
 
\(N(\mathfrak{D})\) = \( 26214400 \) = \( 4^{10} \cdot 5^{2} \)
magma: Norm(Discriminant(E));
 
sage: E.discriminant().norm()
 
gp (2.8): norm(E.disc)
 
\(j\) = \( \frac{132304644}{5} \)
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: \( 1 \)
magma: Rank(E);
 
sage: E.rank()
 

Generator: $\left(-\frac{5}{9} a + \frac{20}{3} : -\frac{65}{27} a + \frac{26}{9} : 1\right)$

Height: 2.1780391255541485

magma: Generators(E); // includes torsion
 
sage: E.gens()
 

Regulator: 2.17803912555

magma: Regulator(Generators(E));
 
sage: E.regulator_of_points(E.gens())
 

Torsion subgroup

Structure: \(\Z/4\Z\)
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: $\left(5 : -4 : 1\right)$
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\((j)_{-}\)
\( \left(-a - 1\right) \) \(5\) \(1\) \(I_{1}\) Split multiplicative \(-1\) \(1\) \(1\) \(1\)
\( \left(a - 2\right) \) \(5\) \(1\) \(I_{1}\) Split multiplicative \(-1\) \(1\) \(1\) \(1\)
\( \left(2\right) \) \(4\) \(4\) \(I_{2}^*\) Additive \(1\) \(4\) \(10\) \(0\)

Galois Representations

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

prime Image of Galois Representation
\(2\) 2B

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2 and 4.
Its isogeny class 6400.2-k consists of curves linked by isogenies of degrees dividing 4.

Base change

This curve is the base-change of elliptic curves 9680.q1, 80.a1, defined over \(\Q\), so it is also a \(\Q\)-curve.