Properties

Base field \(\Q(\sqrt{-2}) \)
Label 2.0.8.1-32.1-a1
Conductor \((4 a)\)
Conductor norm \( 32 \)
CM yes (\(-4\))
base-change yes: 32.a3,64.a3
Q-curve yes
Torsion order \( 4 \)
Rank \( 0 \)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

Base field \(\Q(\sqrt{-2}) \)

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

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^2 + 2)
 
gp (2.8): K = nfinit(a^2 + 2);
 

Weierstrass equation

\( y^2 = x^{3} - x \)
magma: E := ChangeRing(EllipticCurve([0, 0, 0, -1, 0]),K);
 
sage: E = EllipticCurve(K, [0, 0, 0, -1, 0])
 
gp (2.8): E = ellinit([0, 0, 0, -1, 0],K)
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

\(\mathfrak{N} \) = \((4 a)\) = \( \left(a\right)^{5} \)
magma: Conductor(E);
 
sage: E.conductor()
 
\(N(\mathfrak{N}) \) = \( 32 \) = \( 2^{5} \)
magma: Norm(Conductor(E));
 
sage: E.conductor().norm()
 
\(\mathfrak{D}\) = \((64)\) = \( \left(a\right)^{12} \)
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)
 
\(j\) = \( 1728 \)
magma: jInvariant(E);
 
sage: E.j_invariant()
 
gp (2.8): E.j
 
\( \text{End} (E) \) = \(\Z[\sqrt{-1}]\)   ( Complex Multiplication )
magma: HasComplexMultiplication(E);
 
sage: E.has_cm(), E.cm_discriminant()
 
\( \text{ST} (E) \) = $N(\mathrm{U}(1))$

Mordell-Weil group

Rank: \( 0 \)
magma: Rank(E);
 
sage: E.rank()
 
magma: Generators(E); // includes torsion
 
sage: E.gens()
 

Regulator: 1

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

Torsion subgroup

Structure: \(\Z/2\Z\times\Z/2\Z\)
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]
 
Generators: $\left(-1 : 0 : 1\right)$,$\left(0 : 0 : 1\right)$
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\((j)_{-}\)
\( \left(a\right) \) \(2\) \(2\) \(I_{3}^*\) Additive \(-1\) \(5\) \(12\) \(0\)

Galois Representations

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

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

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

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2.
Its isogeny class 32.1-a consists of curves linked by isogenies of degrees dividing 4.

Base change

This curve is the base-change of elliptic curves 32.a3, 64.a3, defined over \(\Q\), so it is also a \(\Q\)-curve.