Properties

Base field \(\Q(\sqrt{-2}) \)
Label 2.0.8.1-32.1-a2
Conductor \((4 a)\)
Conductor norm \( 32 \)
CM yes (\(-4\))
base-change yes: 64.a4,32.a4
Q-curve yes
Torsion order \( 4 \)
Rank \( 0 \)

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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: \(\Z/4\Z\)
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: $\left(-1 : -a : 1\right)$
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\((j)_{-}\)
\( \left(a\right) \) \(2\) \(2\) \(III^*\) Additive \(-1\) \(5\) \(12\) \(0\)

Galois Representations

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

The image is a Borel subgroup if \(p=2\), 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 and 4.
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 64.a4, 32.a4, defined over \(\Q\), so it is also a \(\Q\)-curve.