Properties

Base field \(\Q(\sqrt{2}) \)
Label 2.2.8.1-32.1-a6
Conductor \((4 a)\)
Conductor norm \( 32 \)
CM yes (\(-16\))
base-change yes: 32.a2,64.a2
Q-curve yes
Torsion order \( 8 \)
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

\( y^2 + a x y + a y = x^{3} + x^{2} - 3 x \)
magma: E := ChangeRing(EllipticCurve([a, 1, a, -3, 0]),K);
sage: E = EllipticCurve(K, [a, 1, a, -3, 0])
gp (2.8): E = ellinit([a, 1, a, -3, 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()
\(\mathfrak{D}\) = \((8)\) = \( \left(a\right)^{6} \)
magma: Discriminant(E);
sage: E.discriminant()
gp (2.8): E.disc
\(N(\mathfrak{D})\) = \( 64 \) = \( 2^{6} \)
magma: Norm(Discriminant(E));
sage: E.discriminant().norm()
gp (2.8): norm(E.disc)
\(j\) = \( 287496 \)
magma: jInvariant(E);
sage: E.j_invariant()
gp (2.8): E.j
\( \text{End} (E) \) = \(\Z[\sqrt{-4}]\)   ( 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: \(\Z/2\Z\times\Z/4\Z\)
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]
Generators: $\left(0 : -a : 1\right)$,$\left(a - 1 : -1 : 1\right)$
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 \(1\) \(5\) \(6\) \(0\)

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, 4 and 8.
Its isogeny class 32.1-a consists of curves linked by isogenies of degrees dividing 16.

Base change

This curve is the base-change of elliptic curves 32.a2, 64.a2, defined over \(\Q\), so it is also a \(\Q\)-curve.