Properties

Base field \(\Q(\sqrt{-2}) \)
Label 2.0.8.1-9.1-CMa1
Conductor \((2 a - 1)\)
Conductor norm \( 9 \)
CM yes (\(-8\))
base-change no
Q-curve yes
Torsion order \( 6 \)
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

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

This is a global minimal model.

sage: E.is_global_minimal_model()

Invariants

\(\mathfrak{N} \) = \((2 a - 1)\) = \( \left(-a - 1\right)^{2} \)
magma: Conductor(E);
sage: E.conductor()
\(N(\mathfrak{N}) \) = \( 9 \) = \( 3^{2} \)
magma: Norm(Conductor(E));
sage: E.conductor().norm()
\(\mathfrak{D}\) = \((-10 a + 23)\) = \( \left(-a - 1\right)^{6} \)
magma: Discriminant(E);
sage: E.discriminant()
gp (2.8): E.disc
\(N(\mathfrak{D})\) = \( 729 \) = \( 3^{6} \)
magma: Norm(Discriminant(E));
sage: E.discriminant().norm()
gp (2.8): norm(E.disc)
\(j\) = \( 8000 \)
magma: jInvariant(E);
sage: E.j_invariant()
gp (2.8): E.j
\( \text{End} (E) \) = \(\Z[\sqrt{-2}]\)   ( Complex Multiplication )
magma: HasComplexMultiplication(E);
sage: E.has_cm(), E.cm_discriminant()
\( \text{ST} (E) \) = $\mathrm{U}(1)$

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: \(\Z/6\Z\)
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: $\left(a : -a + 1 : 1\right)$
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\((j)_{-}\)
\( \left(-a - 1\right) \) \(3\) \(2\) \(I_{0}^*\) Additive \(-1\) \(2\) \(6\) \(0\)

Galois Representations

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

prime Image of Galois Representation
\(3\) 3Cs.1.1

For all other primes \(p\), the image is a Borel subgroup if \(p=2\), a split Cartan subgroup if \(\left(\frac{ -2 }{p}\right)=+1\) or a nonsplit Cartan subgroup if \(\left(\frac{ -2 }{p}\right)=-1\).

Isogenies and isogeny class

This curve has no rational isogenies other than endomorphisms. Its isogeny class 9.1-CMa consists of this curve only.

Base change

This curve is not the base-change of an elliptic curve defined over \(\Q\). It is a \(\Q\)-curve.