Base field \(\Q(\sqrt{-1}) \)
Generator \(i\), with minimal polynomial \( x^{2} + 1 \); class number \(1\).
magma: R<x> := PolynomialRing(Rationals()); K<i> := NumberField(R![1, 0, 1]);
sage: x = polygen(QQ); K.<i> = NumberField(x^2 + 1)
gp (2.8): K = nfinit(i^2 + 1);
Weierstrass equation
magma: E := ChangeRing(EllipticCurve([i + 1, i, 0, 2, 3*i]),K);
sage: E = EllipticCurve(K, [i + 1, i, 0, 2, 3*i])
gp (2.8): E = ellinit([i + 1, i, 0, 2, 3*i],K)
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| \(\mathfrak{N} \) | = | \((8)\) | = | \( \left(i + 1\right)^{6} \) |
| magma: Conductor(E);
sage: E.conductor()
| ||||
| \(N(\mathfrak{N}) \) | = | \( 64 \) | = | \( 2^{6} \) |
| magma: Norm(Conductor(E));
sage: E.conductor().norm()
| ||||
| \(\mathfrak{D}\) | = | \((8)\) | = | \( \left(i + 1\right)^{6} \) |
| 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()
| ||||
| \( \text{ST} (E) \) | = | $\mathrm{U}(1)$ | ||
Mordell-Weil group
Rank not available.magma: Rank(E);
sage: E.rank()
magma: Generators(E); // includes torsion
sage: E.gens()
Regulator: not available
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: | $\left(-i : -1 : 1\right)$ |
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \( \left(i + 1\right) \) | \(2\) | \(1\) | \(II\) | Additive | \(-1\) | \(6\) | \(6\) | \(0\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p \) .
The image is a Borel subgroup if \(p=2\), a split Cartan subgroup if \(\left(\frac{ -1 }{p}\right)=+1\) or a nonsplit Cartan subgroup if \(\left(\frac{ -1 }{p}\right)=-1\).
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies (excluding endomorphisms) of degree \(d\) for \(d=\)
2.
Its isogeny class
64.1-CMa
consists of curves linked by isogenies of
degree 2.