Base field 4.4.9909.1
Generator \(a\), with minimal polynomial \( x^{4} - 6 x^{2} - 3 x + 3 \); class number \(1\).
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3, -3, -6, 0, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^4 - 6*x^2 - 3*x + 3)
gp (2.8): K = nfinit(a^4 - 6*a^2 - 3*a + 3);
Weierstrass equation
magma: E := ChangeRing(EllipticCurve([a^3 - 4*a - 2, a, a^3 - 5*a - 1, 9*a^3 - 18*a^2 - 22*a + 14, -46*a^3 + 74*a^2 + 104*a - 66]),K);
sage: E = EllipticCurve(K, [a^3 - 4*a - 2, a, a^3 - 5*a - 1, 9*a^3 - 18*a^2 - 22*a + 14, -46*a^3 + 74*a^2 + 104*a - 66])
gp (2.8): E = ellinit([a^3 - 4*a - 2, a, a^3 - 5*a - 1, 9*a^3 - 18*a^2 - 22*a + 14, -46*a^3 + 74*a^2 + 104*a - 66],K)
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| \(\mathfrak{N} \) | = | \((13,-a^{3} + a^{2} + 3 a - 1)\) | = | \( \left(-a^{2} + 2 a + 2\right) \) |
| magma: Conductor(E);
sage: E.conductor()
| ||||
| \(N(\mathfrak{N}) \) | = | \( 13 \) | = | \( 13 \) |
| magma: Norm(Conductor(E));
sage: E.conductor().norm()
| ||||
| \(\mathfrak{D}\) | = | \((2197,a + 861,a^{3} - a^{2} - 4 a + 232,a^{2} - a + 404)\) | = | \( \left(-a^{2} + 2 a + 2\right)^{3} \) |
| magma: Discriminant(E);
sage: E.discriminant()
gp (2.8): E.disc
| ||||
| \(N(\mathfrak{D})\) | = | \( 2197 \) | = | \( 13^{3} \) |
| magma: Norm(Discriminant(E));
sage: E.discriminant().norm()
gp (2.8): norm(E.disc)
| ||||
| \(j\) | = | \( \frac{1140361876}{2197} a^{3} - \frac{2035650720}{2197} a^{2} - \frac{6076580550}{2197} a + \frac{3481904573}{2197} \) | ||
| magma: jInvariant(E);
sage: E.j_invariant()
gp (2.8): E.j
| ||||
| \( \text{End} (E) \) | = | \(\Z\) | (no Complex Multiplication ) | |
| magma: HasComplexMultiplication(E);
sage: E.has_cm(), E.cm_discriminant()
| ||||
| \( \text{ST} (E) \) | = | $\mathrm{SU}(2)$ | ||
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: | \(\Z/3\Z\) |
|---|---|
| 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: | $\left(a^{2} - 1 : -5 a^{2} - 2 a + 3 : 1\right)$ |
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \( \left(-a^{2} + 2 a + 2\right) \) | \(13\) | \(3\) | \(I_{3}\) | Split multiplicative | \(-1\) | \(1\) | \(3\) | \(3\) |
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 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
3.
Its isogeny class
13.1-b
consists of curves linked by isogenies of
degrees dividing 9.