Base field \(\Q(\zeta_{21})^+\)
Generator \(a\), with minimal polynomial \( x^{6} - x^{5} - 6 x^{4} + 6 x^{3} + 8 x^{2} - 8 x + 1 \); class number \(1\).
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -8, 8, 6, -6, -1, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^6 - x^5 - 6*x^4 + 6*x^3 + 8*x^2 - 8*x + 1)
gp (2.8): K = nfinit(a^6 - a^5 - 6*a^4 + 6*a^3 + 8*a^2 - 8*a + 1);
Weierstrass equation
magma: E := ChangeRing(EllipticCurve([0, -a^5 + 4*a^3 - 2*a^2 - 2*a + 5, a^4 - 3*a^2 + 1, -2*a^5 + 7*a^4 + 4*a^3 - 28*a^2 + 17*a + 4, -6*a^5 + 13*a^4 + 15*a^3 - 52*a^2 + 33*a - 3]),K);
sage: E = EllipticCurve(K, [0, -a^5 + 4*a^3 - 2*a^2 - 2*a + 5, a^4 - 3*a^2 + 1, -2*a^5 + 7*a^4 + 4*a^3 - 28*a^2 + 17*a + 4, -6*a^5 + 13*a^4 + 15*a^3 - 52*a^2 + 33*a - 3])
gp (2.8): E = ellinit([0, -a^5 + 4*a^3 - 2*a^2 - 2*a + 5, a^4 - 3*a^2 + 1, -2*a^5 + 7*a^4 + 4*a^3 - 28*a^2 + 17*a + 4, -6*a^5 + 13*a^4 + 15*a^3 - 52*a^2 + 33*a - 3],K)
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| \(\mathfrak{N} \) | = | \((3,-2 a^{5} + 10 a^{3} - a^{2} - 10 a + 2)\) | = | \( \left(a^{3} + a^{2} - 2 a - 1\right) \) |
| magma: Conductor(E);
sage: E.conductor()
| ||||
| \(N(\mathfrak{N}) \) | = | \( 27 \) | = | \( 27 \) |
| magma: Norm(Conductor(E));
sage: E.conductor().norm()
| ||||
| \(\mathfrak{D}\) | = | \((3,3 a^{3} - 9 a,3 a^{5} - 15 a^{3} + 3 a^{2} + 15 a - 6,3 a^{4} - 12 a^{2} + 6,3 a,3 a^{2} - 6)\) | = | \( \left(a^{3} + a^{2} - 2 a - 1\right)^{2} \) |
| magma: Discriminant(E);
sage: E.discriminant()
gp (2.8): E.disc
| ||||
| \(N(\mathfrak{D})\) | = | \( 729 \) | = | \( 27^{2} \) |
| magma: Norm(Discriminant(E));
sage: E.discriminant().norm()
gp (2.8): norm(E.disc)
| ||||
| \(j\) | = | \( -\frac{28672}{3} \) | ||
| 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: | Trivial |
|---|---|
| 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]
| |
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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \( \left(a^{3} + a^{2} - 2 a - 1\right) \) | \(27\) | \(2\) | \(I_{2}\) | Non-split multiplicative | \(1\) | \(1\) | \(2\) | \(2\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p \) except those listed.
| prime | Image of Galois Representation |
|---|---|
| \(13\) | 13B.12.1 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
13.
Its isogeny class
27.1-a
consists of curves linked by isogenies of
degree 13.