Base field \(\Q(\zeta_{7})^+\)
Generator \(a\), with minimal polynomial \( x^{3} - x^{2} - 2 x + 1 \); class number \(1\).
sage: x = polygen(QQ); K.<a> = NumberField(x^3 - x^2 - 2*x + 1)
gp: K = nfinit(a^3 - a^2 - 2*a + 1);
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -2, -1, 1]);
Weierstrass equation
sage: E = EllipticCurve(K, [1, 0, 1, -1, 0])
gp: E = ellinit([1, 0, 1, -1, 0],K)
magma: E := ChangeRing(EllipticCurve([1, 0, 1, -1, 0]),K);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| \(\mathfrak{N} \) | = | \((14,-2 a^{2} - 2 a + 4)\) | = | \( \left(2\right) \cdot \left(-a^{2} - a + 2\right) \) |
| sage: E.conductor()
magma: Conductor(E);
| ||||
| \(N(\mathfrak{N}) \) | = | \( 56 \) | = | \( 7 \cdot 8 \) |
| sage: E.conductor().norm()
magma: Norm(Conductor(E));
| ||||
| \(\mathfrak{D}\) | = | \((28,28 a,28 a^{2} - 56)\) | = | \( \left(2\right)^{2} \cdot \left(-a^{2} - a + 2\right)^{3} \) |
| sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| ||||
| \(N(\mathfrak{D})\) | = | \( 21952 \) | = | \( 7^{3} \cdot 8^{2} \) |
| sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| ||||
| \(j\) | = | \( -\frac{15625}{28} \) | ||
| sage: E.j_invariant()
gp: E.j
magma: jInvariant(E);
| ||||
| \( \text{End} (E) \) | = | \(\Z\) | (no Complex Multiplication ) | |
| sage: E.has_cm(), E.cm_discriminant()
magma: HasComplexMultiplication(E);
| ||||
| \( \text{ST} (E) \) | = | $\mathrm{SU}(2)$ | ||
Mordell-Weil group
Rank not available.sage: E.rank()
magma: Rank(E);
Regulator: not available
sage: gens = E.gens(); gens
magma: gens := [P:P in Generators(E)|Order(P) eq 0]; gens;
sage: E.regulator_of_points(gens)
magma: Regulator(gens);
Torsion subgroup
| Structure: | \(\Z/18\Z\) |
|---|---|
| sage: T = E.torsion_subgroup(); T.invariants()
gp: T = elltors(E); T[2]
magma: T,piT := TorsionSubgroup(E); Invariants(T);
| |
| Generator: | $\left(2 a^{2} + 2 a - 1 : -8 a^{2} - 6 a + 4 : 1\right)$ |
| sage: T.gens()
gp: T[3]
magma: [piT(P) : P in Generators(T)];
| |
Local data at primes of bad reduction
sage: E.local_data()
magma: LocalInformation(E);
| prime | Norm | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord(\(\mathfrak{N}\)) | ord(\(\mathfrak{D}\)) | ord\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \( \left(-a^{2} - a + 2\right) \) | \(7\) | \(3\) | \(I_{3}\) | Split multiplicative | \(-1\) | \(1\) | \(3\) | \(3\) |
| \( \left(2\right) \) | \(8\) | \(2\) | \(I_{2}\) | Non-split multiplicative | \(1\) | \(1\) | \(2\) | \(2\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.
| prime | Image of Galois Representation |
|---|---|
| \(2\) | 2B |
| \(3\) | 3B.1.1 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2, 3, 6, 9 and 18.
Its isogeny class
56.1-a
consists of curves linked by isogenies of
degrees dividing 18.