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
| Conductor: | \((-2 a^{2} - 2 a + 4)\) | = | \( \left(2\right) \cdot \left(-a^{2} - a + 2\right) \) |
sage: E.conductor()
magma: Conductor(E);
| |||
| Conductor norm: | \( 56 \) | = | \( 7 \cdot 8 \) |
sage: E.conductor().norm()
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((28)\) | = | \( \left(2\right)^{2} \cdot \left(-a^{2} - a + 2\right)^{3} \) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 21952 \) | = | \( 7^{3} \cdot 8^{2} \) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( -\frac{15625}{28} \) | ||
sage: E.j_invariant()
gp: E.j
magma: jInvariant(E);
| |||
| Endomorphism ring: | \(\Z\) | ||
| Geometric endomorphism ring: | \(\Z\) | (no potential complex multiplication) | |
sage: E.has_cm(), E.cm_discriminant()
magma: HasComplexMultiplication(E);
| |||
| Sato-Tate group: | $\mathrm{SU}(2)$ | ||
Mordell-Weil group
| Rank: | \(0\) |
| Torsion structure: | \(\Z/18\Z\) |
| sage: T = E.torsion_subgroup(); T.invariants()
gp: T = elltors(E); T[2]
magma: T,piT := TorsionSubgroup(E); Invariants(T);
| |
| Torsion 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)];
| |
BSD invariants
| Analytic rank: | \( 0 \) | ||
|
sage: E.rank()
magma: Rank(E);
|
|||
| Mordell-Weil rank: | \(0\) | ||
| Regulator: | \( 1 \) | ||
| Period: | \( 210.011014254678 \) | ||
| Tamagawa product: | \( 6 \) = \(3\cdot2\) | ||
| Torsion order: | \(18\) | ||
| Leading coefficient: | \(0.555584693795445\) | ||
| Analytic order of Ш: | \( 1 \) (rounded) | ||
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.
Base change
This curve is the base change of elliptic curves 14.a5, defined over \(\Q\), so it is also a \(\Q\)-curve.