Base field 3.3.148.1
Generator \(a\), with minimal polynomial \( x^{3} - x^{2} - 3 x + 1 \); class number \(1\).
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -3, -1, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^3 - x^2 - 3*x + 1)
gp (2.8): K = nfinit(a^3 - a^2 - 3*a + 1);
Weierstrass equation
magma: E := ChangeRing(EllipticCurve([a^2 - 1, -a^2 + a + 1, 0, -122876240093597390564*a^2 - 143775828060934351842*a + 56622738702290433353, 1339806177584198660985008156380*a^2 + 1567689103089350829411708766103*a - 617397594907508652287429548778]),K);
sage: E = EllipticCurve(K, [a^2 - 1, -a^2 + a + 1, 0, -122876240093597390564*a^2 - 143775828060934351842*a + 56622738702290433353, 1339806177584198660985008156380*a^2 + 1567689103089350829411708766103*a - 617397594907508652287429548778])
gp (2.8): E = ellinit([a^2 - 1, -a^2 + a + 1, 0, -122876240093597390564*a^2 - 143775828060934351842*a + 56622738702290433353, 1339806177584198660985008156380*a^2 + 1567689103089350829411708766103*a - 617397594907508652287429548778],K)
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| \(\mathfrak{N} \) | = | \((2,-2)\) | = | \( \left(a^{2} - a - 2\right)^{3} \) |
| magma: Conductor(E);
sage: E.conductor()
| ||||
| \(N(\mathfrak{N}) \) | = | \( 8 \) | = | \( 2^{3} \) |
| magma: Norm(Conductor(E));
sage: E.conductor().norm()
| ||||
| \(\mathfrak{D}\) | = | \((8,8 a,4 a^{2} - 4)\) | = | \( \left(a^{2} - a - 2\right)^{8} \) |
| magma: Discriminant(E);
sage: E.discriminant()
gp (2.8): E.disc
| ||||
| \(N(\mathfrak{D})\) | = | \( 256 \) | = | \( 2^{8} \) |
| magma: Norm(Discriminant(E));
sage: E.discriminant().norm()
gp (2.8): norm(E.disc)
| ||||
| \(j\) | = | \( 5088 a^{2} + 8224 a + 3904 \) | ||
| 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: | \(\Z/2\Z\times\Z/4\Z\) |
|---|---|
| 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]
| |
| Generators: | $\left(4785876661 a^{2} + 5599889607 a - 2205385219 : 594441845638255 a^{2} + 695548370666361 a - 273925416936940 : 1\right)$,$\left(\frac{4347879635}{2} a^{2} + \frac{5087395205}{2} a - 1001775658 : -\frac{8063741397}{2} a^{2} - 4717637420 a + 1857930881 : 1\right)$ |
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \( \left(a^{2} - a - 2\right) \) | \(2\) | \(2\) | \(I_{1}^*\) | Additive | \(-1\) | \(3\) | \(8\) | \(0\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p \) except those listed.
| prime | Image of Galois Representation |
|---|---|
| \(2\) | 2Cs |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2 and 4.
Its isogeny class
8.1-a
consists of curves linked by isogenies of
degrees dividing 8.