Base field 5.5.36497.1
Generator \(a\), with minimal polynomial \( x^{5} - 2 x^{4} - 3 x^{3} + 5 x^{2} + x - 1 \); class number \(1\).
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 1, 5, -3, -2, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^5 - 2*x^4 - 3*x^3 + 5*x^2 + x - 1)
gp (2.8): K = nfinit(a^5 - 2*a^4 - 3*a^3 + 5*a^2 + a - 1);
Weierstrass equation
magma: E := ChangeRing(EllipticCurve([a^2 - a - 2, a^3 - a^2 - 4*a + 1, a^2 - a - 2, -a^4 - 2*a^3 + 6*a^2 + 8*a - 11, 36*a^4 - 88*a^3 - 60*a^2 + 206*a - 75]),K);
sage: E = EllipticCurve(K, [a^2 - a - 2, a^3 - a^2 - 4*a + 1, a^2 - a - 2, -a^4 - 2*a^3 + 6*a^2 + 8*a - 11, 36*a^4 - 88*a^3 - 60*a^2 + 206*a - 75])
gp (2.8): E = ellinit([a^2 - a - 2, a^3 - a^2 - 4*a + 1, a^2 - a - 2, -a^4 - 2*a^3 + 6*a^2 + 8*a - 11, 36*a^4 - 88*a^3 - 60*a^2 + 206*a - 75],K)
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| \(\mathfrak{N} \) | = | \((9,a^{2} - a - 2)\) | = | \( \left(a^{2} - 1\right)^{2} \) |
| magma: Conductor(E);
sage: E.conductor()
| ||||
| \(N(\mathfrak{N}) \) | = | \( 9 \) | = | \( 3^{2} \) |
| magma: Norm(Conductor(E));
sage: E.conductor().norm()
| ||||
| \(\mathfrak{D}\) | = | \((387420489,a + 306972940,a^{4} - 2 a^{3} - 3 a^{2} + 5 a + 323254835,a^{4} - a^{3} - 4 a^{2} + 2 a + 145780798,a^{2} - a + 69580600)\) | = | \( \left(a^{2} - 1\right)^{18} \) |
| magma: Discriminant(E);
sage: E.discriminant()
gp (2.8): E.disc
| ||||
| \(N(\mathfrak{D})\) | = | \( 387420489 \) | = | \( 3^{18} \) |
| magma: Norm(Discriminant(E));
sage: E.discriminant().norm()
gp (2.8): norm(E.disc)
| ||||
| \(j\) | = | \( \frac{679068463130598740}{531441} a^{4} - \frac{359821343885971159}{177147} a^{3} - \frac{826730295123875959}{177147} a^{2} + \frac{2377533227165763589}{531441} a + \frac{1654749596218451752}{531441} \) | ||
| 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: | \(\Z/2\Z\) |
|---|---|
| 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: | $\left(-2 a^{4} + 4 a^{3} + 4 a^{2} - 8 a + 1 : -a^{4} + 2 a^{3} + 3 a^{2} - 5 a + 1 : 1\right)$ |
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \( \left(a^{2} - 1\right) \) | \(3\) | \(2\) | \(I_{12}^*\) | Additive | \(-1\) | \(2\) | \(18\) | \(12\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p \) except those listed.
| prime | Image of Galois Representation |
|---|---|
| \(2\) | 2B |
| \(3\) | 3B |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2, 3 and 6.
Its isogeny class
9.1-a
consists of curves linked by isogenies of
degrees dividing 6.