Base field 6.6.485125.1
Generator \(a\), with minimal polynomial \( x^{6} - 2 x^{5} - 4 x^{4} + 8 x^{3} + 2 x^{2} - 5 x + 1 \); class number \(1\).
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -5, 2, 8, -4, -2, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^6 - 2*x^5 - 4*x^4 + 8*x^3 + 2*x^2 - 5*x + 1)
gp (2.8): K = nfinit(a^6 - 2*a^5 - 4*a^4 + 8*a^3 + 2*a^2 - 5*a + 1);
Weierstrass equation
magma: E := ChangeRing(EllipticCurve([a + 1, -a^5 + 5*a^3 - 3*a + 1, a^4 - 3*a^2 + 1, -452*a^5 + 1969*a^4 - 1949*a^3 - 1368*a^2 + 2213*a - 432, 32265*a^5 - 121682*a^4 + 90467*a^3 + 89084*a^2 - 96074*a + 17221]),K);
sage: E = EllipticCurve(K, [a + 1, -a^5 + 5*a^3 - 3*a + 1, a^4 - 3*a^2 + 1, -452*a^5 + 1969*a^4 - 1949*a^3 - 1368*a^2 + 2213*a - 432, 32265*a^5 - 121682*a^4 + 90467*a^3 + 89084*a^2 - 96074*a + 17221])
gp (2.8): E = ellinit([a + 1, -a^5 + 5*a^3 - 3*a + 1, a^4 - 3*a^2 + 1, -452*a^5 + 1969*a^4 - 1949*a^3 - 1368*a^2 + 2213*a - 432, 32265*a^5 - 121682*a^4 + 90467*a^3 + 89084*a^2 - 96074*a + 17221],K)
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| \(\mathfrak{N} \) | = | \((31,-a^{4} + 4 a^{2} + a - 3)\) | = | \( \left(2 a^{5} - 3 a^{4} - 8 a^{3} + 10 a^{2} + 5 a - 4\right) \) |
| magma: Conductor(E);
sage: E.conductor()
| ||||
| \(N(\mathfrak{N}) \) | = | \( 31 \) | = | \( 31 \) |
| magma: Norm(Conductor(E));
sage: E.conductor().norm()
| ||||
| \(\mathfrak{D}\) | = | \((887503681,a^{5} - a^{4} - 5 a^{3} + 4 a^{2} + 5 a + 630753110,-a^{5} + 2 a^{4} + 4 a^{3} - 7 a^{2} - 3 a + 390826567,a^{5} - a^{4} - 4 a^{3} + 3 a^{2} + 2 a + 873930656,a + 81304435,-a^{5} + a^{4} + 5 a^{3} - 3 a^{2} - 5 a + 523347266)\) | = | \( \left(2 a^{5} - 3 a^{4} - 8 a^{3} + 10 a^{2} + 5 a - 4\right)^{6} \) |
| magma: Discriminant(E);
sage: E.discriminant()
gp (2.8): E.disc
| ||||
| \(N(\mathfrak{D})\) | = | \( 887503681 \) | = | \( 31^{6} \) |
| magma: Norm(Discriminant(E));
sage: E.discriminant().norm()
gp (2.8): norm(E.disc)
| ||||
| \(j\) | = | \( -\frac{10510572824308508672048934061763533884455273120}{887503681} a^{5} + \frac{3305871471997241704018322895385287159227247726}{887503681} a^{4} + \frac{47614244462683863442287631883194112269531793955}{887503681} a^{3} - \frac{3832115914609593129580786996785582548127284427}{887503681} a^{2} - \frac{27480068985972161720812424402012532444681628853}{887503681} a + \frac{6235982576035498927652588865167121704519444776}{887503681} \) | ||
| 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: | $\left(-a^{5} + 6 a^{4} - 5 a^{3} - \frac{49}{4} a^{2} + \frac{17}{2} a - \frac{5}{4} : -\frac{3}{2} a^{5} + a^{4} + \frac{37}{8} a^{3} + \frac{19}{8} a^{2} - \frac{9}{8} a - \frac{3}{8} : 1\right)$ |
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \( \left(2 a^{5} - 3 a^{4} - 8 a^{3} + 10 a^{2} + 5 a - 4\right) \) | \(31\) | \(2\) | \(I_{6}\) | Non-split multiplicative | \(1\) | \(1\) | \(6\) | \(6\) |
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, 4, 6, 8, 12 and 24.
Its isogeny class
31.1-a
consists of curves linked by isogenies of
degrees dividing 24.