Base field 6.6.300125.1
Generator \(a\), with minimal polynomial \( x^{6} - x^{5} - 7 x^{4} + 2 x^{3} + 7 x^{2} - 2 x - 1 \); class number \(1\).
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, -2, 7, 2, -7, -1, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^6 - x^5 - 7*x^4 + 2*x^3 + 7*x^2 - 2*x - 1)
gp (2.8): K = nfinit(a^6 - a^5 - 7*a^4 + 2*a^3 + 7*a^2 - 2*a - 1);
Weierstrass equation
magma: E := ChangeRing(EllipticCurve([-2*a^5 + a^4 + 14*a^3 + 4*a^2 - 10*a - 2, 7*a^5 - 2*a^4 - 50*a^3 - 22*a^2 + 32*a + 10, -4*a^5 + a^4 + 29*a^3 + 13*a^2 - 18*a - 5, 11*a^5 - 3*a^4 - 79*a^3 - 34*a^2 + 51*a + 16, 7*a^5 + a^4 - 55*a^3 - 37*a^2 + 47*a + 8]),K);
sage: E = EllipticCurve(K, [-2*a^5 + a^4 + 14*a^3 + 4*a^2 - 10*a - 2, 7*a^5 - 2*a^4 - 50*a^3 - 22*a^2 + 32*a + 10, -4*a^5 + a^4 + 29*a^3 + 13*a^2 - 18*a - 5, 11*a^5 - 3*a^4 - 79*a^3 - 34*a^2 + 51*a + 16, 7*a^5 + a^4 - 55*a^3 - 37*a^2 + 47*a + 8])
gp (2.8): E = ellinit([-2*a^5 + a^4 + 14*a^3 + 4*a^2 - 10*a - 2, 7*a^5 - 2*a^4 - 50*a^3 - 22*a^2 + 32*a + 10, -4*a^5 + a^4 + 29*a^3 + 13*a^2 - 18*a - 5, 11*a^5 - 3*a^4 - 79*a^3 - 34*a^2 + 51*a + 16, 7*a^5 + a^4 - 55*a^3 - 37*a^2 + 47*a + 8],K)
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| \(\mathfrak{N} \) | = | \((2,-2)\) | = | \( \left(2\right) \) |
| magma: Conductor(E);
sage: E.conductor()
| ||||
| \(N(\mathfrak{N}) \) | = | \( 64 \) | = | \( 64 \) |
| magma: Norm(Conductor(E));
sage: E.conductor().norm()
| ||||
| \(\mathfrak{D}\) | = | \((256,-512 a^{5} + 3840 a^{3} + 2560 a^{2} - 2560 a - 1280,768 a^{5} - 256 a^{4} - 5376 a^{3} - 2304 a^{2} + 3072 a + 1024,-1024 a^{5} + 256 a^{4} + 7424 a^{3} + 3328 a^{2} - 4864 a - 1280,256 a,-512 a^{5} + 256 a^{4} + 3584 a^{3} + 1024 a^{2} - 2560 a - 512)\) | = | \( \left(2\right)^{8} \) |
| magma: Discriminant(E);
sage: E.discriminant()
gp (2.8): E.disc
| ||||
| \(N(\mathfrak{D})\) | = | \( 281474976710656 \) | = | \( 64^{8} \) |
| magma: Norm(Discriminant(E));
sage: E.discriminant().norm()
gp (2.8): norm(E.disc)
| ||||
| \(j\) | = | \( \frac{1323}{256} \) | ||
| 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: | Trivial |
|---|---|
| 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]
| |
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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \( \left(2\right) \) | \(64\) | \(2\) | \(I_{8}\) | Non-split multiplicative | \(1\) | \(1\) | \(8\) | \(8\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p \) .
Isogenies and isogeny class
This curve has no rational isogenies. Its isogeny class 64.1-b consists of this curve only.