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^4 - 4*a^2 - a + 3, a^5 - 2*a^4 - 4*a^3 + 7*a^2 + 3*a - 2, -a^5 + 2*a^4 + 4*a^3 - 7*a^2 - 2*a + 4, -98*a^5 + 20*a^4 + 286*a^3 - 83*a^2 + 45*a - 74, -14247*a^5 - 1644*a^4 + 52563*a^3 - 926*a^2 - 28214*a + 5435]),K);
sage: E = EllipticCurve(K, [a^4 - 4*a^2 - a + 3, a^5 - 2*a^4 - 4*a^3 + 7*a^2 + 3*a - 2, -a^5 + 2*a^4 + 4*a^3 - 7*a^2 - 2*a + 4, -98*a^5 + 20*a^4 + 286*a^3 - 83*a^2 + 45*a - 74, -14247*a^5 - 1644*a^4 + 52563*a^3 - 926*a^2 - 28214*a + 5435])
gp (2.8): E = ellinit([a^4 - 4*a^2 - a + 3, a^5 - 2*a^4 - 4*a^3 + 7*a^2 + 3*a - 2, -a^5 + 2*a^4 + 4*a^3 - 7*a^2 - 2*a + 4, -98*a^5 + 20*a^4 + 286*a^3 - 83*a^2 + 45*a - 74, -14247*a^5 - 1644*a^4 + 52563*a^3 - 926*a^2 - 28214*a + 5435],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()
| ||||
| \(\mathfrak{D}\) | = | \((128,128 a^{5} - 128 a^{4} - 640 a^{3} + 512 a^{2} + 640 a - 256,-128 a^{5} + 256 a^{4} + 512 a^{3} - 896 a^{2} - 384 a + 384,128 a^{5} - 128 a^{4} - 512 a^{3} + 384 a^{2} + 256 a - 128,128 a,-128 a^{5} + 128 a^{4} + 640 a^{3} - 384 a^{2} - 640 a)\) | = | \( \left(2\right)^{7} \) |
| magma: Discriminant(E);
sage: E.discriminant()
gp (2.8): E.disc
| ||||
| \(N(\mathfrak{D})\) | = | \( 4398046511104 \) | = | \( 64^{7} \) |
| magma: Norm(Discriminant(E));
sage: E.discriminant().norm()
gp (2.8): norm(E.disc)
| ||||
| \(j\) | = | \( -\frac{661423578612904638319}{64} a^{5} + \frac{2517381707375973330273}{64} a^{4} - \frac{950355314310097681067}{32} a^{3} - \frac{3717389947572253097155}{128} a^{2} + \frac{4067931975116471451485}{128} a - \frac{366235659528699810345}{64} \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \( \left(2\right) \) | \(64\) | \(7\) | \(I_{7}\) | Split multiplicative | \(-1\) | \(1\) | \(7\) | \(7\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p \) except those listed.
| prime | Image of Galois Representation |
|---|---|
| \(7\) | 7B.1.3 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
7.
Its isogeny class
64.1-b
consists of curves linked by isogenies of
degree 7.