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([0, 4*a^5 - a^4 - 29*a^3 - 13*a^2 + 19*a + 5, 1, 120*a^5 - 30*a^4 - 870*a^3 - 390*a^2 + 570*a + 91, 408*a^5 - 102*a^4 - 2958*a^3 - 1326*a^2 + 1938*a + 324]),K);
sage: E = EllipticCurve(K, [0, 4*a^5 - a^4 - 29*a^3 - 13*a^2 + 19*a + 5, 1, 120*a^5 - 30*a^4 - 870*a^3 - 390*a^2 + 570*a + 91, 408*a^5 - 102*a^4 - 2958*a^3 - 1326*a^2 + 1938*a + 324])
gp (2.8): E = ellinit([0, 4*a^5 - a^4 - 29*a^3 - 13*a^2 + 19*a + 5, 1, 120*a^5 - 30*a^4 - 870*a^3 - 390*a^2 + 570*a + 91, 408*a^5 - 102*a^4 - 2958*a^3 - 1326*a^2 + 1938*a + 324],K)
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| \(\mathfrak{N} \) | = | \((7,-5 a^{5} + a^{4} + 36 a^{3} + 19 a^{2} - 22 a - 6)\) | = | \( \left(a^{5} - a^{4} - 6 a^{3} + a^{2} + 2 a - 1\right) \) |
| magma: Conductor(E);
sage: E.conductor()
| ||||
| \(N(\mathfrak{N}) \) | = | \( 49 \) | = | \( 49 \) |
| magma: Norm(Conductor(E));
sage: E.conductor().norm()
| ||||
| \(\mathfrak{D}\) | = | \((16807,-33614 a^{5} + 252105 a^{3} + 168070 a^{2} - 168070 a - 84035,50421 a^{5} - 16807 a^{4} - 352947 a^{3} - 151263 a^{2} + 201684 a + 67228,-67228 a^{5} + 16807 a^{4} + 487403 a^{3} + 218491 a^{2} - 319333 a - 84035,16807 a,-33614 a^{5} + 16807 a^{4} + 235298 a^{3} + 67228 a^{2} - 168070 a - 33614)\) | = | \( \left(a^{5} - a^{4} - 6 a^{3} + a^{2} + 2 a - 1\right)^{15} \) |
| magma: Discriminant(E);
sage: E.discriminant()
gp (2.8): E.disc
| ||||
| \(N(\mathfrak{D})\) | = | \( 22539340290692258087863249 \) | = | \( 49^{15} \) |
| magma: Norm(Discriminant(E));
sage: E.discriminant().norm()
gp (2.8): norm(E.disc)
| ||||
| \(j\) | = | \( -\frac{2887553024}{16807} \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \( \left(a^{5} - a^{4} - 6 a^{3} + a^{2} + 2 a - 1\right) \) | \(49\) | \(1\) | \(I_{15}\) | Non-split multiplicative | \(1\) | \(1\) | \(15\) | \(15\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p \) except those listed.
| prime | Image of Galois Representation |
|---|---|
| \(5\) | 5B.1.4[2] |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
5.
Its isogeny class
49.1-a
consists of curves linked by isogenies of
degree 5.