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^5 - a^4 - 4*a^3 + 3*a^2 + 3*a - 1, -a^5 + 6*a^3 - 7*a - 2, a^2 + a - 1, -126*a^5 - 117*a^4 + 741*a^3 + 702*a^2 - 1132*a - 1114, -3781*a^5 - 2012*a^4 + 17879*a^3 + 9699*a^2 - 19472*a - 13392]),K);
sage: E = EllipticCurve(K, [a^5 - a^4 - 4*a^3 + 3*a^2 + 3*a - 1, -a^5 + 6*a^3 - 7*a - 2, a^2 + a - 1, -126*a^5 - 117*a^4 + 741*a^3 + 702*a^2 - 1132*a - 1114, -3781*a^5 - 2012*a^4 + 17879*a^3 + 9699*a^2 - 19472*a - 13392])
gp (2.8): E = ellinit([a^5 - a^4 - 4*a^3 + 3*a^2 + 3*a - 1, -a^5 + 6*a^3 - 7*a - 2, a^2 + a - 1, -126*a^5 - 117*a^4 + 741*a^3 + 702*a^2 - 1132*a - 1114, -3781*a^5 - 2012*a^4 + 17879*a^3 + 9699*a^2 - 19472*a - 13392],K)
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| \(\mathfrak{N} \) | = | \((1)\) | = | \((1)\) |
| magma: Conductor(E);
sage: E.conductor()
| ||||
| \(N(\mathfrak{N}) \) | = | \( 1 \) | = | 1 |
| magma: Norm(Conductor(E));
sage: E.conductor().norm()
| ||||
| \(\mathfrak{D}\) | = | \((1)\) | = | \((1)\) |
| magma: Discriminant(E);
sage: E.discriminant()
gp (2.8): E.disc
| ||||
| \(N(\mathfrak{D})\) | = | \( 1 \) | = | 1 |
| magma: Norm(Discriminant(E));
sage: E.discriminant().norm()
gp (2.8): norm(E.disc)
| ||||
| \(j\) | = | \( 1606219307040269000685664528235176250888 a^{5} - 505201255314545228245706103526492515760 a^{4} - 7276379701134851706189760667243713057765 a^{3} + 585621608998000505707418880591397665054 a^{2} + 4199487325941333963885321109586284787419 a - 952979041145067887798527201046030804711 \) | ||
| 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()
No primes of bad reduction.
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p \) except those listed.
| prime | Image of Galois Representation |
|---|---|
| \(3\) | 3B |
| \(11\) | 11B.1.2 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
3, 11 and 33.
Its isogeny class
1.1-a
consists of curves linked by isogenies of
degrees dividing 33.