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 + 5*a^3 - 3*a^2 - 5*a + 1, -3*a^5 + 3*a^4 + 14*a^3 - 10*a^2 - 12*a + 3, -a^5 + 2*a^4 + 4*a^3 - 7*a^2 - 3*a + 3, -3*a^5 + 2*a^4 + 12*a^3 - 8*a^2 - 4*a + 10, 15*a^5 - 21*a^4 - 72*a^3 + 77*a^2 + 76*a - 28]),K);
sage: E = EllipticCurve(K, [-a^5 + a^4 + 5*a^3 - 3*a^2 - 5*a + 1, -3*a^5 + 3*a^4 + 14*a^3 - 10*a^2 - 12*a + 3, -a^5 + 2*a^4 + 4*a^3 - 7*a^2 - 3*a + 3, -3*a^5 + 2*a^4 + 12*a^3 - 8*a^2 - 4*a + 10, 15*a^5 - 21*a^4 - 72*a^3 + 77*a^2 + 76*a - 28])
gp (2.8): E = ellinit([-a^5 + a^4 + 5*a^3 - 3*a^2 - 5*a + 1, -3*a^5 + 3*a^4 + 14*a^3 - 10*a^2 - 12*a + 3, -a^5 + 2*a^4 + 4*a^3 - 7*a^2 - 3*a + 3, -3*a^5 + 2*a^4 + 12*a^3 - 8*a^2 - 4*a + 10, 15*a^5 - 21*a^4 - 72*a^3 + 77*a^2 + 76*a - 28],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)
| ||||
| \(j\) | = | \( 33995809 a^{5} - 134164325 a^{4} + 105734178 a^{3} + 98860224 a^{2} - 110867202 a + 20070137 \) | ||
| 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: | \(\Z/11\Z\) |
|---|---|
| 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: | $\left(10 a^{5} - 13 a^{4} - 49 a^{3} + 46 a^{2} + 52 a - 14 : 61 a^{5} - 80 a^{4} - 298 a^{3} + 282 a^{2} + 312 a - 90 : 1\right)$ |
| 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()
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.1.1 |
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.