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([-2*a^5 + 3*a^4 + 9*a^3 - 10*a^2 - 8*a + 3, -a^5 + 5*a^3 + a^2 - 5*a - 1, 0, 10*a^5 - 40*a^4 - 45*a^3 + 165*a^2 + 45*a - 104, -98*a^5 + 215*a^4 + 445*a^3 - 840*a^2 - 378*a + 495]),K);
sage: E = EllipticCurve(K, [-2*a^5 + 3*a^4 + 9*a^3 - 10*a^2 - 8*a + 3, -a^5 + 5*a^3 + a^2 - 5*a - 1, 0, 10*a^5 - 40*a^4 - 45*a^3 + 165*a^2 + 45*a - 104, -98*a^5 + 215*a^4 + 445*a^3 - 840*a^2 - 378*a + 495])
gp (2.8): E = ellinit([-2*a^5 + 3*a^4 + 9*a^3 - 10*a^2 - 8*a + 3, -a^5 + 5*a^3 + a^2 - 5*a - 1, 0, 10*a^5 - 40*a^4 - 45*a^3 + 165*a^2 + 45*a - 104, -98*a^5 + 215*a^4 + 445*a^3 - 840*a^2 - 378*a + 495],K)
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| \(\mathfrak{N} \) | = | \((31,-a^{4} + 4 a^{2} + a - 3)\) | = | \( \left(2 a^{5} - 3 a^{4} - 8 a^{3} + 10 a^{2} + 5 a - 4\right) \) |
| magma: Conductor(E);
sage: E.conductor()
| ||||
| \(N(\mathfrak{N}) \) | = | \( 31 \) | = | \( 31 \) |
| magma: Norm(Conductor(E));
sage: E.conductor().norm()
| ||||
| \(\mathfrak{D}\) | = | \((31,a^{5} - a^{4} - 5 a^{3} + 4 a^{2} + 5 a + 16,-a^{5} + 2 a^{4} + 4 a^{3} - 7 a^{2} - 3 a + 19,a^{5} - a^{4} - 4 a^{3} + 3 a^{2} + 2 a + 15,a + 22,-a^{5} + a^{4} + 5 a^{3} - 3 a^{2} - 5 a + 27)\) | = | \( \left(2 a^{5} - 3 a^{4} - 8 a^{3} + 10 a^{2} + 5 a - 4\right) \) |
| magma: Discriminant(E);
sage: E.discriminant()
gp (2.8): E.disc
| ||||
| \(N(\mathfrak{D})\) | = | \( 31 \) | = | \( 31 \) |
| magma: Norm(Discriminant(E));
sage: E.discriminant().norm()
gp (2.8): norm(E.disc)
| ||||
| \(j\) | = | \( -\frac{11771614728650068311718789996606}{31} a^{5} + \frac{20672646131967906688095174594552}{31} a^{4} + \frac{52127615514169524635482470593815}{31} a^{3} - \frac{81461266340971407505611370906367}{31} a^{2} - \frac{43408078775695757231287999254039}{31} a + \frac{48272736797749486398912337205132}{31} \) | ||
| 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: | \(\Z/4\Z\) |
|---|---|
| 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: | $\left(6 : 8 a^{5} - 17 a^{4} - 36 a^{3} + 63 a^{2} + 33 a - 13 : 1\right)$ |
| 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()
| prime | Norm | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord(\(\mathfrak{N}\)) | ord(\(\mathfrak{D}\)) | ord\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \( \left(2 a^{5} - 3 a^{4} - 8 a^{3} + 10 a^{2} + 5 a - 4\right) \) | \(31\) | \(1\) | \(I_{1}\) | Non-split multiplicative | \(1\) | \(1\) | \(1\) | \(1\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p \) except those listed.
| prime | Image of Galois Representation |
|---|---|
| \(2\) | 2B |
| \(3\) | 3B |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2, 3, 4, 6, 8, 12 and 24.
Its isogeny class
31.1-a
consists of curves linked by isogenies of
degrees dividing 24.