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 - 5*a^3 + 4*a + 1, a^5 - 5*a^3 + 5*a + 1, -a^5 + 2*a^4 + 4*a^3 - 7*a^2 - 2*a + 3, -780*a^5 + 1038*a^4 + 3796*a^3 - 3603*a^2 - 3947*a + 909, -18772*a^5 + 25124*a^4 + 91377*a^3 - 88985*a^2 - 95239*a + 28271]),K);
sage: E = EllipticCurve(K, [a^5 - 5*a^3 + 4*a + 1, a^5 - 5*a^3 + 5*a + 1, -a^5 + 2*a^4 + 4*a^3 - 7*a^2 - 2*a + 3, -780*a^5 + 1038*a^4 + 3796*a^3 - 3603*a^2 - 3947*a + 909, -18772*a^5 + 25124*a^4 + 91377*a^3 - 88985*a^2 - 95239*a + 28271])
gp (2.8): E = ellinit([a^5 - 5*a^3 + 4*a + 1, a^5 - 5*a^3 + 5*a + 1, -a^5 + 2*a^4 + 4*a^3 - 7*a^2 - 2*a + 3, -780*a^5 + 1038*a^4 + 3796*a^3 - 3603*a^2 - 3947*a + 909, -18772*a^5 + 25124*a^4 + 91377*a^3 - 88985*a^2 - 95239*a + 28271],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()
| ||||
| \(\mathfrak{D}\) | = | \((961,a^{5} - a^{4} - 5 a^{3} + 4 a^{2} + 5 a + 760,-a^{5} + 2 a^{4} + 4 a^{3} - 7 a^{2} - 3 a + 360,a^{5} - a^{4} - 4 a^{3} + 3 a^{2} + 2 a + 139,a + 952,-a^{5} + a^{4} + 5 a^{3} - 3 a^{2} - 5 a + 120)\) | = | \( \left(2 a^{5} - 3 a^{4} - 8 a^{3} + 10 a^{2} + 5 a - 4\right)^{2} \) |
| magma: Discriminant(E);
sage: E.discriminant()
gp (2.8): E.disc
| ||||
| \(N(\mathfrak{D})\) | = | \( 961 \) | = | \( 31^{2} \) |
| magma: Norm(Discriminant(E));
sage: E.discriminant().norm()
gp (2.8): norm(E.disc)
| ||||
| \(j\) | = | \( \frac{391232997836344118812955063578726935614}{961} a^{5} - \frac{1146049847494581967103997115183815738335}{961} a^{4} - \frac{499875566564262423437058881855118969349}{961} a^{3} + \frac{3594412435730780139116852441515415592364}{961} a^{2} - \frac{2557922789646965076659228190817527550844}{961} a + \frac{420984754490672117422372247124701363035}{961} \) | ||
| 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: | \(\Z/2\Z\) |
|---|---|
| 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: | $\left(\frac{107}{4} a^{5} - \frac{145}{4} a^{4} - \frac{263}{2} a^{3} + 134 a^{2} + \frac{549}{4} a - \frac{115}{2} : -\frac{55}{2} a^{5} + \frac{157}{4} a^{4} + \frac{543}{4} a^{3} - 146 a^{2} - \frac{1145}{8} a + \frac{461}{8} : 1\right)$ |
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \( \left(2 a^{5} - 3 a^{4} - 8 a^{3} + 10 a^{2} + 5 a - 4\right) \) | \(31\) | \(2\) | \(I_{2}\) | Non-split multiplicative | \(1\) | \(1\) | \(2\) | \(2\) |
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.