Base field 4.4.725.1
Generator \(a\), with minimal polynomial \( x^{4} - x^{3} - 3 x^{2} + x + 1 \); class number \(1\).
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 1, -3, -1, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^4 - x^3 - 3*x^2 + x + 1)
gp (2.8): K = nfinit(a^4 - a^3 - 3*a^2 + a + 1);
Weierstrass equation
magma: E := ChangeRing(EllipticCurve([a^3 - a^2 - 2*a + 2, a^3 - a^2 - a + 1, 1, 56*a^3 - 334*a^2 - 2*a + 75, 107*a^3 - 3446*a^2 + 603*a + 1078]),K);
sage: E = EllipticCurve(K, [a^3 - a^2 - 2*a + 2, a^3 - a^2 - a + 1, 1, 56*a^3 - 334*a^2 - 2*a + 75, 107*a^3 - 3446*a^2 + 603*a + 1078])
gp (2.8): E = ellinit([a^3 - a^2 - 2*a + 2, a^3 - a^2 - a + 1, 1, 56*a^3 - 334*a^2 - 2*a + 75, 107*a^3 - 3446*a^2 + 603*a + 1078],K)
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| \(\mathfrak{N} \) | = | \((31,a^{3} - 4 a + 1)\) | = | \( \left(a^{3} - 4 a + 1\right) \) |
| magma: Conductor(E);
sage: E.conductor()
| ||||
| \(N(\mathfrak{N}) \) | = | \( 31 \) | = | \( 31 \) |
| magma: Norm(Conductor(E));
sage: E.conductor().norm()
| ||||
| \(\mathfrak{D}\) | = | \((852891037441,a^{3} - a^{2} - 2 a + 816246635952,a + 317439913159,a^{2} - a + 208075842453)\) | = | \( \left(a^{3} - 4 a + 1\right)^{8} \) |
| magma: Discriminant(E);
sage: E.discriminant()
gp (2.8): E.disc
| ||||
| \(N(\mathfrak{D})\) | = | \( 852891037441 \) | = | \( 31^{8} \) |
| magma: Norm(Discriminant(E));
sage: E.discriminant().norm()
gp (2.8): norm(E.disc)
| ||||
| \(j\) | = | \( \frac{206065140764354323915985}{852891037441} a^{3} + \frac{249452633351127967773393}{852891037441} a^{2} - \frac{151495593455929061592036}{852891037441} a - \frac{106721268617817034471018}{852891037441} \) | ||
| 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: | $\left(-\frac{15}{4} a^{3} - \frac{17}{4} a^{2} + \frac{29}{2} a + \frac{15}{4} : 4 a^{3} + \frac{1}{2} a^{2} - \frac{15}{2} a + \frac{9}{8} : 1\right)$ |
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \( \left(a^{3} - 4 a + 1\right) \) | \(31\) | \(2\) | \(I_{8}\) | Non-split multiplicative | \(1\) | \(1\) | \(8\) | \(8\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p \) except those listed.
| prime | Image of Galois Representation |
|---|---|
| \(2\) | 2B |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2, 4 and 8.
Its isogeny class
31.1-a
consists of curves linked by isogenies of
degrees dividing 8.