Base field 4.4.2777.1
Generator \(a\), with minimal polynomial \( x^{4} - x^{3} - 4 x^{2} + x + 2 \); class number \(1\).
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2, 1, -4, -1, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^4 - x^3 - 4*x^2 + x + 2)
gp (2.8): K = nfinit(a^4 - a^3 - 4*a^2 + a + 2);
Weierstrass equation
magma: E := ChangeRing(EllipticCurve([a + 1, a^3 - a^2 - 3*a, a^3 - a^2 - 2*a + 1, 30*a^3 - 52*a^2 - 65*a - 13, 91*a^3 - 127*a^2 - 206*a - 56]),K);
sage: E = EllipticCurve(K, [a + 1, a^3 - a^2 - 3*a, a^3 - a^2 - 2*a + 1, 30*a^3 - 52*a^2 - 65*a - 13, 91*a^3 - 127*a^2 - 206*a - 56])
gp (2.8): E = ellinit([a + 1, a^3 - a^2 - 3*a, a^3 - a^2 - 2*a + 1, 30*a^3 - 52*a^2 - 65*a - 13, 91*a^3 - 127*a^2 - 206*a - 56],K)
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| \(\mathfrak{N} \) | = | \((2,2)\) | = | \( \left(a^{3} - a^{2} - 4 a + 1\right) \cdot \left(-a\right) \) |
| magma: Conductor(E);
sage: E.conductor()
| ||||
| \(N(\mathfrak{N}) \) | = | \( 16 \) | = | \( 2 \cdot 8 \) |
| magma: Norm(Conductor(E));
sage: E.conductor().norm()
| ||||
| \(\mathfrak{D}\) | = | \((262144,64 a + 162432,64 a^{3} - 64 a^{2} - 192 a + 172928,64 a^{2} - 64 a + 199808)\) | = | \( \left(a^{3} - a^{2} - 4 a + 1\right)^{6} \cdot \left(-a\right)^{18} \) |
| magma: Discriminant(E);
sage: E.discriminant()
gp (2.8): E.disc
| ||||
| \(N(\mathfrak{D})\) | = | \( 68719476736 \) | = | \( 2^{18} \cdot 8^{6} \) |
| magma: Norm(Discriminant(E));
sage: E.discriminant().norm()
gp (2.8): norm(E.disc)
| ||||
| \(j\) | = | \( \frac{5353573950671463885}{262144} a^{3} - \frac{8992096210769676335}{262144} a^{2} - \frac{7651439252430264991}{131072} a + \frac{15754080188450938489}{262144} \) | ||
| 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: | \(\Z/2\Z\times\Z/2\Z\) |
|---|---|
| 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]
| |
| Generators: | $\left(\frac{5}{4} a^{3} - 3 a - 3 : -\frac{7}{4} a^{3} - \frac{1}{2} a^{2} + \frac{37}{8} a + \frac{9}{4} : 1\right)$,$\left(-a^{3} + 3 a^{2} - 4 a - 3 : -a^{3} + 3 a^{2} + 4 a : 1\right)$ |
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \( \left(-a\right) \) | \(2\) | \(2\) | \(I_{18}\) | Non-split multiplicative | \(1\) | \(1\) | \(18\) | \(18\) |
| \( \left(a^{3} - a^{2} - 4 a + 1\right) \) | \(8\) | \(2\) | \(I_{6}\) | Non-split multiplicative | \(1\) | \(1\) | \(6\) | \(6\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p \) except those listed.
| prime | Image of Galois Representation |
|---|---|
| \(2\) | 2Cs |
| \(3\) | 3B.1.2 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2, 3 and 6.
Its isogeny class
16.1-a
consists of curves linked by isogenies of
degrees dividing 12.