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 + 5*a + 2, a^2 - a - 1, -30*a^3 + 42*a^2 + 102*a - 90, -111*a^3 + 178*a^2 + 344*a - 362]),K);
sage: E = EllipticCurve(K, [a + 1, -a^3 + 5*a + 2, a^2 - a - 1, -30*a^3 + 42*a^2 + 102*a - 90, -111*a^3 + 178*a^2 + 344*a - 362])
gp (2.8): E = ellinit([a + 1, -a^3 + 5*a + 2, a^2 - a - 1, -30*a^3 + 42*a^2 + 102*a - 90, -111*a^3 + 178*a^2 + 344*a - 362],K)
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| \(\mathfrak{N} \) | = | \((22,2 a^{3} - 3 a^{2} - 5 a + 4)\) | = | \( \left(-a\right) \cdot \left(-a^{3} + 2 a^{2} + 2 a - 1\right) \) |
| magma: Conductor(E);
sage: E.conductor()
| ||||
| \(N(\mathfrak{N}) \) | = | \( 22 \) | = | \( 2 \cdot 11 \) |
| magma: Norm(Conductor(E));
sage: E.conductor().norm()
| ||||
| \(\mathfrak{D}\) | = | \((52654090776777588736,a + 50885740190649121258,a^{3} - a^{2} - 3 a + 18845939956618783374,a^{2} - a + 43408037767785446450)\) | = | \( \left(-a\right)^{24} \cdot \left(-a^{3} + 2 a^{2} + 2 a - 1\right)^{12} \) |
| magma: Discriminant(E);
sage: E.discriminant()
gp (2.8): E.disc
| ||||
| \(N(\mathfrak{D})\) | = | \( 52654090776777588736 \) | = | \( 2^{24} \cdot 11^{12} \) |
| magma: Norm(Discriminant(E));
sage: E.discriminant().norm()
gp (2.8): norm(E.disc)
| ||||
| \(j\) | = | \( -\frac{2125946353940568480801651}{52654090776777588736} a^{3} + \frac{2559556155064329492768145}{52654090776777588736} a^{2} + \frac{2498155651166365388592929}{26327045388388794368} a + \frac{2080764888610005913150009}{52654090776777588736} \) | ||
| 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: | $\left(\frac{13}{4} a^{3} - 3 a^{2} - 13 a + 7 : -\frac{7}{4} a^{3} + a^{2} + \frac{41}{8} a + \frac{1}{4} : 1\right)$,$\left(-a^{3} + 2 a^{2} + 3 a - 7 : -a^{2} + 2 a + 3 : 1\right)$ |
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \( \left(-a\right) \) | \(2\) | \(2\) | \(I_{24}\) | Non-split multiplicative | \(1\) | \(1\) | \(24\) | \(24\) |
| \( \left(-a^{3} + 2 a^{2} + 2 a - 1\right) \) | \(11\) | \(2\) | \(I_{12}\) | Non-split multiplicative | \(1\) | \(1\) | \(12\) | \(12\) |
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, 4, 6 and 12.
Its isogeny class
22.1-a
consists of curves linked by isogenies of
degrees dividing 24.