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^3 - 4*a - 1, -a^3 + 2*a^2 + 3*a - 4, a^3 - a^2 - 3*a + 2, -25*a^3 + 26*a^2 + 84*a - 75, -255*a^3 - 134*a^2 + 413*a - 111]),K);
sage: E = EllipticCurve(K, [a^3 - 4*a - 1, -a^3 + 2*a^2 + 3*a - 4, a^3 - a^2 - 3*a + 2, -25*a^3 + 26*a^2 + 84*a - 75, -255*a^3 - 134*a^2 + 413*a - 111])
gp (2.8): E = ellinit([a^3 - 4*a - 1, -a^3 + 2*a^2 + 3*a - 4, a^3 - a^2 - 3*a + 2, -25*a^3 + 26*a^2 + 84*a - 75, -255*a^3 - 134*a^2 + 413*a - 111],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()
| ||||
| \(\mathfrak{D}\) | = | \((322102,a + 127936,a^{3} - a^{2} - 3 a + 213304,a^{2} - a + 187200)\) | = | \( \left(-a\right) \cdot \left(-a^{3} + 2 a^{2} + 2 a - 1\right)^{5} \) |
| magma: Discriminant(E);
sage: E.discriminant()
gp (2.8): E.disc
| ||||
| \(N(\mathfrak{D})\) | = | \( 322102 \) | = | \( 2 \cdot 11^{5} \) |
| magma: Norm(Discriminant(E));
sage: E.discriminant().norm()
gp (2.8): norm(E.disc)
| ||||
| \(j\) | = | \( -\frac{776940933840351053858986959}{322102} a^{3} + \frac{135355112947607678458237413}{322102} a^{2} + \frac{1609768948326117754076482158}{161051} a + \frac{1881703725424533772239476133}{322102} \) | ||
| 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: | $\left(\frac{15}{4} a^{3} - \frac{9}{4} a^{2} - \frac{25}{2} a + \frac{31}{4} : -\frac{21}{8} a^{3} + \frac{7}{8} a^{2} + \frac{27}{4} a - \frac{65}{8} : 1\right)$ |
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \( \left(-a\right) \) | \(2\) | \(1\) | \(I_{1}\) | Split multiplicative | \(-1\) | \(1\) | \(1\) | \(1\) |
| \( \left(-a^{3} + 2 a^{2} + 2 a - 1\right) \) | \(11\) | \(5\) | \(I_{5}\) | Split multiplicative | \(-1\) | \(1\) | \(5\) | \(5\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p \) except those listed.
| prime | Image of Galois Representation |
|---|---|
| \(2\) | 2B |
| \(5\) | 5B.1.2 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2, 4, 5, 10 and 20.
Its isogeny class
22.1-b
consists of curves linked by isogenies of
degrees dividing 20.