Properties

Base field 4.4.2777.1
Label 4.4.2777.1-22.1-a7
Conductor \((22,2 a^{3} - 3 a^{2} - 5 a + 4)\)
Conductor norm \( 22 \)
CM no
base-change no
Q-curve no
Torsion order \( 6 \)
Rank not available

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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

\( y^2 + \left(a^{2} - 1\right) x y + \left(a^{3} - a^{2} - 3 a + 2\right) y = x^{3} + \left(a^{3} - 3 a - 1\right) x^{2} + \left(-29 a^{3} - 37 a^{2} + 29 a + 24\right) x + 247 a^{3} + 345 a^{2} - 177 a - 202 \)
magma: E := ChangeRing(EllipticCurve([a^2 - 1, a^3 - 3*a - 1, a^3 - a^2 - 3*a + 2, -29*a^3 - 37*a^2 + 29*a + 24, 247*a^3 + 345*a^2 - 177*a - 202]),K);
sage: E = EllipticCurve(K, [a^2 - 1, a^3 - 3*a - 1, a^3 - a^2 - 3*a + 2, -29*a^3 - 37*a^2 + 29*a + 24, 247*a^3 + 345*a^2 - 177*a - 202])
gp (2.8): E = ellinit([a^2 - 1, a^3 - 3*a - 1, a^3 - a^2 - 3*a + 2, -29*a^3 - 37*a^2 + 29*a + 24, 247*a^3 + 345*a^2 - 177*a - 202],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()
\(\mathfrak{D}\) = \((3429742096,a + 1656698522,a^{3} - a^{2} - 3 a + 3327204862,a^{2} - a + 3204135794)\) = \( \left(-a\right)^{4} \cdot \left(-a^{3} + 2 a^{2} + 2 a - 1\right)^{8} \)
magma: Discriminant(E);
sage: E.discriminant()
gp (2.8): E.disc
\(N(\mathfrak{D})\) = \( 3429742096 \) = \( 2^{4} \cdot 11^{8} \)
magma: Norm(Discriminant(E));
sage: E.discriminant().norm()
gp (2.8): norm(E.disc)
\(j\) = \( \frac{158950703525365495771}{3429742096} a^{3} - \frac{266808256244810209657}{3429742096} a^{2} - \frac{227450195191568192265}{1714871048} a + \frac{468083456791050062719}{3429742096} \)
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: \(\Z/6\Z\)
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: $\left(2 a^{2} + 4 a + 3 : -6 a^{3} - 13 a^{2} - 5 a - 1 : 1\right)$
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\((j)_{-}\)
\( \left(-a\right) \) \(2\) \(2\) \(I_{4}\) Non-split multiplicative \(1\) \(1\) \(4\) \(4\)
\( \left(-a^{3} + 2 a^{2} + 2 a - 1\right) \) \(11\) \(2\) \(I_{8}\) 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
\(3\) 3B.1.1

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 22.1-a consists of curves linked by isogenies of degrees dividing 24.

Base change

This curve is not the base-change of an elliptic curve defined over \(\Q\). It is not a \(\Q\)-curve.