Properties

Base field 4.4.2777.1
Label 4.4.2777.1-22.1-b2
Conductor \((22,2 a^{3} - 3 a^{2} - 5 a + 4)\)
Conductor norm \( 22 \)
CM no
base-change no
Q-curve not determined
Torsion order \( 10 \)
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

\( y^2 + \left(a^{3} - 4 a - 1\right) x y + \left(a^{2} - 2\right) y = x^{3} + \left(-a^{3} + 2 a^{2} + 3 a - 4\right) x^{2} + \left(9 a^{3} + 8 a^{2} - 59 a - 35\right) x - 37 a^{3} + 17 a^{2} + 136 a + 71 \)
magma: E := ChangeRing(EllipticCurve([a^3 - 4*a - 1, -a^3 + 2*a^2 + 3*a - 4, a^2 - 2, 9*a^3 + 8*a^2 - 59*a - 35, -37*a^3 + 17*a^2 + 136*a + 71]),K);
sage: E = EllipticCurve(K, [a^3 - 4*a - 1, -a^3 + 2*a^2 + 3*a - 4, a^2 - 2, 9*a^3 + 8*a^2 - 59*a - 35, -37*a^3 + 17*a^2 + 136*a + 71])
gp (2.8): E = ellinit([a^3 - 4*a - 1, -a^3 + 2*a^2 + 3*a - 4, a^2 - 2, 9*a^3 + 8*a^2 - 59*a - 35, -37*a^3 + 17*a^2 + 136*a + 71],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()
\(\mathfrak{D}\) = \((468512,a + 274346,a^{3} - a^{2} - 3 a + 66894,a^{2} - a + 450738)\) = \( \left(-a\right)^{5} \cdot \left(-a^{3} + 2 a^{2} + 2 a - 1\right)^{4} \)
magma: Discriminant(E);
sage: E.discriminant()
gp (2.8): E.disc
\(N(\mathfrak{D})\) = \( 468512 \) = \( 2^{5} \cdot 11^{4} \)
magma: Norm(Discriminant(E));
sage: E.discriminant().norm()
gp (2.8): norm(E.disc)
\(j\) = \( \frac{19492220557608647265}{468512} a^{3} + \frac{26555029218814954293}{468512} a^{2} - \frac{7618439534802966603}{234256} a - \frac{16502468789812849155}{468512} \)
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 rank and generators

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: \(\Z/10\Z\)
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: $\left(-10 a^{3} + 15 a^{2} + 19 a + 4 : -21 a^{3} + 84 a^{2} - 49 a - 79 : 1\right)$
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 ord(\(\mathfrak{N}\)) ord(\(\mathfrak{D}\)) ord\((j)_{-}\)
\( \left(-a\right) \) 2 \(5\) \( I_{5} \) Split multiplicative 1 5 5
\( \left(-a^{3} + 2 a^{2} + 2 a - 1\right) \) 11 \(4\) \( I_{4} \) Split multiplicative 1 4 4

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.1.1

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.

Base change

This curve is not the base-change of an elliptic curve defined over \(\Q\). It has not yet been determined whether or not it is a \(\Q\)-curve.