# Properties

 Base field 4.4.2777.1 Label 4.4.2777.1-22.1-a3 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

# Related objects

Show commands for: Magma / SageMath / Pari/GP

## 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: K = nfinit(a^4 - a^3 - 4*a^2 + a + 2);

## Weierstrass equation

$$y^2 + \left(a^{2} - a - 1\right) x y + \left(a^{2} - a - 2\right) y = x^{3} + \left(-a^{3} + 2 a^{2} + 3 a - 3\right) x^{2} + \left(-20 a^{3} + 35 a^{2} + 56 a - 64\right) x + 299 a^{3} - 501 a^{2} - 856 a + 876$$
magma: E := ChangeRing(EllipticCurve([a^2 - a - 1, -a^3 + 2*a^2 + 3*a - 3, a^2 - a - 2, -20*a^3 + 35*a^2 + 56*a - 64, 299*a^3 - 501*a^2 - 856*a + 876]),K);

sage: E = EllipticCurve(K, [a^2 - a - 1, -a^3 + 2*a^2 + 3*a - 3, a^2 - a - 2, -20*a^3 + 35*a^2 + 56*a - 64, 299*a^3 - 501*a^2 - 856*a + 876])

gp: E = ellinit([a^2 - a - 1, -a^3 + 2*a^2 + 3*a - 3, a^2 - a - 2, -20*a^3 + 35*a^2 + 56*a - 64, 299*a^3 - 501*a^2 - 856*a + 876],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() $$\mathfrak{D}$$ = $$(7929856,a + 4618730,a^{3} - a^{2} - 3 a + 5991054,a^{2} - a + 5143602)$$ = $$\left(-a\right)^{16} \cdot \left(-a^{3} + 2 a^{2} + 2 a - 1\right)^{2}$$ magma: Discriminant(E);  sage: E.discriminant()  gp: E.disc $$N(\mathfrak{D})$$ = $$7929856$$ = $$2^{16} \cdot 11^{2}$$ magma: Norm(Discriminant(E));  sage: E.discriminant().norm()  gp: norm(E.disc) $$j$$ = $$\frac{22702786953003143813}{7929856} a^{3} + \frac{30928910991628706537}{7929856} a^{2} - \frac{8873274696408279559}{3964928} a - \frac{19220593607745698479}{7929856}$$ magma: jInvariant(E);  sage: E.j_invariant()  gp: 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()

Regulator: not available

magma: gens := [P:P in Generators(E)|Order(P) eq 0]; gens;

sage: gens = E.gens(); gens

magma: Regulator(gens);

sage: E.regulator_of_points(gens)

## Torsion subgroup

Structure: $$\Z/6\Z$$ magma: T,piT := TorsionSubgroup(E); Invariants(T);  sage: T = E.torsion_subgroup(); T.invariants()  gp: T = elltors(E); T[2] $\left(5 a^{3} - 8 a^{2} - 15 a + 14 : -19 a^{3} + 32 a^{2} + 54 a - 56 : 1\right)$ magma: [piT(P) : P in Generators(T)];  sage: T.gens()  gp: T[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$$(j)_{-}$$
$$\left(-a\right)$$ $$2$$ $$2$$ $$I_{16}$$ Non-split multiplicative $$1$$ $$1$$ $$16$$ $$16$$
$$\left(-a^{3} + 2 a^{2} + 2 a - 1\right)$$ $$11$$ $$2$$ $$I_{2}$$ Non-split multiplicative $$1$$ $$1$$ $$2$$ $$2$$

## 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.