# Properties

 Base field 4.4.19821.1 Label 4.4.19821.1-48.1-e1 Conductor $$(6,2 a)$$ Conductor norm $$48$$ CM no base-change no Q-curve no Torsion order $$1$$ Rank not available

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Show commands for: Magma / SageMath / Pari/GP

## Base field 4.4.19821.1

Generator $$a$$, with minimal polynomial $$x^{4} - x^{3} - 8 x^{2} + 6 x + 3$$; class number $$1$$.

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3, 6, -8, -1, 1]);

sage: x = polygen(QQ); K.<a> = NumberField(x^4 - x^3 - 8*x^2 + 6*x + 3)

gp: K = nfinit(a^4 - a^3 - 8*a^2 + 6*a + 3);

## Weierstrass equation

$$y^2 + \left(a^{2} - a - 4\right) x y + \left(a^{2} - a - 4\right) y = x^{3} + \left(-\frac{2}{3} a^{3} + \frac{1}{3} a^{2} + 5 a - 2\right) x^{2} + \left(-\frac{2}{3} a^{3} + \frac{1}{3} a^{2} + 6 a - 5\right) x + \frac{7}{3} a^{3} - \frac{11}{3} a^{2} - 8 a - 4$$
magma: E := ChangeRing(EllipticCurve([a^2 - a - 4, -2/3*a^3 + 1/3*a^2 + 5*a - 2, a^2 - a - 4, -2/3*a^3 + 1/3*a^2 + 6*a - 5, 7/3*a^3 - 11/3*a^2 - 8*a - 4]),K);

sage: E = EllipticCurve(K, [a^2 - a - 4, -2/3*a^3 + 1/3*a^2 + 5*a - 2, a^2 - a - 4, -2/3*a^3 + 1/3*a^2 + 6*a - 5, 7/3*a^3 - 11/3*a^2 - 8*a - 4])

gp: E = ellinit([a^2 - a - 4, -2/3*a^3 + 1/3*a^2 + 5*a - 2, a^2 - a - 4, -2/3*a^3 + 1/3*a^2 + 6*a - 5, 7/3*a^3 - 11/3*a^2 - 8*a - 4],K)

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 $$\mathfrak{N}$$ = $$(6,2 a)$$ = $$\left(2\right) \cdot \left(-\frac{1}{3} a^{3} - \frac{1}{3} a^{2} + 3 a + 2\right)$$ magma: Conductor(E);  sage: E.conductor() $$N(\mathfrak{N})$$ = $$48$$ = $$3 \cdot 16$$ magma: Norm(Conductor(E));  sage: E.conductor().norm() $$\mathfrak{D}$$ = $$(48,16 a,\frac{16}{3} a^{3} + \frac{16}{3} a^{2} - 48 a + 16,-\frac{16}{3} a^{3} + \frac{32}{3} a^{2} + 32 a - 16)$$ = $$\left(2\right)^{4} \cdot \left(-\frac{1}{3} a^{3} - \frac{1}{3} a^{2} + 3 a + 2\right)$$ magma: Discriminant(E);  sage: E.discriminant()  gp: E.disc $$N(\mathfrak{D})$$ = $$196608$$ = $$3 \cdot 16^{4}$$ magma: Norm(Discriminant(E));  sage: E.discriminant().norm()  gp: norm(E.disc) $$j$$ = $$\frac{76926043}{48} a^{3} + \frac{1349417}{8} a^{2} - \frac{303233585}{24} a - \frac{8699767}{2}$$ 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: Trivial magma: T,piT := TorsionSubgroup(E); Invariants(T);  sage: T = E.torsion_subgroup(); T.invariants()  gp: T = elltors(E); T[2]

## 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$$(j)_{-}$$
$$\left(-\frac{1}{3} a^{3} - \frac{1}{3} a^{2} + 3 a + 2\right)$$ $$3$$ $$1$$ $$I_{1}$$ Split multiplicative $$-1$$ $$1$$ $$1$$ $$1$$
$$\left(2\right)$$ $$16$$ $$4$$ $$I_{4}$$ Split multiplicative $$-1$$ $$1$$ $$4$$ $$4$$

## Galois Representations

The mod $$p$$ Galois Representation has maximal image for all primes $$p$$ .

## Isogenies and isogeny class

This curve has no rational isogenies. Its isogeny class 48.1-e consists of this curve only.

## Base change

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