# Properties

 Base field 4.4.19821.1 Label 4.4.19821.1-48.1-b2 Conductor $(6,2 a)$ Conductor norm $48$ CM no base-change no Q-curve not determined Torsion order $3$ Rank not available

# Related objects

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

## Weierstrass equation

$y^2 + \left(\frac{1}{3} a^{3} + \frac{1}{3} a^{2} - 2 a\right) x y + \left(a^{2} - a - 4\right) y = x^{3} + \left(\frac{1}{3} a^{3} + \frac{1}{3} a^{2} - 4 a - 2\right) x^{2} + \left(11 a^{3} - 7 a^{2} - 67 a - 24\right) x - \frac{121}{3} a^{3} - \frac{160}{3} a^{2} + 444 a + 159$
magma: E := ChangeRing(EllipticCurve([1/3*a^3 + 1/3*a^2 - 2*a, 1/3*a^3 + 1/3*a^2 - 4*a - 2, a^2 - a - 4, 11*a^3 - 7*a^2 - 67*a - 24, -121/3*a^3 - 160/3*a^2 + 444*a + 159]),K);
sage: E = EllipticCurve(K, [1/3*a^3 + 1/3*a^2 - 2*a, 1/3*a^3 + 1/3*a^2 - 4*a - 2, a^2 - a - 4, 11*a^3 - 7*a^2 - 67*a - 24, -121/3*a^3 - 160/3*a^2 + 444*a + 159])
gp (2.8): E = ellinit([1/3*a^3 + 1/3*a^2 - 2*a, 1/3*a^3 + 1/3*a^2 - 4*a - 2, a^2 - a - 4, 11*a^3 - 7*a^2 - 67*a - 24, -121/3*a^3 - 160/3*a^2 + 444*a + 159],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() $\mathfrak{D}$ = $(24,24 a,\frac{8}{3} a^{3} + \frac{8}{3} a^{2} - 16 a + 8,-\frac{8}{3} a^{3} + \frac{16}{3} a^{2} + 16 a - 8)$ = $\left(2\right)^{3} \cdot \left(-\frac{1}{3} a^{3} - \frac{1}{3} a^{2} + 3 a + 2\right)^{2}$ magma: Discriminant(E); sage: E.discriminant() gp (2.8): E.disc $N(\mathfrak{D})$ = $36864$ = $3^{2} \cdot 16^{3}$ magma: Norm(Discriminant(E)); sage: E.discriminant().norm() gp (2.8): norm(E.disc) $j$ = $\frac{21224746042177}{24} a^{3} - \frac{9536989987379}{8} a^{2} - \frac{53280470046179}{8} a + \frac{182973377275723}{24}$ 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: $\Z/3\Z$ 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] $\left(-a^{3} + a^{2} + \frac{14}{3} a + \frac{8}{3} : \frac{7}{9} a^{3} - \frac{52}{9} a^{2} + \frac{76}{9} a + 5 : 1\right)$ 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$(j)_{-}$
$\left(-\frac{1}{3} a^{3} - \frac{1}{3} a^{2} + 3 a + 2\right)$ 3 $2$ $I_{2}$ Split multiplicative 1 2 2
$\left(2\right)$ 16 $3$ $I_{3}$ Split multiplicative 1 3 3

## Galois Representations

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

prime Image of Galois Representation
$3$ 3B.1.1

## Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree $d$ for $d=$ 3.
Its isogeny class 48.1-b consists of curves linked by isogenies of degree3.

## 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.