# Properties

 Base field 4.4.19821.1 Label 4.4.19821.1-27.2-f2 Conductor $(9,-\frac{1}{3} a^{3} + \frac{2}{3} a^{2} + 2 a - 1)$ Conductor norm $27$ CM no base-change no Q-curve not determined Torsion order $1$ 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

$y^2 + \left(-\frac{1}{3} a^{3} + \frac{2}{3} a^{2} + 3 a - 3\right) x y + \left(\frac{1}{3} a^{3} + \frac{1}{3} a^{2} - 2 a - 1\right) y = x^{3} + \left(-\frac{1}{3} a^{3} + \frac{2}{3} a^{2} + 3 a - 2\right) x^{2} + \left(-\frac{19}{3} a^{3} - \frac{19}{3} a^{2} + 33 a + 9\right) x - \frac{67}{3} a^{3} - \frac{124}{3} a^{2} + 59 a + 22$
magma: E := ChangeRing(EllipticCurve([-1/3*a^3 + 2/3*a^2 + 3*a - 3, -1/3*a^3 + 2/3*a^2 + 3*a - 2, 1/3*a^3 + 1/3*a^2 - 2*a - 1, -19/3*a^3 - 19/3*a^2 + 33*a + 9, -67/3*a^3 - 124/3*a^2 + 59*a + 22]),K);
sage: E = EllipticCurve(K, [-1/3*a^3 + 2/3*a^2 + 3*a - 3, -1/3*a^3 + 2/3*a^2 + 3*a - 2, 1/3*a^3 + 1/3*a^2 - 2*a - 1, -19/3*a^3 - 19/3*a^2 + 33*a + 9, -67/3*a^3 - 124/3*a^2 + 59*a + 22])
gp (2.8): E = ellinit([-1/3*a^3 + 2/3*a^2 + 3*a - 3, -1/3*a^3 + 2/3*a^2 + 3*a - 2, 1/3*a^3 + 1/3*a^2 - 2*a - 1, -19/3*a^3 - 19/3*a^2 + 33*a + 9, -67/3*a^3 - 124/3*a^2 + 59*a + 22],K)

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 $\mathfrak{N}$ = $(9,-\frac{1}{3} a^{3} + \frac{2}{3} a^{2} + 2 a - 1)$ = $\left(-\frac{1}{3} a^{3} - \frac{1}{3} a^{2} + 3 a + 2\right)^{3}$ magma: Conductor(E); sage: E.conductor() $N(\mathfrak{N})$ = $27$ = $3^{3}$ magma: Norm(Conductor(E)); sage: E.conductor().norm() $\mathfrak{D}$ = $(243,81 a,\frac{1}{3} a^{3} + \frac{1}{3} a^{2} + 37 a + 211,-\frac{1}{3} a^{3} + \frac{2}{3} a^{2} + 35 a + 71)$ = $\left(-\frac{1}{3} a^{3} - \frac{1}{3} a^{2} + 3 a + 2\right)^{9}$ magma: Discriminant(E); sage: E.discriminant() gp (2.8): E.disc $N(\mathfrak{D})$ = $19683$ = $3^{9}$ magma: Norm(Discriminant(E)); sage: E.discriminant().norm() gp (2.8): norm(E.disc) $j$ = $-\frac{2659993414}{3} a^{3} + \frac{3059637059}{3} a^{2} + 7384788197 a - 8211399121$ 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: Trivial 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]

## 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 $3$ $IV^*$ Additive 3 9 0

## Galois Representations

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

prime Image of Galois Representation
$3$ 3B.1.2

## Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree $d$ for $d=$ 3.
Its isogeny class 27.2-f 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.