# Properties

 Base field $\Q(\sqrt{-3})$ Label 2.0.3.1-2268.1-a1 Conductor $(-54 a + 18)$ Conductor norm $2268$ CM no base-change no Q-curve no Torsion order $9$ Rank $0$

# Related objects

Show commands for: Magma / SageMath / Pari/GP

## Base field $\Q(\sqrt{-3})$

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

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -1, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^2 - x + 1)
gp (2.8): K = nfinit(a^2 - a + 1);

## Weierstrass equation

$y^2 + x y = x^{3} - x^{2} - 3 a x - 3 a + 3$
magma: E := ChangeRing(EllipticCurve([1, -1, 0, -3*a, -3*a + 3]),K);
sage: E = EllipticCurve(K, [1, -1, 0, -3*a, -3*a + 3])
gp (2.8): E = ellinit([1, -1, 0, -3*a, -3*a + 3],K)

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 $\mathfrak{N}$ = $(-54 a + 18)$ = $\left(2\right) \cdot \left(-2 a + 1\right)^{4} \cdot \left(-3 a + 1\right)$ magma: Conductor(E); sage: E.conductor() $N(\mathfrak{N})$ = $2268$ = $3^{4} \cdot 4 \cdot 7$ magma: Norm(Conductor(E)); sage: E.conductor().norm() $\mathfrak{D}$ = $(3888 a + 216)$ = $\left(2\right)^{3} \cdot \left(-2 a + 1\right)^{6} \cdot \left(-3 a + 1\right)^{3}$ magma: Discriminant(E); sage: E.discriminant() gp (2.8): E.disc $N(\mathfrak{D})$ = $16003008$ = $3^{6} \cdot 4^{3} \cdot 7^{3}$ magma: Norm(Discriminant(E)); sage: E.discriminant().norm() gp (2.8): norm(E.disc) $j$ = $\frac{1192725}{1372} a - \frac{2098143}{2744}$ 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: $0$
magma: Rank(E);
sage: E.rank()
magma: Generators(E); // includes torsion
sage: E.gens()

Regulator: 1

magma: Regulator(Generators(E));
sage: E.regulator_of_points(E.gens())

## Torsion subgroup

Structure: $\Z/3\Z\times\Z/3\Z$ 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] $\left(-2 : -3 a + 2 : 1\right)$,$\left(-2 a + 1 : -a - 1 : 1\right)$ 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$(j)_{-}$
$\left(-2 a + 1\right)$ 3 $3$ $IV$ Additive 4 6 0
$\left(-3 a + 1\right)$ 7 $3$ $I_{3}$ Split multiplicative 1 3 3
$\left(2\right)$ 4 $3$ $I_{3}$ 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
$3$ 3Cs.1.1[2]

## Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree $d$ for $d=$ 3.
Its isogeny class 2268.1-a consists of curves linked by isogenies of degrees dividing 9.

## Base change

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