Properties

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

Related objects

Learn more about

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: K<a> := NumberField(x^2 - x + 1);
sage: 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

\(\mathfrak{N} \) = \( \left(-54 a + 18\right) \) = \( \left(2\right) \cdot \left(-2 a + 1\right)^{4} \cdot \left(-3 a + 1\right) \)
magma: Conductor(E);
sage: E.conductor()
\(N(\mathfrak{N}) \) = \( 2268 \) = \( 3^{4} \cdot 4 \cdot 7 \)
magma: Norm(Conductor(E));
sage: E.conductor().norm()
\(\mathfrak{D}\) = \( \left(3888 a + 216\right) \) = \( \left(2\right)^{3} \cdot \left(-2 a + 1\right)^{6} \cdot \left(-3 a + 1\right)^{3} \)
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]
Generators: $\left(-2 : -3 a + 2 : 1\right)$,$\left(-2 a + 1 : -a - 1 : 1\right)$
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 5 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.