Properties

Base field \(\Q(\sqrt{-3}) \)
Label 2.0.3.1-2268.1-a5
Conductor \((-54 a + 18)\)
Conductor norm \( 2268 \)
CM no
base-change no
Q-curve no
Torsion order \( 3 \)
Rank \( 0 \)

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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

\( y^2 + x y = x^{3} - x^{2} + \left(-303 a + 30\right) x - 2091 a + 1329 \)
magma: E := ChangeRing(EllipticCurve([1, -1, 0, -303*a + 30, -2091*a + 1329]),K);
sage: E = EllipticCurve(K, [1, -1, 0, -303*a + 30, -2091*a + 1329])
gp (2.8): E = ellinit([1, -1, 0, -303*a + 30, -2091*a + 1329],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()
\(\mathfrak{D}\) = \((-1458 a + 486)\) = \( \left(2\right) \cdot \left(-2 a + 1\right)^{10} \cdot \left(-3 a + 1\right) \)
magma: Discriminant(E);
sage: E.discriminant()
gp (2.8): E.disc
\(N(\mathfrak{D})\) = \( 1653372 \) = \( 3^{10} \cdot 4 \cdot 7 \)
magma: Norm(Discriminant(E));
sage: E.discriminant().norm()
gp (2.8): norm(E.disc)
\(j\) = \( \frac{13989364197}{7} a + \frac{2196790845}{14} \)
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 group

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: \(\Z/3\Z\)
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]
Generator: $\left(-6 a - 5 : 3 a + 2 : 1\right)$
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\((j)_{-}\)
\( \left(-2 a + 1\right) \) \(3\) \(1\) \( IV^* \) Additive \(4\) \(10\) \(0\)
\( \left(-3 a + 1\right) \) \(7\) \(1\) \( I_{1} \) Split multiplicative \(1\) \(1\) \(1\)
\( \left(2\right) \) \(4\) \(1\) \( I_{1} \) Split multiplicative \(1\) \(1\) \(1\)

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[2]

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 3 and 9.
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.