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 \( 0 \)

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

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

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

Weierstrass equation

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

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

\(\mathfrak{N} \) = \((-54 a + 18)\) = \( \left(2\right) \cdot \left(-2 a + 1\right)^{4} \cdot \left(-3 a + 1\right) \)
sage: E.conductor()
 
magma: Conductor(E);
 
\(N(\mathfrak{N}) \) = \( 2268 \) = \( 3^{4} \cdot 4 \cdot 7 \)
sage: E.conductor().norm()
 
magma: Norm(Conductor(E));
 
\(\mathfrak{D}\) = \((3888 a + 216)\) = \( \left(2\right)^{3} \cdot \left(-2 a + 1\right)^{6} \cdot \left(-3 a + 1\right)^{3} \)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
\(N(\mathfrak{D})\) = \( 16003008 \) = \( 3^{6} \cdot 4^{3} \cdot 7^{3} \)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
\(j\) = \( \frac{1192725}{1372} a - \frac{2098143}{2744} \)
sage: E.j_invariant()
 
gp: E.j
 
magma: jInvariant(E);
 
\( \text{End} (E) \) = \(\Z\)   (no Complex Multiplication )
sage: E.has_cm(), E.cm_discriminant()
 
magma: HasComplexMultiplication(E);
 
\( \text{ST} (E) \) = $\mathrm{SU}(2)$

Mordell-Weil group

Rank: \( 0 \)

sage: E.rank()
 
magma: Rank(E);
 

Regulator: 1

sage: gens = E.gens(); gens
 
magma: gens := [P:P in Generators(E)|Order(P) eq 0]; gens;
 
sage: E.regulator_of_points(gens)
 
magma: Regulator(gens);
 

Torsion subgroup

Structure: \(\Z/3\Z\times\Z/3\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Generators: $\left(-2 : -3 a + 2 : 1\right)$,$\left(-2 a + 1 : -a - 1 : 1\right)$
sage: T.gens()
 
gp: T[3]
 
magma: [piT(P) : P in Generators(T)];
 

Local data at primes of bad reduction

sage: E.local_data()
 
magma: LocalInformation(E);
 
prime Norm Tamagawa number Kodaira symbol Reduction type Root number ord(\(\mathfrak{N}\)) ord(\(\mathfrak{D}\)) ord\((j)_{-}\)
\( \left(-2 a + 1\right) \) \(3\) \(3\) \(IV\) Additive \(-1\) \(4\) \(6\) \(0\)
\( \left(-3 a + 1\right) \) \(7\) \(3\) \(I_{3}\) Split multiplicative \(-1\) \(1\) \(3\) \(3\)
\( \left(2\right) \) \(4\) \(3\) \(I_{3}\) Split multiplicative \(-1\) \(1\) \(3\) \(3\)

Galois Representations

The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) 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.