Properties

Base field \(\Q(\sqrt{-3}) \)
Label 2.0.3.1-145152.1-f2
Conductor \((-432 a + 144)\)
Conductor norm \( 145152 \)
CM no
base-change no
Q-curve no
Torsion order \( 1 \)
Rank not available

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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

\( y^2 = x^{3} + \left(9 a - 153\right) x - 420 a - 870 \)
magma: E := ChangeRing(EllipticCurve([0, 0, 0, 9*a - 153, -420*a - 870]),K);
sage: E = EllipticCurve(K, [0, 0, 0, 9*a - 153, -420*a - 870])
gp (2.8): E = ellinit([0, 0, 0, 9*a - 153, -420*a - 870],K)

This is a global minimal model.

sage: E.is_global_minimal_model()

Invariants

\(\mathfrak{N} \) = \((-432 a + 144)\) = \( \left(2\right)^{4} \cdot \left(-2 a + 1\right)^{4} \cdot \left(-3 a + 1\right) \)
magma: Conductor(E);
sage: E.conductor()
\(N(\mathfrak{N}) \) = \( 145152 \) = \( 3^{4} \cdot 4^{4} \cdot 7 \)
magma: Norm(Conductor(E));
sage: E.conductor().norm()
\(\mathfrak{D}\) = \((429981696 a + 23887872)\) = \( \left(2\right)^{15} \cdot \left(-2 a + 1\right)^{12} \cdot \left(-3 a + 1\right)^{3} \)
magma: Discriminant(E);
sage: E.discriminant()
gp (2.8): E.disc
\(N(\mathfrak{D})\) = \( 195726237040115712 \) = \( 3^{12} \cdot 4^{15} \cdot 7^{3} \)
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 group

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 Root number ord(\(\mathfrak{N}\)) ord(\(\mathfrak{D}\)) ord\((j)_{-}\)
\( \left(-2 a + 1\right) \) \(3\) \(1\) \(II^*\) Additive \(1\) \(4\) \(12\) \(0\)
\( \left(-3 a + 1\right) \) \(7\) \(3\) \(I_{3}\) Split multiplicative \(-1\) \(1\) \(3\) \(3\)
\( \left(2\right) \) \(4\) \(4\) \(I_{7}^*\) Additive \(-1\) \(4\) \(15\) \(3\)

Galois Representations

The mod \( p \) Galois Representation has maximal image for all primes \( p \) except those listed.

prime Image of Galois Representation
\(3\) 3Cs[2]

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 3.
Its isogeny class 145152.1-f 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.