Properties

Base field \(\Q(\sqrt{-3}) \)
Label 2.0.3.1-192.1-a3
Conductor \((-16 a + 8)\)
Conductor norm \( 192 \)
CM no
base-change yes: 72.a6,24.a6
Q-curve yes
Torsion order \( 2 \)
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

\( y^2 = x^{3} + a x^{2} + \left(16 a - 16\right) x - 180 \)
magma: E := ChangeRing(EllipticCurve([0, a, 0, 16*a - 16, -180]),K);
sage: E = EllipticCurve(K, [0, a, 0, 16*a - 16, -180])
gp (2.8): E = ellinit([0, a, 0, 16*a - 16, -180],K)

This is a global minimal model.

sage: E.is_global_minimal_model()

Invariants

\(\mathfrak{N} \) = \((-16 a + 8)\) = \( \left(2\right)^{3} \cdot \left(-2 a + 1\right) \)
magma: Conductor(E);
sage: E.conductor()
\(N(\mathfrak{N}) \) = \( 192 \) = \( 3 \cdot 4^{3} \)
magma: Norm(Conductor(E));
sage: E.conductor().norm()
\(\mathfrak{D}\) = \((13436928)\) = \( \left(2\right)^{11} \cdot \left(-2 a + 1\right)^{16} \)
magma: Discriminant(E);
sage: E.discriminant()
gp (2.8): E.disc
\(N(\mathfrak{D})\) = \( 180551034077184 \) = \( 3^{16} \cdot 4^{11} \)
magma: Norm(Discriminant(E));
sage: E.discriminant().norm()
gp (2.8): norm(E.disc)
\(j\) = \( \frac{207646}{6561} \)
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: \(\Z/2\Z\)
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: $\left(-5 a : 0 : 1\right)$
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 Root number ord(\(\mathfrak{N}\)) ord(\(\mathfrak{D}\)) ord\((j)_{-}\)
\( \left(-2 a + 1\right) \) \(3\) \(2\) \(I_{16}\) Non-split multiplicative \(1\) \(1\) \(16\) \(16\)
\( \left(2\right) \) \(4\) \(1\) \(II^*\) Additive \(-1\) \(3\) \(11\) \(0\)

Galois Representations

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

prime Image of Galois Representation
\(2\) 2B

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2, 4, 8 and 16.
Its isogeny class 192.1-a consists of curves linked by isogenies of degrees dividing 16.

Base change

This curve is the base-change of elliptic curves 72.a6, 24.a6, defined over \(\Q\), so it is also a \(\Q\)-curve.