Properties

Base field 4.4.19821.1
Label 4.4.19821.1-48.1-f3
Conductor \((6,2 a)\)
Conductor norm \( 48 \)
CM no
base-change no
Q-curve yes
Torsion order \( 3 \)
Rank not available

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Base field 4.4.19821.1

Generator \(a\), with minimal polynomial \( x^{4} - x^{3} - 8 x^{2} + 6 x + 3 \); class number \(1\).

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3, 6, -8, -1, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^4 - x^3 - 8*x^2 + 6*x + 3)
gp (2.8): K = nfinit(a^4 - a^3 - 8*a^2 + 6*a + 3);

Weierstrass equation

\( y^2 + \left(a + 1\right) x y = x^{3} + \left(a^{2} - 3\right) x^{2} + \left(\frac{4}{3} a^{3} + \frac{1}{3} a^{2} - 3 a + 1\right) x + \frac{13}{3} a^{3} + \frac{28}{3} a^{2} - 11 a - 5 \)
magma: E := ChangeRing(EllipticCurve([a + 1, a^2 - 3, 0, 4/3*a^3 + 1/3*a^2 - 3*a + 1, 13/3*a^3 + 28/3*a^2 - 11*a - 5]),K);
sage: E = EllipticCurve(K, [a + 1, a^2 - 3, 0, 4/3*a^3 + 1/3*a^2 - 3*a + 1, 13/3*a^3 + 28/3*a^2 - 11*a - 5])
gp (2.8): E = ellinit([a + 1, a^2 - 3, 0, 4/3*a^3 + 1/3*a^2 - 3*a + 1, 13/3*a^3 + 28/3*a^2 - 11*a - 5],K)

This is a global minimal model.

sage: E.is_global_minimal_model()

Invariants

\(\mathfrak{N} \) = \((6,2 a)\) = \( \left(2\right) \cdot \left(-\frac{1}{3} a^{3} - \frac{1}{3} a^{2} + 3 a + 2\right) \)
magma: Conductor(E);
sage: E.conductor()
\(N(\mathfrak{N}) \) = \( 48 \) = \( 3 \cdot 16 \)
magma: Norm(Conductor(E));
sage: E.conductor().norm()
\(\mathfrak{D}\) = \((6,6 a,\frac{2}{3} a^{3} + \frac{2}{3} a^{2} - 4 a + 2,-\frac{2}{3} a^{3} + \frac{4}{3} a^{2} + 4 a - 2)\) = \( \left(2\right) \cdot \left(-\frac{1}{3} a^{3} - \frac{1}{3} a^{2} + 3 a + 2\right)^{2} \)
magma: Discriminant(E);
sage: E.discriminant()
gp (2.8): E.disc
\(N(\mathfrak{D})\) = \( 144 \) = \( 3^{2} \cdot 16 \)
magma: Norm(Discriminant(E));
sage: E.discriminant().norm()
gp (2.8): norm(E.disc)
\(j\) = \( \frac{165192620}{3} a^{3} - \frac{1676553739}{6} a^{2} + \frac{895634596}{3} a + \frac{420165101}{3} \)
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: \(\Z/3\Z\)
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: $\left(-\frac{1}{3} a^{2} - \frac{2}{3} a + \frac{5}{3} : \frac{4}{9} a^{3} + \frac{11}{9} a^{2} - \frac{8}{9} a - \frac{5}{3} : 1\right)$
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\((j)_{-}\)
\( \left(-\frac{1}{3} a^{3} - \frac{1}{3} a^{2} + 3 a + 2\right) \) \(3\) \(2\) \( I_{2} \) Split multiplicative \(1\) \(2\) \(2\)
\( \left(2\right) \) \(16\) \(1\) \( I_{1} \) 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

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 3 and 9.
Its isogeny class 48.1-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 a \(\Q\)-curve.