Properties

Base field 4.4.19821.1
Label 4.4.19821.1-9.1-a2
Conductor \((3,\frac{1}{3} a^{3} - \frac{2}{3} a^{2} - 2 a + 4)\)
Conductor norm \( 9 \)
CM no
base-change no
Q-curve no
Torsion order \( 1 \)
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

\( y^2 + \left(a^{2} - 4\right) x y + \left(a^{2} - 4\right) y = x^{3} + \left(\frac{1}{3} a^{3} + \frac{1}{3} a^{2} - 4 a\right) x^{2} + \left(-a^{3} + 3 a^{2} + 8 a + 4\right) x + 3 a^{3} + 8 a^{2} - 11 a - 3 \)
magma: E := ChangeRing(EllipticCurve([a^2 - 4, 1/3*a^3 + 1/3*a^2 - 4*a, a^2 - 4, -a^3 + 3*a^2 + 8*a + 4, 3*a^3 + 8*a^2 - 11*a - 3]),K);
sage: E = EllipticCurve(K, [a^2 - 4, 1/3*a^3 + 1/3*a^2 - 4*a, a^2 - 4, -a^3 + 3*a^2 + 8*a + 4, 3*a^3 + 8*a^2 - 11*a - 3])
gp (2.8): E = ellinit([a^2 - 4, 1/3*a^3 + 1/3*a^2 - 4*a, a^2 - 4, -a^3 + 3*a^2 + 8*a + 4, 3*a^3 + 8*a^2 - 11*a - 3],K)

This is a global minimal model.

sage: E.is_global_minimal_model()

Invariants

\(\mathfrak{N} \) = \((3,\frac{1}{3} a^{3} - \frac{2}{3} a^{2} - 2 a + 4)\) = \( \left(-\frac{1}{3} a^{3} - \frac{1}{3} a^{2} + 3 a + 2\right)^{2} \)
magma: Conductor(E);
sage: E.conductor()
\(N(\mathfrak{N}) \) = \( 9 \) = \( 3^{2} \)
magma: Norm(Conductor(E));
sage: E.conductor().norm()
\(\mathfrak{D}\) = \((19683,6561 a,\frac{1}{3} a^{3} + \frac{1}{3} a^{2} + 4654 a + 8716,-\frac{1}{3} a^{3} + \frac{2}{3} a^{2} + 4976 a + 5903)\) = \( \left(-\frac{1}{3} a^{3} - \frac{1}{3} a^{2} + 3 a + 2\right)^{17} \)
magma: Discriminant(E);
sage: E.discriminant()
gp (2.8): E.disc
\(N(\mathfrak{D})\) = \( 129140163 \) = \( 3^{17} \)
magma: Norm(Discriminant(E));
sage: E.discriminant().norm()
gp (2.8): norm(E.disc)
\(j\) = \( -\frac{73343}{729} a^{3} + \frac{24131}{729} a^{2} + \frac{615892}{729} a + \frac{18719}{27} \)
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\((j)_{-}\)
\( \left(-\frac{1}{3} a^{3} - \frac{1}{3} a^{2} + 3 a + 2\right) \) \(3\) \(2\) \(I_{11}^*\) Additive \(-1\) \(2\) \(17\) \(11\)

Galois Representations

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

prime Image of Galois Representation
\(11\) 11B.10.1

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 11.
Its isogeny class 9.1-a consists of curves linked by isogenies of degree11.

Base change

This curve is not the base-change of an elliptic curve defined over \(\Q\). It is not a \(\Q\)-curve.