Properties

Base field 4.4.19821.1
Label 4.4.19821.1-21.1-a3
Conductor \((21,\frac{2}{3} a^{3} - \frac{1}{3} a^{2} - 5 a - 1)\)
Conductor norm \( 21 \)
CM no
base-change no
Q-curve yes
Torsion order \( 2 \)
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

\( y^2 + \left(a + 1\right) x y + \left(-\frac{1}{3} a^{3} + \frac{2}{3} a^{2} + 3 a - 3\right) y = x^{3} + \left(-a^{2} + 3\right) x^{2} + \left(\frac{28}{3} a^{3} + \frac{1}{3} a^{2} - 75 a - 21\right) x + 37 a^{3} + 4 a^{2} - 292 a - 103 \)
magma: E := ChangeRing(EllipticCurve([a + 1, -a^2 + 3, -1/3*a^3 + 2/3*a^2 + 3*a - 3, 28/3*a^3 + 1/3*a^2 - 75*a - 21, 37*a^3 + 4*a^2 - 292*a - 103]),K);
sage: E = EllipticCurve(K, [a + 1, -a^2 + 3, -1/3*a^3 + 2/3*a^2 + 3*a - 3, 28/3*a^3 + 1/3*a^2 - 75*a - 21, 37*a^3 + 4*a^2 - 292*a - 103])
gp (2.8): E = ellinit([a + 1, -a^2 + 3, -1/3*a^3 + 2/3*a^2 + 3*a - 3, 28/3*a^3 + 1/3*a^2 - 75*a - 21, 37*a^3 + 4*a^2 - 292*a - 103],K)

This is a global minimal model.

sage: E.is_global_minimal_model()

Invariants

\(\mathfrak{N} \) = \((21,\frac{2}{3} a^{3} - \frac{1}{3} a^{2} - 5 a - 1)\) = \( \left(-\frac{1}{3} a^{3} - \frac{1}{3} a^{2} + 3 a + 2\right) \cdot \left(-\frac{1}{3} a^{3} - \frac{1}{3} a^{2} + 2 a\right) \)
magma: Conductor(E);
sage: E.conductor()
\(N(\mathfrak{N}) \) = \( 21 \) = \( 3 \cdot 7 \)
magma: Norm(Conductor(E));
sage: E.conductor().norm()
\(\mathfrak{D}\) = \((189,27 a + 81,\frac{1}{3} a^{3} + \frac{1}{3} a^{2} + 10 a + 22,-\frac{1}{3} a^{3} + \frac{2}{3} a^{2} + 8 a + 44)\) = \( \left(-\frac{1}{3} a^{3} - \frac{1}{3} a^{2} + 3 a + 2\right)^{6} \cdot \left(-\frac{1}{3} a^{3} - \frac{1}{3} a^{2} + 2 a\right) \)
magma: Discriminant(E);
sage: E.discriminant()
gp (2.8): E.disc
\(N(\mathfrak{D})\) = \( 5103 \) = \( 3^{6} \cdot 7 \)
magma: Norm(Discriminant(E));
sage: E.discriminant().norm()
gp (2.8): norm(E.disc)
\(j\) = \( -\frac{3858185}{189} a^{3} + \frac{5279251}{63} a^{2} - \frac{2609822}{63} a - \frac{4497131}{189} \)
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: \(\Z/2\Z\)
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: $\left(\frac{1}{3} a^{3} + \frac{1}{3} a^{2} - 3 a - 2 : -\frac{1}{3} a^{3} - \frac{1}{3} a^{2} + 2 a + 3 : 1\right)$
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\((j)_{-}\)
\( \left(-\frac{1}{3} a^{3} - \frac{1}{3} a^{2} + 3 a + 2\right) \) \(3\) \(6\) \( I_{6} \) Split multiplicative \(1\) \(6\) \(6\)
\( \left(-\frac{1}{3} a^{3} - \frac{1}{3} a^{2} + 2 a\right) \) \(7\) \(1\) \( I_{1} \) 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
\(2\) 2B
\(3\) 3B.1.2

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2, 3 and 6.
Its isogeny class 21.1-a consists of curves linked by isogenies of degrees dividing 6.

Base change

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