Properties

Base field 4.4.2525.1
Label 4.4.2525.1-976.2-b1
Conductor \((122,2 a^{3} + 2 a^{2} - 10 a - 6)\)
Conductor norm \( 976 \)
CM no
base-change no
Q-curve no
Torsion order \( 1 \)
Rank not available

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Base field 4.4.2525.1

Generator \(a\), with minimal polynomial \( x^{4} - 2 x^{3} - 4 x^{2} + 5 x + 5 \); class number \(1\).

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![5, 5, -4, -2, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^4 - 2*x^3 - 4*x^2 + 5*x + 5)
gp (2.8): K = nfinit(a^4 - 2*a^3 - 4*a^2 + 5*a + 5);

Weierstrass equation

\( y^2 + \left(a^{3} - a^{2} - 3 a + 1\right) x y + \left(a^{2} - a - 3\right) y = x^{3} + \left(a^{3} - 4 a - 3\right) x^{2} + \left(-2 a^{3} + 6 a^{2} + 4 a - 12\right) x - 2 a^{3} + 8 a^{2} + 2 a - 20 \)
magma: E := ChangeRing(EllipticCurve([a^3 - a^2 - 3*a + 1, a^3 - 4*a - 3, a^2 - a - 3, -2*a^3 + 6*a^2 + 4*a - 12, -2*a^3 + 8*a^2 + 2*a - 20]),K);
sage: E = EllipticCurve(K, [a^3 - a^2 - 3*a + 1, a^3 - 4*a - 3, a^2 - a - 3, -2*a^3 + 6*a^2 + 4*a - 12, -2*a^3 + 8*a^2 + 2*a - 20])
gp (2.8): E = ellinit([a^3 - a^2 - 3*a + 1, a^3 - 4*a - 3, a^2 - a - 3, -2*a^3 + 6*a^2 + 4*a - 12, -2*a^3 + 8*a^2 + 2*a - 20],K)

This is a global minimal model.

sage: E.is_global_minimal_model()

Invariants

\(\mathfrak{N} \) = \((122,2 a^{3} + 2 a^{2} - 10 a - 6)\) = \( \left(2\right) \cdot \left(a^{3} - 2 a^{2} - 3 a + 1\right) \)
magma: Conductor(E);
sage: E.conductor()
\(N(\mathfrak{N}) \) = \( 976 \) = \( 16 \cdot 61 \)
magma: Norm(Conductor(E));
sage: E.conductor().norm()
\(\mathfrak{D}\) = \((244,4 a^{2} - 4 a + 164,4 a + 224,4 a^{3} - 4 a^{2} - 12 a + 148)\) = \( \left(2\right)^{2} \cdot \left(a^{3} - 2 a^{2} - 3 a + 1\right) \)
magma: Discriminant(E);
sage: E.discriminant()
gp (2.8): E.disc
\(N(\mathfrak{D})\) = \( 15616 \) = \( 16^{2} \cdot 61 \)
magma: Norm(Discriminant(E));
sage: E.discriminant().norm()
gp (2.8): norm(E.disc)
\(j\) = \( \frac{65719}{122} a^{3} - \frac{301635}{244} a^{2} - \frac{342275}{244} a + \frac{712131}{244} \)
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\((j)_{-}\)
\( \left(a^{3} - 2 a^{2} - 3 a + 1\right) \) \(61\) \(1\) \(I_{1}\) Non-split multiplicative \(1\) \(1\) \(1\) \(1\)
\( \left(2\right) \) \(16\) \(2\) \(I_{2}\) Split multiplicative \(-1\) \(1\) \(2\) \(2\)

Galois Representations

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

Isogenies and isogeny class

This curve has no rational isogenies. Its isogeny class 976.2-b consists of this curve only.

Base change

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