Properties

Base field 4.4.2777.1
Label 4.4.2777.1-8.1-a1
Conductor \((2,-a^{3} + a^{2} + 4 a - 1)\)
Conductor norm \( 8 \)
CM no
base-change no
Q-curve yes
Torsion order \( 1 \)
Rank not available

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Base field 4.4.2777.1

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

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

Weierstrass equation

\( y^2 + \left(a^{3} - a^{2} - 3 a + 2\right) x y + \left(a^{2} - a - 1\right) y = x^{3} + \left(a^{3} - a^{2} - 3 a\right) x^{2} + \left(21 a^{3} - 17 a^{2} - 68 a - 26\right) x + 51 a^{3} - 31 a^{2} - 177 a - 87 \)
magma: E := ChangeRing(EllipticCurve([a^3 - a^2 - 3*a + 2, a^3 - a^2 - 3*a, a^2 - a - 1, 21*a^3 - 17*a^2 - 68*a - 26, 51*a^3 - 31*a^2 - 177*a - 87]),K);
sage: E = EllipticCurve(K, [a^3 - a^2 - 3*a + 2, a^3 - a^2 - 3*a, a^2 - a - 1, 21*a^3 - 17*a^2 - 68*a - 26, 51*a^3 - 31*a^2 - 177*a - 87])
gp (2.8): E = ellinit([a^3 - a^2 - 3*a + 2, a^3 - a^2 - 3*a, a^2 - a - 1, 21*a^3 - 17*a^2 - 68*a - 26, 51*a^3 - 31*a^2 - 177*a - 87],K)

This is a global minimal model.

sage: E.is_global_minimal_model()

Invariants

\(\mathfrak{N} \) = \((2,-a^{3} + a^{2} + 4 a - 1)\) = \( \left(a^{3} - a^{2} - 4 a + 1\right) \)
magma: Conductor(E);
sage: E.conductor()
\(N(\mathfrak{N}) \) = \( 8 \) = \( 8 \)
magma: Norm(Conductor(E));
sage: E.conductor().norm()
\(\mathfrak{D}\) = \((8,8 a,4 a^{3} - 4 a^{2} - 8 a + 4,3 a^{3} - a^{2} - 10 a - 1)\) = \( \left(a^{3} - a^{2} - 4 a + 1\right)^{3} \)
magma: Discriminant(E);
sage: E.discriminant()
gp (2.8): E.disc
\(N(\mathfrak{D})\) = \( 512 \) = \( 8^{3} \)
magma: Norm(Discriminant(E));
sage: E.discriminant().norm()
gp (2.8): norm(E.disc)
\(j\) = \( 49366998461727605 a^{3} - \frac{17200987518233623}{2} a^{2} - \frac{1636561189853588101}{8} a - \frac{478255480551161905}{4} \)
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 ord(\(\mathfrak{N}\)) ord(\(\mathfrak{D}\)) ord\((j)_{-}\)
\( \left(a^{3} - a^{2} - 4 a + 1\right) \) \(8\) \(1\) \( I_{3} \) Non-split multiplicative \(1\) \(3\) \(3\)

Galois Representations

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

prime Image of Galois Representation
\(3\) 3B.1.2

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 3.
Its isogeny class 8.1-a consists of curves linked by isogenies of degree3.

Base change

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