Properties

Base field 4.4.2777.1
Label 4.4.2777.1-16.1-a7
Conductor \((2,2)\)
Conductor norm \( 16 \)
CM no
base-change no
Q-curve yes
Torsion order \( 4 \)
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

\( y^2 + \left(a + 1\right) x y + \left(a^{3} - a^{2} - 2 a + 1\right) y = x^{3} + \left(a^{3} - a^{2} - 3 a\right) x^{2} + \left(30 a^{3} - 52 a^{2} - 65 a - 13\right) x + 91 a^{3} - 127 a^{2} - 206 a - 56 \)
magma: E := ChangeRing(EllipticCurve([a + 1, a^3 - a^2 - 3*a, a^3 - a^2 - 2*a + 1, 30*a^3 - 52*a^2 - 65*a - 13, 91*a^3 - 127*a^2 - 206*a - 56]),K);
sage: E = EllipticCurve(K, [a + 1, a^3 - a^2 - 3*a, a^3 - a^2 - 2*a + 1, 30*a^3 - 52*a^2 - 65*a - 13, 91*a^3 - 127*a^2 - 206*a - 56])
gp (2.8): E = ellinit([a + 1, a^3 - a^2 - 3*a, a^3 - a^2 - 2*a + 1, 30*a^3 - 52*a^2 - 65*a - 13, 91*a^3 - 127*a^2 - 206*a - 56],K)

This is a global minimal model.

sage: E.is_global_minimal_model()

Invariants

\(\mathfrak{N} \) = \((2,2)\) = \( \left(a^{3} - a^{2} - 4 a + 1\right) \cdot \left(-a\right) \)
magma: Conductor(E);
sage: E.conductor()
\(N(\mathfrak{N}) \) = \( 16 \) = \( 2 \cdot 8 \)
magma: Norm(Conductor(E));
sage: E.conductor().norm()
\(\mathfrak{D}\) = \((262144,64 a + 162432,64 a^{3} - 64 a^{2} - 192 a + 172928,64 a^{2} - 64 a + 199808)\) = \( \left(a^{3} - a^{2} - 4 a + 1\right)^{6} \cdot \left(-a\right)^{18} \)
magma: Discriminant(E);
sage: E.discriminant()
gp (2.8): E.disc
\(N(\mathfrak{D})\) = \( 68719476736 \) = \( 2^{18} \cdot 8^{6} \)
magma: Norm(Discriminant(E));
sage: E.discriminant().norm()
gp (2.8): norm(E.disc)
\(j\) = \( \frac{5353573950671463885}{262144} a^{3} - \frac{8992096210769676335}{262144} a^{2} - \frac{7651439252430264991}{131072} a + \frac{15754080188450938489}{262144} \)
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: \(\Z/2\Z\times\Z/2\Z\)
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]
Generators: $\left(\frac{5}{4} a^{3} - 3 a - 3 : -\frac{7}{4} a^{3} - \frac{1}{2} a^{2} + \frac{37}{8} a + \frac{9}{4} : 1\right)$,$\left(-a^{3} + 3 a^{2} - 4 a - 3 : -a^{3} + 3 a^{2} + 4 a : 1\right)$
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\((j)_{-}\)
\( \left(-a\right) \) \(2\) \(2\) \( I_{18} \) Non-split multiplicative \(1\) \(18\) \(18\)
\( \left(a^{3} - a^{2} - 4 a + 1\right) \) \(8\) \(2\) \( I_{6} \) Non-split multiplicative \(1\) \(6\) \(6\)

Galois Representations

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

prime Image of Galois Representation
\(2\) 2Cs
\(3\) 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 16.1-a consists of curves linked by isogenies of degrees dividing 12.

Base change

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