Properties

Base field 4.4.2777.1
Label 4.4.2777.1-16.1-a4
Conductor \((2,2)\)
Conductor norm \( 16 \)
CM no
base-change no
Q-curve yes
Torsion order \( 2 \)
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

\( y^2 + x y + \left(a^{3} - 4 a\right) y = x^{3} + \left(-a^{3} + 2 a^{2} + 2 a - 3\right) x^{2} + \left(248 a^{3} + 94 a^{2} - 178 a - 86\right) x - 61 a^{3} + 1602 a^{2} - 240 a - 924 \)
magma: E := ChangeRing(EllipticCurve([1, -a^3 + 2*a^2 + 2*a - 3, a^3 - 4*a, 248*a^3 + 94*a^2 - 178*a - 86, -61*a^3 + 1602*a^2 - 240*a - 924]),K);
sage: E = EllipticCurve(K, [1, -a^3 + 2*a^2 + 2*a - 3, a^3 - 4*a, 248*a^3 + 94*a^2 - 178*a - 86, -61*a^3 + 1602*a^2 - 240*a - 924])
gp (2.8): E = ellinit([1, -a^3 + 2*a^2 + 2*a - 3, a^3 - 4*a, 248*a^3 + 94*a^2 - 178*a - 86, -61*a^3 + 1602*a^2 - 240*a - 924],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()
\(\mathfrak{D}\) = \((68719476736,8 a + 19925815120,8 a^{3} - 8 a^{2} - 24 a + 57123492976,8 a^{2} - 8 a + 6656090512)\) = \( \left(a^{3} - a^{2} - 4 a + 1\right)^{3} \cdot \left(-a\right)^{36} \)
magma: Discriminant(E);
sage: E.discriminant()
gp (2.8): E.disc
\(N(\mathfrak{D})\) = \( 35184372088832 \) = \( 2^{36} \cdot 8^{3} \)
magma: Norm(Discriminant(E));
sage: E.discriminant().norm()
gp (2.8): norm(E.disc)
\(j\) = \( -\frac{11596855953503300897687859}{68719476736} a^{3} + \frac{2020351217336280011001041}{68719476736} a^{2} + \frac{24027899520071110551507553}{34359738368} a + \frac{28086881208189762512317305}{68719476736} \)
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: $\left(-\frac{3}{4} a^{3} + \frac{13}{4} a^{2} - \frac{7}{2} a - 2 : -\frac{1}{8} a^{3} - \frac{13}{8} a^{2} + \frac{15}{4} a + 1 : 1\right)$
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\((j)_{-}\)
\( \left(-a\right) \) \(2\) \(2\) \( I_{36} \) Non-split multiplicative \(1\) \(36\) \(36\)
\( \left(a^{3} - a^{2} - 4 a + 1\right) \) \(8\) \(1\) \( I_{3} \) 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
\(2\) 2B
\(3\) 3B.1.2

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2, 3, 4, 6 and 12.
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.