Properties

Base field \(\Q(\sqrt{5}) \)
Label 2.2.5.1-55.2-a4
Conductor \((-\phi + 8)\)
Conductor norm \( 55 \)
CM no
base-change no
Q-curve no
Torsion order \( 2 \)
Rank \( 0 \)

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Base field \(\Q(\sqrt{5}) \)

Generator \(\phi\), with minimal polynomial \( x^{2} - x - 1 \); class number \(1\).

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

Weierstrass equation

\( y^2 + x y + y = x^{3} + \phi x^{2} + \left(-54 \phi + 54\right) x + 374 \phi - 572 \)
magma: E := ChangeRing(EllipticCurve([1, phi, 1, -54*phi + 54, 374*phi - 572]),K);
 
sage: E = EllipticCurve(K, [1, phi, 1, -54*phi + 54, 374*phi - 572])
 
gp (2.8): E = ellinit([1, phi, 1, -54*phi + 54, 374*phi - 572],K)
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

\(\mathfrak{N} \) = \((-\phi + 8)\) = \( \left(-2 \phi + 1\right) \cdot \left(-3 \phi + 2\right) \)
magma: Conductor(E);
 
sage: E.conductor()
 
\(N(\mathfrak{N}) \) = \( 55 \) = \( 5 \cdot 11 \)
magma: Norm(Conductor(E));
 
sage: E.conductor().norm()
 
\(\mathfrak{D}\) = \((-17848835 \phi + 6490780)\) = \( \left(-2 \phi + 1\right)^{3} \cdot \left(-3 \phi + 2\right)^{12} \)
magma: Discriminant(E);
 
sage: E.discriminant()
 
gp (2.8): E.disc
 
\(N(\mathfrak{D})\) = \( 392303547090125 \) = \( 5^{3} \cdot 11^{12} \)
magma: Norm(Discriminant(E));
 
sage: E.discriminant().norm()
 
gp (2.8): norm(E.disc)
 
\(j\) = \( \frac{114278307303626907}{78460709418025} \phi + \frac{89325070732461329}{78460709418025} \)
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: \( 0 \)
magma: Rank(E);
 
sage: E.rank()
 
magma: Generators(E); // includes torsion
 
sage: E.gens()
 

Regulator: 1

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: $\left(-\phi + \frac{23}{4} : \frac{1}{2} \phi - \frac{27}{8} : 1\right)$
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 Root number ord(\(\mathfrak{N}\)) ord(\(\mathfrak{D}\)) ord\((j)_{-}\)
\( \left(-2 \phi + 1\right) \) \(5\) \(1\) \(I_{3}\) Non-split multiplicative \(1\) \(1\) \(3\) \(3\)
\( \left(-3 \phi + 2\right) \) \(11\) \(2\) \(I_{12}\) Non-split multiplicative \(1\) \(1\) \(12\) \(12\)

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 55.2-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 not a \(\Q\)-curve.