Properties

Base field \(\Q(\sqrt{5}) \)
Label 2.2.5.1-55.2-a6
Conductor \((-\phi + 8)\)
Conductor norm \( 55 \)
CM no
base-change no
Q-curve no
Torsion order \( 12 \)
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

\( y^2 + \phi x y + \phi y = x^{3} + \left(-\phi + 1\right) x^{2} + \left(-6 \phi - 5\right) x + 8 \phi + 5 \)
magma: E := ChangeRing(EllipticCurve([phi, -phi + 1, phi, -6*phi - 5, 8*phi + 5]),K);
sage: E = EllipticCurve(K, [phi, -phi + 1, phi, -6*phi - 5, 8*phi + 5])
gp (2.8): E = ellinit([phi, -phi + 1, phi, -6*phi - 5, 8*phi + 5],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()
\(\mathfrak{D}\) = \((50 \phi - 15)\) = \( \left(-2 \phi + 1\right)^{2} \cdot \left(-3 \phi + 2\right)^{2} \)
magma: Discriminant(E);
sage: E.discriminant()
gp (2.8): E.disc
\(N(\mathfrak{D})\) = \( 3025 \) = \( 5^{2} \cdot 11^{2} \)
magma: Norm(Discriminant(E));
sage: E.discriminant().norm()
gp (2.8): norm(E.disc)
\(j\) = \( -\frac{132583563}{605} \phi + \frac{59730809}{121} \)
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: \(\Z/2\Z\times\Z/6\Z\)
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: $\left(\phi : -1 : 1\right)$,$\left(\frac{3}{4} \phi + \frac{3}{4} : -\frac{5}{4} \phi - \frac{3}{8} : 1\right)$
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\((j)_{-}\)
\( \left(-2 \phi + 1\right) \) \(5\) \(2\) \(I_{2}\) Non-split multiplicative \(1\) \(1\) \(2\) \(2\)
\( \left(-3 \phi + 2\right) \) \(11\) \(2\) \(I_{2}\) Non-split multiplicative \(1\) \(1\) \(2\) \(2\)

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.1.1

Isogenies and isogeny class

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