Properties

Base field \(\Q(\sqrt{5}) \)
Label 2.2.5.1-2420.1-e1
Conductor \((-44 \phi + 22)\)
Conductor norm \( 2420 \)
CM no
base-change yes: 110.a1,550.i1
Q-curve yes
Torsion order \( 3 \)
Rank \( 1 \)

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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

\( y^2 + x y + y = x^{3} - 89 x + 316 \)
magma: E := ChangeRing(EllipticCurve([1, 0, 1, -89, 316]),K);
 
sage: E = EllipticCurve(K, [1, 0, 1, -89, 316])
 
gp (2.8): E = ellinit([1, 0, 1, -89, 316],K)
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

\(\mathfrak{N} \) = \((-44 \phi + 22)\) = \( \left(2\right) \cdot \left(-2 \phi + 1\right) \cdot \left(-3 \phi + 2\right) \cdot \left(-3 \phi + 1\right) \)
magma: Conductor(E);
 
sage: E.conductor()
 
\(N(\mathfrak{N}) \) = \( 2420 \) = \( 4 \cdot 5 \cdot 11^{2} \)
magma: Norm(Conductor(E));
 
sage: E.conductor().norm()
 
\(\mathfrak{D}\) = \((851840)\) = \( \left(2\right)^{7} \cdot \left(-2 \phi + 1\right)^{2} \cdot \left(-3 \phi + 2\right)^{3} \cdot \left(-3 \phi + 1\right)^{3} \)
magma: Discriminant(E);
 
sage: E.discriminant()
 
gp (2.8): E.disc
 
\(N(\mathfrak{D})\) = \( 725631385600 \) = \( 4^{7} \cdot 5^{2} \cdot 11^{6} \)
magma: Norm(Discriminant(E));
 
sage: E.discriminant().norm()
 
gp (2.8): norm(E.disc)
 
\(j\) = \( -\frac{76711450249}{851840} \)
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: \( 1 \)
magma: Rank(E);
 
sage: E.rank()
 

Generator: $\left(2 \phi + 7 : -10 \phi - 12 : 1\right)$

Height: 0.02140014800682888

magma: Generators(E); // includes torsion
 
sage: E.gens()
 

Regulator: 0.0214001480068

magma: Regulator(Generators(E));
 
sage: E.regulator_of_points(E.gens())
 

Torsion subgroup

Structure: \(\Z/3\Z\)
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: $\left(4 : -8 : 1\right)$
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 \(1\) \(1\) \(2\) \(2\)
\( \left(-3 \phi + 2\right) \) \(11\) \(3\) \(I_{3}\) Split multiplicative \(-1\) \(1\) \(3\) \(3\)
\( \left(-3 \phi + 1\right) \) \(11\) \(3\) \(I_{3}\) Split multiplicative \(-1\) \(1\) \(3\) \(3\)
\( \left(2\right) \) \(4\) \(7\) \(I_{7}\) Split multiplicative \(-1\) \(1\) \(7\) \(7\)

Galois Representations

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

prime Image of Galois Representation
\(3\) 3B.1.1

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 3.
Its isogeny class 2420.1-e consists of curves linked by isogenies of degree 3.

Base change

This curve is the base-change of elliptic curves 110.a1, 550.i1, defined over \(\Q\), so it is also a \(\Q\)-curve.