Properties

Base field \(\Q(\sqrt{5}) \)
Label 2.2.5.1-45.1-a3
Conductor \((-6 \phi + 3)\)
Conductor norm \( 45 \)
CM no
base-change yes: 15.a7,75.b7
Q-curve yes
Torsion order \( 8 \)
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

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

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

\(\mathfrak{N} \) = \((-6 \phi + 3)\) = \( \left(3\right) \cdot \left(-2 \phi + 1\right) \)
magma: Conductor(E);
 
sage: E.conductor()
 
\(N(\mathfrak{N}) \) = \( 45 \) = \( 5 \cdot 9 \)
magma: Norm(Conductor(E));
 
sage: E.conductor().norm()
 
\(\mathfrak{D}\) = \((15)\) = \( \left(3\right) \cdot \left(-2 \phi + 1\right)^{2} \)
magma: Discriminant(E);
 
sage: E.discriminant()
 
gp (2.8): E.disc
 
\(N(\mathfrak{D})\) = \( 225 \) = \( 5^{2} \cdot 9 \)
magma: Norm(Discriminant(E));
 
sage: E.discriminant().norm()
 
gp (2.8): norm(E.disc)
 
\(j\) = \( -\frac{1}{15} \)
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: \(\Z/8\Z\)
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: $\left(-\phi + 1 : 2 \phi - 3 : 1\right)$
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\((j)_{-}\)
\( \left(-2 \phi + 1\right) \) \(5\) \(2\) \(I_{2}\) Split multiplicative \(-1\) \(1\) \(2\) \(2\)
\( \left(3\right) \) \(9\) \(1\) \(I_{1}\) Split multiplicative \(-1\) \(1\) \(1\) \(1\)

Galois Representations

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

prime Image of Galois Representation
\(2\) 2B

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2, 4, 8, 16 and 32.
Its isogeny class 45.1-a consists of curves linked by isogenies of degrees dividing 32.

Base change

This curve is the base-change of elliptic curves 15.a7, 75.b7, defined over \(\Q\), so it is also a \(\Q\)-curve.