Properties

Base field \(\Q(\sqrt{5}) \)
Label 2.2.5.1-1280.1-i6
Conductor \((-32 \phi + 16)\)
Conductor norm \( 1280 \)
CM no
base-change yes: 80.a1,400.e1
Q-curve yes
Torsion order \( 8 \)
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: K = nfinit(phi^2 - phi - 1);
 

Weierstrass equation

\( y^2 = x^{3} - 107 x + 426 \)
magma: E := ChangeRing(EllipticCurve([0, 0, 0, -107, 426]),K);
 
sage: E = EllipticCurve(K, [0, 0, 0, -107, 426])
 
gp: E = ellinit([0, 0, 0, -107, 426],K)
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

\(\mathfrak{N} \) = \((-32 \phi + 16)\) = \( \left(2\right)^{4} \cdot \left(-2 \phi + 1\right) \)
magma: Conductor(E);
 
sage: E.conductor()
 
\(N(\mathfrak{N}) \) = \( 1280 \) = \( 4^{4} \cdot 5 \)
magma: Norm(Conductor(E));
 
sage: E.conductor().norm()
 
\(\mathfrak{D}\) = \((5120)\) = \( \left(2\right)^{10} \cdot \left(-2 \phi + 1\right)^{2} \)
magma: Discriminant(E);
 
sage: E.discriminant()
 
gp: E.disc
 
\(N(\mathfrak{D})\) = \( 26214400 \) = \( 4^{10} \cdot 5^{2} \)
magma: Norm(Discriminant(E));
 
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
\(j\) = \( \frac{132304644}{5} \)
magma: jInvariant(E);
 
sage: E.j_invariant()
 
gp: 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: $\left(-3 \phi + 11 : -15 \phi + 25 : 1\right)$

magma: gens := [P:P in Generators(E)|Order(P) eq 0]; gens;
 
sage: gens = E.gens(); gens
 

Height: 0.9333935029996572

magma: [Height(P):P in gens];
 
sage: [P.height() for P in gens]
 

Regulator: 0.933393503

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

Torsion subgroup

Structure: \(\Z/2\Z\times\Z/4\Z\)
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
Generators: $\left(5 : -4 : 1\right)$,$\left(8 \phi - 7 : 0 : 1\right)$
magma: [piT(P) : P in Generators(T)];
 
sage: T.gens()
 
gp: T[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 \(-1\) \(1\) \(2\) \(2\)
\( \left(2\right) \) \(4\) \(4\) \(I_{2}^*\) Additive \(1\) \(4\) \(10\) \(0\)

Galois Representations

The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.

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

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2, 4 and 8.
Its isogeny class 1280.1-i consists of curves linked by isogenies of degrees dividing 16.

Base change

This curve is the base-change of elliptic curves 80.a1, 400.e1, defined over \(\Q\), so it is also a \(\Q\)-curve.