Properties

Label 2.2.5.1-2880.1-d8
Base field \(\Q(\sqrt{5}) \)
Conductor \((-48 \phi + 24)\)
Conductor norm \( 2880 \)
CM no
Base change no
Q-curve yes
Torsion order \( 4 \)
Rank \( 1 \)

Related objects

Downloads

Learn more about

Show commands for: Magma / Pari/GP / SageMath

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

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

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-1, -1, 1]))
 
gp: K = nfinit(Pol(Vecrev([-1, -1, 1])));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, -1, 1]);
 

Weierstrass equation

\({y}^2={x}^{3}+\left(-\phi-1\right){x}^{2}+\left(17\phi-61\right){x}-259\phi+532\)
sage: E = EllipticCurve([K([0,0]),K([-1,-1]),K([0,0]),K([-61,17]),K([532,-259])])
 
gp: E = ellinit([Pol(Vecrev([0,0])),Pol(Vecrev([-1,-1])),Pol(Vecrev([0,0])),Pol(Vecrev([-61,17])),Pol(Vecrev([532,-259]))], K);
 
magma: E := EllipticCurve([K![0,0],K![-1,-1],K![0,0],K![-61,17],K![532,-259]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((-48 \phi + 24)\) = \( \left(2\right)^{3} \cdot \left(3\right) \cdot \left(-2 \phi + 1\right) \)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 2880 \) = \( 4^{3} \cdot 5 \cdot 9 \)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((-1536 \phi + 768)\) = \( \left(2\right)^{8} \cdot \left(3\right) \cdot \left(-2 \phi + 1\right) \)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 2949120 \) = \( 4^{8} \cdot 5 \cdot 9 \)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{2051350000672}{15} \phi + \frac{1267804019824}{15} \)
sage: E.j_invariant()
 
gp: E.j
 
magma: jInvariant(E);
 
Endomorphism ring: \(\Z\)
Geometric endomorphism ring: \(\Z\) (no potential complex multiplication)
sage: E.has_cm(), E.cm_discriminant()
 
magma: HasComplexMultiplication(E);
 
Sato-Tate group: $\mathrm{SU}(2)$

Mordell-Weil group

Rank: \(1\)
Generator $\left(\frac{7}{4} \phi + \frac{3}{2} : -\frac{23}{2} \phi + \frac{149}{8} : 1\right)$
Height \(1.60129868018092\)
Torsion structure: \(\Z/4\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Torsion generator: $\left(-\phi + 6 : -10 \phi + 16 : 1\right)$
sage: T.gens()
 
gp: T[3]
 
magma: [piT(P) : P in Generators(T)];
 

BSD invariants

Analytic rank: \( 1 \)
sage: E.rank()
 
magma: Rank(E);
 
Mordell-Weil rank: \(1\)
Regulator: \( 1.60129868018092 \)
Period: \( 12.8929257742532 \)
Tamagawa product: \( 2 \)  =  \(1\cdot2\cdot1\)
Torsion order: \(4\)
Leading coefficient: \(2.30822868912357\)
Analytic order of Ш: \( 1 \) (rounded)

Local data at primes of bad reduction

sage: E.local_data()
 
magma: LocalInformation(E);
 
prime Norm Tamagawa number Kodaira symbol Reduction type Root number ord(\(\mathfrak{N}\)) ord(\(\mathfrak{D}\)) ord\((j)_{-}\)
\( \left(-2 \phi + 1\right) \) \(5\) \(1\) \(I_{1}\) Split multiplicative \(-1\) \(1\) \(1\) \(1\)
\( \left(2\right) \) \(4\) \(2\) \(I_{1}^*\) Additive \(-1\) \(3\) \(8\) \(0\)
\( \left(3\right) \) \(9\) \(1\) \(I_{1}\) Split multiplicative \(-1\) \(1\) \(1\) \(1\)

Galois Representations

The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) 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 and 16.
Its isogeny class 2880.1-d consists of curves linked by isogenies of degrees dividing 16.

Base change

This curve is not the base change of an elliptic curve defined over \(\Q\). It is a \(\Q\)-curve.