Properties

Label 2.2.5.1-605.1-b5
Base field \(\Q(\sqrt{5}) \)
Conductor norm \( 605 \)
CM no
Base change no
Q-curve no
Torsion order \( 4 \)
Rank \( 0 \)

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / 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(Polrev([-1, -1, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, -1, 1]);
 

Weierstrass equation

\({y}^2+\left(\phi+1\right){x}{y}={x}^{3}+\phi{x}^{2}+\left(-35\phi-54\right){x}-238\phi-211\)
sage: E = EllipticCurve([K([1,1]),K([0,1]),K([0,0]),K([-54,-35]),K([-211,-238])])
 
gp: E = ellinit([Polrev([1,1]),Polrev([0,1]),Polrev([0,0]),Polrev([-54,-35]),Polrev([-211,-238])], K);
 
magma: E := EllipticCurve([K![1,1],K![0,1],K![0,0],K![-54,-35],K![-211,-238]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((-22\phi+11)\) = \((-2\phi+1)\cdot(-3\phi+2)\cdot(-3\phi+1)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 605 \) = \(5\cdot11\cdot11\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((30250\phi-21175)\) = \((-2\phi+1)^{4}\cdot(-3\phi+2)^{2}\cdot(-3\phi+1)^{4}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 1107225625 \) = \(5^{4}\cdot11^{2}\cdot11^{4}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{1566703575423}{366025} \phi + \frac{989779352762}{366025} \)
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: \(0\)
Torsion structure: \(\Z/2\Z\oplus\Z/2\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Torsion generators: $\left(-\frac{7}{4} \phi - \frac{9}{2} : 4 \phi + \frac{25}{8} : 1\right)$ $\left(-2 \phi - 4 : 4 \phi + 3 : 1\right)$
sage: T.gens()
 
gp: T[3]
 
magma: [piT(P) : P in Generators(T)];
 

BSD invariants

Analytic rank: \( 0 \)
sage: E.rank()
 
magma: Rank(E);
 
Mordell-Weil rank: \(0\)
Regulator: \( 1 \)
Period: \( 2.9228066635483768159544414640506020915 \)
Tamagawa product: \( 16 \)  =  \(2^{2}\cdot2\cdot2\)
Torsion order: \(4\)
Leading coefficient: \( 1.3071188769567054487045004608680353170 \)
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\((j)_{-}\)
\((-2\phi+1)\) \(5\) \(4\) \(I_{4}\) Split multiplicative \(-1\) \(1\) \(4\) \(4\)
\((-3\phi+2)\) \(11\) \(2\) \(I_{2}\) Split multiplicative \(-1\) \(1\) \(2\) \(2\)
\((-3\phi+1)\) \(11\) \(2\) \(I_{4}\) Non-split multiplicative \(1\) \(1\) \(4\) \(4\)

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 and 4.
Its isogeny class 605.1-b consists of curves linked by isogenies of degrees dividing 8.

Base change

This elliptic curve is not a \(\Q\)-curve.

It is not the base change of an elliptic curve defined over any subfield.