Properties

Label 2.2.5.1-605.2-a2
Base field \(\Q(\sqrt{5}) \)
Conductor norm \( 605 \)
CM no
Base change no
Q-curve no
Torsion order \( 2 \)
Rank \( 0 \)

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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

\({y}^2+\phi{x}{y}={x}^{3}+\phi{x}^{2}+\left(3\phi-195\right){x}+847\phi-500\)
sage: E = EllipticCurve([K([0,1]),K([0,1]),K([0,0]),K([-195,3]),K([-500,847])])
 
gp: E = ellinit([Pol(Vecrev([0,1])),Pol(Vecrev([0,1])),Pol(Vecrev([0,0])),Pol(Vecrev([-195,3])),Pol(Vecrev([-500,847]))], K);
 
magma: E := EllipticCurve([K![0,1],K![0,1],K![0,0],K![-195,3],K![-500,847]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((-4\phi-23)\) = \((-2\phi+1)\cdot(-3\phi+2)^{2}\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 605 \) = \(5\cdot11^{2}\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((499750\phi+5845625)\) = \((-2\phi+1)^{6}\cdot(-3\phi+2)^{9}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 36842932671875 \) = \(5^{6}\cdot11^{9}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( -\frac{393466}{25} \phi + \frac{17856881}{125} \)
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: \(\Z/2\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Torsion generator: $\left(\frac{27}{4} \phi - \frac{17}{4} : -\frac{5}{4} \phi - \frac{27}{8} : 1\right)$
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: \( 3.03830961336023 \)
Tamagawa product: \( 4 \)  =  \(2\cdot2\)
Torsion order: \(2\)
Leading coefficient: \( 1.35877336643291 \)
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\((j)_{-}\)
\((-2\phi+1)\) \(5\) \(2\) \(I_{6}\) Non-split multiplicative \(1\) \(1\) \(6\) \(6\)
\((-3\phi+2)\) \(11\) \(2\) \(III^{*}\) Additive \(1\) \(2\) \(9\) \(0\)

Galois Representations

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

prime Image of Galois Representation
\(2\) 2B
\(3\) 3Ns

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2.
Its isogeny class 605.2-a consists of curves linked by isogenies of degree 2.

Base change

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

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