Properties

Label 2.2.5.1-361.1-b4
Base field \(\Q(\sqrt{5}) \)
Conductor norm \( 361 \)
CM no
Base change no
Q-curve yes
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(Polrev([-1, -1, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, -1, 1]);
 

Weierstrass equation

\({y}^2+\phi{x}{y}+\phi{y}={x}^{3}+\left(-\phi-1\right){x}^{2}-30{x}-38\phi+17\)
sage: E = EllipticCurve([K([0,1]),K([-1,-1]),K([0,1]),K([-30,0]),K([17,-38])])
 
gp: E = ellinit([Polrev([0,1]),Polrev([-1,-1]),Polrev([0,1]),Polrev([-30,0]),Polrev([17,-38])], K);
 
magma: E := EllipticCurve([K![0,1],K![-1,-1],K![0,1],K![-30,0],K![17,-38]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((19)\) = \((4\phi-3)\cdot(-4\phi+1)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 361 \) = \(19\cdot19\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((285\phi+1463)\) = \((4\phi-3)\cdot(-4\phi+1)^{4}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 2476099 \) = \(19\cdot19^{4}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{76062245858406}{130321} \phi + \frac{47009406073653}{130321} \)
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: $\left(-\frac{13}{4} \phi + \frac{11}{4} : -\frac{1}{4} \phi + \frac{13}{8} : 1\right)$
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: \( 5.1458099663463501177043100795359611056 \)
Tamagawa product: \( 2 \)  =  \(1\cdot2\)
Torsion order: \(2\)
Leading coefficient: \( 1.1506380884046343989888331022289588177 \)
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\((j)_{-}\)
\((4\phi-3)\) \(19\) \(1\) \(I_{1}\) Non-split multiplicative \(1\) \(1\) \(1\) \(1\)
\((-4\phi+1)\) \(19\) \(2\) \(I_{4}\) 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
\(2\) 2B

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2 and 4.
Its isogeny class 361.1-b consists of curves linked by isogenies of degrees dividing 4.

Base change

This elliptic curve is a \(\Q\)-curve.

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