Properties

Label 2.2.5.1-3249.1-d2
Base field \(\Q(\sqrt{5}) \)
Conductor norm \( 3249 \)
CM no
Base change no
Q-curve no
Torsion order \( 2 \)
Rank \( 1 \)

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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

\({y}^2+\left(\phi+1\right){x}{y}+\phi{y}={x}^{3}+\left(-\phi-1\right){x}^{2}+\left(5\phi-26\right){x}+38\phi-34\)
sage: E = EllipticCurve([K([1,1]),K([-1,-1]),K([0,1]),K([-26,5]),K([-34,38])])
 
gp: E = ellinit([Polrev([1,1]),Polrev([-1,-1]),Polrev([0,1]),Polrev([-26,5]),Polrev([-34,38])], K);
 
magma: E := EllipticCurve([K![1,1],K![-1,-1],K![0,1],K![-26,5],K![-34,38]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((57)\) = \((3)\cdot(4\phi-3)\cdot(-4\phi+1)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 3249 \) = \(9\cdot19\cdot19\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((10602\phi+13167)\) = \((3)^{2}\cdot(4\phi-3)^{4}\cdot(-4\phi+1)\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 200564019 \) = \(9^{2}\cdot19^{4}\cdot19\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( -\frac{23498946430}{130321} \phi + \frac{359933949035}{1172889} \)
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 $\left(-3 \phi + 6 : -5 \phi + 9 : 1\right)$
Height \(0.13735599744251621411569869705985157834\)
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: $\left(\frac{9}{4} \phi - \frac{3}{2} : -2 \phi - \frac{3}{8} : 1\right)$
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: \( 0.13735599744251621411569869705985157834 \)
Period: \( 7.4905310388054956268188218177711909104 \)
Tamagawa product: \( 8 \)  =  \(2\cdot2^{2}\cdot1\)
Torsion order: \(2\)
Leading coefficient: \( 1.8404974670933996571054388263113053369 \)
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\((j)_{-}\)
\((3)\) \(9\) \(2\) \(I_{2}\) Non-split multiplicative \(1\) \(1\) \(2\) \(2\)
\((4\phi-3)\) \(19\) \(4\) \(I_{4}\) Split multiplicative \(-1\) \(1\) \(4\) \(4\)
\((-4\phi+1)\) \(19\) \(1\) \(I_{1}\) Non-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.
Its isogeny class 3249.1-d 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.