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Properties

Label 4.4.4205.1-16.1-a2
Base field 4.4.4205.1
Conductor norm \( 16 \)
CM no
Base change yes
Q-curve yes
Torsion order \( 1 \)
Rank \( 1 \)

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Base field 4.4.4205.1

Generator \(a\), with minimal polynomial \( x^{4} - x^{3} - 5 x^{2} - x + 1 \); class number \(1\).

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([1, -1, -5, -1, 1]))
 
gp: K = nfinit(Polrev([1, -1, -5, -1, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -1, -5, -1, 1]);
 

Weierstrass equation

\({y}^2+\left(-a^{3}+2a^{2}+4a\right){x}{y}+\left(-a^{3}+2a^{2}+4a-1\right){y}={x}^{3}+a{x}^{2}+\left(14a^{2}+2a-1\right){x}-a^{3}-88a^{2}-13a+14\)
sage: E = EllipticCurve([K([0,4,2,-1]),K([0,1,0,0]),K([-1,4,2,-1]),K([-1,2,14,0]),K([14,-13,-88,-1])])
 
gp: E = ellinit([Polrev([0,4,2,-1]),Polrev([0,1,0,0]),Polrev([-1,4,2,-1]),Polrev([-1,2,14,0]),Polrev([14,-13,-88,-1])], K);
 
magma: E := EllipticCurve([K![0,4,2,-1],K![0,1,0,0],K![-1,4,2,-1],K![-1,2,14,0],K![14,-13,-88,-1]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((2)\) = \((2)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 16 \) = \(16\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((-1048576)\) = \((2)^{20}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 1208925819614629174706176 \) = \(16^{20}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{237176659}{1048576} \)
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 $\left(\frac{11}{5} a^{2} - \frac{24}{5} a + \frac{11}{5} : \frac{52}{25} a^{3} + \frac{14}{5} a^{2} - \frac{119}{5} a + \frac{148}{25} : 1\right)$
Height \(0.36167522492860718305654285883953767877\)
Torsion structure: trivial
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 

BSD invariants

Analytic rank: \( 1 \)
sage: E.rank()
 
magma: Rank(E);
 
Mordell-Weil rank: \(1\)
Regulator: \( 0.36167522492860718305654285883953767877 \)
Period: \( 3.8118452576281646697640441271269360156 \)
Tamagawa product: \( 20 \)
Torsion order: \(1\)
Leading coefficient: \( 1.70083039830846 \)
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\((j)_{-}\)
\((2)\) \(16\) \(20\) \(I_{20}\) Split multiplicative \(-1\) \(1\) \(20\) \(20\)

Galois Representations

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

prime Image of Galois Representation
\(5\) 5B.1.2

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 5.
Its isogeny class 16.1-a consists of curves linked by isogenies of degree 5.

Base change

This elliptic curve is a \(\Q\)-curve. It is the base change of the following 2 elliptic curves:

Base field Curve
\(\Q(\sqrt{29}) \) 2.2.29.1-4.1-a2
\(\Q(\sqrt{29}) \) 2.2.29.1-100.2-e2