Properties

Label 4.4.2624.1-16.1-a3
Base field 4.4.2624.1
Conductor norm \( 16 \)
CM no
Base change no
Q-curve no
Torsion order \( 2 \)
Rank \( 0 \)

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Base field 4.4.2624.1

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

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([1, 2, -3, -2, 1]))
 
gp: K = nfinit(Polrev([1, 2, -3, -2, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 2, -3, -2, 1]);
 

Weierstrass equation

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

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((2)\) = \((a^3-2a^2-2a+1)^{2}\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 16 \) = \(4^{2}\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((-4)\) = \((a^3-2a^2-2a+1)^{4}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 256 \) = \(4^{4}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( 54722560 a^{3} - 129179680 a^{2} - 117596128 a + 151876368 \)
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: $\left(\frac{1}{2} a^{2} - a - 1 : \frac{1}{4} a^{3} - a^{2} + \frac{1}{4} : 1\right)$
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: \( 203.33970644621227653745346657857448845 \)
Tamagawa product: \( 1 \)
Torsion order: \(2\)
Leading coefficient: \( 0.992385215532182 \)
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\((j)_{-}\)
\((a^3-2a^2-2a+1)\) \(4\) \(1\) \(IV\) Additive \(1\) \(2\) \(4\) \(0\)

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

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2, 4 and 8.
Its isogeny class 16.1-a 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.