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Properties

Label 4.4.1957.1-48.1-a2
Base field 4.4.1957.1
Conductor norm \( 48 \)
CM no
Base change no
Q-curve no
Torsion order \( 1 \)
Rank \( 0 \)

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Base field 4.4.1957.1

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

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([1, -1, -4, 0, 1]))
 
gp: K = nfinit(Polrev([1, -1, -4, 0, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -1, -4, 0, 1]);
 

Weierstrass equation

\({y}^2+\left(-a^{3}+a^{2}+4a\right){x}{y}+\left(a^{2}-a-2\right){y}={x}^{3}+\left(-a^{2}+2a+2\right){x}^{2}+\left(-255a^{3}+230a^{2}+886a-555\right){x}-3441a^{3}+2714a^{2}+11994a-6178\)
sage: E = EllipticCurve([K([0,4,1,-1]),K([2,2,-1,0]),K([-2,-1,1,0]),K([-555,886,230,-255]),K([-6178,11994,2714,-3441])])
 
gp: E = ellinit([Polrev([0,4,1,-1]),Polrev([2,2,-1,0]),Polrev([-2,-1,1,0]),Polrev([-555,886,230,-255]),Polrev([-6178,11994,2714,-3441])], K);
 
magma: E := EllipticCurve([K![0,4,1,-1],K![2,2,-1,0],K![-2,-1,1,0],K![-555,886,230,-255],K![-6178,11994,2714,-3441]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((2a^3-8a)\) = \((a^3-4a)\cdot(2)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 48 \) = \(3\cdot16\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((-32a^3+128a)\) = \((a^3-4a)\cdot(2)^{5}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 3145728 \) = \(3\cdot16^{5}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{2808717352846811}{48} a^{3} + \frac{2225925010621747}{96} a^{2} - \frac{7195553538165561}{32} a - \frac{14172658068106189}{96} \)
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: 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: \( 0 \)
sage: E.rank()
 
magma: Rank(E);
 
Mordell-Weil rank: \(0\)
Regulator: \( 1 \)
Period: \( 0.51176903168764297302304089740525219452 \)
Tamagawa product: \( 5 \)  =  \(1\cdot5\)
Torsion order: \(1\)
Leading coefficient: \( 1.44606762255545 \)
Analytic order of Ш: \( 25 \) (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\((j)_{-}\)
\((a^3-4a)\) \(3\) \(1\) \(I_{1}\) Split multiplicative \(-1\) \(1\) \(1\) \(1\)
\((2)\) \(16\) \(5\) \(I_{5}\) Split multiplicative \(-1\) \(1\) \(5\) \(5\)

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 48.1-a consists of curves linked by isogenies of degree 5.

Base change

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

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