Properties

Label 4.4.1957.1-48.1-a1
Base field 4.4.1957.1
Conductor norm \( 48 \)
CM no
Base change no
Q-curve no
Torsion order \( 5 \)
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

\({y}^2+\left(-a^{3}+a^{2}+3a-1\right){x}{y}+\left(-a^{3}+a^{2}+4a\right){y}={x}^{3}+\left(-a^{3}+5a+2\right){x}^{2}+\left(-2a^{3}+a^{2}+7a+2\right){x}+2a^{3}-5a^{2}+2\)
sage: E = EllipticCurve([K([-1,3,1,-1]),K([2,5,0,-1]),K([0,4,1,-1]),K([2,7,1,-2]),K([2,0,-5,2])])
 
gp: E = ellinit([Polrev([-1,3,1,-1]),Polrev([2,5,0,-1]),Polrev([0,4,1,-1]),Polrev([2,7,1,-2]),Polrev([2,0,-5,2])], K);
 
magma: E := EllipticCurve([K![-1,3,1,-1],K![2,5,0,-1],K![0,4,1,-1],K![2,7,1,-2],K![2,0,-5,2]]);
 

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: \((-4a^3+4a^2+10a-10)\) = \((a^3-4a)^{5}\cdot(2)\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 3888 \) = \(3^{5}\cdot16\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( -\frac{256787}{486} a^{3} - \frac{917825}{486} a^{2} - \frac{221027}{162} a + \frac{155944}{243} \)
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: \(\Z/5\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Torsion generator: $\left(-a + 1 : 2 a^{3} - 3 a^{2} - 5 a + 4 : 1\right)$
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: \( 319.85564480477685813940056087828262158 \)
Tamagawa product: \( 5 \)  =  \(5\cdot1\)
Torsion order: \(5\)
Leading coefficient: \( 1.44606762255545 \)
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\((j)_{-}\)
\((a^3-4a)\) \(3\) \(5\) \(I_{5}\) Split multiplicative \(-1\) \(1\) \(5\) \(5\)
\((2)\) \(16\) \(1\) \(I_{1}\) 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
\(5\) 5B.1.1

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.