Properties

Label 4.4.2304.1-47.1-b1
Base field \(\Q(\sqrt{2}, \sqrt{3})\)
Conductor norm \( 47 \)
CM no
Base change no
Q-curve no
Torsion order \( 2 \)
Rank \( 1 \)

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Base field \(\Q(\sqrt{2}, \sqrt{3})\)

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

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

Weierstrass equation

\({y}^2+\left(a^{2}-1\right){x}{y}+\left(a^{2}+a-1\right){y}={x}^{3}+\left(a^{3}+a^{2}-5a-1\right){x}^{2}+\left(-3a^{3}+a^{2}+8a-1\right){x}-2a\)
sage: E = EllipticCurve([K([-1,0,1,0]),K([-1,-5,1,1]),K([-1,1,1,0]),K([-1,8,1,-3]),K([0,-2,0,0])])
 
gp: E = ellinit([Polrev([-1,0,1,0]),Polrev([-1,-5,1,1]),Polrev([-1,1,1,0]),Polrev([-1,8,1,-3]),Polrev([0,-2,0,0])], K);
 
magma: E := EllipticCurve([K![-1,0,1,0],K![-1,-5,1,1],K![-1,1,1,0],K![-1,8,1,-3],K![0,-2,0,0]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((a^3-a^2-5a)\) = \((a^3-a^2-5a)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 47 \) = \(47\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((a^3-a^2-5a)\) = \((a^3-a^2-5a)\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( -47 \) = \(-47\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( -\frac{189051392}{47} a^{3} + \frac{365769728}{47} a^{2} + \frac{49299456}{47} a - \frac{97220672}{47} \)
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 $\left(a + 1 : 1 : 1\right)$
Height \(0.021467440490993194556956848342212736556\)
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: $\left(-\frac{1}{2} a^{2} + 1 : -\frac{1}{4} a^{2} - \frac{1}{2} a + \frac{3}{4} : 1\right)$
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: \( 0.021467440490993194556956848342212736556 \)
Period: \( 2021.4079571782359854749306215477437152 \)
Tamagawa product: \( 1 \)
Torsion order: \(2\)
Leading coefficient: \( 0.904051146432165 \)
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\((j)_{-}\)
\((a^3-a^2-5a)\) \(47\) \(1\) \(I_{1}\) 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
\(2\) 2B

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2 and 4.
Its isogeny class 47.1-b consists of curves linked by isogenies of degrees dividing 4.

Base change

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

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