Properties

Label 4.4.2304.1-47.2-b2
Base field \(\Q(\sqrt{2}, \sqrt{3})\)
Conductor norm \( 47 \)
CM no
Base change no
Q-curve no
Torsion order \( 4 \)
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

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

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((a^3+2a^2-2)\) = \((a^3+2a^2-2)\)
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: \((4a^3-a^2-18a+4)\) = \((a^3+2a^2-2)^{2}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 2209 \) = \(47^{2}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{27756544}{2209} a^{3} + \frac{44907520}{2209} a^{2} - \frac{20659712}{2209} a - \frac{948672}{2209} \)
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 $\left(-a^{3} + 3 a^{2} - 3 a : 12 a^{3} - 22 a^{2} - 4 a + 6 : 1\right)$
Height \(0.042934880981986389113913696684425473113\)
Torsion structure: \(\Z/2\Z\oplus\Z/2\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Torsion generators: $\left(-\frac{1}{2} a^{2} : -\frac{1}{2} a^{3} + \frac{3}{4} a^{2} + 2 a - \frac{1}{4} : 1\right)$ $\left(-\frac{1}{2} a^{2} - a + \frac{3}{2} : \frac{3}{2} a + \frac{1}{2} : 1\right)$
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: \( 0.042934880981986389113913696684425473113 \)
Period: \( 2021.4079571782359854749306215477437152 \)
Tamagawa product: \( 2 \)
Torsion order: \(4\)
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\((j)_{-}\)
\((a^3+2a^2-2)\) \(47\) \(2\) \(I_{2}\) Split multiplicative \(-1\) \(1\) \(2\) \(2\)

Galois Representations

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

prime Image of Galois Representation
\(2\) 2Cs

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2.
Its isogeny class 47.2-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.