Properties

Label 4.4.1957.1-47.1-b2
Base field 4.4.1957.1
Conductor norm \( 47 \)
CM no
Base change no
Q-curve no
Torsion order \( 4 \)
Rank not available

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

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

\({y}^2+\left(a^{2}-2\right){x}{y}+\left(-a^{3}+a^{2}+3a\right){y}={x}^{3}+\left(-2a^{3}+a^{2}+8a+1\right){x}^{2}+\left(-13a^{3}-19a^{2}+12a+7\right){x}-65a^{3}-133a^{2}-13a+31\)
sage: E = EllipticCurve([K([-2,0,1,0]),K([1,8,1,-2]),K([0,3,1,-1]),K([7,12,-19,-13]),K([31,-13,-133,-65])])
 
gp: E = ellinit([Polrev([-2,0,1,0]),Polrev([1,8,1,-2]),Polrev([0,3,1,-1]),Polrev([7,12,-19,-13]),Polrev([31,-13,-133,-65])], K);
 
magma: E := EllipticCurve([K![-2,0,1,0],K![1,8,1,-2],K![0,3,1,-1],K![7,12,-19,-13],K![31,-13,-133,-65]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((2a^3-6a-1)\) = \((2a^3-6a-1)\)
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: \((5a^3-a^2-14a-4)\) = \((2a^3-6a-1)^{2}\)
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: \( -\frac{8324109193351}{2209} a^{3} + \frac{14771368925300}{2209} a^{2} + \frac{7178823816030}{2209} a - \frac{4647340584154}{2209} \)
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: \(0 \le r \le 2\)
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: $\left(-a^{2} - \frac{9}{4} a - \frac{7}{4} : \frac{13}{8} a^{3} + \frac{11}{8} a^{2} - \frac{13}{4} a - \frac{9}{4} : 1\right)$ $\left(a^{3} + 2 a^{2} - a - 2 : -2 a^{2} - 3 a - 1 : 1\right)$
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: \(0 \le r \le 2\)
Regulator*: \( 1 \)
Period: \( 86.293805775748154364662897174831717783 \)
Tamagawa product: \( 2 \)
Torsion order: \(4\)
Leading coefficient: \( 0.975335910013100 \)
Analytic order of Ш*: \( 4 \) (rounded)

* Conditional on BSD: assuming rank = analytic rank.

Note: We expect that the nontriviality of Ш explains the discrepancy between the upper bound on the rank and the analytic rank. The application of further descents should suffice to establish the weak BSD conjecture for this curve.

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\((j)_{-}\)
\((2a^3-6a-1)\) \(47\) \(2\) \(I_{2}\) Non-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 47.1-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.