Properties

Label 4.4.2777.1-22.1-b2
Base field 4.4.2777.1
Conductor norm \( 22 \)
CM no
Base change no
Q-curve no
Torsion order \( 10 \)
Rank \( 0 \)

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Base field 4.4.2777.1

Generator \(a\), with minimal polynomial \( x^{4} - x^{3} - 4 x^{2} + x + 2 \); class number \(1\).

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([2, 1, -4, -1, 1]))
 
gp: K = nfinit(Polrev([2, 1, -4, -1, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2, 1, -4, -1, 1]);
 

Weierstrass equation

\({y}^2+\left(a^{3}-4a-1\right){x}{y}+\left(a^{2}-2\right){y}={x}^{3}+\left(-a^{3}+2a^{2}+3a-4\right){x}^{2}+\left(9a^{3}+8a^{2}-59a-35\right){x}-37a^{3}+17a^{2}+136a+71\)
sage: E = EllipticCurve([K([-1,-4,0,1]),K([-4,3,2,-1]),K([-2,0,1,0]),K([-35,-59,8,9]),K([71,136,17,-37])])
 
gp: E = ellinit([Polrev([-1,-4,0,1]),Polrev([-4,3,2,-1]),Polrev([-2,0,1,0]),Polrev([-35,-59,8,9]),Polrev([71,136,17,-37])], K);
 
magma: E := EllipticCurve([K![-1,-4,0,1],K![-4,3,2,-1],K![-2,0,1,0],K![-35,-59,8,9],K![71,136,17,-37]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((-a^3+3a)\) = \((-a)\cdot(-a^3+2a^2+2a-1)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 22 \) = \(2\cdot11\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((3a^3+9a^2-9a-38)\) = \((-a)^{5}\cdot(-a^3+2a^2+2a-1)^{4}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( -468512 \) = \(-2^{5}\cdot11^{4}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{19492220557608647265}{468512} a^{3} + \frac{26555029218814954293}{468512} a^{2} - \frac{7618439534802966603}{234256} a - \frac{16502468789812849155}{468512} \)
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: \(\Z/10\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Torsion generator: $\left(-10 a^{3} + 15 a^{2} + 19 a + 4 : -21 a^{3} + 84 a^{2} - 49 a - 79 : 1\right)$
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: \( 424.47369924821109617300796673406217367 \)
Tamagawa product: \( 20 \)  =  \(5\cdot2^{2}\)
Torsion order: \(10\)
Leading coefficient: \( 1.61098999014251 \)
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\((j)_{-}\)
\((-a)\) \(2\) \(5\) \(I_{5}\) Split multiplicative \(-1\) \(1\) \(5\) \(5\)
\((-a^3+2a^2+2a-1)\) \(11\) \(4\) \(I_{4}\) Split multiplicative \(-1\) \(1\) \(4\) \(4\)

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
\(5\) 5B.1.1

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2, 4, 5, 10 and 20.
Its isogeny class 22.1-b consists of curves linked by isogenies of degrees dividing 20.

Base change

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

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