Properties

Label 4.4.2777.1-22.1-a7
Base field 4.4.2777.1
Conductor norm \( 22 \)
CM no
Base change no
Q-curve no
Torsion order \( 6 \)
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

\({y}^2+\left(a^{2}-1\right){x}{y}+\left(a^{3}-a^{2}-3a+2\right){y}={x}^{3}+\left(a^{3}-3a-1\right){x}^{2}+\left(-29a^{3}-37a^{2}+29a+24\right){x}+247a^{3}+345a^{2}-177a-202\)
sage: E = EllipticCurve([K([-1,0,1,0]),K([-1,-3,0,1]),K([2,-3,-1,1]),K([24,29,-37,-29]),K([-202,-177,345,247])])
 
gp: E = ellinit([Polrev([-1,0,1,0]),Polrev([-1,-3,0,1]),Polrev([2,-3,-1,1]),Polrev([24,29,-37,-29]),Polrev([-202,-177,345,247])], K);
 
magma: E := EllipticCurve([K![-1,0,1,0],K![-1,-3,0,1],K![2,-3,-1,1],K![24,29,-37,-29],K![-202,-177,345,247]]);
 

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: \((-46a^3+187a^2-115a-330)\) = \((-a)^{4}\cdot(-a^3+2a^2+2a-1)^{8}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 3429742096 \) = \(2^{4}\cdot11^{8}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{158950703525365495771}{3429742096} a^{3} - \frac{266808256244810209657}{3429742096} a^{2} - \frac{227450195191568192265}{1714871048} a + \frac{468083456791050062719}{3429742096} \)
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: \(\Z/6\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Torsion generator: $\left(2 a^{2} + 4 a + 3 : -6 a^{3} - 13 a^{2} - 5 a - 1 : 1\right)$
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: \(0\)
Regulator: \( 1 \)
Period: \( 399.16273111820358219567736182149329260 \)
Tamagawa product: \( 4 \)  =  \(2\cdot2\)
Torsion order: \(6\)
Leading coefficient: \( 0.841626760920221 \)
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\((j)_{-}\)
\((-a)\) \(2\) \(2\) \(I_{4}\) Non-split multiplicative \(1\) \(1\) \(4\) \(4\)
\((-a^3+2a^2+2a-1)\) \(11\) \(2\) \(I_{8}\) Non-split multiplicative \(1\) \(1\) \(8\) \(8\)

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
\(3\) 3B.1.1

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2, 3, 4, 6, 8, 12 and 24.
Its isogeny class 22.1-a consists of curves linked by isogenies of degrees dividing 24.

Base change

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

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