Properties

Label 4.4.2777.1-22.1-c1
Base field 4.4.2777.1
Conductor norm \( 22 \)
CM no
Base change no
Q-curve no
Torsion order \( 4 \)
Rank \( 1 \)

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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

\({y}^2+\left(a^{3}-4a-1\right){x}{y}+\left(a^{3}-a^{2}-3a+1\right){y}={x}^{3}+\left(-a^{2}+2a+2\right){x}^{2}+\left(-5a^{3}+4a^{2}+18a-5\right){x}-4a+10\)
sage: E = EllipticCurve([K([-1,-4,0,1]),K([2,2,-1,0]),K([1,-3,-1,1]),K([-5,18,4,-5]),K([10,-4,0,0])])
 
gp: E = ellinit([Polrev([-1,-4,0,1]),Polrev([2,2,-1,0]),Polrev([1,-3,-1,1]),Polrev([-5,18,4,-5]),Polrev([10,-4,0,0])], K);
 
magma: E := EllipticCurve([K![-1,-4,0,1],K![2,2,-1,0],K![1,-3,-1,1],K![-5,18,4,-5],K![10,-4,0,0]]);
 

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: \((3a^2+2a-8)\) = \((-a)^{4}\cdot(-a^3+2a^2+2a-1)^{2}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 1936 \) = \(2^{4}\cdot11^{2}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{52878458037}{1936} a^{3} - \frac{27483917095}{1936} a^{2} - \frac{55514275415}{968} a + \frac{80654280017}{1936} \)
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 $\left(a^{3} - 4 a : -a^{3} + a - 4 : 1\right)$
Height \(0.027949886221301609546704933021313245522\)
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: $\left(a^{3} - 2 a^{2} - 3 a + 1 : 2 a^{2} - a - 4 : 1\right)$ $\left(\frac{3}{4} a^{3} - \frac{15}{4} a - \frac{1}{2} : \frac{1}{2} a^{3} + \frac{1}{8} a^{2} - \frac{17}{8} a - \frac{15}{4} : 1\right)$
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: \( 0.027949886221301609546704933021313245522 \)
Period: \( 2051.3223854071491467004683628793383508 \)
Tamagawa product: \( 4 \)  =  \(2\cdot2\)
Torsion order: \(4\)
Leading coefficient: \( 1.08799280611160 \)
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 \(1\) \(1\) \(4\) \(4\)
\((-a^3+2a^2+2a-1)\) \(11\) \(2\) \(I_{2}\) 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 and 4.
Its isogeny class 22.1-c consists of curves linked by isogenies of degrees dividing 8.

Base change

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

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