Properties

Label 4.4.5125.1-55.1-e1
Base field 4.4.5125.1
Conductor \((a^3-4a)\)
Conductor norm \( 55 \)
CM no
Base change no
Q-curve no
Torsion order \( 2 \)
Rank \( 1 \)

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Base field 4.4.5125.1

Generator \(a\), with minimal polynomial \( x^{4} - 2 x^{3} - 6 x^{2} + 7 x + 11 \); class number \(1\).

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([11, 7, -6, -2, 1]))
 
gp: K = nfinit(Pol(Vecrev([11, 7, -6, -2, 1])));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![11, 7, -6, -2, 1]);
 

Weierstrass equation

\({y}^2+\left(a^{2}-3\right){x}{y}={x}^{3}+\left(a^{3}-2a^{2}-4a+4\right){x}^{2}+\left(22a^{3}-20a^{2}-95a-42\right){x}-107a^{3}+34a^{2}+540a+416\)
sage: E = EllipticCurve([K([-3,0,1,0]),K([4,-4,-2,1]),K([0,0,0,0]),K([-42,-95,-20,22]),K([416,540,34,-107])])
 
gp: E = ellinit([Pol(Vecrev([-3,0,1,0])),Pol(Vecrev([4,-4,-2,1])),Pol(Vecrev([0,0,0,0])),Pol(Vecrev([-42,-95,-20,22])),Pol(Vecrev([416,540,34,-107]))], K);
 
magma: E := EllipticCurve([K![-3,0,1,0],K![4,-4,-2,1],K![0,0,0,0],K![-42,-95,-20,22],K![416,540,34,-107]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((a^3-4a)\) = \((-a^3+a^2+4a+1)\cdot(-a)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 55 \) = \(5\cdot11\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((25a^3+50a^2-175a-275)\) = \((-a^3+a^2+4a+1)^{8}\cdot(-a)^{2}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 47265625 \) = \(5^{8}\cdot11^{2}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( -\frac{196392867540579}{3025} a^{3} + \frac{727903424191802}{3025} a^{2} - \frac{12741819493871}{605} a - \frac{1266039682540334}{3025} \)
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 $\left(a^{3} + 2 a^{2} - 3 a - 2 : 6 a^{3} + 8 a^{2} - 25 a - 36 : 1\right)$
Height \(0.0473985038471800\)
Torsion structure: \(\Z/2\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Torsion generator: $\left(-\frac{3}{2} a^{3} + 2 a^{2} + \frac{23}{4} a + \frac{1}{2} : \frac{3}{8} a^{3} + \frac{1}{2} a^{2} - \frac{25}{8} a - \frac{19}{4} : 1\right)$
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: \( 0.0473985038471800 \)
Period: \( 856.409689243170 \)
Tamagawa product: \( 4 \)  =  \(2\cdot2\)
Torsion order: \(2\)
Leading coefficient: \( 2.26808466293632 \)
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\((j)_{-}\)
\((-a^3+a^2+4a+1)\) \(5\) \(2\) \(I_{8}\) Non-split multiplicative \(1\) \(1\) \(8\) \(8\)
\((-a)\) \(11\) \(2\) \(I_{2}\) 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
\(2\) 2B

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2.
Its isogeny class 55.1-e consists of curves linked by isogenies of degree 2.

Base change

This curve is not the base change of an elliptic curve defined over \(\Q\). It is not a \(\Q\)-curve.