Properties

Label 4.4.8069.1-19.1-c3
Base field 4.4.8069.1
Conductor norm \( 19 \)
CM no
Base change no
Q-curve no
Torsion order \( 6 \)
Rank not available

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Base field 4.4.8069.1

Generator \(a\), with minimal polynomial \( x^{4} - x^{3} - 5 x^{2} + 5 x + 1 \); class number \(1\).

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([1, 5, -5, -1, 1]))
 
gp: K = nfinit(Polrev([1, 5, -5, -1, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 5, -5, -1, 1]);
 

Weierstrass equation

\({y}^2+\left(a^{3}+a^{2}-4a-1\right){x}{y}+\left(a^{3}+a^{2}-3a-1\right){y}={x}^{3}+\left(a^{3}+a^{2}-3a-1\right){x}^{2}+\left(-a^{3}+11a^{2}+19a-45\right){x}-13a^{3}+30a^{2}+74a-109\)
sage: E = EllipticCurve([K([-1,-4,1,1]),K([-1,-3,1,1]),K([-1,-3,1,1]),K([-45,19,11,-1]),K([-109,74,30,-13])])
 
gp: E = ellinit([Polrev([-1,-4,1,1]),Polrev([-1,-3,1,1]),Polrev([-1,-3,1,1]),Polrev([-45,19,11,-1]),Polrev([-109,74,30,-13])], K);
 
magma: E := EllipticCurve([K![-1,-4,1,1],K![-1,-3,1,1],K![-1,-3,1,1],K![-45,19,11,-1],K![-109,74,30,-13]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((a^2+a-4)\) = \((a^2+a-4)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 19 \) = \(19\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((a^2+a-4)\) = \((a^2+a-4)\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 19 \) = \(19\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( -\frac{140184437194705}{19} a^{3} - \frac{32674219622507}{19} a^{2} + \frac{660633828065334}{19} a + \frac{113692420421896}{19} \)
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: \(0 \le r \le 1\)
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: $\left(-a^{3} + 5 a - 3 : -a^{3} + a^{2} + 2 a : 1\right)$
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: \(0 \le r \le 1\)
Regulator: not available
Period: \( 622.70598152227703014998759867729480546 \)
Tamagawa product: \( 1 \)
Torsion order: \(6\)
Leading coefficient: \( 2.40202085800390 \)
Analytic order of Ш: not available

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\((j)_{-}\)
\((a^2+a-4)\) \(19\) \(1\) \(I_{1}\) Split multiplicative \(-1\) \(1\) \(1\) \(1\)

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 and 6.
Its isogeny class 19.1-c consists of curves linked by isogenies of degrees dividing 6.

Base change

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

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