Properties

Label 4.4.9909.1-3.1-a1
Base field 4.4.9909.1
Conductor norm \( 3 \)
CM no
Base change no
Q-curve no
Torsion order \( 2 \)
Rank \( 0 \)

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Base field 4.4.9909.1

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

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([3, -3, -6, 0, 1]))
 
gp: K = nfinit(Polrev([3, -3, -6, 0, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3, -3, -6, 0, 1]);
 

Weierstrass equation

\({y}^2+\left(a^{3}-4a-2\right){x}{y}+\left(a^{3}-5a-1\right){y}={x}^{3}+\left(a^{2}-2a-2\right){x}^{2}+\left(13a^{3}-28a^{2}-26a+32\right){x}-85a^{3}+160a^{2}+209a-133\)
sage: E = EllipticCurve([K([-2,-4,0,1]),K([-2,-2,1,0]),K([-1,-5,0,1]),K([32,-26,-28,13]),K([-133,209,160,-85])])
 
gp: E = ellinit([Polrev([-2,-4,0,1]),Polrev([-2,-2,1,0]),Polrev([-1,-5,0,1]),Polrev([32,-26,-28,13]),Polrev([-133,209,160,-85])], K);
 
magma: E := EllipticCurve([K![-2,-4,0,1],K![-2,-2,1,0],K![-1,-5,0,1],K![32,-26,-28,13],K![-133,209,160,-85]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((-a^3+a^2+4a)\) = \((-a^3+a^2+4a)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 3 \) = \(3\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((9)\) = \((-a^3+a^2+4a)^{8}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 6561 \) = \(3^{8}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( -357531767 a^{3} + \frac{3885538150}{9} a^{2} + \frac{14614849117}{9} a - \frac{7994350247}{9} \)
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: \(\Z/2\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Torsion generator: $\left(2 a^{3} - 4 a^{2} - 5 a + 3 : a^{3} - 2 a^{2} - a + 2 : 1\right)$
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: \( 140.02828576123659971595553912401769629 \)
Tamagawa product: \( 2 \)
Torsion order: \(2\)
Leading coefficient: \( 0.703348980524783 \)
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\((j)_{-}\)
\((-a^3+a^2+4a)\) \(3\) \(2\) \(I_{8}\) 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

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2.
Its isogeny class 3.1-a consists of curves linked by isogenies of degree 2.

Base change

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

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