Properties

Label 4.4.19821.1-27.2-c1
Base field 4.4.19821.1
Conductor norm \( 27 \)
CM no
Base change no
Q-curve no
Torsion order \( 3 \)
Rank \( 0 \)

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Base field 4.4.19821.1

Generator \(a\), with minimal polynomial \( x^{4} - x^{3} - 8 x^{2} + 6 x + 3 \); class number \(1\).

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([3, 6, -8, -1, 1]))
 
gp: K = nfinit(Polrev([3, 6, -8, -1, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3, 6, -8, -1, 1]);
 

Weierstrass equation

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

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((1/3a^3-2/3a^2-2a+1)\) = \((-1/3a^3-1/3a^2+3a+2)^{3}\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 27 \) = \(3^{3}\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((-1/3a^3+2/3a^2+2a-1)\) = \((-1/3a^3-1/3a^2+3a+2)^{3}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 27 \) = \(3^{3}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{18512}{3} a^{3} + \frac{1769}{3} a^{2} - 48583 a - 15892 \)
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: \(\Z/3\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Torsion generator: $\left(-a + 1 : \frac{1}{3} a^{3} - \frac{2}{3} a^{2} : 1\right)$
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: \( 241.75049406015903070893950008655198795 \)
Tamagawa product: \( 1 \)
Torsion order: \(3\)
Leading coefficient: \( 1.71713558954464 \)
Analytic order of Ш: \( 9 \) (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\((j)_{-}\)
\((-1/3a^3-1/3a^2+3a+2)\) \(3\) \(1\) \(II\) Additive \(1\) \(3\) \(3\) \(0\)

Galois Representations

The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.

prime Image of Galois Representation
\(3\) 3B.1.1

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 3.
Its isogeny class 27.2-c consists of curves linked by isogenies of degree 3.

Base change

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

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