Properties

Label 3.3.321.1-27.2-a1
Base field 3.3.321.1
Conductor norm \( 27 \)
CM no
Base change no
Q-curve no
Torsion order \( 2 \)
Rank \( 0 \)

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Base field 3.3.321.1

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

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([1, -4, -1, 1]))
 
gp: K = nfinit(Polrev([1, -4, -1, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -4, -1, 1]);
 

Weierstrass equation

\({y}^2+\left(a^{2}-a-3\right){x}{y}+\left(a^{2}-2\right){y}={x}^{3}+\left(-a^{2}+3\right){x}^{2}+\left(48a^{2}-17a-244\right){x}+487a^{2}-421a-1924\)
sage: E = EllipticCurve([K([-3,-1,1]),K([3,0,-1]),K([-2,0,1]),K([-244,-17,48]),K([-1924,-421,487])])
 
gp: E = ellinit([Polrev([-3,-1,1]),Polrev([3,0,-1]),Polrev([-2,0,1]),Polrev([-244,-17,48]),Polrev([-1924,-421,487])], K);
 
magma: E := EllipticCurve([K![-3,-1,1],K![3,0,-1],K![-2,0,1],K![-244,-17,48],K![-1924,-421,487]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((a^2-7)\) = \((a+1)^{2}\cdot(-a^2+a+3)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 27 \) = \(3^{2}\cdot3\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((-2187a^2-8748a+24057)\) = \((a+1)^{7}\cdot(-a^2+a+3)^{20}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( -7625597484987 \) = \(-3^{7}\cdot3^{20}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( -\frac{56237456370572}{19683} a^{2} + \frac{128369230367570}{59049} a + \frac{705545800652999}{59049} \)
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: $\left(\frac{3}{2} a^{2} + \frac{7}{4} a - \frac{23}{2} : \frac{9}{2} a^{2} - \frac{47}{8} a - \frac{123}{8} : 1\right)$
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: \( 5.8590638786133734492931702311769675346 \)
Tamagawa product: \( 4 \)  =  \(2\cdot2\)
Torsion order: \(2\)
Leading coefficient: \( 1.3080842244010714462633965744987198100 \)
Analytic order of Ш: \( 4 \) (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\((j)_{-}\)
\((a+1)\) \(3\) \(2\) \(I_{1}^{*}\) Additive \(-1\) \(2\) \(7\) \(1\)
\((-a^2+a+3)\) \(3\) \(2\) \(I_{20}\) Non-split multiplicative \(1\) \(1\) \(20\) \(20\)

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 and 4.
Its isogeny class 27.2-a consists of curves linked by isogenies of degrees dividing 4.

Base change

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

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