Properties

Label 3.3.361.1-27.1-b2
Base field 3.3.361.1
Conductor norm \( 27 \)
CM no
Base change yes
Q-curve yes
Torsion order \( 7 \)
Rank \( 1 \)

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Base field 3.3.361.1

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

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

Weierstrass equation

\({y}^2+\left(a+1\right){x}{y}+\left(a^{2}-4\right){y}={x}^{3}+\left(-a^{2}-a+4\right){x}^{2}+\left(-2a^{2}-a+12\right){x}-a^{2}+5\)
sage: E = EllipticCurve([K([1,1,0]),K([4,-1,-1]),K([-4,0,1]),K([12,-1,-2]),K([5,0,-1])])
 
gp: E = ellinit([Polrev([1,1,0]),Polrev([4,-1,-1]),Polrev([-4,0,1]),Polrev([12,-1,-2]),Polrev([5,0,-1])], K);
 
magma: E := EllipticCurve([K![1,1,0],K![4,-1,-1],K![-4,0,1],K![12,-1,-2],K![5,0,-1]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((3)\) = \((3)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 27 \) = \(27\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((-3)\) = \((3)\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( -27 \) = \(-27\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{2375}{3} \)
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 $\left(\frac{12849}{2401} a^{2} + \frac{17928}{2401} a - \frac{5179}{343} : -\frac{5027331}{117649} a^{2} - \frac{6269846}{117649} a + \frac{2247488}{16807} : 1\right)$
Height \(3.5712293443622126639241266809074098791\)
Torsion structure: \(\Z/7\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Torsion generator: $\left(0 : -1 : 1\right)$
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: \( 3.5712293443622126639241266809074098791 \)
Period: \( 155.64397066533495533947782968372936928 \)
Tamagawa product: \( 1 \)
Torsion order: \(7\)
Leading coefficient: \( 1.7911073533182458002224241964941132866 \)
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\((j)_{-}\)
\((3)\) \(27\) \(1\) \(I_{1}\) Non-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
\(7\) 7B.1.1

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 7.
Its isogeny class 27.1-b consists of curves linked by isogenies of degree 7.

Base change

This elliptic curve is a \(\Q\)-curve. It is the base change of the following elliptic curve:

Base field Curve
\(\Q\) 1083.c2