Properties

Label 3.3.81.1-27.1-a4
Base field \(\Q(\zeta_{9})^+\)
Conductor \((3)\)
Conductor norm \( 27 \)
CM yes (\(-3\))
Base change yes: 27.a4
Q-curve yes
Torsion order \( 9 \)
Rank \( 0 \)

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Base field \(\Q(\zeta_{9})^+\)

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

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

Weierstrass equation

\({y}^2+{y}={x}^{3}\)
sage: E = EllipticCurve([K([0,0,0]),K([0,0,0]),K([1,0,0]),K([0,0,0]),K([0,0,0])])
 
gp: E = ellinit([Pol(Vecrev([0,0,0])),Pol(Vecrev([0,0,0])),Pol(Vecrev([1,0,0])),Pol(Vecrev([0,0,0])),Pol(Vecrev([0,0,0]))], K);
 
magma: E := EllipticCurve([K![0,0,0],K![0,0,0],K![1,0,0],K![0,0,0],K![0,0,0]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((3)\) = \((-a^2+1)^{3}\)
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: \((-27)\) = \((-a^2+1)^{9}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( -19683 \) = \(3^{9}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( 0 \)
sage: E.j_invariant()
 
gp: E.j
 
magma: jInvariant(E);
 
Endomorphism ring: \(\Z\)
Geometric endomorphism ring: \(\Z[(1+\sqrt{-3})/2]\) (potential complex multiplication)
sage: E.has_cm(), E.cm_discriminant()
 
magma: HasComplexMultiplication(E);
 
Sato-Tate group: $N(\mathrm{U}(1))$

Mordell-Weil group

Rank: \(0\)
Torsion structure: \(\Z/9\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Torsion generator: $\left(a + 1 : a^{2} + a - 1 : 1\right)$
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: \( 148.869942571185 \)
Tamagawa product: \( 3 \)
Torsion order: \(9\)
Leading coefficient: \( 0.612633508523396 \)
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\((j)_{-}\)
\((-a^2+1)\) \(3\) \(3\) \(IV^{*}\) Additive \(-1\) \(3\) \(9\) \(0\)

Galois Representations

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

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

For all other primes \(p\), the image is the normalizer of a split Cartan subgroup if \(\left(\frac{ -3 }{p}\right)=+1\) or the normalizer of a nonsplit Cartan subgroup if \(\left(\frac{ -3 }{p}\right)=-1\).

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 3 and 9.
Its isogeny class 27.1-a consists of curves linked by isogenies of degrees dividing 27.

Base change

This curve is the base change of elliptic curves 27.a4, defined over \(\Q\), so it is also a \(\Q\)-curve.