Properties

Label 3.3.81.1-216.1-a1
Base field \(\Q(\zeta_{9})^+\)
Conductor norm \( 216 \)
CM no
Base change yes
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(Polrev([-1, -3, 0, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, -3, 0, 1]);
 

Weierstrass equation

\({y}^2+a{x}{y}+\left(a^{2}-2\right){y}={x}^{3}+\left(-a^{2}+3\right){x}^{2}+\left(-87a^{2}+175a-23\right){x}+910a^{2}-1519a-360\)
sage: E = EllipticCurve([K([0,1,0]),K([3,0,-1]),K([-2,0,1]),K([-23,175,-87]),K([-360,-1519,910])])
 
gp: E = ellinit([Polrev([0,1,0]),Polrev([3,0,-1]),Polrev([-2,0,1]),Polrev([-23,175,-87]),Polrev([-360,-1519,910])], K);
 
magma: E := EllipticCurve([K![0,1,0],K![3,0,-1],K![-2,0,1],K![-23,175,-87],K![-360,-1519,910]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((6)\) = \((-a^2+1)^{3}\cdot(2)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 216 \) = \(3^{3}\cdot8\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((-54)\) = \((-a^2+1)^{9}\cdot(2)\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( -157464 \) = \(-3^{9}\cdot8\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( -\frac{132651}{2} \)
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: \(\Z/9\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Torsion generator: $\left(6 a^{2} - 6 a - 10 : -10 a^{2} + 3 a + 29 : 1\right)$
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: \( 251.73842965572625019843711770180742397 \)
Tamagawa product: \( 3 \)  =  \(3\cdot1\)
Torsion order: \(9\)
Leading coefficient: \( 1.0359606158671862148083832004189605924 \)
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 \(-1\) \(3\) \(9\) \(0\)
\((2)\) \(8\) \(1\) \(I_{1}\) 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
\(3\) 3B.1.1

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 3 and 9.
Its isogeny class 216.1-a consists of curves linked by isogenies of degrees dividing 9.

Base change

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

Base field Curve
\(\Q\) 54.a2