Properties

Label 2.0.4.1-5184.1-CMb1
Base field \(\Q(\sqrt{-1}) \)
Conductor \((72)\)
Conductor norm \( 5184 \)
CM yes (\(-4\))
Base change yes: 288.d4,288.d3
Q-curve yes
Torsion order \( 4 \)
Rank \( 0 \)

Related objects

Downloads

Learn more about

Show commands for: Magma / Pari/GP / SageMath

Base field \(\Q(\sqrt{-1}) \)

Generator \(i\), with minimal polynomial \( x^{2} + 1 \); class number \(1\).

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

Weierstrass equation

\({y}^2={x}^{3}-9{x}\)
sage: E = EllipticCurve([K([0,0]),K([0,0]),K([0,0]),K([-9,0]),K([0,0])])
 
gp: E = ellinit([Pol(Vecrev([0,0])),Pol(Vecrev([0,0])),Pol(Vecrev([0,0])),Pol(Vecrev([-9,0])),Pol(Vecrev([0,0]))], K);
 
magma: E := EllipticCurve([K![0,0],K![0,0],K![0,0],K![-9,0],K![0,0]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((72)\) = \((i+1)^{6}\cdot(3)^{2}\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 5184 \) = \(2^{6}\cdot9^{2}\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((46656)\) = \((i+1)^{12}\cdot(3)^{6}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 2176782336 \) = \(2^{12}\cdot9^{6}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( 1728 \)
sage: E.j_invariant()
 
gp: E.j
 
magma: jInvariant(E);
 
Endomorphism ring: \(\Z[\sqrt{-1}]\) (complex multiplication)
Geometric endomorphism ring: \(\Z[\sqrt{-1}]\)
sage: E.has_cm(), E.cm_discriminant()
 
magma: HasComplexMultiplication(E);
 
Sato-Tate group: $\mathrm{U}(1)$

Mordell-Weil group

Rank: \(0\)
Torsion structure: \(\Z/2\Z\times\Z/2\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Torsion generators: $\left(-3 : 0 : 1\right)$ $\left(0 : 0 : 1\right)$
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: \( 2.29172860600679 \)
Tamagawa product: \( 16 \)  =  \(2^{2}\cdot2^{2}\)
Torsion order: \(4\)
Leading coefficient: \( 2.29172860600679 \)
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\((j)_{-}\)
\((i+1)\) \(2\) \(4\) \(I_2^{*}\) Additive \(-1\) \(6\) \(12\) \(0\)
\((3)\) \(9\) \(4\) \(I_0^{*}\) Additive \(1\) \(2\) \(6\) \(0\)

Galois Representations

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

prime Image of Galois Representation
\(2\) 2Cs

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

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies (excluding endomorphisms) of degree \(d\) for \(d=\) 2.
Its isogeny class 5184.1-CMb consists of curves linked by isogenies of degree 2.

Base change

This curve is the base change of elliptic curves 288.d4, 288.d3, defined over \(\Q\), so it is also a \(\Q\)-curve.