Properties

Label 288.d3
Conductor $288$
Discriminant $46656$
j-invariant \( 1728 \)
CM yes (\(D=-4\))
Rank $0$
Torsion structure \(\Z/{2}\Z \times \Z/{2}\Z\)

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Minimal Weierstrass equation

sage: E = EllipticCurve([0, 0, 0, -9, 0])
 
gp: E = ellinit([0, 0, 0, -9, 0])
 
magma: E := EllipticCurve([0, 0, 0, -9, 0]);
 

\(y^2=x^3-9x\)

Mordell-Weil group structure

\(\Z/{2}\Z \times \Z/{2}\Z\)

Torsion generators

sage: E.torsion_subgroup().gens()
 
gp: elltors(E)
 
magma: TorsionSubgroup(E);
 

\( \left(0, 0\right) \), \( \left(3, 0\right) \)

Integral points

sage: E.integral_points()
 
magma: IntegralPoints(E);
 

\( \left(-3, 0\right) \), \( \left(0, 0\right) \), \( \left(3, 0\right) \)

Invariants

sage: E.conductor().factor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor: \( 288 \)  =  \(2^{5} \cdot 3^{2}\)
sage: E.discriminant().factor()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant: \(46656 \)  =  \(2^{6} \cdot 3^{6} \)
sage: E.j_invariant().factor()
 
gp: E.j
 
magma: jInvariant(E);
 
j-invariant: \( 1728 \)  =  \(2^{6} \cdot 3^{3}\)
Endomorphism ring: \(\Z\)
Geometric endomorphism ring: \(\Z[\sqrt{-1}]\) (potential complex multiplication)
Sato-Tate group: $N(\mathrm{U}(1))$

BSD invariants

sage: E.rank()
 
magma: Rank(E);
 
Analytic rank: \(0\)
sage: E.regulator()
 
magma: Regulator(E);
 
Regulator: \(1\)
sage: E.period_lattice().omega()
 
gp: E.omega[1]
 
magma: RealPeriod(E);
 
Real period: \(3.0276912696024942909434725134\)
sage: E.tamagawa_numbers()
 
gp: gr=ellglobalred(E); [[gr[4][i,1],gr[5][i][4]] | i<-[1..#gr[4][,1]]]
 
magma: TamagawaNumbers(E);
 
Tamagawa product: \( 8 \)  = \( 2\cdot2^{2} \)
sage: E.torsion_order()
 
gp: elltors(E)[1]
 
magma: Order(TorsionSubgroup(E));
 
Torsion order: \(4\)
sage: E.sha().an_numerical()
 
magma: MordellWeilShaInformation(E);
 
Analytic order of Ш: \(1\) (exact)

Modular invariants

Modular form   288.2.a.d

sage: E.q_eigenform(20)
 
gp: xy = elltaniyama(E);
 
gp: x*deriv(xy[1])/(2*xy[2]+E.a1*xy[1]+E.a3)
 
magma: ModularForm(E);
 

\( q + 2q^{5} + 6q^{13} - 2q^{17} + O(q^{20}) \)

For more coefficients, see the Downloads section to the right.

sage: E.modular_degree()
 
magma: ModularDegree(E);
 
Modular degree: 16
\( \Gamma_0(N) \)-optimal: yes
Manin constant: 1

Special L-value

sage: r = E.rank();
 
sage: E.lseries().dokchitser().derivative(1,r)/r.factorial()
 
gp: ar = ellanalyticrank(E);
 
gp: ar[2]/factorial(ar[1])
 
magma: Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12);
 

\( L(E,1) \) ≈ \( 1.5138456348012471454717362566780873986 \)

Local data

This elliptic curve is not semistable. There are 2 primes of bad reduction:

sage: E.local_data()
 
gp: ellglobalred(E)[5]
 
magma: [LocalInformation(E,p) : p in BadPrimes(E)];
 
prime Tamagawa number Kodaira symbol Reduction type Root number ord(\(N\)) ord(\(\Delta\)) ord\((j)_{-}\)
\(2\) \(2\) \(III\) Additive 1 5 6 0
\(3\) \(4\) \(I_0^{*}\) Additive -1 2 6 0

Galois representations

sage: rho = E.galois_representation();
 
sage: [rho.image_type(p) for p in rho.non_surjective()]
 
magma: [GaloisRepresentation(E,p): p in PrimesUpTo(20)];
 

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

prime Image of Galois representation
\(2\) Cs

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

$p$-adic data

$p$-adic regulators

sage: [E.padic_regulator(p) for p in primes(5,20) if E.conductor().valuation(p)<2]
 

All \(p\)-adic regulators are identically \(1\) since the rank is \(0\).

Iwasawa invariants

$p$ 2 3
Reduction type add add
$\lambda$-invariant(s) - -
$\mu$-invariant(s) - -

All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 3$ of good reduction are zero.

An entry - indicates that the invariants are not computed because the reduction is additive.

Isogenies

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2.
Its isogeny class 288.d consists of 4 curves linked by isogenies of degrees dividing 4.

Growth of torsion in number fields

The number fields $K$ of degree less than 24 such that $E(K)_{\rm tors}$ is strictly larger than $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z \times \Z/{2}\Z$ are as follows:

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base change curve
$4$ \(\Q(\sqrt{2}, \sqrt{3})\) \(\Z/2\Z \times \Z/4\Z\) 4.4.2304.1-64.1-b2
$4$ \(\Q(\zeta_{12})\) \(\Z/2\Z \times \Z/4\Z\) Not in database
$4$ \(\Q(\sqrt{2}, \sqrt{-3})\) \(\Z/2\Z \times \Z/4\Z\) Not in database
$8$ \(\Q(\zeta_{24})\) \(\Z/4\Z \times \Z/4\Z\) Not in database
$8$ 8.2.143327232.1 \(\Z/2\Z \times \Z/6\Z\) Not in database
$8$ 8.0.663552000.1 \(\Z/2\Z \times \Z/10\Z\) Not in database
$16$ 16.8.118192468620711297024.2 \(\Z/2\Z \times \Z/8\Z\) Not in database
$16$ 16.0.115422332637413376.2 \(\Z/4\Z \times \Z/8\Z\) Not in database
$16$ 16.0.29548117155177824256.4 \(\Z/2\Z \times \Z/8\Z\) Not in database
$16$ 16.0.20542695432781824.1 \(\Z/6\Z \times \Z/12\Z\) Not in database
$16$ Deg 16 \(\Z/2\Z \times \Z/10\Z\) Not in database
$16$ 16.4.5258930030792146944.1 \(\Z/2\Z \times \Z/12\Z\) Not in database
$16$ Deg 16 \(\Z/2\Z \times \Z/20\Z\) Not in database

We only show fields where the torsion growth is primitive. For fields not in the database, click on the degree shown to reveal the defining polynomial.