Properties

Label 1728m2
Conductor $1728$
Discriminant $-322486272$
j-invariant \( 0 \)
CM yes (\(D=-3\))
Rank $0$
Torsion structure trivial

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Show commands: Magma / Pari/GP / SageMath

Minimal Weierstrass equation

sage: E = EllipticCurve([0, 0, 0, 0, 864])
 
gp: E = ellinit([0, 0, 0, 0, 864])
 
magma: E := EllipticCurve([0, 0, 0, 0, 864]);
 

\(y^2=x^3+864\)  Toggle raw display

Mordell-Weil group structure

trivial

Integral points

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

None

Invariants

sage: E.conductor().factor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor: \( 1728 \)  =  $2^{6} \cdot 3^{3}$
sage: E.discriminant().factor()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant: $-322486272 $  =  $-1 \cdot 2^{14} \cdot 3^{9} $
sage: E.j_invariant().factor()
 
gp: E.j
 
magma: jInvariant(E);
 
j-invariant: \( 0 \)  =  $0$
Endomorphism ring: $\Z$
Geometric endomorphism ring: \(\Z[(1+\sqrt{-3})/2]\) (potential complex multiplication)
Sato-Tate group: $N(\mathrm{U}(1))$
Faltings height: $0.31151349872631388121000220687\dots$
Stable Faltings height: $-1.3211174284280379149898691959\dots$

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: $1.3630340904278903107840077772\dots$
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: $ 1 $
sage: E.torsion_order()
 
gp: elltors(E)[1]
 
magma: Order(TorsionSubgroup(E));
 
Torsion order: $1$
sage: E.sha().an_numerical()
 
magma: MordellWeilShaInformation(E);
 
Analytic order of Ш: $1$ (exact)
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);
 
Special value: $ L(E,1) $ ≈ $ 1.3630340904278903107840077772330627599 $

Modular invariants

Modular form   1728.2.a.m

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);
 

\( q - 5q^{7} + 7q^{13} - q^{19} + O(q^{20}) \)  Toggle raw display

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

sage: E.modular_degree()
 
magma: ModularDegree(E);
 
Modular degree: 576
$ \Gamma_0(N) $-optimal: no
Manin constant: 1

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$(j)_{-}$
$2$ $1$ $II^{*}$ Additive -1 6 14 0
$3$ $1$ $IV^{*}$ Additive 1 3 9 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 $\ell$-adic Galois representation has maximal image for all primes $\ell$.

$p$-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 5 7 11 13 17 19 23 29 31 37 41 43 47
Reduction type add add ss ordinary ss ordinary ss ordinary ss ss ordinary ordinary ss ordinary ss
$\lambda$-invariant(s) - - 0,0 0 0,0 0 0,0 0 0,0 0,0 0 2 0,0 0 0,0
$\mu$-invariant(s) - - 0,0 0 0,0 0 0,0 0 0,0 0,0 0 0 0,0 0 0,0

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 $d=$ 3.
Its isogeny class 1728m consists of 2 curves linked by isogenies of degree 3.

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 $E(\Q)_{\rm tors}$ (which is trivial) are as follows:

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base change curve
$2$ \(\Q(\sqrt{6}) \) \(\Z/3\Z\) 2.2.24.1-324.1-a2
$3$ 3.1.108.1 \(\Z/2\Z\) Not in database
$6$ 6.0.34992.1 \(\Z/2\Z \times \Z/2\Z\) Not in database
$6$ 6.0.1492992.1 \(\Z/6\Z\) Not in database
$6$ 6.2.4478976.4 \(\Z/6\Z\) Not in database
$12$ 12.2.962938848411648.4 \(\Z/4\Z\) Not in database
$12$ 12.0.20061226008576.4 \(\Z/6\Z \times \Z/6\Z\) Not in database
$12$ Deg 12 \(\Z/7\Z\) Not in database
$18$ 18.6.5305774073297600589594624.1 \(\Z/18\Z\) 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.