Properties

Label 1728bb3
Conductor $1728$
Discriminant $-2.378\times 10^{13}$
j-invariant \( -\frac{1167051}{512} \)
CM no
Rank $1$
Torsion structure trivial

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Show commands: Magma / Pari/GP / SageMath

Minimal Weierstrass equation

sage: E = EllipticCurve([0, 0, 0, -7884, 357264])
 
gp: E = ellinit([0, 0, 0, -7884, 357264])
 
magma: E := EllipticCurve([0, 0, 0, -7884, 357264]);
 

\(y^2=x^3-7884x+357264\)  Toggle raw display

Mordell-Weil group structure

$\Z$

Infinite order Mordell-Weil generator and height

sage: E.gens()
 
magma: Generators(E);
 

$P$ =  \(\left(82, 512\right)\)  Toggle raw display
$\hat{h}(P)$ ≈  $1.0156816950072237676174813689$

Integral points

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

\((-20,\pm 712)\), \((82,\pm 512)\)  Toggle raw display

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: $-23776267862016 $  =  $-1 \cdot 2^{27} \cdot 3^{11} $
sage: E.j_invariant().factor()
 
gp: E.j
 
magma: jInvariant(E);
 
j-invariant: \( -\frac{1167051}{512} \)  =  $-1 \cdot 2^{-9} \cdot 3 \cdot 73^{3}$
Endomorphism ring: $\Z$
Geometric endomorphism ring: \(\Z\) (no potential complex multiplication)
Sato-Tate group: $\mathrm{SU}(2)$
Faltings height: $1.2738224870740616427996011157\dots$
Stable Faltings height: $-0.77295954837829020510522186700\dots$

BSD invariants

sage: E.rank()
 
magma: Rank(E);
 
Analytic rank: $1$
sage: E.regulator()
 
magma: Regulator(E);
 
Regulator: $1.0156816950072237676174813689\dots$
sage: E.period_lattice().omega()
 
gp: E.omega[1]
 
magma: RealPeriod(E);
 
Real period: $0.63106317513913839488058876442\dots$
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: $ 4 $  = $ 2^{2}\cdot1 $
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) $ ≈ $ 2.5638372615278423979242032018322979018 $

Modular invariants

Modular form   1728.2.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);
 

\( q - 3 q^{5} + q^{7} - 3 q^{11} + 4 q^{13} + 2 q^{19} + O(q^{20}) \)  Toggle raw display

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

sage: E.modular_degree()
 
magma: ModularDegree(E);
 
Modular degree: 3456
$ \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$(j)_{-}$
$2$ $4$ $I_{17}^{*}$ Additive -1 6 27 9
$3$ $1$ $II^{*}$ Additive -1 3 11 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 $\ell$ except those listed in the table below.

prime $\ell$ mod-$\ell$ image $\ell$-adic image
$3$ 3B 9.36.0.4

$p$-adic regulators

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

$p$-adic regulators are not yet computed for curves that are not $\Gamma_0$-optimal.

Iwasawa invariants

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47
Reduction type add add ordinary ordinary ordinary ordinary ss ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary
$\lambda$-invariant(s) - - 1 1 1 1 1,1 1 1 1 1 1 1 1 1
$\mu$-invariant(s) - - 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 and 9.
Its isogeny class 1728bb consists of 3 curves linked by isogenies of degrees dividing 9.

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
$2$ \(\Q(\sqrt{6}) \) \(\Z/9\Z\) 2.2.24.1-162.1-f2
$3$ 3.1.216.1 \(\Z/2\Z\) Not in database
$6$ 6.0.1119744.1 \(\Z/2\Z \times \Z/2\Z\) Not in database
$6$ 6.0.30233088.2 \(\Z/3\Z\) Not in database
$6$ 6.2.4478976.1 \(\Z/18\Z\) Not in database
$12$ 12.2.1925877696823296.6 \(\Z/4\Z\) Not in database
$12$ 12.0.8226356490141696.17 \(\Z/3\Z \times \Z/9\Z\) Not in database
$12$ 12.0.80244904034304.3 \(\Z/2\Z \times \Z/18\Z\) Not in database
$18$ 18.6.34811183694905557468330328064.6 \(\Z/27\Z\) Not in database
$18$ 18.0.1289303099811316943271493632.2 \(\Z/6\Z\) Not in database

We only show fields where the torsion growth is primitive.