Properties

Label 221760es4
Conductor $221760$
Discriminant $2.630\times 10^{20}$
j-invariant \( \frac{52702650535889104}{22020583921875} \)
CM no
Rank $0$
Torsion structure \(\Z/{2}\Z\)

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Show commands: Magma / Pari/GP / SageMath

Minimal Weierstrass equation

sage: E = EllipticCurve([0, 0, 0, -1785468, 484166608])
 
gp: E = ellinit([0, 0, 0, -1785468, 484166608])
 
magma: E := EllipticCurve([0, 0, 0, -1785468, 484166608]);
 

\(y^2=x^3-1785468x+484166608\)  Toggle raw display

Mordell-Weil group structure

$\Z/{2}\Z$

Torsion generators

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

\( \left(284, 0\right) \)  Toggle raw display

Integral points

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

\( \left(284, 0\right) \)  Toggle raw display

Invariants

sage: E.conductor().factor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor: \( 221760 \)  =  $2^{6} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 11$
sage: E.discriminant().factor()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant: $263012445045504000000 $  =  $2^{14} \cdot 3^{8} \cdot 5^{6} \cdot 7^{6} \cdot 11^{3} $
sage: E.j_invariant().factor()
 
gp: E.j
 
magma: jInvariant(E);
 
j-invariant: \( \frac{52702650535889104}{22020583921875} \)  =  $2^{4} \cdot 3^{-2} \cdot 5^{-6} \cdot 7^{-6} \cdot 11^{-3} \cdot 19^{3} \cdot 41^{3} \cdot 191^{3}$
Endomorphism ring: $\Z$
Geometric endomorphism ring: \(\Z\) (no potential complex multiplication)
Sato-Tate group: $\mathrm{SU}(2)$
Faltings height: $2.6153004641600859623763761520\dots$
Stable Faltings height: $1.2573226091727615890253160585\dots$

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: $0.15789378933437483254588716031\dots$
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: $ 48 $  = $ 2\cdot2\cdot2\cdot2\cdot3 $
sage: E.torsion_order()
 
gp: elltors(E)[1]
 
magma: Order(TorsionSubgroup(E));
 
Torsion order: $2$
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) $ ≈ $ 1.8947254720124979905506459237281223767 $

Modular invariants

Modular form 221760.2.a.cl

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);
 

\( q - q^{5} - q^{7} + q^{11} - 2q^{13} + 6q^{17} + 8q^{19} + O(q^{20}) \)  Toggle raw display

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

sage: E.modular_degree()
 
magma: ModularDegree(E);
 
Modular degree: 7962624
$ \Gamma_0(N) $-optimal: no
Manin constant: 1

Local data

This elliptic curve is not semistable. There are 5 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$(j)_{-}$
$2$ $2$ $I_4^{*}$ Additive -1 6 14 0
$3$ $2$ $I_2^{*}$ Additive -1 2 8 2
$5$ $2$ $I_{6}$ Non-split multiplicative 1 1 6 6
$7$ $2$ $I_{6}$ Non-split multiplicative 1 1 6 6
$11$ $3$ $I_{3}$ Split multiplicative -1 1 3 3

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
$2$ 2B 2.3.0.1
$3$ 3B 3.4.0.1

$p$-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 $0$.

No Iwasawa invariant data is available for this curve.

Isogenies

This curve has non-trivial cyclic isogenies of degree $d$ for $d=$ 2, 3 and 6.
Its isogeny class 221760es consists of 4 curves linked by isogenies of degrees dividing 6.

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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z$ are as follows:

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base change curve
$2$ \(\Q(\sqrt{11}) \) \(\Z/2\Z \times \Z/2\Z\) Not in database
$2$ \(\Q(\sqrt{-2}) \) \(\Z/6\Z\) Not in database
$4$ 4.0.7761600.4 \(\Z/4\Z\) Not in database
$4$ \(\Q(\sqrt{-2}, \sqrt{11})\) \(\Z/2\Z \times \Z/6\Z\) Not in database
$6$ 6.2.362797056.9 \(\Z/6\Z\) Not in database
$8$ Deg 8 \(\Z/2\Z \times \Z/4\Z\) Not in database
$8$ Deg 8 \(\Z/2\Z \times \Z/4\Z\) Not in database
$8$ Deg 8 \(\Z/12\Z\) Not in database
$12$ 12.0.131621703842267136.38 \(\Z/3\Z \times \Z/6\Z\) Not in database
$12$ Deg 12 \(\Z/2\Z \times \Z/6\Z\) Not in database
$16$ Deg 16 \(\Z/8\Z\) Not in database
$16$ Deg 16 \(\Z/2\Z \times \Z/12\Z\) Not in database
$16$ Deg 16 \(\Z/2\Z \times \Z/12\Z\) Not in database
$18$ 18.0.18756692241762077604603442737214284846071808000000000000.1 \(\Z/18\Z\) 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.