Properties

Label 1568.e4
Conductor $1568$
Discriminant $-481890304$
j-invariant \( 1728 \)
CM yes (\(D=-4\))
Rank $1$
Torsion structure \(\Z/{2}\Z\)

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Show commands for: Magma / Pari/GP / SageMath

Minimal Weierstrass equation

sage: E = EllipticCurve([0, 0, 0, 196, 0])
 
gp: E = ellinit([0, 0, 0, 196, 0])
 
magma: E := EllipticCurve([0, 0, 0, 196, 0]);
 

\(y^2=x^3+196x\)  Toggle raw display

Mordell-Weil group structure

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

Infinite order Mordell-Weil generator and height

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

\(P\) =  \(\left(2, 20\right)\)  Toggle raw display
\(\hat{h}(P)\) ≈  $1.4944407538016703049134574706$

Torsion generators

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

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

Integral points

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

\( \left(0, 0\right) \), \((2,\pm 20)\), \((98,\pm 980)\)  Toggle raw display

Invariants

sage: E.conductor().factor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor: \( 1568 \)  =  \(2^{5} \cdot 7^{2}\)
sage: E.discriminant().factor()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant: \(-481890304 \)  =  \(-1 \cdot 2^{12} \cdot 7^{6} \)
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: $N(\mathrm{U}(1))$
Faltings height: \(0.35556932917609244371763340988\dots\)
Stable Faltings height: \(-1.3105329259115095182522750833\dots\)

BSD invariants

sage: E.rank()
 
magma: Rank(E);
 
Analytic rank: \(1\)
sage: E.regulator()
 
magma: Regulator(E);
 
Regulator: \(1.4944407538016703049134574706\dots\)
sage: E.period_lattice().omega()
 
gp: E.omega[1]
 
magma: RealPeriod(E);
 
Real period: \(0.99104460170788427303819589878\dots\)
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: \( 8 \)  = \( 2^{2}\cdot2 \)
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)

Modular invariants

Modular form   1568.2.a.e

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);
 

\( q + 2q^{5} - 3q^{9} - 6q^{13} - 2q^{17} + O(q^{20}) \)  Toggle raw display

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

sage: E.modular_degree()
 
magma: ModularDegree(E);
 
Modular degree: 384
\( \Gamma_0(N) \)-optimal: no
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);
 

\( L'(E,1) \) ≈ \( 2.9621148832548133741401093523738541723 \)

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\((j)_{-}\)
\(2\) \(4\) \(I_3^{*}\) Additive -1 5 12 0
\(7\) \(2\) \(I_0^{*}\) 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 \( p < 1000 \) .

The image is a Borel subgroup if \(p=2\), 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]
 

\(p\)-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 ss ordinary add ss ordinary ordinary ss ss ordinary ss ordinary ordinary ss ss
$\lambda$-invariant(s) - 1,1 1 - 1,1 1 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 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 \(d=\) 2 and 4.
Its isogeny class 1568.e consists of 3 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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z$ are as follows:

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base change curve
$2$ \(\Q(\sqrt{-1}) \) \(\Z/2\Z \times \Z/2\Z\) Not in database
$2$ \(\Q(\sqrt{7}) \) \(\Z/4\Z\) 2.2.28.1-64.1-b2
$2$ \(\Q(\sqrt{-7}) \) \(\Z/4\Z\) 2.0.7.1-1024.6-a2
$4$ \(\Q(i, \sqrt{7})\) \(\Z/2\Z \times \Z/4\Z\) Not in database
$8$ 8.0.157351936.1 \(\Z/4\Z \times \Z/4\Z\) Not in database
$8$ 8.4.10070523904.1 \(\Z/8\Z\) Not in database
$8$ 8.0.2517630976.6 \(\Z/8\Z\) Not in database
$8$ 8.2.344128684032.1 \(\Z/6\Z\) Not in database
$8$ 8.0.19668992000.1 \(\Z/2\Z \times \Z/10\Z\) Not in database
$16$ 16.0.101415451701035401216.11 \(\Z/4\Z \times \Z/8\Z\) Not in database
$16$ Deg 16 \(\Z/6\Z \times \Z/6\Z\) Not in database
$16$ Deg 16 \(\Z/10\Z\) Not in database
$16$ Deg 16 \(\Z/12\Z\) Not in database
$16$ Deg 16 \(\Z/2\Z \times \Z/20\Z\) Not in database
$16$ Deg 16 \(\Z/12\Z\) 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.