Properties

Label 816.b5
Conductor $816$
Discriminant $1443889152$
j-invariant \( \frac{4354703137}{352512} \)
CM no
Rank $1$
Torsion structure \(\Z/{2}\Z\)

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Show commands: Magma / Pari/GP / SageMath

Minimal Weierstrass equation

sage: E = EllipticCurve([0, -1, 0, -544, -4352])
 
gp: E = ellinit([0, -1, 0, -544, -4352])
 
magma: E := EllipticCurve([0, -1, 0, -544, -4352]);
 

\(y^2=x^3-x^2-544x-4352\)  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);
 

$P$ =  \(\left(-14, 18\right)\)  Toggle raw display
$\hat{h}(P)$ ≈  $0.95674544237211927834287572036$

Torsion generators

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

\( \left(-16, 0\right) \)  Toggle raw display

Integral points

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

\( \left(-16, 0\right) \), \((-14,\pm 18)\), \((33,\pm 112)\), \((112,\pm 1152)\)  Toggle raw display

Invariants

sage: E.conductor().factor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor: \( 816 \)  =  $2^{4} \cdot 3 \cdot 17$
sage: E.discriminant().factor()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant: $1443889152 $  =  $2^{20} \cdot 3^{4} \cdot 17 $
sage: E.j_invariant().factor()
 
gp: E.j
 
magma: jInvariant(E);
 
j-invariant: \( \frac{4354703137}{352512} \)  =  $2^{-8} \cdot 3^{-4} \cdot 17^{-1} \cdot 23^{3} \cdot 71^{3}$
Endomorphism ring: $\Z$
Geometric endomorphism ring: \(\Z\) (no potential complex multiplication)
Sato-Tate group: $\mathrm{SU}(2)$
Faltings height: $0.50050469536149959952921099595\dots$
Stable Faltings height: $-0.19264248519844570988802112551\dots$

BSD invariants

sage: E.rank()
 
magma: Rank(E);
 
Analytic rank: $1$
sage: E.regulator()
 
magma: Regulator(E);
 
Regulator: $0.95674544237211927834287572036\dots$
sage: E.period_lattice().omega()
 
gp: E.omega[1]
 
magma: RealPeriod(E);
 
Real period: $0.99348185850601324739329990214\dots$
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\cdot1 $
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) $ ≈ $ 1.9010184804100217125383093577879546832 $

Modular invariants

Modular form   816.2.a.b

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);
 

\( q - q^{3} - 2q^{5} + q^{9} + 4q^{11} - 2q^{13} + 2q^{15} + q^{17} - 4q^{19} + O(q^{20}) \)  Toggle raw display

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

sage: E.modular_degree()
 
magma: ModularDegree(E);
 
Modular degree: 384
$ \Gamma_0(N) $-optimal: yes
Manin constant: 1

Local data

This elliptic curve is not semistable. There are 3 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$(j)_{-}$
$2$ $4$ $I_{12}^{*}$ Additive -1 4 20 8
$3$ $2$ $I_{4}$ Non-split multiplicative 1 1 4 4
$17$ $1$ $I_{1}$ Split multiplicative -1 1 1 1

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 16.96.0.145

$p$-adic regulators

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

Note: $p$-adic regulator data only exists for primes $p\ge 5$ of good ordinary reduction.

Iwasawa invariants

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47
Reduction type add nonsplit ordinary ss ordinary ordinary split ordinary ss ordinary ordinary ordinary ordinary ordinary ss
$\lambda$-invariant(s) - 1 1 1,3 1 1 2 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

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, 4 and 8.
Its isogeny class 816.b consists of 6 curves linked by isogenies of degrees dividing 8.

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
$2$ \(\Q(\sqrt{17}) \) \(\Z/2\Z \times \Z/2\Z\) Not in database
$2$ \(\Q(\sqrt{-17}) \) \(\Z/4\Z\) Not in database
$2$ \(\Q(\sqrt{-1}) \) \(\Z/8\Z\) 2.0.4.1-5202.2-e4
$4$ \(\Q(i, \sqrt{17})\) \(\Z/2\Z \times \Z/8\Z\) Not in database
$4$ 4.2.19652.1 \(\Z/2\Z \times \Z/4\Z\) Not in database
$4$ 4.0.78608.2 \(\Z/8\Z\) Not in database
$8$ 8.0.6179217664.3 \(\Z/4\Z \times \Z/8\Z\) Not in database
$8$ 8.0.1534132224.14 \(\Z/16\Z\) Not in database
$8$ 8.2.15150586457088.10 \(\Z/6\Z\) Not in database
$16$ Deg 16 \(\Z/2\Z \times \Z/8\Z\) Not in database
$16$ Deg 16 \(\Z/2\Z \times \Z/16\Z\) Not in database
$16$ Deg 16 \(\Z/2\Z \times \Z/16\Z\) Not in database
$16$ Deg 16 \(\Z/4\Z \times \Z/16\Z\) Not in database
$16$ Deg 16 \(\Z/2\Z \times \Z/6\Z\) Not in database
$16$ Deg 16 \(\Z/12\Z\) Not in database
$16$ Deg 16 \(\Z/24\Z\) 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.