Properties

Label 116886.m3
Conductor $116886$
Discriminant $8.267\times 10^{25}$
j-invariant \( \frac{8856076866003496152467137}{46664863048067576004} \)
CM no
Rank $1$
Torsion structure \(\Z/{2}\Z \times \Z/{2}\Z\)

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Show commands: Magma / Pari/GP / SageMath

Minimal Weierstrass equation

sage: E = EllipticCurve([1, 0, 1, -521544367, 4563463980590])
 
gp: E = ellinit([1, 0, 1, -521544367, 4563463980590])
 
magma: E := EllipticCurve([1, 0, 1, -521544367, 4563463980590]);
 

\(y^2+xy+y=x^3-521544367x+4563463980590\)  Toggle raw display

Mordell-Weil group structure

$\Z\times \Z/{2}\Z \times \Z/{2}\Z$

Infinite order Mordell-Weil generator and height

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

$P$ =  \(\left(\frac{60247}{4}, \frac{2768209}{8}\right)\)  Toggle raw display
$\hat{h}(P)$ ≈  $9.7258368990451003830986242132$

Torsion generators

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

\( \left(-26357, 13178\right) \), \( \left(12451, -6226\right) \)  Toggle raw display

Integral points

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

\( \left(-26357, 13178\right) \), \( \left(12451, -6226\right) \)  Toggle raw display

Invariants

sage: E.conductor().factor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor: \( 116886 \)  =  $2 \cdot 3 \cdot 7 \cdot 11^{2} \cdot 23$
sage: E.discriminant().factor()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant: $82669651446297643013222244 $  =  $2^{2} \cdot 3^{4} \cdot 7^{4} \cdot 11^{14} \cdot 23^{4} $
sage: E.j_invariant().factor()
 
gp: E.j
 
magma: jInvariant(E);
 
j-invariant: \( \frac{8856076866003496152467137}{46664863048067576004} \)  =  $2^{-2} \cdot 3^{-4} \cdot 7^{-4} \cdot 11^{-8} \cdot 23^{-4} \cdot 206893633^{3}$
Endomorphism ring: $\Z$
Geometric endomorphism ring: \(\Z\) (no potential complex multiplication)
Sato-Tate group: $\mathrm{SU}(2)$
Faltings height: $3.8179689274420920208130282787\dots$
Stable Faltings height: $2.6190212910429067487820564897\dots$

BSD invariants

sage: E.rank()
 
magma: Rank(E);
 
Analytic rank: $1$
sage: E.regulator()
 
magma: Regulator(E);
 
Regulator: $9.7258368990451003830986242132\dots$
sage: E.period_lattice().omega()
 
gp: E.omega[1]
 
magma: RealPeriod(E);
 
Real period: $0.061108245146086653869616941008\dots$
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: $ 128 $  = $ 2\cdot2^{2}\cdot2\cdot2^{2}\cdot2 $
sage: E.torsion_order()
 
gp: elltors(E)[1]
 
magma: Order(TorsionSubgroup(E));
 
Torsion order: $4$
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) $ ≈ $ 4.7546306038216258313850778519296944060 $

Modular invariants

Modular form 116886.2.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);
 

\( q - q^{2} + q^{3} + q^{4} - 2q^{5} - q^{6} - q^{7} - q^{8} + q^{9} + 2q^{10} + q^{12} + 2q^{13} + q^{14} - 2q^{15} + q^{16} - 2q^{17} - q^{18} + 4q^{19} + O(q^{20}) \)  Toggle raw display

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

sage: E.modular_degree()
 
magma: ModularDegree(E);
 
Modular degree: 49152000
$ \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$(j)_{-}$
$2$ $2$ $I_{2}$ Non-split multiplicative 1 1 2 2
$3$ $4$ $I_{4}$ Split multiplicative -1 1 4 4
$7$ $2$ $I_{4}$ Non-split multiplicative 1 1 4 4
$11$ $4$ $I_8^{*}$ Additive -1 2 14 8
$23$ $2$ $I_{4}$ Non-split multiplicative 1 1 4 4

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 $\GL(2,\Z_\ell)$ for all primes $\ell$ except those listed in the table below.

prime $\ell$ mod-$\ell$ image $\ell$-adic image
$2$ 2Cs 8.48.0.96

$p$-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 nonsplit split ordinary nonsplit add ordinary ordinary ordinary nonsplit ordinary ss ordinary ordinary ordinary ss
$\lambda$-invariant(s) 5 2 1 3 - 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

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 116886.m consists of 3 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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z \times \Z/{2}\Z$ are as follows:

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base change curve
$2$ \(\Q(\sqrt{11}) \) \(\Z/2\Z \times \Z/4\Z\) Not in database
$4$ \(\Q(\sqrt{2}, \sqrt{-11})\) \(\Z/2\Z \times \Z/4\Z\) Not in database
$4$ \(\Q(\sqrt{-2}, \sqrt{-11})\) \(\Z/2\Z \times \Z/4\Z\) Not in database
$4$ \(\Q(\sqrt{11}, \sqrt{23})\) \(\Z/2\Z \times \Z/8\Z\) Not in database
$8$ 8.0.959512576.1 \(\Z/4\Z \times \Z/4\Z\) Not in database
$8$ 8.0.11942845778755584.331 \(\Z/2\Z \times \Z/8\Z\) Not in database
$8$ 8.0.268510958780416.12 \(\Z/2\Z \times \Z/8\Z\) Not in database
$8$ Deg 8 \(\Z/2\Z \times \Z/6\Z\) Not in database
$16$ Deg 16 \(\Z/2\Z \times \Z/8\Z\) Not in database
$16$ Deg 16 \(\Z/4\Z \times \Z/8\Z\) Not in database
$16$ 16.0.72098134985178260206105133056.1 \(\Z/4\Z \times \Z/8\Z\) Not in database
$16$ Deg 16 \(\Z/2\Z \times \Z/16\Z\) Not in database
$16$ Deg 16 \(\Z/2\Z \times \Z/12\Z\) 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.