Properties

Label 189618.bb1
Conductor $189618$
Discriminant $1.470\times 10^{18}$
j-invariant \( \frac{3047678972871625}{304559880768} \)
CM no
Rank $2$
Torsion structure \(\Z/{2}\Z\)

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Show commands: Magma / Pari/GP / SageMath

Minimal Weierstrass equation

sage: E = EllipticCurve([1, 1, 1, -510468, -127896891])
 
gp: E = ellinit([1, 1, 1, -510468, -127896891])
 
magma: E := EllipticCurve([1, 1, 1, -510468, -127896891]);
 

\(y^2+xy+y=x^3+x^2-510468x-127896891\)  Toggle raw display

Mordell-Weil group structure

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

Infinite order Mordell-Weil generators and heights

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

$P$ =  \(\left(-359, 3221\right)\)  Toggle raw display\(\left(993, 18093\right)\)  Toggle raw display
$\hat{h}(P)$ ≈  $1.6041475606442772154913147729$$2.5545271685516342827933922341$

Torsion generators

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

\( \left(-307, 153\right) \)  Toggle raw display

Integral points

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

\( \left(-503, 1637\right) \), \( \left(-503, -1135\right) \), \( \left(-359, 3221\right) \), \( \left(-359, -2863\right) \), \( \left(-307, 153\right) \), \( \left(993, 18093\right) \), \( \left(993, -19087\right) \), \( \left(1137, 27057\right) \), \( \left(1137, -28195\right) \), \( \left(4087, 255005\right) \), \( \left(4087, -259093\right) \)  Toggle raw display

Invariants

sage: E.conductor().factor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor: \( 189618 \)  =  $2 \cdot 3 \cdot 11 \cdot 13^{2} \cdot 17$
sage: E.discriminant().factor()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant: $1470052373529909312 $  =  $2^{6} \cdot 3^{4} \cdot 11^{2} \cdot 13^{10} \cdot 17 $
sage: E.j_invariant().factor()
 
gp: E.j
 
magma: jInvariant(E);
 
j-invariant: \( \frac{3047678972871625}{304559880768} \)  =  $2^{-6} \cdot 3^{-4} \cdot 5^{3} \cdot 11^{-2} \cdot 13^{-4} \cdot 17^{-1} \cdot 107^{3} \cdot 271^{3}$
Endomorphism ring: $\Z$
Geometric endomorphism ring: \(\Z\) (no potential complex multiplication)
Sato-Tate group: $\mathrm{SU}(2)$
Faltings height: $2.2223552154528744781555377856\dots$
Stable Faltings height: $0.93988053672210611012879406482\dots$

BSD invariants

sage: E.rank()
 
magma: Rank(E);
 
Analytic rank: $2$
sage: E.regulator()
 
magma: Regulator(E);
 
Regulator: $3.7783622598555203767902349218\dots$
sage: E.period_lattice().omega()
 
gp: E.omega[1]
 
magma: RealPeriod(E);
 
Real period: $0.17976031074037294968890387851\dots$
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: $ 96 $  = $ ( 2 \cdot 3 )\cdot2\cdot2\cdot2^{2}\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: $1$ (rounded)
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^{(2)}(E,1)/2! $ ≈ $ 16.300789774111826627174250840596761785 $

Modular invariants

Modular form 189618.2.a.bb

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);
 

\( q + q^{2} - q^{3} + q^{4} - q^{6} - 2 q^{7} + q^{8} + q^{9} - q^{11} - q^{12} - 2 q^{14} + q^{16} - q^{17} + q^{18} - 4 q^{19} + O(q^{20}) \)  Toggle raw display

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

sage: E.modular_degree()
 
magma: ModularDegree(E);
 
Modular degree: 3612672
$ \Gamma_0(N) $-optimal: yes
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$(j)_{-}$
$2$ $6$ $I_{6}$ Split multiplicative -1 1 6 6
$3$ $2$ $I_{4}$ Non-split multiplicative 1 1 4 4
$11$ $2$ $I_{2}$ Non-split multiplicative 1 1 2 2
$13$ $4$ $I_{4}^{*}$ Additive 1 2 10 4
$17$ $1$ $I_{1}$ Non-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 8.6.0.4

$p$-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.

No Iwasawa invariant data is available for this curve.

Isogenies

This curve has non-trivial cyclic isogenies of degree $d$ for $d=$ 2.
Its isogeny class 189618.bb consists of 2 curves linked by isogenies of degree 2.

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
$4$ 4.0.183872.4 \(\Z/4\Z\) Not in database
$8$ 8.0.9770775678976.36 \(\Z/2\Z \times \Z/4\Z\) Not in database
$8$ Deg 8 \(\Z/2\Z \times \Z/4\Z\) Not in database
$8$ Deg 8 \(\Z/6\Z\) Not in database
$16$ Deg 16 \(\Z/8\Z\) Not in database
$16$ Deg 16 \(\Z/2\Z \times \Z/6\Z\) 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.