Properties

Label 116886.t2
Conductor $116886$
Discriminant $8.308\times 10^{14}$
j-invariant \( \frac{7110352307247726625}{6866458324992} \)
CM no
Rank $2$
Torsion structure trivial

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

Minimal Weierstrass equation

sage: E = EllipticCurve([1, 0, 1, -198146, 33903956])
 
gp: E = ellinit([1, 0, 1, -198146, 33903956])
 
magma: E := EllipticCurve([1, 0, 1, -198146, 33903956]);
 

\(y^2+xy+y=x^3-198146x+33903956\)  Toggle raw display

Mordell-Weil group structure

$\Z^2$

Infinite order Mordell-Weil generators and heights

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

$P$ =  \(\left(333, 2041\right)\)  Toggle raw display\(\left(-150, 7837\right)\)  Toggle raw display
$\hat{h}(P)$ ≈  $0.20019401278715076104148666239$$1.9259645209467094210250306052$

Integral points

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

\( \left(-462, 5416\right) \), \( \left(-462, -4955\right) \), \( \left(-150, 7837\right) \), \( \left(-150, -7688\right) \), \( \left(-51, 6649\right) \), \( \left(-51, -6599\right) \), \( \left(228, 655\right) \), \( \left(228, -884\right) \), \( \left(264, -29\right) \), \( \left(264, -236\right) \), \( \left(333, 2041\right) \), \( \left(333, -2375\right) \), \( \left(525, 8377\right) \), \( \left(525, -8903\right) \), \( \left(2334, 109681\right) \), \( \left(2334, -112016\right) \)  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: $830841457324032 $  =  $2^{12} \cdot 3^{9} \cdot 7 \cdot 11^{2} \cdot 23^{3} $
sage: E.j_invariant().factor()
 
gp: E.j
 
magma: jInvariant(E);
 
j-invariant: \( \frac{7110352307247726625}{6866458324992} \)  =  $2^{-12} \cdot 3^{-9} \cdot 5^{3} \cdot 7^{-1} \cdot 11 \cdot 23^{-3} \cdot 29^{3} \cdot 67^{3} \cdot 89^{3}$
Endomorphism ring: $\Z$
Geometric endomorphism ring: \(\Z\) (no potential complex multiplication)
Sato-Tate group: $\mathrm{SU}(2)$
Faltings height: $1.7838613857216645808712192726\dots$
Stable Faltings height: $1.3842121735886028235275620096\dots$

BSD invariants

sage: E.rank()
 
magma: Rank(E);
 
Analytic rank: $2$
sage: E.regulator()
 
magma: Regulator(E);
 
Regulator: $0.38177688188022575219221997154\dots$
sage: E.period_lattice().omega()
 
gp: E.omega[1]
 
magma: RealPeriod(E);
 
Real period: $0.49881443128703802571949454423\dots$
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: $ 54 $  = $ 2\cdot3^{2}\cdot1\cdot1\cdot3 $
sage: E.torsion_order()
 
gp: elltors(E)[1]
 
magma: Order(TorsionSubgroup(E));
 
Torsion order: $1$
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! $ ≈ $ 10.283534183535669063263909402424732939 $

Modular invariants

Modular form 116886.2.a.t

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);
 

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

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

sage: E.modular_degree()
 
magma: ModularDegree(E);
 
Modular degree: 829440
$ \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$(j)_{-}$
$2$ $2$ $I_{12}$ Non-split multiplicative 1 1 12 12
$3$ $9$ $I_{9}$ Split multiplicative -1 1 9 9
$7$ $1$ $I_{1}$ Non-split multiplicative 1 1 1 1
$11$ $1$ $II$ Additive -1 2 2 0
$23$ $3$ $I_{3}$ 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 $\GL(2,\Z_\ell)$ for all primes $\ell$ except those listed in the table below.

prime $\ell$ mod-$\ell$ image $\ell$-adic image
$3$ 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]
 

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 nonsplit split ss nonsplit add ordinary ordinary ordinary split ss ordinary ordinary ordinary ordinary ordinary
$\lambda$-invariant(s) 3 3 2,2 2 - 2 2 2 3 2,2 2 2 2 2 2
$\mu$-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 $d=$ 3.
Its isogeny class 116886.t consists of 2 curves linked by isogenies of degree 3.

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 $E(\Q)_{\rm tors}$ (which is trivial) are as follows:

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base change curve
$2$ \(\Q(\sqrt{-11}) \) \(\Z/3\Z\) Not in database
$3$ 3.3.233772.1 \(\Z/2\Z\) Not in database
$6$ 6.6.105582540305088.1 \(\Z/2\Z \times \Z/2\Z\) Not in database
$6$ 6.2.10440453177.1 \(\Z/3\Z\) Not in database
$6$ 6.0.601142827824.1 \(\Z/6\Z\) Not in database
$12$ Deg 12 \(\Z/4\Z\) Not in database
$12$ Deg 12 \(\Z/3\Z \times \Z/3\Z\) Not in database
$12$ Deg 12 \(\Z/2\Z \times \Z/6\Z\) Not in database
$18$ 18.0.837435105392897779905772911050703358087442813005824.1 \(\Z/9\Z\) Not in database
$18$ 18.6.33812796763489799274131827370131381291880448.1 \(\Z/6\Z\) 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.