Properties

Label 116886.v1
Conductor $116886$
Discriminant $-5.171\times 10^{13}$
j-invariant \( -\frac{39402364010111991625}{3532128768} \)
CM no
Rank $0$
Torsion structure trivial

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Show commands: Magma / Pari/GP / SageMath

Minimal Weierstrass equation

sage: E = EllipticCurve([1, 0, 1, -1734296, -879233434])
 
gp: E = ellinit([1, 0, 1, -1734296, -879233434])
 
magma: E := EllipticCurve([1, 0, 1, -1734296, -879233434]);
 

\(y^2+xy+y=x^3-1734296x-879233434\)  Toggle raw display

Mordell-Weil group structure

trivial

Integral points

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

None

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: $-51713897292288 $  =  $-1 \cdot 2^{9} \cdot 3^{4} \cdot 7 \cdot 11^{4} \cdot 23^{3} $
sage: E.j_invariant().factor()
 
gp: E.j
 
magma: jInvariant(E);
 
j-invariant: \( -\frac{39402364010111991625}{3532128768} \)  =  $-1 \cdot 2^{-9} \cdot 3^{-4} \cdot 5^{3} \cdot 7^{-1} \cdot 11^{2} \cdot 23^{-3} \cdot 137597^{3}$
Endomorphism ring: $\Z$
Geometric endomorphism ring: \(\Z\) (no potential complex multiplication)
Sato-Tate group: $\mathrm{SU}(2)$
Faltings height: $2.0697122485123406573457496292\dots$
Stable Faltings height: $1.2704138242462171426584351032\dots$

BSD invariants

sage: E.rank()
 
magma: Rank(E);
 
Analytic rank: $0$
sage: E.regulator()
 
magma: Regulator(E);
 
Regulator: $1$
sage: E.period_lattice().omega()
 
gp: E.omega[1]
 
magma: RealPeriod(E);
 
Real period: $0.065779229343451429695682629643\dots$
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: $ 4 $  = $ 1\cdot2^{2}\cdot1\cdot1\cdot1 $
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: $9$ = $3^2$ (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) $ ≈ $ 2.3680522563642514690445746671514125053 $

Modular invariants

Modular form 116886.2.a.v

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);
 

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

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

sage: E.modular_degree()
 
magma: ModularDegree(E);
 
Modular degree: 1804032
$ \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$(j)_{-}$
$2$ $1$ $I_{9}$ Non-split multiplicative 1 1 9 9
$3$ $4$ $I_{4}$ Split multiplicative -1 1 4 4
$7$ $1$ $I_{1}$ Split multiplicative -1 1 1 1
$11$ $1$ $IV$ Additive -1 2 4 0
$23$ $1$ $I_{3}$ Non-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.1.2 3.8.0.2

$p$-adic regulators

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

All $p$-adic regulators are identically $1$ since the rank is $0$.

Iwasawa invariants

$p$ 2 3 7 11 23
Reduction type nonsplit split split add nonsplit
$\lambda$-invariant(s) 3 3 1 - 0
$\mu$-invariant(s) 0 1 0 - 0

All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 5$ of good reduction are zero.

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.v 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
$2$ \(\Q(\sqrt{-3}) \) \(\Z/3\Z\) Not in database
$3$ 3.1.155848.1 \(\Z/2\Z\) Not in database
$3$ 3.1.1440747.4 \(\Z/3\Z\) Not in database
$6$ 6.0.31283715645952.1 \(\Z/2\Z \times \Z/2\Z\) Not in database
$6$ 6.0.6227255754027.2 \(\Z/3\Z \times \Z/3\Z\) Not in database
$6$ 6.0.655792175808.1 \(\Z/6\Z\) Not in database
$9$ 9.1.130411136854144838095455744.1 \(\Z/6\Z\) Not in database
$12$ Deg 12 \(\Z/4\Z\) Not in database
$12$ Deg 12 \(\Z/2\Z \times \Z/6\Z\) Not in database
$18$ 18.0.244900386504383336210569103511422796897019629559227.2 \(\Z/9\Z\) Not in database
$18$ 18.0.459190744620943338504692760707352197978690062690025472.1 \(\Z/3\Z \times \Z/6\Z\) Not in database
$18$ 18.0.741619039357556113959939370414164941741455245259790780203008.1 \(\Z/2\Z \times \Z/6\Z\) 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.