Properties

 Label 116886.u1 Conductor $116886$ Discriminant $-2.721\times 10^{20}$ j-invariant $$-\frac{3133382230165522315000208250857964625}{153574604080128}$$ CM no Rank $1$ Torsion structure trivial

Related objects

Show commands: Magma / Pari/GP / SageMath

Minimal Weierstrass equation

sage: E = EllipticCurve([1, 0, 1, -3688772954056, 2726910926533633046])

gp: E = ellinit([1, 0, 1, -3688772954056, 2726910926533633046])

magma: E := EllipticCurve([1, 0, 1, -3688772954056, 2726910926533633046]);

$$y^2+xy+y=x^3-3688772954056x+2726910926533633046$$

Mordell-Weil group structure

$\Z$

Infinite order Mordell-Weil generator and height

sage: E.gens()

magma: Generators(E);

 $P$ = $$\left(\frac{40540448905}{36481}, \frac{26723536344757}{6967871}\right)$$ $\hat{h}(P)$ ≈ $17.271061848354840408395142436$

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: $-272066779178795639808$ = $-1 \cdot 2^{10} \cdot 3^{3} \cdot 7^{3} \cdot 11^{9} \cdot 23^{3}$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$-\frac{3133382230165522315000208250857964625}{153574604080128}$$ = $-1 \cdot 2^{-10} \cdot 3^{-3} \cdot 5^{3} \cdot 7^{-3} \cdot 11^{-3} \cdot 23^{-3} \cdot 127^{3} \cdot 347^{3} \cdot 997^{3} \cdot 6661^{3}$ Endomorphism ring: $\Z$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $5.3134135977425850350969950550\dots$ Stable Faltings height: $4.1144659613433997630660232660\dots$

BSD invariants

 sage: E.rank()  magma: Rank(E); Analytic rank: $1$ sage: E.regulator()  magma: Regulator(E); Regulator: $17.271061848354840408395142436\dots$ sage: E.period_lattice().omega()  gp: E.omega[1]  magma: RealPeriod(E); Real period: $0.028164919011763628349927444696\dots$ 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: $12$  = $2\cdot3\cdot1\cdot2\cdot1$ 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: $1$ (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: $L'(E,1)$ ≈ $5.8372566984728965938364008124189889558$

Modular invariants

Modular form 116886.2.a.u

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);

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

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 1077753600 $\Gamma_0(N)$-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$(j)_{-}$
$2$ $2$ $I_{10}$ Non-split multiplicative 1 1 10 10
$3$ $3$ $I_{3}$ Split multiplicative -1 1 3 3
$7$ $1$ $I_{3}$ Non-split multiplicative 1 1 3 3
$11$ $2$ $I_3^{*}$ Additive -1 2 9 3
$23$ $1$ $I_{3}$ 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 3.4.0.1

$p$-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

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

 $[K:\Q]$ $E(K)_{\rm tors}$ Base change curve $K$ $2$ $$\Q(\sqrt{33})$$ $$\Z/3\Z$$ Not in database $3$ 3.1.21252.1 $$\Z/2\Z$$ Not in database $6$ 6.0.9598412755008.1 $$\Z/2\Z \times \Z/2\Z$$ Not in database $6$ 6.0.15524784.3 $$\Z/3\Z$$ Not in database $6$ 6.2.14904367632.1 $$\Z/6\Z$$ Not in database $12$ Deg 12 $$\Z/4\Z$$ Not in database $12$ 12.0.2169170264219904.1 $$\Z/3\Z \times \Z/3\Z$$ Not in database $12$ Deg 12 $$\Z/2\Z \times \Z/6\Z$$ Not in database $18$ 18.6.178705596032807626424196258121171422452777824512.1 $$\Z/9\Z$$ Not in database $18$ 18.0.150146229396195040055829928397453131776.1 $$\Z/6\Z$$ 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.