Properties

 Label 116688.p3 Conductor $116688$ Discriminant $3.937\times 10^{28}$ j-invariant $$\frac{433744050935826360922067531137}{9612122270219882316693504}$$ CM no Rank $0$ Torsion structure $$\Z/{2}\Z \times \Z/{2}\Z$$

Related objects

Show commands: Magma / Pari/GP / SageMath

Minimal Weierstrass equation

sage: E = EllipticCurve([0, 1, 0, -2523228544, 47840592467636])

gp: E = ellinit([0, 1, 0, -2523228544, 47840592467636])

magma: E := EllipticCurve([0, 1, 0, -2523228544, 47840592467636]);

$$y^2=x^3+x^2-2523228544x+47840592467636$$

Mordell-Weil group structure

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

Torsion generators

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

$$\left(25643, 0\right)$$, $$\left(32234, 0\right)$$

Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$\left(-57878, 0\right)$$, $$\left(25643, 0\right)$$, $$\left(32234, 0\right)$$

Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor: $$116688$$ = $2^{4} \cdot 3 \cdot 11 \cdot 13 \cdot 17$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $39371252818820637969176592384$ = $2^{30} \cdot 3^{2} \cdot 11^{2} \cdot 13^{6} \cdot 17^{8}$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$\frac{433744050935826360922067531137}{9612122270219882316693504}$$ = $2^{-18} \cdot 3^{-2} \cdot 11^{-2} \cdot 13^{-6} \cdot 17^{-8} \cdot 19^{3} \cdot 739^{3} \cdot 539113^{3}$ Endomorphism ring: $\Z$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $4.2745709419901743485760358334\dots$ Stable Faltings height: $3.5814237614302290391588037119\dots$

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: $0.036326094113909577128424188272\dots$ 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: $192$  = $2^{2}\cdot2\cdot2\cdot( 2 \cdot 3 )\cdot2$ 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: $L(E,1)$ ≈ $0.43591312936691492554109025926763474306$

Modular invariants

Modular form 116688.2.a.p

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);

$$q + q^{3} - 2q^{5} + q^{9} - q^{11} + q^{13} - 2q^{15} - q^{17} - 8q^{19} + O(q^{20})$$

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 113467392 $\Gamma_0(N)$-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$(j)_{-}$
$2$ $4$ $I_{22}^{*}$ Additive -1 4 30 18
$3$ $2$ $I_{2}$ Split multiplicative -1 1 2 2
$11$ $2$ $I_{2}$ Non-split multiplicative 1 1 2 2
$13$ $6$ $I_{6}$ Split multiplicative -1 1 6 6
$17$ $2$ $I_{8}$ Non-split multiplicative 1 1 8 8

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
$2$ 2Cs 4.12.0.1

$p$-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

 $p$ Reduction type $\lambda$-invariant(s) $\mu$-invariant(s) 2 3 11 13 17 add split nonsplit split nonsplit - 3 0 1 0 - 0 0 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 $d=$ 2.
Its isogeny class 116688.p consists of 2 curves linked by isogenies of degrees dividing 4.

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:

 $[K:\Q]$ $E(K)_{\rm tors}$ Base change curve $K$ $2$ $$\Q(\sqrt{-39})$$ $$\Z/2\Z \times \Z/4\Z$$ Not in database $4$ $$\Q(\sqrt{22}, \sqrt{39})$$ $$\Z/2\Z \times \Z/4\Z$$ Not in database $4$ $$\Q(i, \sqrt{22})$$ $$\Z/2\Z \times \Z/4\Z$$ Not in database $8$ 8.0.2219775733334016.225 $$\Z/4\Z \times \Z/4\Z$$ Not in database $8$ Deg 8 $$\Z/2\Z \times \Z/8\Z$$ 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 $$\Z/2\Z \times \Z/8\Z$$ 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.