# Properties

 Label 189618.a2 Conductor $189618$ Discriminant $6.617\times 10^{15}$ j-invariant $$\frac{7457162887153}{1370924676}$$ CM no Rank $2$ Torsion structure $$\Z/{2}\Z \times \Z/{2}\Z$$

# Related objects

Show commands for: Magma / Pari/GP / SageMath

## Minimal Weierstrass equation

sage: E = EllipticCurve([1, 1, 0, -68786, 5707200])

gp: E = ellinit([1, 1, 0, -68786, 5707200])

magma: E := EllipticCurve([1, 1, 0, -68786, 5707200]);

$$y^2+xy=x^3+x^2-68786x+5707200$$

## Mordell-Weil group structure

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

### Infinite order Mordell-Weil generators and heights

sage: E.gens()

magma: Generators(E);

 $$P$$ = $$\left(83, 719\right)$$ $$\left(-25, 2735\right)$$ $$\hat{h}(P)$$ ≈ $1.6944024364256929761347957466$ $2.1614553477875153194373041824$

## Torsion generators

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

$$\left(96, -48\right)$$, $$\left(200, -100\right)$$

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$\left(-242, 2994\right)$$, $$\left(-242, -2752\right)$$, $$\left(-138, 3618\right)$$, $$\left(-138, -3480\right)$$, $$\left(-25, 2735\right)$$, $$\left(-25, -2710\right)$$, $$\left(47, 1583\right)$$, $$\left(47, -1630\right)$$, $$\left(83, 719\right)$$, $$\left(83, -802\right)$$, $$\left(96, -48\right)$$, $$\left(200, -100\right)$$, $$\left(217, 920\right)$$, $$\left(217, -1137\right)$$, $$\left(421, 6972\right)$$, $$\left(421, -7393\right)$$, $$\left(434, 7388\right)$$, $$\left(434, -7822\right)$$, $$\left(825, 22200\right)$$, $$\left(825, -23025\right)$$, $$\left(3242, 182420\right)$$, $$\left(3242, -185662\right)$$, $$\left(1293050, 1469711780\right)$$, $$\left(1293050, -1471004830\right)$$

## 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: $$6617191564438884$$ = $$2^{2} \cdot 3^{4} \cdot 11^{4} \cdot 13^{6} \cdot 17^{2}$$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$\frac{7457162887153}{1370924676}$$ = $$2^{-2} \cdot 3^{-4} \cdot 7^{3} \cdot 11^{-4} \cdot 17^{-2} \cdot 2791^{3}$$ Endomorphism ring: $$\Z$$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $$1.7541604449833395783710678192\dots$$ Stable Faltings height: $$0.47168576625257121034432409842\dots$$

## BSD invariants

 sage: E.rank()  magma: Rank(E); Analytic rank: $$2$$ sage: E.regulator()  magma: Regulator(E); Regulator: $$3.4615215840457233088757741789\dots$$ sage: E.period_lattice().omega()  gp: E.omega[1]  magma: RealPeriod(E); Real period: $$0.40125466216697867189185693629\dots$$ 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: $$64$$  = $$2\cdot2\cdot2\cdot2^{2}\cdot2$$ 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: $$1$$ (rounded)

## Modular invariants

Modular form 189618.2.a.a

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);

$$q - q^{2} - q^{3} + q^{4} - 2q^{5} + q^{6} - 4q^{7} - q^{8} + q^{9} + 2q^{10} - q^{11} - q^{12} + 4q^{14} + 2q^{15} + q^{16} + q^{17} - q^{18} - 4q^{19} + O(q^{20})$$

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 1769472 $$\Gamma_0(N)$$-optimal: no Manin constant: 1

#### Special L-value

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);

$$L^{(2)}(E,1)/2!$$ ≈ $$5.5558066951598863027096883373833371331$$

## 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$$(j)_{-}$$
$$2$$ $$2$$ $$I_{2}$$ Non-split multiplicative 1 1 2 2
$$3$$ $$2$$ $$I_{4}$$ Non-split multiplicative 1 1 4 4
$$11$$ $$2$$ $$I_{4}$$ Non-split multiplicative 1 1 4 4
$$13$$ $$4$$ $$I_0^{*}$$ Additive 1 2 6 0
$$17$$ $$2$$ $$I_{2}$$ Split multiplicative -1 1 2 2

## Galois representations

The image of the 2-adic representation attached to this elliptic curve is the subgroup of $\GL(2,\Z_2)$ with Rouse label X46.

This subgroup is the pull-back of the subgroup of $\GL(2,\Z_2/2^3\Z_2)$ generated by $\left(\begin{array}{rr} 3 & 0 \\ 0 & 5 \end{array}\right),\left(\begin{array}{rr} 3 & 0 \\ 6 & 7 \end{array}\right),\left(\begin{array}{rr} 3 & 0 \\ 0 & 3 \end{array}\right),\left(\begin{array}{rr} 3 & 6 \\ 6 & 7 \end{array}\right)$ and has index 12.

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 mod $$p$$ Galois representation has maximal image $$\GL(2,\F_p)$$ for all primes $$p$$ except those listed.

prime Image of Galois representation
$$2$$ Cs

## $p$-adic data

### $p$-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.a 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 $K$ $4$ $$\Q(\sqrt{-2}, \sqrt{13})$$ $$\Z/2\Z \times \Z/4\Z$$ Not in database $4$ $$\Q(\sqrt{-13}, \sqrt{17})$$ $$\Z/2\Z \times \Z/4\Z$$ Not in database $4$ $$\Q(\sqrt{26}, \sqrt{34})$$ $$\Z/2\Z \times \Z/4\Z$$ Not in database $8$ Deg 8 $$\Z/2\Z \times \Z/6\Z$$ Not in database $16$ 16.0.24439822686430441934952595456.3 $$\Z/4\Z \times \Z/4\Z$$ 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 $$\Z/2\Z \times \Z/8\Z$$ 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.