# Properties

 Label 189618.r1 Conductor $189618$ Discriminant $-314077377042$ j-invariant $$-\frac{672937149625}{10996722}$$ CM no Rank $2$ Torsion structure trivial

# Related objects

Show commands for: Magma / Pari/GP / SageMath

## Minimal Weierstrass equation

sage: E = EllipticCurve([1, 0, 1, -5581, 162242])

gp: E = ellinit([1, 0, 1, -5581, 162242])

magma: E := EllipticCurve([1, 0, 1, -5581, 162242]);

$$y^2+xy+y=x^3-5581x+162242$$

## Mordell-Weil group structure

$$\Z^2$$

### Infinite order Mordell-Weil generators and heights

sage: E.gens()

magma: Generators(E);

 $$P$$ = $$\left(40, 38\right)$$ $$\left(118, 1013\right)$$ $$\hat{h}(P)$$ ≈ $0.49784620556766830192422508905$ $0.55781405818641041312529394748$

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$\left(-80, 353\right)$$, $$\left(-80, -274\right)$$, $$\left(-38, 584\right)$$, $$\left(-38, -547\right)$$, $$\left(30, 133\right)$$, $$\left(30, -164\right)$$, $$\left(40, 38\right)$$, $$\left(40, -79\right)$$, $$\left(52, 89\right)$$, $$\left(52, -142\right)$$, $$\left(118, 1013\right)$$, $$\left(118, -1132\right)$$, $$\left(5812, 440153\right)$$, $$\left(5812, -445966\right)$$

## 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: $$-314077377042$$ = $$-1 \cdot 2 \cdot 3^{5} \cdot 11^{3} \cdot 13^{4} \cdot 17$$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$-\frac{672937149625}{10996722}$$ = $$-1 \cdot 2^{-1} \cdot 3^{-5} \cdot 5^{3} \cdot 11^{-3} \cdot 13^{2} \cdot 17^{-1} \cdot 317^{3}$$ Endomorphism ring: $$\Z$$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $$1.0061079502832900628346293671\dots$$ Stable Faltings height: $$0.15112483112944448415013355324\dots$$

## BSD invariants

 sage: E.rank()  magma: Rank(E); Analytic rank: $$2$$ sage: E.regulator()  magma: Regulator(E); Regulator: $$0.26895443631848715819901563272\dots$$ sage: E.period_lattice().omega()  gp: E.omega[1]  magma: RealPeriod(E); Real period: $$0.96907265491325672600765888664\dots$$ 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: $$45$$  = $$1\cdot5\cdot3\cdot3\cdot1$$ 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: $$1$$ (rounded)

## Modular invariants

Modular form 189618.2.a.r

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);

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

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 288000 $$\Gamma_0(N)$$-optimal: yes 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);

$$L^{(2)}(E,1)/2!$$ ≈ $$11.728637534423465442823270783098275227$$

## 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$$(j)_{-}$$
$$2$$ $$1$$ $$I_{1}$$ Non-split multiplicative 1 1 1 1
$$3$$ $$5$$ $$I_{5}$$ Split multiplicative -1 1 5 5
$$11$$ $$3$$ $$I_{3}$$ Split multiplicative -1 1 3 3
$$13$$ $$3$$ $$IV$$ Additive 1 2 4 0
$$17$$ $$1$$ $$I_{1}$$ Split multiplicative -1 1 1 1

## Galois representations

The 2-adic representation attached to this elliptic curve is surjective.

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 $$p$$ .

## $p$-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 no rational isogenies. Its isogeny class 189618.r consists of this curve only.

## 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 $K$ $3$ 3.1.758472.1 $$\Z/2\Z$$ Not in database $6$ 6.0.2581855629230592.1 $$\Z/2\Z \times \Z/2\Z$$ Not in database $8$ 8.2.6761185934388912.11 $$\Z/3\Z$$ Not in database $12$ Deg 12 $$\Z/4\Z$$ 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.