# Properties

 Label 189618.bo1 Conductor $189618$ Discriminant $-1.357\times 10^{19}$ j-invariant $$-\frac{1449073218392281}{2811246362496}$$ CM no Rank $2$ Torsion structure trivial

# Related objects

Show commands: Magma / Pari/GP / SageMath

## Minimal Weierstrass equation

sage: E = EllipticCurve([1, 0, 0, -398421, 201907809])

gp: E = ellinit([1, 0, 0, -398421, 201907809])

magma: E := EllipticCurve([1, 0, 0, -398421, 201907809]);

$$y^2+xy=x^3-398421x+201907809$$

## Mordell-Weil group structure

$$\Z^2$$

### Infinite order Mordell-Weil generators and heights

sage: E.gens()

magma: Generators(E);

 $$P$$ = $$\left(2250, 102303\right)$$ $$\left(-792, 4959\right)$$ $$\hat{h}(P)$$ ≈ $0.19410218863023820498837034160$ $0.84967828326253243504600717975$

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$\left(-792, 4959\right)$$, $$\left(-792, -4167\right)$$, $$\left(-522, 16623\right)$$, $$\left(-522, -16101\right)$$, $$\left(-402, 17439\right)$$, $$\left(-402, -17037\right)$$, $$\left(-198, 16623\right)$$, $$\left(-198, -16425\right)$$, $$\left(222, 11043\right)$$, $$\left(222, -11265\right)$$, $$\left(468, 10629\right)$$, $$\left(468, -11097\right)$$, $$\left(612, 13383\right)$$, $$\left(612, -13995\right)$$, $$\left(720, 16623\right)$$, $$\left(720, -17343\right)$$, $$\left(810, 19863\right)$$, $$\left(810, -20673\right)$$, $$\left(2250, 102303\right)$$, $$\left(2250, -104553\right)$$, $$\left(7320, 620457\right)$$, $$\left(7320, -627777\right)$$, $$\left(11022, 1149843\right)$$, $$\left(11022, -1160865\right)$$, $$\left(27990, 4667643\right)$$, $$\left(27990, -4695633\right)$$

## 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: $$-13569349243712955264$$ = $$-1 \cdot 2^{7} \cdot 3^{12} \cdot 11 \cdot 13^{7} \cdot 17^{2}$$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$-\frac{1449073218392281}{2811246362496}$$ = $$-1 \cdot 2^{-7} \cdot 3^{-12} \cdot 11^{-1} \cdot 13^{-1} \cdot 17^{-2} \cdot 113161^{3}$$ Endomorphism ring: $$\Z$$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $$2.3608213722390901946013097566\dots$$ Stable Faltings height: $$1.0783466935083218265745660358\dots$$

## BSD invariants

 sage: E.rank()  magma: Rank(E); Analytic rank: $$2$$ sage: E.regulator()  magma: Regulator(E); Regulator: $$0.16445394688238565910884552852\dots$$ sage: E.period_lattice().omega()  gp: E.omega[1]  magma: RealPeriod(E); Real period: $$0.19916558832237595030861795732\dots$$ 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: $$672$$  = $$7\cdot( 2^{2} \cdot 3 )\cdot1\cdot2^{2}\cdot2$$ 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.bo

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);

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

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 5870592 $$\Gamma_0(N)$$-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);

$$L^{(2)}(E,1)/2!$$ ≈ $$22.010397079619493919847983084157489833$$

## 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$$(j)_{-}$$
$$2$$ $$7$$ $$I_{7}$$ Split multiplicative -1 1 7 7
$$3$$ $$12$$ $$I_{12}$$ Split multiplicative -1 1 12 12
$$11$$ $$1$$ $$I_{1}$$ Split multiplicative -1 1 1 1
$$13$$ $$4$$ $$I_1^{*}$$ Additive 1 2 7 1
$$17$$ $$2$$ $$I_{2}$$ Non-split multiplicative 1 1 2 2

## 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.bo 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 $K$ $3$ 3.1.1144.1 $$\Z/2\Z$$ Not in database $6$ 6.0.1497193984.1 $$\Z/2\Z \times \Z/2\Z$$ Not in database $8$ Deg 8 $$\Z/3\Z$$ 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.