# Properties

 Label 22386.w2 Conductor 22386 Discriminant -40036908053495808 j-invariant $$-\frac{19645130164017251655217}{40036908053495808}$$ CM no Rank 1 Torsion Structure $$\Z/{2}\Z$$

# Related objects

Show commands for: Magma / SageMath / Pari/GP

## Minimal Weierstrass equation

magma: E := EllipticCurve([1, 0, 0, -562139, 162462033]); // or

magma: E := EllipticCurve("22386v1");

sage: E = EllipticCurve([1, 0, 0, -562139, 162462033]) # or

sage: E = EllipticCurve("22386v1")

gp: E = ellinit([1, 0, 0, -562139, 162462033]) \\ or

gp: E = ellinit("22386v1")

$$y^2 + x y = x^{3} - 562139 x + 162462033$$

## Mordell-Weil group structure

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

### Infinite order Mordell-Weil generator and height

magma: Generators(E);

sage: E.gens()

 $$P$$ = $$\left(382, 1681\right)$$ $$\hat{h}(P)$$ ≈ 0.485392274793

## Torsion generators

magma: TorsionSubgroup(E);

sage: E.torsion_subgroup().gens()

gp: elltors(E)

$$\left(-866, 433\right)$$

## Integral points

magma: IntegralPoints(E);

sage: E.integral_points()

$$\left(-866, 433\right)$$, $$\left(-738, 13617\right)$$, $$\left(-738, -12879\right)$$, $$\left(382, 1681\right)$$, $$\left(382, -2063\right)$$, $$\left(486, 1785\right)$$, $$\left(486, -2271\right)$$, $$\left(538, 3709\right)$$, $$\left(538, -4247\right)$$, $$\left(12316, 1358179\right)$$, $$\left(12316, -1370495\right)$$

## Invariants

 magma: Conductor(E);  sage: E.conductor().factor()  gp: ellglobalred(E)[1] Conductor: $$22386$$ = $$2 \cdot 3 \cdot 7 \cdot 13 \cdot 41$$ magma: Discriminant(E);  sage: E.discriminant().factor()  gp: E.disc Discriminant: $$-40036908053495808$$ = $$-1 \cdot 2^{16} \cdot 3^{2} \cdot 7^{3} \cdot 13^{6} \cdot 41$$ magma: jInvariant(E);  sage: E.j_invariant().factor()  gp: E.j j-invariant: $$-\frac{19645130164017251655217}{40036908053495808}$$ = $$-1 \cdot 2^{-16} \cdot 3^{-2} \cdot 7^{-3} \cdot 13^{-6} \cdot 29^{3} \cdot 41^{-1} \cdot 930437^{3}$$ Endomorphism ring: $$\Z$$ (no Complex Multiplication) Sato-Tate Group: $\mathrm{SU}(2)$

## BSD invariants

 magma: Rank(E);  sage: E.rank() Rank: $$1$$ magma: Regulator(E);  sage: E.regulator() Regulator: $$0.485392274793$$ magma: RealPeriod(E);  sage: E.period_lattice().omega()  gp: E.omega[1] Real period: $$0.363617255782$$ magma: TamagawaNumbers(E);  sage: E.tamagawa_numbers()  gp: gr=ellglobalred(E); [[gr[4][i,1],gr[5][i][4]] | i<-[1..#gr[4][,1]]] Tamagawa product: $$192$$  = $$2^{4}\cdot2\cdot1\cdot( 2 \cdot 3 )\cdot1$$ magma: Order(TorsionSubgroup(E));  sage: E.torsion_order()  gp: elltors(E)[1] Torsion order: $$2$$ magma: MordellWeilShaInformation(E);  sage: E.sha().an_numerical() Analytic order of Ш: $$1$$ (exact)

## Modular invariants

#### Modular form 22386.2.a.w

magma: ModularForm(E);

sage: E.q_eigenform(20)

gp: xy = elltaniyama(E);

gp: x*deriv(xy[1])/(2*xy[2]+E.a1*xy[1]+E.a3)

$$q + q^{2} + q^{3} + q^{4} - 2q^{5} + q^{6} - q^{7} + q^{8} + q^{9} - 2q^{10} + 2q^{11} + q^{12} + q^{13} - q^{14} - 2q^{15} + q^{16} + 2q^{17} + q^{18} - 4q^{19} + O(q^{20})$$

 magma: ModularDegree(E);  sage: E.modular_degree() Modular degree: 276480 $$\Gamma_0(N)$$-optimal: yes Manin constant: 1

#### Special L-value

magma: Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12);

sage: r = E.rank();

sage: E.lseries().dokchitser().derivative(1,r)/r.factorial()

gp: ar = ellanalyticrank(E);

gp: ar[2]/factorial(ar[1])

$$L'(E,1)$$ ≈ $$8.47185633303$$

## Local data

magma: [LocalInformation(E,p) : p in BadPrimes(E)];

sage: E.local_data()

gp: ellglobalred(E)[5]

prime Tamagawa number Kodaira symbol Reduction type Root number ord($$N$$) ord($$\Delta$$) ord$$(j)_{-}$$
$$2$$ $$16$$ $$I_{16}$$ Split multiplicative -1 1 16 16
$$3$$ $$2$$ $$I_{2}$$ Split multiplicative -1 1 2 2
$$7$$ $$1$$ $$I_{3}$$ Non-split multiplicative 1 1 3 3
$$13$$ $$6$$ $$I_{6}$$ Split multiplicative -1 1 6 6
$$41$$ $$1$$ $$I_{1}$$ Split multiplicative -1 1 1 1

## 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 X6.

This subgroup is the pull-back of the subgroup of $\GL(2,\Z_2/2^1\Z_2)$ generated by $\left(\begin{array}{rr} 1 & 1 \\ 0 & 1 \end{array}\right)$ and has index 3.

magma: [GaloisRepresentation(E,p): p in PrimesUpTo(20)];

sage: rho = E.galois_representation();

sage: [rho.image_type(p) for p in rho.non_surjective()]

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

prime Image of Galois representation
$$2$$ B

## $p$-adic data

### $p$-adic regulators

sage: [E.padic_regulator(p) for p in primes(3,20) if E.conductor().valuation(p)<2]

Note: $$p$$-adic regulator data only exists for primes $$p\ge5$$ of good ordinary reduction.

## 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 split split ordinary nonsplit ordinary split ordinary ordinary ss ss ordinary ordinary split ordinary ss 4 4 1 3 1 2 1 1 1,1 1,1 1 1 2 1 1,1 0 0 0 0 0 0 0 0 0,0 0,0 0 0 0 0 0,0

## Isogenies

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 2.
Its isogeny class 22386.w consists of 2 curves linked by isogenies of degree 2.

## Growth of torsion in number fields

The number fields $K$ of degree up to 7 such that $E(K)_{\rm tors}$ is strictly larger than $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z$ are as follows:

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base-change curve
2 $$\Q(\sqrt{-287})$$ $$\Z/2\Z \times \Z/2\Z$$ Not in database
4 4.2.6984432.6 $$\Z/4\Z$$ Not in database

We only show fields where the torsion growth is primitive. For each field $K$ we either show its label, or a defining polynomial when $K$ is not in the database.