# Properties

 Label 22386.u1 Conductor 22386 Discriminant 21429653098766238144 j-invariant $$\frac{120986373702456846135875233}{21429653098766238144}$$ CM no Rank 1 Torsion Structure $$\mathrm{Trivial}$$

# Related objects

Show commands for: Magma / SageMath / Pari/GP

## Minimal Weierstrass equation

magma: E := EllipticCurve([1, 0, 0, -10303962, -12729674844]); // or

magma: E := EllipticCurve("22386bc2");

sage: E = EllipticCurve([1, 0, 0, -10303962, -12729674844]) # or

sage: E = EllipticCurve("22386bc2")

gp: E = ellinit([1, 0, 0, -10303962, -12729674844]) \\ or

gp: E = ellinit("22386bc2")

$$y^2 + x y = x^{3} - 10303962 x - 12729674844$$

## Mordell-Weil group structure

$$\Z$$

### Infinite order Mordell-Weil generator and height

magma: Generators(E);

sage: E.gens()

 $$P$$ = $$\left(-1866, 1962\right)$$ $$\hat{h}(P)$$ ≈ 0.435839725398

## Integral points

magma: IntegralPoints(E);

sage: E.integral_points()

$$\left(-1866, 1962\right)$$, $$\left(-1866, -96\right)$$, $$\left(3744, 32442\right)$$, $$\left(3744, -36186\right)$$, $$\left(8424, 701682\right)$$, $$\left(8424, -710106\right)$$

## 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: $$21429653098766238144$$ = $$2^{6} \cdot 3^{3} \cdot 7^{12} \cdot 13 \cdot 41^{3}$$ magma: jInvariant(E);  sage: E.j_invariant().factor()  gp: E.j j-invariant: $$\frac{120986373702456846135875233}{21429653098766238144}$$ = $$2^{-6} \cdot 3^{-3} \cdot 7^{-12} \cdot 13^{-1} \cdot 41^{-3} \cdot 89^{3} \cdot 919^{3} \cdot 6047^{3}$$ 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: $$0.435839725398$$ magma: RealPeriod(E);  sage: E.period_lattice().omega()  gp: E.omega[1] Real period: $$0.0842661415116$$ 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: $$216$$  = $$( 2 \cdot 3 )\cdot3\cdot( 2^{2} \cdot 3 )\cdot1\cdot1$$ magma: Order(TorsionSubgroup(E));  sage: E.torsion_order()  gp: elltors(E)[1] Torsion order: $$1$$ magma: MordellWeilShaInformation(E);  sage: E.sha().an_numerical() Analytic order of Ш: $$1$$ (exact)

## Modular invariants

#### Modular form 22386.2.a.u

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)

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

 magma: ModularDegree(E);  sage: E.modular_degree() Modular degree: 1306368 $$\Gamma_0(N)$$-optimal: no 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])

$$L'(E,1)$$ ≈ $$7.93293090699$$

## 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$$(j)_{-}$$
$$2$$ $$6$$ $$I_{6}$$ Split multiplicative -1 1 6 6
$$3$$ $$3$$ $$I_{3}$$ Split multiplicative -1 1 3 3
$$7$$ $$12$$ $$I_{12}$$ Split multiplicative -1 1 12 12
$$13$$ $$1$$ $$I_{1}$$ Split multiplicative -1 1 1 1
$$41$$ $$1$$ $$I_{3}$$ Non-split multiplicative 1 1 3 3

## Galois representations

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

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
$$3$$ B.1.2

## $p$-adic data

### $p$-adic regulators

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

$$p$$-adic regulators are not yet computed for curves that are not $$\Gamma_0$$-optimal.

## 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 split ordinary split ordinary ordinary ordinary ordinary ordinary ordinary nonsplit ordinary ordinary 4 2 1 4 1 2 1 1 1 1 1 1 1 1 1 0 1 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 $$d=$$ 3.
Its isogeny class 22386.u consists of 2 curves linked by isogenies of degree 3.

## 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 $E(\Q)_{\rm tors}$ (which is trivial) are as follows:

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base-change curve
2 $$\Q(\sqrt{-3})$$ $$\Z/3\Z$$ Not in database
3 3.3.6396.1 $$\Z/2\Z$$ Not in database
3.1.4563.1 $$\Z/3\Z$$ Not in database
6 6.0.122726448.1 $$\Z/6\Z$$ Not in database
6.0.62462907.1 $$\Z/3\Z \times \Z/3\Z$$ Not in database
6.6.261652787136.1 $$\Z/2\Z \times \Z/2\Z$$ 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.