# Properties

 Label 22386.v2 Conductor 22386 Discriminant -81108488685750967074816 j-invariant $$-\frac{4128223528775369483123266513}{81108488685750967074816}$$ CM no Rank 1 Torsion Structure $$\Z/{3}\Z$$

# Related objects

Show commands for: Magma / SageMath / Pari/GP

## Minimal Weierstrass equation

magma: E := EllipticCurve([1, 0, 0, -33420517, -75619489279]); // or

magma: E := EllipticCurve("22386bb1");

sage: E = EllipticCurve([1, 0, 0, -33420517, -75619489279]) # or

sage: E = EllipticCurve("22386bb1")

gp: E = ellinit([1, 0, 0, -33420517, -75619489279]) \\ or

gp: E = ellinit("22386bb1")

$$y^2 + x y = x^{3} - 33420517 x - 75619489279$$

## Mordell-Weil group structure

$$\Z\times \Z/{3}\Z$$

### Infinite order Mordell-Weil generator and height

magma: Generators(E);

sage: E.gens()

 $$P$$ = $$\left(7010, 182471\right)$$ $$\hat{h}(P)$$ ≈ 1.16207864859

## Torsion generators

magma: TorsionSubgroup(E);

sage: E.torsion_subgroup().gens()

gp: elltors(E)

$$\left(10034, 769127\right)$$

## Integral points

magma: IntegralPoints(E);

sage: E.integral_points()

$$\left(7010, 182471\right)$$, $$\left(10034, 769127\right)$$, $$\left(20786, 2855015\right)$$, $$\left(21266, 2961095\right)$$, $$\left(79922, 22493543\right)$$, $$\left(83762, 24140903\right)$$

Note: only one of each pair $\pm P$ is listed.

## 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: $$-81108488685750967074816$$ = $$-1 \cdot 2^{39} \cdot 3^{9} \cdot 7^{3} \cdot 13 \cdot 41^{2}$$ magma: jInvariant(E);  sage: E.j_invariant().factor()  gp: E.j j-invariant: $$-\frac{4128223528775369483123266513}{81108488685750967074816}$$ = $$-1 \cdot 2^{-39} \cdot 3^{-9} \cdot 7^{-3} \cdot 13^{-1} \cdot 41^{-2} \cdot 347^{3} \cdot 1087^{3} \cdot 4253^{3}$$ 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: $$1.16207864859$$ magma: RealPeriod(E);  sage: E.period_lattice().omega()  gp: E.omega[1] Real period: $$0.0313588808906$$ 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: $$2106$$  = $$( 3 \cdot 13 )\cdot3^{2}\cdot3\cdot1\cdot2$$ magma: Order(TorsionSubgroup(E));  sage: E.torsion_order()  gp: elltors(E)[1] Torsion order: $$3$$ magma: MordellWeilShaInformation(E);  sage: E.sha().an_numerical() Analytic order of Ш: $$1$$ (exact)

## Modular invariants

#### Modular form 22386.2.a.v

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)

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

 magma: ModularDegree(E);  sage: E.modular_degree() Modular degree: 3133728 $$\Gamma_0(N)$$-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])

$$L'(E,1)$$ ≈ $$8.52730770684$$

## 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$$(j)_{-}$$
$$2$$ $$39$$ $$I_{39}$$ Split multiplicative -1 1 39 39
$$3$$ $$9$$ $$I_{9}$$ Split multiplicative -1 1 9 9
$$7$$ $$3$$ $$I_{3}$$ Split multiplicative -1 1 3 3
$$13$$ $$1$$ $$I_{1}$$ Split multiplicative -1 1 1 1
$$41$$ $$2$$ $$I_{2}$$ Non-split multiplicative 1 1 2 2

## 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.1.1

## $p$-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 split ordinary split ordinary ordinary ordinary ordinary ordinary ordinary nonsplit ordinary ss 4 2 1 2 1 2 1 1 1 1 1 1 1 1 1,1 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 $$d=$$ 3.
Its isogeny class 22386.v 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 $E(\Q)_{\rm tors}$ $\cong \Z/{3}\Z$ are as follows:

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base-change curve
3 3.1.2184.1 $$\Z/6\Z$$ Not in database
6 6.0.10417365504.1 $$\Z/2\Z \times \Z/6\Z$$ Not in database
6.0.2179077117867.1 $$\Z/3\Z \times \Z/3\Z$$ 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.