# Properties

 Label 22050bo2 Conductor $22050$ Discriminant $-3.163\times 10^{20}$ j-invariant $$-\frac{16591834777}{98304}$$ CM no Rank $1$ Torsion structure trivial

# Related objects

Show commands for: Magma / Pari/GP / SageMath

## Minimal Weierstrass equation

sage: E = EllipticCurve([1, -1, 0, -7844517, -8497850859]) # or

sage: E = EllipticCurve("22050bo2")

gp: E = ellinit([1, -1, 0, -7844517, -8497850859]) \\ or

gp: E = ellinit("22050bo2")

magma: E := EllipticCurve([1, -1, 0, -7844517, -8497850859]); // or

magma: E := EllipticCurve("22050bo2");

$$y^2 + x y = x^{3} - x^{2} - 7844517 x - 8497850859$$

## Mordell-Weil group structure

$$\Z$$

### Infinite order Mordell-Weil generator and height

sage: E.gens()

magma: Generators(E);

 $$P$$ = $$\left(\frac{272988195}{75076}, -\frac{2199227456091}{20570824}\right)$$ $$\hat{h}(P)$$ ≈ $18.419879049559864$

## Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

None

## Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor: $$22050$$ = $$2 \cdot 3^{2} \cdot 5^{2} \cdot 7^{2}$$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $$-316299965216256000000$$ = $$-1 \cdot 2^{15} \cdot 3^{7} \cdot 5^{6} \cdot 7^{10}$$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$-\frac{16591834777}{98304}$$ = $$-1 \cdot 2^{-15} \cdot 3^{-1} \cdot 7^{2} \cdot 17^{3} \cdot 41^{3}$$ Endomorphism ring: $$\Z$$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$

## BSD invariants

 sage: E.rank()  magma: Rank(E); Rank: $$1$$ sage: E.regulator()  magma: Regulator(E); Regulator: $$18.4198790496$$ sage: E.period_lattice().omega()  gp: E.omega[1]  magma: RealPeriod(E); Real period: $$0.0450893726182$$ 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: $$4$$  = $$1\cdot2^{2}\cdot1\cdot1$$ 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: $$1$$ (exact)

## Modular invariants

Modular form 22050.2.a.k

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);

$$q - q^{2} + q^{4} - q^{8} - 3q^{11} - 4q^{13} + q^{16} + 4q^{19} + O(q^{20})$$

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 1088640 $$\Gamma_0(N)$$-optimal: no 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);

$$L'(E,1)$$ ≈ $$3.32216316019$$

## Local data

This elliptic curve is not semistable.

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$$(j)_{-}$$
$$2$$ $$1$$ $$I_{15}$$ Non-split multiplicative 1 1 15 15
$$3$$ $$4$$ $$I_1^{*}$$ Additive -1 2 7 1
$$5$$ $$1$$ $$I_0^{*}$$ Additive 1 2 6 0
$$7$$ $$1$$ $$II^{*}$$ Additive -1 2 10 0

## 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 $$p$$ except those listed.

prime Image of Galois representation
$$3$$ B

## $p$-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 nonsplit add add add ordinary ordinary ss ordinary ss ordinary ordinary ordinary ss ordinary ordinary 2 - - - 3 1 1,1 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

An entry - indicates that the invariants are not computed because the reduction is additive.

## Isogenies

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 3.
Its isogeny class 22050bo 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
$2$ $$\Q(\sqrt{-35})$$ $$\Z/3\Z$$ Not in database
$3$ 3.1.1176.1 $$\Z/2\Z$$ Not in database
$6$ 6.0.33191424.2 $$\Z/2\Z \times \Z/2\Z$$ Not in database
$6$ 6.0.1210104000.1 $$\Z/6\Z$$ Not in database
$6$ 6.2.372163703625.2 $$\Z/3\Z$$ 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.