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Properties

Label 20.0.310...401.1
Degree $20$
Signature $[0, 10]$
Discriminant $3.106\times 10^{21}$
Root discriminant \(11.87\)
Ramified primes $3,47$
Class number $1$
Class group trivial
Galois group $D_{10}$ (as 20T4)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^20 - x^19 - 5*x^18 + 22*x^16 + x^15 - 38*x^14 - 11*x^13 + 46*x^12 + 11*x^11 - 51*x^10 + 11*x^9 + 46*x^8 - 11*x^7 - 38*x^6 + x^5 + 22*x^4 - 5*x^2 - x + 1)
 
gp: K = bnfinit(y^20 - y^19 - 5*y^18 + 22*y^16 + y^15 - 38*y^14 - 11*y^13 + 46*y^12 + 11*y^11 - 51*y^10 + 11*y^9 + 46*y^8 - 11*y^7 - 38*y^6 + y^5 + 22*y^4 - 5*y^2 - y + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 - x^19 - 5*x^18 + 22*x^16 + x^15 - 38*x^14 - 11*x^13 + 46*x^12 + 11*x^11 - 51*x^10 + 11*x^9 + 46*x^8 - 11*x^7 - 38*x^6 + x^5 + 22*x^4 - 5*x^2 - x + 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^20 - x^19 - 5*x^18 + 22*x^16 + x^15 - 38*x^14 - 11*x^13 + 46*x^12 + 11*x^11 - 51*x^10 + 11*x^9 + 46*x^8 - 11*x^7 - 38*x^6 + x^5 + 22*x^4 - 5*x^2 - x + 1)
 

\( x^{20} - x^{19} - 5 x^{18} + 22 x^{16} + x^{15} - 38 x^{14} - 11 x^{13} + 46 x^{12} + 11 x^{11} + \cdots + 1 \) Copy content Toggle raw display

sage: K.defining_polynomial()
 
gp: K.pol
 
magma: DefiningPolynomial(K);
 
oscar: defining_polynomial(K)
 

Invariants

Degree:  $20$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[0, 10]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(3105926159393528563401\) \(\medspace = 3^{10}\cdot 47^{10}\) Copy content Toggle raw display
sage: K.disc()
 
gp: K.disc
 
magma: OK := Integers(K); Discriminant(OK);
 
oscar: OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(11.87\)
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(OK))^(1/Degree(K));
 
oscar: (1.0 * dK)^(1/degree(K))
 
Galois root discriminant:  $3^{1/2}47^{1/2}\approx 11.874342087037917$
Ramified primes:   \(3\), \(47\) Copy content Toggle raw display
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(OK));
 
oscar: prime_divisors(discriminant((OK)))
 
Discriminant root field:  \(\Q\)
$\card{ \Gal(K/\Q) }$:  $20$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{5}a^{16}-\frac{2}{5}a^{14}-\frac{1}{5}a^{13}-\frac{1}{5}a^{12}-\frac{1}{5}a^{9}+\frac{1}{5}a^{8}-\frac{1}{5}a^{7}-\frac{1}{5}a^{4}-\frac{1}{5}a^{3}-\frac{2}{5}a^{2}+\frac{1}{5}$, $\frac{1}{5}a^{17}-\frac{2}{5}a^{15}-\frac{1}{5}a^{14}-\frac{1}{5}a^{13}-\frac{1}{5}a^{10}+\frac{1}{5}a^{9}-\frac{1}{5}a^{8}-\frac{1}{5}a^{5}-\frac{1}{5}a^{4}-\frac{2}{5}a^{3}+\frac{1}{5}a$, $\frac{1}{158455}a^{18}-\frac{7114}{158455}a^{17}-\frac{8651}{158455}a^{16}+\frac{61137}{158455}a^{15}+\frac{68476}{158455}a^{14}+\frac{23128}{158455}a^{13}-\frac{38306}{158455}a^{12}+\frac{5754}{14405}a^{11}-\frac{243}{31691}a^{10}-\frac{72631}{158455}a^{9}-\frac{243}{31691}a^{8}+\frac{5754}{14405}a^{7}-\frac{38306}{158455}a^{6}+\frac{23128}{158455}a^{5}+\frac{68476}{158455}a^{4}+\frac{61137}{158455}a^{3}-\frac{8651}{158455}a^{2}-\frac{7114}{158455}a+\frac{1}{158455}$, $\frac{1}{158455}a^{19}-\frac{1424}{31691}a^{17}-\frac{1537}{158455}a^{16}+\frac{13962}{31691}a^{15}+\frac{1468}{31691}a^{14}-\frac{4126}{14405}a^{13}-\frac{12289}{31691}a^{12}-\frac{56809}{158455}a^{11}-\frac{64498}{158455}a^{10}-\frac{6497}{14405}a^{9}+\frac{71427}{158455}a^{8}+\frac{12911}{31691}a^{7}+\frac{56844}{158455}a^{6}+\frac{61396}{158455}a^{5}+\frac{45349}{158455}a^{4}-\frac{7317}{158455}a^{3}-\frac{69788}{158455}a^{2}+\frac{1532}{158455}a+\frac{7114}{158455}$ Copy content Toggle raw display

sage: K.integral_basis()
 
gp: K.zk
 
magma: IntegralBasis(K);
 
oscar: basis(OK)
 

Monogenic:  Not computed
Index:  $1$
Inessential primes:  None

Class group and class number

Trivial group, which has order $1$

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 
oscar: class_group(K)
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, fUK := UnitGroup(K);
 
oscar: UK, fUK = unit_group(OK)
 
Rank:  $9$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
oscar: rank(UK)
 
Torsion generator:   \( -\frac{32919}{31691} a^{19} + \frac{45216}{158455} a^{18} + \frac{867594}{158455} a^{17} + \frac{625479}{158455} a^{16} - \frac{3248754}{158455} a^{15} - \frac{230742}{14405} a^{14} + \frac{943475}{31691} a^{13} + \frac{5405924}{158455} a^{12} - \frac{4343186}{158455} a^{11} - \frac{5238968}{158455} a^{10} + \frac{5266152}{158455} a^{9} + \frac{37736}{2365} a^{8} - \frac{99718}{2365} a^{7} - \frac{3087386}{158455} a^{6} + \frac{984228}{31691} a^{5} + \frac{3353628}{158455} a^{4} - \frac{1884874}{158455} a^{3} - \frac{149566}{14405} a^{2} + \frac{192919}{158455} a + \frac{415746}{158455} \)  (order $6$) Copy content Toggle raw display
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
oscar: torsion_units_generator(OK)
 
Fundamental units:   $\frac{79389}{158455}a^{19}-\frac{61071}{158455}a^{18}-\frac{382121}{158455}a^{17}-\frac{101056}{158455}a^{16}+\frac{1585998}{158455}a^{15}+\frac{376317}{158455}a^{14}-\frac{2431643}{158455}a^{13}-\frac{240940}{31691}a^{12}+\frac{542419}{31691}a^{11}+\frac{1195113}{158455}a^{10}-\frac{3237738}{158455}a^{9}+\frac{281944}{158455}a^{8}+\frac{558145}{31691}a^{7}+\frac{104477}{158455}a^{6}-\frac{2267799}{158455}a^{5}-\frac{504786}{158455}a^{4}+\frac{1378559}{158455}a^{3}+\frac{269322}{158455}a^{2}-\frac{94008}{158455}a-\frac{306564}{158455}$, $\frac{121917}{158455}a^{19}-\frac{2443}{31691}a^{18}-\frac{697273}{158455}a^{17}-\frac{522122}{158455}a^{16}+\frac{2509286}{158455}a^{15}+\frac{2311679}{158455}a^{14}-\frac{3989301}{158455}a^{13}-\frac{4422977}{158455}a^{12}+\frac{3378487}{158455}a^{11}+\frac{4404977}{158455}a^{10}-\frac{101333}{3685}a^{9}-\frac{1842426}{158455}a^{8}+\frac{5974183}{158455}a^{7}+\frac{1953703}{158455}a^{6}-\frac{809525}{31691}a^{5}-\frac{2610266}{158455}a^{4}+\frac{256174}{31691}a^{3}+\frac{172506}{31691}a^{2}-\frac{72504}{158455}a-\frac{2793}{31691}$, $\frac{18105}{31691}a^{19}-\frac{106041}{158455}a^{18}-\frac{478697}{158455}a^{17}+\frac{82952}{158455}a^{16}+\frac{434098}{31691}a^{15}-\frac{43147}{158455}a^{14}-\frac{4057543}{158455}a^{13}-\frac{206512}{31691}a^{12}+\frac{4695331}{158455}a^{11}+\frac{1383621}{158455}a^{10}-\frac{4807871}{158455}a^{9}+\frac{1202912}{158455}a^{8}+\frac{869081}{31691}a^{7}-\frac{1436499}{158455}a^{6}-\frac{3815667}{158455}a^{5}-\frac{413966}{158455}a^{4}+\frac{4358}{335}a^{3}+\frac{360374}{158455}a^{2}-\frac{182552}{158455}a-\frac{39815}{31691}$, $\frac{5481}{14405}a^{19}-\frac{17642}{31691}a^{18}-\frac{58019}{31691}a^{17}+\frac{1633}{2365}a^{16}+\frac{6613}{737}a^{15}-\frac{340836}{158455}a^{14}-\frac{2492584}{158455}a^{13}-\frac{395193}{158455}a^{12}+\frac{281881}{14405}a^{11}+\frac{371717}{158455}a^{10}-\frac{585570}{31691}a^{9}+\frac{235819}{31691}a^{8}+\frac{217612}{14405}a^{7}-\frac{24952}{3685}a^{6}-\frac{2272144}{158455}a^{5}+\frac{95556}{158455}a^{4}+\frac{224948}{31691}a^{3}+\frac{137956}{158455}a^{2}-\frac{127768}{158455}a-\frac{179103}{158455}$, $\frac{1383}{31691}a^{19}+\frac{43027}{158455}a^{18}+\frac{4461}{31691}a^{17}-\frac{249489}{158455}a^{16}-\frac{35872}{14405}a^{15}+\frac{562138}{158455}a^{14}+\frac{327816}{31691}a^{13}+\frac{9142}{31691}a^{12}-\frac{2416262}{158455}a^{11}-\frac{1144758}{158455}a^{10}+\frac{1566863}{158455}a^{9}+\frac{68545}{31691}a^{8}-\frac{301245}{31691}a^{7}+\frac{798328}{158455}a^{6}+\frac{2682113}{158455}a^{5}+\frac{41286}{14405}a^{4}-\frac{282814}{31691}a^{3}-\frac{928293}{158455}a^{2}+\frac{54361}{31691}a+\frac{42072}{31691}$, $\frac{87209}{158455}a^{19}-\frac{314}{3685}a^{18}-\frac{420493}{158455}a^{17}-\frac{438201}{158455}a^{16}+\frac{1470533}{158455}a^{15}+\frac{1594074}{158455}a^{14}-\frac{1374084}{158455}a^{13}-\frac{3242878}{158455}a^{12}+\frac{267086}{158455}a^{11}+\frac{2680544}{158455}a^{10}-\frac{576377}{158455}a^{9}-\frac{1257601}{158455}a^{8}+\frac{1566077}{158455}a^{7}+\frac{2747948}{158455}a^{6}-\frac{789176}{158455}a^{5}-\frac{535495}{31691}a^{4}-\frac{100326}{31691}a^{3}+\frac{122938}{31691}a^{2}+\frac{99959}{31691}a+\frac{597}{2365}$, $\frac{4484}{31691}a^{19}-\frac{101161}{158455}a^{18}-\frac{46089}{158455}a^{17}+\frac{9242}{3685}a^{16}+\frac{564269}{158455}a^{15}-\frac{1643278}{158455}a^{14}-\frac{238442}{31691}a^{13}+\frac{222176}{14405}a^{12}+\frac{2187186}{158455}a^{11}-\frac{2633957}{158455}a^{10}-\frac{1965017}{158455}a^{9}+\frac{325248}{14405}a^{8}-\frac{39564}{158455}a^{7}-\frac{3440854}{158455}a^{6}-\frac{25436}{31691}a^{5}+\frac{34461}{2365}a^{4}+\frac{437889}{158455}a^{3}-\frac{857719}{158455}a^{2}+\frac{141286}{158455}a-\frac{1001}{14405}$, $\frac{182152}{158455}a^{19}-\frac{238}{737}a^{18}-\frac{834859}{158455}a^{17}-\frac{138950}{31691}a^{16}+\frac{2961408}{158455}a^{15}+\frac{200086}{14405}a^{14}-\frac{3047967}{158455}a^{13}-\frac{3760729}{158455}a^{12}+\frac{2156942}{158455}a^{11}+\frac{1974783}{158455}a^{10}-\frac{4076662}{158455}a^{9}+\frac{300027}{158455}a^{8}+\frac{4253006}{158455}a^{7}+\frac{2095898}{158455}a^{6}-\frac{1291494}{158455}a^{5}-\frac{1275702}{158455}a^{4}+\frac{16472}{31691}a^{3}-\frac{25139}{14405}a^{2}+\frac{52000}{31691}a+\frac{59932}{158455}$, $\frac{13892}{158455}a^{19}-\frac{2038}{3685}a^{18}-\frac{32196}{158455}a^{17}+\frac{334272}{158455}a^{16}+\frac{500196}{158455}a^{15}-\frac{111526}{14405}a^{14}-\frac{1120299}{158455}a^{13}+\frac{1246232}{158455}a^{12}+\frac{1669156}{158455}a^{11}-\frac{836989}{158455}a^{10}-\frac{1051869}{158455}a^{9}+\frac{1881993}{158455}a^{8}-\frac{268198}{158455}a^{7}-\frac{1608593}{158455}a^{6}-\frac{780173}{158455}a^{5}+\frac{311659}{158455}a^{4}+\frac{155630}{31691}a^{3}+\frac{3409}{14405}a^{2}+\frac{210953}{158455}a-\frac{230939}{158455}$ Copy content Toggle raw display
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K|fUK(g): g in Generators(UK)];
 
oscar: [K(fUK(a)) for a in gens(UK)]
 
Regulator:  \( 630.365304677 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{10}\cdot 630.365304677 \cdot 1}{6\cdot\sqrt{3105926159393528563401}}\cr\approx \mathstrut & 0.180777414796 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^20 - x^19 - 5*x^18 + 22*x^16 + x^15 - 38*x^14 - 11*x^13 + 46*x^12 + 11*x^11 - 51*x^10 + 11*x^9 + 46*x^8 - 11*x^7 - 38*x^6 + x^5 + 22*x^4 - 5*x^2 - x + 1)
 
DK = K.disc(); r1,r2 = K.signature(); RK = K.regulator(); RR = RK.parent()
 
hK = K.class_number(); wK = K.unit_group().torsion_generator().order();
 
2^r1 * (2*RR(pi))^r2 * RK * hK / (wK * RR(sqrt(abs(DK))))
 
# self-contained Pari/GP code snippet to compute the analytic class number formula
 
K = bnfinit(x^20 - x^19 - 5*x^18 + 22*x^16 + x^15 - 38*x^14 - 11*x^13 + 46*x^12 + 11*x^11 - 51*x^10 + 11*x^9 + 46*x^8 - 11*x^7 - 38*x^6 + x^5 + 22*x^4 - 5*x^2 - x + 1, 1);
 
[polcoeff (lfunrootres (lfuncreate (K))[1][1][2], -1), 2^K.r1 * (2*Pi)^K.r2 * K.reg * K.no / (K.tu[1] * sqrt (abs (K.disc)))]
 
/* self-contained Magma code snippet to compute the analytic class number formula */
 
Qx<x> := PolynomialRing(QQ); K<a> := NumberField(x^20 - x^19 - 5*x^18 + 22*x^16 + x^15 - 38*x^14 - 11*x^13 + 46*x^12 + 11*x^11 - 51*x^10 + 11*x^9 + 46*x^8 - 11*x^7 - 38*x^6 + x^5 + 22*x^4 - 5*x^2 - x + 1);
 
OK := Integers(K); DK := Discriminant(OK);
 
UK, fUK := UnitGroup(OK); clK, fclK := ClassGroup(OK);
 
r1,r2 := Signature(K); RK := Regulator(K); RR := Parent(RK);
 
hK := #clK; wK := #TorsionSubgroup(UK);
 
2^r1 * (2*Pi(RR))^r2 * RK * hK / (wK * Sqrt(RR!Abs(DK)));
 
# self-contained Oscar code snippet to compute the analytic class number formula
 
Qx, x = PolynomialRing(QQ); K, a = NumberField(x^20 - x^19 - 5*x^18 + 22*x^16 + x^15 - 38*x^14 - 11*x^13 + 46*x^12 + 11*x^11 - 51*x^10 + 11*x^9 + 46*x^8 - 11*x^7 - 38*x^6 + x^5 + 22*x^4 - 5*x^2 - x + 1);
 
OK = ring_of_integers(K); DK = discriminant(OK);
 
UK, fUK = unit_group(OK); clK, fclK = class_group(OK);
 
r1,r2 = signature(K); RK = regulator(K); RR = parent(RK);
 
hK = order(clK); wK = torsion_units_order(K);
 
2^r1 * (2*pi)^r2 * RK * hK / (wK * sqrt(RR(abs(DK))))
 

Galois group

$D_{10}$ (as 20T4):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: G = GaloisGroup(K);
 
oscar: G, Gtx = galois_group(K); G, transitive_group_identification(G)
 
A solvable group of order 20
The 8 conjugacy class representatives for $D_{10}$
Character table for $D_{10}$

Intermediate fields

\(\Q(\sqrt{-47}) \), \(\Q(\sqrt{141}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-3}, \sqrt{-47})\), 5.1.2209.1 x5, 10.0.229345007.1, 10.2.55730836701.1 x5, 10.0.1185762483.1 x5

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

sage: K.subfields()[1:-1]
 
gp: L = nfsubfields(K); L[2..length(b)]
 
magma: L := Subfields(K); L[2..#L];
 
oscar: subfields(K)[2:end-1]
 

Sibling fields

Degree 10 siblings: 10.2.55730836701.1, 10.0.1185762483.1
Minimal sibling: 10.0.1185762483.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/padicField/2.10.0.1}{10} }^{2}$ R ${\href{/padicField/5.2.0.1}{2} }^{10}$ ${\href{/padicField/7.5.0.1}{5} }^{4}$ ${\href{/padicField/11.2.0.1}{2} }^{10}$ ${\href{/padicField/13.2.0.1}{2} }^{10}$ ${\href{/padicField/17.10.0.1}{10} }^{2}$ ${\href{/padicField/19.2.0.1}{2} }^{10}$ ${\href{/padicField/23.2.0.1}{2} }^{10}$ ${\href{/padicField/29.2.0.1}{2} }^{10}$ ${\href{/padicField/31.2.0.1}{2} }^{10}$ ${\href{/padicField/37.5.0.1}{5} }^{4}$ ${\href{/padicField/41.2.0.1}{2} }^{10}$ ${\href{/padicField/43.2.0.1}{2} }^{10}$ R ${\href{/padicField/53.10.0.1}{10} }^{2}$ ${\href{/padicField/59.10.0.1}{10} }^{2}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Sage:
 
p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
\\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Pari:
 
p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
 
// to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7 in Magma:
 
p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];
 
# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Oscar:
 
p = 7; pfac = factor(ideal(ring_of_integers(K), p)); [(e, valuation(norm(pr),p)) for (pr,e) in pfac]
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
\(3\) Copy content Toggle raw display 3.10.5.2$x^{10} + 15 x^{8} + 94 x^{6} + 2 x^{5} + 210 x^{4} - 60 x^{3} + 229 x^{2} + 94 x + 364$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
3.10.5.2$x^{10} + 15 x^{8} + 94 x^{6} + 2 x^{5} + 210 x^{4} - 60 x^{3} + 229 x^{2} + 94 x + 364$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
\(47\) Copy content Toggle raw display 47.4.2.1$x^{4} + 90 x^{3} + 2129 x^{2} + 4680 x + 96939$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
47.4.2.1$x^{4} + 90 x^{3} + 2129 x^{2} + 4680 x + 96939$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
47.4.2.1$x^{4} + 90 x^{3} + 2129 x^{2} + 4680 x + 96939$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
47.4.2.1$x^{4} + 90 x^{3} + 2129 x^{2} + 4680 x + 96939$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
47.4.2.1$x^{4} + 90 x^{3} + 2129 x^{2} + 4680 x + 96939$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$