Properties

Label 15.3.726...941.2
Degree $15$
Signature $[3, 6]$
Discriminant $7.269\times 10^{18}$
Root discriminant \(18.09\)
Ramified primes $3,47$
Class number $1$
Class group trivial
Galois group $C_3^4:D_{10}$ (as 15T43)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^15 + 6*x^13 - 4*x^12 + 9*x^11 + 3*x^10 - 18*x^9 + 45*x^8 - 126*x^7 + 216*x^6 - 324*x^5 + 243*x^4 - 205*x^3 + 90*x^2 - 30*x + 11)
 
gp: K = bnfinit(y^15 + 6*y^13 - 4*y^12 + 9*y^11 + 3*y^10 - 18*y^9 + 45*y^8 - 126*y^7 + 216*y^6 - 324*y^5 + 243*y^4 - 205*y^3 + 90*y^2 - 30*y + 11, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^15 + 6*x^13 - 4*x^12 + 9*x^11 + 3*x^10 - 18*x^9 + 45*x^8 - 126*x^7 + 216*x^6 - 324*x^5 + 243*x^4 - 205*x^3 + 90*x^2 - 30*x + 11);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^15 + 6*x^13 - 4*x^12 + 9*x^11 + 3*x^10 - 18*x^9 + 45*x^8 - 126*x^7 + 216*x^6 - 324*x^5 + 243*x^4 - 205*x^3 + 90*x^2 - 30*x + 11)
 

\( x^{15} + 6 x^{13} - 4 x^{12} + 9 x^{11} + 3 x^{10} - 18 x^{9} + 45 x^{8} - 126 x^{7} + 216 x^{6} - 324 x^{5} + 243 x^{4} - 205 x^{3} + 90 x^{2} - 30 x + 11 \) Copy content Toggle raw display

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

Invariants

Degree:  $15$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[3, 6]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(7269488039573383941\) \(\medspace = 3^{15}\cdot 47^{7}\) 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:  \(18.09\)
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:  $3^{241/162}47^{1/2}\approx 35.143128879205776$
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:  \(\Q(\sqrt{141}) \)
$\card{ \Aut(K/\Q) }$:  $1$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{25\!\cdots\!31}a^{14}-\frac{258750044004950}{25\!\cdots\!31}a^{13}+\frac{110910697476049}{25\!\cdots\!31}a^{12}-\frac{10\!\cdots\!27}{25\!\cdots\!31}a^{11}-\frac{533127928511868}{25\!\cdots\!31}a^{10}-\frac{234487009736970}{25\!\cdots\!31}a^{9}-\frac{433602086469113}{25\!\cdots\!31}a^{8}-\frac{47648693997309}{25\!\cdots\!31}a^{7}-\frac{706262700091971}{25\!\cdots\!31}a^{6}+\frac{185251441205242}{25\!\cdots\!31}a^{5}+\frac{10\!\cdots\!17}{25\!\cdots\!31}a^{4}+\frac{164469805584897}{25\!\cdots\!31}a^{3}-\frac{460854777520338}{25\!\cdots\!31}a^{2}-\frac{734868361836927}{25\!\cdots\!31}a-\frac{11\!\cdots\!74}{25\!\cdots\!31}$ 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:  $8$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
oscar: rank(UK)
 
Torsion generator:   \( -1 \)  (order $2$) 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:   $\frac{38043001556496}{25\!\cdots\!31}a^{14}-\frac{48296423944521}{25\!\cdots\!31}a^{13}+\frac{174067564617403}{25\!\cdots\!31}a^{12}-\frac{453065650852761}{25\!\cdots\!31}a^{11}+\frac{228271763832279}{25\!\cdots\!31}a^{10}-\frac{163788039386616}{25\!\cdots\!31}a^{9}-\frac{11\!\cdots\!55}{25\!\cdots\!31}a^{8}+\frac{22\!\cdots\!64}{25\!\cdots\!31}a^{7}-\frac{59\!\cdots\!13}{25\!\cdots\!31}a^{6}+\frac{12\!\cdots\!47}{25\!\cdots\!31}a^{5}-\frac{16\!\cdots\!48}{25\!\cdots\!31}a^{4}+\frac{15\!\cdots\!02}{25\!\cdots\!31}a^{3}-\frac{52\!\cdots\!32}{25\!\cdots\!31}a^{2}+\frac{67\!\cdots\!10}{25\!\cdots\!31}a-\frac{10\!\cdots\!09}{25\!\cdots\!31}$, $\frac{165444378283260}{25\!\cdots\!31}a^{14}+\frac{118218925645920}{25\!\cdots\!31}a^{13}+\frac{994029282971125}{25\!\cdots\!31}a^{12}+\frac{25592013915471}{25\!\cdots\!31}a^{11}+\frac{10\!\cdots\!04}{25\!\cdots\!31}a^{10}+\frac{14\!\cdots\!88}{25\!\cdots\!31}a^{9}-\frac{24\!\cdots\!98}{25\!\cdots\!31}a^{8}+\frac{51\!\cdots\!50}{25\!\cdots\!31}a^{7}-\frac{15\!\cdots\!34}{25\!\cdots\!31}a^{6}+\frac{21\!\cdots\!57}{25\!\cdots\!31}a^{5}-\frac{29\!\cdots\!29}{25\!\cdots\!31}a^{4}+\frac{46\!\cdots\!43}{25\!\cdots\!31}a^{3}-\frac{11\!\cdots\!26}{25\!\cdots\!31}a^{2}-\frac{19\!\cdots\!32}{25\!\cdots\!31}a+\frac{517043990670607}{25\!\cdots\!31}$, $\frac{74039452196061}{25\!\cdots\!31}a^{14}+\frac{18940358034794}{25\!\cdots\!31}a^{13}+\frac{360630633517514}{25\!\cdots\!31}a^{12}-\frac{221792748300561}{25\!\cdots\!31}a^{11}+\frac{118616125900557}{25\!\cdots\!31}a^{10}+\frac{513002582316782}{25\!\cdots\!31}a^{9}-\frac{16\!\cdots\!54}{25\!\cdots\!31}a^{8}+\frac{23\!\cdots\!73}{25\!\cdots\!31}a^{7}-\frac{69\!\cdots\!57}{25\!\cdots\!31}a^{6}+\frac{10\!\cdots\!90}{25\!\cdots\!31}a^{5}-\frac{11\!\cdots\!77}{25\!\cdots\!31}a^{4}-\frac{355338767148790}{25\!\cdots\!31}a^{3}+\frac{30\!\cdots\!31}{25\!\cdots\!31}a^{2}+\frac{15\!\cdots\!31}{25\!\cdots\!31}a+\frac{556871188999820}{25\!\cdots\!31}$, $\frac{167490929200191}{25\!\cdots\!31}a^{14}+\frac{2685719722084}{25\!\cdots\!31}a^{13}+\frac{981533778688417}{25\!\cdots\!31}a^{12}-\frac{658746539489490}{25\!\cdots\!31}a^{11}+\frac{13\!\cdots\!05}{25\!\cdots\!31}a^{10}+\frac{597140581676474}{25\!\cdots\!31}a^{9}-\frac{31\!\cdots\!54}{25\!\cdots\!31}a^{8}+\frac{74\!\cdots\!05}{25\!\cdots\!31}a^{7}-\frac{20\!\cdots\!03}{25\!\cdots\!31}a^{6}+\frac{35\!\cdots\!61}{25\!\cdots\!31}a^{5}-\frac{51\!\cdots\!48}{25\!\cdots\!31}a^{4}+\frac{35\!\cdots\!37}{25\!\cdots\!31}a^{3}-\frac{27\!\cdots\!52}{25\!\cdots\!31}a^{2}+\frac{98\!\cdots\!57}{25\!\cdots\!31}a-\frac{20\!\cdots\!31}{25\!\cdots\!31}$, $\frac{68851530394218}{25\!\cdots\!31}a^{14}+\frac{9838036202774}{25\!\cdots\!31}a^{13}+\frac{392241993717571}{25\!\cdots\!31}a^{12}-\frac{202638486436899}{25\!\cdots\!31}a^{11}+\frac{474757950087996}{25\!\cdots\!31}a^{10}+\frac{469812445896634}{25\!\cdots\!31}a^{9}-\frac{13\!\cdots\!10}{25\!\cdots\!31}a^{8}+\frac{29\!\cdots\!16}{25\!\cdots\!31}a^{7}-\frac{76\!\cdots\!12}{25\!\cdots\!31}a^{6}+\frac{12\!\cdots\!69}{25\!\cdots\!31}a^{5}-\frac{17\!\cdots\!81}{25\!\cdots\!31}a^{4}+\frac{72\!\cdots\!08}{25\!\cdots\!31}a^{3}-\frac{48\!\cdots\!86}{25\!\cdots\!31}a^{2}-\frac{17\!\cdots\!90}{25\!\cdots\!31}a-\frac{319902043572305}{25\!\cdots\!31}$, $\frac{55065280286527}{25\!\cdots\!31}a^{14}+\frac{17721662610740}{25\!\cdots\!31}a^{13}+\frac{368960415844164}{25\!\cdots\!31}a^{12}-\frac{55407559622651}{25\!\cdots\!31}a^{11}+\frac{675663729475804}{25\!\cdots\!31}a^{10}+\frac{428482794515119}{25\!\cdots\!31}a^{9}-\frac{817335203135992}{25\!\cdots\!31}a^{8}+\frac{22\!\cdots\!80}{25\!\cdots\!31}a^{7}-\frac{66\!\cdots\!21}{25\!\cdots\!31}a^{6}+\frac{10\!\cdots\!16}{25\!\cdots\!31}a^{5}-\frac{17\!\cdots\!08}{25\!\cdots\!31}a^{4}+\frac{10\!\cdots\!36}{25\!\cdots\!31}a^{3}-\frac{11\!\cdots\!55}{25\!\cdots\!31}a^{2}+\frac{17\!\cdots\!16}{25\!\cdots\!31}a-\frac{11\!\cdots\!16}{25\!\cdots\!31}$, $\frac{118218925645920}{25\!\cdots\!31}a^{14}+\frac{1363013271565}{25\!\cdots\!31}a^{13}+\frac{687369527048511}{25\!\cdots\!31}a^{12}-\frac{458407366349136}{25\!\cdots\!31}a^{11}+\frac{941386107916708}{25\!\cdots\!31}a^{10}+\frac{497934527851782}{25\!\cdots\!31}a^{9}-\frac{22\!\cdots\!50}{25\!\cdots\!31}a^{8}+\frac{52\!\cdots\!26}{25\!\cdots\!31}a^{7}-\frac{14\!\cdots\!03}{25\!\cdots\!31}a^{6}+\frac{23\!\cdots\!11}{25\!\cdots\!31}a^{5}-\frac{35\!\cdots\!37}{25\!\cdots\!31}a^{4}+\frac{22\!\cdots\!74}{25\!\cdots\!31}a^{3}-\frac{16\!\cdots\!32}{25\!\cdots\!31}a^{2}+\frac{54\!\cdots\!07}{25\!\cdots\!31}a+\frac{689442018940871}{25\!\cdots\!31}$, $\frac{110597019265331}{25\!\cdots\!31}a^{14}+\frac{76870185919336}{25\!\cdots\!31}a^{13}+\frac{710278662309861}{25\!\cdots\!31}a^{12}+\frac{53215248499520}{25\!\cdots\!31}a^{11}+\frac{965539067417517}{25\!\cdots\!31}a^{10}+\frac{10\!\cdots\!03}{25\!\cdots\!31}a^{9}-\frac{14\!\cdots\!36}{25\!\cdots\!31}a^{8}+\frac{41\!\cdots\!59}{25\!\cdots\!31}a^{7}-\frac{10\!\cdots\!69}{25\!\cdots\!31}a^{6}+\frac{15\!\cdots\!49}{25\!\cdots\!31}a^{5}-\frac{23\!\cdots\!73}{25\!\cdots\!31}a^{4}+\frac{75\!\cdots\!21}{25\!\cdots\!31}a^{3}-\frac{10\!\cdots\!80}{25\!\cdots\!31}a^{2}-\frac{18\!\cdots\!60}{25\!\cdots\!31}a-\frac{19\!\cdots\!85}{25\!\cdots\!31}$ 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:  \( 2652.22907153 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{6}\cdot 2652.22907153 \cdot 1}{2\cdot\sqrt{7269488039573383941}}\cr\approx \mathstrut & 0.242101967819 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^15 + 6*x^13 - 4*x^12 + 9*x^11 + 3*x^10 - 18*x^9 + 45*x^8 - 126*x^7 + 216*x^6 - 324*x^5 + 243*x^4 - 205*x^3 + 90*x^2 - 30*x + 11)
 
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^15 + 6*x^13 - 4*x^12 + 9*x^11 + 3*x^10 - 18*x^9 + 45*x^8 - 126*x^7 + 216*x^6 - 324*x^5 + 243*x^4 - 205*x^3 + 90*x^2 - 30*x + 11, 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^15 + 6*x^13 - 4*x^12 + 9*x^11 + 3*x^10 - 18*x^9 + 45*x^8 - 126*x^7 + 216*x^6 - 324*x^5 + 243*x^4 - 205*x^3 + 90*x^2 - 30*x + 11);
 
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^15 + 6*x^13 - 4*x^12 + 9*x^11 + 3*x^10 - 18*x^9 + 45*x^8 - 126*x^7 + 216*x^6 - 324*x^5 + 243*x^4 - 205*x^3 + 90*x^2 - 30*x + 11);
 
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

$C_3^4:D_{10}$ (as 15T43):

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 1620
The 24 conjugacy class representatives for $C_3^4:D_{10}$
Character table for $C_3^4:D_{10}$ is not computed

Intermediate fields

5.1.2209.1

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 15 siblings: data not computed
Degree 30 siblings: data not computed
Degree 45 siblings: data not computed
Minimal sibling: 15.1.154669958288795403.3

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/padicField/2.10.0.1}{10} }{,}\,{\href{/padicField/2.5.0.1}{5} }$ R ${\href{/padicField/5.6.0.1}{6} }^{2}{,}\,{\href{/padicField/5.3.0.1}{3} }$ ${\href{/padicField/7.5.0.1}{5} }^{3}$ ${\href{/padicField/11.6.0.1}{6} }^{2}{,}\,{\href{/padicField/11.1.0.1}{1} }^{3}$ ${\href{/padicField/13.6.0.1}{6} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }{,}\,{\href{/padicField/13.1.0.1}{1} }$ ${\href{/padicField/17.10.0.1}{10} }{,}\,{\href{/padicField/17.5.0.1}{5} }$ ${\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.2.0.1}{2} }^{4}{,}\,{\href{/padicField/19.1.0.1}{1} }$ ${\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.3.0.1}{3} }{,}\,{\href{/padicField/23.2.0.1}{2} }^{3}$ ${\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.3.0.1}{3} }{,}\,{\href{/padicField/29.2.0.1}{2} }^{3}$ ${\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.2.0.1}{2} }^{4}{,}\,{\href{/padicField/31.1.0.1}{1} }$ ${\href{/padicField/37.5.0.1}{5} }^{3}$ ${\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.3.0.1}{3} }{,}\,{\href{/padicField/41.2.0.1}{2} }^{3}$ ${\href{/padicField/43.6.0.1}{6} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ R ${\href{/padicField/53.10.0.1}{10} }{,}\,{\href{/padicField/53.5.0.1}{5} }$ ${\href{/padicField/59.10.0.1}{10} }{,}\,{\href{/padicField/59.5.0.1}{5} }$

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 3.15.15.38$x^{15} + 30 x^{14} + 432 x^{13} + 4335 x^{12} + 32373 x^{11} + 162027 x^{10} + 501129 x^{9} + 945432 x^{8} + 1104921 x^{7} + 951939 x^{6} + 550233 x^{5} + 242271 x^{4} + 60669 x^{3} + 11664 x^{2} - 729 x + 243$$3$$5$$15$15T33$[3/2, 3/2, 3/2, 3/2]_{2}^{5}$
\(47\) Copy content Toggle raw display $\Q_{47}$$x + 42$$1$$1$$0$Trivial$[\ ]$
47.2.1.2$x^{2} + 47$$2$$1$$1$$C_2$$[\ ]_{2}$
47.2.1.2$x^{2} + 47$$2$$1$$1$$C_2$$[\ ]_{2}$
47.2.1.1$x^{2} + 235$$2$$1$$1$$C_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}$