Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^17 + 20*x^16 - 45*x^15 + 68*x^14 - 71*x^13 + 52*x^12 - 53*x^11 + 123*x^10 - 205*x^9 + 218*x^8 - 156*x^7 + 48*x^6 - 26*x^5 + 43*x^4 - 20*x^3 - x + 1)
gp: K = bnfinit(y^18 - 6*y^17 + 20*y^16 - 45*y^15 + 68*y^14 - 71*y^13 + 52*y^12 - 53*y^11 + 123*y^10 - 205*y^9 + 218*y^8 - 156*y^7 + 48*y^6 - 26*y^5 + 43*y^4 - 20*y^3 - y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 6*x^17 + 20*x^16 - 45*x^15 + 68*x^14 - 71*x^13 + 52*x^12 - 53*x^11 + 123*x^10 - 205*x^9 + 218*x^8 - 156*x^7 + 48*x^6 - 26*x^5 + 43*x^4 - 20*x^3 - x + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 - 6*x^17 + 20*x^16 - 45*x^15 + 68*x^14 - 71*x^13 + 52*x^12 - 53*x^11 + 123*x^10 - 205*x^9 + 218*x^8 - 156*x^7 + 48*x^6 - 26*x^5 + 43*x^4 - 20*x^3 - x + 1)
\( x^{18} - 6 x^{17} + 20 x^{16} - 45 x^{15} + 68 x^{14} - 71 x^{13} + 52 x^{12} - 53 x^{11} + 123 x^{10} + \cdots + 1 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $18$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[2, 8]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(3104175012088106186109\)
\(\medspace = 3^{6}\cdot 11^{5}\cdot 31^{9}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(15.63\) | 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: | $3^{1/2}11^{5/6}31^{3/4}\approx 167.845280953044$ | ||
| Ramified primes: |
\(3\), \(11\), \(31\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{341}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{3}a^{16}-\frac{1}{3}a^{15}-\frac{1}{3}a^{14}-\frac{1}{3}a^{13}+\frac{1}{3}a^{12}+\frac{1}{3}a^{11}-\frac{1}{3}a^{10}+\frac{1}{3}a^{9}+\frac{1}{3}a^{7}+\frac{1}{3}a^{6}+\frac{1}{3}a^{5}+\frac{1}{3}a^{4}-\frac{1}{3}a^{3}+\frac{1}{3}a^{2}+\frac{1}{3}a+\frac{1}{3}$, $\frac{1}{50498052683613}a^{17}-\frac{5504233345793}{50498052683613}a^{16}-\frac{3192546477560}{16832684227871}a^{15}+\frac{5892461591799}{16832684227871}a^{14}+\frac{9284667055922}{50498052683613}a^{13}+\frac{2111148659286}{16832684227871}a^{12}-\frac{18289738712984}{50498052683613}a^{11}+\frac{20988376068104}{50498052683613}a^{10}-\frac{17226026279662}{50498052683613}a^{9}+\frac{3992049681286}{50498052683613}a^{8}+\frac{8206426812809}{16832684227871}a^{7}-\frac{3717842830645}{16832684227871}a^{6}+\frac{7499473760341}{16832684227871}a^{5}-\frac{4875307209911}{50498052683613}a^{4}-\frac{15072925630114}{50498052683613}a^{3}-\frac{1054062842694}{16832684227871}a^{2}+\frac{6724216071751}{16832684227871}a-\frac{12683203953364}{50498052683613}$
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: |
\( -1 \)
(order $2$)
| 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: |
$\frac{1663776933}{4391516887}a^{17}-\frac{26758986059}{13174550661}a^{16}+\frac{80500200023}{13174550661}a^{15}-\frac{160094061001}{13174550661}a^{14}+\frac{193886405606}{13174550661}a^{13}-\frac{133864709108}{13174550661}a^{12}+\frac{28986440407}{13174550661}a^{11}-\frac{94393931581}{13174550661}a^{10}+\frac{442057945507}{13174550661}a^{9}-\frac{208646391485}{4391516887}a^{8}+\frac{426172900987}{13174550661}a^{7}-\frac{77280486005}{13174550661}a^{6}-\frac{270366157100}{13174550661}a^{5}+\frac{33655740577}{13174550661}a^{4}+\frac{134766951131}{13174550661}a^{3}+\frac{47127130441}{13174550661}a^{2}-\frac{45943479482}{13174550661}a-\frac{8001223262}{13174550661}$, $\frac{14912596816587}{16832684227871}a^{17}-\frac{245615481617930}{50498052683613}a^{16}+\frac{766768471985975}{50498052683613}a^{15}-\frac{16\!\cdots\!21}{50498052683613}a^{14}+\frac{21\!\cdots\!39}{50498052683613}a^{13}-\frac{19\!\cdots\!93}{50498052683613}a^{12}+\frac{11\!\cdots\!47}{50498052683613}a^{11}-\frac{15\!\cdots\!78}{50498052683613}a^{10}+\frac{45\!\cdots\!02}{50498052683613}a^{9}-\frac{22\!\cdots\!99}{16832684227871}a^{8}+\frac{60\!\cdots\!55}{50498052683613}a^{7}-\frac{33\!\cdots\!86}{50498052683613}a^{6}-\frac{257238204795395}{50498052683613}a^{5}-\frac{738316273109297}{50498052683613}a^{4}+\frac{13\!\cdots\!02}{50498052683613}a^{3}-\frac{164899708006064}{50498052683613}a^{2}-\frac{124986039467177}{50498052683613}a-\frac{44225194837727}{50498052683613}$, $\frac{12901105883861}{50498052683613}a^{17}-\frac{18702482931025}{16832684227871}a^{16}+\frac{136903824209672}{50498052683613}a^{15}-\frac{189968696520307}{50498052683613}a^{14}+\frac{8820621174893}{16832684227871}a^{13}+\frac{312908851465468}{50498052683613}a^{12}-\frac{185732868207722}{16832684227871}a^{11}+\frac{59694935746172}{16832684227871}a^{10}+\frac{601468955165846}{50498052683613}a^{9}-\frac{262143149958040}{50498052683613}a^{8}-\frac{891425837176022}{50498052683613}a^{7}+\frac{17\!\cdots\!04}{50498052683613}a^{6}-\frac{19\!\cdots\!23}{50498052683613}a^{5}+\frac{87843460960086}{16832684227871}a^{4}-\frac{10542085823271}{16832684227871}a^{3}+\frac{440045456643619}{50498052683613}a^{2}-\frac{179359221720776}{50498052683613}a-\frac{28745918576731}{50498052683613}$, $\frac{3699054196799}{50498052683613}a^{17}-\frac{4049617765832}{16832684227871}a^{16}+\frac{21280376778884}{50498052683613}a^{15}-\frac{9787913019058}{50498052683613}a^{14}-\frac{18663489030174}{16832684227871}a^{13}+\frac{101012283542116}{50498052683613}a^{12}-\frac{16816218018693}{16832684227871}a^{11}-\frac{50506467170002}{16832684227871}a^{10}+\frac{261571593805784}{50498052683613}a^{9}+\frac{121517605786400}{50498052683613}a^{8}-\frac{396912539994032}{50498052683613}a^{7}+\frac{236550803446987}{50498052683613}a^{6}+\frac{44719867331629}{50498052683613}a^{5}-\frac{211990808092512}{16832684227871}a^{4}+\frac{48502855149947}{16832684227871}a^{3}+\frac{237051496546369}{50498052683613}a^{2}-\frac{22028947978439}{50498052683613}a+\frac{4831015681697}{50498052683613}$, $\frac{43321006654585}{50498052683613}a^{17}-\frac{251481508045508}{50498052683613}a^{16}+\frac{272955091740059}{16832684227871}a^{15}-\frac{598160299601960}{16832684227871}a^{14}+\frac{26\!\cdots\!29}{50498052683613}a^{13}-\frac{851702599915676}{16832684227871}a^{12}+\frac{16\!\cdots\!02}{50498052683613}a^{11}-\frac{18\!\cdots\!65}{50498052683613}a^{10}+\frac{47\!\cdots\!63}{50498052683613}a^{9}-\frac{78\!\cdots\!37}{50498052683613}a^{8}+\frac{26\!\cdots\!00}{16832684227871}a^{7}-\frac{16\!\cdots\!36}{16832684227871}a^{6}+\frac{216928098186539}{16832684227871}a^{5}-\frac{388401737831375}{50498052683613}a^{4}+\frac{12\!\cdots\!78}{50498052683613}a^{3}-\frac{163141085448407}{16832684227871}a^{2}-\frac{19448416011424}{16832684227871}a+\frac{13141816913417}{50498052683613}$, $a$, $\frac{47157172070393}{50498052683613}a^{17}-\frac{256024114586995}{50498052683613}a^{16}+\frac{266958332485534}{16832684227871}a^{15}-\frac{563115463563712}{16832684227871}a^{14}+\frac{23\!\cdots\!26}{50498052683613}a^{13}-\frac{736882480292494}{16832684227871}a^{12}+\frac{14\!\cdots\!75}{50498052683613}a^{11}-\frac{19\!\cdots\!08}{50498052683613}a^{10}+\frac{48\!\cdots\!91}{50498052683613}a^{9}-\frac{70\!\cdots\!01}{50498052683613}a^{8}+\frac{22\!\cdots\!72}{16832684227871}a^{7}-\frac{14\!\cdots\!77}{16832684227871}a^{6}+\frac{227023544284714}{16832684227871}a^{5}-\frac{14\!\cdots\!93}{50498052683613}a^{4}+\frac{13\!\cdots\!90}{50498052683613}a^{3}-\frac{42406191930063}{16832684227871}a^{2}-\frac{12333751581055}{16832684227871}a-\frac{52037793333095}{50498052683613}$, $\frac{6181594203418}{16832684227871}a^{17}-\frac{27424340743958}{16832684227871}a^{16}+\frac{73308616132454}{16832684227871}a^{15}-\frac{126402417423488}{16832684227871}a^{14}+\frac{114603543744649}{16832684227871}a^{13}-\frac{51798932736048}{16832684227871}a^{12}+\frac{277514792977}{16832684227871}a^{11}-\frac{158848499338650}{16832684227871}a^{10}+\frac{454320149144776}{16832684227871}a^{9}-\frac{364197785406649}{16832684227871}a^{8}+\frac{140249323284641}{16832684227871}a^{7}+\frac{24828861943988}{16832684227871}a^{6}-\frac{186742435512738}{16832684227871}a^{5}-\frac{304028378766621}{16832684227871}a^{4}+\frac{29874717849845}{16832684227871}a^{3}+\frac{99773591361121}{16832684227871}a^{2}+\frac{12969159625722}{16832684227871}a-\frac{21363813041461}{16832684227871}$, $\frac{19787171143187}{50498052683613}a^{17}-\frac{102192761720764}{50498052683613}a^{16}+\frac{99911758638108}{16832684227871}a^{15}-\frac{191834469116785}{16832684227871}a^{14}+\frac{646511679611839}{50498052683613}a^{13}-\frac{121516083245014}{16832684227871}a^{12}-\frac{55293181057720}{50498052683613}a^{11}-\frac{237527357789066}{50498052683613}a^{10}+\frac{15\!\cdots\!43}{50498052683613}a^{9}-\frac{20\!\cdots\!09}{50498052683613}a^{8}+\frac{391943996837123}{16832684227871}a^{7}+\frac{75058563215293}{16832684227871}a^{6}-\frac{491508092489370}{16832684227871}a^{5}+\frac{292078188183782}{50498052683613}a^{4}+\frac{263799015616681}{50498052683613}a^{3}+\frac{86780522097860}{16832684227871}a^{2}-\frac{97260693812178}{16832684227871}a+\frac{35636041126333}{50498052683613}$
| 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: | \( 2687.87073054 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{8}\cdot 2687.87073054 \cdot 1}{2\cdot\sqrt{3104175012088106186109}}\cr\approx \mathstrut & 0.234371187940 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^17 + 20*x^16 - 45*x^15 + 68*x^14 - 71*x^13 + 52*x^12 - 53*x^11 + 123*x^10 - 205*x^9 + 218*x^8 - 156*x^7 + 48*x^6 - 26*x^5 + 43*x^4 - 20*x^3 - 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^18 - 6*x^17 + 20*x^16 - 45*x^15 + 68*x^14 - 71*x^13 + 52*x^12 - 53*x^11 + 123*x^10 - 205*x^9 + 218*x^8 - 156*x^7 + 48*x^6 - 26*x^5 + 43*x^4 - 20*x^3 - 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^18 - 6*x^17 + 20*x^16 - 45*x^15 + 68*x^14 - 71*x^13 + 52*x^12 - 53*x^11 + 123*x^10 - 205*x^9 + 218*x^8 - 156*x^7 + 48*x^6 - 26*x^5 + 43*x^4 - 20*x^3 - 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^18 - 6*x^17 + 20*x^16 - 45*x^15 + 68*x^14 - 71*x^13 + 52*x^12 - 53*x^11 + 123*x^10 - 205*x^9 + 218*x^8 - 156*x^7 + 48*x^6 - 26*x^5 + 43*x^4 - 20*x^3 - 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
$C_6^3:S_4$ (as 18T485):
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 5184 |
| The 58 conjugacy class representatives for $C_6^3:S_4$ |
| Character table for $C_6^3:S_4$ |
Intermediate fields
| 3.1.31.1, 6.2.2949309.1, 9.1.1005714369.1 |
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 18 siblings: | data not computed |
| Degree 36 siblings: | data not computed |
| Minimal sibling: | 18.0.11126075312143749771.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | $18$ | R | ${\href{/padicField/5.9.0.1}{9} }^{2}$ | $18$ | R | ${\href{/padicField/13.6.0.1}{6} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }^{2}{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ | ${\href{/padicField/17.4.0.1}{4} }^{2}{,}\,{\href{/padicField/17.3.0.1}{3} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }^{2}$ | $18$ | ${\href{/padicField/23.6.0.1}{6} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }^{3}$ | ${\href{/padicField/29.4.0.1}{4} }^{2}{,}\,{\href{/padicField/29.3.0.1}{3} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}$ | R | ${\href{/padicField/37.12.0.1}{12} }{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | $18$ | ${\href{/padicField/43.4.0.1}{4} }^{2}{,}\,{\href{/padicField/43.3.0.1}{3} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }^{2}$ | ${\href{/padicField/47.2.0.1}{2} }^{8}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | ${\href{/padicField/53.12.0.1}{12} }{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.3.0.1}{3} }^{6}$ |
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
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
|
\(3\)
| 3.4.2.1 | $x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| 3.6.0.1 | $x^{6} + 2 x^{4} + x^{2} + 2 x + 2$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
| 3.8.4.1 | $x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
|
\(11\)
| 11.6.5.2 | $x^{6} + 11$ | $6$ | $1$ | $5$ | $D_{6}$ | $[\ ]_{6}^{2}$ |
| 11.12.0.1 | $x^{12} + x^{8} + x^{7} + 4 x^{6} + 2 x^{5} + 5 x^{4} + 5 x^{3} + 6 x^{2} + 5 x + 2$ | $1$ | $12$ | $0$ | $C_{12}$ | $[\ ]^{12}$ | |
|
\(31\)
| 31.2.0.1 | $x^{2} + 29 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 31.2.0.1 | $x^{2} + 29 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 31.2.0.1 | $x^{2} + 29 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 31.4.3.1 | $x^{4} + 31$ | $4$ | $1$ | $3$ | $D_{4}$ | $[\ ]_{4}^{2}$ | |
| 31.8.6.1 | $x^{8} + 116 x^{7} + 5058 x^{6} + 98600 x^{5} + 737673 x^{4} + 299396 x^{3} + 200832 x^{2} + 2995004 x + 21614459$ | $4$ | $2$ | $6$ | $D_4$ | $[\ ]_{4}^{2}$ |