Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 - 8*x^14 + 160*x^13 - 441*x^12 - 134*x^11 + 4342*x^10 - 16579*x^9 + 43548*x^8 - 89962*x^7 + 151269*x^6 - 210738*x^5 + 244446*x^4 - 223845*x^3 + 154504*x^2 - 70325*x + 18419)
gp: K = bnfinit(y^16 - 6*y^15 - 8*y^14 + 160*y^13 - 441*y^12 - 134*y^11 + 4342*y^10 - 16579*y^9 + 43548*y^8 - 89962*y^7 + 151269*y^6 - 210738*y^5 + 244446*y^4 - 223845*y^3 + 154504*y^2 - 70325*y + 18419, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 6*x^15 - 8*x^14 + 160*x^13 - 441*x^12 - 134*x^11 + 4342*x^10 - 16579*x^9 + 43548*x^8 - 89962*x^7 + 151269*x^6 - 210738*x^5 + 244446*x^4 - 223845*x^3 + 154504*x^2 - 70325*x + 18419);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 6*x^15 - 8*x^14 + 160*x^13 - 441*x^12 - 134*x^11 + 4342*x^10 - 16579*x^9 + 43548*x^8 - 89962*x^7 + 151269*x^6 - 210738*x^5 + 244446*x^4 - 223845*x^3 + 154504*x^2 - 70325*x + 18419)
\( x^{16} - 6 x^{15} - 8 x^{14} + 160 x^{13} - 441 x^{12} - 134 x^{11} + 4342 x^{10} - 16579 x^{9} + \cdots + 18419 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $16$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[0, 8]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(14816104373013890157094140625\)
\(\medspace = 5^{8}\cdot 41^{14}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(57.63\) | 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: | $5^{1/2}41^{7/8}\approx 57.632953563318466$ | ||
| Ramified primes: |
\(5\), \(41\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Aut(K/\Q) }$: | $8$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is not Galois over $\Q$. | |||
| This is a CM field. | |||
| Reflex fields: | unavailable$^{128}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2}a^{12}-\frac{1}{2}a^{9}-\frac{1}{2}a^{6}-\frac{1}{2}a^{3}-\frac{1}{2}$, $\frac{1}{2}a^{13}-\frac{1}{2}a^{10}-\frac{1}{2}a^{7}-\frac{1}{2}a^{4}-\frac{1}{2}a$, $\frac{1}{118}a^{14}-\frac{1}{118}a^{13}+\frac{15}{118}a^{12}-\frac{29}{118}a^{11}+\frac{11}{118}a^{10}+\frac{53}{118}a^{9}-\frac{41}{118}a^{8}-\frac{19}{118}a^{7}-\frac{57}{118}a^{6}-\frac{37}{118}a^{5}+\frac{41}{118}a^{4}+\frac{5}{118}a^{3}-\frac{57}{118}a^{2}+\frac{33}{118}a+\frac{27}{118}$, $\frac{1}{31\!\cdots\!66}a^{15}+\frac{31\!\cdots\!10}{15\!\cdots\!83}a^{14}+\frac{21\!\cdots\!82}{15\!\cdots\!83}a^{13}+\frac{31\!\cdots\!41}{15\!\cdots\!83}a^{12}+\frac{40\!\cdots\!07}{15\!\cdots\!83}a^{11}-\frac{68\!\cdots\!51}{15\!\cdots\!83}a^{10}-\frac{41\!\cdots\!21}{15\!\cdots\!83}a^{9}+\frac{70\!\cdots\!77}{15\!\cdots\!83}a^{8}-\frac{32\!\cdots\!36}{15\!\cdots\!83}a^{7}-\frac{15\!\cdots\!58}{15\!\cdots\!83}a^{6}-\frac{22\!\cdots\!53}{15\!\cdots\!83}a^{5}-\frac{72\!\cdots\!67}{15\!\cdots\!83}a^{4}+\frac{40\!\cdots\!08}{15\!\cdots\!83}a^{3}+\frac{37\!\cdots\!73}{15\!\cdots\!83}a^{2}+\frac{39\!\cdots\!69}{15\!\cdots\!83}a+\frac{83\!\cdots\!49}{31\!\cdots\!66}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $2$ |
Class group and class number
$C_{3}\times C_{3}\times C_{6}$, which has order $54$ (assuming GRH)
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: | $7$ | 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: |
$\frac{22\!\cdots\!10}{42\!\cdots\!71}a^{15}-\frac{13\!\cdots\!31}{42\!\cdots\!71}a^{14}-\frac{21\!\cdots\!57}{42\!\cdots\!71}a^{13}+\frac{36\!\cdots\!59}{42\!\cdots\!71}a^{12}-\frac{92\!\cdots\!09}{42\!\cdots\!71}a^{11}-\frac{64\!\cdots\!38}{42\!\cdots\!71}a^{10}+\frac{10\!\cdots\!90}{42\!\cdots\!71}a^{9}-\frac{35\!\cdots\!79}{42\!\cdots\!71}a^{8}+\frac{88\!\cdots\!49}{42\!\cdots\!71}a^{7}-\frac{17\!\cdots\!76}{42\!\cdots\!71}a^{6}+\frac{28\!\cdots\!12}{42\!\cdots\!71}a^{5}-\frac{37\!\cdots\!76}{42\!\cdots\!71}a^{4}+\frac{40\!\cdots\!05}{42\!\cdots\!71}a^{3}-\frac{34\!\cdots\!69}{42\!\cdots\!71}a^{2}+\frac{18\!\cdots\!94}{42\!\cdots\!71}a-\frac{57\!\cdots\!46}{42\!\cdots\!71}$, $\frac{68\!\cdots\!50}{42\!\cdots\!71}a^{15}-\frac{40\!\cdots\!54}{42\!\cdots\!71}a^{14}-\frac{60\!\cdots\!66}{42\!\cdots\!71}a^{13}+\frac{11\!\cdots\!87}{42\!\cdots\!71}a^{12}-\frac{29\!\cdots\!64}{42\!\cdots\!71}a^{11}-\frac{17\!\cdots\!32}{42\!\cdots\!71}a^{10}+\frac{31\!\cdots\!31}{42\!\cdots\!71}a^{9}-\frac{11\!\cdots\!56}{42\!\cdots\!71}a^{8}+\frac{27\!\cdots\!36}{42\!\cdots\!71}a^{7}-\frac{54\!\cdots\!93}{42\!\cdots\!71}a^{6}+\frac{87\!\cdots\!50}{42\!\cdots\!71}a^{5}-\frac{11\!\cdots\!37}{42\!\cdots\!71}a^{4}+\frac{12\!\cdots\!67}{42\!\cdots\!71}a^{3}-\frac{10\!\cdots\!68}{42\!\cdots\!71}a^{2}+\frac{56\!\cdots\!45}{42\!\cdots\!71}a-\frac{17\!\cdots\!01}{42\!\cdots\!71}$, $\frac{65\!\cdots\!28}{15\!\cdots\!83}a^{15}-\frac{34\!\cdots\!98}{15\!\cdots\!83}a^{14}-\frac{73\!\cdots\!97}{15\!\cdots\!83}a^{13}+\frac{98\!\cdots\!05}{15\!\cdots\!83}a^{12}-\frac{22\!\cdots\!58}{15\!\cdots\!83}a^{11}-\frac{20\!\cdots\!14}{15\!\cdots\!83}a^{10}+\frac{26\!\cdots\!47}{15\!\cdots\!83}a^{9}-\frac{92\!\cdots\!30}{15\!\cdots\!83}a^{8}+\frac{22\!\cdots\!79}{15\!\cdots\!83}a^{7}-\frac{44\!\cdots\!03}{15\!\cdots\!83}a^{6}+\frac{71\!\cdots\!85}{15\!\cdots\!83}a^{5}-\frac{92\!\cdots\!95}{15\!\cdots\!83}a^{4}+\frac{97\!\cdots\!87}{15\!\cdots\!83}a^{3}-\frac{77\!\cdots\!25}{15\!\cdots\!83}a^{2}+\frac{41\!\cdots\!03}{15\!\cdots\!83}a-\frac{10\!\cdots\!71}{15\!\cdots\!83}$, $\frac{64\!\cdots\!05}{15\!\cdots\!83}a^{15}-\frac{40\!\cdots\!72}{15\!\cdots\!83}a^{14}-\frac{42\!\cdots\!25}{15\!\cdots\!83}a^{13}+\frac{10\!\cdots\!56}{15\!\cdots\!83}a^{12}-\frac{31\!\cdots\!09}{15\!\cdots\!83}a^{11}-\frac{83\!\cdots\!77}{15\!\cdots\!83}a^{10}+\frac{31\!\cdots\!30}{15\!\cdots\!83}a^{9}-\frac{19\!\cdots\!77}{26\!\cdots\!37}a^{8}+\frac{28\!\cdots\!04}{15\!\cdots\!83}a^{7}-\frac{55\!\cdots\!03}{15\!\cdots\!83}a^{6}+\frac{89\!\cdots\!82}{15\!\cdots\!83}a^{5}-\frac{11\!\cdots\!84}{15\!\cdots\!83}a^{4}+\frac{12\!\cdots\!08}{15\!\cdots\!83}a^{3}-\frac{92\!\cdots\!56}{15\!\cdots\!83}a^{2}+\frac{47\!\cdots\!05}{15\!\cdots\!83}a-\frac{13\!\cdots\!42}{15\!\cdots\!83}$, $\frac{31\!\cdots\!72}{15\!\cdots\!83}a^{15}-\frac{18\!\cdots\!24}{15\!\cdots\!83}a^{14}-\frac{28\!\cdots\!22}{15\!\cdots\!83}a^{13}+\frac{50\!\cdots\!43}{15\!\cdots\!83}a^{12}-\frac{13\!\cdots\!08}{15\!\cdots\!83}a^{11}-\frac{80\!\cdots\!62}{15\!\cdots\!83}a^{10}+\frac{14\!\cdots\!61}{15\!\cdots\!83}a^{9}-\frac{49\!\cdots\!79}{15\!\cdots\!83}a^{8}+\frac{12\!\cdots\!14}{15\!\cdots\!83}a^{7}-\frac{24\!\cdots\!63}{15\!\cdots\!83}a^{6}+\frac{39\!\cdots\!46}{15\!\cdots\!83}a^{5}-\frac{51\!\cdots\!64}{15\!\cdots\!83}a^{4}+\frac{94\!\cdots\!61}{26\!\cdots\!37}a^{3}-\frac{45\!\cdots\!68}{15\!\cdots\!83}a^{2}+\frac{24\!\cdots\!65}{15\!\cdots\!83}a-\frac{66\!\cdots\!65}{15\!\cdots\!83}$, $\frac{49\!\cdots\!32}{15\!\cdots\!83}a^{15}-\frac{25\!\cdots\!18}{15\!\cdots\!83}a^{14}-\frac{58\!\cdots\!32}{15\!\cdots\!83}a^{13}+\frac{73\!\cdots\!17}{15\!\cdots\!83}a^{12}-\frac{16\!\cdots\!70}{15\!\cdots\!83}a^{11}-\frac{17\!\cdots\!58}{15\!\cdots\!83}a^{10}+\frac{20\!\cdots\!29}{15\!\cdots\!83}a^{9}-\frac{67\!\cdots\!30}{15\!\cdots\!83}a^{8}+\frac{16\!\cdots\!36}{15\!\cdots\!83}a^{7}-\frac{32\!\cdots\!23}{15\!\cdots\!83}a^{6}+\frac{51\!\cdots\!52}{15\!\cdots\!83}a^{5}-\frac{66\!\cdots\!96}{15\!\cdots\!83}a^{4}+\frac{71\!\cdots\!89}{15\!\cdots\!83}a^{3}-\frac{57\!\cdots\!55}{15\!\cdots\!83}a^{2}+\frac{30\!\cdots\!49}{15\!\cdots\!83}a-\frac{95\!\cdots\!88}{15\!\cdots\!83}$, $\frac{22\!\cdots\!50}{42\!\cdots\!71}a^{15}-\frac{24\!\cdots\!29}{42\!\cdots\!71}a^{14}+\frac{38\!\cdots\!41}{42\!\cdots\!71}a^{13}+\frac{53\!\cdots\!85}{42\!\cdots\!71}a^{12}-\frac{28\!\cdots\!79}{42\!\cdots\!71}a^{11}+\frac{25\!\cdots\!66}{42\!\cdots\!71}a^{10}+\frac{19\!\cdots\!14}{42\!\cdots\!71}a^{9}-\frac{90\!\cdots\!76}{42\!\cdots\!71}a^{8}+\frac{21\!\cdots\!39}{42\!\cdots\!71}a^{7}-\frac{40\!\cdots\!52}{42\!\cdots\!71}a^{6}+\frac{65\!\cdots\!42}{42\!\cdots\!71}a^{5}-\frac{82\!\cdots\!88}{42\!\cdots\!71}a^{4}+\frac{72\!\cdots\!55}{42\!\cdots\!71}a^{3}-\frac{40\!\cdots\!74}{42\!\cdots\!71}a^{2}+\frac{12\!\cdots\!10}{42\!\cdots\!71}a-\frac{13\!\cdots\!66}{42\!\cdots\!71}$
| 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: | \( 645179.367444 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 645179.367444 \cdot 54}{2\cdot\sqrt{14816104373013890157094140625}}\cr\approx \mathstrut & 0.347629134405 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 - 8*x^14 + 160*x^13 - 441*x^12 - 134*x^11 + 4342*x^10 - 16579*x^9 + 43548*x^8 - 89962*x^7 + 151269*x^6 - 210738*x^5 + 244446*x^4 - 223845*x^3 + 154504*x^2 - 70325*x + 18419)
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^16 - 6*x^15 - 8*x^14 + 160*x^13 - 441*x^12 - 134*x^11 + 4342*x^10 - 16579*x^9 + 43548*x^8 - 89962*x^7 + 151269*x^6 - 210738*x^5 + 244446*x^4 - 223845*x^3 + 154504*x^2 - 70325*x + 18419, 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^16 - 6*x^15 - 8*x^14 + 160*x^13 - 441*x^12 - 134*x^11 + 4342*x^10 - 16579*x^9 + 43548*x^8 - 89962*x^7 + 151269*x^6 - 210738*x^5 + 244446*x^4 - 223845*x^3 + 154504*x^2 - 70325*x + 18419);
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^16 - 6*x^15 - 8*x^14 + 160*x^13 - 441*x^12 - 134*x^11 + 4342*x^10 - 16579*x^9 + 43548*x^8 - 89962*x^7 + 151269*x^6 - 210738*x^5 + 244446*x^4 - 223845*x^3 + 154504*x^2 - 70325*x + 18419);
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
$C_2^2:C_8$ (as 16T24):
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 32 |
| The 20 conjugacy class representatives for $C_2^2 : C_8$ |
| Character table for $C_2^2 : C_8$ |
Intermediate fields
| \(\Q(\sqrt{41}) \), 4.0.8405.1, 4.4.68921.1, 4.0.344605.1, 8.8.4868856847025.1, 8.0.121721421175625.1, 8.0.118752606025.1 |
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
| Galois closure: | deg 32 |
| Degree 16 sibling: | 16.0.23705766996822224251350625.1 |
| Minimal sibling: | 16.0.23705766996822224251350625.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.4.0.1}{4} }^{4}$ | ${\href{/padicField/3.8.0.1}{8} }^{2}$ | R | ${\href{/padicField/7.8.0.1}{8} }^{2}$ | ${\href{/padicField/11.8.0.1}{8} }^{2}$ | ${\href{/padicField/13.8.0.1}{8} }^{2}$ | ${\href{/padicField/17.8.0.1}{8} }^{2}$ | ${\href{/padicField/19.8.0.1}{8} }^{2}$ | ${\href{/padicField/23.2.0.1}{2} }^{4}{,}\,{\href{/padicField/23.1.0.1}{1} }^{8}$ | ${\href{/padicField/29.8.0.1}{8} }^{2}$ | ${\href{/padicField/31.2.0.1}{2} }^{8}$ | ${\href{/padicField/37.2.0.1}{2} }^{8}$ | R | ${\href{/padicField/43.4.0.1}{4} }^{4}$ | ${\href{/padicField/47.8.0.1}{8} }^{2}$ | ${\href{/padicField/53.8.0.1}{8} }^{2}$ | ${\href{/padicField/59.1.0.1}{1} }^{16}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(5\)
| 5.4.2.2 | $x^{4} - 20 x^{2} + 50$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ |
| 5.4.2.2 | $x^{4} - 20 x^{2} + 50$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ | |
| 5.8.4.1 | $x^{8} + 80 x^{7} + 2428 x^{6} + 33688 x^{5} + 195810 x^{4} + 305952 x^{3} + 870132 x^{2} + 1037416 x + 503089$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
|
\(41\)
| 41.16.14.1 | $x^{16} + 304 x^{15} + 40480 x^{14} + 3085600 x^{13} + 147416080 x^{12} + 4529584192 x^{11} + 87831092608 x^{10} + 996302227840 x^{9} + 5391168776882 x^{8} + 5977813379504 x^{7} + 3161920977824 x^{6} + 978514601120 x^{5} + 196936323920 x^{4} + 202153692608 x^{3} + 3372805705856 x^{2} + 36445904670848 x + 172395305267889$ | $8$ | $2$ | $14$ | $C_8\times C_2$ | $[\ ]_{8}^{2}$ |