Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + 34*x^14 - 86*x^13 + 539*x^12 - 1176*x^11 + 4498*x^10 - 8465*x^9 + 22378*x^8 - 36402*x^7 + 69763*x^6 - 83044*x^5 + 129016*x^4 - 87451*x^3 + 105638*x^2 - 40729*x + 17407)
gp: K = bnfinit(y^16 - 2*y^15 + 34*y^14 - 86*y^13 + 539*y^12 - 1176*y^11 + 4498*y^10 - 8465*y^9 + 22378*y^8 - 36402*y^7 + 69763*y^6 - 83044*y^5 + 129016*y^4 - 87451*y^3 + 105638*y^2 - 40729*y + 17407, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 2*x^15 + 34*x^14 - 86*x^13 + 539*x^12 - 1176*x^11 + 4498*x^10 - 8465*x^9 + 22378*x^8 - 36402*x^7 + 69763*x^6 - 83044*x^5 + 129016*x^4 - 87451*x^3 + 105638*x^2 - 40729*x + 17407);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 2*x^15 + 34*x^14 - 86*x^13 + 539*x^12 - 1176*x^11 + 4498*x^10 - 8465*x^9 + 22378*x^8 - 36402*x^7 + 69763*x^6 - 83044*x^5 + 129016*x^4 - 87451*x^3 + 105638*x^2 - 40729*x + 17407)
\( x^{16} - 2 x^{15} + 34 x^{14} - 86 x^{13} + 539 x^{12} - 1176 x^{11} + 4498 x^{10} - 8465 x^{9} + \cdots + 17407 \)
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: |
\(137350965859713069141239809\)
\(\medspace = 13^{8}\cdot 17^{14}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(43.01\) | 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: | $13^{1/2}17^{7/8}\approx 43.014460711706995$ | ||
| Ramified primes: |
\(13\), \(17\)
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2}a^{12}-\frac{1}{2}a^{9}-\frac{1}{2}a^{6}-\frac{1}{2}a^{3}-\frac{1}{2}$, $\frac{1}{2}a^{13}-\frac{1}{2}a^{10}-\frac{1}{2}a^{7}-\frac{1}{2}a^{4}-\frac{1}{2}a$, $\frac{1}{2678}a^{14}-\frac{131}{1339}a^{13}+\frac{186}{1339}a^{12}-\frac{671}{2678}a^{11}+\frac{367}{1339}a^{10}-\frac{315}{1339}a^{9}+\frac{63}{206}a^{8}-\frac{599}{1339}a^{7}+\frac{295}{1339}a^{6}-\frac{483}{2678}a^{5}+\frac{191}{1339}a^{4}-\frac{4}{103}a^{3}+\frac{251}{2678}a^{2}-\frac{22}{103}a$, $\frac{1}{14\!\cdots\!74}a^{15}-\frac{69\!\cdots\!03}{71\!\cdots\!37}a^{14}-\frac{41\!\cdots\!88}{71\!\cdots\!37}a^{13}-\frac{28\!\cdots\!25}{13\!\cdots\!58}a^{12}+\frac{18\!\cdots\!84}{70\!\cdots\!37}a^{11}-\frac{26\!\cdots\!17}{71\!\cdots\!37}a^{10}+\frac{62\!\cdots\!97}{14\!\cdots\!74}a^{9}-\frac{30\!\cdots\!49}{71\!\cdots\!37}a^{8}+\frac{10\!\cdots\!76}{71\!\cdots\!37}a^{7}+\frac{92\!\cdots\!21}{14\!\cdots\!74}a^{6}-\frac{10\!\cdots\!99}{71\!\cdots\!37}a^{5}+\frac{56\!\cdots\!71}{15\!\cdots\!71}a^{4}-\frac{18\!\cdots\!81}{14\!\cdots\!74}a^{3}-\frac{60\!\cdots\!42}{71\!\cdots\!37}a^{2}+\frac{10\!\cdots\!50}{54\!\cdots\!49}a+\frac{13\!\cdots\!50}{53\!\cdots\!83}$
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
$C_{2}\times C_{2}\times C_{4}$, which has order $16$ (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: |
$\frac{22\!\cdots\!54}{56\!\cdots\!49}a^{15}-\frac{19\!\cdots\!79}{86\!\cdots\!46}a^{14}+\frac{28\!\cdots\!00}{56\!\cdots\!49}a^{13}-\frac{77\!\cdots\!81}{10\!\cdots\!66}a^{12}+\frac{20\!\cdots\!79}{11\!\cdots\!98}a^{11}-\frac{56\!\cdots\!16}{56\!\cdots\!49}a^{10}+\frac{24\!\cdots\!31}{11\!\cdots\!98}a^{9}-\frac{80\!\cdots\!01}{11\!\cdots\!98}a^{8}+\frac{79\!\cdots\!36}{56\!\cdots\!49}a^{7}-\frac{33\!\cdots\!09}{11\!\cdots\!98}a^{6}+\frac{61\!\cdots\!77}{11\!\cdots\!98}a^{5}-\frac{93\!\cdots\!05}{12\!\cdots\!67}a^{4}+\frac{11\!\cdots\!25}{11\!\cdots\!98}a^{3}-\frac{15\!\cdots\!83}{11\!\cdots\!98}a^{2}+\frac{26\!\cdots\!21}{43\!\cdots\!73}a-\frac{60\!\cdots\!91}{84\!\cdots\!82}$, $\frac{51\!\cdots\!62}{56\!\cdots\!49}a^{15}-\frac{13\!\cdots\!39}{43\!\cdots\!73}a^{14}+\frac{37\!\cdots\!22}{56\!\cdots\!49}a^{13}-\frac{54\!\cdots\!41}{54\!\cdots\!83}a^{12}+\frac{13\!\cdots\!11}{55\!\cdots\!49}a^{11}-\frac{83\!\cdots\!22}{56\!\cdots\!49}a^{10}+\frac{17\!\cdots\!95}{56\!\cdots\!49}a^{9}-\frac{69\!\cdots\!98}{56\!\cdots\!49}a^{8}+\frac{13\!\cdots\!34}{56\!\cdots\!49}a^{7}-\frac{34\!\cdots\!57}{56\!\cdots\!49}a^{6}+\frac{58\!\cdots\!87}{56\!\cdots\!49}a^{5}-\frac{21\!\cdots\!70}{12\!\cdots\!67}a^{4}+\frac{11\!\cdots\!21}{56\!\cdots\!49}a^{3}-\frac{17\!\cdots\!84}{56\!\cdots\!49}a^{2}+\frac{58\!\cdots\!46}{43\!\cdots\!73}a-\frac{79\!\cdots\!15}{42\!\cdots\!91}$, $\frac{80\!\cdots\!28}{56\!\cdots\!49}a^{15}-\frac{83\!\cdots\!89}{86\!\cdots\!46}a^{14}+\frac{32\!\cdots\!50}{56\!\cdots\!49}a^{13}-\frac{37\!\cdots\!77}{10\!\cdots\!66}a^{12}+\frac{13\!\cdots\!09}{11\!\cdots\!98}a^{11}-\frac{27\!\cdots\!84}{56\!\cdots\!49}a^{10}+\frac{13\!\cdots\!99}{11\!\cdots\!98}a^{9}-\frac{40\!\cdots\!47}{11\!\cdots\!98}a^{8}+\frac{38\!\cdots\!78}{56\!\cdots\!49}a^{7}-\frac{17\!\cdots\!81}{11\!\cdots\!98}a^{6}+\frac{27\!\cdots\!87}{11\!\cdots\!98}a^{5}-\frac{48\!\cdots\!96}{12\!\cdots\!67}a^{4}+\frac{49\!\cdots\!61}{11\!\cdots\!98}a^{3}-\frac{70\!\cdots\!09}{11\!\cdots\!98}a^{2}+\frac{11\!\cdots\!96}{43\!\cdots\!73}a-\frac{37\!\cdots\!31}{84\!\cdots\!82}$, $\frac{10\!\cdots\!77}{14\!\cdots\!74}a^{15}-\frac{66\!\cdots\!58}{71\!\cdots\!37}a^{14}+\frac{41\!\cdots\!47}{14\!\cdots\!74}a^{13}-\frac{39\!\cdots\!91}{13\!\cdots\!58}a^{12}+\frac{45\!\cdots\!50}{70\!\cdots\!37}a^{11}-\frac{50\!\cdots\!91}{14\!\cdots\!74}a^{10}+\frac{74\!\cdots\!39}{14\!\cdots\!74}a^{9}-\frac{14\!\cdots\!63}{71\!\cdots\!37}a^{8}+\frac{29\!\cdots\!61}{14\!\cdots\!74}a^{7}-\frac{89\!\cdots\!43}{14\!\cdots\!74}a^{6}+\frac{22\!\cdots\!55}{71\!\cdots\!37}a^{5}-\frac{16\!\cdots\!13}{30\!\cdots\!42}a^{4}-\frac{21\!\cdots\!07}{14\!\cdots\!74}a^{3}+\frac{39\!\cdots\!29}{71\!\cdots\!37}a^{2}-\frac{57\!\cdots\!17}{10\!\cdots\!98}a+\frac{61\!\cdots\!70}{53\!\cdots\!83}$, $\frac{90\!\cdots\!98}{71\!\cdots\!37}a^{15}-\frac{15\!\cdots\!67}{14\!\cdots\!74}a^{14}+\frac{69\!\cdots\!35}{14\!\cdots\!74}a^{13}-\frac{48\!\cdots\!91}{13\!\cdots\!58}a^{12}+\frac{14\!\cdots\!41}{14\!\cdots\!74}a^{11}-\frac{62\!\cdots\!91}{14\!\cdots\!74}a^{10}+\frac{12\!\cdots\!75}{14\!\cdots\!74}a^{9}-\frac{37\!\cdots\!65}{14\!\cdots\!74}a^{8}+\frac{55\!\cdots\!57}{14\!\cdots\!74}a^{7}-\frac{12\!\cdots\!39}{14\!\cdots\!74}a^{6}+\frac{14\!\cdots\!97}{14\!\cdots\!74}a^{5}-\frac{43\!\cdots\!77}{30\!\cdots\!42}a^{4}+\frac{18\!\cdots\!39}{14\!\cdots\!74}a^{3}-\frac{18\!\cdots\!23}{14\!\cdots\!74}a^{2}-\frac{10\!\cdots\!53}{10\!\cdots\!98}a+\frac{24\!\cdots\!25}{10\!\cdots\!66}$, $\frac{94\!\cdots\!78}{71\!\cdots\!37}a^{15}-\frac{24\!\cdots\!89}{71\!\cdots\!37}a^{14}+\frac{32\!\cdots\!04}{71\!\cdots\!37}a^{13}-\frac{94\!\cdots\!28}{69\!\cdots\!79}a^{12}+\frac{51\!\cdots\!37}{70\!\cdots\!37}a^{11}-\frac{12\!\cdots\!73}{71\!\cdots\!37}a^{10}+\frac{42\!\cdots\!47}{71\!\cdots\!37}a^{9}-\frac{85\!\cdots\!56}{71\!\cdots\!37}a^{8}+\frac{19\!\cdots\!78}{71\!\cdots\!37}a^{7}-\frac{33\!\cdots\!71}{71\!\cdots\!37}a^{6}+\frac{55\!\cdots\!68}{71\!\cdots\!37}a^{5}-\frac{13\!\cdots\!79}{15\!\cdots\!71}a^{4}+\frac{76\!\cdots\!46}{71\!\cdots\!37}a^{3}-\frac{40\!\cdots\!35}{71\!\cdots\!37}a^{2}+\frac{11\!\cdots\!40}{54\!\cdots\!49}a-\frac{12\!\cdots\!22}{53\!\cdots\!83}$, $\frac{79\!\cdots\!55}{71\!\cdots\!37}a^{15}-\frac{32\!\cdots\!02}{71\!\cdots\!37}a^{14}+\frac{26\!\cdots\!58}{71\!\cdots\!37}a^{13}-\frac{11\!\cdots\!57}{69\!\cdots\!79}a^{12}+\frac{43\!\cdots\!76}{70\!\cdots\!37}a^{11}-\frac{14\!\cdots\!36}{71\!\cdots\!37}a^{10}+\frac{33\!\cdots\!37}{71\!\cdots\!37}a^{9}-\frac{92\!\cdots\!21}{71\!\cdots\!37}a^{8}+\frac{14\!\cdots\!58}{71\!\cdots\!37}a^{7}-\frac{30\!\cdots\!44}{71\!\cdots\!37}a^{6}+\frac{31\!\cdots\!21}{71\!\cdots\!37}a^{5}-\frac{77\!\cdots\!91}{15\!\cdots\!71}a^{4}-\frac{10\!\cdots\!38}{71\!\cdots\!37}a^{3}+\frac{16\!\cdots\!33}{71\!\cdots\!37}a^{2}-\frac{13\!\cdots\!98}{54\!\cdots\!49}a+\frac{31\!\cdots\!33}{53\!\cdots\!83}$
| 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: | \( 232376.238125 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 232376.238125 \cdot 16}{2\cdot\sqrt{137350965859713069141239809}}\cr\approx \mathstrut & 0.385305130384 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + 34*x^14 - 86*x^13 + 539*x^12 - 1176*x^11 + 4498*x^10 - 8465*x^9 + 22378*x^8 - 36402*x^7 + 69763*x^6 - 83044*x^5 + 129016*x^4 - 87451*x^3 + 105638*x^2 - 40729*x + 17407)
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 - 2*x^15 + 34*x^14 - 86*x^13 + 539*x^12 - 1176*x^11 + 4498*x^10 - 8465*x^9 + 22378*x^8 - 36402*x^7 + 69763*x^6 - 83044*x^5 + 129016*x^4 - 87451*x^3 + 105638*x^2 - 40729*x + 17407, 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 - 2*x^15 + 34*x^14 - 86*x^13 + 539*x^12 - 1176*x^11 + 4498*x^10 - 8465*x^9 + 22378*x^8 - 36402*x^7 + 69763*x^6 - 83044*x^5 + 129016*x^4 - 87451*x^3 + 105638*x^2 - 40729*x + 17407);
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 - 2*x^15 + 34*x^14 - 86*x^13 + 539*x^12 - 1176*x^11 + 4498*x^10 - 8465*x^9 + 22378*x^8 - 36402*x^7 + 69763*x^6 - 83044*x^5 + 129016*x^4 - 87451*x^3 + 105638*x^2 - 40729*x + 17407);
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
| \(\Q(\sqrt{17}) \), 4.4.4913.1, 4.0.3757.1, 4.0.63869.1, 8.8.11719682839553.1, 8.0.69347235737.1, 8.0.4079249161.1 |
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: | 16.0.4809039104363049933169.2 |
| Minimal sibling: | 16.0.4809039104363049933169.2 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.4.0.1}{4} }^{4}$ | ${\href{/padicField/3.8.0.1}{8} }^{2}$ | ${\href{/padicField/5.8.0.1}{8} }^{2}$ | ${\href{/padicField/7.8.0.1}{8} }^{2}$ | ${\href{/padicField/11.8.0.1}{8} }^{2}$ | R | R | ${\href{/padicField/19.4.0.1}{4} }^{4}$ | ${\href{/padicField/23.8.0.1}{8} }^{2}$ | ${\href{/padicField/29.8.0.1}{8} }^{2}$ | ${\href{/padicField/31.8.0.1}{8} }^{2}$ | ${\href{/padicField/37.8.0.1}{8} }^{2}$ | ${\href{/padicField/41.8.0.1}{8} }^{2}$ | ${\href{/padicField/43.4.0.1}{4} }^{4}$ | ${\href{/padicField/47.2.0.1}{2} }^{4}{,}\,{\href{/padicField/47.1.0.1}{1} }^{8}$ | ${\href{/padicField/53.4.0.1}{4} }^{4}$ | ${\href{/padicField/59.4.0.1}{4} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(13\)
| 13.2.1.2 | $x^{2} + 26$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 13.2.1.2 | $x^{2} + 26$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 13.2.1.2 | $x^{2} + 26$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 13.2.1.2 | $x^{2} + 26$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 13.4.2.1 | $x^{4} + 284 x^{3} + 21754 x^{2} + 225780 x + 59193$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 13.4.2.1 | $x^{4} + 284 x^{3} + 21754 x^{2} + 225780 x + 59193$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
|
\(17\)
| 17.8.7.3 | $x^{8} + 17$ | $8$ | $1$ | $7$ | $C_8$ | $[\ ]_{8}$ |
| 17.8.7.3 | $x^{8} + 17$ | $8$ | $1$ | $7$ | $C_8$ | $[\ ]_{8}$ |