Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - x^11 - 100*x^10 + 99*x^9 + 3340*x^8 - 2877*x^7 - 46644*x^6 + 33812*x^5 + 266073*x^4 - 180321*x^3 - 517506*x^2 + 369428*x + 16999)
gp: K = bnfinit(y^12 - y^11 - 100*y^10 + 99*y^9 + 3340*y^8 - 2877*y^7 - 46644*y^6 + 33812*y^5 + 266073*y^4 - 180321*y^3 - 517506*y^2 + 369428*y + 16999, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - x^11 - 100*x^10 + 99*x^9 + 3340*x^8 - 2877*x^7 - 46644*x^6 + 33812*x^5 + 266073*x^4 - 180321*x^3 - 517506*x^2 + 369428*x + 16999);
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^12 - x^11 - 100*x^10 + 99*x^9 + 3340*x^8 - 2877*x^7 - 46644*x^6 + 33812*x^5 + 266073*x^4 - 180321*x^3 - 517506*x^2 + 369428*x + 16999)
\( x^{12} - x^{11} - 100 x^{10} + 99 x^{9} + 3340 x^{8} - 2877 x^{7} - 46644 x^{6} + 33812 x^{5} + \cdots + 16999 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $12$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[12, 0]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(22734163157293282124804657\)
\(\medspace = 17^{9}\cdot 61^{8}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(129.73\) | 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: | $17^{3/4}61^{2/3}\approx 129.73482819609035$ | ||
| Ramified primes: |
\(17\), \(61\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{17}) \) | ||
| $\card{ \Gal(K/\Q) }$: | $12$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(1037=17\cdot 61\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{1037}(1,·)$, $\chi_{1037}(611,·)$, $\chi_{1037}(135,·)$, $\chi_{1037}(169,·)$, $\chi_{1037}(684,·)$, $\chi_{1037}(13,·)$, $\chi_{1037}(718,·)$, $\chi_{1037}(47,·)$, $\chi_{1037}(562,·)$, $\chi_{1037}(596,·)$, $\chi_{1037}(489,·)$, $\chi_{1037}(123,·)$$\rbrace$ | ||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3}a^{8}-\frac{1}{3}a^{6}-\frac{1}{3}a^{5}+\frac{1}{3}a^{4}+\frac{1}{3}a^{3}+\frac{1}{3}a-\frac{1}{3}$, $\frac{1}{6}a^{9}-\frac{1}{6}a^{8}-\frac{1}{6}a^{7}-\frac{1}{2}a^{6}+\frac{1}{3}a^{5}-\frac{1}{2}a^{4}-\frac{1}{6}a^{3}+\frac{1}{6}a^{2}+\frac{1}{6}a+\frac{1}{6}$, $\frac{1}{6}a^{10}+\frac{1}{3}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{3}a^{4}+\frac{1}{3}a^{3}+\frac{1}{3}a^{2}-\frac{1}{3}a-\frac{1}{6}$, $\frac{1}{57\!\cdots\!86}a^{11}+\frac{18\!\cdots\!63}{57\!\cdots\!86}a^{10}+\frac{41\!\cdots\!15}{57\!\cdots\!86}a^{9}-\frac{57\!\cdots\!93}{57\!\cdots\!86}a^{8}-\frac{10\!\cdots\!76}{95\!\cdots\!81}a^{7}-\frac{64\!\cdots\!13}{57\!\cdots\!86}a^{6}+\frac{67\!\cdots\!77}{19\!\cdots\!62}a^{5}-\frac{99\!\cdots\!17}{57\!\cdots\!86}a^{4}+\frac{13\!\cdots\!61}{57\!\cdots\!86}a^{3}-\frac{23\!\cdots\!03}{57\!\cdots\!86}a^{2}-\frac{24\!\cdots\!04}{28\!\cdots\!43}a-\frac{48\!\cdots\!21}{32\!\cdots\!87}$
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 $1$ (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: | $11$ | 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: |
$\frac{98\!\cdots\!63}{19\!\cdots\!62}a^{11}-\frac{12\!\cdots\!81}{19\!\cdots\!62}a^{10}-\frac{45\!\cdots\!32}{95\!\cdots\!81}a^{9}+\frac{51\!\cdots\!93}{95\!\cdots\!81}a^{8}+\frac{25\!\cdots\!21}{19\!\cdots\!62}a^{7}-\frac{90\!\cdots\!40}{95\!\cdots\!81}a^{6}-\frac{21\!\cdots\!19}{19\!\cdots\!62}a^{5}-\frac{31\!\cdots\!54}{95\!\cdots\!81}a^{4}-\frac{10\!\cdots\!45}{95\!\cdots\!81}a^{3}+\frac{10\!\cdots\!62}{95\!\cdots\!81}a^{2}+\frac{41\!\cdots\!13}{19\!\cdots\!62}a-\frac{60\!\cdots\!23}{21\!\cdots\!58}$, $\frac{20\!\cdots\!36}{95\!\cdots\!81}a^{11}-\frac{15\!\cdots\!66}{95\!\cdots\!81}a^{10}-\frac{19\!\cdots\!74}{95\!\cdots\!81}a^{9}+\frac{14\!\cdots\!22}{95\!\cdots\!81}a^{8}+\frac{63\!\cdots\!16}{95\!\cdots\!81}a^{7}-\frac{36\!\cdots\!84}{95\!\cdots\!81}a^{6}-\frac{83\!\cdots\!60}{95\!\cdots\!81}a^{5}+\frac{31\!\cdots\!63}{95\!\cdots\!81}a^{4}+\frac{43\!\cdots\!12}{95\!\cdots\!81}a^{3}-\frac{11\!\cdots\!47}{95\!\cdots\!81}a^{2}-\frac{72\!\cdots\!47}{95\!\cdots\!81}a+\frac{18\!\cdots\!85}{10\!\cdots\!29}$, $\frac{50\!\cdots\!35}{19\!\cdots\!62}a^{11}-\frac{44\!\cdots\!13}{19\!\cdots\!62}a^{10}-\frac{24\!\cdots\!06}{95\!\cdots\!81}a^{9}+\frac{20\!\cdots\!15}{95\!\cdots\!81}a^{8}+\frac{15\!\cdots\!53}{19\!\cdots\!62}a^{7}-\frac{45\!\cdots\!24}{95\!\cdots\!81}a^{6}-\frac{18\!\cdots\!39}{19\!\cdots\!62}a^{5}+\frac{28\!\cdots\!09}{95\!\cdots\!81}a^{4}+\frac{42\!\cdots\!67}{95\!\cdots\!81}a^{3}-\frac{96\!\cdots\!85}{95\!\cdots\!81}a^{2}-\frac{10\!\cdots\!81}{19\!\cdots\!62}a-\frac{18\!\cdots\!95}{21\!\cdots\!58}$, $\frac{89\!\cdots\!22}{28\!\cdots\!43}a^{11}+\frac{23\!\cdots\!45}{57\!\cdots\!86}a^{10}-\frac{10\!\cdots\!10}{28\!\cdots\!43}a^{9}-\frac{36\!\cdots\!95}{95\!\cdots\!81}a^{8}+\frac{46\!\cdots\!52}{28\!\cdots\!43}a^{7}+\frac{66\!\cdots\!83}{57\!\cdots\!86}a^{6}-\frac{16\!\cdots\!73}{57\!\cdots\!86}a^{5}-\frac{37\!\cdots\!74}{28\!\cdots\!43}a^{4}+\frac{60\!\cdots\!32}{28\!\cdots\!43}a^{3}+\frac{12\!\cdots\!92}{28\!\cdots\!43}a^{2}-\frac{42\!\cdots\!18}{95\!\cdots\!81}a-\frac{12\!\cdots\!83}{64\!\cdots\!74}$, $\frac{28\!\cdots\!95}{57\!\cdots\!86}a^{11}-\frac{33\!\cdots\!24}{28\!\cdots\!43}a^{10}-\frac{48\!\cdots\!69}{95\!\cdots\!81}a^{9}+\frac{32\!\cdots\!46}{28\!\cdots\!43}a^{8}+\frac{10\!\cdots\!57}{57\!\cdots\!86}a^{7}-\frac{62\!\cdots\!69}{19\!\cdots\!62}a^{6}-\frac{75\!\cdots\!07}{28\!\cdots\!43}a^{5}+\frac{10\!\cdots\!37}{28\!\cdots\!43}a^{4}+\frac{15\!\cdots\!18}{95\!\cdots\!81}a^{3}-\frac{49\!\cdots\!78}{28\!\cdots\!43}a^{2}-\frac{19\!\cdots\!37}{57\!\cdots\!86}a+\frac{92\!\cdots\!77}{32\!\cdots\!87}$, $\frac{15\!\cdots\!37}{57\!\cdots\!86}a^{11}-\frac{10\!\cdots\!73}{28\!\cdots\!43}a^{10}-\frac{13\!\cdots\!99}{57\!\cdots\!86}a^{9}+\frac{68\!\cdots\!81}{19\!\cdots\!62}a^{8}+\frac{16\!\cdots\!81}{28\!\cdots\!43}a^{7}-\frac{10\!\cdots\!03}{95\!\cdots\!81}a^{6}-\frac{10\!\cdots\!63}{28\!\cdots\!43}a^{5}+\frac{68\!\cdots\!25}{57\!\cdots\!86}a^{4}-\frac{11\!\cdots\!61}{57\!\cdots\!86}a^{3}-\frac{22\!\cdots\!01}{57\!\cdots\!86}a^{2}+\frac{21\!\cdots\!37}{95\!\cdots\!81}a+\frac{13\!\cdots\!21}{21\!\cdots\!58}$, $\frac{71\!\cdots\!32}{95\!\cdots\!81}a^{11}-\frac{48\!\cdots\!09}{95\!\cdots\!81}a^{10}-\frac{13\!\cdots\!87}{19\!\cdots\!62}a^{9}+\frac{11\!\cdots\!47}{19\!\cdots\!62}a^{8}+\frac{43\!\cdots\!77}{19\!\cdots\!62}a^{7}+\frac{12\!\cdots\!25}{19\!\cdots\!62}a^{6}-\frac{26\!\cdots\!80}{95\!\cdots\!81}a^{5}-\frac{84\!\cdots\!33}{19\!\cdots\!62}a^{4}+\frac{20\!\cdots\!53}{19\!\cdots\!62}a^{3}+\frac{43\!\cdots\!25}{19\!\cdots\!62}a^{2}-\frac{12\!\cdots\!75}{19\!\cdots\!62}a-\frac{18\!\cdots\!75}{21\!\cdots\!58}$, $\frac{50\!\cdots\!01}{57\!\cdots\!86}a^{11}+\frac{13\!\cdots\!74}{28\!\cdots\!43}a^{10}-\frac{24\!\cdots\!26}{28\!\cdots\!43}a^{9}-\frac{13\!\cdots\!01}{28\!\cdots\!43}a^{8}+\frac{15\!\cdots\!53}{57\!\cdots\!86}a^{7}+\frac{10\!\cdots\!39}{57\!\cdots\!86}a^{6}-\frac{33\!\cdots\!86}{95\!\cdots\!81}a^{5}-\frac{23\!\cdots\!59}{95\!\cdots\!81}a^{4}+\frac{15\!\cdots\!68}{95\!\cdots\!81}a^{3}+\frac{20\!\cdots\!76}{28\!\cdots\!43}a^{2}-\frac{15\!\cdots\!13}{57\!\cdots\!86}a-\frac{26\!\cdots\!20}{32\!\cdots\!87}$, $\frac{36\!\cdots\!07}{57\!\cdots\!86}a^{11}-\frac{11\!\cdots\!94}{28\!\cdots\!43}a^{10}-\frac{11\!\cdots\!57}{19\!\cdots\!62}a^{9}+\frac{73\!\cdots\!57}{57\!\cdots\!86}a^{8}+\frac{58\!\cdots\!91}{28\!\cdots\!43}a^{7}-\frac{20\!\cdots\!44}{28\!\cdots\!43}a^{6}-\frac{80\!\cdots\!04}{28\!\cdots\!43}a^{5}+\frac{24\!\cdots\!97}{19\!\cdots\!62}a^{4}+\frac{92\!\cdots\!33}{57\!\cdots\!86}a^{3}-\frac{53\!\cdots\!31}{57\!\cdots\!86}a^{2}-\frac{30\!\cdots\!58}{95\!\cdots\!81}a+\frac{15\!\cdots\!97}{64\!\cdots\!74}$, $\frac{45\!\cdots\!15}{57\!\cdots\!86}a^{11}-\frac{44\!\cdots\!21}{95\!\cdots\!81}a^{10}-\frac{14\!\cdots\!93}{19\!\cdots\!62}a^{9}+\frac{24\!\cdots\!89}{57\!\cdots\!86}a^{8}+\frac{21\!\cdots\!55}{95\!\cdots\!81}a^{7}-\frac{11\!\cdots\!56}{95\!\cdots\!81}a^{6}-\frac{64\!\cdots\!40}{28\!\cdots\!43}a^{5}+\frac{65\!\cdots\!03}{57\!\cdots\!86}a^{4}+\frac{18\!\cdots\!67}{57\!\cdots\!86}a^{3}-\frac{19\!\cdots\!03}{57\!\cdots\!86}a^{2}+\frac{53\!\cdots\!41}{28\!\cdots\!43}a+\frac{18\!\cdots\!53}{21\!\cdots\!58}$, $\frac{19\!\cdots\!05}{57\!\cdots\!86}a^{11}-\frac{78\!\cdots\!89}{95\!\cdots\!81}a^{10}-\frac{88\!\cdots\!27}{28\!\cdots\!43}a^{9}+\frac{22\!\cdots\!68}{28\!\cdots\!43}a^{8}+\frac{50\!\cdots\!25}{57\!\cdots\!86}a^{7}-\frac{41\!\cdots\!79}{19\!\cdots\!62}a^{6}-\frac{84\!\cdots\!89}{95\!\cdots\!81}a^{5}+\frac{63\!\cdots\!61}{28\!\cdots\!43}a^{4}+\frac{52\!\cdots\!74}{28\!\cdots\!43}a^{3}-\frac{18\!\cdots\!00}{28\!\cdots\!43}a^{2}+\frac{60\!\cdots\!83}{19\!\cdots\!62}a+\frac{47\!\cdots\!60}{32\!\cdots\!87}$
| 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: | \( 319721807.832 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{12}\cdot(2\pi)^{0}\cdot 319721807.832 \cdot 1}{2\cdot\sqrt{22734163157293282124804657}}\cr\approx \mathstrut & 0.137329148115 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^12 - x^11 - 100*x^10 + 99*x^9 + 3340*x^8 - 2877*x^7 - 46644*x^6 + 33812*x^5 + 266073*x^4 - 180321*x^3 - 517506*x^2 + 369428*x + 16999)
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^12 - x^11 - 100*x^10 + 99*x^9 + 3340*x^8 - 2877*x^7 - 46644*x^6 + 33812*x^5 + 266073*x^4 - 180321*x^3 - 517506*x^2 + 369428*x + 16999, 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^12 - x^11 - 100*x^10 + 99*x^9 + 3340*x^8 - 2877*x^7 - 46644*x^6 + 33812*x^5 + 266073*x^4 - 180321*x^3 - 517506*x^2 + 369428*x + 16999);
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^12 - x^11 - 100*x^10 + 99*x^9 + 3340*x^8 - 2877*x^7 - 46644*x^6 + 33812*x^5 + 266073*x^4 - 180321*x^3 - 517506*x^2 + 369428*x + 16999);
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
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 cyclic group of order 12 |
| The 12 conjugacy class representatives for $C_{12}$ |
| Character table for $C_{12}$ |
Intermediate fields
| \(\Q(\sqrt{17}) \), 3.3.3721.1, 4.4.4913.1, 6.6.68024616833.1 |
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]
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.6.0.1}{6} }^{2}$ | ${\href{/padicField/3.4.0.1}{4} }^{3}$ | ${\href{/padicField/5.12.0.1}{12} }$ | ${\href{/padicField/7.12.0.1}{12} }$ | ${\href{/padicField/11.4.0.1}{4} }^{3}$ | ${\href{/padicField/13.3.0.1}{3} }^{4}$ | R | ${\href{/padicField/19.6.0.1}{6} }^{2}$ | ${\href{/padicField/23.4.0.1}{4} }^{3}$ | ${\href{/padicField/29.12.0.1}{12} }$ | ${\href{/padicField/31.12.0.1}{12} }$ | ${\href{/padicField/37.4.0.1}{4} }^{3}$ | ${\href{/padicField/41.4.0.1}{4} }^{3}$ | ${\href{/padicField/43.6.0.1}{6} }^{2}$ | ${\href{/padicField/47.3.0.1}{3} }^{4}$ | ${\href{/padicField/53.2.0.1}{2} }^{6}$ | ${\href{/padicField/59.6.0.1}{6} }^{2}$ |
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
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
|
\(17\)
| 17.12.9.1 | $x^{12} + 4 x^{10} + 56 x^{9} + 57 x^{8} + 168 x^{7} + 1044 x^{6} - 11256 x^{5} + 3356 x^{4} + 10080 x^{3} + 97736 x^{2} + 58576 x + 57252$ | $4$ | $3$ | $9$ | $C_{12}$ | $[\ ]_{4}^{3}$ |
|
\(61\)
| 61.12.8.1 | $x^{12} + 9 x^{10} + 364 x^{9} + 33 x^{8} + 720 x^{7} - 16731 x^{6} - 1002 x^{5} - 64041 x^{4} - 1125646 x^{3} + 428523 x^{2} - 4549998 x + 62837938$ | $3$ | $4$ | $8$ | $C_{12}$ | $[\ ]_{3}^{4}$ |