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Properties

Label 21.17.185...661.1
Degree $21$
Signature $[17, 2]$
Discriminant $1.859\times 10^{39}$
Root discriminant \(74.13\)
Ramified primes see page
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $S_7\wr C_3.C_2$ (as 21T162)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 21*x^19 - 3*x^18 + 186*x^17 + 51*x^16 - 901*x^15 - 354*x^14 + 2589*x^13 + 1292*x^12 - 4480*x^11 - 2669*x^10 + 4528*x^9 + 3147*x^8 - 2428*x^7 - 2028*x^6 + 502*x^5 + 620*x^4 + 33*x^3 - 62*x^2 - 15*x - 1)
 
gp: K = bnfinit(y^21 - 21*y^19 - 3*y^18 + 186*y^17 + 51*y^16 - 901*y^15 - 354*y^14 + 2589*y^13 + 1292*y^12 - 4480*y^11 - 2669*y^10 + 4528*y^9 + 3147*y^8 - 2428*y^7 - 2028*y^6 + 502*y^5 + 620*y^4 + 33*y^3 - 62*y^2 - 15*y - 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^21 - 21*x^19 - 3*x^18 + 186*x^17 + 51*x^16 - 901*x^15 - 354*x^14 + 2589*x^13 + 1292*x^12 - 4480*x^11 - 2669*x^10 + 4528*x^9 + 3147*x^8 - 2428*x^7 - 2028*x^6 + 502*x^5 + 620*x^4 + 33*x^3 - 62*x^2 - 15*x - 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^21 - 21*x^19 - 3*x^18 + 186*x^17 + 51*x^16 - 901*x^15 - 354*x^14 + 2589*x^13 + 1292*x^12 - 4480*x^11 - 2669*x^10 + 4528*x^9 + 3147*x^8 - 2428*x^7 - 2028*x^6 + 502*x^5 + 620*x^4 + 33*x^3 - 62*x^2 - 15*x - 1)
 

\( x^{21} - 21 x^{19} - 3 x^{18} + 186 x^{17} + 51 x^{16} - 901 x^{15} - 354 x^{14} + 2589 x^{13} + \cdots - 1 \) Copy content Toggle raw display

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

Invariants

Degree:  $21$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[17, 2]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(1859129620558936360977052492338516111661\) \(\medspace = 67\cdot 229^{7}\cdot 4447\cdot 13291\cdot 2106919\cdot 6747049\) 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:  \(74.13\)
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:  $67^{1/2}229^{1/2}4447^{1/2}13291^{1/2}2106919^{1/2}6747049^{1/2}\approx 3590445907221.6016$
Ramified primes:   \(67\), \(229\), \(4447\), \(13291\), \(2106919\), \(6747049\) 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:  $\Q(\sqrt{12891\!\cdots\!40741}$)
$\card{ \Aut(K/\Q) }$:  $1$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$ Copy content Toggle raw display

sage: K.integral_basis()
 
gp: K.zk
 
magma: IntegralBasis(K);
 
oscar: basis(OK)
 

Monogenic:  Yes
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:  $18$
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:   $a$, $a+1$, $a^{19}-a^{18}-20a^{17}+17a^{16}+169a^{15}-118a^{14}-783a^{13}+429a^{12}+2160a^{11}-868a^{10}-3612a^{9}+943a^{8}+3585a^{7}-438a^{6}-1991a^{5}-37a^{4}+546a^{3}+75a^{2}-55a-10$, $a^{19}-a^{18}-20a^{17}+17a^{16}+169a^{15}-118a^{14}-783a^{13}+429a^{12}+2160a^{11}-868a^{10}-3612a^{9}+943a^{8}+3585a^{7}-438a^{6}-1990a^{5}-38a^{4}+541a^{3}+78a^{2}-50a-9$, $a^{19}-a^{18}-20a^{17}+17a^{16}+169a^{15}-118a^{14}-783a^{13}+429a^{12}+2160a^{11}-868a^{10}-3612a^{9}+943a^{8}+3585a^{7}-438a^{6}-1991a^{5}-37a^{4}+546a^{3}+75a^{2}-55a-8$, $a^{19}-a^{18}-20a^{17}+17a^{16}+169a^{15}-118a^{14}-783a^{13}+429a^{12}+2160a^{11}-868a^{10}-3612a^{9}+943a^{8}+3585a^{7}-438a^{6}-1990a^{5}-37a^{4}+540a^{3}+74a^{2}-47a-7$, $a^{20}-6a^{19}-15a^{18}+118a^{17}+82a^{16}-981a^{15}-158a^{14}+4478a^{13}-237a^{12}-12199a^{11}+1688a^{10}+20202a^{9}-3197a^{8}-19895a^{7}+2677a^{6}+10958a^{5}-772a^{4}-2980a^{3}-73a^{2}+288a+39$, $42a^{20}-27a^{19}-861a^{18}+424a^{17}+7466a^{16}-2598a^{15}-35545a^{14}+7567a^{13}+100930a^{12}-9092a^{11}-174052a^{10}-3384a^{9}+178220a^{8}+21297a^{7}-101251a^{6}-22211a^{5}+27118a^{4}+8907a^{3}-2031a^{2}-1141a-125$, $17a^{20}-3a^{19}-356a^{18}+11a^{17}+3152a^{16}+324a^{15}-15318a^{14}-3400a^{13}+44410a^{12}+14405a^{11}-78305a^{10}-32031a^{9}+82243a^{8}+39371a^{7}-48119a^{6}-26060a^{5}+13232a^{4}+8157a^{3}-956a^{2}-876a-89$, $2a^{20}-2a^{19}-39a^{18}+32a^{17}+321a^{16}-204a^{15}-1446a^{14}+652a^{13}+3863a^{12}-1056a^{11}-6201a^{10}+675a^{9}+5792a^{8}+234a^{7}-2883a^{6}-501a^{5}+613a^{4}+163a^{3}-30a^{2}-6a+1$, $10a^{20}-13a^{19}-198a^{18}+233a^{17}+1655a^{16}-1740a^{15}-7575a^{14}+7025a^{13}+20592a^{12}-16651a^{11}-33755a^{10}+23532a^{9}+32486a^{8}-19143a^{7}-17157a^{6}+8131a^{5}+4386a^{4}-1540a^{3}-431a^{2}+88a+19$, $2a^{20}-a^{19}-41a^{18}+15a^{17}+354a^{16}-86a^{15}-1668a^{14}+225a^{13}+4647a^{12}-209a^{11}-7764a^{10}-200a^{9}+7560a^{8}+634a^{7}-3963a^{6}-600a^{5}+905a^{4}+253a^{3}-21a^{2}-39a-9$, $6a^{20}+5a^{19}-132a^{18}-118a^{17}+1223a^{16}+1152a^{15}-6200a^{14}-6055a^{13}+18715a^{12}+18652a^{11}-34353a^{10}-34380a^{9}+37627a^{8}+37278a^{7}-22975a^{6}-22468a^{5}+6488a^{4}+6564a^{3}-391a^{2}-665a-71$, $34a^{20}-27a^{19}-691a^{18}+444a^{17}+5943a^{16}-2940a^{15}-28080a^{14}+9954a^{13}+79186a^{12}-17849a^{11}-135734a^{10}+14801a^{9}+138335a^{8}-291a^{7}-78515a^{6}-8139a^{5}+21337a^{4}+4505a^{3}-1807a^{2}-661a-55$, $16a^{20}-8a^{19}-331a^{18}+116a^{17}+2899a^{16}-608a^{15}-13959a^{14}+1137a^{13}+40176a^{12}+1249a^{11}-70491a^{10}-8883a^{9}+73916a^{8}+15164a^{7}-43484a^{6}-11921a^{5}+12329a^{4}+4149a^{3}-1083a^{2}-482a-42$, $7a^{20}-8a^{19}-141a^{18}+142a^{17}+1204a^{16}-1048a^{15}-5664a^{14}+4173a^{13}+15972a^{12}-9742a^{11}-27557a^{10}+13564a^{9}+28564a^{8}-10876a^{7}-16817a^{6}+4479a^{5}+4994a^{4}-701a^{3}-564a^{2}-14a+6$, $2a^{20}-a^{19}-41a^{18}+14a^{17}+355a^{16}-67a^{15}-1684a^{14}+75a^{13}+4749a^{12}+424a^{11}-8092a^{10}-1726a^{9}+8113a^{8}+2710a^{7}-4420a^{6}-2073a^{5}+1055a^{4}+714a^{3}-36a^{2}-84a-10$, $23a^{20}-4a^{19}-483a^{18}+16a^{17}+4288a^{16}+411a^{15}-20893a^{14}-4401a^{13}+60725a^{12}+18784a^{11}-107323a^{10}-42001a^{9}+112940a^{8}+51945a^{7}-66153a^{6}-34666a^{5}+18182a^{4}+10972a^{3}-1295a^{2}-1195a-124$ Copy content Toggle raw display (assuming GRH)
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:  \( 5933753579790 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{17}\cdot(2\pi)^{2}\cdot 5933753579790 \cdot 1}{2\cdot\sqrt{1859129620558936360977052492338516111661}}\cr\approx \mathstrut & 0.356052745483159 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^21 - 21*x^19 - 3*x^18 + 186*x^17 + 51*x^16 - 901*x^15 - 354*x^14 + 2589*x^13 + 1292*x^12 - 4480*x^11 - 2669*x^10 + 4528*x^9 + 3147*x^8 - 2428*x^7 - 2028*x^6 + 502*x^5 + 620*x^4 + 33*x^3 - 62*x^2 - 15*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^21 - 21*x^19 - 3*x^18 + 186*x^17 + 51*x^16 - 901*x^15 - 354*x^14 + 2589*x^13 + 1292*x^12 - 4480*x^11 - 2669*x^10 + 4528*x^9 + 3147*x^8 - 2428*x^7 - 2028*x^6 + 502*x^5 + 620*x^4 + 33*x^3 - 62*x^2 - 15*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^21 - 21*x^19 - 3*x^18 + 186*x^17 + 51*x^16 - 901*x^15 - 354*x^14 + 2589*x^13 + 1292*x^12 - 4480*x^11 - 2669*x^10 + 4528*x^9 + 3147*x^8 - 2428*x^7 - 2028*x^6 + 502*x^5 + 620*x^4 + 33*x^3 - 62*x^2 - 15*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^21 - 21*x^19 - 3*x^18 + 186*x^17 + 51*x^16 - 901*x^15 - 354*x^14 + 2589*x^13 + 1292*x^12 - 4480*x^11 - 2669*x^10 + 4528*x^9 + 3147*x^8 - 2428*x^7 - 2028*x^6 + 502*x^5 + 620*x^4 + 33*x^3 - 62*x^2 - 15*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

$S_7\wr C_3.C_2$ (as 21T162):

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 non-solvable group of order 768144384000
The 920 conjugacy class representatives for $S_7\wr C_3.C_2$
Character table for $S_7\wr C_3.C_2$

Intermediate fields

3.3.229.1

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 42 siblings: data not computed
Minimal sibling: This field is its own minimal sibling

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/padicField/2.8.0.1}{8} }{,}\,{\href{/padicField/2.7.0.1}{7} }{,}\,{\href{/padicField/2.4.0.1}{4} }{,}\,{\href{/padicField/2.2.0.1}{2} }$ $15{,}\,{\href{/padicField/3.3.0.1}{3} }^{2}$ $15{,}\,{\href{/padicField/5.3.0.1}{3} }^{2}$ ${\href{/padicField/7.6.0.1}{6} }^{2}{,}\,{\href{/padicField/7.3.0.1}{3} }{,}\,{\href{/padicField/7.2.0.1}{2} }^{3}$ ${\href{/padicField/11.9.0.1}{9} }{,}\,{\href{/padicField/11.6.0.1}{6} }^{2}$ ${\href{/padicField/13.10.0.1}{10} }{,}\,{\href{/padicField/13.5.0.1}{5} }{,}\,{\href{/padicField/13.2.0.1}{2} }^{2}{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ $18{,}\,{\href{/padicField/17.3.0.1}{3} }$ $21$ ${\href{/padicField/23.12.0.1}{12} }{,}\,{\href{/padicField/23.5.0.1}{5} }{,}\,{\href{/padicField/23.2.0.1}{2} }{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ ${\href{/padicField/29.14.0.1}{14} }{,}\,{\href{/padicField/29.4.0.1}{4} }{,}\,{\href{/padicField/29.2.0.1}{2} }{,}\,{\href{/padicField/29.1.0.1}{1} }$ ${\href{/padicField/31.8.0.1}{8} }{,}\,{\href{/padicField/31.4.0.1}{4} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }$ ${\href{/padicField/37.5.0.1}{5} }{,}\,{\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.3.0.1}{3} }{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }^{5}$ ${\href{/padicField/41.6.0.1}{6} }^{2}{,}\,{\href{/padicField/41.4.0.1}{4} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }$ $15{,}\,{\href{/padicField/43.6.0.1}{6} }$ ${\href{/padicField/47.10.0.1}{10} }{,}\,{\href{/padicField/47.5.0.1}{5} }{,}\,{\href{/padicField/47.4.0.1}{4} }{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ ${\href{/padicField/53.7.0.1}{7} }{,}\,{\href{/padicField/53.5.0.1}{5} }{,}\,{\href{/padicField/53.4.0.1}{4} }{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }$ ${\href{/padicField/59.10.0.1}{10} }{,}\,{\href{/padicField/59.7.0.1}{7} }{,}\,{\href{/padicField/59.4.0.1}{4} }$

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
\(67\) Copy content Toggle raw display $\Q_{67}$$x + 65$$1$$1$$0$Trivial$[\ ]$
$\Q_{67}$$x + 65$$1$$1$$0$Trivial$[\ ]$
67.2.1.2$x^{2} + 67$$2$$1$$1$$C_2$$[\ ]_{2}$
67.2.0.1$x^{2} + 63 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
67.3.0.1$x^{3} + 6 x + 65$$1$$3$$0$$C_3$$[\ ]^{3}$
67.12.0.1$x^{12} + 3 x^{8} + 57 x^{7} + 27 x^{6} + 4 x^{5} + 55 x^{4} + 64 x^{3} + 21 x^{2} + 27 x + 2$$1$$12$$0$$C_{12}$$[\ ]^{12}$
\(229\) Copy content Toggle raw display $\Q_{229}$$x$$1$$1$$0$Trivial$[\ ]$
Deg $3$$1$$3$$0$$C_3$$[\ ]^{3}$
Deg $3$$1$$3$$0$$C_3$$[\ ]^{3}$
Deg $14$$2$$7$$7$
\(4447\) Copy content Toggle raw display $\Q_{4447}$$x$$1$$1$$0$Trivial$[\ ]$
$\Q_{4447}$$x$$1$$1$$0$Trivial$[\ ]$
$\Q_{4447}$$x$$1$$1$$0$Trivial$[\ ]$
Deg $2$$1$$2$$0$$C_2$$[\ ]^{2}$
Deg $2$$1$$2$$0$$C_2$$[\ ]^{2}$
Deg $2$$1$$2$$0$$C_2$$[\ ]^{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $4$$1$$4$$0$$C_4$$[\ ]^{4}$
Deg $6$$1$$6$$0$$C_6$$[\ ]^{6}$
\(13291\) Copy content Toggle raw display $\Q_{13291}$$x$$1$$1$$0$Trivial$[\ ]$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $3$$1$$3$$0$$C_3$$[\ ]^{3}$
Deg $3$$1$$3$$0$$C_3$$[\ ]^{3}$
Deg $3$$1$$3$$0$$C_3$$[\ ]^{3}$
Deg $4$$1$$4$$0$$C_4$$[\ ]^{4}$
Deg $5$$1$$5$$0$$C_5$$[\ ]^{5}$
\(2106919\) Copy content Toggle raw display $\Q_{2106919}$$x$$1$$1$$0$Trivial$[\ ]$
Deg $2$$1$$2$$0$$C_2$$[\ ]^{2}$
Deg $2$$1$$2$$0$$C_2$$[\ ]^{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $14$$1$$14$$0$$C_{14}$$[\ ]^{14}$
\(6747049\) Copy content Toggle raw display $\Q_{6747049}$$x$$1$$1$$0$Trivial$[\ ]$
Deg $2$$1$$2$$0$$C_2$$[\ ]^{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $4$$1$$4$$0$$C_4$$[\ ]^{4}$
Deg $4$$1$$4$$0$$C_4$$[\ ]^{4}$
Deg $8$$1$$8$$0$$C_8$$[\ ]^{8}$