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Properties

Label 16.10.608...875.1
Degree $16$
Signature $[10, 3]$
Discriminant $-6.083\times 10^{19}$
Root discriminant \(17.24\)
Ramified primes $5,139,439,2411$
Class number $1$
Class group trivial
Galois group $C_2^6.S_4^2:D_4$ (as 16T1905)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 - 8*x^14 + 16*x^13 + 12*x^12 - 15*x^11 - 30*x^10 - 15*x^9 + 94*x^8 - 21*x^7 - 89*x^6 + 58*x^5 + 13*x^4 - 17*x^3 + 7*x^2 - 4*x + 1)
 
gp: K = bnfinit(y^16 - 2*y^15 - 8*y^14 + 16*y^13 + 12*y^12 - 15*y^11 - 30*y^10 - 15*y^9 + 94*y^8 - 21*y^7 - 89*y^6 + 58*y^5 + 13*y^4 - 17*y^3 + 7*y^2 - 4*y + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 2*x^15 - 8*x^14 + 16*x^13 + 12*x^12 - 15*x^11 - 30*x^10 - 15*x^9 + 94*x^8 - 21*x^7 - 89*x^6 + 58*x^5 + 13*x^4 - 17*x^3 + 7*x^2 - 4*x + 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 2*x^15 - 8*x^14 + 16*x^13 + 12*x^12 - 15*x^11 - 30*x^10 - 15*x^9 + 94*x^8 - 21*x^7 - 89*x^6 + 58*x^5 + 13*x^4 - 17*x^3 + 7*x^2 - 4*x + 1)
 

\( x^{16} - 2 x^{15} - 8 x^{14} + 16 x^{13} + 12 x^{12} - 15 x^{11} - 30 x^{10} - 15 x^{9} + 94 x^{8} + \cdots + 1 \) Copy content Toggle raw display

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:  $[10, 3]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(-60827265928788671875\) \(\medspace = -\,5^{8}\cdot 139\cdot 439^{2}\cdot 2411^{2}\) 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:  \(17.24\)
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:  $5^{1/2}139^{1/2}439^{1/2}2411^{1/2}\approx 27122.09717186339$
Ramified primes:   \(5\), \(139\), \(439\), \(2411\) 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:  \(\Q(\sqrt{-139}) \)
$\card{ \Aut(K/\Q) }$:  $2$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{118802669}a^{15}-\frac{9600594}{118802669}a^{14}-\frac{1554644}{118802669}a^{13}-\frac{32965213}{118802669}a^{12}+\frac{2096868}{118802669}a^{11}+\frac{56918848}{118802669}a^{10}-\frac{19804119}{118802669}a^{9}-\frac{49818491}{118802669}a^{8}+\frac{53835680}{118802669}a^{7}+\frac{58039320}{118802669}a^{6}+\frac{10946341}{118802669}a^{5}-\frac{57271111}{118802669}a^{4}-\frac{2434625}{118802669}a^{3}+\frac{10185578}{118802669}a^{2}-\frac{32584248}{118802669}a-\frac{28466594}{118802669}$ Copy content Toggle raw display

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:  $12$
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:   $\frac{39248373}{118802669}a^{15}-\frac{81039572}{118802669}a^{14}-\frac{315598481}{118802669}a^{13}+\frac{658346937}{118802669}a^{12}+\frac{484499394}{118802669}a^{11}-\frac{695409843}{118802669}a^{10}-\frac{1226051656}{118802669}a^{9}-\frac{472966359}{118802669}a^{8}+\frac{3919982590}{118802669}a^{7}-\frac{938796179}{118802669}a^{6}-\frac{3937839584}{118802669}a^{5}+\frac{2507239323}{118802669}a^{4}+\frac{886628300}{118802669}a^{3}-\frac{884159019}{118802669}a^{2}+\frac{54438535}{118802669}a-\frac{35415969}{118802669}$, $a$, $\frac{79554296}{118802669}a^{15}-\frac{156565766}{118802669}a^{14}-\frac{634822871}{118802669}a^{13}+\frac{1242495767}{118802669}a^{12}+\frac{941132634}{118802669}a^{11}-\frac{1086630192}{118802669}a^{10}-\frac{2338028414}{118802669}a^{9}-\frac{1309073676}{118802669}a^{8}+\frac{7247468296}{118802669}a^{7}-\frac{1556059870}{118802669}a^{6}-\frac{6635597957}{118802669}a^{5}+\frac{4383315479}{118802669}a^{4}+\frac{657806397}{118802669}a^{3}-\frac{1135486354}{118802669}a^{2}+\frac{777180148}{118802669}a-\frac{320992038}{118802669}$, $\frac{496536314}{118802669}a^{15}-\frac{806621167}{118802669}a^{14}-\frac{4266431802}{118802669}a^{13}+\frac{6317880522}{118802669}a^{12}+\frac{8277082352}{118802669}a^{11}-\frac{4158777860}{118802669}a^{10}-\frac{16457757736}{118802669}a^{9}-\frac{13722738336}{118802669}a^{8}+\frac{41204719881}{118802669}a^{7}+\frac{4988472819}{118802669}a^{6}-\frac{41378109734}{118802669}a^{5}+\frac{12728520111}{118802669}a^{4}+\frac{11091257530}{118802669}a^{3}-\frac{4137934822}{118802669}a^{2}+\frac{1804540205}{118802669}a-\frac{1015269921}{118802669}$, $\frac{303921353}{118802669}a^{15}-\frac{484736397}{118802669}a^{14}-\frac{2620705495}{118802669}a^{13}+\frac{3775323540}{118802669}a^{12}+\frac{5127850005}{118802669}a^{11}-\frac{2268504635}{118802669}a^{10}-\frac{9991042301}{118802669}a^{9}-\frac{8886852925}{118802669}a^{8}+\frac{24598202785}{118802669}a^{7}+\frac{3661459176}{118802669}a^{6}-\frac{24450538688}{118802669}a^{5}+\frac{7223840934}{118802669}a^{4}+\frac{6040127779}{118802669}a^{3}-\frac{2025936152}{118802669}a^{2}+\frac{1498348440}{118802669}a-\frac{744735264}{118802669}$, $\frac{334455223}{118802669}a^{15}-\frac{564151057}{118802669}a^{14}-\frac{2886466797}{118802669}a^{13}+\frac{4454175075}{118802669}a^{12}+\frac{5725997023}{118802669}a^{11}-\frac{3234486334}{118802669}a^{10}-\frac{11730688379}{118802669}a^{9}-\frac{9177529729}{118802669}a^{8}+\frac{29063199335}{118802669}a^{7}+\frac{3674877525}{118802669}a^{6}-\frac{29488828280}{118802669}a^{5}+\frac{8393220745}{118802669}a^{4}+\frac{8405808448}{118802669}a^{3}-\frac{2458805569}{118802669}a^{2}+\frac{1236833376}{118802669}a-\frac{770885358}{118802669}$, $\frac{187396624}{118802669}a^{15}-\frac{272509892}{118802669}a^{14}-\frac{1667277282}{118802669}a^{13}+\frac{2125806474}{118802669}a^{12}+\frac{3542797414}{118802669}a^{11}-\frac{1147534024}{118802669}a^{10}-\frac{6321903059}{118802669}a^{9}-\frac{6170679215}{118802669}a^{8}+\frac{14656194532}{118802669}a^{7}+\frac{4199863326}{118802669}a^{6}-\frac{15805714714}{118802669}a^{5}+\frac{3092051804}{118802669}a^{4}+\frac{4868622847}{118802669}a^{3}-\frac{1220023849}{118802669}a^{2}+\frac{761167781}{118802669}a-\frac{352262030}{118802669}$, $\frac{196879724}{118802669}a^{15}-\frac{309307163}{118802669}a^{14}-\frac{1694584127}{118802669}a^{13}+\frac{2424308683}{118802669}a^{12}+\frac{3279793663}{118802669}a^{11}-\frac{1591189148}{118802669}a^{10}-\frac{6315610069}{118802669}a^{9}-\frac{5288647838}{118802669}a^{8}+\frac{16053775557}{118802669}a^{7}+\frac{1930278676}{118802669}a^{6}-\frac{16767469744}{118802669}a^{5}+\frac{5147135936}{118802669}a^{4}+\frac{4830570800}{118802669}a^{3}-\frac{1924901147}{118802669}a^{2}+\frac{643289193}{118802669}a-\frac{406740138}{118802669}$, $\frac{339107378}{118802669}a^{15}-\frac{508528584}{118802669}a^{14}-\frac{2981243904}{118802669}a^{13}+\frac{3975023090}{118802669}a^{12}+\frac{6150203980}{118802669}a^{11}-\frac{2329193565}{118802669}a^{10}-\frac{11379289310}{118802669}a^{9}-\frac{10490802599}{118802669}a^{8}+\frac{27144437385}{118802669}a^{7}+\frac{6504525114}{118802669}a^{6}-\frac{28633445578}{118802669}a^{5}+\frac{6163038038}{118802669}a^{4}+\frac{8324193357}{118802669}a^{3}-\frac{2372123636}{118802669}a^{2}+\frac{1480510838}{118802669}a-\frac{780123762}{118802669}$, $\frac{466799936}{118802669}a^{15}-\frac{767310381}{118802669}a^{14}-\frac{3992169664}{118802669}a^{13}+\frac{5988006201}{118802669}a^{12}+\frac{7648021546}{118802669}a^{11}-\frac{3822257148}{118802669}a^{10}-\frac{15430580360}{118802669}a^{9}-\frac{12935111117}{118802669}a^{8}+\frac{38662144030}{118802669}a^{7}+\frac{4160570820}{118802669}a^{6}-\frac{37884996187}{118802669}a^{5}+\frac{11873581069}{118802669}a^{4}+\frac{9436451842}{118802669}a^{3}-\frac{3326574559}{118802669}a^{2}+\frac{1740331616}{118802669}a-\frac{986885534}{118802669}$, $\frac{248198950}{118802669}a^{15}-\frac{378928008}{118802669}a^{14}-\frac{2176186707}{118802669}a^{13}+\frac{2977178207}{118802669}a^{12}+\frac{4435064356}{118802669}a^{11}-\frac{1884401095}{118802669}a^{10}-\frac{8152804819}{118802669}a^{9}-\frac{7439332925}{118802669}a^{8}+\frac{19913542010}{118802669}a^{7}+\frac{3970289422}{118802669}a^{6}-\frac{21241869586}{118802669}a^{5}+\frac{6015586513}{118802669}a^{4}+\frac{5814141836}{118802669}a^{3}-\frac{2482436887}{118802669}a^{2}+\frac{1382670491}{118802669}a-\frac{584125364}{118802669}$, $\frac{247697714}{118802669}a^{15}-\frac{436504338}{118802669}a^{14}-\frac{2159352694}{118802669}a^{13}+\frac{3491082293}{118802669}a^{12}+\frac{4456548151}{118802669}a^{11}-\frac{2862374239}{118802669}a^{10}-\frac{9360448720}{118802669}a^{9}-\frac{6919556476}{118802669}a^{8}+\frac{22746931570}{118802669}a^{7}+\frac{3426839925}{118802669}a^{6}-\frac{23298030008}{118802669}a^{5}+\frac{6544479915}{118802669}a^{4}+\frac{6864046389}{118802669}a^{3}-\frac{2197308351}{118802669}a^{2}+\frac{1146274906}{118802669}a-\frac{540567418}{118802669}$ Copy content Toggle raw display
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:  \( 10086.9602538 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{10}\cdot(2\pi)^{3}\cdot 10086.9602538 \cdot 1}{2\cdot\sqrt{60827265928788671875}}\cr\approx \mathstrut & 0.164255805196 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 - 8*x^14 + 16*x^13 + 12*x^12 - 15*x^11 - 30*x^10 - 15*x^9 + 94*x^8 - 21*x^7 - 89*x^6 + 58*x^5 + 13*x^4 - 17*x^3 + 7*x^2 - 4*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^16 - 2*x^15 - 8*x^14 + 16*x^13 + 12*x^12 - 15*x^11 - 30*x^10 - 15*x^9 + 94*x^8 - 21*x^7 - 89*x^6 + 58*x^5 + 13*x^4 - 17*x^3 + 7*x^2 - 4*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^16 - 2*x^15 - 8*x^14 + 16*x^13 + 12*x^12 - 15*x^11 - 30*x^10 - 15*x^9 + 94*x^8 - 21*x^7 - 89*x^6 + 58*x^5 + 13*x^4 - 17*x^3 + 7*x^2 - 4*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^16 - 2*x^15 - 8*x^14 + 16*x^13 + 12*x^12 - 15*x^11 - 30*x^10 - 15*x^9 + 94*x^8 - 21*x^7 - 89*x^6 + 58*x^5 + 13*x^4 - 17*x^3 + 7*x^2 - 4*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

$C_2^6.S_4^2:D_4$ (as 16T1905):

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 294912
The 230 conjugacy class representatives for $C_2^6.S_4^2:D_4$
Character table for $C_2^6.S_4^2:D_4$

Intermediate fields

\(\Q(\sqrt{5}) \), 8.8.661518125.1

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 16 siblings: data not computed
Degree 32 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 $16$ $16$ R ${\href{/padicField/7.12.0.1}{12} }{,}\,{\href{/padicField/7.4.0.1}{4} }$ ${\href{/padicField/11.4.0.1}{4} }^{2}{,}\,{\href{/padicField/11.3.0.1}{3} }^{2}{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ ${\href{/padicField/13.8.0.1}{8} }^{2}$ ${\href{/padicField/17.8.0.1}{8} }{,}\,{\href{/padicField/17.4.0.1}{4} }^{2}$ ${\href{/padicField/19.4.0.1}{4} }^{2}{,}\,{\href{/padicField/19.3.0.1}{3} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }$ $16$ ${\href{/padicField/29.4.0.1}{4} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }^{4}$ ${\href{/padicField/31.4.0.1}{4} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }^{4}$ ${\href{/padicField/37.4.0.1}{4} }^{4}$ ${\href{/padicField/41.8.0.1}{8} }{,}\,{\href{/padicField/41.3.0.1}{3} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }$ ${\href{/padicField/43.8.0.1}{8} }{,}\,{\href{/padicField/43.4.0.1}{4} }^{2}$ ${\href{/padicField/47.8.0.1}{8} }^{2}$ ${\href{/padicField/53.8.0.1}{8} }{,}\,{\href{/padicField/53.4.0.1}{4} }^{2}$ ${\href{/padicField/59.3.0.1}{3} }^{4}{,}\,{\href{/padicField/59.2.0.1}{2} }{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$

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

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
\(5\) Copy content Toggle raw display 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$$[\ ]_{2}^{4}$
5.8.4.1$x^{8} + 80 x^{7} + 2428 x^{6} + 33688 x^{5} + 195810 x^{4} + 305952 x^{3} + 870132 x^{2} + 1037416 x + 503089$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
\(139\) Copy content Toggle raw display 139.2.1.2$x^{2} + 139$$2$$1$$1$$C_2$$[\ ]_{2}$
139.2.0.1$x^{2} + 138 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
139.3.0.1$x^{3} + 6 x + 137$$1$$3$$0$$C_3$$[\ ]^{3}$
139.3.0.1$x^{3} + 6 x + 137$$1$$3$$0$$C_3$$[\ ]^{3}$
139.6.0.1$x^{6} + 4 x^{4} + 46 x^{3} + 10 x^{2} + 118 x + 2$$1$$6$$0$$C_6$$[\ ]^{6}$
\(439\) Copy content Toggle raw display $\Q_{439}$$x$$1$$1$$0$Trivial$[\ ]$
$\Q_{439}$$x$$1$$1$$0$Trivial$[\ ]$
Deg $3$$1$$3$$0$$C_3$$[\ ]^{3}$
Deg $3$$1$$3$$0$$C_3$$[\ ]^{3}$
Deg $4$$1$$4$$0$$C_4$$[\ ]^{4}$
Deg $4$$2$$2$$2$
\(2411\) Copy content Toggle raw display $\Q_{2411}$$x$$1$$1$$0$Trivial$[\ ]$
$\Q_{2411}$$x$$1$$1$$0$Trivial$[\ ]$
$\Q_{2411}$$x$$1$$1$$0$Trivial$[\ ]$
$\Q_{2411}$$x$$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$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $4$$1$$4$$0$$C_4$$[\ ]^{4}$