Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 5*x^19 + 11*x^18 - 11*x^17 - x^16 + 16*x^15 - 20*x^14 + 11*x^13 + 5*x^12 - 20*x^11 + 27*x^10 - 2*x^9 - 22*x^8 + 40*x^7 - 9*x^6 - 5*x^5 + 17*x^4 - 7*x^3 + 5*x^2 - 3*x + 1)
gp: K = bnfinit(y^20 - 5*y^19 + 11*y^18 - 11*y^17 - y^16 + 16*y^15 - 20*y^14 + 11*y^13 + 5*y^12 - 20*y^11 + 27*y^10 - 2*y^9 - 22*y^8 + 40*y^7 - 9*y^6 - 5*y^5 + 17*y^4 - 7*y^3 + 5*y^2 - 3*y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 - 5*x^19 + 11*x^18 - 11*x^17 - x^16 + 16*x^15 - 20*x^14 + 11*x^13 + 5*x^12 - 20*x^11 + 27*x^10 - 2*x^9 - 22*x^8 + 40*x^7 - 9*x^6 - 5*x^5 + 17*x^4 - 7*x^3 + 5*x^2 - 3*x + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^20 - 5*x^19 + 11*x^18 - 11*x^17 - x^16 + 16*x^15 - 20*x^14 + 11*x^13 + 5*x^12 - 20*x^11 + 27*x^10 - 2*x^9 - 22*x^8 + 40*x^7 - 9*x^6 - 5*x^5 + 17*x^4 - 7*x^3 + 5*x^2 - 3*x + 1)
\( x^{20} - 5 x^{19} + 11 x^{18} - 11 x^{17} - x^{16} + 16 x^{15} - 20 x^{14} + 11 x^{13} + 5 x^{12} + \cdots + 1 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $20$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[0, 10]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(1430869177844446526464\)
\(\medspace = 2^{10}\cdot 31^{3}\cdot 2617^{4}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(11.42\) | 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: | $2^{31/16}31^{3/4}2617^{1/2}\approx 2574.3568552858005$ | ||
| Ramified primes: |
\(2\), \(31\), \(2617\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{31}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{272085514565}a^{19}-\frac{92853578442}{272085514565}a^{18}+\frac{15888810470}{54417102913}a^{17}+\frac{69553206389}{272085514565}a^{16}+\frac{65420503271}{272085514565}a^{15}+\frac{7720682619}{272085514565}a^{14}-\frac{44089421458}{272085514565}a^{13}+\frac{88201761702}{272085514565}a^{12}+\frac{78281952096}{272085514565}a^{11}+\frac{40337167528}{272085514565}a^{10}+\frac{117992989721}{272085514565}a^{9}-\frac{128966888769}{272085514565}a^{8}+\frac{81784740281}{272085514565}a^{7}-\frac{12603299272}{272085514565}a^{6}-\frac{26314477145}{54417102913}a^{5}+\frac{21533092220}{54417102913}a^{4}-\frac{127958118368}{272085514565}a^{3}+\frac{103342545059}{272085514565}a^{2}-\frac{23029136343}{272085514565}a-\frac{99283461742}{272085514565}$
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: | $9$ | 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: |
$\frac{60133888873}{272085514565}a^{19}-\frac{298778534621}{272085514565}a^{18}+\frac{125749937816}{54417102913}a^{17}-\frac{565943696058}{272085514565}a^{16}-\frac{154497353552}{272085514565}a^{15}+\frac{880885769627}{272085514565}a^{14}-\frac{902910373719}{272085514565}a^{13}+\frac{330553090031}{272085514565}a^{12}+\frac{537516957093}{272085514565}a^{11}-\frac{1245219308116}{272085514565}a^{10}+\frac{1311304839963}{272085514565}a^{9}+\frac{291306858373}{272085514565}a^{8}-\frac{1453679251817}{272085514565}a^{7}+\frac{1815058065489}{272085514565}a^{6}-\frac{53766486376}{54417102913}a^{5}-\frac{188743143152}{54417102913}a^{4}+\frac{771328825651}{272085514565}a^{3}-\frac{362791202978}{272085514565}a^{2}-\frac{395377999964}{272085514565}a-\frac{71740030276}{272085514565}$, $\frac{73550967561}{272085514565}a^{19}-\frac{358776658262}{272085514565}a^{18}+\frac{142299674631}{54417102913}a^{17}-\frac{436406983611}{272085514565}a^{16}-\frac{744471731989}{272085514565}a^{15}+\frac{1616685326159}{272085514565}a^{14}-\frac{1020982344778}{272085514565}a^{13}-\frac{252445519393}{272085514565}a^{12}+\frac{1082197170616}{272085514565}a^{11}-\frac{1500089405487}{272085514565}a^{10}+\frac{1410734894131}{272085514565}a^{9}+\frac{1042918876741}{272085514565}a^{8}-\frac{2656861552999}{272085514565}a^{7}+\frac{2221579545398}{272085514565}a^{6}+\frac{272466210976}{54417102913}a^{5}-\frac{311271020603}{54417102913}a^{4}+\frac{933714806707}{272085514565}a^{3}+\frac{450012416949}{272085514565}a^{2}+\frac{254296271707}{272085514565}a-\frac{173184072872}{272085514565}$, $\frac{127984771237}{272085514565}a^{19}-\frac{532733281509}{272085514565}a^{18}+\frac{193034343081}{54417102913}a^{17}-\frac{707180442757}{272085514565}a^{16}-\frac{269254360793}{272085514565}a^{15}+\frac{983362978913}{272085514565}a^{14}-\frac{1092387527761}{272085514565}a^{13}+\frac{697932733944}{272085514565}a^{12}+\frac{472860699732}{272085514565}a^{11}-\frac{1556997933119}{272085514565}a^{10}+\frac{1885044705792}{272085514565}a^{9}+\frac{911584962807}{272085514565}a^{8}-\frac{783999526338}{272085514565}a^{7}+\frac{3024738593601}{272085514565}a^{6}+\frac{133067493646}{54417102913}a^{5}+\frac{232176916716}{54417102913}a^{4}+\frac{1645466705109}{272085514565}a^{3}+\frac{110315921928}{272085514565}a^{2}+\frac{517400587054}{272085514565}a-\frac{42132770904}{272085514565}$, $\frac{116662206199}{272085514565}a^{19}-\frac{487288654348}{272085514565}a^{18}+\frac{165299068902}{54417102913}a^{17}-\frac{321741318959}{272085514565}a^{16}-\frac{1032002613231}{272085514565}a^{15}+\frac{1773593005816}{272085514565}a^{14}-\frac{1108933642372}{272085514565}a^{13}-\frac{147476332167}{272085514565}a^{12}+\frac{1228052637539}{272085514565}a^{11}-\frac{1687633199978}{272085514565}a^{10}+\frac{1470718218004}{272085514565}a^{9}+\frac{1831636664369}{272085514565}a^{8}-\frac{2309326354126}{272085514565}a^{7}+\frac{2863526312377}{272085514565}a^{6}+\frac{444385326395}{54417102913}a^{5}-\frac{143037990508}{54417102913}a^{4}+\frac{1568461147963}{272085514565}a^{3}+\frac{482651736191}{272085514565}a^{2}+\frac{329093784468}{272085514565}a-\frac{89249173098}{272085514565}$, $a$, $\frac{121043780821}{272085514565}a^{19}-\frac{511287704607}{272085514565}a^{18}+\frac{184359513833}{54417102913}a^{17}-\frac{590104834066}{272085514565}a^{16}-\frac{576575495099}{272085514565}a^{15}+\frac{1431971428944}{272085514565}a^{14}-\frac{1265065125243}{272085514565}a^{13}+\frac{302517690527}{272085514565}a^{12}+\frac{1068191710581}{272085514565}a^{11}-\frac{1848794657447}{272085514565}a^{10}+\frac{1779139461261}{272085514565}a^{9}+\frac{1253660237991}{272085514565}a^{8}-\frac{1710299971159}{272085514565}a^{7}+\frac{3351618338858}{272085514565}a^{6}+\frac{257732785208}{54417102913}a^{5}-\frac{61669399591}{54417102913}a^{4}+\frac{1803374116507}{272085514565}a^{3}+\frac{350551941274}{272085514565}a^{2}+\frac{214119567437}{272085514565}a-\frac{78800251327}{272085514565}$, $\frac{26690513153}{272085514565}a^{19}-\frac{57891189996}{272085514565}a^{18}-\frac{8758705072}{54417102913}a^{17}+\frac{396983275937}{272085514565}a^{16}-\frac{738211666602}{272085514565}a^{15}+\frac{640294231932}{272085514565}a^{14}-\frac{137582305099}{272085514565}a^{13}-\frac{463344785969}{272085514565}a^{12}+\frac{779464091073}{272085514565}a^{11}-\frac{440618909511}{272085514565}a^{10}-\frac{142025019307}{272085514565}a^{9}+\frac{1278257683113}{272085514565}a^{8}-\frac{600593342562}{272085514565}a^{7}+\frac{774984880204}{272085514565}a^{6}+\frac{350800358432}{54417102913}a^{5}-\frac{65585334078}{54417102913}a^{4}+\frac{1186031476246}{272085514565}a^{3}+\frac{538173228647}{272085514565}a^{2}+\frac{253466725256}{272085514565}a-\frac{52034880576}{272085514565}$, $\frac{8626170245}{54417102913}a^{19}-\frac{38281173852}{54417102913}a^{18}+\frac{72495427401}{54417102913}a^{17}-\frac{51763341159}{54417102913}a^{16}-\frac{40385565630}{54417102913}a^{15}+\frac{119512886590}{54417102913}a^{14}-\frac{123743377065}{54417102913}a^{13}+\frac{72331623399}{54417102913}a^{12}+\frac{23242877453}{54417102913}a^{11}-\frac{127694611450}{54417102913}a^{10}+\frac{189634705473}{54417102913}a^{9}+\frac{13392752130}{54417102913}a^{8}-\frac{126382811347}{54417102913}a^{7}+\frac{262934825390}{54417102913}a^{6}-\frac{3479156481}{54417102913}a^{5}+\frac{50243519052}{54417102913}a^{4}+\frac{71244877364}{54417102913}a^{3}-\frac{74663797617}{54417102913}a^{2}+\frac{163365589771}{54417102913}a-\frac{31640793947}{54417102913}$, $\frac{279239715169}{272085514565}a^{19}-\frac{1352613605393}{272085514565}a^{18}+\frac{561297320065}{54417102913}a^{17}-\frac{2396198118229}{272085514565}a^{16}-\frac{1045895317166}{272085514565}a^{15}+\frac{4395077301911}{272085514565}a^{14}-\frac{4286021689687}{272085514565}a^{13}+\frac{1557249414303}{272085514565}a^{12}+\frac{1962963697954}{272085514565}a^{11}-\frac{5058150276318}{272085514565}a^{10}+\frac{6143952111429}{272085514565}a^{9}+\frac{1301238085734}{272085514565}a^{8}-\frac{6570577392841}{272085514565}a^{7}+\frac{9029785200267}{272085514565}a^{6}+\frac{67101723915}{54417102913}a^{5}-\frac{427626138192}{54417102913}a^{4}+\frac{3454698653708}{272085514565}a^{3}-\frac{643793005269}{272085514565}a^{2}+\frac{1119472424748}{272085514565}a-\frac{636043584023}{272085514565}$
| 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: | \( 130.791344738 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{10}\cdot 130.791344738 \cdot 1}{2\cdot\sqrt{1430869177844446526464}}\cr\approx \mathstrut & 0.165785952416 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^20 - 5*x^19 + 11*x^18 - 11*x^17 - x^16 + 16*x^15 - 20*x^14 + 11*x^13 + 5*x^12 - 20*x^11 + 27*x^10 - 2*x^9 - 22*x^8 + 40*x^7 - 9*x^6 - 5*x^5 + 17*x^4 - 7*x^3 + 5*x^2 - 3*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^20 - 5*x^19 + 11*x^18 - 11*x^17 - x^16 + 16*x^15 - 20*x^14 + 11*x^13 + 5*x^12 - 20*x^11 + 27*x^10 - 2*x^9 - 22*x^8 + 40*x^7 - 9*x^6 - 5*x^5 + 17*x^4 - 7*x^3 + 5*x^2 - 3*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^20 - 5*x^19 + 11*x^18 - 11*x^17 - x^16 + 16*x^15 - 20*x^14 + 11*x^13 + 5*x^12 - 20*x^11 + 27*x^10 - 2*x^9 - 22*x^8 + 40*x^7 - 9*x^6 - 5*x^5 + 17*x^4 - 7*x^3 + 5*x^2 - 3*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^20 - 5*x^19 + 11*x^18 - 11*x^17 - x^16 + 16*x^15 - 20*x^14 + 11*x^13 + 5*x^12 - 20*x^11 + 27*x^10 - 2*x^9 - 22*x^8 + 40*x^7 - 9*x^6 - 5*x^5 + 17*x^4 - 7*x^3 + 5*x^2 - 3*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
$C_2^{10}.C_2\wr S_5$ (as 20T1015):
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 3932160 |
| The 506 conjugacy class representatives for $C_2^{10}.C_2\wr S_5$ |
| Character table for $C_2^{10}.C_2\wr S_5$ |
Intermediate fields
| 5.1.2617.1, 10.0.212309359.1 |
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 20 siblings: | data not computed |
| Degree 40 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 | R | ${\href{/padicField/3.10.0.1}{10} }^{2}$ | ${\href{/padicField/5.8.0.1}{8} }^{2}{,}\,{\href{/padicField/5.2.0.1}{2} }^{2}$ | $16{,}\,{\href{/padicField/7.2.0.1}{2} }^{2}$ | ${\href{/padicField/11.8.0.1}{8} }{,}\,{\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.3.0.1}{3} }^{2}$ | $20$ | ${\href{/padicField/17.6.0.1}{6} }^{2}{,}\,{\href{/padicField/17.4.0.1}{4} }{,}\,{\href{/padicField/17.2.0.1}{2} }^{2}$ | ${\href{/padicField/19.8.0.1}{8} }^{2}{,}\,{\href{/padicField/19.4.0.1}{4} }$ | ${\href{/padicField/23.10.0.1}{10} }^{2}$ | ${\href{/padicField/29.4.0.1}{4} }^{4}{,}\,{\href{/padicField/29.2.0.1}{2} }{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | R | $20$ | ${\href{/padicField/41.8.0.1}{8} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }^{2}$ | ${\href{/padicField/43.4.0.1}{4} }^{4}{,}\,{\href{/padicField/43.2.0.1}{2} }^{2}$ | ${\href{/padicField/47.8.0.1}{8} }{,}\,{\href{/padicField/47.4.0.1}{4} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}$ | ${\href{/padicField/53.4.0.1}{4} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }^{5}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.3.0.1}{3} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.10.10.14 | $x^{10} + 4 x^{9} - 6 x^{8} + 176 x^{7} + 848 x^{6} + 2256 x^{5} + 1216 x^{4} - 1088 x^{3} - 5392 x^{2} - 5120 x - 3616$ | $2$ | $5$ | $10$ | $C_2 \times (C_2^4 : C_5)$ | $[2, 2, 2, 2, 2]^{5}$ |
| 2.10.0.1 | $x^{10} + x^{6} + x^{5} + x^{3} + x^{2} + x + 1$ | $1$ | $10$ | $0$ | $C_{10}$ | $[\ ]^{10}$ | |
|
\(31\)
| 31.2.0.1 | $x^{2} + 29 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 31.2.0.1 | $x^{2} + 29 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 31.4.3.2 | $x^{4} + 93$ | $4$ | $1$ | $3$ | $D_{4}$ | $[\ ]_{4}^{2}$ | |
| 31.4.0.1 | $x^{4} + 3 x^{2} + 16 x + 3$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 31.4.0.1 | $x^{4} + 3 x^{2} + 16 x + 3$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 31.4.0.1 | $x^{4} + 3 x^{2} + 16 x + 3$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
|
\(2617\)
| 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 $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
| Deg $4$ | $2$ | $2$ | $2$ | ||||
| Deg $4$ | $2$ | $2$ | $2$ |