Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 + 6*x^16 - 18*x^15 + 17*x^14 + 4*x^13 + 25*x^12 - 92*x^11 + 92*x^10 - 34*x^9 + 44*x^8 - 140*x^7 + 188*x^6 - 144*x^5 + 88*x^4 - 54*x^3 + 27*x^2 - 8*x + 1)
gp: K = bnfinit(y^20 - 2*y^19 + 6*y^16 - 18*y^15 + 17*y^14 + 4*y^13 + 25*y^12 - 92*y^11 + 92*y^10 - 34*y^9 + 44*y^8 - 140*y^7 + 188*y^6 - 144*y^5 + 88*y^4 - 54*y^3 + 27*y^2 - 8*y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 - 2*x^19 + 6*x^16 - 18*x^15 + 17*x^14 + 4*x^13 + 25*x^12 - 92*x^11 + 92*x^10 - 34*x^9 + 44*x^8 - 140*x^7 + 188*x^6 - 144*x^5 + 88*x^4 - 54*x^3 + 27*x^2 - 8*x + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^20 - 2*x^19 + 6*x^16 - 18*x^15 + 17*x^14 + 4*x^13 + 25*x^12 - 92*x^11 + 92*x^10 - 34*x^9 + 44*x^8 - 140*x^7 + 188*x^6 - 144*x^5 + 88*x^4 - 54*x^3 + 27*x^2 - 8*x + 1)
\( x^{20} - 2 x^{19} + 6 x^{16} - 18 x^{15} + 17 x^{14} + 4 x^{13} + 25 x^{12} - 92 x^{11} + 92 x^{10} + \cdots + 1 \)
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: |
\(267622853577068773376\)
\(\medspace = 2^{20}\cdot 761^{5}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(10.50\) | 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: | $2\cdot 761^{1/2}\approx 55.17245689653489$ | ||
| Ramified primes: |
\(2\), \(761\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{761}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $10$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{3}a^{18}+\frac{1}{3}a^{16}-\frac{1}{3}a^{15}-\frac{1}{3}a^{14}+\frac{1}{3}a^{12}-\frac{1}{3}a^{10}-\frac{1}{3}a^{9}-\frac{1}{3}a^{8}-\frac{1}{3}a^{7}-\frac{1}{3}a^{6}+\frac{1}{3}a^{5}+\frac{1}{3}a^{3}+\frac{1}{3}a-\frac{1}{3}$, $\frac{1}{4379139123}a^{19}-\frac{119017966}{4379139123}a^{18}+\frac{1685921074}{4379139123}a^{17}-\frac{2114361461}{4379139123}a^{16}+\frac{33058990}{1459713041}a^{15}+\frac{365282}{257596419}a^{14}+\frac{114632567}{257596419}a^{13}-\frac{973911976}{4379139123}a^{12}-\frac{1115738719}{4379139123}a^{11}+\frac{537527623}{1459713041}a^{10}-\frac{379823704}{1459713041}a^{9}+\frac{493709918}{1459713041}a^{8}+\frac{230256856}{1459713041}a^{7}-\frac{1268264551}{4379139123}a^{6}+\frac{160458095}{4379139123}a^{5}+\frac{746923456}{4379139123}a^{4}-\frac{934160479}{4379139123}a^{3}-\frac{270645920}{4379139123}a^{2}+\frac{754334224}{4379139123}a+\frac{1554696913}{4379139123}$
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: |
\( -\frac{3778880}{270869} a^{19} + \frac{11227566}{270869} a^{18} - \frac{1330507}{270869} a^{17} - \frac{6019962}{270869} a^{16} - \frac{29575666}{270869} a^{15} + \frac{83845502}{270869} a^{14} - \frac{99234599}{270869} a^{13} - \frac{28701735}{270869} a^{12} - \frac{57842432}{270869} a^{11} + \frac{495643623}{270869} a^{10} - \frac{479848985}{270869} a^{9} + \frac{109683743}{270869} a^{8} - \frac{126499126}{270869} a^{7} + \frac{696261730}{270869} a^{6} - \frac{967858891}{270869} a^{5} + \frac{644490779}{270869} a^{4} - \frac{358982021}{270869} a^{3} + \frac{241607430}{270869} a^{2} - \frac{112651583}{270869} a + \frac{21204039}{270869} \)
(order $4$)
| 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: |
$\frac{22484982706}{257596419}a^{19}-\frac{10267563344}{85865473}a^{18}-\frac{18681268559}{257596419}a^{17}-\frac{12403016392}{257596419}a^{16}+\frac{126129587750}{257596419}a^{15}-\frac{108706852533}{85865473}a^{14}+\frac{180482208595}{257596419}a^{13}+\frac{65016364200}{85865473}a^{12}+\frac{686360073866}{257596419}a^{11}-\frac{1628730592246}{257596419}a^{10}+\frac{1068399829907}{257596419}a^{9}-\frac{131443441969}{257596419}a^{8}+\frac{919263914954}{257596419}a^{7}-\frac{2564824514810}{257596419}a^{6}+\frac{880920894703}{85865473}a^{5}-\frac{1640266238186}{257596419}a^{4}+\frac{331721523585}{85865473}a^{3}-\frac{612024660836}{257596419}a^{2}+\frac{239459512025}{257596419}a-\frac{13024885462}{85865473}$, $\frac{69263414900}{1459713041}a^{19}-\frac{98990575245}{1459713041}a^{18}-\frac{58325263366}{1459713041}a^{17}-\frac{31661277263}{1459713041}a^{16}+\frac{399712883294}{1459713041}a^{15}-\frac{59794785242}{85865473}a^{14}+\frac{34595791711}{85865473}a^{13}+\frac{635014217009}{1459713041}a^{12}+\frac{2089633010243}{1459713041}a^{11}-\frac{5196500477196}{1459713041}a^{10}+\frac{3342715150435}{1459713041}a^{9}-\frac{345642874025}{1459713041}a^{8}+\frac{2808495426073}{1459713041}a^{7}-\frac{8099927885639}{1459713041}a^{6}+\frac{8320217357093}{1459713041}a^{5}-\frac{5054769095606}{1459713041}a^{4}+\frac{3072299576199}{1459713041}a^{3}-\frac{1912378527199}{1459713041}a^{2}+\frac{729363743977}{1459713041}a-\frac{110382905066}{1459713041}$, $\frac{274186238164}{4379139123}a^{19}-\frac{310114549069}{4379139123}a^{18}-\frac{289820724893}{4379139123}a^{17}-\frac{236659651055}{4379139123}a^{16}+\frac{488919187375}{1459713041}a^{15}-\frac{213678048997}{257596419}a^{14}+\frac{82738018919}{257596419}a^{13}+\frac{2550755438822}{4379139123}a^{12}+\frac{9055463962178}{4379139123}a^{11}-\frac{5850080813225}{1459713041}a^{10}+\frac{3075300799592}{1459713041}a^{9}-\frac{96825647013}{1459713041}a^{8}+\frac{3838817430477}{1459713041}a^{7}-\frac{28462645152622}{4379139123}a^{6}+\frac{25918129239158}{4379139123}a^{5}-\frac{15187044858281}{4379139123}a^{4}+\frac{9684955053521}{4379139123}a^{3}-\frac{5733087900083}{4379139123}a^{2}+\frac{1940777253562}{4379139123}a-\frac{253833225818}{4379139123}$, $\frac{256082710069}{4379139123}a^{19}-\frac{243715209961}{4379139123}a^{18}-\frac{302541568538}{4379139123}a^{17}-\frac{282098082329}{4379139123}a^{16}+\frac{434567912183}{1459713041}a^{15}-\frac{186927350236}{257596419}a^{14}+\frac{46814430611}{257596419}a^{13}+\frac{2395193759594}{4379139123}a^{12}+\frac{8877549763871}{4379139123}a^{11}-\frac{4900271844177}{1459713041}a^{10}+\frac{2140885146371}{1459713041}a^{9}+\frac{123470246761}{1459713041}a^{8}+\frac{3655388509456}{1459713041}a^{7}-\frac{24560345888668}{4379139123}a^{6}+\frac{20332936581794}{4379139123}a^{5}-\frac{11475972626885}{4379139123}a^{4}+\frac{7619314371542}{4379139123}a^{3}-\frac{4327311695096}{4379139123}a^{2}+\frac{1283844712198}{4379139123}a-\frac{132245407610}{4379139123}$, $\frac{622498112728}{4379139123}a^{19}-\frac{284365108256}{1459713041}a^{18}-\frac{530709946850}{4379139123}a^{17}-\frac{338622627874}{4379139123}a^{16}+\frac{3513221362835}{4379139123}a^{15}-\frac{176515499718}{85865473}a^{14}+\frac{290883354725}{257596419}a^{13}+\frac{1843851157256}{1459713041}a^{12}+\frac{19047338827496}{4379139123}a^{11}-\frac{45218319105220}{4379139123}a^{10}+\frac{29036826545753}{4379139123}a^{9}-\frac{3191303072413}{4379139123}a^{8}+\frac{25466279930543}{4379139123}a^{7}-\frac{71087980248518}{4379139123}a^{6}+\frac{24183797989670}{1459713041}a^{5}-\frac{44502536818610}{4379139123}a^{4}+\frac{9044874356995}{1459713041}a^{3}-\frac{16724636160620}{4379139123}a^{2}+\frac{6417441948464}{4379139123}a-\frac{336293108530}{1459713041}$, $\frac{12725910013}{85865473}a^{19}-\frac{51347100616}{257596419}a^{18}-\frac{11139794757}{85865473}a^{17}-\frac{22015499563}{257596419}a^{16}+\frac{214361547709}{257596419}a^{15}-\frac{547133431781}{257596419}a^{14}+\frac{97184943316}{85865473}a^{13}+\frac{341342269517}{257596419}a^{12}+\frac{392634562768}{85865473}a^{11}-\frac{2738969182184}{257596419}a^{10}+\frac{1726274751118}{257596419}a^{9}-\frac{176343190334}{257596419}a^{8}+\frac{1566953837377}{257596419}a^{7}-\frac{4317828222095}{257596419}a^{6}+\frac{4358348967530}{257596419}a^{5}-\frac{886503265379}{85865473}a^{4}+\frac{1628808176507}{257596419}a^{3}-\frac{333736532892}{85865473}a^{2}+\frac{379440591935}{257596419}a-\frac{58843172087}{257596419}$, $\frac{18442726496}{4379139123}a^{19}-\frac{28354740428}{1459713041}a^{18}+\frac{30422980973}{4379139123}a^{17}+\frac{69273759073}{4379139123}a^{16}+\frac{183771051676}{4379139123}a^{15}-\frac{10931905844}{85865473}a^{14}+\frac{49167233317}{257596419}a^{13}+\frac{34882336640}{1459713041}a^{12}-\frac{9595592843}{4379139123}a^{11}-\frac{3514809195605}{4379139123}a^{10}+\frac{3908271100021}{4379139123}a^{9}-\frac{1031329410416}{4379139123}a^{8}+\frac{472927679311}{4379139123}a^{7}-\frac{4719920728990}{4379139123}a^{6}+\frac{2473909747831}{1459713041}a^{5}-\frac{5100166634713}{4379139123}a^{4}+\frac{920844359084}{1459713041}a^{3}-\frac{1907247864193}{4379139123}a^{2}+\frac{944966254459}{4379139123}a-\frac{63404147661}{1459713041}$, $\frac{86929084073}{1459713041}a^{19}-\frac{365572249892}{4379139123}a^{18}-\frac{73619934518}{1459713041}a^{17}-\frac{130524656357}{4379139123}a^{16}+\frac{1489169676638}{4379139123}a^{15}-\frac{223555268912}{257596419}a^{14}+\frac{42160947849}{85865473}a^{13}+\frac{2355354812725}{4379139123}a^{12}+\frac{2642107206061}{1459713041}a^{11}-\frac{19268860492204}{4379139123}a^{10}+\frac{12389354382080}{4379139123}a^{9}-\frac{1350072886834}{4379139123}a^{8}+\frac{10617772774694}{4379139123}a^{7}-\frac{30166750940914}{4379139123}a^{6}+\frac{30895127472859}{4379139123}a^{5}-\frac{6294115831162}{1459713041}a^{4}+\frac{11501845134472}{4379139123}a^{3}-\frac{2373386563199}{1459713041}a^{2}+\frac{2734532475973}{4379139123}a-\frac{424558247398}{4379139123}$, $\frac{72693729430}{4379139123}a^{19}-\frac{6629273768}{4379139123}a^{18}-\frac{128570458199}{4379139123}a^{17}-\frac{55594378598}{1459713041}a^{16}+\frac{277424279866}{4379139123}a^{15}-\frac{35700099206}{257596419}a^{14}-\frac{27481249561}{257596419}a^{13}+\frac{684883114930}{4379139123}a^{12}+\frac{3117542290250}{4379139123}a^{11}-\frac{1832402393063}{4379139123}a^{10}-\frac{1140985615001}{4379139123}a^{9}+\frac{821328236032}{4379139123}a^{8}+\frac{3404469309307}{4379139123}a^{7}-\frac{1403976336323}{1459713041}a^{6}+\frac{511178507752}{4379139123}a^{5}+\frac{231512870782}{4379139123}a^{4}+\frac{350501921356}{4379139123}a^{3}+\frac{134265351574}{4379139123}a^{2}-\frac{102433391125}{1459713041}a+\frac{86468861213}{4379139123}$
| 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: | \( 99.9526828541 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{10}\cdot 99.9526828541 \cdot 1}{4\cdot\sqrt{267622853577068773376}}\cr\approx \mathstrut & 0.146477712569 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 + 6*x^16 - 18*x^15 + 17*x^14 + 4*x^13 + 25*x^12 - 92*x^11 + 92*x^10 - 34*x^9 + 44*x^8 - 140*x^7 + 188*x^6 - 144*x^5 + 88*x^4 - 54*x^3 + 27*x^2 - 8*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 - 2*x^19 + 6*x^16 - 18*x^15 + 17*x^14 + 4*x^13 + 25*x^12 - 92*x^11 + 92*x^10 - 34*x^9 + 44*x^8 - 140*x^7 + 188*x^6 - 144*x^5 + 88*x^4 - 54*x^3 + 27*x^2 - 8*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 - 2*x^19 + 6*x^16 - 18*x^15 + 17*x^14 + 4*x^13 + 25*x^12 - 92*x^11 + 92*x^10 - 34*x^9 + 44*x^8 - 140*x^7 + 188*x^6 - 144*x^5 + 88*x^4 - 54*x^3 + 27*x^2 - 8*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 - 2*x^19 + 6*x^16 - 18*x^15 + 17*x^14 + 4*x^13 + 25*x^12 - 92*x^11 + 92*x^10 - 34*x^9 + 44*x^8 - 140*x^7 + 188*x^6 - 144*x^5 + 88*x^4 - 54*x^3 + 27*x^2 - 8*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
$D_5\wr C_2$ (as 20T48):
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 200 |
| The 14 conjugacy class representatives for $D_5\wr C_2$ |
| Character table for $D_5\wr C_2$ |
Intermediate fields
| \(\Q(\sqrt{-1}) \), 4.0.12176.1, 10.0.593019904.1 x5 |
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 10 siblings: | data not computed |
| Degree 20 siblings: | data not computed |
| Degree 25 sibling: | data not computed |
| Degree 40 siblings: | data not computed |
| Minimal sibling: | 10.0.261350442946356224.2 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/padicField/3.4.0.1}{4} }^{5}$ | ${\href{/padicField/5.5.0.1}{5} }^{4}$ | ${\href{/padicField/7.4.0.1}{4} }^{5}$ | ${\href{/padicField/11.4.0.1}{4} }^{5}$ | ${\href{/padicField/13.5.0.1}{5} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }^{5}$ | ${\href{/padicField/17.2.0.1}{2} }^{10}$ | ${\href{/padicField/19.10.0.1}{10} }^{2}$ | ${\href{/padicField/23.10.0.1}{10} }^{2}$ | ${\href{/padicField/29.2.0.1}{2} }^{10}$ | ${\href{/padicField/31.4.0.1}{4} }^{5}$ | ${\href{/padicField/37.5.0.1}{5} }^{4}$ | ${\href{/padicField/41.2.0.1}{2} }^{10}$ | ${\href{/padicField/43.4.0.1}{4} }^{5}$ | ${\href{/padicField/47.10.0.1}{10} }^{2}$ | ${\href{/padicField/53.5.0.1}{5} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }^{5}$ | ${\href{/padicField/59.2.0.1}{2} }^{10}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.10.10.7 | $x^{10} + 10 x^{9} + 50 x^{8} + 160 x^{7} + 360 x^{6} + 592 x^{5} + 656 x^{4} + 384 x^{3} - 112 x^{2} - 352 x - 1248$ | $2$ | $5$ | $10$ | $C_{10}$ | $[2]^{5}$ |
| 2.10.10.7 | $x^{10} + 10 x^{9} + 50 x^{8} + 160 x^{7} + 360 x^{6} + 592 x^{5} + 656 x^{4} + 384 x^{3} - 112 x^{2} - 352 x - 1248$ | $2$ | $5$ | $10$ | $C_{10}$ | $[2]^{5}$ | |
|
\(761\)
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $5$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | ||
| Deg $5$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ |