Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - x^21 - 21*x^20 + 19*x^19 + 181*x^18 - 145*x^17 - 833*x^16 + 575*x^15 + 2241*x^14 - 1289*x^13 - 3653*x^12 + 1683*x^11 + 3653*x^10 - 1289*x^9 - 2241*x^8 + 575*x^7 + 833*x^6 - 145*x^5 - 181*x^4 + 19*x^3 + 21*x^2 - x - 1)
gp: K = bnfinit(y^22 - y^21 - 21*y^20 + 19*y^19 + 181*y^18 - 145*y^17 - 833*y^16 + 575*y^15 + 2241*y^14 - 1289*y^13 - 3653*y^12 + 1683*y^11 + 3653*y^10 - 1289*y^9 - 2241*y^8 + 575*y^7 + 833*y^6 - 145*y^5 - 181*y^4 + 19*y^3 + 21*y^2 - y - 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^22 - x^21 - 21*x^20 + 19*x^19 + 181*x^18 - 145*x^17 - 833*x^16 + 575*x^15 + 2241*x^14 - 1289*x^13 - 3653*x^12 + 1683*x^11 + 3653*x^10 - 1289*x^9 - 2241*x^8 + 575*x^7 + 833*x^6 - 145*x^5 - 181*x^4 + 19*x^3 + 21*x^2 - x - 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^22 - x^21 - 21*x^20 + 19*x^19 + 181*x^18 - 145*x^17 - 833*x^16 + 575*x^15 + 2241*x^14 - 1289*x^13 - 3653*x^12 + 1683*x^11 + 3653*x^10 - 1289*x^9 - 2241*x^8 + 575*x^7 + 833*x^6 - 145*x^5 - 181*x^4 + 19*x^3 + 21*x^2 - x - 1)
\( x^{22} - x^{21} - 21 x^{20} + 19 x^{19} + 181 x^{18} - 145 x^{17} - 833 x^{16} + 575 x^{15} + 2241 x^{14} + \cdots - 1 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $22$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[22, 0]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(386235767790620472019065537319289881\)
\(\medspace = 23^{20}\cdot 229\cdot 982789\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(41.46\) | 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: | $23^{10/11}229^{1/2}982789^{1/2}\approx 259467.08994206376$ | ||
| Ramified primes: |
\(23\), \(229\), \(982789\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | $\Q(\sqrt{225058681}$) | ||
| $\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$
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: | $21$ | 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: |
$16a^{21}-30a^{20}-321a^{19}+596a^{18}+2610a^{17}-4814a^{16}-11136a^{15}+20536a^{14}+27118a^{13}-50590a^{12}-38736a^{11}+74540a^{10}+32522a^{9}-66430a^{8}-15840a^{7}+35632a^{6}+4326a^{5}-11167a^{4}-608a^{3}+1868a^{2}+34a-126$, $a^{21}-2a^{20}-20a^{19}+40a^{18}+162a^{17}-326a^{16}-688a^{15}+1408a^{14}+1666a^{13}-3530a^{12}-2364a^{11}+5336a^{10}+1970a^{9}-4942a^{8}-952a^{7}+2816a^{6}+258a^{5}-978a^{4}-36a^{3}+199a^{2}+2a-18$, $a^{21}-a^{20}-21a^{19}+19a^{18}+181a^{17}-145a^{16}-833a^{15}+575a^{14}+2241a^{13}-1289a^{12}-3653a^{11}+1683a^{10}+3653a^{9}-1289a^{8}-2241a^{7}+575a^{6}+833a^{5}-145a^{4}-181a^{3}+19a^{2}+20a-1$, $264a^{21}-410a^{20}-5350a^{19}+8004a^{18}+44040a^{17}-63178a^{16}-190785a^{15}+261304a^{14}+473354a^{13}-617098a^{12}-691512a^{11}+857820a^{10}+595790a^{9}-705611a^{8}-298400a^{7}+339392a^{6}+83826a^{5}-92058a^{4}-12104a^{3}+12868a^{2}+694a-714$, $96a^{21}-110a^{20}-1990a^{19}+2105a^{18}+16860a^{17}-16206a^{16}-75806a^{15}+64960a^{14}+197416a^{13}-147530a^{12}-307158a^{11}+195576a^{10}+286820a^{9}-152326a^{8}-158590a^{7}+69136a^{6}+49967a^{5}-17730a^{4}-8156a^{3}+2360a^{2}+526a-125$, $304a^{21}-394a^{20}-6226a^{19}+7576a^{18}+51924a^{17}-58638a^{16}-228618a^{15}+236352a^{14}+578761a^{13}-539410a^{12}-866556a^{11}+716743a^{10}+768522a^{9}-556226a^{8}-397464a^{7}+248928a^{6}+115434a^{5}-62022a^{4}-17220a^{3}+7880a^{2}+1018a-395$, $80a^{21}-149a^{20}-1600a^{19}+2948a^{18}+12949a^{17}-23672a^{16}-54854a^{15}+100112a^{14}+132054a^{13}-243420a^{12}-185086a^{11}+351516a^{10}+150486a^{9}-303664a^{8}-69350a^{7}+155255a^{6}+17174a^{5}-45272a^{4}-2013a^{3}+6852a^{2}+80a-413$, $76a^{20}-76a^{19}-1576a^{18}+1424a^{17}+13341a^{16}-10645a^{15}-59792a^{14}+40894a^{13}+154540a^{12}-87170a^{11}-236792a^{10}+104858a^{9}+214941a^{8}-70175a^{7}-113296a^{6}+25060a^{5}+33196a^{4}-4352a^{3}-4920a^{2}+284a+285$, $304a^{21}-305a^{20}-6315a^{19}+5729a^{18}+53593a^{17}-42985a^{16}-241111a^{15}+166073a^{14}+626858a^{13}-357242a^{12}-969460a^{11}+436272a^{10}+893129a^{9}-299533a^{8}-481871a^{7}+111721a^{6}+146233a^{5}-20909a^{4}-22755a^{3}+1592a^{2}+1394a-18$, $90a^{20}-90a^{19}-1868a^{18}+1688a^{17}+15834a^{16}-12638a^{15}-71112a^{14}+48672a^{13}+184409a^{12}-104193a^{11}-284124a^{10}+126290a^{9}+260346a^{8}-85696a^{7}-139448a^{6}+31374a^{5}+41946a^{4}-5680a^{3}-6468a^{2}+394a+394$, $a$, $a-1$, $264a^{21}-334a^{20}-5426a^{19}+6428a^{18}+45464a^{17}-49837a^{16}-201430a^{15}+201512a^{14}+514248a^{13}-462558a^{12}-778682a^{11}+621028a^{10}+700648a^{9}-490670a^{8}-368575a^{7}+226096a^{6}+108886a^{5}-58862a^{4}-16456a^{3}+7948a^{2}+977a-430$, $410a^{21}-528a^{20}-8414a^{19}+10172a^{18}+70362a^{17}-78964a^{16}-310934a^{15}+319782a^{14}+791050a^{13}-735438a^{12}-1192190a^{11}+989630a^{10}+1065963a^{9}-783894a^{8}-556167a^{7}+362182a^{6}+162664a^{5}-94542a^{4}-24308a^{3}+12798a^{2}+1428a-693$, $264a^{21}-410a^{20}-5350a^{19}+8004a^{18}+44040a^{17}-63178a^{16}-190785a^{15}+261304a^{14}+473354a^{13}-617098a^{12}-691512a^{11}+857820a^{10}+595790a^{9}-705611a^{8}-298400a^{7}+339392a^{6}+83826a^{5}-92058a^{4}-12104a^{3}+12868a^{2}+693a-714$, $437a^{21}-664a^{20}-8869a^{19}+12942a^{18}+73145a^{17}-101951a^{16}-317640a^{15}+420604a^{14}+790599a^{13}-990161a^{12}-1159822a^{11}+1371143a^{10}+1004827a^{9}-1123010a^{8}-506926a^{7}+537899a^{6}+143712a^{5}-145449a^{4}-20959a^{3}+20311a^{2}+1209a-1128$, $90a^{20}-90a^{19}-1868a^{18}+1688a^{17}+15834a^{16}-12638a^{15}-71112a^{14}+48672a^{13}+184409a^{12}-104193a^{11}-284124a^{10}+126290a^{9}+260346a^{8}-85696a^{7}-139448a^{6}+31374a^{5}+41946a^{4}-5680a^{3}-6468a^{2}+395a+394$, $430a^{21}-582a^{20}-8808a^{19}+11271a^{18}+73503a^{17}-88103a^{16}-324086a^{15}+360146a^{14}+822716a^{13}-838952a^{12}-1237926a^{11}+1148926a^{10}+1106760a^{9}-931800a^{8}-579046a^{7}+443785a^{6}+170502a^{5}-120156a^{4}-25756a^{3}+16934a^{2}+1534a-956$, $17a^{21}-18a^{20}-355a^{19}+343a^{18}+3037a^{17}-2627a^{16}-13835a^{15}+10463a^{14}+36689a^{13}-23579a^{12}-58571a^{11}+30975a^{10}+56765a^{9}-23883a^{8}-33155a^{7}+10727a^{6}+11345a^{5}-2723a^{4}-2100a^{3}+359a^{2}+163a-19$, $314a^{21}-458a^{20}-6384a^{19}+8887a^{18}+52762a^{17}-69586a^{16}-229700a^{15}+284687a^{14}+573385a^{13}-662274a^{12}-843868a^{11}+901496a^{10}+733419a^{9}-720237a^{8}-370946a^{7}+332942a^{6}+105321a^{5}-85692a^{4}-15396a^{3}+11202a^{2}+898a-573$, $146a^{21}-237a^{20}-2945a^{19}+4633a^{18}+24095a^{17}-36621a^{16}-103530a^{15}+151658a^{14}+254014a^{13}-358445a^{12}-365443a^{11}+498130a^{10}+308389a^{9}-408863a^{8}-150303a^{7}+195791a^{6}+40789a^{5}-52810a^{4}-5636a^{3}+7358a^{2}+304a-409$
| 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: | \( 47528575548.1 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{22}\cdot(2\pi)^{0}\cdot 47528575548.1 \cdot 1}{2\cdot\sqrt{386235767790620472019065537319289881}}\cr\approx \mathstrut & 0.160383047948 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^22 - x^21 - 21*x^20 + 19*x^19 + 181*x^18 - 145*x^17 - 833*x^16 + 575*x^15 + 2241*x^14 - 1289*x^13 - 3653*x^12 + 1683*x^11 + 3653*x^10 - 1289*x^9 - 2241*x^8 + 575*x^7 + 833*x^6 - 145*x^5 - 181*x^4 + 19*x^3 + 21*x^2 - 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^22 - x^21 - 21*x^20 + 19*x^19 + 181*x^18 - 145*x^17 - 833*x^16 + 575*x^15 + 2241*x^14 - 1289*x^13 - 3653*x^12 + 1683*x^11 + 3653*x^10 - 1289*x^9 - 2241*x^8 + 575*x^7 + 833*x^6 - 145*x^5 - 181*x^4 + 19*x^3 + 21*x^2 - 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^22 - x^21 - 21*x^20 + 19*x^19 + 181*x^18 - 145*x^17 - 833*x^16 + 575*x^15 + 2241*x^14 - 1289*x^13 - 3653*x^12 + 1683*x^11 + 3653*x^10 - 1289*x^9 - 2241*x^8 + 575*x^7 + 833*x^6 - 145*x^5 - 181*x^4 + 19*x^3 + 21*x^2 - 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^22 - x^21 - 21*x^20 + 19*x^19 + 181*x^18 - 145*x^17 - 833*x^16 + 575*x^15 + 2241*x^14 - 1289*x^13 - 3653*x^12 + 1683*x^11 + 3653*x^10 - 1289*x^9 - 2241*x^8 + 575*x^7 + 833*x^6 - 145*x^5 - 181*x^4 + 19*x^3 + 21*x^2 - 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
$C_{15}\times C_{420}$ (as 22T28):
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 22528 |
| The 208 conjugacy class representatives for $C_{15}\times C_{420}$ |
| Character table for $C_{15}\times C_{420}$ |
Intermediate fields
| \(\Q(\zeta_{23})^+\) |
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 22 siblings: | data not computed |
| Degree 44 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 | ${\href{/padicField/2.11.0.1}{11} }^{2}$ | ${\href{/padicField/3.11.0.1}{11} }^{2}$ | ${\href{/padicField/5.11.0.1}{11} }^{2}$ | ${\href{/padicField/7.11.0.1}{11} }^{2}$ | ${\href{/padicField/11.11.0.1}{11} }^{2}$ | ${\href{/padicField/13.11.0.1}{11} }^{2}$ | ${\href{/padicField/17.11.0.1}{11} }^{2}$ | $22$ | R | ${\href{/padicField/29.11.0.1}{11} }^{2}$ | ${\href{/padicField/31.11.0.1}{11} }^{2}$ | $22$ | ${\href{/padicField/41.11.0.1}{11} }^{2}$ | ${\href{/padicField/43.11.0.1}{11} }^{2}$ | ${\href{/padicField/47.2.0.1}{2} }^{6}{,}\,{\href{/padicField/47.1.0.1}{1} }^{10}$ | ${\href{/padicField/53.11.0.1}{11} }^{2}$ | $22$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(23\)
| 23.11.10.10 | $x^{11} + 23$ | $11$ | $1$ | $10$ | $C_{11}$ | $[\ ]_{11}$ |
| 23.11.10.10 | $x^{11} + 23$ | $11$ | $1$ | $10$ | $C_{11}$ | $[\ ]_{11}$ | |
|
\(229\)
| $\Q_{229}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{229}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{229}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{229}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{229}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{229}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{229}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{229}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{229}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{229}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{229}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{229}$ | $x$ | $1$ | $1$ | $0$ | 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 $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
|
\(982789\)
| $\Q_{982789}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{982789}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{982789}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{982789}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{982789}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{982789}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{982789}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{982789}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{982789}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{982789}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{982789}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{982789}$ | $x$ | $1$ | $1$ | $0$ | 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 $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |