Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - 7*x^20 - 72*x^18 - 36*x^16 + 711*x^14 + 1469*x^12 - 353*x^10 - 3136*x^8 - 2011*x^6 + 858*x^4 + 1172*x^2 + 289)
gp: K = bnfinit(y^22 - 7*y^20 - 72*y^18 - 36*y^16 + 711*y^14 + 1469*y^12 - 353*y^10 - 3136*y^8 - 2011*y^6 + 858*y^4 + 1172*y^2 + 289, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^22 - 7*x^20 - 72*x^18 - 36*x^16 + 711*x^14 + 1469*x^12 - 353*x^10 - 3136*x^8 - 2011*x^6 + 858*x^4 + 1172*x^2 + 289);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^22 - 7*x^20 - 72*x^18 - 36*x^16 + 711*x^14 + 1469*x^12 - 353*x^10 - 3136*x^8 - 2011*x^6 + 858*x^4 + 1172*x^2 + 289)
\( x^{22} - 7 x^{20} - 72 x^{18} - 36 x^{16} + 711 x^{14} + 1469 x^{12} - 353 x^{10} - 3136 x^{8} + \cdots + 289 \)
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: | $[8, 7]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(-56501459388151144478039723653407440896\)
\(\medspace = -\,2^{22}\cdot 1297^{10}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(52.00\) | 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: | not computed | ||
| Ramified primes: |
\(2\), \(1297\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-1}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $\frac{1}{842353788145}a^{20}+\frac{204011153886}{842353788145}a^{18}+\frac{318441489531}{842353788145}a^{16}-\frac{149381160433}{842353788145}a^{14}-\frac{270218048098}{842353788145}a^{12}-\frac{50186867455}{168470757629}a^{10}-\frac{295176283943}{842353788145}a^{8}-\frac{56331632028}{168470757629}a^{6}+\frac{134320968534}{842353788145}a^{4}+\frac{11939414678}{168470757629}a^{2}-\frac{361572309978}{842353788145}$, $\frac{1}{14320014398465}a^{21}-\frac{4850111574984}{14320014398465}a^{19}+\frac{6214918006546}{14320014398465}a^{17}+\frac{2377680204002}{14320014398465}a^{15}-\frac{4481986988823}{14320014398465}a^{13}-\frac{1397952928487}{2864002879693}a^{11}+\frac{5601300233072}{14320014398465}a^{9}-\frac{1235626935431}{2864002879693}a^{7}-\frac{6604509336626}{14320014398465}a^{5}+\frac{348880929936}{2864002879693}a^{3}+\frac{5534904207037}{14320014398465}a$
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 $1$ (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: | $14$ | 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: |
$\frac{1783009437588}{14320014398465}a^{21}-\frac{13700496303282}{14320014398465}a^{19}-\frac{118894981020892}{14320014398465}a^{17}+\frac{16180060452966}{14320014398465}a^{15}+\frac{12\!\cdots\!16}{14320014398465}a^{13}+\frac{354231508069724}{2864002879693}a^{11}-\frac{17\!\cdots\!54}{14320014398465}a^{9}-\frac{866690424084732}{2864002879693}a^{7}-\frac{757253984120973}{14320014398465}a^{5}+\frac{383476194528130}{2864002879693}a^{3}+\frac{833174358686926}{14320014398465}a$, $\frac{613731285957}{2864002879693}a^{21}-\frac{4806668530694}{2864002879693}a^{19}-\frac{40189005476503}{2864002879693}a^{17}+\frac{11337156576018}{2864002879693}a^{15}+\frac{426803105714540}{2864002879693}a^{13}+\frac{546043062717176}{2864002879693}a^{11}-\frac{670332074647651}{2864002879693}a^{9}-\frac{13\!\cdots\!98}{2864002879693}a^{7}-\frac{93164858147206}{2864002879693}a^{5}+\frac{598021042446439}{2864002879693}a^{3}+\frac{204052839167994}{2864002879693}a$, $\frac{865438946124}{14320014398465}a^{21}-\frac{6395691143001}{14320014398465}a^{19}-\frac{60244656235841}{14320014398465}a^{17}-\frac{3304662496347}{14320014398465}a^{15}+\frac{634324300800253}{14320014398465}a^{13}+\frac{195445963568089}{2864002879693}a^{11}-\frac{871809024746852}{14320014398465}a^{9}-\frac{471680423998375}{2864002879693}a^{7}-\frac{429313738764344}{14320014398465}a^{5}+\frac{212206801169491}{2864002879693}a^{3}+\frac{437424574918543}{14320014398465}a$, $\frac{53974734792}{842353788145}a^{20}-\frac{429694421193}{842353788145}a^{18}-\frac{3450019105003}{842353788145}a^{16}+\frac{1146160173489}{842353788145}a^{14}+\frac{36211605778939}{842353788145}a^{12}+\frac{9340326147155}{168470757629}a^{10}-\frac{52779665563606}{842353788145}a^{8}-\frac{23235882358021}{168470757629}a^{6}-\frac{19290602668037}{842353788145}a^{4}+\frac{10074670197567}{168470757629}a^{2}+\frac{23279399045199}{842353788145}$, $\frac{63839295041}{842353788145}a^{20}-\frac{511646121354}{842353788145}a^{18}-\frac{4054019377239}{842353788145}a^{16}+\frac{1599242100377}{842353788145}a^{14}+\frac{42557610201652}{842353788145}a^{12}+\frac{10537708886713}{168470757629}a^{10}-\frac{63199914530108}{842353788145}a^{8}-\frac{26555426010228}{168470757629}a^{6}-\frac{19483878752216}{842353788145}a^{4}+\frac{11764587950165}{168470757629}a^{2}+\frac{25206733754197}{842353788145}$, $\frac{322933346181}{2864002879693}a^{21}-\frac{2419132171358}{2864002879693}a^{19}-\frac{22087283080614}{2864002879693}a^{17}-\frac{531052157132}{2864002879693}a^{15}+\frac{230851056653645}{2864002879693}a^{13}+\frac{358338099539537}{2864002879693}a^{11}-\frac{300683848423044}{2864002879693}a^{9}-\frac{864343083125138}{2864002879693}a^{7}-\frac{197354767457676}{2864002879693}a^{5}+\frac{388786233879465}{2864002879693}a^{3}+\frac{166516954075572}{2864002879693}a$, $\frac{3333853761384}{14320014398465}a^{21}-\frac{25181702786076}{14320014398465}a^{19}-\frac{226129362014851}{14320014398465}a^{17}+\frac{5554401695218}{14320014398465}a^{15}+\frac{23\!\cdots\!38}{14320014398465}a^{13}+\frac{715394999437861}{2864002879693}a^{11}-\frac{31\!\cdots\!67}{14320014398465}a^{9}-\frac{17\!\cdots\!12}{2864002879693}a^{7}-\frac{18\!\cdots\!49}{14320014398465}a^{5}+\frac{760653364661482}{2864002879693}a^{3}+\frac{17\!\cdots\!68}{14320014398465}a$, $\frac{6266834167008}{14320014398465}a^{21}-\frac{49255187783752}{14320014398465}a^{19}-\frac{408376485158967}{14320014398465}a^{17}+\frac{120586610611281}{14320014398465}a^{15}+\frac{43\!\cdots\!21}{14320014398465}a^{13}+\frac{11\!\cdots\!62}{2864002879693}a^{11}-\frac{67\!\cdots\!94}{14320014398465}a^{9}-\frac{27\!\cdots\!79}{2864002879693}a^{7}-\frac{11\!\cdots\!03}{14320014398465}a^{5}+\frac{12\!\cdots\!24}{2864002879693}a^{3}+\frac{22\!\cdots\!46}{14320014398465}a$, $\frac{3204333184262}{14320014398465}a^{21}-\frac{25236678412598}{14320014398465}a^{19}-\frac{208707771641848}{14320014398465}a^{17}+\frac{68569322496709}{14320014398465}a^{15}+\frac{22\!\cdots\!59}{14320014398465}a^{13}+\frac{549517860340798}{2864002879693}a^{11}-\frac{35\!\cdots\!71}{14320014398465}a^{9}-\frac{13\!\cdots\!34}{2864002879693}a^{7}-\frac{326826875505257}{14320014398465}a^{5}+\frac{609823198752526}{2864002879693}a^{3}+\frac{10\!\cdots\!79}{14320014398465}a$, $\frac{613731285957}{2864002879693}a^{21}-\frac{4806668530694}{2864002879693}a^{19}-\frac{40189005476503}{2864002879693}a^{17}+\frac{11337156576018}{2864002879693}a^{15}+\frac{426803105714540}{2864002879693}a^{13}+\frac{546043062717176}{2864002879693}a^{11}-\frac{670332074647651}{2864002879693}a^{9}-\frac{13\!\cdots\!98}{2864002879693}a^{7}-\frac{93164858147206}{2864002879693}a^{5}+\frac{598021042446439}{2864002879693}a^{3}+\frac{201188836288301}{2864002879693}a$, $\frac{1801076525808}{14320014398465}a^{21}-\frac{130632097167}{842353788145}a^{20}-\frac{14008965752047}{14320014398465}a^{19}+\frac{1016054390623}{842353788145}a^{18}-\frac{118777198527847}{14320014398465}a^{17}+\frac{8602590648548}{842353788145}a^{16}+\frac{27654512952571}{14320014398465}a^{15}-\frac{1876525990139}{842353788145}a^{14}+\frac{12\!\cdots\!81}{14320014398465}a^{13}-\frac{90764907469884}{842353788145}a^{12}+\frac{332547714892763}{2864002879693}a^{11}-\frac{24443261429565}{168470757629}a^{10}-\frac{19\!\cdots\!64}{14320014398465}a^{9}+\frac{134535184557061}{842353788145}a^{8}-\frac{822945116782279}{2864002879693}a^{7}+\frac{60360294843183}{168470757629}a^{6}-\frac{384171784549073}{14320014398465}a^{5}+\frac{38843951132082}{842353788145}a^{4}+\frac{364295266051083}{2864002879693}a^{3}-\frac{26936250737045}{168470757629}a^{2}+\frac{639041521278306}{14320014398465}a-\frac{50985708244719}{842353788145}$, $\frac{3454000080804}{14320014398465}a^{21}-\frac{126885933573}{842353788145}a^{20}-\frac{26650885545986}{14320014398465}a^{19}+\frac{978241412792}{842353788145}a^{18}-\frac{229738032756801}{14320014398465}a^{17}+\frac{8424072753377}{842353788145}a^{16}+\frac{40775594034748}{14320014398465}a^{15}-\frac{1266921439386}{842353788145}a^{14}+\frac{24\!\cdots\!93}{14320014398465}a^{13}-\frac{88242337689991}{842353788145}a^{12}+\frac{669274607259716}{2864002879693}a^{11}-\frac{24837182581365}{168470757629}a^{10}-\frac{37\!\cdots\!82}{14320014398465}a^{9}+\frac{122287526689594}{842353788145}a^{8}-\frac{16\!\cdots\!66}{2864002879693}a^{7}+\frac{59378466218675}{168470757629}a^{6}-\frac{995530332249954}{14320014398465}a^{5}+\frac{53814226149378}{842353788145}a^{4}+\frac{816534777305657}{2864002879693}a^{3}-\frac{23857534351597}{168470757629}a^{2}+\frac{16\!\cdots\!63}{14320014398465}a-\frac{47087161205556}{842353788145}$, $\frac{5659328402881}{14320014398465}a^{21}-\frac{20716912288}{168470757629}a^{20}-\frac{42122536942494}{14320014398465}a^{19}+\frac{161789635698}{168470757629}a^{18}-\frac{388087933000414}{14320014398465}a^{17}+\frac{1379986490734}{168470757629}a^{16}-\frac{35578151482098}{14320014398465}a^{15}-\frac{502693098736}{168470757629}a^{14}+\frac{39\!\cdots\!72}{14320014398465}a^{13}-\frac{15687262187893}{168470757629}a^{12}+\frac{12\!\cdots\!52}{2864002879693}a^{11}-\frac{19147898018373}{168470757629}a^{10}-\frac{43\!\cdots\!98}{14320014398465}a^{9}+\frac{32830173414607}{168470757629}a^{8}-\frac{28\!\cdots\!37}{2864002879693}a^{7}+\frac{67193395816812}{168470757629}a^{6}-\frac{35\!\cdots\!86}{14320014398465}a^{5}+\frac{9513974279564}{168470757629}a^{4}+\frac{12\!\cdots\!15}{2864002879693}a^{3}-\frac{33257768616040}{168470757629}a^{2}+\frac{28\!\cdots\!02}{14320014398465}a-\frac{14157180837506}{168470757629}$, $\frac{326507827854}{2864002879693}a^{21}+\frac{48290717082}{842353788145}a^{20}-\frac{2552848127735}{2864002879693}a^{19}-\frac{378309744318}{842353788145}a^{18}-\frac{21528430015294}{2864002879693}a^{17}-\frac{3237552071753}{842353788145}a^{16}+\frac{6402008863460}{2864002879693}a^{15}+\frac{1495609889604}{842353788145}a^{14}+\frac{236187200937411}{2864002879693}a^{13}+\frac{38122534851024}{842353788145}a^{12}+\frac{307288716600154}{2864002879693}a^{11}+\frac{8954613788660}{168470757629}a^{10}-\frac{406445387142505}{2864002879693}a^{9}-\frac{79713311211376}{842353788145}a^{8}-\frac{876867539087793}{2864002879693}a^{7}-\frac{29981535524553}{168470757629}a^{6}-\frac{103283976746958}{2864002879693}a^{5}-\frac{5563837296517}{842353788145}a^{4}+\frac{441269353302526}{2864002879693}a^{3}+\frac{16279637114986}{168470757629}a^{2}+\frac{183868972491829}{2864002879693}a+\frac{30197461269329}{842353788145}$
| 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: | \( 43490276888.5 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{8}\cdot(2\pi)^{7}\cdot 43490276888.5 \cdot 1}{2\cdot\sqrt{56501459388151144478039723653407440896}}\cr\approx \mathstrut & 0.286306618810 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^22 - 7*x^20 - 72*x^18 - 36*x^16 + 711*x^14 + 1469*x^12 - 353*x^10 - 3136*x^8 - 2011*x^6 + 858*x^4 + 1172*x^2 + 289)
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 - 7*x^20 - 72*x^18 - 36*x^16 + 711*x^14 + 1469*x^12 - 353*x^10 - 3136*x^8 - 2011*x^6 + 858*x^4 + 1172*x^2 + 289, 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 - 7*x^20 - 72*x^18 - 36*x^16 + 711*x^14 + 1469*x^12 - 353*x^10 - 3136*x^8 - 2011*x^6 + 858*x^4 + 1172*x^2 + 289);
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 - 7*x^20 - 72*x^18 - 36*x^16 + 711*x^14 + 1469*x^12 - 353*x^10 - 3136*x^8 - 2011*x^6 + 858*x^4 + 1172*x^2 + 289);
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
$C_2^{10}.D_{22}$ (as 22T32):
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 45056 |
| The 200 conjugacy class representatives for $C_2^{10}.D_{22}$ |
| Character table for $C_2^{10}.D_{22}$ |
Intermediate fields
| 11.11.3670285774226257.1 |
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 22 siblings: | data not computed |
| Degree 44 siblings: | data not computed |
| Minimal sibling: | 22.12.56501459388151144478039723653407440896.13 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | $22$ | ${\href{/padicField/5.4.0.1}{4} }^{2}{,}\,{\href{/padicField/5.2.0.1}{2} }^{6}{,}\,{\href{/padicField/5.1.0.1}{1} }^{2}$ | $22$ | ${\href{/padicField/11.4.0.1}{4} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }^{7}$ | ${\href{/padicField/13.11.0.1}{11} }^{2}$ | ${\href{/padicField/17.4.0.1}{4} }^{4}{,}\,{\href{/padicField/17.2.0.1}{2} }^{2}{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ | $22$ | $22$ | ${\href{/padicField/29.4.0.1}{4} }^{3}{,}\,{\href{/padicField/29.2.0.1}{2} }^{5}$ | ${\href{/padicField/31.4.0.1}{4} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }^{7}$ | ${\href{/padicField/37.4.0.1}{4} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }^{6}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.4.0.1}{4} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }^{6}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.4.0.1}{4} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }^{7}$ | $22$ | ${\href{/padicField/53.11.0.1}{11} }^{2}$ | ${\href{/padicField/59.4.0.1}{4} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }^{7}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| Deg $22$ | $2$ | $11$ | $22$ | |||
|
\(1297\)
| $\Q_{1297}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{1297}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 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 $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $4$ | $2$ | $2$ | $2$ | ||||
| Deg $4$ | $2$ | $2$ | $2$ |