Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - x^20 - 23*x^18 + 27*x^16 + 145*x^14 - 193*x^12 - 240*x^10 + 421*x^8 - 162*x^6 + 9*x^2 - 1)
gp: K = bnfinit(y^22 - y^20 - 23*y^18 + 27*y^16 + 145*y^14 - 193*y^12 - 240*y^10 + 421*y^8 - 162*y^6 + 9*y^2 - 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^22 - x^20 - 23*x^18 + 27*x^16 + 145*x^14 - 193*x^12 - 240*x^10 + 421*x^8 - 162*x^6 + 9*x^2 - 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^22 - x^20 - 23*x^18 + 27*x^16 + 145*x^14 - 193*x^12 - 240*x^10 + 421*x^8 - 162*x^6 + 9*x^2 - 1)
\( x^{22} - x^{20} - 23x^{18} + 27x^{16} + 145x^{14} - 193x^{12} - 240x^{10} + 421x^{8} - 162x^{6} + 9x^{2} - 1 \)
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: | $[14, 4]$ | 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: | \(\Q\) | ||
| $\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{5}a^{18}+\frac{1}{5}a^{14}-\frac{2}{5}a^{12}+\frac{2}{5}a^{10}+\frac{1}{5}a^{8}-\frac{1}{5}a^{6}-\frac{1}{5}a^{4}-\frac{2}{5}a^{2}-\frac{1}{5}$, $\frac{1}{5}a^{19}+\frac{1}{5}a^{15}-\frac{2}{5}a^{13}+\frac{2}{5}a^{11}+\frac{1}{5}a^{9}-\frac{1}{5}a^{7}-\frac{1}{5}a^{5}-\frac{2}{5}a^{3}-\frac{1}{5}a$, $\frac{1}{24335}a^{20}-\frac{1822}{24335}a^{18}-\frac{11189}{24335}a^{16}+\frac{6801}{24335}a^{14}+\frac{6906}{24335}a^{12}-\frac{4558}{24335}a^{10}+\frac{367}{785}a^{8}-\frac{3144}{24335}a^{6}+\frac{294}{4867}a^{4}-\frac{4887}{24335}a^{2}+\frac{7227}{24335}$, $\frac{1}{24335}a^{21}-\frac{1822}{24335}a^{19}-\frac{11189}{24335}a^{17}+\frac{6801}{24335}a^{15}+\frac{6906}{24335}a^{13}-\frac{4558}{24335}a^{11}+\frac{367}{785}a^{9}-\frac{3144}{24335}a^{7}+\frac{294}{4867}a^{5}-\frac{4887}{24335}a^{3}+\frac{7227}{24335}a$
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: | $17$ | 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: |
$a^{21}-a^{19}-23a^{17}+27a^{15}+145a^{13}-193a^{11}-240a^{9}+421a^{7}-162a^{5}+9a$, $\frac{622504}{24335}a^{21}-\frac{78167}{4867}a^{19}-\frac{14464211}{24335}a^{17}+\frac{11424997}{24335}a^{15}+\frac{94542733}{24335}a^{13}-\frac{84972791}{24335}a^{11}-\frac{5845634}{785}a^{9}+\frac{194772006}{24335}a^{7}-\frac{27968508}{24335}a^{5}-\frac{10805034}{24335}a^{3}+\frac{322320}{4867}a$, $\frac{135212}{24335}a^{20}-\frac{81197}{24335}a^{18}-\frac{3143668}{24335}a^{16}+\frac{2395529}{24335}a^{14}+\frac{20593463}{24335}a^{12}-\frac{17888912}{24335}a^{10}-\frac{1283504}{785}a^{8}+\frac{8234308}{4867}a^{6}-\frac{5121857}{24335}a^{4}-\frac{2358683}{24335}a^{2}+\frac{341022}{24335}$, $\frac{646839}{24335}a^{21}-\frac{83034}{4867}a^{19}-\frac{15023916}{24335}a^{17}+\frac{12082042}{24335}a^{15}+\frac{98071308}{24335}a^{13}-\frac{89669446}{24335}a^{11}-\frac{6034034}{785}a^{9}+\frac{205017041}{24335}a^{7}-\frac{31910778}{24335}a^{5}-\frac{10805034}{24335}a^{3}+\frac{366123}{4867}a$, $\frac{13142}{24335}a^{20}-\frac{8818}{24335}a^{18}-\frac{305788}{24335}a^{16}+\frac{254238}{24335}a^{14}+\frac{400808}{4867}a^{12}-\frac{1881729}{24335}a^{10}-\frac{25011}{157}a^{8}+\frac{4319411}{24335}a^{6}-\frac{480216}{24335}a^{4}-\frac{350446}{24335}a^{2}+\frac{7463}{24335}$, $\frac{96457}{24335}a^{20}-\frac{65422}{24335}a^{18}-\frac{2238943}{24335}a^{16}+\frac{1884124}{24335}a^{14}+\frac{14577018}{24335}a^{12}-\frac{13924782}{24335}a^{10}-\frac{887534}{785}a^{8}+\frac{6341120}{4867}a^{6}-\frac{5683177}{24335}a^{4}-\frac{1632253}{24335}a^{2}+\frac{305817}{24335}$, $\frac{113252}{24335}a^{20}-\frac{67083}{24335}a^{18}-\frac{2632688}{24335}a^{16}+\frac{1985503}{24335}a^{14}+\frac{3448012}{4867}a^{12}-\frac{14848079}{24335}a^{10}-\frac{214507}{157}a^{8}+\frac{34181506}{24335}a^{6}-\frac{4487536}{24335}a^{4}-\frac{1817276}{24335}a^{2}+\frac{266233}{24335}$, $\frac{175324}{24335}a^{20}-\frac{111591}{24335}a^{18}-\frac{4071161}{24335}a^{16}+\frac{3253616}{24335}a^{14}+\frac{5312741}{4867}a^{12}-\frac{24168983}{24335}a^{10}-\frac{326566}{157}a^{8}+\frac{55302017}{24335}a^{6}-\frac{8722502}{24335}a^{4}-\frac{2878637}{24335}a^{2}+\frac{497936}{24335}$, $\frac{171883}{24335}a^{21}-\frac{22157}{4867}a^{19}-\frac{3994777}{24335}a^{17}+\frac{3222709}{24335}a^{15}+\frac{26117956}{24335}a^{13}-\frac{23942497}{24335}a^{11}-\frac{1615813}{785}a^{9}+\frac{54965357}{24335}a^{7}-\frac{7731091}{24335}a^{5}-\frac{3700478}{24335}a^{3}+\frac{127295}{4867}a$, $\frac{175324}{24335}a^{20}-\frac{111591}{24335}a^{18}-\frac{4071161}{24335}a^{16}+\frac{3253616}{24335}a^{14}+\frac{5312741}{4867}a^{12}-\frac{24168983}{24335}a^{10}-\frac{326566}{157}a^{8}+\frac{55302017}{24335}a^{6}-\frac{8722502}{24335}a^{4}-\frac{2878637}{24335}a^{2}+\frac{522271}{24335}$, $\frac{31798}{24335}a^{21}-\frac{126394}{24335}a^{20}-\frac{18656}{24335}a^{19}+\frac{75901}{24335}a^{18}-\frac{740172}{24335}a^{17}+\frac{2938476}{24335}a^{16}+\frac{552758}{24335}a^{15}-\frac{2239741}{24335}a^{14}+\frac{4864948}{24335}a^{13}-\frac{3848820}{4867}a^{12}-\frac{4132974}{24335}a^{11}+\frac{16729808}{24335}a^{10}-\frac{306879}{785}a^{9}+\frac{239518}{157}a^{8}+\frac{9510253}{24335}a^{7}-\frac{38500917}{24335}a^{6}-\frac{166373}{4867}a^{5}+\frac{4967752}{24335}a^{4}-\frac{480216}{24335}a^{3}+\frac{2143387}{24335}a^{2}-\frac{39929}{24335}a-\frac{273696}{24335}$, $\frac{290568}{24335}a^{21}-\frac{192119}{24335}a^{20}-\frac{177316}{24335}a^{19}+\frac{113252}{24335}a^{18}-\frac{6750147}{24335}a^{17}+\frac{4464906}{24335}a^{16}+\frac{5212648}{24335}a^{15}-\frac{670999}{4867}a^{14}+\frac{44117863}{24335}a^{13}-\frac{29226747}{24335}a^{12}-\frac{38839564}{24335}a^{11}+\frac{5018456}{4867}a^{10}-\frac{2728529}{785}a^{9}+\frac{1817831}{785}a^{8}+\frac{89080543}{24335}a^{7}-\frac{57777923}{24335}a^{6}-\frac{2578102}{4867}a^{5}+\frac{7526861}{24335}a^{4}-\frac{4487536}{24335}a^{3}+\frac{607865}{4867}a^{2}+\frac{822171}{24335}a-\frac{482687}{24335}$, $\frac{720953}{24335}a^{21}+\frac{338681}{24335}a^{20}-\frac{454899}{24335}a^{19}-\frac{218601}{24335}a^{18}-\frac{16749452}{24335}a^{17}-\frac{7867044}{24335}a^{16}+\frac{2656530}{4867}a^{15}+\frac{6354762}{24335}a^{14}+\frac{109433849}{24335}a^{13}+\frac{51363114}{24335}a^{12}-\frac{19744015}{4867}a^{11}-\frac{47149301}{24335}a^{10}-\frac{6756332}{785}a^{9}-\frac{3163267}{785}a^{8}+\frac{226074626}{24335}a^{7}+\frac{21565463}{4867}a^{6}-\frac{33332157}{24335}a^{5}-\frac{16376416}{24335}a^{4}-\frac{2450649}{4867}a^{3}-\frac{5907294}{24335}a^{2}+\frac{1926749}{24335}a+\frac{943416}{24335}$, $\frac{345121}{24335}a^{21}+\frac{131021}{24335}a^{20}-\frac{227678}{24335}a^{19}-\frac{71784}{24335}a^{18}-\frac{8014279}{24335}a^{17}-\frac{3046774}{24335}a^{16}+\frac{1317737}{4867}a^{15}+\frac{2160074}{24335}a^{14}+\frac{52263503}{24335}a^{13}+\frac{3999165}{4867}a^{12}-\frac{9754131}{4867}a^{11}-\frac{16253997}{24335}a^{10}-\frac{3203614}{785}a^{9}-\frac{251325}{157}a^{8}+\frac{111214107}{24335}a^{7}+\frac{37663968}{24335}a^{6}-\frac{18220524}{24335}a^{5}-\frac{3826383}{24335}a^{4}-\frac{1156602}{4867}a^{3}-\frac{2153188}{24335}a^{2}+\frac{1107653}{24335}a+\frac{237799}{24335}$, $\frac{49329}{785}a^{21}+\frac{224877}{24335}a^{20}-\frac{32461}{785}a^{19}-\frac{162977}{24335}a^{18}-\frac{1145361}{785}a^{17}-\frac{5214783}{24335}a^{16}+\frac{940096}{785}a^{15}+\frac{4635249}{24335}a^{14}+\frac{1493403}{157}a^{13}+\frac{33829783}{24335}a^{12}-\frac{6962948}{785}a^{11}-\frac{34058432}{24335}a^{10}-\frac{2834465}{157}a^{9}-\frac{2031754}{785}a^{8}+\frac{15888747}{785}a^{7}+\frac{15420843}{4867}a^{6}-\frac{2654202}{785}a^{5}-\frac{15941742}{24335}a^{4}-\frac{829737}{785}a^{3}-\frac{3835628}{24335}a^{2}+\frac{159981}{785}a+\frac{965972}{24335}$, $\frac{391887}{24335}a^{21}+\frac{162182}{24335}a^{20}-\frac{52566}{4867}a^{19}-\frac{102773}{24335}a^{18}-\frac{9098623}{24335}a^{17}-\frac{3765373}{24335}a^{16}+\frac{7583536}{24335}a^{15}+\frac{2999378}{24335}a^{14}+\frac{59288194}{24335}a^{13}+\frac{4911933}{4867}a^{12}-\frac{56080318}{24335}a^{11}-\frac{22287254}{24335}a^{10}-\frac{3622137}{785}a^{9}-\frac{301555}{157}a^{8}+\frac{127786008}{24335}a^{7}+\frac{50982606}{24335}a^{6}-\frac{21895464}{24335}a^{5}-\frac{8242286}{24335}a^{4}-\frac{6674527}{24335}a^{3}-\frac{2552526}{24335}a^{2}+\frac{248546}{4867}a+\frac{466138}{24335}$, $\frac{1623748}{24335}a^{21}-\frac{237799}{24335}a^{20}-\frac{943833}{24335}a^{19}+\frac{106778}{24335}a^{18}-\frac{37762652}{24335}a^{17}+\frac{5541161}{24335}a^{16}+\frac{28037876}{24335}a^{15}-\frac{3373799}{24335}a^{14}+\frac{247678382}{24335}a^{13}-\frac{36640929}{24335}a^{12}-\frac{209963278}{24335}a^{11}+\frac{25899382}{24335}a^{10}-\frac{15512506}{785}a^{9}+\frac{2365347}{785}a^{8}+\frac{96894328}{4867}a^{7}-\frac{61158004}{24335}a^{6}-\frac{53618388}{24335}a^{5}+\frac{171894}{4867}a^{4}-\frac{27758837}{24335}a^{3}+\frac{3826383}{24335}a^{2}+\frac{2952498}{24335}a-\frac{11338}{24335}$
| 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: | \( 135405112524 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{14}\cdot(2\pi)^{4}\cdot 135405112524 \cdot 1}{2\cdot\sqrt{56501459388151144478039723653407440896}}\cr\approx \mathstrut & 0.229993015147823 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^22 - x^20 - 23*x^18 + 27*x^16 + 145*x^14 - 193*x^12 - 240*x^10 + 421*x^8 - 162*x^6 + 9*x^2 - 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^20 - 23*x^18 + 27*x^16 + 145*x^14 - 193*x^12 - 240*x^10 + 421*x^8 - 162*x^6 + 9*x^2 - 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^20 - 23*x^18 + 27*x^16 + 145*x^14 - 193*x^12 - 240*x^10 + 421*x^8 - 162*x^6 + 9*x^2 - 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^20 - 23*x^18 + 27*x^16 + 145*x^14 - 193*x^12 - 240*x^10 + 421*x^8 - 162*x^6 + 9*x^2 - 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
$C_2^{10}.D_{11}$ (as 22T30):
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 100 conjugacy class representatives for $C_2^{10}.D_{11}$ |
| Character table for $C_2^{10}.D_{11}$ |
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: | 22.10.73282392826432034388017521578469450842112.6 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/padicField/3.11.0.1}{11} }^{2}$ | ${\href{/padicField/5.4.0.1}{4} }{,}\,{\href{/padicField/5.2.0.1}{2} }^{9}$ | ${\href{/padicField/7.11.0.1}{11} }^{2}$ | ${\href{/padicField/11.4.0.1}{4} }{,}\,{\href{/padicField/11.2.0.1}{2} }^{9}$ | ${\href{/padicField/13.11.0.1}{11} }^{2}$ | ${\href{/padicField/17.4.0.1}{4} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }^{6}{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ | ${\href{/padicField/19.11.0.1}{11} }^{2}$ | ${\href{/padicField/23.11.0.1}{11} }^{2}$ | ${\href{/padicField/29.4.0.1}{4} }{,}\,{\href{/padicField/29.2.0.1}{2} }^{9}$ | ${\href{/padicField/31.4.0.1}{4} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }^{6}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.4.0.1}{4} }^{3}{,}\,{\href{/padicField/37.2.0.1}{2} }^{5}$ | ${\href{/padicField/41.4.0.1}{4} }^{4}{,}\,{\href{/padicField/41.2.0.1}{2} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.4.0.1}{4} }^{3}{,}\,{\href{/padicField/43.2.0.1}{2} }^{5}$ | ${\href{/padicField/47.11.0.1}{11} }^{2}$ | ${\href{/padicField/53.11.0.1}{11} }^{2}$ | ${\href{/padicField/59.4.0.1}{4} }^{3}{,}\,{\href{/padicField/59.2.0.1}{2} }^{5}$ |
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 $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $4$ | $2$ | $2$ | $2$ |