Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 2*x^17 + 4*x^15 - 7*x^14 + x^13 + 7*x^12 + 2*x^11 - 10*x^10 + x^9 + 20*x^8 - 22*x^7 - 2*x^6 + 7*x^5 + 2*x^4 - 10*x^3 + 11*x^2 - 5*x + 1)
gp: K = bnfinit(y^18 - 2*y^17 + 4*y^15 - 7*y^14 + y^13 + 7*y^12 + 2*y^11 - 10*y^10 + y^9 + 20*y^8 - 22*y^7 - 2*y^6 + 7*y^5 + 2*y^4 - 10*y^3 + 11*y^2 - 5*y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 2*x^17 + 4*x^15 - 7*x^14 + x^13 + 7*x^12 + 2*x^11 - 10*x^10 + x^9 + 20*x^8 - 22*x^7 - 2*x^6 + 7*x^5 + 2*x^4 - 10*x^3 + 11*x^2 - 5*x + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 - 2*x^17 + 4*x^15 - 7*x^14 + x^13 + 7*x^12 + 2*x^11 - 10*x^10 + x^9 + 20*x^8 - 22*x^7 - 2*x^6 + 7*x^5 + 2*x^4 - 10*x^3 + 11*x^2 - 5*x + 1)
\( x^{18} - 2 x^{17} + 4 x^{15} - 7 x^{14} + x^{13} + 7 x^{12} + 2 x^{11} - 10 x^{10} + x^{9} + 20 x^{8} + \cdots + 1 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $18$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[2, 8]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(10910724785335819009\)
\(\medspace = 43^{2}\cdot 83\cdot 107\cdot 311^{2}\cdot 2621^{2}\)
| 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: | $43^{1/2}83^{1/2}107^{1/2}311^{1/2}2621^{1/2}\approx 557928.9127415785$ | ||
| Ramified primes: |
\(43\), \(83\), \(107\), \(311\), \(2621\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{8881}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{2536123627}a^{17}+\frac{333588062}{2536123627}a^{16}-\frac{456392292}{2536123627}a^{15}+\frac{355601592}{2536123627}a^{14}+\frac{455066414}{2536123627}a^{13}-\frac{6296606}{2536123627}a^{12}+\frac{777290417}{2536123627}a^{11}-\frac{998120764}{2536123627}a^{10}+\frac{139136720}{2536123627}a^{9}-\frac{134210511}{2536123627}a^{8}+\frac{239163599}{2536123627}a^{7}-\frac{708830024}{2536123627}a^{6}+\frac{877414092}{2536123627}a^{5}+\frac{1027607358}{2536123627}a^{4}-\frac{109031228}{2536123627}a^{3}-\frac{1066153715}{2536123627}a^{2}-\frac{1205708747}{2536123627}a+\frac{168602202}{2536123627}$
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: |
$\frac{1083118396}{2536123627}a^{17}-\frac{1183964211}{2536123627}a^{16}-\frac{1517579809}{2536123627}a^{15}+\frac{3430132757}{2536123627}a^{14}-\frac{3867406756}{2536123627}a^{13}-\frac{3661889466}{2536123627}a^{12}+\frac{5794509444}{2536123627}a^{11}+\frac{8604894738}{2536123627}a^{10}-\frac{4780863761}{2536123627}a^{9}-\frac{7116079219}{2536123627}a^{8}+\frac{16443708488}{2536123627}a^{7}-\frac{5626051194}{2536123627}a^{6}-\frac{12845253676}{2536123627}a^{5}-\frac{1909889562}{2536123627}a^{4}+\frac{3087827905}{2536123627}a^{3}-\frac{3318867465}{2536123627}a^{2}+\frac{6133863175}{2536123627}a+\frac{809186095}{2536123627}$, $\frac{147413056}{2536123627}a^{17}-\frac{1109225995}{2536123627}a^{16}+\frac{1097422941}{2536123627}a^{15}+\frac{1387359224}{2536123627}a^{14}-\frac{3921847727}{2536123627}a^{13}+\frac{4097852302}{2536123627}a^{12}+\frac{3012121435}{2536123627}a^{11}-\frac{4760663814}{2536123627}a^{10}-\frac{6112521924}{2536123627}a^{9}+\frac{5068598702}{2536123627}a^{8}+\frac{5712258616}{2536123627}a^{7}-\frac{18652715257}{2536123627}a^{6}+\frac{7991174168}{2536123627}a^{5}+\frac{10830941746}{2536123627}a^{4}-\frac{1363643689}{2536123627}a^{3}-\frac{3796841817}{2536123627}a^{2}+\frac{8583636394}{2536123627}a-\frac{2939115562}{2536123627}$, $\frac{1762172871}{2536123627}a^{17}-\frac{2401810859}{2536123627}a^{16}-\frac{1923409908}{2536123627}a^{15}+\frac{6428923864}{2536123627}a^{14}-\frac{7746453945}{2536123627}a^{13}-\frac{5092538698}{2536123627}a^{12}+\frac{11251286610}{2536123627}a^{11}+\frac{12071635564}{2536123627}a^{10}-\frac{14101060916}{2536123627}a^{9}-\frac{8491019686}{2536123627}a^{8}+\frac{34219460779}{2536123627}a^{7}-\frac{15453058002}{2536123627}a^{6}-\frac{22365707582}{2536123627}a^{5}+\frac{2933824776}{2536123627}a^{4}+\frac{12448100469}{2536123627}a^{3}-\frac{16180969586}{2536123627}a^{2}+\frac{7019377771}{2536123627}a+\frac{308231335}{2536123627}$, $\frac{1594588243}{2536123627}a^{17}-\frac{1528984308}{2536123627}a^{16}-\frac{2097508841}{2536123627}a^{15}+\frac{4649523839}{2536123627}a^{14}-\frac{5664160352}{2536123627}a^{13}-\frac{5820346528}{2536123627}a^{12}+\frac{7027392323}{2536123627}a^{11}+\frac{12491308747}{2536123627}a^{10}-\frac{5533672401}{2536123627}a^{9}-\frac{7389691532}{2536123627}a^{8}+\frac{25877800880}{2536123627}a^{7}-\frac{7195103546}{2536123627}a^{6}-\frac{18612347609}{2536123627}a^{5}-\frac{5044162678}{2536123627}a^{4}+\frac{3451518412}{2536123627}a^{3}-\frac{12083702652}{2536123627}a^{2}+\frac{6753113846}{2536123627}a+\frac{1471512250}{2536123627}$, $a$, $\frac{108329068}{2536123627}a^{17}-\frac{39580205}{2536123627}a^{16}-\frac{823865411}{2536123627}a^{15}+\frac{782859870}{2536123627}a^{14}+\frac{521335225}{2536123627}a^{13}-\frac{2329443423}{2536123627}a^{12}+\frac{2331370078}{2536123627}a^{11}+\frac{3305483114}{2536123627}a^{10}-\frac{1787133582}{2536123627}a^{9}-\frac{5325087427}{2536123627}a^{8}+\frac{2878524887}{2536123627}a^{7}+\frac{3937720658}{2536123627}a^{6}-\frac{11806568359}{2536123627}a^{5}-\frac{839422476}{2536123627}a^{4}+\frac{6468641539}{2536123627}a^{3}+\frac{1956941922}{2536123627}a^{2}-\frac{1220297730}{2536123627}a+\frac{3755278621}{2536123627}$, $\frac{1634111690}{2536123627}a^{17}-\frac{2412626748}{2536123627}a^{16}-\frac{1611676476}{2536123627}a^{15}+\frac{6708393435}{2536123627}a^{14}-\frac{8032473109}{2536123627}a^{13}-\frac{4654597154}{2536123627}a^{12}+\frac{12094558334}{2536123627}a^{11}+\frac{9070366555}{2536123627}a^{10}-\frac{15988120487}{2536123627}a^{9}-\frac{7066383948}{2536123627}a^{8}+\frac{36880608297}{2536123627}a^{7}-\frac{16917828440}{2536123627}a^{6}-\frac{23820386950}{2536123627}a^{5}+\frac{10682869992}{2536123627}a^{4}+\frac{13601357832}{2536123627}a^{3}-\frac{17595703715}{2536123627}a^{2}+\frac{3944952937}{2536123627}a+\frac{752691234}{2536123627}$, $\frac{797401991}{2536123627}a^{17}-\frac{1845181748}{2536123627}a^{16}+\frac{642812823}{2536123627}a^{15}+\frac{3288893602}{2536123627}a^{14}-\frac{6790583042}{2536123627}a^{13}+\frac{2631465109}{2536123627}a^{12}+\frac{5195224123}{2536123627}a^{11}-\frac{826069961}{2536123627}a^{10}-\frac{9278351814}{2536123627}a^{9}+\frac{4154533530}{2536123627}a^{8}+\frac{17751394367}{2536123627}a^{7}-\frac{23435584234}{2536123627}a^{6}+\frac{5264267125}{2536123627}a^{5}+\frac{8758951661}{2536123627}a^{4}+\frac{586968859}{2536123627}a^{3}-\frac{9716083390}{2536123627}a^{2}+\frac{10545144915}{2536123627}a-\frac{6379915334}{2536123627}$, $\frac{532301530}{2536123627}a^{17}+\frac{1034412700}{2536123627}a^{16}-\frac{2413742495}{2536123627}a^{15}-\frac{396739271}{2536123627}a^{14}+\frac{3225412477}{2536123627}a^{13}-\frac{7816883774}{2536123627}a^{12}-\frac{3060855368}{2536123627}a^{11}+\frac{12505559928}{2536123627}a^{10}+\frac{9435167019}{2536123627}a^{9}-\frac{10943751153}{2536123627}a^{8}+\frac{209594279}{2536123627}a^{7}+\frac{24747288166}{2536123627}a^{6}-\frac{14051259987}{2536123627}a^{5}-\frac{20733662495}{2536123627}a^{4}+\frac{518886070}{2536123627}a^{3}+\frac{3199135220}{2536123627}a^{2}-\frac{7771445813}{2536123627}a+\frac{5893201583}{2536123627}$
| 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: | \( 106.811954559 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{8}\cdot 106.811954559 \cdot 1}{2\cdot\sqrt{10910724785335819009}}\cr\approx \mathstrut & 0.157094870386 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^18 - 2*x^17 + 4*x^15 - 7*x^14 + x^13 + 7*x^12 + 2*x^11 - 10*x^10 + x^9 + 20*x^8 - 22*x^7 - 2*x^6 + 7*x^5 + 2*x^4 - 10*x^3 + 11*x^2 - 5*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^18 - 2*x^17 + 4*x^15 - 7*x^14 + x^13 + 7*x^12 + 2*x^11 - 10*x^10 + x^9 + 20*x^8 - 22*x^7 - 2*x^6 + 7*x^5 + 2*x^4 - 10*x^3 + 11*x^2 - 5*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^18 - 2*x^17 + 4*x^15 - 7*x^14 + x^13 + 7*x^12 + 2*x^11 - 10*x^10 + x^9 + 20*x^8 - 22*x^7 - 2*x^6 + 7*x^5 + 2*x^4 - 10*x^3 + 11*x^2 - 5*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^18 - 2*x^17 + 4*x^15 - 7*x^14 + x^13 + 7*x^12 + 2*x^11 - 10*x^10 + x^9 + 20*x^8 - 22*x^7 - 2*x^6 + 7*x^5 + 2*x^4 - 10*x^3 + 11*x^2 - 5*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
$C_2^9.S_9$ (as 18T968):
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 185794560 |
| The 300 conjugacy class representatives for $C_2^9.S_9$ |
| Character table for $C_2^9.S_9$ |
Intermediate fields
| 9.1.35050633.1 |
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 18 sibling: | data not computed |
| Degree 36 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 | ${\href{/padicField/2.9.0.1}{9} }^{2}$ | ${\href{/padicField/3.9.0.1}{9} }^{2}$ | ${\href{/padicField/5.5.0.1}{5} }^{2}{,}\,{\href{/padicField/5.4.0.1}{4} }^{2}$ | $18$ | ${\href{/padicField/11.14.0.1}{14} }{,}\,{\href{/padicField/11.4.0.1}{4} }$ | ${\href{/padicField/13.12.0.1}{12} }{,}\,{\href{/padicField/13.3.0.1}{3} }^{2}$ | ${\href{/padicField/17.4.0.1}{4} }^{4}{,}\,{\href{/padicField/17.2.0.1}{2} }$ | ${\href{/padicField/19.8.0.1}{8} }{,}\,{\href{/padicField/19.3.0.1}{3} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }^{2}$ | ${\href{/padicField/23.9.0.1}{9} }^{2}$ | ${\href{/padicField/29.8.0.1}{8} }{,}\,{\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}$ | ${\href{/padicField/31.12.0.1}{12} }{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.6.0.1}{6} }^{2}{,}\,{\href{/padicField/37.3.0.1}{3} }^{2}$ | ${\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.4.0.1}{4} }^{3}$ | R | ${\href{/padicField/47.10.0.1}{10} }{,}\,{\href{/padicField/47.3.0.1}{3} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | ${\href{/padicField/53.5.0.1}{5} }^{2}{,}\,{\href{/padicField/53.3.0.1}{3} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }$ | ${\href{/padicField/59.8.0.1}{8} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(43\)
| $\Q_{43}$ | $x + 40$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{43}$ | $x + 40$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 43.2.0.1 | $x^{2} + 42 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 43.2.0.1 | $x^{2} + 42 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 43.4.0.1 | $x^{4} + 5 x^{2} + 42 x + 3$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 43.4.2.1 | $x^{4} + 84 x^{3} + 1856 x^{2} + 3864 x + 77452$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 43.4.0.1 | $x^{4} + 5 x^{2} + 42 x + 3$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
|
\(83\)
| 83.2.1.2 | $x^{2} + 83$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 83.2.0.1 | $x^{2} + 82 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 83.2.0.1 | $x^{2} + 82 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 83.6.0.1 | $x^{6} + x^{4} + 76 x^{3} + 32 x^{2} + 17 x + 2$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
| 83.6.0.1 | $x^{6} + x^{4} + 76 x^{3} + 32 x^{2} + 17 x + 2$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
|
\(107\)
| 107.2.1.2 | $x^{2} + 107$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 107.4.0.1 | $x^{4} + 13 x^{2} + 79 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 107.4.0.1 | $x^{4} + 13 x^{2} + 79 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 107.8.0.1 | $x^{8} + 2 x^{4} + 105 x^{3} + 24 x^{2} + 95 x + 2$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | |
|
\(311\)
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $7$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | ||
| Deg $7$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | ||
|
\(2621\)
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $7$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | ||
| Deg $7$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ |