Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 5*x^17 + 16*x^16 - 35*x^15 + 59*x^14 - 73*x^13 + 59*x^12 - 5*x^11 - 79*x^10 + 163*x^9 - 224*x^8 + 229*x^7 - 192*x^6 + 127*x^5 - 59*x^4 + 24*x^3 - 4*x - 1)
gp: K = bnfinit(y^18 - 5*y^17 + 16*y^16 - 35*y^15 + 59*y^14 - 73*y^13 + 59*y^12 - 5*y^11 - 79*y^10 + 163*y^9 - 224*y^8 + 229*y^7 - 192*y^6 + 127*y^5 - 59*y^4 + 24*y^3 - 4*y - 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 5*x^17 + 16*x^16 - 35*x^15 + 59*x^14 - 73*x^13 + 59*x^12 - 5*x^11 - 79*x^10 + 163*x^9 - 224*x^8 + 229*x^7 - 192*x^6 + 127*x^5 - 59*x^4 + 24*x^3 - 4*x - 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 - 5*x^17 + 16*x^16 - 35*x^15 + 59*x^14 - 73*x^13 + 59*x^12 - 5*x^11 - 79*x^10 + 163*x^9 - 224*x^8 + 229*x^7 - 192*x^6 + 127*x^5 - 59*x^4 + 24*x^3 - 4*x - 1)
\( x^{18} - 5 x^{17} + 16 x^{16} - 35 x^{15} + 59 x^{14} - 73 x^{13} + 59 x^{12} - 5 x^{11} - 79 x^{10} + \cdots - 1 \)
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: |
\(254891514450553129009\)
\(\medspace = 7^{12}\cdot 97^{2}\cdot 1399^{2}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(13.60\) | 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: | $7^{2/3}97^{1/2}1399^{1/2}\approx 1348.010942860009$ | ||
| Ramified primes: |
\(7\), \(97\), \(1399\)
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{8612553139}a^{17}+\frac{631857562}{8612553139}a^{16}+\frac{3426518164}{8612553139}a^{15}-\frac{3203053914}{8612553139}a^{14}-\frac{12917791}{296984591}a^{13}-\frac{2000258920}{8612553139}a^{12}+\frac{3004105579}{8612553139}a^{11}-\frac{2169053281}{8612553139}a^{10}-\frac{7712333}{296984591}a^{9}+\frac{1417698542}{8612553139}a^{8}+\frac{3912680889}{8612553139}a^{7}-\frac{1649932402}{8612553139}a^{6}-\frac{334580478}{8612553139}a^{5}+\frac{2026585257}{8612553139}a^{4}+\frac{406796564}{8612553139}a^{3}-\frac{1377289051}{8612553139}a^{2}-\frac{3270739425}{8612553139}a+\frac{2210145592}{8612553139}$
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: |
$\frac{2032544064}{8612553139}a^{17}-\frac{7868245202}{8612553139}a^{16}+\frac{23061949809}{8612553139}a^{15}-\frac{43779250763}{8612553139}a^{14}+\frac{2397363882}{296984591}a^{13}-\frac{76619989985}{8612553139}a^{12}+\frac{58856805421}{8612553139}a^{11}+\frac{241016035}{8612553139}a^{10}-\frac{2741642974}{296984591}a^{9}+\frac{167729162183}{8612553139}a^{8}-\frac{251518953279}{8612553139}a^{7}+\frac{265131047054}{8612553139}a^{6}-\frac{258558223537}{8612553139}a^{5}+\frac{180428912920}{8612553139}a^{4}-\frac{108608975835}{8612553139}a^{3}+\frac{66395694127}{8612553139}a^{2}-\frac{4057851507}{8612553139}a+\frac{1819654986}{8612553139}$, $\frac{723919132}{8612553139}a^{17}-\frac{4563309330}{8612553139}a^{16}+\frac{15662062021}{8612553139}a^{15}-\frac{37681843805}{8612553139}a^{14}+\frac{2279977766}{296984591}a^{13}-\frac{87374760978}{8612553139}a^{12}+\frac{73082305907}{8612553139}a^{11}-\frac{11523639592}{8612553139}a^{10}-\frac{3286453247}{296984591}a^{9}+\frac{196664549667}{8612553139}a^{8}-\frac{263651112688}{8612553139}a^{7}+\frac{261306083366}{8612553139}a^{6}-\frac{203151073564}{8612553139}a^{5}+\frac{127667073370}{8612553139}a^{4}-\frac{42663137648}{8612553139}a^{3}+\frac{13782802278}{8612553139}a^{2}+\frac{7766584103}{8612553139}a-\frac{4368537603}{8612553139}$, $\frac{1984900030}{8612553139}a^{17}-\frac{10306247961}{8612553139}a^{16}+\frac{33330617956}{8612553139}a^{15}-\frac{74758015556}{8612553139}a^{14}+\frac{4429690654}{296984591}a^{13}-\frac{165488236544}{8612553139}a^{12}+\frac{142364789486}{8612553139}a^{11}-\frac{33071385778}{8612553139}a^{10}-\frac{5319924449}{296984591}a^{9}+\frac{349204326186}{8612553139}a^{8}-\frac{506200881989}{8612553139}a^{7}+\frac{533800211080}{8612553139}a^{6}-\frac{460881077246}{8612553139}a^{5}+\frac{317253753120}{8612553139}a^{4}-\frac{158403182415}{8612553139}a^{3}+\frac{70587370826}{8612553139}a^{2}-\frac{5295954418}{8612553139}a+\frac{2798979149}{8612553139}$, $\frac{4368537603}{8612553139}a^{17}-\frac{22566607147}{8612553139}a^{16}+\frac{74459910978}{8612553139}a^{15}-\frac{168560878126}{8612553139}a^{14}+\frac{10187088358}{296984591}a^{13}-\frac{385022600233}{8612553139}a^{12}+\frac{345118479555}{8612553139}a^{11}-\frac{94924993922}{8612553139}a^{10}-\frac{11503132105}{296984591}a^{9}+\frac{807378773452}{8612553139}a^{8}-\frac{1175216972739}{8612553139}a^{7}+\frac{1264046223775}{8612553139}a^{6}-\frac{1100065303142}{8612553139}a^{5}+\frac{757955349145}{8612553139}a^{4}-\frac{385410791947}{8612553139}a^{3}+\frac{147508040120}{8612553139}a^{2}-\frac{13782802278}{8612553139}a-\frac{25240734515}{8612553139}$, $\frac{2803618405}{8612553139}a^{17}-\frac{14385625605}{8612553139}a^{16}+\frac{47273405381}{8612553139}a^{15}-\frac{105651148342}{8612553139}a^{14}+\frac{6261641701}{296984591}a^{13}-\frac{228131209437}{8612553139}a^{12}+\frac{187695288007}{8612553139}a^{11}-\frac{16367291765}{8612553139}a^{10}-\frac{8736756813}{296984591}a^{9}+\frac{521406656753}{8612553139}a^{8}-\frac{700450675624}{8612553139}a^{7}+\frac{694063329339}{8612553139}a^{6}-\frac{561907754517}{8612553139}a^{5}+\frac{348296216181}{8612553139}a^{4}-\frac{158856749805}{8612553139}a^{3}+\frac{60190639734}{8612553139}a^{2}-\frac{1605909689}{8612553139}a-\frac{2056353742}{8612553139}$, $\frac{1493661585}{8612553139}a^{17}-\frac{7567802543}{8612553139}a^{16}+\frac{24457034949}{8612553139}a^{15}-\frac{55041225761}{8612553139}a^{14}+\frac{3313903271}{296984591}a^{13}-\frac{127132835533}{8612553139}a^{12}+\frac{117435639181}{8612553139}a^{11}-\frac{43857803329}{8612553139}a^{10}-\frac{3184149479}{296984591}a^{9}+\frac{249377171436}{8612553139}a^{8}-\frac{388826570619}{8612553139}a^{7}+\frac{438257731190}{8612553139}a^{6}-\frac{401538249411}{8612553139}a^{5}+\frac{293715917154}{8612553139}a^{4}-\frac{153555395072}{8612553139}a^{3}+\frac{66319549397}{8612553139}a^{2}-\frac{3630661574}{8612553139}a-\frac{15937064634}{8612553139}$, $\frac{835805931}{8612553139}a^{17}-\frac{5921004908}{8612553139}a^{16}+\frac{20212206380}{8612553139}a^{15}-\frac{49448938920}{8612553139}a^{14}+\frac{3043056774}{296984591}a^{13}-\frac{122004896878}{8612553139}a^{12}+\frac{113108677170}{8612553139}a^{11}-\frac{41995522561}{8612553139}a^{10}-\frac{3208511808}{296984591}a^{9}+\frac{242425811086}{8612553139}a^{8}-\frac{360311194339}{8612553139}a^{7}+\frac{402375477085}{8612553139}a^{6}-\frac{337032872229}{8612553139}a^{5}+\frac{247312549672}{8612553139}a^{4}-\frac{127762181235}{8612553139}a^{3}+\frac{45173113284}{8612553139}a^{2}-\frac{15064326211}{8612553139}a-\frac{12123075905}{8612553139}$, $\frac{193392198}{8612553139}a^{17}-\frac{2269298659}{8612553139}a^{16}+\frac{8971489289}{8612553139}a^{15}-\frac{25106611047}{8612553139}a^{14}+\frac{1716392372}{296984591}a^{13}-\frac{77385049862}{8612553139}a^{12}+\frac{86326969587}{8612553139}a^{11}-\frac{58422472787}{8612553139}a^{10}-\frac{600056783}{296984591}a^{9}+\frac{121951209181}{8612553139}a^{8}-\frac{218792933762}{8612553139}a^{7}+\frac{282946879793}{8612553139}a^{6}-\frac{270559464853}{8612553139}a^{5}+\frac{221188052101}{8612553139}a^{4}-\frac{128060153523}{8612553139}a^{3}+\frac{54892032719}{8612553139}a^{2}-\frac{25675473102}{8612553139}a-\frac{10102415132}{8612553139}$, $\frac{2890222961}{8612553139}a^{17}-\frac{13291147074}{8612553139}a^{16}+\frac{41710270508}{8612553139}a^{15}-\frac{87716943199}{8612553139}a^{14}+\frac{4989021586}{296984591}a^{13}-\frac{170440526946}{8612553139}a^{12}+\frac{129048376220}{8612553139}a^{11}+\frac{8227994602}{8612553139}a^{10}-\frac{7033125959}{296984591}a^{9}+\frac{391286694677}{8612553139}a^{8}-\frac{522524195191}{8612553139}a^{7}+\frac{520191080835}{8612553139}a^{6}-\frac{449331425932}{8612553139}a^{5}+\frac{295977697009}{8612553139}a^{4}-\frac{146794066231}{8612553139}a^{3}+\frac{73693491322}{8612553139}a^{2}-\frac{7219685899}{8612553139}a-\frac{3934824827}{8612553139}$
| 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: | \( 661.987726773 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{8}\cdot 661.987726773 \cdot 1}{2\cdot\sqrt{254891514450553129009}}\cr\approx \mathstrut & 0.201437902853 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^18 - 5*x^17 + 16*x^16 - 35*x^15 + 59*x^14 - 73*x^13 + 59*x^12 - 5*x^11 - 79*x^10 + 163*x^9 - 224*x^8 + 229*x^7 - 192*x^6 + 127*x^5 - 59*x^4 + 24*x^3 - 4*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 - 5*x^17 + 16*x^16 - 35*x^15 + 59*x^14 - 73*x^13 + 59*x^12 - 5*x^11 - 79*x^10 + 163*x^9 - 224*x^8 + 229*x^7 - 192*x^6 + 127*x^5 - 59*x^4 + 24*x^3 - 4*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 - 5*x^17 + 16*x^16 - 35*x^15 + 59*x^14 - 73*x^13 + 59*x^12 - 5*x^11 - 79*x^10 + 163*x^9 - 224*x^8 + 229*x^7 - 192*x^6 + 127*x^5 - 59*x^4 + 24*x^3 - 4*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 - 5*x^17 + 16*x^16 - 35*x^15 + 59*x^14 - 73*x^13 + 59*x^12 - 5*x^11 - 79*x^10 + 163*x^9 - 224*x^8 + 229*x^7 - 192*x^6 + 127*x^5 - 59*x^4 + 24*x^3 - 4*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
$S_4^3.A_4$ (as 18T838):
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 165888 |
| The 180 conjugacy class representatives for $S_4^3.A_4$ |
| Character table for $S_4^3.A_4$ |
Intermediate fields
| \(\Q(\zeta_{7})^+\), 9.3.164590951.1 |
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 siblings: | data not computed |
| Degree 24 siblings: | 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 | ${\href{/padicField/2.9.0.1}{9} }^{2}$ | ${\href{/padicField/3.6.0.1}{6} }^{2}{,}\,{\href{/padicField/3.3.0.1}{3} }^{2}$ | ${\href{/padicField/5.6.0.1}{6} }^{2}{,}\,{\href{/padicField/5.3.0.1}{3} }^{2}$ | R | ${\href{/padicField/11.6.0.1}{6} }^{2}{,}\,{\href{/padicField/11.3.0.1}{3} }^{2}$ | ${\href{/padicField/13.6.0.1}{6} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }^{2}{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ | ${\href{/padicField/17.6.0.1}{6} }^{2}{,}\,{\href{/padicField/17.3.0.1}{3} }^{2}$ | ${\href{/padicField/19.6.0.1}{6} }^{2}{,}\,{\href{/padicField/19.3.0.1}{3} }^{2}$ | ${\href{/padicField/23.9.0.1}{9} }^{2}$ | ${\href{/padicField/29.4.0.1}{4} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }^{6}$ | ${\href{/padicField/31.12.0.1}{12} }{,}\,{\href{/padicField/31.6.0.1}{6} }$ | ${\href{/padicField/37.6.0.1}{6} }^{2}{,}\,{\href{/padicField/37.3.0.1}{3} }^{2}$ | ${\href{/padicField/41.4.0.1}{4} }{,}\,{\href{/padicField/41.3.0.1}{3} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }^{3}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.6.0.1}{6} }^{2}{,}\,{\href{/padicField/43.4.0.1}{4} }{,}\,{\href{/padicField/43.2.0.1}{2} }$ | ${\href{/padicField/47.6.0.1}{6} }^{2}{,}\,{\href{/padicField/47.3.0.1}{3} }^{2}$ | ${\href{/padicField/53.6.0.1}{6} }^{2}{,}\,{\href{/padicField/53.3.0.1}{3} }^{2}$ | ${\href{/padicField/59.9.0.1}{9} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(7\)
| 7.9.6.1 | $x^{9} + 18 x^{8} + 108 x^{7} + 249 x^{6} + 396 x^{5} + 1944 x^{4} + 2631 x^{3} - 2358 x^{2} - 756 x + 11915$ | $3$ | $3$ | $6$ | $C_3^2$ | $[\ ]_{3}^{3}$ |
| 7.9.6.1 | $x^{9} + 18 x^{8} + 108 x^{7} + 249 x^{6} + 396 x^{5} + 1944 x^{4} + 2631 x^{3} - 2358 x^{2} - 756 x + 11915$ | $3$ | $3$ | $6$ | $C_3^2$ | $[\ ]_{3}^{3}$ | |
|
\(97\)
| 97.2.1.1 | $x^{2} + 97$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 97.2.1.1 | $x^{2} + 97$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 97.2.0.1 | $x^{2} + 96 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 97.2.0.1 | $x^{2} + 96 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 97.4.0.1 | $x^{4} + 6 x^{2} + 80 x + 5$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 97.6.0.1 | $x^{6} + 92 x^{3} + 58 x^{2} + 88 x + 5$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
|
\(1399\)
| 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 $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
| Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
| Deg $4$ | $2$ | $2$ | $2$ |