Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 - 7*x^14 + 88*x^13 - 6*x^12 - 542*x^11 + 166*x^10 + 1764*x^9 - 522*x^8 - 3120*x^7 + 745*x^6 + 2830*x^5 - 598*x^4 - 1092*x^3 + 256*x^2 + 78*x - 19)
gp: K = bnfinit(y^16 - 6*y^15 - 7*y^14 + 88*y^13 - 6*y^12 - 542*y^11 + 166*y^10 + 1764*y^9 - 522*y^8 - 3120*y^7 + 745*y^6 + 2830*y^5 - 598*y^4 - 1092*y^3 + 256*y^2 + 78*y - 19, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 6*x^15 - 7*x^14 + 88*x^13 - 6*x^12 - 542*x^11 + 166*x^10 + 1764*x^9 - 522*x^8 - 3120*x^7 + 745*x^6 + 2830*x^5 - 598*x^4 - 1092*x^3 + 256*x^2 + 78*x - 19);
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^16 - 6*x^15 - 7*x^14 + 88*x^13 - 6*x^12 - 542*x^11 + 166*x^10 + 1764*x^9 - 522*x^8 - 3120*x^7 + 745*x^6 + 2830*x^5 - 598*x^4 - 1092*x^3 + 256*x^2 + 78*x - 19)
\( x^{16} - 6 x^{15} - 7 x^{14} + 88 x^{13} - 6 x^{12} - 542 x^{11} + 166 x^{10} + 1764 x^{9} - 522 x^{8} + \cdots - 19 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $16$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[16, 0]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(248438446096000000000000\)
\(\medspace = 2^{16}\cdot 5^{12}\cdot 353^{4}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(28.99\) | 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: | $2\cdot 5^{3/4}353^{1/2}\approx 125.64489612056876$ | ||
| Ramified primes: |
\(2\), \(5\), \(353\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Aut(K/\Q) }$: | $4$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{6}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{7}-\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{4}a^{12}-\frac{1}{4}a^{10}-\frac{1}{4}a^{8}-\frac{1}{2}a^{7}+\frac{1}{4}a^{6}+\frac{1}{4}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}+\frac{1}{4}$, $\frac{1}{4}a^{13}-\frac{1}{4}a^{11}-\frac{1}{4}a^{9}-\frac{1}{2}a^{8}+\frac{1}{4}a^{7}+\frac{1}{4}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}+\frac{1}{4}a$, $\frac{1}{8}a^{14}-\frac{1}{4}a^{10}-\frac{1}{4}a^{9}-\frac{1}{4}a^{7}-\frac{1}{4}a^{6}-\frac{1}{4}a^{5}+\frac{3}{8}a^{4}+\frac{1}{4}a^{3}-\frac{1}{8}a^{2}-\frac{1}{2}a+\frac{1}{8}$, $\frac{1}{328}a^{15}-\frac{1}{41}a^{14}-\frac{4}{41}a^{13}-\frac{3}{82}a^{12}+\frac{9}{164}a^{11}+\frac{39}{164}a^{10}-\frac{9}{41}a^{9}-\frac{71}{164}a^{8}+\frac{45}{164}a^{7}+\frac{31}{164}a^{6}-\frac{117}{328}a^{5}+\frac{15}{164}a^{4}-\frac{125}{328}a^{3}-\frac{13}{41}a^{2}+\frac{13}{328}a-\frac{14}{41}$
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: | $15$ | 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: |
$\frac{119}{164}a^{15}-\frac{1945}{328}a^{14}+\frac{278}{41}a^{13}+\frac{9273}{164}a^{12}-\frac{10655}{82}a^{11}-\frac{15859}{82}a^{10}+\frac{106437}{164}a^{9}+\frac{39805}{164}a^{8}-\frac{243859}{164}a^{7}+\frac{3956}{41}a^{6}+\frac{135821}{82}a^{5}-\frac{162805}{328}a^{4}-\frac{63177}{82}a^{3}+\frac{112229}{328}a^{2}+\frac{9665}{164}a-\frac{8247}{328}$, $\frac{1135}{328}a^{15}-\frac{8301}{328}a^{14}+\frac{380}{41}a^{13}+\frac{47679}{164}a^{12}-\frac{66127}{164}a^{11}-\frac{217397}{164}a^{10}+\frac{377545}{164}a^{9}+\frac{243981}{82}a^{8}-\frac{230720}{41}a^{7}-\frac{512903}{164}a^{6}+\frac{2126223}{328}a^{5}+\frac{297885}{328}a^{4}-\frac{1001317}{328}a^{3}+\frac{143909}{328}a^{2}+\frac{72483}{328}a-\frac{14575}{328}$, $\frac{1135}{328}a^{15}-\frac{8301}{328}a^{14}+\frac{380}{41}a^{13}+\frac{47679}{164}a^{12}-\frac{66127}{164}a^{11}-\frac{217397}{164}a^{10}+\frac{377545}{164}a^{9}+\frac{243981}{82}a^{8}-\frac{230720}{41}a^{7}-\frac{512903}{164}a^{6}+\frac{2126223}{328}a^{5}+\frac{297885}{328}a^{4}-\frac{1001317}{328}a^{3}+\frac{143909}{328}a^{2}+\frac{72483}{328}a-\frac{14247}{328}$, $\frac{1143}{328}a^{15}-\frac{4203}{164}a^{14}+\frac{1843}{164}a^{13}+\frac{11672}{41}a^{12}-\frac{33089}{82}a^{11}-\frac{207983}{164}a^{10}+\frac{359791}{164}a^{9}+\frac{466771}{164}a^{8}-\frac{420875}{82}a^{7}-\frac{518723}{164}a^{6}+\frac{1853047}{328}a^{5}+\frac{53517}{41}a^{4}-\frac{832823}{328}a^{3}+\frac{16783}{164}a^{2}+\frac{59549}{328}a-\frac{3041}{164}$, $\frac{5}{328}a^{15}-\frac{61}{164}a^{14}+\frac{289}{164}a^{13}+\frac{339}{164}a^{12}-\frac{2089}{82}a^{11}+\frac{979}{82}a^{10}+\frac{22821}{164}a^{9}-\frac{10161}{82}a^{8}-\frac{15206}{41}a^{7}+\frac{15342}{41}a^{6}+\frac{157757}{328}a^{5}-\frac{80941}{164}a^{4}-\frac{80657}{328}a^{3}+\frac{42093}{164}a^{2}+\frac{2935}{328}a-\frac{1329}{82}$, $\frac{183}{164}a^{15}-\frac{2723}{328}a^{14}+\frac{311}{82}a^{13}+\frac{15639}{164}a^{12}-\frac{5962}{41}a^{11}-\frac{17403}{41}a^{10}+\frac{137909}{164}a^{9}+\frac{145435}{164}a^{8}-\frac{344863}{164}a^{7}-\frac{61321}{82}a^{6}+\frac{204237}{82}a^{5}-\frac{25879}{328}a^{4}-\frac{49640}{41}a^{3}+\frac{111955}{328}a^{2}+\frac{15007}{164}a-\frac{9217}{328}$, $\frac{319}{328}a^{15}-\frac{1317}{164}a^{14}+\frac{364}{41}a^{13}+\frac{13543}{164}a^{12}-\frac{31651}{164}a^{11}-\frac{24427}{82}a^{10}+\frac{86303}{82}a^{9}+\frac{28917}{82}a^{8}-\frac{429675}{164}a^{7}+\frac{14352}{41}a^{6}+\frac{1036057}{328}a^{5}-\frac{191769}{164}a^{4}-\frac{519575}{328}a^{3}+\frac{129003}{164}a^{2}+\frac{43999}{328}a-\frac{2416}{41}$, $\frac{731}{328}a^{15}-\frac{5397}{328}a^{14}+\frac{315}{41}a^{13}+\frac{29767}{164}a^{12}-\frac{42949}{164}a^{11}-\frac{131719}{164}a^{10}+\frac{231697}{164}a^{9}+\frac{147131}{82}a^{8}-\frac{269835}{82}a^{7}-\frac{327233}{164}a^{6}+\frac{1184243}{328}a^{5}+\frac{273957}{328}a^{4}-\frac{531141}{328}a^{3}+\frac{18481}{328}a^{2}+\frac{37875}{328}a-\frac{3603}{328}$, $\frac{113}{82}a^{15}-\frac{3329}{328}a^{14}+\frac{681}{164}a^{13}+\frac{9509}{82}a^{12}-\frac{27297}{164}a^{11}-\frac{85979}{164}a^{10}+\frac{38859}{41}a^{9}+\frac{47573}{41}a^{8}-\frac{190689}{82}a^{7}-\frac{194391}{164}a^{6}+\frac{110670}{41}a^{5}+\frac{96749}{328}a^{4}-\frac{211069}{164}a^{3}+\frac{67997}{328}a^{2}+\frac{16017}{164}a-\frac{7041}{328}$, $\frac{915}{328}a^{15}-\frac{1707}{82}a^{14}+\frac{440}{41}a^{13}+\frac{9472}{41}a^{12}-\frac{58103}{164}a^{11}-\frac{165625}{164}a^{10}+\frac{80120}{41}a^{9}+\frac{352005}{164}a^{8}-\frac{769559}{164}a^{7}-\frac{337273}{164}a^{6}+\frac{1753033}{328}a^{5}+\frac{58169}{164}a^{4}-\frac{820395}{328}a^{3}+\frac{38161}{82}a^{2}+\frac{60931}{328}a-\frac{1781}{41}$, $\frac{419}{328}a^{15}-\frac{879}{82}a^{14}+\frac{1117}{82}a^{13}+\frac{16715}{164}a^{12}-\frac{42231}{164}a^{11}-\frac{27069}{82}a^{10}+\frac{108119}{82}a^{9}+\frac{23153}{82}a^{8}-\frac{512013}{164}a^{7}+\frac{26853}{41}a^{6}+\frac{1183765}{328}a^{5}-\frac{63629}{41}a^{4}-\frac{573075}{328}a^{3}+\frac{79429}{82}a^{2}+\frac{46283}{328}a-\frac{11943}{164}$, $\frac{423}{82}a^{15}-\frac{1610}{41}a^{14}+\frac{1022}{41}a^{13}+\frac{70495}{164}a^{12}-\frac{29772}{41}a^{11}-\frac{297969}{164}a^{10}+\frac{164147}{41}a^{9}+\frac{585601}{164}a^{8}-\frac{798207}{82}a^{7}-\frac{439671}{164}a^{6}+\frac{926391}{82}a^{5}-\frac{124229}{164}a^{4}-\frac{221700}{41}a^{3}+\frac{64268}{41}a^{2}+\frac{34035}{82}a-\frac{21075}{164}$, $\frac{21}{164}a^{15}-\frac{295}{328}a^{14}-\frac{57}{164}a^{13}+\frac{593}{41}a^{12}-\frac{2943}{164}a^{11}-\frac{12753}{164}a^{10}+\frac{6059}{41}a^{9}+\frac{7106}{41}a^{8}-\frac{37185}{82}a^{7}-\frac{19321}{164}a^{6}+\frac{101601}{164}a^{5}-\frac{34205}{328}a^{4}-\frac{13479}{41}a^{3}+\frac{45939}{328}a^{2}+\frac{878}{41}a-\frac{4499}{328}$, $\frac{2737}{328}a^{15}-\frac{2532}{41}a^{14}+\frac{4629}{164}a^{13}+\frac{56897}{82}a^{12}-\frac{83767}{82}a^{11}-\frac{505715}{164}a^{10}+\frac{933971}{164}a^{9}+\frac{1101027}{164}a^{8}-\frac{562981}{41}a^{7}-\frac{1106531}{164}a^{6}+\frac{5136789}{328}a^{5}+\frac{266637}{164}a^{4}-\frac{2402949}{328}a^{3}+\frac{96241}{82}a^{2}+\frac{177195}{328}a-\frac{9437}{82}$, $\frac{219}{41}a^{15}-\frac{12991}{328}a^{14}+\frac{823}{41}a^{13}+\frac{70709}{164}a^{12}-\frac{26111}{41}a^{11}-\frac{77136}{41}a^{10}+\frac{555495}{164}a^{9}+\frac{678839}{164}a^{8}-\frac{1281171}{164}a^{7}-\frac{369437}{82}a^{6}+\frac{1394459}{164}a^{5}+\frac{582813}{328}a^{4}-\frac{620401}{164}a^{3}+\frac{68343}{328}a^{2}+\frac{21561}{82}a-\frac{11601}{328}$
| 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: | \( 2794821.00742 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{16}\cdot(2\pi)^{0}\cdot 2794821.00742 \cdot 1}{2\cdot\sqrt{248438446096000000000000}}\cr\approx \mathstrut & 0.183736116113 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 - 7*x^14 + 88*x^13 - 6*x^12 - 542*x^11 + 166*x^10 + 1764*x^9 - 522*x^8 - 3120*x^7 + 745*x^6 + 2830*x^5 - 598*x^4 - 1092*x^3 + 256*x^2 + 78*x - 19)
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^16 - 6*x^15 - 7*x^14 + 88*x^13 - 6*x^12 - 542*x^11 + 166*x^10 + 1764*x^9 - 522*x^8 - 3120*x^7 + 745*x^6 + 2830*x^5 - 598*x^4 - 1092*x^3 + 256*x^2 + 78*x - 19, 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^16 - 6*x^15 - 7*x^14 + 88*x^13 - 6*x^12 - 542*x^11 + 166*x^10 + 1764*x^9 - 522*x^8 - 3120*x^7 + 745*x^6 + 2830*x^5 - 598*x^4 - 1092*x^3 + 256*x^2 + 78*x - 19);
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^16 - 6*x^15 - 7*x^14 + 88*x^13 - 6*x^12 - 542*x^11 + 166*x^10 + 1764*x^9 - 522*x^8 - 3120*x^7 + 745*x^6 + 2830*x^5 - 598*x^4 - 1092*x^3 + 256*x^2 + 78*x - 19);
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
$C_4\times S_4$ (as 16T181):
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 96 |
| The 20 conjugacy class representatives for $C_4\times S_4$ |
| Character table for $C_4\times S_4$ |
Intermediate fields
| \(\Q(\sqrt{5}) \), \(\Q(\zeta_{20})^+\), 4.4.706000.1, 8.8.498436000000.1 |
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 12 siblings: | data not computed |
| Degree 16 sibling: | data not computed |
| Degree 24 siblings: | data not computed |
| Degree 32 sibling: | data not computed |
| Minimal sibling: | 12.12.19409253601250000.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/padicField/3.12.0.1}{12} }{,}\,{\href{/padicField/3.4.0.1}{4} }$ | R | ${\href{/padicField/7.12.0.1}{12} }{,}\,{\href{/padicField/7.4.0.1}{4} }$ | ${\href{/padicField/11.6.0.1}{6} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }^{2}$ | ${\href{/padicField/13.4.0.1}{4} }^{4}$ | ${\href{/padicField/17.4.0.1}{4} }^{4}$ | ${\href{/padicField/19.3.0.1}{3} }^{4}{,}\,{\href{/padicField/19.1.0.1}{1} }^{4}$ | ${\href{/padicField/23.4.0.1}{4} }^{4}$ | ${\href{/padicField/29.6.0.1}{6} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}$ | ${\href{/padicField/31.2.0.1}{2} }^{8}$ | ${\href{/padicField/37.12.0.1}{12} }{,}\,{\href{/padicField/37.4.0.1}{4} }$ | ${\href{/padicField/41.3.0.1}{3} }^{4}{,}\,{\href{/padicField/41.1.0.1}{1} }^{4}$ | ${\href{/padicField/43.4.0.1}{4} }^{4}$ | ${\href{/padicField/47.4.0.1}{4} }^{4}$ | ${\href{/padicField/53.12.0.1}{12} }{,}\,{\href{/padicField/53.4.0.1}{4} }$ | ${\href{/padicField/59.4.0.1}{4} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.8.8.1 | $x^{8} + 8 x^{7} + 32 x^{6} + 82 x^{5} + 148 x^{4} + 184 x^{3} + 137 x^{2} + 44 x + 5$ | $2$ | $4$ | $8$ | $C_4\times C_2$ | $[2]^{4}$ |
| 2.8.8.1 | $x^{8} + 8 x^{7} + 32 x^{6} + 82 x^{5} + 148 x^{4} + 184 x^{3} + 137 x^{2} + 44 x + 5$ | $2$ | $4$ | $8$ | $C_4\times C_2$ | $[2]^{4}$ | |
|
\(5\)
| 5.16.12.1 | $x^{16} + 16 x^{14} + 16 x^{13} + 124 x^{12} + 192 x^{11} + 368 x^{10} - 16 x^{9} + 1462 x^{8} + 2688 x^{7} + 2544 x^{6} + 5232 x^{5} + 14108 x^{4} + 4800 x^{3} + 8592 x^{2} - 6512 x + 13041$ | $4$ | $4$ | $12$ | $C_4^2$ | $[\ ]_{4}^{4}$ |
|
\(353\)
| Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
| Deg $8$ | $2$ | $4$ | $4$ |