Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 3*x^19 + 3*x^18 - 3*x^17 + 8*x^16 - 13*x^15 + 16*x^14 - 18*x^13 + 22*x^12 - 30*x^11 + 32*x^10 - 25*x^9 + 18*x^8 - 17*x^7 + 25*x^6 - 31*x^5 + 29*x^4 - 21*x^3 + 11*x^2 - 4*x + 1)
gp: K = bnfinit(y^20 - 3*y^19 + 3*y^18 - 3*y^17 + 8*y^16 - 13*y^15 + 16*y^14 - 18*y^13 + 22*y^12 - 30*y^11 + 32*y^10 - 25*y^9 + 18*y^8 - 17*y^7 + 25*y^6 - 31*y^5 + 29*y^4 - 21*y^3 + 11*y^2 - 4*y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 - 3*x^19 + 3*x^18 - 3*x^17 + 8*x^16 - 13*x^15 + 16*x^14 - 18*x^13 + 22*x^12 - 30*x^11 + 32*x^10 - 25*x^9 + 18*x^8 - 17*x^7 + 25*x^6 - 31*x^5 + 29*x^4 - 21*x^3 + 11*x^2 - 4*x + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^20 - 3*x^19 + 3*x^18 - 3*x^17 + 8*x^16 - 13*x^15 + 16*x^14 - 18*x^13 + 22*x^12 - 30*x^11 + 32*x^10 - 25*x^9 + 18*x^8 - 17*x^7 + 25*x^6 - 31*x^5 + 29*x^4 - 21*x^3 + 11*x^2 - 4*x + 1)
\( x^{20} - 3 x^{19} + 3 x^{18} - 3 x^{17} + 8 x^{16} - 13 x^{15} + 16 x^{14} - 18 x^{13} + 22 x^{12} + \cdots + 1 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $20$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[0, 10]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(2904089910121808158573\)
\(\medspace = 67\cdot 83^{2}\cdot 631\cdot 1777^{4}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(11.83\) | 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: | $67^{1/2}83^{1/2}631^{1/2}1777^{1/2}\approx 78965.03661114836$ | ||
| Ramified primes: |
\(67\), \(83\), \(631\), \(1777\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{42277}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{17}a^{18}-\frac{7}{17}a^{17}+\frac{5}{17}a^{16}+\frac{6}{17}a^{15}+\frac{7}{17}a^{14}+\frac{7}{17}a^{13}-\frac{7}{17}a^{12}-\frac{2}{17}a^{11}+\frac{8}{17}a^{10}+\frac{7}{17}a^{9}-\frac{3}{17}a^{7}-\frac{4}{17}a^{6}-\frac{8}{17}a^{5}+\frac{8}{17}a^{4}-\frac{8}{17}a^{3}+\frac{6}{17}a^{2}-\frac{7}{17}a+\frac{2}{17}$, $\frac{1}{188207}a^{19}-\frac{4904}{188207}a^{18}-\frac{634}{188207}a^{17}+\frac{84848}{188207}a^{16}-\frac{2209}{188207}a^{15}+\frac{3189}{11071}a^{14}+\frac{61458}{188207}a^{13}-\frac{63405}{188207}a^{12}+\frac{7099}{188207}a^{11}-\frac{84644}{188207}a^{10}-\frac{89733}{188207}a^{9}-\frac{47280}{188207}a^{8}+\frac{36481}{188207}a^{7}+\frac{25394}{188207}a^{6}-\frac{84355}{188207}a^{5}+\frac{54826}{188207}a^{4}-\frac{64382}{188207}a^{3}+\frac{34803}{188207}a^{2}+\frac{89973}{188207}a+\frac{253}{188207}$
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: |
$\frac{143988}{188207}a^{19}-\frac{252334}{188207}a^{18}-\frac{63352}{188207}a^{17}-\frac{108948}{188207}a^{16}+\frac{908160}{188207}a^{15}-\frac{561354}{188207}a^{14}+\frac{341340}{188207}a^{13}-\frac{822167}{188207}a^{12}+\frac{1158908}{188207}a^{11}-\frac{1361306}{188207}a^{10}+\frac{32666}{11071}a^{9}-\frac{117243}{188207}a^{8}+\frac{455982}{188207}a^{7}-\frac{785010}{188207}a^{6}+\frac{1380945}{188207}a^{5}-\frac{1041846}{188207}a^{4}+\frac{509274}{188207}a^{3}-\frac{206638}{188207}a^{2}-\frac{251876}{188207}a+\frac{5526}{11071}$, $\frac{549968}{188207}a^{19}-\frac{1420637}{188207}a^{18}+\frac{909255}{188207}a^{17}-\frac{51709}{11071}a^{16}+\frac{3738871}{188207}a^{15}-\frac{5319706}{188207}a^{14}+\frac{5584063}{188207}a^{13}-\frac{6038550}{188207}a^{12}+\frac{7906163}{188207}a^{11}-\frac{11443012}{188207}a^{10}+\frac{10494571}{188207}a^{9}-\frac{6206958}{188207}a^{8}+\frac{4103412}{188207}a^{7}-\frac{5581756}{188207}a^{6}+\frac{10228411}{188207}a^{5}-\frac{11168902}{188207}a^{4}+\frac{8343857}{188207}a^{3}-\frac{4654706}{188207}a^{2}+\frac{1421683}{188207}a-\frac{264128}{188207}$, $a^{19}-3a^{18}+3a^{17}-3a^{16}+8a^{15}-13a^{14}+16a^{13}-18a^{12}+22a^{11}-30a^{10}+32a^{9}-25a^{8}+18a^{7}-17a^{6}+25a^{5}-31a^{4}+29a^{3}-21a^{2}+11a-4$, $\frac{261450}{188207}a^{19}-\frac{793260}{188207}a^{18}+\frac{682114}{188207}a^{17}-\frac{416277}{188207}a^{16}+\frac{1835593}{188207}a^{15}-\frac{3334472}{188207}a^{14}+\frac{3342775}{188207}a^{13}-\frac{3285990}{188207}a^{12}+\frac{4552723}{188207}a^{11}-\frac{6463092}{188207}a^{10}+\frac{6583347}{188207}a^{9}-\frac{3872587}{188207}a^{8}+\frac{1940529}{188207}a^{7}-\frac{3117280}{188207}a^{6}+\frac{5902690}{188207}a^{5}-\frac{6761428}{188207}a^{4}+\frac{5099290}{188207}a^{3}-\frac{2556880}{188207}a^{2}+\frac{1020002}{188207}a-\frac{13446}{188207}$, $\frac{13771}{11071}a^{19}-\frac{518365}{188207}a^{18}-\frac{50296}{188207}a^{17}+\frac{177169}{188207}a^{16}+\frac{1445419}{188207}a^{15}-\frac{1480129}{188207}a^{14}+\frac{197465}{188207}a^{13}-\frac{542854}{188207}a^{12}+\frac{1474871}{188207}a^{11}-\frac{2283125}{188207}a^{10}+\frac{663690}{188207}a^{9}+\frac{103340}{11071}a^{8}-\frac{885459}{188207}a^{7}-\frac{1287729}{188207}a^{6}+\frac{2556944}{188207}a^{5}-\frac{1530195}{188207}a^{4}-\frac{826379}{188207}a^{3}+\frac{1351630}{188207}a^{2}-\frac{1128317}{188207}a+\frac{597055}{188207}$, $\frac{288317}{188207}a^{19}-\frac{615921}{188207}a^{18}+\frac{210652}{188207}a^{17}-\frac{368245}{188207}a^{16}+\frac{1771595}{188207}a^{15}-\frac{2010325}{188207}a^{14}+\frac{2077401}{188207}a^{13}-\frac{2197326}{188207}a^{12}+\frac{3122209}{188207}a^{11}-\frac{4606189}{188207}a^{10}+\frac{3458987}{188207}a^{9}-\frac{1676820}{188207}a^{8}+\frac{1328879}{188207}a^{7}-\frac{2166022}{188207}a^{6}+\frac{4419943}{188207}a^{5}-\frac{3836163}{188207}a^{4}+\frac{148455}{11071}a^{3}-\frac{1199606}{188207}a^{2}+\frac{52850}{188207}a+\frac{8453}{188207}$, $\frac{16372}{11071}a^{19}-\frac{743347}{188207}a^{18}+\frac{601087}{188207}a^{17}-\frac{625336}{188207}a^{16}+\frac{1947694}{188207}a^{15}-\frac{2931413}{188207}a^{14}+\frac{3475207}{188207}a^{13}-\frac{3712289}{188207}a^{12}+\frac{4663739}{188207}a^{11}-\frac{6531593}{188207}a^{10}+\frac{6476523}{188207}a^{9}-\frac{260615}{11071}a^{8}+\frac{3058281}{188207}a^{7}-\frac{3121979}{188207}a^{6}+\frac{5429908}{188207}a^{5}-\frac{6753477}{188207}a^{4}+\frac{5701415}{188207}a^{3}-\frac{3504359}{188207}a^{2}+\frac{1412446}{188207}a-\frac{95227}{188207}$, $\frac{349051}{188207}a^{19}-\frac{1010900}{188207}a^{18}+\frac{874494}{188207}a^{17}-\frac{682816}{188207}a^{16}+\frac{2454969}{188207}a^{15}-\frac{4211867}{188207}a^{14}+\frac{4570898}{188207}a^{13}-\frac{4557718}{188207}a^{12}+\frac{6135163}{188207}a^{11}-\frac{8774086}{188207}a^{10}+\frac{8772718}{188207}a^{9}-\frac{5470281}{188207}a^{8}+\frac{3419122}{188207}a^{7}-\frac{260897}{11071}a^{6}+\frac{7530284}{188207}a^{5}-\frac{8818357}{188207}a^{4}+\frac{7374006}{188207}a^{3}-\frac{4337972}{188207}a^{2}+\frac{1786999}{188207}a-\frac{280239}{188207}$, $\frac{403487}{188207}a^{19}-\frac{976808}{188207}a^{18}+\frac{405395}{188207}a^{17}-\frac{319732}{188207}a^{16}+\frac{2569757}{188207}a^{15}-\frac{3383789}{188207}a^{14}+\frac{2859233}{188207}a^{13}-\frac{3060611}{188207}a^{12}+\frac{4460280}{188207}a^{11}-\frac{397422}{11071}a^{10}+\frac{5426024}{188207}a^{9}-\frac{1709496}{188207}a^{8}+\frac{1124374}{188207}a^{7}-\frac{3405064}{188207}a^{6}+\frac{6382003}{188207}a^{5}-\frac{6317284}{188207}a^{4}+\frac{3381009}{188207}a^{3}-\frac{1182575}{188207}a^{2}-\frac{57746}{188207}a+\frac{162585}{188207}$
| 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: | \( 194.247728439 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{10}\cdot 194.247728439 \cdot 1}{2\cdot\sqrt{2904089910121808158573}}\cr\approx \mathstrut & 0.172830190148 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^20 - 3*x^19 + 3*x^18 - 3*x^17 + 8*x^16 - 13*x^15 + 16*x^14 - 18*x^13 + 22*x^12 - 30*x^11 + 32*x^10 - 25*x^9 + 18*x^8 - 17*x^7 + 25*x^6 - 31*x^5 + 29*x^4 - 21*x^3 + 11*x^2 - 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^20 - 3*x^19 + 3*x^18 - 3*x^17 + 8*x^16 - 13*x^15 + 16*x^14 - 18*x^13 + 22*x^12 - 30*x^11 + 32*x^10 - 25*x^9 + 18*x^8 - 17*x^7 + 25*x^6 - 31*x^5 + 29*x^4 - 21*x^3 + 11*x^2 - 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^20 - 3*x^19 + 3*x^18 - 3*x^17 + 8*x^16 - 13*x^15 + 16*x^14 - 18*x^13 + 22*x^12 - 30*x^11 + 32*x^10 - 25*x^9 + 18*x^8 - 17*x^7 + 25*x^6 - 31*x^5 + 29*x^4 - 21*x^3 + 11*x^2 - 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^20 - 3*x^19 + 3*x^18 - 3*x^17 + 8*x^16 - 13*x^15 + 16*x^14 - 18*x^13 + 22*x^12 - 30*x^11 + 32*x^10 - 25*x^9 + 18*x^8 - 17*x^7 + 25*x^6 - 31*x^5 + 29*x^4 - 21*x^3 + 11*x^2 - 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
$C_2^{10}.C_2\wr S_5$ (as 20T1015):
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 3932160 |
| The 506 conjugacy class representatives for $C_2^{10}.C_2\wr S_5$ |
| Character table for $C_2^{10}.C_2\wr S_5$ |
Intermediate fields
| 5.1.1777.1, 10.0.262091507.1 |
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 20 siblings: | data not computed |
| Degree 40 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 | $20$ | ${\href{/padicField/3.10.0.1}{10} }^{2}$ | ${\href{/padicField/5.6.0.1}{6} }^{2}{,}\,{\href{/padicField/5.4.0.1}{4} }{,}\,{\href{/padicField/5.2.0.1}{2} }^{2}$ | $16{,}\,{\href{/padicField/7.4.0.1}{4} }$ | $16{,}\,{\href{/padicField/11.4.0.1}{4} }$ | ${\href{/padicField/13.10.0.1}{10} }^{2}$ | ${\href{/padicField/17.12.0.1}{12} }{,}\,{\href{/padicField/17.2.0.1}{2} }^{3}{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ | ${\href{/padicField/19.4.0.1}{4} }^{5}$ | ${\href{/padicField/23.8.0.1}{8} }{,}\,{\href{/padicField/23.4.0.1}{4} }^{3}$ | ${\href{/padicField/29.4.0.1}{4} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }^{6}$ | ${\href{/padicField/31.10.0.1}{10} }{,}\,{\href{/padicField/31.5.0.1}{5} }^{2}$ | ${\href{/padicField/37.10.0.1}{10} }{,}\,{\href{/padicField/37.5.0.1}{5} }^{2}$ | ${\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.3.0.1}{3} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }^{4}$ | $20$ | ${\href{/padicField/47.8.0.1}{8} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}$ | ${\href{/padicField/53.10.0.1}{10} }^{2}$ | ${\href{/padicField/59.10.0.1}{10} }{,}\,{\href{/padicField/59.5.0.1}{5} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(67\)
| $\Q_{67}$ | $x + 65$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{67}$ | $x + 65$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{67}$ | $x + 65$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{67}$ | $x + 65$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{67}$ | $x + 65$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{67}$ | $x + 65$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 67.2.1.1 | $x^{2} + 134$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 67.12.0.1 | $x^{12} + 3 x^{8} + 57 x^{7} + 27 x^{6} + 4 x^{5} + 55 x^{4} + 64 x^{3} + 21 x^{2} + 27 x + 2$ | $1$ | $12$ | $0$ | $C_{12}$ | $[\ ]^{12}$ | |
|
\(83\)
| 83.4.0.1 | $x^{4} + 4 x^{2} + 42 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |
| 83.4.0.1 | $x^{4} + 4 x^{2} + 42 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 83.4.0.1 | $x^{4} + 4 x^{2} + 42 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 83.4.2.1 | $x^{4} + 164 x^{3} + 6894 x^{2} + 13940 x + 564653$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 83.4.0.1 | $x^{4} + 4 x^{2} + 42 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
|
\(631\)
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
| Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
| Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
| Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
|
\(1777\)
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
| Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
| Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
| Deg $4$ | $2$ | $2$ | $2$ |