Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - 2*x^21 + 6*x^20 - 12*x^19 + 22*x^18 - 40*x^17 + 62*x^16 - 97*x^15 + 138*x^14 - 177*x^13 + 216*x^12 - 233*x^11 + 238*x^10 - 226*x^9 + 195*x^8 - 164*x^7 + 124*x^6 - 85*x^5 + 55*x^4 - 29*x^3 + 13*x^2 - 5*x + 1)
gp: K = bnfinit(y^22 - 2*y^21 + 6*y^20 - 12*y^19 + 22*y^18 - 40*y^17 + 62*y^16 - 97*y^15 + 138*y^14 - 177*y^13 + 216*y^12 - 233*y^11 + 238*y^10 - 226*y^9 + 195*y^8 - 164*y^7 + 124*y^6 - 85*y^5 + 55*y^4 - 29*y^3 + 13*y^2 - 5*y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^22 - 2*x^21 + 6*x^20 - 12*x^19 + 22*x^18 - 40*x^17 + 62*x^16 - 97*x^15 + 138*x^14 - 177*x^13 + 216*x^12 - 233*x^11 + 238*x^10 - 226*x^9 + 195*x^8 - 164*x^7 + 124*x^6 - 85*x^5 + 55*x^4 - 29*x^3 + 13*x^2 - 5*x + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^22 - 2*x^21 + 6*x^20 - 12*x^19 + 22*x^18 - 40*x^17 + 62*x^16 - 97*x^15 + 138*x^14 - 177*x^13 + 216*x^12 - 233*x^11 + 238*x^10 - 226*x^9 + 195*x^8 - 164*x^7 + 124*x^6 - 85*x^5 + 55*x^4 - 29*x^3 + 13*x^2 - 5*x + 1)
\( x^{22} - 2 x^{21} + 6 x^{20} - 12 x^{19} + 22 x^{18} - 40 x^{17} + 62 x^{16} - 97 x^{15} + 138 x^{14} + \cdots + 1 \)
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: | $[0, 11]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(-1163485434252431102394067\)
\(\medspace = -\,971^{2}\cdot 1867\cdot 25709231^{2}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(12.41\) | 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: | $971^{1/2}1867^{1/2}25709231^{1/2}\approx 6826943.634084509$ | ||
| Ramified primes: |
\(971\), \(1867\), \(25709231\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-1867}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $\frac{1}{50213}a^{21}-\frac{3597}{50213}a^{20}-\frac{23733}{50213}a^{19}+\frac{8236}{50213}a^{18}+\frac{17272}{50213}a^{17}+\frac{20601}{50213}a^{16}+\frac{3642}{50213}a^{15}+\frac{12506}{50213}a^{14}-\frac{18297}{50213}a^{13}-\frac{1492}{50213}a^{12}-\frac{8835}{50213}a^{11}-\frac{23237}{50213}a^{10}-\frac{17179}{50213}a^{9}-\frac{3711}{50213}a^{8}-\frac{15418}{50213}a^{7}-\frac{7606}{50213}a^{6}-\frac{22391}{50213}a^{5}+\frac{4121}{50213}a^{4}-\frac{2105}{50213}a^{3}-\frac{14717}{50213}a^{2}-\frac{16874}{50213}a+\frac{4721}{50213}$
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: | $10$ | 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: |
$\frac{12666}{50213}a^{21}-\frac{66624}{50213}a^{20}+\frac{173692}{50213}a^{19}-\frac{326716}{50213}a^{18}+\frac{692093}{50213}a^{17}-\frac{1079168}{50213}a^{16}+\frac{1841706}{50213}a^{15}-\frac{2732521}{50213}a^{14}+\frac{3799381}{50213}a^{13}-\frac{5139310}{50213}a^{12}+\frac{5594310}{50213}a^{11}-\frac{5745731}{50213}a^{10}+\frac{5356506}{50213}a^{9}-\frac{4322476}{50213}a^{8}+\frac{3860370}{50213}a^{7}-\frac{2489499}{50213}a^{6}+\frac{1705860}{50213}a^{5}-\frac{1029194}{50213}a^{4}+\frac{151812}{50213}a^{3}-\frac{14866}{50213}a^{2}-\frac{69769}{50213}a+\frac{92929}{50213}$, $a$, $\frac{109947}{50213}a^{21}-\frac{303049}{50213}a^{20}+\frac{699589}{50213}a^{19}-\frac{1524140}{50213}a^{18}+\frac{2610213}{50213}a^{17}-\frac{4609453}{50213}a^{16}+\frac{7208971}{50213}a^{15}-\frac{10630553}{50213}a^{14}+\frac{15197699}{50213}a^{13}-\frac{18624076}{50213}a^{12}+\frac{21178626}{50213}a^{11}-\frac{21743228}{50213}a^{10}+\frac{20017469}{50213}a^{9}-\frac{18209798}{50213}a^{8}+\frac{14539591}{50213}a^{7}-\frac{11056440}{50213}a^{6}+\frac{8053767}{50213}a^{5}-\frac{4399056}{50213}a^{4}+\frac{2303080}{50213}a^{3}-\frac{879908}{50213}a^{2}+\frac{124672}{50213}a-\frac{42207}{50213}$, $\frac{6821}{50213}a^{21}+\frac{119446}{50213}a^{20}+\frac{3919}{50213}a^{19}+\frac{290687}{50213}a^{18}-\frac{188238}{50213}a^{17}+\frac{274512}{50213}a^{16}-\frac{615909}{50213}a^{15}-\frac{8461}{50213}a^{14}-\frac{526662}{50213}a^{13}-\frac{485823}{50213}a^{12}+\frac{2251650}{50213}a^{11}-\frac{2889490}{50213}a^{10}+\frac{5592826}{50213}a^{9}-\frac{5327957}{50213}a^{8}+\frac{5804552}{50213}a^{7}-\frac{6136483}{50213}a^{6}+\frac{4287040}{50213}a^{5}-\frac{4278044}{50213}a^{4}+\frac{3166132}{50213}a^{3}-\frac{1615686}{50213}a^{2}+\frac{1095328}{50213}a-\frac{386296}{50213}$, $\frac{88354}{50213}a^{21}-\frac{212113}{50213}a^{20}+\frac{692380}{50213}a^{19}-\frac{1409216}{50213}a^{18}+\frac{2738507}{50213}a^{17}-\frac{5011583}{50213}a^{16}+\frac{7954018}{50213}a^{15}-\frac{12736043}{50213}a^{14}+\frac{18271929}{50213}a^{13}-\frac{24468774}{50213}a^{12}+\frac{30583425}{50213}a^{11}-\frac{33565251}{50213}a^{10}+\frac{35003659}{50213}a^{9}-\frac{33030958}{50213}a^{8}+\frac{28858980}{50213}a^{7}-\frac{24473676}{50213}a^{6}+\frac{18134466}{50213}a^{5}-\frac{12992796}{50213}a^{4}+\frac{8138288}{50213}a^{3}-\frac{3906584}{50213}a^{2}+\frac{1846668}{50213}a-\frac{502287}{50213}$, $\frac{131584}{50213}a^{21}-\frac{250975}{50213}a^{20}+\frac{817445}{50213}a^{19}-\frac{1577958}{50213}a^{18}+\frac{2990822}{50213}a^{17}-\frac{5409825}{50213}a^{16}+\frac{8381627}{50213}a^{15}-\frac{13297377}{50213}a^{14}+\frac{18749925}{50213}a^{13}-\frac{24394016}{50213}a^{12}+\frac{29913684}{50213}a^{11}-\frac{32183732}{50213}a^{10}+\frac{33198091}{50213}a^{9}-\frac{31018433}{50213}a^{8}+\frac{26757256}{50213}a^{7}-\frac{22427599}{50213}a^{6}+\frac{16520721}{50213}a^{5}-\frac{11541513}{50213}a^{4}+\frac{7221260}{50213}a^{3}-\frac{3572293}{50213}a^{2}+\frac{1576834}{50213}a-\frac{428876}{50213}$, $\frac{249258}{50213}a^{21}-\frac{429615}{50213}a^{20}+\frac{1359380}{50213}a^{19}-\frac{2530054}{50213}a^{18}+\frac{4591365}{50213}a^{17}-\frac{8253106}{50213}a^{16}+\frac{12349207}{50213}a^{15}-\frac{19334497}{50213}a^{14}+\frac{26685628}{50213}a^{13}-\frac{33206251}{50213}a^{12}+\frac{39514742}{50213}a^{11}-\frac{40410274}{50213}a^{10}+\frac{40281645}{50213}a^{9}-\frac{36778681}{50213}a^{8}+\frac{30421189}{50213}a^{7}-\frac{25221246}{50213}a^{6}+\frac{17764261}{50213}a^{5}-\frac{11513900}{50213}a^{4}+\frac{7067580}{50213}a^{3}-\frac{3082264}{50213}a^{2}+\frac{1267352}{50213}a-\frac{446554}{50213}$, $\frac{336061}{50213}a^{21}-\frac{586211}{50213}a^{20}+\frac{1955088}{50213}a^{19}-\frac{3607713}{50213}a^{18}+\frac{6802399}{50213}a^{17}-\frac{12176686}{50213}a^{16}+\frac{18520884}{50213}a^{15}-\frac{29323413}{50213}a^{14}+\frac{40647541}{50213}a^{13}-\frac{51996662}{50213}a^{12}+\frac{62711792}{50213}a^{11}-\frac{65652514}{50213}a^{10}+\frac{66881259}{50213}a^{9}-\frac{61191737}{50213}a^{8}+\frac{51961001}{50213}a^{7}-\frac{43069955}{50213}a^{6}+\frac{30858472}{50213}a^{5}-\frac{21006406}{50213}a^{4}+\frac{12595802}{50213}a^{3}-\frac{5854797}{50213}a^{2}+\frac{2471852}{50213}a-\frac{588523}{50213}$, $\frac{71889}{50213}a^{21}+\frac{12217}{50213}a^{20}+\frac{196529}{50213}a^{19}-\frac{84105}{50213}a^{18}+\frac{200596}{50213}a^{17}-\frac{398637}{50213}a^{16}+\frac{9156}{50213}a^{15}-\frac{522061}{50213}a^{14}-\frac{224350}{50213}a^{13}+\frac{950627}{50213}a^{12}-\frac{1049551}{50213}a^{11}+\frac{2662680}{50213}a^{10}-\frac{1950703}{50213}a^{9}+\frac{2512240}{50213}a^{8}-\frac{2543703}{50213}a^{7}+\frac{1387587}{50213}a^{6}-\frac{1896552}{50213}a^{5}+\frac{1002129}{50213}a^{4}-\frac{486493}{50213}a^{3}+\frac{399201}{50213}a^{2}+\frac{40881}{50213}a-\frac{51911}{50213}$, $\frac{202027}{50213}a^{21}-\frac{209435}{50213}a^{20}+\frac{735135}{50213}a^{19}-\frac{1168708}{50213}a^{18}+\frac{1866429}{50213}a^{17}-\frac{3461188}{50213}a^{16}+\frac{4329563}{50213}a^{15}-\frac{6846827}{50213}a^{14}+\frac{8478186}{50213}a^{13}-\frac{8331003}{50213}a^{12}+\frac{9101519}{50213}a^{11}-\frac{6716145}{50213}a^{10}+\frac{6126287}{50213}a^{9}-\frac{5013194}{50213}a^{8}+\frac{2320541}{50213}a^{7}-\frac{2359147}{50213}a^{6}+\frac{604743}{50213}a^{5}+\frac{423431}{50213}a^{4}+\frac{37275}{50213}a^{3}+\frac{282075}{50213}a^{2}-\frac{93241}{50213}a-\frac{76681}{50213}$
| 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: | \( 584.201894616 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{11}\cdot 584.201894616 \cdot 1}{2\cdot\sqrt{1163485434252431102394067}}\cr\approx \mathstrut & 0.163166516610 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^22 - 2*x^21 + 6*x^20 - 12*x^19 + 22*x^18 - 40*x^17 + 62*x^16 - 97*x^15 + 138*x^14 - 177*x^13 + 216*x^12 - 233*x^11 + 238*x^10 - 226*x^9 + 195*x^8 - 164*x^7 + 124*x^6 - 85*x^5 + 55*x^4 - 29*x^3 + 13*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^22 - 2*x^21 + 6*x^20 - 12*x^19 + 22*x^18 - 40*x^17 + 62*x^16 - 97*x^15 + 138*x^14 - 177*x^13 + 216*x^12 - 233*x^11 + 238*x^10 - 226*x^9 + 195*x^8 - 164*x^7 + 124*x^6 - 85*x^5 + 55*x^4 - 29*x^3 + 13*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^22 - 2*x^21 + 6*x^20 - 12*x^19 + 22*x^18 - 40*x^17 + 62*x^16 - 97*x^15 + 138*x^14 - 177*x^13 + 216*x^12 - 233*x^11 + 238*x^10 - 226*x^9 + 195*x^8 - 164*x^7 + 124*x^6 - 85*x^5 + 55*x^4 - 29*x^3 + 13*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^22 - 2*x^21 + 6*x^20 - 12*x^19 + 22*x^18 - 40*x^17 + 62*x^16 - 97*x^15 + 138*x^14 - 177*x^13 + 216*x^12 - 233*x^11 + 238*x^10 - 226*x^9 + 195*x^8 - 164*x^7 + 124*x^6 - 85*x^5 + 55*x^4 - 29*x^3 + 13*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
$C_2^{10}.(C_2\times S_{11})$ (as 22T53):
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 81749606400 |
| The 752 conjugacy class representatives for $C_2^{10}.(C_2\times S_{11})$ are not computed |
| Character table for $C_2^{10}.(C_2\times S_{11})$ is not computed |
Intermediate fields
| 11.3.24963663301.1 |
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 sibling: | data not computed |
| Degree 44 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 | ${\href{/padicField/2.10.0.1}{10} }{,}\,{\href{/padicField/2.6.0.1}{6} }^{2}$ | $22$ | $22$ | $16{,}\,{\href{/padicField/7.6.0.1}{6} }$ | ${\href{/padicField/11.11.0.1}{11} }^{2}$ | ${\href{/padicField/13.8.0.1}{8} }^{2}{,}\,{\href{/padicField/13.6.0.1}{6} }$ | $18{,}\,{\href{/padicField/17.2.0.1}{2} }^{2}$ | ${\href{/padicField/19.12.0.1}{12} }{,}\,{\href{/padicField/19.3.0.1}{3} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }^{2}$ | ${\href{/padicField/23.12.0.1}{12} }{,}\,{\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.4.0.1}{4} }$ | ${\href{/padicField/29.8.0.1}{8} }{,}\,{\href{/padicField/29.7.0.1}{7} }^{2}$ | ${\href{/padicField/31.14.0.1}{14} }{,}\,{\href{/padicField/31.4.0.1}{4} }^{2}$ | $18{,}\,{\href{/padicField/37.1.0.1}{1} }^{4}$ | ${\href{/padicField/41.9.0.1}{9} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.10.0.1}{10} }{,}\,{\href{/padicField/43.5.0.1}{5} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }$ | ${\href{/padicField/47.10.0.1}{10} }{,}\,{\href{/padicField/47.5.0.1}{5} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | ${\href{/padicField/53.14.0.1}{14} }{,}\,{\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.2.0.1}{2} }$ | ${\href{/padicField/59.14.0.1}{14} }{,}\,{\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.1.0.1}{1} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(971\)
| $\Q_{971}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{971}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $4$ | $2$ | $2$ | $2$ | ||||
| Deg $14$ | $1$ | $14$ | $0$ | $C_{14}$ | $[\ ]^{14}$ | ||
|
\(1867\)
| $\Q_{1867}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{1867}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{1867}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{1867}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 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 $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
|
\(25709231\)
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
| Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
| Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
| Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |