Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 + 9*x^18 - 14*x^17 + 19*x^16 - 29*x^15 + 45*x^14 - 60*x^13 + 64*x^12 - 62*x^11 + 65*x^10 - 52*x^9 + 42*x^8 - 47*x^7 + 35*x^6 - 24*x^5 + 22*x^4 - 11*x^3 + 4*x^2 - 3*x + 1)
gp: K = bnfinit(y^20 - 4*y^19 + 9*y^18 - 14*y^17 + 19*y^16 - 29*y^15 + 45*y^14 - 60*y^13 + 64*y^12 - 62*y^11 + 65*y^10 - 52*y^9 + 42*y^8 - 47*y^7 + 35*y^6 - 24*y^5 + 22*y^4 - 11*y^3 + 4*y^2 - 3*y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 - 4*x^19 + 9*x^18 - 14*x^17 + 19*x^16 - 29*x^15 + 45*x^14 - 60*x^13 + 64*x^12 - 62*x^11 + 65*x^10 - 52*x^9 + 42*x^8 - 47*x^7 + 35*x^6 - 24*x^5 + 22*x^4 - 11*x^3 + 4*x^2 - 3*x + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^20 - 4*x^19 + 9*x^18 - 14*x^17 + 19*x^16 - 29*x^15 + 45*x^14 - 60*x^13 + 64*x^12 - 62*x^11 + 65*x^10 - 52*x^9 + 42*x^8 - 47*x^7 + 35*x^6 - 24*x^5 + 22*x^4 - 11*x^3 + 4*x^2 - 3*x + 1)
\( x^{20} - 4 x^{19} + 9 x^{18} - 14 x^{17} + 19 x^{16} - 29 x^{15} + 45 x^{14} - 60 x^{13} + 64 x^{12} + \cdots + 1 \)
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: |
\(2086514456522375390625\)
\(\medspace = 3^{10}\cdot 5^{8}\cdot 67^{6}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(11.64\) | 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: | $3^{1/2}5^{2/3}67^{3/4}\approx 118.60328889512358$ | ||
| Ramified primes: |
\(3\), \(5\), \(67\)
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{3}a^{16}-\frac{1}{3}a^{15}+\frac{1}{3}a^{14}+\frac{1}{3}a^{12}-\frac{1}{3}a^{11}-\frac{1}{3}a^{9}+\frac{1}{3}a^{7}-\frac{1}{3}a^{6}+\frac{1}{3}a^{5}-\frac{1}{3}a^{4}+\frac{1}{3}a^{3}-\frac{1}{3}a^{2}+\frac{1}{3}$, $\frac{1}{21}a^{17}+\frac{1}{7}a^{16}+\frac{3}{7}a^{15}+\frac{10}{21}a^{14}+\frac{1}{3}a^{13}-\frac{3}{7}a^{12}+\frac{5}{21}a^{11}-\frac{4}{21}a^{10}+\frac{2}{21}a^{9}-\frac{5}{21}a^{8}-\frac{1}{7}a^{7}+\frac{1}{7}a^{6}+\frac{2}{7}a^{5}+\frac{3}{7}a^{3}+\frac{8}{21}a^{2}+\frac{1}{21}a-\frac{5}{21}$, $\frac{1}{105}a^{18}-\frac{1}{15}a^{16}-\frac{52}{105}a^{15}+\frac{11}{35}a^{14}-\frac{2}{7}a^{13}-\frac{38}{105}a^{12}-\frac{4}{35}a^{11}+\frac{2}{15}a^{10}-\frac{4}{105}a^{9}-\frac{2}{7}a^{8}+\frac{47}{105}a^{7}+\frac{5}{21}a^{6}-\frac{4}{105}a^{5}-\frac{26}{105}a^{4}+\frac{16}{105}a^{3}+\frac{47}{105}a^{2}-\frac{29}{105}a-\frac{34}{105}$, $\frac{1}{207795}a^{19}-\frac{338}{207795}a^{18}+\frac{14}{29685}a^{17}-\frac{7617}{69265}a^{16}+\frac{23498}{69265}a^{15}-\frac{66169}{207795}a^{14}-\frac{86078}{207795}a^{13}-\frac{1626}{69265}a^{12}-\frac{4050}{13853}a^{11}-\frac{55661}{207795}a^{10}-\frac{22391}{69265}a^{9}-\frac{89248}{207795}a^{8}-\frac{103711}{207795}a^{7}-\frac{60359}{207795}a^{6}-\frac{15964}{207795}a^{5}-\frac{14997}{69265}a^{4}-\frac{49006}{207795}a^{3}+\frac{7457}{41559}a^{2}+\frac{18691}{69265}a-\frac{70373}{207795}$
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: |
\( -\frac{446380}{41559} a^{19} + \frac{507763}{13853} a^{18} - \frac{3138139}{41559} a^{17} + \frac{637280}{5937} a^{16} - \frac{5973073}{41559} a^{15} + \frac{1369838}{5937} a^{14} - \frac{4883015}{13853} a^{13} + \frac{18497107}{41559} a^{12} - \frac{18213646}{41559} a^{11} + \frac{5870395}{13853} a^{10} - \frac{19242005}{41559} a^{9} + \frac{12423695}{41559} a^{8} - \frac{574441}{1979} a^{7} + \frac{4774813}{13853} a^{6} - \frac{2508248}{13853} a^{5} + \frac{6669826}{41559} a^{4} - \frac{6152458}{41559} a^{3} + \frac{477231}{13853} a^{2} - \frac{1074751}{41559} a + \frac{788960}{41559} \)
(order $6$)
| 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: |
$\frac{2899322}{207795}a^{19}-\frac{9845246}{207795}a^{18}+\frac{20174536}{207795}a^{17}-\frac{28457162}{207795}a^{16}+\frac{37905823}{207795}a^{15}-\frac{61020278}{207795}a^{14}+\frac{93281894}{207795}a^{13}-\frac{117304126}{207795}a^{12}+\frac{7619062}{13853}a^{11}-\frac{36592994}{69265}a^{10}+\frac{120386464}{207795}a^{9}-\frac{76059866}{207795}a^{8}+\frac{74263358}{207795}a^{7}-\frac{89398313}{207795}a^{6}+\frac{6564116}{29685}a^{5}-\frac{41089732}{207795}a^{4}+\frac{1783043}{9895}a^{3}-\frac{537746}{13853}a^{2}+\frac{6174596}{207795}a-\frac{1496592}{69265}$, $\frac{1317224}{207795}a^{19}-\frac{4516223}{207795}a^{18}+\frac{9249607}{207795}a^{17}-\frac{13083317}{207795}a^{16}+\frac{17427853}{207795}a^{15}-\frac{4022147}{29685}a^{14}+\frac{43179023}{207795}a^{13}-\frac{7760152}{29685}a^{12}+\frac{17691524}{69265}a^{11}-\frac{17050801}{69265}a^{10}+\frac{8096341}{29685}a^{9}-\frac{36870662}{207795}a^{8}+\frac{34932419}{207795}a^{7}-\frac{6071063}{29685}a^{6}+\frac{21936608}{207795}a^{5}-\frac{19318948}{207795}a^{4}+\frac{1214813}{13853}a^{3}-\frac{200902}{9895}a^{2}+\frac{445313}{29685}a-\frac{778206}{69265}$, $\frac{8769}{1979}a^{19}-\frac{644008}{41559}a^{18}+\frac{1349779}{41559}a^{17}-\frac{1952813}{41559}a^{16}+\frac{2616919}{41559}a^{15}-\frac{592973}{5937}a^{14}+\frac{6363352}{41559}a^{13}-\frac{8142433}{41559}a^{12}+\frac{8149271}{41559}a^{11}-\frac{2612830}{13853}a^{10}+\frac{2817980}{13853}a^{9}-\frac{5738996}{41559}a^{8}+\frac{5407840}{41559}a^{7}-\frac{6225883}{41559}a^{6}+\frac{497350}{5937}a^{5}-\frac{3001924}{41559}a^{4}+\frac{2673971}{41559}a^{3}-\frac{240308}{13853}a^{2}+\frac{161004}{13853}a-\frac{48940}{5937}$, $\frac{1444658}{207795}a^{19}-\frac{4795019}{207795}a^{18}+\frac{9666239}{207795}a^{17}-\frac{13361098}{207795}a^{16}+\frac{5912294}{69265}a^{15}-\frac{9650114}{69265}a^{14}+\frac{14751432}{69265}a^{13}-\frac{2617974}{9895}a^{12}+\frac{1497728}{5937}a^{11}-\frac{7204264}{29685}a^{10}+\frac{8056598}{29685}a^{9}-\frac{34323439}{207795}a^{8}+\frac{11596194}{69265}a^{7}-\frac{14046944}{69265}a^{6}+\frac{973178}{9895}a^{5}-\frac{2741534}{29685}a^{4}+\frac{17548102}{207795}a^{3}-\frac{97315}{5937}a^{2}+\frac{992793}{69265}a-\frac{2282344}{207795}$, $\frac{1048673}{207795}a^{19}-\frac{520577}{29685}a^{18}+\frac{2529983}{69265}a^{17}-\frac{520693}{9895}a^{16}+\frac{2094251}{29685}a^{15}-\frac{23299147}{207795}a^{14}+\frac{35569861}{207795}a^{13}-\frac{6459857}{29685}a^{12}+\frac{9017263}{41559}a^{11}-\frac{43477073}{207795}a^{10}+\frac{2228896}{9895}a^{9}-\frac{1456524}{9895}a^{8}+\frac{28822027}{207795}a^{7}-\frac{34675147}{207795}a^{6}+\frac{18779008}{207795}a^{5}-\frac{5307916}{69265}a^{4}+\frac{2163701}{29685}a^{3}-\frac{99320}{5937}a^{2}+\frac{2292079}{207795}a-\frac{1988309}{207795}$, $\frac{267857}{69265}a^{19}-\frac{2633192}{207795}a^{18}+\frac{5378533}{207795}a^{17}-\frac{1081109}{29685}a^{16}+\frac{3421189}{69265}a^{15}-\frac{5528172}{69265}a^{14}+\frac{3581401}{29685}a^{13}-\frac{10478207}{69265}a^{12}+\frac{10283781}{69265}a^{11}-\frac{30393887}{207795}a^{10}+\frac{33145358}{207795}a^{9}-\frac{20368778}{207795}a^{8}+\frac{21687151}{207795}a^{7}-\frac{3466777}{29685}a^{6}+\frac{1736581}{29685}a^{5}-\frac{12210862}{207795}a^{4}+\frac{703335}{13853}a^{3}-\frac{2215468}{207795}a^{2}+\frac{312512}{29685}a-\frac{1223632}{207795}$, $\frac{365954}{29685}a^{19}-\frac{8684873}{207795}a^{18}+\frac{5913378}{69265}a^{17}-\frac{4998884}{41559}a^{16}+\frac{2223355}{13853}a^{15}-\frac{53920049}{207795}a^{14}+\frac{82434571}{207795}a^{13}-\frac{34480169}{69265}a^{12}+\frac{4797933}{9895}a^{11}-\frac{97290014}{207795}a^{10}+\frac{107494627}{207795}a^{9}-\frac{3251694}{9895}a^{8}+\frac{22134648}{69265}a^{7}-\frac{3819167}{9895}a^{6}+\frac{13566158}{69265}a^{5}-\frac{36844124}{207795}a^{4}+\frac{34191013}{207795}a^{3}-\frac{2529851}{69265}a^{2}+\frac{172262}{5937}a-\frac{1395206}{69265}$, $\frac{66007}{9895}a^{19}-\frac{4735078}{207795}a^{18}+\frac{3246842}{69265}a^{17}-\frac{4607171}{69265}a^{16}+\frac{18477932}{207795}a^{15}-\frac{29672989}{207795}a^{14}+\frac{15105798}{69265}a^{13}-\frac{11412400}{41559}a^{12}+\frac{55966849}{207795}a^{11}-\frac{10799233}{41559}a^{10}+\frac{59038247}{207795}a^{9}-\frac{12528787}{69265}a^{8}+\frac{12000208}{69265}a^{7}-\frac{14511536}{69265}a^{6}+\frac{1525084}{13853}a^{5}-\frac{1330927}{13853}a^{4}+\frac{2680903}{29685}a^{3}-\frac{4251379}{207795}a^{2}+\frac{3098924}{207795}a-\frac{736991}{69265}$, $\frac{2057777}{207795}a^{19}-\frac{65695}{1979}a^{18}+\frac{14112316}{207795}a^{17}-\frac{2837207}{29685}a^{16}+\frac{1266951}{9895}a^{15}-\frac{2861921}{13853}a^{14}+\frac{65423474}{207795}a^{13}-\frac{3911159}{9895}a^{12}+\frac{80058613}{207795}a^{11}-\frac{77550698}{207795}a^{10}+\frac{810337}{1979}a^{9}-\frac{7615658}{29685}a^{8}+\frac{10694281}{41559}a^{7}-\frac{62523728}{207795}a^{6}+\frac{32017343}{207795}a^{5}-\frac{29639083}{207795}a^{4}+\frac{3738397}{29685}a^{3}-\frac{841879}{29685}a^{2}+\frac{4793287}{207795}a-\frac{197447}{13853}$
| 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: | \( 527.926443922 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{10}\cdot 527.926443922 \cdot 1}{6\cdot\sqrt{2086514456522375390625}}\cr\approx \mathstrut & 0.184718396785 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 + 9*x^18 - 14*x^17 + 19*x^16 - 29*x^15 + 45*x^14 - 60*x^13 + 64*x^12 - 62*x^11 + 65*x^10 - 52*x^9 + 42*x^8 - 47*x^7 + 35*x^6 - 24*x^5 + 22*x^4 - 11*x^3 + 4*x^2 - 3*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 - 4*x^19 + 9*x^18 - 14*x^17 + 19*x^16 - 29*x^15 + 45*x^14 - 60*x^13 + 64*x^12 - 62*x^11 + 65*x^10 - 52*x^9 + 42*x^8 - 47*x^7 + 35*x^6 - 24*x^5 + 22*x^4 - 11*x^3 + 4*x^2 - 3*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 - 4*x^19 + 9*x^18 - 14*x^17 + 19*x^16 - 29*x^15 + 45*x^14 - 60*x^13 + 64*x^12 - 62*x^11 + 65*x^10 - 52*x^9 + 42*x^8 - 47*x^7 + 35*x^6 - 24*x^5 + 22*x^4 - 11*x^3 + 4*x^2 - 3*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 - 4*x^19 + 9*x^18 - 14*x^17 + 19*x^16 - 29*x^15 + 45*x^14 - 60*x^13 + 64*x^12 - 62*x^11 + 65*x^10 - 52*x^9 + 42*x^8 - 47*x^7 + 35*x^6 - 24*x^5 + 22*x^4 - 11*x^3 + 4*x^2 - 3*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
$C_2\wr S_5$ (as 20T288):
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 3840 |
| The 36 conjugacy class representatives for $C_2\wr S_5$ |
| Character table for $C_2\wr S_5$ is not computed |
Intermediate fields
| \(\Q(\sqrt{-3}) \), 5.1.5025.1, 10.0.681766875.1, 10.0.1691791875.1, 10.2.45678380625.1 |
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 10 siblings: | data not computed |
| Degree 20 siblings: | data not computed |
| Degree 30 siblings: | data not computed |
| Degree 32 siblings: | data not computed |
| Degree 40 siblings: | data not computed |
| Minimal sibling: | 10.0.1691791875.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.10.0.1}{10} }^{2}$ | R | R | ${\href{/padicField/7.4.0.1}{4} }^{2}{,}\,{\href{/padicField/7.3.0.1}{3} }^{4}$ | ${\href{/padicField/11.10.0.1}{10} }^{2}$ | ${\href{/padicField/13.4.0.1}{4} }^{4}{,}\,{\href{/padicField/13.2.0.1}{2} }^{2}$ | ${\href{/padicField/17.8.0.1}{8} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }^{2}$ | ${\href{/padicField/19.5.0.1}{5} }^{4}$ | ${\href{/padicField/23.4.0.1}{4} }^{4}{,}\,{\href{/padicField/23.2.0.1}{2} }^{2}$ | ${\href{/padicField/29.8.0.1}{8} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}$ | ${\href{/padicField/31.6.0.1}{6} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }^{4}$ | ${\href{/padicField/37.5.0.1}{5} }^{4}$ | ${\href{/padicField/41.10.0.1}{10} }^{2}$ | ${\href{/padicField/43.4.0.1}{4} }^{4}{,}\,{\href{/padicField/43.2.0.1}{2} }^{2}$ | ${\href{/padicField/47.4.0.1}{4} }^{4}{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}$ | ${\href{/padicField/53.6.0.1}{6} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }^{4}$ | ${\href{/padicField/59.8.0.1}{8} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(3\)
| 3.4.2.1 | $x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| 3.4.2.1 | $x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 3.12.6.2 | $x^{12} + 22 x^{10} + 177 x^{8} + 4 x^{7} + 644 x^{6} - 100 x^{5} + 876 x^{4} - 224 x^{3} + 1076 x^{2} + 344 x + 112$ | $2$ | $6$ | $6$ | $C_6\times C_2$ | $[\ ]_{2}^{6}$ | |
|
\(5\)
| 5.4.0.1 | $x^{4} + 4 x^{2} + 4 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |
| 5.4.0.1 | $x^{4} + 4 x^{2} + 4 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 5.6.4.1 | $x^{6} + 12 x^{5} + 54 x^{4} + 122 x^{3} + 168 x^{2} + 228 x + 233$ | $3$ | $2$ | $4$ | $S_3$ | $[\ ]_{3}^{2}$ | |
| 5.6.4.1 | $x^{6} + 12 x^{5} + 54 x^{4} + 122 x^{3} + 168 x^{2} + 228 x + 233$ | $3$ | $2$ | $4$ | $S_3$ | $[\ ]_{3}^{2}$ | |
|
\(67\)
| 67.2.0.1 | $x^{2} + 63 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 67.2.0.1 | $x^{2} + 63 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 67.4.0.1 | $x^{4} + 8 x^{2} + 54 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 67.4.3.2 | $x^{4} + 134$ | $4$ | $1$ | $3$ | $D_{4}$ | $[\ ]_{4}^{2}$ | |
| 67.4.3.2 | $x^{4} + 134$ | $4$ | $1$ | $3$ | $D_{4}$ | $[\ ]_{4}^{2}$ | |
| 67.4.0.1 | $x^{4} + 8 x^{2} + 54 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |