Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 + 5*x^16 - 22*x^15 + 69*x^14 - 125*x^13 + 204*x^12 - 417*x^11 + 873*x^10 - 1568*x^9 + 1977*x^8 - 1405*x^7 + 2837*x^6 - 7788*x^5 + 7654*x^4 - 1164*x^3 - 48*x^2 - 144*x + 36)
gp: K = bnfinit(y^18 - 3*y^17 + 5*y^16 - 22*y^15 + 69*y^14 - 125*y^13 + 204*y^12 - 417*y^11 + 873*y^10 - 1568*y^9 + 1977*y^8 - 1405*y^7 + 2837*y^6 - 7788*y^5 + 7654*y^4 - 1164*y^3 - 48*y^2 - 144*y + 36, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 3*x^17 + 5*x^16 - 22*x^15 + 69*x^14 - 125*x^13 + 204*x^12 - 417*x^11 + 873*x^10 - 1568*x^9 + 1977*x^8 - 1405*x^7 + 2837*x^6 - 7788*x^5 + 7654*x^4 - 1164*x^3 - 48*x^2 - 144*x + 36);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 - 3*x^17 + 5*x^16 - 22*x^15 + 69*x^14 - 125*x^13 + 204*x^12 - 417*x^11 + 873*x^10 - 1568*x^9 + 1977*x^8 - 1405*x^7 + 2837*x^6 - 7788*x^5 + 7654*x^4 - 1164*x^3 - 48*x^2 - 144*x + 36)
\( x^{18} - 3 x^{17} + 5 x^{16} - 22 x^{15} + 69 x^{14} - 125 x^{13} + 204 x^{12} - 417 x^{11} + 873 x^{10} + \cdots + 36 \)
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: | $[0, 9]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(-272438055977283000000000000\)
\(\medspace = -\,2^{12}\cdot 3^{9}\cdot 5^{12}\cdot 7^{12}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(29.42\) | 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: | $2^{2/3}3^{1/2}5^{2/3}7^{2/3}\approx 29.418870015005623$ | ||
| Ramified primes: |
\(2\), \(3\), \(5\), \(7\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-3}) \) | ||
| $\card{ \Gal(K/\Q) }$: | $18$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is Galois over $\Q$. | |||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3}a^{6}+\frac{1}{3}a^{4}+\frac{1}{3}a^{2}$, $\frac{1}{3}a^{7}+\frac{1}{3}a^{5}+\frac{1}{3}a^{3}$, $\frac{1}{3}a^{8}-\frac{1}{3}a^{2}$, $\frac{1}{9}a^{9}-\frac{1}{3}a^{5}-\frac{4}{9}a^{3}-\frac{1}{3}a$, $\frac{1}{9}a^{10}-\frac{1}{9}a^{4}$, $\frac{1}{9}a^{11}-\frac{1}{9}a^{5}$, $\frac{1}{54}a^{12}+\frac{1}{27}a^{11}-\frac{1}{27}a^{10}-\frac{1}{18}a^{9}+\frac{1}{9}a^{8}-\frac{1}{54}a^{6}-\frac{1}{27}a^{5}+\frac{10}{27}a^{4}+\frac{1}{18}a^{3}+\frac{2}{9}a^{2}+\frac{1}{3}$, $\frac{1}{54}a^{13}+\frac{1}{54}a^{10}+\frac{1}{9}a^{8}-\frac{1}{54}a^{7}+\frac{17}{54}a^{4}+\frac{2}{9}a^{2}+\frac{1}{3}$, $\frac{1}{324}a^{14}+\frac{1}{162}a^{13}-\frac{5}{324}a^{11}+\frac{1}{162}a^{10}-\frac{43}{324}a^{8}-\frac{19}{162}a^{7}-\frac{1}{9}a^{6}-\frac{157}{324}a^{5}-\frac{55}{162}a^{4}+\frac{1}{3}a^{3}+\frac{7}{54}a^{2}-\frac{2}{9}a+\frac{4}{9}$, $\frac{1}{972}a^{15}+\frac{1}{486}a^{13}-\frac{5}{972}a^{12}+\frac{4}{81}a^{11}+\frac{1}{486}a^{10}-\frac{43}{972}a^{9}-\frac{2}{81}a^{8}-\frac{37}{486}a^{7}+\frac{23}{972}a^{6}-\frac{22}{81}a^{5}+\frac{107}{486}a^{4}-\frac{47}{162}a^{3}+\frac{38}{81}a^{2}-\frac{10}{27}a-\frac{5}{27}$, $\frac{1}{235224}a^{16}+\frac{95}{235224}a^{15}+\frac{37}{117612}a^{14}-\frac{1885}{235224}a^{13}+\frac{527}{235224}a^{12}-\frac{5153}{117612}a^{11}+\frac{4303}{78408}a^{10}-\frac{1139}{235224}a^{9}+\frac{3863}{117612}a^{8}-\frac{18869}{235224}a^{7}+\frac{2263}{235224}a^{6}+\frac{33071}{117612}a^{5}-\frac{28109}{117612}a^{4}-\frac{1669}{6534}a^{3}+\frac{422}{9801}a^{2}+\frac{58}{3267}a-\frac{2221}{6534}$, $\frac{1}{1245703728456}a^{17}+\frac{405827}{311425932114}a^{16}+\frac{31024321}{415234576152}a^{15}-\frac{152754853}{1245703728456}a^{14}+\frac{688199147}{622851864228}a^{13}+\frac{3193652749}{415234576152}a^{12}+\frac{14391917831}{415234576152}a^{11}-\frac{1885291235}{207617288076}a^{10}-\frac{8123234341}{415234576152}a^{9}+\frac{8595908287}{1245703728456}a^{8}-\frac{64432407581}{622851864228}a^{7}-\frac{42956607355}{415234576152}a^{6}-\frac{205887487193}{622851864228}a^{5}-\frac{67497975307}{622851864228}a^{4}-\frac{13262172139}{34602881346}a^{3}+\frac{1888394191}{4513419306}a^{2}+\frac{647455883}{3145716486}a-\frac{11813699273}{34602881346}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $2$, $3$ |
Class group and class number
$C_{3}$, which has order $3$
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: | $8$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
| Torsion generator: |
\( \frac{80041913}{3431690712} a^{17} - \frac{214977701}{3431690712} a^{16} + \frac{55203773}{571948452} a^{15} - \frac{1651709819}{3431690712} a^{14} + \frac{4996308431}{3431690712} a^{13} - \frac{1400370739}{571948452} a^{12} + \frac{4524989513}{1143896904} a^{11} - \frac{9636851881}{1143896904} a^{10} + \frac{10078465345}{571948452} a^{9} - \frac{105829846579}{3431690712} a^{8} + \frac{123579584359}{3431690712} a^{7} - \frac{11843138939}{571948452} a^{6} + \frac{50363151085}{857922678} a^{5} - \frac{69710401220}{428961339} a^{4} + \frac{12028943263}{95324742} a^{3} + \frac{205439405}{12433662} a^{2} - \frac{1102633}{95324742} a - \frac{116868029}{47662371} \)
(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: |
$\frac{29736126967}{1245703728456}a^{17}-\frac{81451533085}{1245703728456}a^{16}+\frac{21476802739}{207617288076}a^{15}-\frac{627082610065}{1245703728456}a^{14}+\frac{1898255012767}{1245703728456}a^{13}-\frac{543295616441}{207617288076}a^{12}+\frac{1785011262347}{415234576152}a^{11}-\frac{3755728099337}{415234576152}a^{10}+\frac{3908897423891}{207617288076}a^{9}-\frac{41366240617481}{1245703728456}a^{8}+\frac{49778706304919}{1245703728456}a^{7}-\frac{5355672877153}{207617288076}a^{6}+\frac{10074504491065}{155712966057}a^{5}-\frac{26769577319212}{155712966057}a^{4}+\frac{4910703895771}{34602881346}a^{3}-\frac{18043006739}{4513419306}a^{2}+\frac{426625243129}{34602881346}a-\frac{7965125057}{1572858243}$, $\frac{525777184}{155712966057}a^{17}-\frac{5950576591}{622851864228}a^{16}+\frac{531248699}{34602881346}a^{15}-\frac{22082302025}{311425932114}a^{14}+\frac{137115880357}{622851864228}a^{13}-\frac{13261333529}{34602881346}a^{12}+\frac{63582981571}{103808644038}a^{11}-\frac{9818963497}{7689529188}a^{10}+\frac{93025060969}{34602881346}a^{9}-\frac{1484925960055}{311425932114}a^{8}+\frac{3568638605333}{622851864228}a^{7}-\frac{4486865827}{1281588198}a^{6}+\frac{238702951703}{28311448374}a^{5}-\frac{3793409782385}{155712966057}a^{4}+\frac{41379990200}{1922382297}a^{3}+\frac{2470063876}{2256709653}a^{2}-\frac{37096562468}{17301440673}a-\frac{54823397}{17301440673}$, $\frac{606672829}{155712966057}a^{17}-\frac{4103308661}{311425932114}a^{16}+\frac{2554830973}{103808644038}a^{15}-\frac{29341587125}{311425932114}a^{14}+\frac{8604865405}{28311448374}a^{13}-\frac{31286435281}{51904322019}a^{12}+\frac{35000324611}{34602881346}a^{11}-\frac{205619281877}{103808644038}a^{10}+\frac{19435332437}{4718574729}a^{9}-\frac{2371603942495}{311425932114}a^{8}+\frac{3246944032789}{311425932114}a^{7}-\frac{469699943981}{51904322019}a^{6}+\frac{4307569733779}{311425932114}a^{5}-\frac{495569188876}{14155724187}a^{4}+\frac{1490712132337}{34602881346}a^{3}-\frac{41649274103}{2256709653}a^{2}+\frac{51399306907}{17301440673}a-\frac{5589495872}{17301440673}$, $\frac{847929449}{46137175128}a^{17}-\frac{23276520637}{415234576152}a^{16}+\frac{19884209545}{207617288076}a^{15}-\frac{171051615971}{415234576152}a^{14}+\frac{178956738473}{138411525384}a^{13}-\frac{495065712197}{207617288076}a^{12}+\frac{1633844200363}{415234576152}a^{11}-\frac{3308364962143}{415234576152}a^{10}+\frac{3450854668703}{207617288076}a^{9}-\frac{12470851489891}{415234576152}a^{8}+\frac{5343101233321}{138411525384}a^{7}-\frac{6030271096837}{207617288076}a^{6}+\frac{5720039038121}{103808644038}a^{5}-\frac{15224426255539}{103808644038}a^{4}+\frac{2603206030069}{17301440673}a^{3}-\frac{54356119207}{1504473102}a^{2}+\frac{2025386147}{349524054}a-\frac{5083677371}{5767146891}$, $\frac{496004009}{622851864228}a^{17}-\frac{2387733121}{1245703728456}a^{16}+\frac{453342985}{138411525384}a^{15}-\frac{10196958059}{622851864228}a^{14}+\frac{56900708545}{1245703728456}a^{13}-\frac{3711187141}{46137175128}a^{12}+\frac{26619098045}{207617288076}a^{11}-\frac{112193350051}{415234576152}a^{10}+\frac{28050725761}{46137175128}a^{9}-\frac{592398970825}{622851864228}a^{8}+\frac{1340100163553}{1245703728456}a^{7}-\frac{130058181511}{138411525384}a^{6}+\frac{308966538649}{155712966057}a^{5}-\frac{2980707326125}{622851864228}a^{4}+\frac{146744291165}{34602881346}a^{3}+\frac{433201765}{4513419306}a^{2}+\frac{1443738698}{17301440673}a-\frac{6219157}{3145716486}$, $\frac{44162104301}{1245703728456}a^{17}-\frac{2754814739}{56622896748}a^{16}-\frac{7234560227}{415234576152}a^{15}-\frac{527212532915}{1245703728456}a^{14}+\frac{330997329317}{311425932114}a^{13}+\frac{14383904077}{415234576152}a^{12}-\frac{70135706231}{46137175128}a^{11}-\frac{8469152167}{51904322019}a^{10}+\frac{953406847691}{415234576152}a^{9}+\frac{5022319396937}{1245703728456}a^{8}-\frac{12387844897013}{311425932114}a^{7}+\frac{41316748261733}{415234576152}a^{6}-\frac{3968391177019}{155712966057}a^{5}-\frac{50075411780777}{622851864228}a^{4}-\frac{374420026325}{1572858243}a^{3}+\frac{1293727353511}{2256709653}a^{2}-\frac{8749749448705}{34602881346}a+\frac{1109045843417}{34602881346}$, $\frac{130720067}{138411525384}a^{17}-\frac{485506829}{138411525384}a^{16}+\frac{381398383}{207617288076}a^{15}-\frac{3058704505}{138411525384}a^{14}+\frac{28381232905}{415234576152}a^{13}-\frac{18276289871}{207617288076}a^{12}+\frac{806128739}{5126352792}a^{11}-\frac{142215613613}{415234576152}a^{10}+\frac{157103968565}{207617288076}a^{9}-\frac{57799854491}{46137175128}a^{8}+\frac{546501160937}{415234576152}a^{7}-\frac{73780323835}{207617288076}a^{6}+\frac{7999492571}{3145716486}a^{5}-\frac{948899355661}{103808644038}a^{4}+\frac{3811211936}{17301440673}a^{3}-\frac{443697229}{1504473102}a^{2}+\frac{1220272459}{11534293782}a+\frac{114336382}{5767146891}$, $\frac{816605701}{1245703728456}a^{17}-\frac{2089834723}{1245703728456}a^{16}+\frac{15896869}{6291432972}a^{15}-\frac{16073760187}{1245703728456}a^{14}+\frac{47554529713}{1245703728456}a^{13}-\frac{4177387943}{69205762692}a^{12}+\frac{36536431889}{415234576152}a^{11}-\frac{8788519607}{46137175128}a^{10}+\frac{26966180833}{69205762692}a^{9}-\frac{802415797979}{1245703728456}a^{8}+\frac{77415797467}{113245793496}a^{7}-\frac{1511197067}{7689529188}a^{6}+\frac{242387016601}{155712966057}a^{5}-\frac{739027766098}{155712966057}a^{4}+\frac{18241904803}{5767146891}a^{3}+\frac{4399064713}{4513419306}a^{2}-\frac{8181605753}{34602881346}a-\frac{586743400}{17301440673}$
| 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: | \( 25716902.8477 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{9}\cdot 25716902.8477 \cdot 3}{6\cdot\sqrt{272438055977283000000000000}}\cr\approx \mathstrut & 11.8897739394 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 + 5*x^16 - 22*x^15 + 69*x^14 - 125*x^13 + 204*x^12 - 417*x^11 + 873*x^10 - 1568*x^9 + 1977*x^8 - 1405*x^7 + 2837*x^6 - 7788*x^5 + 7654*x^4 - 1164*x^3 - 48*x^2 - 144*x + 36)
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 - 3*x^17 + 5*x^16 - 22*x^15 + 69*x^14 - 125*x^13 + 204*x^12 - 417*x^11 + 873*x^10 - 1568*x^9 + 1977*x^8 - 1405*x^7 + 2837*x^6 - 7788*x^5 + 7654*x^4 - 1164*x^3 - 48*x^2 - 144*x + 36, 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 - 3*x^17 + 5*x^16 - 22*x^15 + 69*x^14 - 125*x^13 + 204*x^12 - 417*x^11 + 873*x^10 - 1568*x^9 + 1977*x^8 - 1405*x^7 + 2837*x^6 - 7788*x^5 + 7654*x^4 - 1164*x^3 - 48*x^2 - 144*x + 36);
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 - 3*x^17 + 5*x^16 - 22*x^15 + 69*x^14 - 125*x^13 + 204*x^12 - 417*x^11 + 873*x^10 - 1568*x^9 + 1977*x^8 - 1405*x^7 + 2837*x^6 - 7788*x^5 + 7654*x^4 - 1164*x^3 - 48*x^2 - 144*x + 36);
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
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 18 |
| The 6 conjugacy class representatives for $C_3^2 : C_2$ |
| Character table for $C_3^2 : C_2$ |
Intermediate fields
| \(\Q(\sqrt{-3}) \), 3.1.300.1 x3, 3.1.14700.1 x3, 3.1.3675.1 x3, 3.1.588.1 x3, 6.0.270000.1, 6.0.648270000.1, 6.0.40516875.1, 6.0.1037232.1, 9.1.9529569000000.1 x9 |
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
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | R | R | R | ${\href{/padicField/11.2.0.1}{2} }^{9}$ | ${\href{/padicField/13.3.0.1}{3} }^{6}$ | ${\href{/padicField/17.2.0.1}{2} }^{9}$ | ${\href{/padicField/19.3.0.1}{3} }^{6}$ | ${\href{/padicField/23.2.0.1}{2} }^{9}$ | ${\href{/padicField/29.2.0.1}{2} }^{9}$ | ${\href{/padicField/31.3.0.1}{3} }^{6}$ | ${\href{/padicField/37.3.0.1}{3} }^{6}$ | ${\href{/padicField/41.2.0.1}{2} }^{9}$ | ${\href{/padicField/43.3.0.1}{3} }^{6}$ | ${\href{/padicField/47.2.0.1}{2} }^{9}$ | ${\href{/padicField/53.2.0.1}{2} }^{9}$ | ${\href{/padicField/59.2.0.1}{2} }^{9}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.6.4.1 | $x^{6} + 3 x^{5} + 10 x^{4} + 19 x^{3} + 22 x^{2} + 11 x + 7$ | $3$ | $2$ | $4$ | $S_3$ | $[\ ]_{3}^{2}$ |
| 2.6.4.1 | $x^{6} + 3 x^{5} + 10 x^{4} + 19 x^{3} + 22 x^{2} + 11 x + 7$ | $3$ | $2$ | $4$ | $S_3$ | $[\ ]_{3}^{2}$ | |
| 2.6.4.1 | $x^{6} + 3 x^{5} + 10 x^{4} + 19 x^{3} + 22 x^{2} + 11 x + 7$ | $3$ | $2$ | $4$ | $S_3$ | $[\ ]_{3}^{2}$ | |
|
\(3\)
| 3.2.1.2 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 3.2.1.2 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 3.2.1.2 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 3.2.1.2 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 3.2.1.2 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 3.2.1.2 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 3.2.1.2 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 3.2.1.2 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 3.2.1.2 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
|
\(5\)
| 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 | $x^{6} + 12 x^{5} + 54 x^{4} + 122 x^{3} + 168 x^{2} + 228 x + 233$ | $3$ | $2$ | $4$ | $S_3$ | $[\ ]_{3}^{2}$ | |
| 5.6.4.1 | $x^{6} + 12 x^{5} + 54 x^{4} + 122 x^{3} + 168 x^{2} + 228 x + 233$ | $3$ | $2$ | $4$ | $S_3$ | $[\ ]_{3}^{2}$ | |
|
\(7\)
| 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 | $x^{9} + 18 x^{8} + 108 x^{7} + 249 x^{6} + 396 x^{5} + 1944 x^{4} + 2631 x^{3} - 2358 x^{2} - 756 x + 11915$ | $3$ | $3$ | $6$ | $C_3^2$ | $[\ ]_{3}^{3}$ |