Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^16 + 12*x^14 - 126*x^12 - 386*x^10 + 708*x^8 + 2277*x^6 - 2571*x^4 + 796*x^2 - 108)
gp: K = bnfinit(y^18 - 3*y^16 + 12*y^14 - 126*y^12 - 386*y^10 + 708*y^8 + 2277*y^6 - 2571*y^4 + 796*y^2 - 108, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 3*x^16 + 12*x^14 - 126*x^12 - 386*x^10 + 708*x^8 + 2277*x^6 - 2571*x^4 + 796*x^2 - 108);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 - 3*x^16 + 12*x^14 - 126*x^12 - 386*x^10 + 708*x^8 + 2277*x^6 - 2571*x^4 + 796*x^2 - 108)
\( x^{18} - 3x^{16} + 12x^{14} - 126x^{12} - 386x^{10} + 708x^{8} + 2277x^{6} - 2571x^{4} + 796x^{2} - 108 \)
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: | $[6, 6]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(2534788041928904776561145413632\)
\(\medspace = 2^{18}\cdot 3^{9}\cdot 53^{12}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(48.88\) | 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: | $2\cdot 3^{1/2}53^{2/3}\approx 48.877374517097394$ | ||
| Ramified primes: |
\(2\), \(3\), \(53\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{3}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $6$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{4}a^{8}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}+\frac{1}{4}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{4}a^{9}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}+\frac{1}{4}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{4}a^{10}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}+\frac{1}{4}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}$, $\frac{1}{4}a^{11}-\frac{1}{2}a^{7}+\frac{1}{4}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}$, $\frac{1}{28}a^{12}-\frac{1}{14}a^{10}-\frac{3}{28}a^{8}-\frac{1}{4}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}+\frac{1}{4}a^{2}-\frac{1}{2}a+\frac{1}{14}$, $\frac{1}{56}a^{13}+\frac{5}{56}a^{11}-\frac{3}{56}a^{9}-\frac{3}{8}a^{7}-\frac{1}{8}a^{5}-\frac{1}{8}a^{3}-\frac{1}{2}a^{2}-\frac{13}{28}a-\frac{1}{2}$, $\frac{1}{280}a^{14}-\frac{3}{280}a^{12}+\frac{13}{280}a^{10}+\frac{31}{280}a^{8}-\frac{9}{40}a^{6}+\frac{7}{40}a^{4}-\frac{1}{2}a^{3}+\frac{57}{140}a^{2}-\frac{1}{2}a-\frac{9}{35}$, $\frac{1}{280}a^{15}+\frac{1}{140}a^{13}-\frac{4}{35}a^{11}+\frac{2}{35}a^{9}-\frac{1}{10}a^{7}-\frac{1}{5}a^{5}-\frac{61}{280}a^{3}-\frac{1}{2}a^{2}+\frac{39}{140}a-\frac{1}{2}$, $\frac{1}{3250018240}a^{16}+\frac{59993}{50781535}a^{14}-\frac{1128733}{812504560}a^{12}-\frac{75151357}{1625009120}a^{10}-\frac{5469369}{101563070}a^{8}-\frac{330397}{116072080}a^{6}-\frac{1}{2}a^{5}-\frac{1369349791}{3250018240}a^{4}-\frac{8331999}{406252280}a^{2}-\frac{1}{2}a+\frac{72038861}{162500912}$, $\frac{1}{9750054720}a^{17}+\frac{18389}{11607208}a^{15}+\frac{6394627}{812504560}a^{13}-\frac{7718439}{325001824}a^{11}-\frac{4085315}{243751368}a^{9}-\frac{11435781}{23214416}a^{7}-\frac{1}{2}a^{6}+\frac{1358143587}{3250018240}a^{5}-\frac{9841233}{81250456}a^{3}-\frac{1}{2}a^{2}+\frac{1021805161}{2437513680}a$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $2$ |
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: | $11$ | 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: |
$\frac{68941301}{4875027360}a^{17}-\frac{6226917}{1625009120}a^{16}-\frac{7868629}{203126140}a^{15}+\frac{1681657}{203126140}a^{14}+\frac{64748001}{406252280}a^{13}-\frac{14972099}{406252280}a^{12}-\frac{202069349}{116072080}a^{11}+\frac{363829129}{812504560}a^{10}-\frac{128745809}{21763515}a^{9}+\frac{381752407}{203126140}a^{8}+\frac{492492977}{58036040}a^{7}-\frac{80957991}{58036040}a^{6}+\frac{56286964999}{1625009120}a^{5}-\frac{17862486853}{1625009120}a^{4}-\frac{193342727}{7254505}a^{3}+\frac{144011253}{203126140}a^{2}+\frac{971503247}{174108120}a+\frac{79869527}{81250456}$, $\frac{30830113}{1950010944}a^{17}+\frac{8392313}{650003648}a^{16}-\frac{2274547}{58036040}a^{15}-\frac{1570916}{50781535}a^{14}+\frac{137706619}{812504560}a^{13}+\frac{15840609}{116072080}a^{12}-\frac{3096020703}{1625009120}a^{11}-\frac{2515078481}{1625009120}a^{10}-\frac{8641717967}{1218756840}a^{9}-\frac{299896181}{50781535}a^{8}+\frac{865944607}{116072080}a^{7}+\frac{637442159}{116072080}a^{6}+\frac{129784610927}{3250018240}a^{5}+\frac{107357047949}{3250018240}a^{4}-\frac{7881900021}{406252280}a^{3}-\frac{722017521}{58036040}a^{2}+\frac{6855102533}{2437513680}a+\frac{1917963757}{812504560}$, $\frac{675481}{174108120}a^{17}-\frac{68361}{5803604}a^{15}+\frac{691507}{14509010}a^{13}-\frac{2847967}{5803604}a^{11}-\frac{12852823}{8705406}a^{9}+\frac{3983074}{1450901}a^{7}+\frac{477089787}{58036040}a^{5}-\frac{60940909}{5803604}a^{3}+\frac{112491908}{21763515}a+2$, $\frac{5992591}{650003648}a^{17}-\frac{29371087}{3250018240}a^{16}-\frac{4826371}{203126140}a^{15}+\frac{8495849}{406252280}a^{14}+\frac{82472617}{812504560}a^{13}-\frac{76050411}{812504560}a^{12}-\frac{1821759479}{1625009120}a^{11}+\frac{1742937143}{1625009120}a^{10}-\frac{817515371}{203126140}a^{9}+\frac{1717348639}{406252280}a^{8}+\frac{546726761}{116072080}a^{7}-\frac{415120487}{116072080}a^{6}+\frac{73341299131}{3250018240}a^{5}-\frac{75583555559}{3250018240}a^{4}-\frac{5611355713}{406252280}a^{3}+\frac{3038067691}{406252280}a^{2}+\frac{3188721163}{812504560}a-\frac{420118063}{812504560}$, $\frac{73662651}{3250018240}a^{17}+\frac{4129697}{406252280}a^{16}-\frac{1908169}{29018020}a^{15}-\frac{2553975}{81250456}a^{14}+\frac{215871057}{812504560}a^{13}+\frac{49154573}{406252280}a^{12}-\frac{4596327087}{1625009120}a^{11}-\frac{14972799}{11607208}a^{10}-\frac{1833181403}{203126140}a^{9}-\frac{311935955}{81250456}a^{8}+\frac{1757877113}{116072080}a^{7}+\frac{92093803}{11607208}a^{6}+\frac{171859547739}{3250018240}a^{5}+\frac{5073613851}{203126140}a^{4}-\frac{22006982589}{406252280}a^{3}-\frac{261540430}{10156307}a^{2}+\frac{1776477343}{162500912}a+\frac{284837926}{50781535}$, $\frac{16007861}{9750054720}a^{17}+\frac{53359}{232144160}a^{16}-\frac{1113411}{203126140}a^{15}-\frac{9733}{81250456}a^{14}+\frac{16514429}{812504560}a^{13}+\frac{35908}{7254505}a^{12}-\frac{348871179}{1625009120}a^{11}-\frac{795233}{23214416}a^{10}-\frac{349835533}{609378420}a^{9}-\frac{7007087}{81250456}a^{8}+\frac{171156941}{116072080}a^{7}-\frac{3601173}{5803604}a^{6}+\frac{13386213063}{3250018240}a^{5}-\frac{68728981}{232144160}a^{4}-\frac{1174497073}{406252280}a^{3}+\frac{15540387}{20312614}a^{2}+\frac{107726953}{487502736}a-\frac{23724047}{406252280}$, $\frac{36986289}{3250018240}a^{17}+\frac{17137059}{3250018240}a^{16}-\frac{8546551}{203126140}a^{15}-\frac{103062}{10156307}a^{14}+\frac{127751051}{812504560}a^{13}+\frac{39186009}{812504560}a^{12}-\frac{2468865989}{1625009120}a^{11}-\frac{27914733}{46428832}a^{10}-\frac{696856151}{203126140}a^{9}-\frac{110424333}{40625228}a^{8}+\frac{1337342251}{116072080}a^{7}+\frac{29779061}{23214416}a^{6}+\frac{14029353677}{650003648}a^{5}+\frac{49566472099}{3250018240}a^{4}-\frac{20276577623}{406252280}a^{3}+\frac{35500501}{81250456}a^{2}+\frac{17599975537}{812504560}a-\frac{5908840421}{812504560}$, $\frac{644108321}{9750054720}a^{17}-\frac{142763493}{3250018240}a^{16}-\frac{41370057}{203126140}a^{15}+\frac{49423923}{406252280}a^{14}+\frac{92594883}{116072080}a^{13}-\frac{408050221}{812504560}a^{12}-\frac{13592326303}{1625009120}a^{11}+\frac{1260211443}{232144160}a^{10}-\frac{3796031854}{152344605}a^{9}+\frac{1054324929}{58036040}a^{8}+\frac{5826346397}{116072080}a^{7}-\frac{3078024289}{116072080}a^{6}+\frac{494749023659}{3250018240}a^{5}-\frac{340892787357}{3250018240}a^{4}-\frac{10880239473}{58036040}a^{3}+\frac{5015677381}{58036040}a^{2}+\frac{88352873729}{2437513680}a-\frac{425839439}{23214416}$, $\frac{175223353}{696432480}a^{17}-\frac{59623073}{406252280}a^{16}-\frac{276218753}{406252280}a^{15}+\frac{165144199}{406252280}a^{14}+\frac{142541372}{50781535}a^{13}-\frac{675725759}{406252280}a^{12}-\frac{25057691663}{812504560}a^{11}+\frac{7351525137}{406252280}a^{10}-\frac{129682197119}{1218756840}a^{9}+\frac{24727545019}{406252280}a^{8}+\frac{4284678771}{29018020}a^{7}-\frac{5249215301}{58036040}a^{6}+\frac{144193137407}{232144160}a^{5}-\frac{72446680221}{203126140}a^{4}-\frac{95091406311}{203126140}a^{3}+\frac{15163041162}{50781535}a^{2}+\frac{45823743739}{1218756840}a-\frac{394930645}{10156307}$, $\frac{69643877}{4875027360}a^{17}+\frac{21638097}{1625009120}a^{16}-\frac{15894999}{406252280}a^{15}-\frac{734131}{20312614}a^{14}+\frac{16403339}{101563070}a^{13}+\frac{61413567}{406252280}a^{12}-\frac{1428603509}{812504560}a^{11}-\frac{38055851}{23214416}a^{10}-\frac{7280344957}{1218756840}a^{9}-\frac{226933927}{40625228}a^{8}+\frac{248530893}{29018020}a^{7}+\frac{89665359}{11607208}a^{6}+\frac{11247228199}{325001824}a^{5}+\frac{51294387497}{1625009120}a^{4}-\frac{5774643633}{203126140}a^{3}-\frac{267825157}{10156307}a^{2}+\frac{4017218861}{1218756840}a+\frac{2010338917}{406252280}$, $\frac{4717115741}{9750054720}a^{17}+\frac{497584709}{3250018240}a^{16}-\frac{578096599}{406252280}a^{15}-\frac{17841343}{40625228}a^{14}+\frac{4636627291}{812504560}a^{13}+\frac{1429673279}{812504560}a^{12}-\frac{98484425623}{1625009120}a^{11}-\frac{879415683}{46428832}a^{10}-\frac{232028027627}{1218756840}a^{9}-\frac{2525356587}{40625228}a^{8}+\frac{38630277807}{116072080}a^{7}+\frac{2415525859}{23214416}a^{6}+\frac{3678609097239}{3250018240}a^{5}+\frac{1169342813749}{3250018240}a^{4}-\frac{481734770511}{406252280}a^{3}-\frac{28573393715}{81250456}a^{2}+\frac{623499309989}{2437513680}a+\frac{53925472229}{812504560}$
| 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: | \( 11538528860.450172 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{6}\cdot(2\pi)^{6}\cdot 11538528860.450172 \cdot 1}{2\cdot\sqrt{2534788041928904776561145413632}}\cr\approx \mathstrut & 14.2695015939181 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^16 + 12*x^14 - 126*x^12 - 386*x^10 + 708*x^8 + 2277*x^6 - 2571*x^4 + 796*x^2 - 108)
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^16 + 12*x^14 - 126*x^12 - 386*x^10 + 708*x^8 + 2277*x^6 - 2571*x^4 + 796*x^2 - 108, 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^16 + 12*x^14 - 126*x^12 - 386*x^10 + 708*x^8 + 2277*x^6 - 2571*x^4 + 796*x^2 - 108);
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^16 + 12*x^14 - 126*x^12 - 386*x^10 + 708*x^8 + 2277*x^6 - 2571*x^4 + 796*x^2 - 108);
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 36 |
| The 9 conjugacy class representatives for $S_3^2$ |
| Character table for $S_3^2$ |
Intermediate fields
| \(\Q(\sqrt{3}) \), 3.3.33708.1 x3, 3.1.8427.1, 6.2.13634751168.1, 6.6.13634751168.1, 9.3.114900048092736.1 x3 |
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
| Galois closure: | data not computed |
| Degree 6 sibling: | 6.0.1617984.1 |
| Degree 9 sibling: | 9.3.114900048092736.1 |
| Degree 12 sibling: | deg 12 |
| Degree 18 siblings: | 18.0.39606063155139137133767897088.1, deg 18 |
| Minimal sibling: | 6.0.1617984.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | R | ${\href{/padicField/5.2.0.1}{2} }^{9}$ | ${\href{/padicField/7.6.0.1}{6} }^{3}$ | ${\href{/padicField/11.6.0.1}{6} }^{2}{,}\,{\href{/padicField/11.3.0.1}{3} }^{2}$ | ${\href{/padicField/13.3.0.1}{3} }^{6}$ | ${\href{/padicField/17.2.0.1}{2} }^{9}$ | ${\href{/padicField/19.6.0.1}{6} }^{3}$ | ${\href{/padicField/23.6.0.1}{6} }^{2}{,}\,{\href{/padicField/23.3.0.1}{3} }^{2}$ | ${\href{/padicField/29.2.0.1}{2} }^{9}$ | ${\href{/padicField/31.6.0.1}{6} }^{3}$ | ${\href{/padicField/37.3.0.1}{3} }^{6}$ | ${\href{/padicField/41.2.0.1}{2} }^{9}$ | ${\href{/padicField/43.6.0.1}{6} }^{3}$ | ${\href{/padicField/47.6.0.1}{6} }^{2}{,}\,{\href{/padicField/47.3.0.1}{3} }^{2}$ | R | ${\href{/padicField/59.6.0.1}{6} }^{2}{,}\,{\href{/padicField/59.3.0.1}{3} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.2.2.2 | $x^{2} + 2 x + 6$ | $2$ | $1$ | $2$ | $C_2$ | $[2]$ |
| 2.2.2.2 | $x^{2} + 2 x + 6$ | $2$ | $1$ | $2$ | $C_2$ | $[2]$ | |
| 2.2.2.2 | $x^{2} + 2 x + 6$ | $2$ | $1$ | $2$ | $C_2$ | $[2]$ | |
| 2.4.4.1 | $x^{4} + 6 x^{3} + 17 x^{2} + 24 x + 13$ | $2$ | $2$ | $4$ | $C_2^2$ | $[2]^{2}$ | |
| 2.4.4.1 | $x^{4} + 6 x^{3} + 17 x^{2} + 24 x + 13$ | $2$ | $2$ | $4$ | $C_2^2$ | $[2]^{2}$ | |
| 2.4.4.1 | $x^{4} + 6 x^{3} + 17 x^{2} + 24 x + 13$ | $2$ | $2$ | $4$ | $C_2^2$ | $[2]^{2}$ | |
|
\(3\)
| 3.2.1.1 | $x^{2} + 6$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 3.2.1.1 | $x^{2} + 6$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 3.2.1.1 | $x^{2} + 6$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 3.4.2.1 | $x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 3.4.2.1 | $x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 3.4.2.1 | $x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
|
\(53\)
| 53.6.4.1 | $x^{6} + 147 x^{5} + 7209 x^{4} + 118343 x^{3} + 22209 x^{2} + 381711 x + 6222632$ | $3$ | $2$ | $4$ | $S_3$ | $[\ ]_{3}^{2}$ |
| 53.6.4.1 | $x^{6} + 147 x^{5} + 7209 x^{4} + 118343 x^{3} + 22209 x^{2} + 381711 x + 6222632$ | $3$ | $2$ | $4$ | $S_3$ | $[\ ]_{3}^{2}$ | |
| 53.6.4.1 | $x^{6} + 147 x^{5} + 7209 x^{4} + 118343 x^{3} + 22209 x^{2} + 381711 x + 6222632$ | $3$ | $2$ | $4$ | $S_3$ | $[\ ]_{3}^{2}$ |