Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - x^21 - x^19 + 3*x^18 - 3*x^17 - x^16 + 5*x^14 - 3*x^13 - 2*x^12 + x^11 + 6*x^10 - 4*x^9 - 2*x^8 - x^7 + 5*x^6 - 2*x^5 - x^4 - 2*x^3 + x^2 + x + 1)
gp: K = bnfinit(y^22 - y^21 - y^19 + 3*y^18 - 3*y^17 - y^16 + 5*y^14 - 3*y^13 - 2*y^12 + y^11 + 6*y^10 - 4*y^9 - 2*y^8 - y^7 + 5*y^6 - 2*y^5 - y^4 - 2*y^3 + y^2 + y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^22 - x^21 - x^19 + 3*x^18 - 3*x^17 - x^16 + 5*x^14 - 3*x^13 - 2*x^12 + x^11 + 6*x^10 - 4*x^9 - 2*x^8 - x^7 + 5*x^6 - 2*x^5 - x^4 - 2*x^3 + x^2 + x + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^22 - x^21 - x^19 + 3*x^18 - 3*x^17 - x^16 + 5*x^14 - 3*x^13 - 2*x^12 + x^11 + 6*x^10 - 4*x^9 - 2*x^8 - x^7 + 5*x^6 - 2*x^5 - x^4 - 2*x^3 + x^2 + x + 1)
\( x^{22} - x^{21} - x^{19} + 3 x^{18} - 3 x^{17} - x^{16} + 5 x^{14} - 3 x^{13} - 2 x^{12} + x^{11} + \cdots + 1 \)
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: |
\(-6293360649840402636051867\)
\(\medspace = -\,3^{11}\cdot 64661^{2}\cdot 92179^{2}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(13.40\) | 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: | $3^{1/2}64661^{1/2}92179^{1/2}\approx 133720.45078072388$ | ||
| Ramified primes: |
\(3\), \(64661\), \(92179\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-3}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $\frac{1}{5981}a^{21}-\frac{1485}{5981}a^{20}+\frac{2732}{5981}a^{19}+\frac{829}{5981}a^{18}+\frac{1853}{5981}a^{17}+\frac{1405}{5981}a^{16}+\frac{2348}{5981}a^{15}+\frac{2491}{5981}a^{14}-\frac{381}{5981}a^{13}-\frac{2794}{5981}a^{12}+\frac{1461}{5981}a^{11}+\frac{2980}{5981}a^{10}-\frac{2355}{5981}a^{9}+\frac{1912}{5981}a^{8}-\frac{2416}{5981}a^{7}+\frac{2724}{5981}a^{6}+\frac{745}{5981}a^{5}+\frac{903}{5981}a^{4}-\frac{309}{5981}a^{3}-\frac{1983}{5981}a^{2}+\frac{121}{5981}a-\frac{133}{5981}$
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: |
\( \frac{975}{5981} a^{21} - \frac{473}{5981} a^{20} + \frac{2155}{5981} a^{19} - \frac{5141}{5981} a^{18} + \frac{6394}{5981} a^{17} - \frac{5755}{5981} a^{16} + \frac{4558}{5981} a^{15} - \frac{11523}{5981} a^{14} + \frac{11309}{5981} a^{13} - \frac{2795}{5981} a^{12} + \frac{997}{5981} a^{11} - \frac{7247}{5981} a^{10} + \frac{12541}{5981} a^{9} - \frac{1872}{5981} a^{8} + \frac{914}{5981} a^{7} - \frac{11626}{5981} a^{6} + \frac{14636}{5981} a^{5} - \frac{4763}{5981} a^{4} + \frac{3756}{5981} a^{3} - \frac{7543}{5981} a^{2} + \frac{4336}{5981} a + \frac{1907}{5981} \)
(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: |
$\frac{502}{5981}a^{21}+\frac{2155}{5981}a^{20}-\frac{4166}{5981}a^{19}+\frac{3469}{5981}a^{18}-\frac{2830}{5981}a^{17}+\frac{5533}{5981}a^{16}-\frac{11523}{5981}a^{15}+\frac{6434}{5981}a^{14}+\frac{130}{5981}a^{13}+\frac{2947}{5981}a^{12}-\frac{8222}{5981}a^{11}+\frac{6691}{5981}a^{10}+\frac{2028}{5981}a^{9}+\frac{2864}{5981}a^{8}-\frac{10651}{5981}a^{7}+\frac{9761}{5981}a^{6}-\frac{2813}{5981}a^{5}+\frac{4731}{5981}a^{4}-\frac{5593}{5981}a^{3}+\frac{3361}{5981}a^{2}+\frac{932}{5981}a-\frac{975}{5981}$, $\frac{2234}{5981}a^{21}-\frac{4016}{5981}a^{20}+\frac{2668}{5981}a^{19}-\frac{2124}{5981}a^{18}+\frac{6731}{5981}a^{17}-\frac{13217}{5981}a^{16}+\frac{6076}{5981}a^{15}+\frac{2564}{5981}a^{14}+\frac{4129}{5981}a^{13}-\frac{15575}{5981}a^{12}+\frac{10210}{5981}a^{11}+\frac{6448}{5981}a^{10}+\frac{2210}{5981}a^{9}-\frac{16969}{5981}a^{8}+\frac{15461}{5981}a^{7}-\frac{3242}{5981}a^{6}+\frac{1612}{5981}a^{5}-\frac{10257}{5981}a^{4}+\frac{9471}{5981}a^{3}-\frac{4082}{5981}a^{2}+\frac{1169}{5981}a+\frac{1928}{5981}$, $\frac{4899}{5981}a^{21}-\frac{8100}{5981}a^{20}+\frac{4571}{5981}a^{19}-\frac{5809}{5981}a^{18}+\frac{16632}{5981}a^{17}-\frac{24960}{5981}a^{16}+\frac{7370}{5981}a^{15}+\frac{2169}{5981}a^{14}+\frac{17496}{5981}a^{13}-\frac{27202}{5981}a^{12}+\frac{4163}{5981}a^{11}+\frac{11361}{5981}a^{10}+\frac{18147}{5981}a^{9}-\frac{35244}{5981}a^{8}+\frac{12377}{5981}a^{7}+\frac{1265}{5981}a^{6}+\frac{19288}{5981}a^{5}-\frac{26067}{5981}a^{4}+\frac{5383}{5981}a^{3}-\frac{1573}{5981}a^{2}+\frac{660}{5981}a+\frac{362}{5981}$, $\frac{4857}{5981}a^{21}-\frac{5540}{5981}a^{20}-\frac{2515}{5981}a^{19}+\frac{1240}{5981}a^{18}+\frac{10578}{5981}a^{17}-\frac{12198}{5981}a^{16}-\frac{13493}{5981}a^{15}+\frac{17167}{5981}a^{14}+\frac{15555}{5981}a^{13}-\frac{17512}{5981}a^{12}-\frac{15332}{5981}a^{11}+\frac{23764}{5981}a^{10}+\frac{21361}{5981}a^{9}-\frac{25833}{5981}a^{8}-\frac{17733}{5981}a^{7}+\frac{18439}{5981}a^{6}+\frac{11922}{5981}a^{5}-\frac{10164}{5981}a^{4}-\frac{17525}{5981}a^{3}+\frac{3960}{5981}a^{2}+\frac{1559}{5981}a+\frac{5948}{5981}$, $\frac{270}{5981}a^{21}-\frac{223}{5981}a^{20}+\frac{1977}{5981}a^{19}-\frac{3448}{5981}a^{18}+\frac{3887}{5981}a^{17}-\frac{3434}{5981}a^{16}+\frac{5955}{5981}a^{15}-\frac{9264}{5981}a^{14}+\frac{4788}{5981}a^{13}-\frac{774}{5981}a^{12}+\frac{5705}{5981}a^{11}-\frac{8816}{5981}a^{10}+\frac{4117}{5981}a^{9}+\frac{1874}{5981}a^{8}+\frac{5590}{5981}a^{7}-\frac{12145}{5981}a^{6}+\frac{3777}{5981}a^{5}-\frac{1411}{5981}a^{4}+\frac{6285}{5981}a^{3}-\frac{9082}{5981}a^{2}+\frac{2765}{5981}a-\frac{24}{5981}$, $\frac{623}{5981}a^{21}-\frac{4081}{5981}a^{20}+\frac{3432}{5981}a^{19}+\frac{2101}{5981}a^{18}+\frac{86}{5981}a^{17}-\frac{9873}{5981}a^{16}+\frac{9421}{5981}a^{15}+\frac{8795}{5981}a^{14}-\frac{10085}{5981}a^{13}-\frac{18134}{5981}a^{12}+\frac{19034}{5981}a^{11}+\frac{8411}{5981}a^{10}-\frac{13782}{5981}a^{9}-\frac{16986}{5981}a^{8}+\frac{25968}{5981}a^{7}+\frac{10410}{5981}a^{6}-\frac{14345}{5981}a^{5}-\frac{11607}{5981}a^{4}+\frac{10847}{5981}a^{3}+\frac{2658}{5981}a^{2}-\frac{2370}{5981}a-\frac{5106}{5981}$, $\frac{1858}{5981}a^{21}-\frac{1889}{5981}a^{20}-\frac{1813}{5981}a^{19}+\frac{3165}{5981}a^{18}+\frac{3799}{5981}a^{17}-\frac{9188}{5981}a^{16}-\frac{3546}{5981}a^{15}+\frac{10946}{5981}a^{14}+\frac{3841}{5981}a^{13}-\frac{17687}{5981}a^{12}-\frac{836}{5981}a^{11}+\frac{22358}{5981}a^{10}-\frac{3479}{5981}a^{9}-\frac{18161}{5981}a^{8}+\frac{2803}{5981}a^{7}+\frac{19209}{5981}a^{6}-\frac{3382}{5981}a^{5}-\frac{14849}{5981}a^{4}+\frac{6035}{5981}a^{3}+\frac{5863}{5981}a^{2}-\frac{2460}{5981}a-\frac{1893}{5981}$, $\frac{1431}{5981}a^{21}-\frac{1780}{5981}a^{20}+\frac{3899}{5981}a^{19}-\frac{3920}{5981}a^{18}+\frac{8041}{5981}a^{17}-\frac{11023}{5981}a^{16}+\frac{10628}{5981}a^{15}-\frac{12017}{5981}a^{14}+\frac{11022}{5981}a^{13}-\frac{14868}{5981}a^{12}+\frac{9303}{5981}a^{11}-\frac{6054}{5981}a^{10}+\frac{9260}{5981}a^{9}-\frac{9207}{5981}a^{8}+\frac{11684}{5981}a^{7}-\frac{7549}{5981}a^{6}+\frac{13439}{5981}a^{5}-\frac{17646}{5981}a^{4}+\frac{12377}{5981}a^{3}-\frac{8660}{5981}a^{2}+\frac{5683}{5981}a-\frac{4912}{5981}$, $\frac{1808}{5981}a^{21}+\frac{589}{5981}a^{20}-\frac{850}{5981}a^{19}-\frac{2399}{5981}a^{18}+\frac{864}{5981}a^{17}+\frac{4296}{5981}a^{16}-\frac{7307}{5981}a^{15}-\frac{5946}{5981}a^{14}+\frac{4948}{5981}a^{13}+\frac{14355}{5981}a^{12}-\frac{8095}{5981}a^{11}-\frac{13003}{5981}a^{10}+\frac{12594}{5981}a^{9}+\frac{17821}{5981}a^{8}-\frac{13960}{5981}a^{7}-\frac{15314}{5981}a^{6}+\frac{7216}{5981}a^{5}+\frac{17754}{5981}a^{4}-\frac{8420}{5981}a^{3}-\frac{8626}{5981}a^{2}-\frac{2529}{5981}a+\frac{4757}{5981}$, $\frac{3992}{5981}a^{21}-\frac{6930}{5981}a^{20}+\frac{8762}{5981}a^{19}-\frac{10087}{5981}a^{18}+\frac{16622}{5981}a^{17}-\frac{25342}{5981}a^{16}+\frac{18932}{5981}a^{15}-\frac{14293}{5981}a^{14}+\frac{16165}{5981}a^{13}-\frac{23007}{5981}a^{12}+\frac{18780}{5981}a^{11}-\frac{6030}{5981}a^{10}+\frac{12934}{5981}a^{9}-\frac{22976}{5981}a^{8}+\frac{26605}{5981}a^{7}-\frac{17193}{5981}a^{6}+\frac{19426}{5981}a^{5}-\frac{19710}{5981}a^{4}+\frac{22482}{5981}a^{3}-\frac{15235}{5981}a^{2}+\frac{4552}{5981}a-\frac{4608}{5981}$
| 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: | \( 4402.38521796 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{11}\cdot 4402.38521796 \cdot 1}{6\cdot\sqrt{6293360649840402636051867}}\cr\approx \mathstrut & 0.176227643111 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^22 - x^21 - x^19 + 3*x^18 - 3*x^17 - x^16 + 5*x^14 - 3*x^13 - 2*x^12 + x^11 + 6*x^10 - 4*x^9 - 2*x^8 - x^7 + 5*x^6 - 2*x^5 - x^4 - 2*x^3 + x^2 + 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 - x^21 - x^19 + 3*x^18 - 3*x^17 - x^16 + 5*x^14 - 3*x^13 - 2*x^12 + x^11 + 6*x^10 - 4*x^9 - 2*x^8 - x^7 + 5*x^6 - 2*x^5 - x^4 - 2*x^3 + x^2 + 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 - x^21 - x^19 + 3*x^18 - 3*x^17 - x^16 + 5*x^14 - 3*x^13 - 2*x^12 + x^11 + 6*x^10 - 4*x^9 - 2*x^8 - x^7 + 5*x^6 - 2*x^5 - x^4 - 2*x^3 + x^2 + 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 - x^21 - x^19 + 3*x^18 - 3*x^17 - x^16 + 5*x^14 - 3*x^13 - 2*x^12 + x^11 + 6*x^10 - 4*x^9 - 2*x^8 - x^7 + 5*x^6 - 2*x^5 - x^4 - 2*x^3 + x^2 + 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
$C_2\times S_{11}$ (as 22T47):
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 79833600 |
| The 112 conjugacy class representatives for $C_2\times S_{11}$ |
| Character table for $C_2\times S_{11}$ |
Intermediate fields
| \(\Q(\sqrt{-3}) \), 11.1.5960386319.1 |
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 sibling: | 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 | $22$ | R | ${\href{/padicField/5.10.0.1}{10} }{,}\,{\href{/padicField/5.6.0.1}{6} }^{2}$ | ${\href{/padicField/7.11.0.1}{11} }^{2}$ | ${\href{/padicField/11.10.0.1}{10} }{,}\,{\href{/padicField/11.4.0.1}{4} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }^{2}$ | ${\href{/padicField/13.5.0.1}{5} }^{2}{,}\,{\href{/padicField/13.4.0.1}{4} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }^{2}$ | $22$ | ${\href{/padicField/19.6.0.1}{6} }^{2}{,}\,{\href{/padicField/19.3.0.1}{3} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }^{2}$ | $18{,}\,{\href{/padicField/23.2.0.1}{2} }^{2}$ | ${\href{/padicField/29.8.0.1}{8} }^{2}{,}\,{\href{/padicField/29.6.0.1}{6} }$ | ${\href{/padicField/31.5.0.1}{5} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }^{4}{,}\,{\href{/padicField/31.1.0.1}{1} }^{4}$ | ${\href{/padicField/37.10.0.1}{10} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.14.0.1}{14} }{,}\,{\href{/padicField/41.2.0.1}{2} }^{4}$ | ${\href{/padicField/43.5.0.1}{5} }^{2}{,}\,{\href{/padicField/43.4.0.1}{4} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }^{4}$ | ${\href{/padicField/47.8.0.1}{8} }^{2}{,}\,{\href{/padicField/47.6.0.1}{6} }$ | ${\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.4.0.1}{4} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }^{4}$ | $22$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(3\)
| 3.6.3.2 | $x^{6} + 13 x^{4} + 2 x^{3} + 31 x^{2} - 14 x + 4$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ |
| 3.8.4.1 | $x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
| 3.8.4.1 | $x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
|
\(64661\)
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| Deg $4$ | $2$ | $2$ | $2$ | ||||
| Deg $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | ||
| Deg $10$ | $1$ | $10$ | $0$ | $C_{10}$ | $[\ ]^{10}$ | ||
|
\(92179\)
| 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 $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |