Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 10*x^19 + 33*x^18 - 12*x^17 - 160*x^16 + 260*x^15 + 251*x^14 - 799*x^13 - 123*x^12 + 1297*x^11 - 52*x^10 - 1396*x^9 + 42*x^8 + 1031*x^7 + 58*x^6 - 475*x^5 - 69*x^4 + 110*x^3 + 20*x^2 - 7*x - 1)
gp: K = bnfinit(y^20 - 10*y^19 + 33*y^18 - 12*y^17 - 160*y^16 + 260*y^15 + 251*y^14 - 799*y^13 - 123*y^12 + 1297*y^11 - 52*y^10 - 1396*y^9 + 42*y^8 + 1031*y^7 + 58*y^6 - 475*y^5 - 69*y^4 + 110*y^3 + 20*y^2 - 7*y - 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 - 10*x^19 + 33*x^18 - 12*x^17 - 160*x^16 + 260*x^15 + 251*x^14 - 799*x^13 - 123*x^12 + 1297*x^11 - 52*x^10 - 1396*x^9 + 42*x^8 + 1031*x^7 + 58*x^6 - 475*x^5 - 69*x^4 + 110*x^3 + 20*x^2 - 7*x - 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^20 - 10*x^19 + 33*x^18 - 12*x^17 - 160*x^16 + 260*x^15 + 251*x^14 - 799*x^13 - 123*x^12 + 1297*x^11 - 52*x^10 - 1396*x^9 + 42*x^8 + 1031*x^7 + 58*x^6 - 475*x^5 - 69*x^4 + 110*x^3 + 20*x^2 - 7*x - 1)
\( x^{20} - 10 x^{19} + 33 x^{18} - 12 x^{17} - 160 x^{16} + 260 x^{15} + 251 x^{14} - 799 x^{13} + \cdots - 1 \)
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: | $[8, 6]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(22993730893173532944527721\)
\(\medspace = 3^{6}\cdot 61\cdot 11119^{4}\cdot 33829\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(18.54\) | 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: | $3^{1/2}61^{1/2}11119^{1/2}33829^{1/2}\approx 262363.2427246622$ | ||
| Ramified primes: |
\(3\), \(61\), \(11119\), \(33829\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{2063569}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{11}a^{18}+\frac{2}{11}a^{17}-\frac{3}{11}a^{16}-\frac{3}{11}a^{15}-\frac{5}{11}a^{14}-\frac{5}{11}a^{13}-\frac{4}{11}a^{12}+\frac{3}{11}a^{11}-\frac{1}{11}a^{10}+\frac{5}{11}a^{9}+\frac{2}{11}a^{8}-\frac{1}{11}a^{6}-\frac{4}{11}a^{5}+\frac{4}{11}a^{4}-\frac{1}{11}a^{2}-\frac{1}{11}a+\frac{2}{11}$, $\frac{1}{11}a^{19}+\frac{4}{11}a^{17}+\frac{3}{11}a^{16}+\frac{1}{11}a^{15}+\frac{5}{11}a^{14}-\frac{5}{11}a^{13}+\frac{4}{11}a^{11}-\frac{4}{11}a^{10}+\frac{3}{11}a^{9}-\frac{4}{11}a^{8}-\frac{1}{11}a^{7}-\frac{2}{11}a^{6}+\frac{1}{11}a^{5}+\frac{3}{11}a^{4}-\frac{1}{11}a^{3}+\frac{1}{11}a^{2}+\frac{4}{11}a-\frac{4}{11}$
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: | $13$ | 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: |
$\frac{1}{11}a^{18}-\frac{9}{11}a^{17}+\frac{30}{11}a^{16}-\frac{36}{11}a^{15}-\frac{38}{11}a^{14}+\frac{182}{11}a^{13}-\frac{191}{11}a^{12}-\frac{206}{11}a^{11}+\frac{648}{11}a^{10}-\frac{17}{11}a^{9}-\frac{1065}{11}a^{8}+18a^{7}+\frac{1154}{11}a^{6}-\frac{59}{11}a^{5}-\frac{799}{11}a^{4}-11a^{3}+\frac{263}{11}a^{2}+\frac{65}{11}a-\frac{20}{11}$, $a^{2}-a-1$, $\frac{3}{11}a^{18}-\frac{27}{11}a^{17}+\frac{68}{11}a^{16}+\frac{68}{11}a^{15}-\frac{521}{11}a^{14}+\frac{315}{11}a^{13}+\frac{1418}{11}a^{12}-\frac{1553}{11}a^{11}-\frac{2214}{11}a^{10}+\frac{2831}{11}a^{9}+\frac{2569}{11}a^{8}-259a^{7}-\frac{2302}{11}a^{6}+\frac{1605}{11}a^{5}+\frac{1332}{11}a^{4}-40a^{3}-\frac{366}{11}a^{2}+\frac{63}{11}a+\frac{28}{11}$, $\frac{8}{11}a^{18}-\frac{72}{11}a^{17}+\frac{196}{11}a^{16}+\frac{64}{11}a^{15}-\frac{1118}{11}a^{14}+\frac{994}{11}a^{13}+\frac{2443}{11}a^{12}-\frac{3452}{11}a^{11}-\frac{3209}{11}a^{10}+\frac{5408}{11}a^{9}+\frac{3437}{11}a^{8}-452a^{7}-\frac{3088}{11}a^{6}+\frac{2663}{11}a^{5}+\frac{1792}{11}a^{4}-61a^{3}-\frac{481}{11}a^{2}+\frac{58}{11}a+\frac{27}{11}$, $a^{18}-9a^{17}+25a^{16}+4a^{15}-130a^{14}+126a^{13}+266a^{12}-400a^{11}-342a^{10}+610a^{9}+367a^{8}-558a^{7}-327a^{6}+295a^{5}+190a^{4}-72a^{3}-50a^{2}+4a+2$, $\frac{6}{11}a^{18}-\frac{54}{11}a^{17}+\frac{147}{11}a^{16}+\frac{48}{11}a^{15}-\frac{833}{11}a^{14}+\frac{707}{11}a^{13}+\frac{1901}{11}a^{12}-\frac{2501}{11}a^{11}-\frac{2756}{11}a^{10}+\frac{4078}{11}a^{9}+\frac{3213}{11}a^{8}-351a^{7}-\frac{3009}{11}a^{6}+\frac{2011}{11}a^{5}+\frac{1850}{11}a^{4}-38a^{3}-\frac{523}{11}a^{2}-\frac{6}{11}a+\frac{23}{11}$, $\frac{14}{11}a^{19}-\frac{126}{11}a^{18}+\frac{343}{11}a^{17}+\frac{112}{11}a^{16}-\frac{1951}{11}a^{15}+\frac{1701}{11}a^{14}+\frac{4344}{11}a^{13}-\frac{5953}{11}a^{12}-\frac{5965}{11}a^{11}+\frac{9486}{11}a^{10}+\frac{6650}{11}a^{9}-803a^{8}-\frac{6097}{11}a^{7}+\frac{4674}{11}a^{6}+\frac{3631}{11}a^{5}-97a^{4}-\frac{982}{11}a^{3}+\frac{19}{11}a^{2}+\frac{39}{11}a$, $\frac{1}{11}a^{19}-\frac{10}{11}a^{18}+\frac{39}{11}a^{17}-6a^{16}-\frac{2}{11}a^{15}+20a^{14}-\frac{384}{11}a^{13}+\frac{62}{11}a^{12}+\frac{700}{11}a^{11}-\frac{742}{11}a^{10}-\frac{498}{11}a^{9}+\frac{1054}{11}a^{8}+\frac{186}{11}a^{7}-\frac{872}{11}a^{6}+\frac{19}{11}a^{5}+\frac{480}{11}a^{4}-\frac{100}{11}a^{3}-15a^{2}+\frac{36}{11}a+\frac{20}{11}$, $\frac{4}{11}a^{19}-\frac{35}{11}a^{18}+\frac{89}{11}a^{17}+\frac{62}{11}a^{16}-\frac{584}{11}a^{15}+\frac{382}{11}a^{14}+\frac{1541}{11}a^{13}-\frac{1741}{11}a^{12}-\frac{2498}{11}a^{11}+\frac{3341}{11}a^{10}+\frac{3027}{11}a^{9}-\frac{3705}{11}a^{8}-\frac{2908}{11}a^{7}+\frac{2425}{11}a^{6}+\frac{1981}{11}a^{5}-\frac{843}{11}a^{4}-\frac{730}{11}a^{3}+\frac{116}{11}a^{2}+\frac{95}{11}a-\frac{9}{11}$, $\frac{2}{11}a^{19}-a^{18}-\frac{14}{11}a^{17}+\frac{193}{11}a^{16}-\frac{273}{11}a^{15}-\frac{584}{11}a^{14}+\frac{1431}{11}a^{13}+68a^{12}-\frac{3105}{11}a^{11}-\frac{811}{11}a^{10}+\frac{4076}{11}a^{9}+\frac{1268}{11}a^{8}-\frac{3379}{11}a^{7}-\frac{1588}{11}a^{6}+\frac{1443}{11}a^{5}+\frac{1062}{11}a^{4}-\frac{123}{11}a^{3}-\frac{251}{11}a^{2}-\frac{58}{11}a+\frac{3}{11}$, $\frac{4}{11}a^{19}-\frac{30}{11}a^{18}+4a^{17}+\frac{190}{11}a^{16}-\frac{599}{11}a^{15}-\frac{138}{11}a^{14}+\frac{2077}{11}a^{13}-\frac{870}{11}a^{12}-\frac{3682}{11}a^{11}+\frac{2148}{11}a^{10}+\frac{4383}{11}a^{9}-\frac{2177}{11}a^{8}-\frac{3832}{11}a^{7}+95a^{6}+\frac{2192}{11}a^{5}-\frac{42}{11}a^{4}-\frac{631}{11}a^{3}-\frac{131}{11}a^{2}+\frac{13}{11}a+\frac{12}{11}$, $\frac{4}{11}a^{19}-\frac{46}{11}a^{18}+\frac{188}{11}a^{17}-\frac{213}{11}a^{16}-\frac{639}{11}a^{15}+\frac{1900}{11}a^{14}-\frac{54}{11}a^{13}-\frac{4744}{11}a^{12}+\frac{2837}{11}a^{11}+\frac{6487}{11}a^{10}-\frac{5289}{11}a^{9}-\frac{6257}{11}a^{8}+\frac{5045}{11}a^{7}+\frac{4537}{11}a^{6}-\frac{2870}{11}a^{5}-\frac{2152}{11}a^{4}+\frac{920}{11}a^{3}+\frac{501}{11}a^{2}-\frac{147}{11}a-\frac{20}{11}$, $\frac{8}{11}a^{19}-\frac{76}{11}a^{18}+\frac{232}{11}a^{17}-\frac{34}{11}a^{16}-\frac{1150}{11}a^{15}+\frac{1553}{11}a^{14}+\frac{1946}{11}a^{13}-\frac{4679}{11}a^{12}-\frac{1461}{11}a^{11}+639a^{10}+\frac{579}{11}a^{9}-\frac{6630}{11}a^{8}-\frac{250}{11}a^{7}+\frac{4020}{11}a^{6}+\frac{48}{11}a^{5}-\frac{1457}{11}a^{4}+\frac{168}{11}a^{3}+\frac{304}{11}a^{2}-\frac{90}{11}a-\frac{19}{11}$
| 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: | \( 99704.5727669 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{8}\cdot(2\pi)^{6}\cdot 99704.5727669 \cdot 1}{2\cdot\sqrt{22993730893173532944527721}}\cr\approx \mathstrut & 0.163756872178 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^20 - 10*x^19 + 33*x^18 - 12*x^17 - 160*x^16 + 260*x^15 + 251*x^14 - 799*x^13 - 123*x^12 + 1297*x^11 - 52*x^10 - 1396*x^9 + 42*x^8 + 1031*x^7 + 58*x^6 - 475*x^5 - 69*x^4 + 110*x^3 + 20*x^2 - 7*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 - 10*x^19 + 33*x^18 - 12*x^17 - 160*x^16 + 260*x^15 + 251*x^14 - 799*x^13 - 123*x^12 + 1297*x^11 - 52*x^10 - 1396*x^9 + 42*x^8 + 1031*x^7 + 58*x^6 - 475*x^5 - 69*x^4 + 110*x^3 + 20*x^2 - 7*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 - 10*x^19 + 33*x^18 - 12*x^17 - 160*x^16 + 260*x^15 + 251*x^14 - 799*x^13 - 123*x^12 + 1297*x^11 - 52*x^10 - 1396*x^9 + 42*x^8 + 1031*x^7 + 58*x^6 - 475*x^5 - 69*x^4 + 110*x^3 + 20*x^2 - 7*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 - 10*x^19 + 33*x^18 - 12*x^17 - 160*x^16 + 260*x^15 + 251*x^14 - 799*x^13 - 123*x^12 + 1297*x^11 - 52*x^10 - 1396*x^9 + 42*x^8 + 1031*x^7 + 58*x^6 - 475*x^5 - 69*x^4 + 110*x^3 + 20*x^2 - 7*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
$C_2^{10}.C_2\wr S_5$ (as 20T1015):
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 3932160 |
| The 506 conjugacy class representatives for $C_2^{10}.C_2\wr S_5$ |
| Character table for $C_2^{10}.C_2\wr S_5$ |
Intermediate fields
| 5.3.11119.1, 10.4.3338068347.1 |
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 20 siblings: | data not computed |
| Degree 40 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 | ${\href{/padicField/2.10.0.1}{10} }^{2}$ | R | ${\href{/padicField/5.10.0.1}{10} }^{2}$ | ${\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.4.0.1}{4} }{,}\,{\href{/padicField/7.3.0.1}{3} }^{2}{,}\,{\href{/padicField/7.2.0.1}{2} }^{2}$ | ${\href{/padicField/11.12.0.1}{12} }{,}\,{\href{/padicField/11.2.0.1}{2} }^{4}$ | ${\href{/padicField/13.8.0.1}{8} }{,}\,{\href{/padicField/13.4.0.1}{4} }^{3}$ | $20$ | ${\href{/padicField/19.4.0.1}{4} }^{2}{,}\,{\href{/padicField/19.3.0.1}{3} }^{4}$ | ${\href{/padicField/23.10.0.1}{10} }^{2}$ | $16{,}\,{\href{/padicField/29.2.0.1}{2} }{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.4.0.1}{4} }^{3}{,}\,{\href{/padicField/31.2.0.1}{2} }^{4}$ | ${\href{/padicField/37.12.0.1}{12} }{,}\,{\href{/padicField/37.4.0.1}{4} }^{2}$ | ${\href{/padicField/41.10.0.1}{10} }^{2}$ | ${\href{/padicField/43.4.0.1}{4} }{,}\,{\href{/padicField/43.3.0.1}{3} }^{4}{,}\,{\href{/padicField/43.2.0.1}{2} }^{2}$ | ${\href{/padicField/47.4.0.1}{4} }^{4}{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}$ | ${\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.3.0.1}{3} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }^{4}$ | ${\href{/padicField/59.8.0.1}{8} }^{2}{,}\,{\href{/padicField/59.4.0.1}{4} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(3\)
| 3.4.0.1 | $x^{4} + 2 x^{3} + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |
| 3.4.0.1 | $x^{4} + 2 x^{3} + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 3.12.6.2 | $x^{12} + 22 x^{10} + 177 x^{8} + 4 x^{7} + 644 x^{6} - 100 x^{5} + 876 x^{4} - 224 x^{3} + 1076 x^{2} + 344 x + 112$ | $2$ | $6$ | $6$ | $C_6\times C_2$ | $[\ ]_{2}^{6}$ | |
|
\(61\)
| $\Q_{61}$ | $x + 59$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{61}$ | $x + 59$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 61.2.1.2 | $x^{2} + 122$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 61.2.0.1 | $x^{2} + 60 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 61.2.0.1 | $x^{2} + 60 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 61.4.0.1 | $x^{4} + 3 x^{2} + 40 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 61.8.0.1 | $x^{8} + 57 x^{3} + x^{2} + 56 x + 2$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | |
|
\(11119\)
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $4$ | $2$ | $2$ | $2$ | ||||
| Deg $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | ||
| Deg $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | ||
|
\(33829\)
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 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 $8$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ |