Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - 2*x^21 + 4*x^20 - 2*x^18 + 9*x^17 - 4*x^16 + 6*x^15 + 4*x^14 - 2*x^13 + 10*x^12 + 7*x^10 + 4*x^9 + 2*x^8 + 11*x^7 + x^6 + 3*x^5 + 4*x^4 + 2*x^3 + 3*x^2 - x + 1)
gp: K = bnfinit(y^22 - 2*y^21 + 4*y^20 - 2*y^18 + 9*y^17 - 4*y^16 + 6*y^15 + 4*y^14 - 2*y^13 + 10*y^12 + 7*y^10 + 4*y^9 + 2*y^8 + 11*y^7 + y^6 + 3*y^5 + 4*y^4 + 2*y^3 + 3*y^2 - y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^22 - 2*x^21 + 4*x^20 - 2*x^18 + 9*x^17 - 4*x^16 + 6*x^15 + 4*x^14 - 2*x^13 + 10*x^12 + 7*x^10 + 4*x^9 + 2*x^8 + 11*x^7 + x^6 + 3*x^5 + 4*x^4 + 2*x^3 + 3*x^2 - x + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^22 - 2*x^21 + 4*x^20 - 2*x^18 + 9*x^17 - 4*x^16 + 6*x^15 + 4*x^14 - 2*x^13 + 10*x^12 + 7*x^10 + 4*x^9 + 2*x^8 + 11*x^7 + x^6 + 3*x^5 + 4*x^4 + 2*x^3 + 3*x^2 - x + 1)
\( x^{22} - 2 x^{21} + 4 x^{20} - 2 x^{18} + 9 x^{17} - 4 x^{16} + 6 x^{15} + 4 x^{14} - 2 x^{13} + \cdots + 1 \)
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: |
\(-3101889720909924494411835627\)
\(\medspace = -\,3^{11}\cdot 211441^{2}\cdot 625831^{2}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(17.77\) | 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: | $3^{1/2}211441^{1/2}625831^{1/2}\approx 630062.6932401251$ | ||
| Ramified primes: |
\(3\), \(211441\), \(625831\)
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $\frac{1}{226307}a^{21}+\frac{58004}{226307}a^{20}+\frac{73859}{226307}a^{19}+\frac{47337}{226307}a^{18}+\frac{47189}{226307}a^{17}+\frac{61978}{226307}a^{16}-\frac{17138}{226307}a^{15}+\frac{59829}{226307}a^{14}+\frac{23133}{226307}a^{13}+\frac{78593}{226307}a^{12}-\frac{88947}{226307}a^{11}-\frac{112696}{226307}a^{10}+\frac{59833}{226307}a^{9}+\frac{28850}{226307}a^{8}-\frac{67163}{226307}a^{7}+\frac{18038}{226307}a^{6}+\frac{94968}{226307}a^{5}-\frac{51183}{226307}a^{4}+\frac{439}{226307}a^{3}-\frac{108055}{226307}a^{2}-\frac{39655}{226307}a-\frac{43583}{226307}$
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: |
\( \frac{174996}{226307} a^{21} - \frac{306194}{226307} a^{20} + \frac{636794}{226307} a^{19} + \frac{44224}{226307} a^{18} - \frac{56186}{226307} a^{17} + \frac{1044341}{226307} a^{16} - \frac{61084}{226307} a^{15} + \frac{873864}{226307} a^{14} + \frac{455466}{226307} a^{13} + \frac{331624}{226307} a^{12} + \frac{931476}{226307} a^{11} + \frac{626913}{226307} a^{10} + \frac{1121234}{226307} a^{9} + \frac{630658}{226307} a^{8} + \frac{676618}{226307} a^{7} + \frac{1405654}{226307} a^{6} + \frac{844504}{226307} a^{5} + \frac{410792}{226307} a^{4} + \frac{331478}{226307} a^{3} + \frac{567526}{226307} a^{2} + \frac{237775}{226307} a + \frac{147846}{226307} \)
(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: |
$\frac{78461}{226307}a^{21}+\frac{18074}{226307}a^{20}+\frac{7650}{226307}a^{19}+\frac{636794}{226307}a^{18}-\frac{112698}{226307}a^{17}+\frac{649963}{226307}a^{16}+\frac{730497}{226307}a^{15}+\frac{409682}{226307}a^{14}+\frac{1187708}{226307}a^{13}+\frac{298544}{226307}a^{12}+\frac{1116234}{226307}a^{11}+\frac{931476}{226307}a^{10}+\frac{1176140}{226307}a^{9}+\frac{1435078}{226307}a^{8}+\frac{787580}{226307}a^{7}+\frac{1539689}{226307}a^{6}+\frac{1484115}{226307}a^{5}+\frac{1079887}{226307}a^{4}+\frac{724636}{226307}a^{3}+\frac{488400}{226307}a^{2}+\frac{802909}{226307}a+\frac{159314}{226307}$, $\frac{129464}{226307}a^{21}-\frac{115325}{226307}a^{20}+\frac{158212}{226307}a^{19}+\frac{722729}{226307}a^{18}-\frac{559690}{226307}a^{17}+\frac{884028}{226307}a^{16}+\frac{865024}{226307}a^{15}-\frac{334340}{226307}a^{14}+\frac{1301716}{226307}a^{13}+\frac{201432}{226307}a^{12}+\frac{423594}{226307}a^{11}+\frac{1295188}{226307}a^{10}+\frac{636130}{226307}a^{9}+\frac{744593}{226307}a^{8}+\frac{655843}{226307}a^{7}+\frac{1367541}{226307}a^{6}+\frac{1035684}{226307}a^{5}-\frac{86952}{226307}a^{4}+\frac{484253}{226307}a^{3}+\frac{387299}{226307}a^{2}+\frac{331989}{226307}a-\frac{143388}{226307}$, $\frac{129464}{226307}a^{21}-\frac{115325}{226307}a^{20}+\frac{158212}{226307}a^{19}+\frac{722729}{226307}a^{18}-\frac{559690}{226307}a^{17}+\frac{884028}{226307}a^{16}+\frac{865024}{226307}a^{15}-\frac{334340}{226307}a^{14}+\frac{1301716}{226307}a^{13}+\frac{201432}{226307}a^{12}+\frac{423594}{226307}a^{11}+\frac{1295188}{226307}a^{10}+\frac{636130}{226307}a^{9}+\frac{744593}{226307}a^{8}+\frac{655843}{226307}a^{7}+\frac{1367541}{226307}a^{6}+\frac{1035684}{226307}a^{5}-\frac{86952}{226307}a^{4}+\frac{484253}{226307}a^{3}+\frac{613606}{226307}a^{2}+\frac{331989}{226307}a-\frac{143388}{226307}$, $\frac{297946}{226307}a^{21}-\frac{553192}{226307}a^{20}+\frac{1032469}{226307}a^{19}+\frac{191255}{226307}a^{18}-\frac{680116}{226307}a^{17}+\frac{2161672}{226307}a^{16}-\frac{712628}{226307}a^{15}+\frac{966686}{226307}a^{14}+\frac{1110361}{226307}a^{13}-\frac{873154}{226307}a^{12}+\frac{2088829}{226307}a^{11}+\frac{299788}{226307}a^{10}+\frac{1253242}{226307}a^{9}+\frac{1054854}{226307}a^{8}-\frac{203337}{226307}a^{7}+\frac{2500689}{226307}a^{6}+\frac{171518}{226307}a^{5}-\frac{72923}{226307}a^{4}+\frac{445462}{226307}a^{3}+\frac{105097}{226307}a^{2}+\frac{666147}{226307}a-\frac{337472}{226307}$, $\frac{165075}{226307}a^{21}-\frac{265177}{226307}a^{20}+\frac{437414}{226307}a^{19}+\frac{453486}{226307}a^{18}-\frac{667993}{226307}a^{17}+\frac{1263029}{226307}a^{16}+\frac{461071}{226307}a^{15}-\frac{444226}{226307}a^{14}+\frac{1786113}{226307}a^{13}-\frac{214528}{226307}a^{12}+\frac{551056}{226307}a^{11}+\frac{1858884}{226307}a^{10}-\frac{10233}{226307}a^{9}+\frac{1140777}{226307}a^{8}+\frac{752933}{226307}a^{7}+\frac{1233186}{226307}a^{6}+\frac{1235631}{226307}a^{5}-\frac{88187}{226307}a^{4}+\frac{954913}{226307}a^{3}+\frac{791229}{226307}a^{2}+\frac{107157}{226307}a+\frac{288419}{226307}$, $\frac{43798}{226307}a^{21}-\frac{63190}{226307}a^{20}+\frac{44224}{226307}a^{19}+\frac{293806}{226307}a^{18}-\frac{530623}{226307}a^{17}+\frac{638900}{226307}a^{16}-\frac{176112}{226307}a^{15}-\frac{244518}{226307}a^{14}+\frac{681616}{226307}a^{13}-\frac{818484}{226307}a^{12}+\frac{626913}{226307}a^{11}-\frac{103738}{226307}a^{10}-\frac{69326}{226307}a^{9}+\frac{326626}{226307}a^{8}-\frac{519302}{226307}a^{7}+\frac{669508}{226307}a^{6}-\frac{114196}{226307}a^{5}-\frac{368506}{226307}a^{4}+\frac{217534}{226307}a^{3}-\frac{287213}{226307}a^{2}+\frac{96535}{226307}a+\frac{51311}{226307}$, $\frac{21515}{226307}a^{21}-\frac{127045}{226307}a^{20}+\frac{174938}{226307}a^{19}-\frac{152252}{226307}a^{18}-\frac{394481}{226307}a^{17}+\frac{508440}{226307}a^{16}-\frac{748888}{226307}a^{15}-\frac{13281}{226307}a^{14}+\frac{57402}{226307}a^{13}-\frac{716430}{226307}a^{12}+\frac{409901}{226307}a^{11}-\frac{453856}{226307}a^{10}-\frac{379835}{226307}a^{9}-\frac{278658}{226307}a^{8}-\frac{494364}{226307}a^{7}-\frac{28935}{226307}a^{6}-\frac{541997}{226307}a^{5}-\frac{444997}{226307}a^{4}+\frac{392805}{226307}a^{3}-\frac{177821}{226307}a^{2}+\frac{65}{226307}a-\frac{98344}{226307}$, $\frac{98695}{226307}a^{21}-\frac{183399}{226307}a^{20}+\frac{391842}{226307}a^{19}+\frac{43507}{226307}a^{18}-\frac{79705}{226307}a^{17}+\frac{745728}{226307}a^{16}+\frac{209915}{226307}a^{15}+\frac{247218}{226307}a^{14}+\frac{805340}{226307}a^{13}+\frac{290017}{226307}a^{12}+\frac{503286}{226307}a^{11}+\frac{909944}{226307}a^{10}+\frac{415691}{226307}a^{9}+\frac{634997}{226307}a^{8}+\frac{784973}{226307}a^{7}+\frac{1034776}{226307}a^{6}+\frac{814969}{226307}a^{5}+\frac{344976}{226307}a^{4}+\frac{555082}{226307}a^{3}+\frac{681764}{226307}a^{2}+\frac{3033}{226307}a-\frac{7036}{226307}$, $\frac{169704}{226307}a^{21}-\frac{164763}{226307}a^{20}+\frac{380848}{226307}a^{19}+\frac{511283}{226307}a^{18}-\frac{163753}{226307}a^{17}+\frac{1201915}{226307}a^{16}+\frac{110412}{226307}a^{15}+\frac{862289}{226307}a^{14}+\frac{1146638}{226307}a^{13}-\frac{309187}{226307}a^{12}+\frac{1599361}{226307}a^{11}+\frac{695200}{226307}a^{10}+\frac{1314798}{226307}a^{9}+\frac{1618911}{226307}a^{8}+\frac{348610}{226307}a^{7}+\frac{1676419}{226307}a^{6}+\frac{1128002}{226307}a^{5}+\frac{608056}{226307}a^{4}+\frac{497667}{226307}a^{3}+\frac{64183}{226307}a^{2}+\frac{758060}{226307}a+\frac{182249}{226307}$, $\frac{273375}{226307}a^{21}-\frac{487990}{226307}a^{20}+\frac{998813}{226307}a^{19}+\frac{65501}{226307}a^{18}-\frac{337660}{226307}a^{17}+\frac{2120037}{226307}a^{16}-\frac{998464}{226307}a^{15}+\frac{1677520}{226307}a^{14}+\frac{966295}{226307}a^{13}-\frac{677819}{226307}a^{12}+\frac{2385174}{226307}a^{11}+\frac{260752}{226307}a^{10}+\frac{1639485}{226307}a^{9}+\frac{1201335}{226307}a^{8}+\frac{280706}{226307}a^{7}+\frac{2171790}{226307}a^{6}+\frac{616881}{226307}a^{5}+\frac{635492}{226307}a^{4}+\frac{747836}{226307}a^{3}+\frac{90778}{226307}a^{2}+\frac{551210}{226307}a-\frac{117996}{226307}$
| 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: | \( 73023.7380593 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{11}\cdot 73023.7380593 \cdot 1}{6\cdot\sqrt{3101889720909924494411835627}}\cr\approx \mathstrut & 0.131667377430 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^22 - 2*x^21 + 4*x^20 - 2*x^18 + 9*x^17 - 4*x^16 + 6*x^15 + 4*x^14 - 2*x^13 + 10*x^12 + 7*x^10 + 4*x^9 + 2*x^8 + 11*x^7 + x^6 + 3*x^5 + 4*x^4 + 2*x^3 + 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 - 2*x^21 + 4*x^20 - 2*x^18 + 9*x^17 - 4*x^16 + 6*x^15 + 4*x^14 - 2*x^13 + 10*x^12 + 7*x^10 + 4*x^9 + 2*x^8 + 11*x^7 + x^6 + 3*x^5 + 4*x^4 + 2*x^3 + 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 - 2*x^21 + 4*x^20 - 2*x^18 + 9*x^17 - 4*x^16 + 6*x^15 + 4*x^14 - 2*x^13 + 10*x^12 + 7*x^10 + 4*x^9 + 2*x^8 + 11*x^7 + x^6 + 3*x^5 + 4*x^4 + 2*x^3 + 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 - 2*x^21 + 4*x^20 - 2*x^18 + 9*x^17 - 4*x^16 + 6*x^15 + 4*x^14 - 2*x^13 + 10*x^12 + 7*x^10 + 4*x^9 + 2*x^8 + 11*x^7 + x^6 + 3*x^5 + 4*x^4 + 2*x^3 + 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
| \(\Q(\sqrt{-3}) \), 11.5.132326332471.1 |
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 | $22$ | ${\href{/padicField/7.7.0.1}{7} }^{2}{,}\,{\href{/padicField/7.4.0.1}{4} }^{2}$ | ${\href{/padicField/11.10.0.1}{10} }{,}\,{\href{/padicField/11.6.0.1}{6} }^{2}$ | ${\href{/padicField/13.8.0.1}{8} }^{2}{,}\,{\href{/padicField/13.3.0.1}{3} }^{2}$ | $22$ | ${\href{/padicField/19.8.0.1}{8} }^{2}{,}\,{\href{/padicField/19.3.0.1}{3} }^{2}$ | $18{,}\,{\href{/padicField/23.2.0.1}{2} }^{2}$ | ${\href{/padicField/29.10.0.1}{10} }{,}\,{\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.2.0.1}{2} }^{3}$ | ${\href{/padicField/31.4.0.1}{4} }^{2}{,}\,{\href{/padicField/31.3.0.1}{3} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }^{4}$ | ${\href{/padicField/37.6.0.1}{6} }^{2}{,}\,{\href{/padicField/37.3.0.1}{3} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}$ | ${\href{/padicField/41.10.0.1}{10} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }$ | ${\href{/padicField/43.10.0.1}{10} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | ${\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.4.0.1}{4} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }^{4}$ | ${\href{/padicField/53.10.0.1}{10} }{,}\,{\href{/padicField/53.2.0.1}{2} }^{6}$ | ${\href{/padicField/59.4.0.1}{4} }^{4}{,}\,{\href{/padicField/59.2.0.1}{2} }^{3}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(3\)
| 3.8.4.1 | $x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |
| 3.14.7.2 | $x^{14} + 21 x^{12} + 189 x^{10} + 4 x^{9} + 945 x^{8} - 94 x^{7} + 2835 x^{6} - 630 x^{5} + 5107 x^{4} + 630 x^{3} + 5131 x^{2} + 1242 x + 2212$ | $2$ | $7$ | $7$ | $C_{14}$ | $[\ ]_{2}^{7}$ | |
|
\(211441\)
| $\Q_{211441}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{211441}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $8$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | ||
| Deg $8$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | ||
|
\(625831\)
| $\Q_{625831}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{625831}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 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 $5$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | ||
| Deg $5$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ |