Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 6*x^18 - 6*x^16 + 8*x^15 - 8*x^14 + 17*x^13 - 3*x^12 + 23*x^11 - 28*x^10 - 11*x^9 - 45*x^8 - 14*x^7 - 44*x^6 - 17*x^5 - 21*x^4 - 4*x^3 - 8*x^2 - 1)
gp: K = bnfinit(y^21 - 6*y^18 - 6*y^16 + 8*y^15 - 8*y^14 + 17*y^13 - 3*y^12 + 23*y^11 - 28*y^10 - 11*y^9 - 45*y^8 - 14*y^7 - 44*y^6 - 17*y^5 - 21*y^4 - 4*y^3 - 8*y^2 - 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^21 - 6*x^18 - 6*x^16 + 8*x^15 - 8*x^14 + 17*x^13 - 3*x^12 + 23*x^11 - 28*x^10 - 11*x^9 - 45*x^8 - 14*x^7 - 44*x^6 - 17*x^5 - 21*x^4 - 4*x^3 - 8*x^2 - 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^21 - 6*x^18 - 6*x^16 + 8*x^15 - 8*x^14 + 17*x^13 - 3*x^12 + 23*x^11 - 28*x^10 - 11*x^9 - 45*x^8 - 14*x^7 - 44*x^6 - 17*x^5 - 21*x^4 - 4*x^3 - 8*x^2 - 1)
\( x^{21} - 6 x^{18} - 6 x^{16} + 8 x^{15} - 8 x^{14} + 17 x^{13} - 3 x^{12} + 23 x^{11} - 28 x^{10} + \cdots - 1 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $21$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[1, 10]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(238583305854552412020101\)
\(\medspace = 23^{7}\cdot 41227^{3}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(12.98\) | 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: | $23^{1/2}41227^{1/2}\approx 973.7663990916918$ | ||
| Ramified primes: |
\(23\), \(41227\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{948221}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $\frac{1}{518266524713}a^{20}+\frac{60217880587}{518266524713}a^{19}-\frac{48116156212}{518266524713}a^{18}-\frac{79153041266}{518266524713}a^{17}-\frac{92621406219}{518266524713}a^{16}+\frac{42692262758}{518266524713}a^{15}+\frac{31180083129}{74038074959}a^{14}+\frac{101440730908}{518266524713}a^{13}+\frac{139464783058}{518266524713}a^{12}-\frac{15199703780}{74038074959}a^{11}+\frac{129938744042}{518266524713}a^{10}+\frac{216947069004}{518266524713}a^{9}-\frac{16279227377}{74038074959}a^{8}-\frac{173697961391}{518266524713}a^{7}+\frac{66140149252}{518266524713}a^{6}-\frac{227637798004}{518266524713}a^{5}-\frac{36081591520}{74038074959}a^{4}-\frac{2528948491}{74038074959}a^{3}+\frac{239167134524}{518266524713}a^{2}-\frac{104948707685}{518266524713}a-\frac{254047246773}{518266524713}$
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: |
\( -1 \)
(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: |
$\frac{1840141022}{2263172597}a^{20}+\frac{877143788}{2263172597}a^{19}+\frac{449344201}{2263172597}a^{18}-\frac{11487780082}{2263172597}a^{17}-\frac{5472735332}{2263172597}a^{16}-\frac{13500248291}{2263172597}a^{15}+\frac{1667617199}{323310371}a^{14}-\frac{8650682375}{2263172597}a^{13}+\frac{29084376644}{2263172597}a^{12}+\frac{840462758}{323310371}a^{11}+\frac{45513354493}{2263172597}a^{10}-\frac{34338086854}{2263172597}a^{9}-\frac{6107916974}{323310371}a^{8}-\frac{106230878737}{2263172597}a^{7}-\frac{69811258792}{2263172597}a^{6}-\frac{94229005548}{2263172597}a^{5}-\frac{9132197950}{323310371}a^{4}-\frac{7106189621}{323310371}a^{3}-\frac{19268887963}{2263172597}a^{2}-\frac{10469768775}{2263172597}a-\frac{3954802245}{2263172597}$, $a$, $\frac{384420544792}{518266524713}a^{20}+\frac{229288999410}{518266524713}a^{19}-\frac{98828140774}{518266524713}a^{18}-\frac{2292108019564}{518266524713}a^{17}-\frac{1441783757438}{518266524713}a^{16}-\frac{1718212929210}{518266524713}a^{15}+\frac{236405694242}{74038074959}a^{14}-\frac{320576439327}{518266524713}a^{13}+\frac{3965937182329}{518266524713}a^{12}+\frac{531696392713}{74038074959}a^{11}+\frac{6253413636198}{518266524713}a^{10}-\frac{5270878609937}{518266524713}a^{9}-\frac{1866388343097}{74038074959}a^{8}-\frac{17907267077096}{518266524713}a^{7}-\frac{14574063616349}{518266524713}a^{6}-\frac{15895094456798}{518266524713}a^{5}-\frac{1836528670722}{74038074959}a^{4}-\frac{1022434666669}{74038074959}a^{3}-\frac{2311040908101}{518266524713}a^{2}-\frac{1889977628712}{518266524713}a-\frac{216748597701}{518266524713}$, $\frac{129583014030}{518266524713}a^{20}-\frac{61396266508}{518266524713}a^{19}-\frac{97778237562}{518266524713}a^{18}-\frac{719603998937}{518266524713}a^{17}+\frac{379119510204}{518266524713}a^{16}-\frac{134991559537}{518266524713}a^{15}+\frac{144459214909}{74038074959}a^{14}-\frac{1048529565014}{518266524713}a^{13}+\frac{1323468656208}{518266524713}a^{12}-\frac{15019801145}{74038074959}a^{11}+\frac{1164025539168}{518266524713}a^{10}-\frac{3666341008126}{518266524713}a^{9}-\frac{270633178779}{74038074959}a^{8}-\frac{1083276687986}{518266524713}a^{7}+\frac{1114707308206}{518266524713}a^{6}-\frac{2118736104108}{518266524713}a^{5}-\frac{188722385709}{74038074959}a^{4}+\frac{3873798615}{74038074959}a^{3}+\frac{703454848603}{518266524713}a^{2}-\frac{724391249543}{518266524713}a+\frac{110541671508}{518266524713}$, $\frac{146692774465}{518266524713}a^{20}+\frac{57273645278}{518266524713}a^{19}+\frac{102932512331}{518266524713}a^{18}-\frac{831663862087}{518266524713}a^{17}-\frac{422579544990}{518266524713}a^{16}-\frac{1427025741545}{518266524713}a^{15}+\frac{61856372089}{74038074959}a^{14}-\frac{791130313938}{518266524713}a^{13}+\frac{2171091176479}{518266524713}a^{12}+\frac{150742608670}{74038074959}a^{11}+\frac{3328264393648}{518266524713}a^{10}-\frac{777686482826}{518266524713}a^{9}-\frac{468259935241}{74038074959}a^{8}-\frac{7441333256753}{518266524713}a^{7}-\frac{9720901805547}{518266524713}a^{6}-\frac{9415301771196}{518266524713}a^{5}-\frac{1357358193704}{74038074959}a^{4}-\frac{797639305011}{74038074959}a^{3}-\frac{4992374975297}{518266524713}a^{2}-\frac{1634311571813}{518266524713}a-\frac{900034280910}{518266524713}$, $\frac{342658115757}{518266524713}a^{20}-\frac{29164090029}{518266524713}a^{19}-\frac{148978858352}{518266524713}a^{18}-\frac{2019307180683}{518266524713}a^{17}+\frac{248846541579}{518266524713}a^{16}-\frac{1257867105703}{518266524713}a^{15}+\frac{404556124694}{74038074959}a^{14}-\frac{2626257870620}{518266524713}a^{13}+\frac{5139274261850}{518266524713}a^{12}-\frac{148288599575}{74038074959}a^{11}+\frac{6634564584394}{518266524713}a^{10}-\frac{10710744719361}{518266524713}a^{9}-\frac{597339481924}{74038074959}a^{8}-\frac{12026049292373}{518266524713}a^{7}-\frac{372104367154}{518266524713}a^{6}-\frac{11463821981282}{518266524713}a^{5}-\frac{443835315600}{74038074959}a^{4}-\frac{607634361899}{74038074959}a^{3}+\frac{384092017446}{518266524713}a^{2}-\frac{1420410121057}{518266524713}a+\frac{101815443378}{518266524713}$, $\frac{49532839703}{518266524713}a^{20}-\frac{98637379062}{518266524713}a^{19}-\frac{84757897573}{518266524713}a^{18}-\frac{281167121115}{518266524713}a^{17}+\frac{652537267386}{518266524713}a^{16}+\frac{134644762265}{518266524713}a^{15}+\frac{135924433745}{74038074959}a^{14}-\frac{1062224303935}{518266524713}a^{13}+\frac{1282214878633}{518266524713}a^{12}-\frac{234551938079}{74038074959}a^{11}+\frac{843007913294}{518266524713}a^{10}-\frac{4327064129902}{518266524713}a^{9}+\frac{265476494677}{74038074959}a^{8}+\frac{127695891864}{518266524713}a^{7}+\frac{6043653516237}{518266524713}a^{6}+\frac{315110168187}{518266524713}a^{5}+\frac{736408507281}{74038074959}a^{4}+\frac{324045093290}{74038074959}a^{3}+\frac{3458330660972}{518266524713}a^{2}+\frac{280069450495}{518266524713}a+\frac{1075626096735}{518266524713}$, $\frac{520689878469}{518266524713}a^{20}-\frac{260720939655}{518266524713}a^{19}-\frac{173353578808}{518266524713}a^{18}-\frac{3075841947147}{518266524713}a^{17}+\frac{1622646755728}{518266524713}a^{16}-\frac{2153230681472}{518266524713}a^{15}+\frac{781355432950}{74038074959}a^{14}-\frac{5515603227664}{518266524713}a^{13}+\frac{9570833124449}{518266524713}a^{12}-\frac{662591672832}{74038074959}a^{11}+\frac{10003814915664}{518266524713}a^{10}-\frac{19898640160002}{518266524713}a^{9}-\frac{222044200854}{74038074959}a^{8}-\frac{15152097329055}{518266524713}a^{7}+\frac{5846937344860}{518266524713}a^{6}-\frac{13549677273830}{518266524713}a^{5}+\frac{440039205951}{74038074959}a^{4}-\frac{75616626630}{74038074959}a^{3}+\frac{3902308119900}{518266524713}a^{2}-\frac{1643082222160}{518266524713}a+\frac{1564036279270}{518266524713}$, $\frac{28733056595}{518266524713}a^{20}-\frac{256920674214}{518266524713}a^{19}-\frac{82287906782}{518266524713}a^{18}-\frac{174908951690}{518266524713}a^{17}+\frac{1630660017887}{518266524713}a^{16}+\frac{332093357390}{518266524713}a^{15}+\frac{250042292697}{74038074959}a^{14}-\frac{2264035138580}{518266524713}a^{13}+\frac{1725482969100}{518266524713}a^{12}-\frac{582291430617}{74038074959}a^{11}+\frac{377488133718}{518266524713}a^{10}-\frac{6384157574237}{518266524713}a^{9}+\frac{799398178696}{74038074959}a^{8}+\frac{4673308977042}{518266524713}a^{7}+\frac{12619896246464}{518266524713}a^{6}+\frac{5150939228374}{518266524713}a^{5}+\frac{1299509641957}{74038074959}a^{4}+\frac{671292739123}{74038074959}a^{3}+\frac{3948212468992}{518266524713}a^{2}+\frac{1027403093975}{518266524713}a+\frac{972550743419}{518266524713}$, $\frac{499161549793}{518266524713}a^{20}-\frac{256651413884}{518266524713}a^{19}-\frac{124824071854}{518266524713}a^{18}-\frac{2879785562485}{518266524713}a^{17}+\frac{1421818684003}{518266524713}a^{16}-\frac{2160024917205}{518266524713}a^{15}+\frac{691815038620}{74038074959}a^{14}-\frac{4692477340541}{518266524713}a^{13}+\frac{8525283089095}{518266524713}a^{12}-\frac{487999908292}{74038074959}a^{11}+\frac{8397026408898}{518266524713}a^{10}-\frac{16903888642686}{518266524713}a^{9}-\frac{404055561453}{74038074959}a^{8}-\frac{13525264522173}{518266524713}a^{7}+\frac{2043893609276}{518266524713}a^{6}-\frac{11931927272010}{518266524713}a^{5}+\frac{160183524736}{74038074959}a^{4}-\frac{111700673438}{74038074959}a^{3}+\frac{2550239630545}{518266524713}a^{2}-\frac{889941322891}{518266524713}a+\frac{1547778934122}{518266524713}$
| 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: | \( 952.218112417 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{10}\cdot 952.218112417 \cdot 1}{2\cdot\sqrt{238583305854552412020101}}\cr\approx \mathstrut & 0.186945534449 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^21 - 6*x^18 - 6*x^16 + 8*x^15 - 8*x^14 + 17*x^13 - 3*x^12 + 23*x^11 - 28*x^10 - 11*x^9 - 45*x^8 - 14*x^7 - 44*x^6 - 17*x^5 - 21*x^4 - 4*x^3 - 8*x^2 - 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^21 - 6*x^18 - 6*x^16 + 8*x^15 - 8*x^14 + 17*x^13 - 3*x^12 + 23*x^11 - 28*x^10 - 11*x^9 - 45*x^8 - 14*x^7 - 44*x^6 - 17*x^5 - 21*x^4 - 4*x^3 - 8*x^2 - 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^21 - 6*x^18 - 6*x^16 + 8*x^15 - 8*x^14 + 17*x^13 - 3*x^12 + 23*x^11 - 28*x^10 - 11*x^9 - 45*x^8 - 14*x^7 - 44*x^6 - 17*x^5 - 21*x^4 - 4*x^3 - 8*x^2 - 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^21 - 6*x^18 - 6*x^16 + 8*x^15 - 8*x^14 + 17*x^13 - 3*x^12 + 23*x^11 - 28*x^10 - 11*x^9 - 45*x^8 - 14*x^7 - 44*x^6 - 17*x^5 - 21*x^4 - 4*x^3 - 8*x^2 - 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
$D_7\wr S_3$ (as 21T62):
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 16464 |
| The 65 conjugacy class representatives for $D_7\wr S_3$ |
| Character table for $D_7\wr S_3$ |
Intermediate fields
| 3.1.23.1 |
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 28 sibling: | data not computed |
| Degree 42 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 | ${\href{/padicField/2.6.0.1}{6} }^{3}{,}\,{\href{/padicField/2.3.0.1}{3} }$ | ${\href{/padicField/3.6.0.1}{6} }^{3}{,}\,{\href{/padicField/3.3.0.1}{3} }$ | ${\href{/padicField/5.7.0.1}{7} }{,}\,{\href{/padicField/5.4.0.1}{4} }^{3}{,}\,{\href{/padicField/5.2.0.1}{2} }$ | ${\href{/padicField/7.7.0.1}{7} }{,}\,{\href{/padicField/7.4.0.1}{4} }^{3}{,}\,{\href{/padicField/7.2.0.1}{2} }$ | ${\href{/padicField/11.14.0.1}{14} }{,}\,{\href{/padicField/11.7.0.1}{7} }$ | $21$ | ${\href{/padicField/17.4.0.1}{4} }^{3}{,}\,{\href{/padicField/17.2.0.1}{2} }^{4}{,}\,{\href{/padicField/17.1.0.1}{1} }$ | ${\href{/padicField/19.7.0.1}{7} }{,}\,{\href{/padicField/19.4.0.1}{4} }^{3}{,}\,{\href{/padicField/19.2.0.1}{2} }$ | R | ${\href{/padicField/29.6.0.1}{6} }^{3}{,}\,{\href{/padicField/29.3.0.1}{3} }$ | ${\href{/padicField/31.6.0.1}{6} }^{3}{,}\,{\href{/padicField/31.3.0.1}{3} }$ | ${\href{/padicField/37.14.0.1}{14} }{,}\,{\href{/padicField/37.7.0.1}{7} }$ | ${\href{/padicField/41.6.0.1}{6} }^{3}{,}\,{\href{/padicField/41.3.0.1}{3} }$ | ${\href{/padicField/43.4.0.1}{4} }^{3}{,}\,{\href{/padicField/43.2.0.1}{2} }^{4}{,}\,{\href{/padicField/43.1.0.1}{1} }$ | ${\href{/padicField/47.6.0.1}{6} }^{3}{,}\,{\href{/padicField/47.3.0.1}{3} }$ | ${\href{/padicField/53.4.0.1}{4} }^{3}{,}\,{\href{/padicField/53.2.0.1}{2} }^{4}{,}\,{\href{/padicField/53.1.0.1}{1} }$ | ${\href{/padicField/59.7.0.1}{7} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }^{3}{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(23\)
| 23.2.1.2 | $x^{2} + 23$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 23.4.2.1 | $x^{4} + 42 x^{3} + 497 x^{2} + 1176 x + 10467$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 23.4.2.1 | $x^{4} + 42 x^{3} + 497 x^{2} + 1176 x + 10467$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 23.4.2.1 | $x^{4} + 42 x^{3} + 497 x^{2} + 1176 x + 10467$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 23.7.0.1 | $x^{7} + 21 x + 18$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | |
|
\(41227\)
| $\Q_{41227}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| 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 $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
| Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
| Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |