Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 9*x^18 + 18*x^16 + 26*x^14 - 80*x^12 + 22*x^10 - 93*x^8 - 105*x^6 + 780*x^4 - 210*x^2 - 241)
gp: K = bnfinit(y^20 - 9*y^18 + 18*y^16 + 26*y^14 - 80*y^12 + 22*y^10 - 93*y^8 - 105*y^6 + 780*y^4 - 210*y^2 - 241, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 - 9*x^18 + 18*x^16 + 26*x^14 - 80*x^12 + 22*x^10 - 93*x^8 - 105*x^6 + 780*x^4 - 210*x^2 - 241);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^20 - 9*x^18 + 18*x^16 + 26*x^14 - 80*x^12 + 22*x^10 - 93*x^8 - 105*x^6 + 780*x^4 - 210*x^2 - 241)
\( x^{20} - 9x^{18} + 18x^{16} + 26x^{14} - 80x^{12} + 22x^{10} - 93x^{8} - 105x^{6} + 780x^{4} - 210x^{2} - 241 \)
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: | $[10, 5]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(-658618684118710741619590144\)
\(\medspace = -\,2^{10}\cdot 11^{16}\cdot 241^{3}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(21.92\) | 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: | $2^{31/16}11^{4/5}241^{1/2}\approx 404.91912156279443$ | ||
| Ramified primes: |
\(2\), \(11\), \(241\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-241}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{2}a^{15}-\frac{1}{2}a^{12}-\frac{1}{2}a^{11}-\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{16}-\frac{1}{2}a^{13}-\frac{1}{2}a^{12}-\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{2}a^{17}-\frac{1}{2}a^{14}-\frac{1}{2}a^{13}-\frac{1}{2}a^{8}-\frac{1}{2}a^{6}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}$, $\frac{1}{1884100816354}a^{18}+\frac{196056870279}{1884100816354}a^{16}+\frac{44137292369}{1884100816354}a^{14}-\frac{1}{2}a^{13}-\frac{221044887606}{942050408177}a^{12}-\frac{1}{2}a^{11}+\frac{11466341879}{40958713399}a^{10}-\frac{1}{2}a^{9}-\frac{286577714801}{942050408177}a^{8}-\frac{328046848113}{1884100816354}a^{6}-\frac{1}{2}a^{5}+\frac{9722235910}{40958713399}a^{4}-\frac{1}{2}a^{3}+\frac{863293794789}{1884100816354}a^{2}-\frac{562584574043}{1884100816354}$, $\frac{1}{1884100816354}a^{19}+\frac{196056870279}{1884100816354}a^{17}+\frac{44137292369}{1884100816354}a^{15}-\frac{1}{2}a^{14}-\frac{221044887606}{942050408177}a^{13}-\frac{1}{2}a^{12}+\frac{11466341879}{40958713399}a^{11}-\frac{1}{2}a^{10}-\frac{286577714801}{942050408177}a^{9}-\frac{328046848113}{1884100816354}a^{7}-\frac{1}{2}a^{6}+\frac{9722235910}{40958713399}a^{5}-\frac{1}{2}a^{4}+\frac{863293794789}{1884100816354}a^{3}-\frac{562584574043}{1884100816354}a$
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: | $14$ | 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: |
$\frac{2125382437}{942050408177}a^{18}-\frac{13682990507}{942050408177}a^{16}+\frac{11867604853}{942050408177}a^{14}+\frac{23975543518}{942050408177}a^{12}-\frac{3096794910}{40958713399}a^{10}+\frac{215391397923}{942050408177}a^{8}+\frac{7721309560}{942050408177}a^{6}-\frac{5757868697}{40958713399}a^{4}+\frac{544244545450}{942050408177}a^{2}-\frac{1832225582468}{942050408177}$, $\frac{642902041}{942050408177}a^{18}-\frac{1065531950}{942050408177}a^{16}-\frac{6802741362}{942050408177}a^{14}-\frac{48124031727}{942050408177}a^{12}+\frac{4001790363}{40958713399}a^{10}+\frac{275230010723}{942050408177}a^{8}+\frac{61407509504}{942050408177}a^{6}-\frac{8499509068}{40958713399}a^{4}-\frac{1003529206152}{942050408177}a^{2}-\frac{1177343817341}{942050408177}$, $\frac{14023785104}{942050408177}a^{18}-\frac{89433586980}{942050408177}a^{16}+\frac{6581336796}{942050408177}a^{14}+\frac{470692728950}{942050408177}a^{12}+\frac{170898316}{40958713399}a^{10}-\frac{59581646122}{942050408177}a^{8}-\frac{968964121207}{942050408177}a^{6}-\frac{169705474660}{40958713399}a^{4}+\frac{1429610452801}{942050408177}a^{2}+\frac{3076706617222}{942050408177}$, $\frac{5174126977}{942050408177}a^{18}-\frac{38959298078}{942050408177}a^{16}+\frac{40819572636}{942050408177}a^{14}+\frac{168562600657}{942050408177}a^{12}-\frac{8850524556}{40958713399}a^{10}+\frac{10770224019}{942050408177}a^{8}-\frac{329539136297}{942050408177}a^{6}-\frac{50824161645}{40958713399}a^{4}+\frac{2704754430009}{942050408177}a^{2}+\frac{574471844795}{942050408177}$, $\frac{13086532009}{942050408177}a^{18}-\frac{98709965826}{942050408177}a^{16}+\frac{113249738020}{942050408177}a^{14}+\frac{360546840900}{942050408177}a^{12}-\frac{19653523616}{40958713399}a^{10}+\frac{273298915177}{942050408177}a^{8}-\frac{1078207742041}{942050408177}a^{6}-\frac{119167635768}{40958713399}a^{4}+\frac{5102899372665}{942050408177}a^{2}-\frac{696077854044}{942050408177}$, $\frac{966536997}{942050408177}a^{18}-\frac{12824459987}{942050408177}a^{16}+\frac{36583735074}{942050408177}a^{14}+\frac{61251950671}{942050408177}a^{12}-\frac{7877330766}{40958713399}a^{10}-\frac{140847088757}{942050408177}a^{8}-\frac{432550787336}{942050408177}a^{6}-\frac{2051625005}{40958713399}a^{4}+\frac{2462594951666}{942050408177}a^{2}+\frac{1156886484649}{942050408177}$, $\frac{13376361884}{942050408177}a^{18}-\frac{92261168476}{942050408177}a^{16}+\frac{67871812663}{942050408177}a^{14}+\frac{334582671185}{942050408177}a^{12}-\frac{7829114326}{40958713399}a^{10}+\frac{333351266579}{942050408177}a^{8}-\frac{1019998365222}{942050408177}a^{6}-\frac{118608417328}{40958713399}a^{4}+\frac{2734619493653}{942050408177}a^{2}-\frac{416896431395}{942050408177}$, $\frac{3586932141}{942050408177}a^{19}+\frac{1209582139}{942050408177}a^{18}-\frac{59470483343}{1884100816354}a^{17}-\frac{5139328187}{1884100816354}a^{16}+\frac{92306514425}{1884100816354}a^{15}-\frac{66263618193}{1884100816354}a^{14}+\frac{101487735463}{942050408177}a^{13}+\frac{49139873340}{942050408177}a^{12}-\frac{13281133725}{81917426798}a^{11}+\frac{6962434452}{40958713399}a^{10}-\frac{60199668616}{942050408177}a^{9}-\frac{327356805}{1884100816354}a^{8}-\frac{847226312033}{1884100816354}a^{7}+\frac{32227780952}{942050408177}a^{6}-\frac{37147041931}{81917426798}a^{5}-\frac{66564744805}{81917426798}a^{4}+\frac{4177878442391}{1884100816354}a^{3}-\frac{1566081535847}{942050408177}a^{2}+\frac{412576405651}{942050408177}a-\frac{450355387843}{1884100816354}$, $\frac{3538673928}{942050408177}a^{19}+\frac{7203870705}{1884100816354}a^{18}-\frac{44305627695}{1884100816354}a^{17}-\frac{26713818065}{942050408177}a^{16}-\frac{3765328349}{942050408177}a^{15}+\frac{32880392584}{942050408177}a^{14}+\frac{255994722763}{1884100816354}a^{13}+\frac{155405343131}{1884100816354}a^{12}+\frac{4295598855}{81917426798}a^{11}-\frac{5211831884}{40958713399}a^{10}-\frac{70565608743}{1884100816354}a^{9}+\frac{118463604275}{1884100816354}a^{8}-\frac{643948161785}{1884100816354}a^{7}-\frac{179484011761}{942050408177}a^{6}-\frac{61091085913}{40958713399}a^{5}-\frac{16942553609}{40958713399}a^{4}-\frac{227318924899}{942050408177}a^{3}+\frac{3083152860865}{1884100816354}a^{2}+\frac{2877935114905}{1884100816354}a-\frac{339395044873}{1884100816354}$, $\frac{30006423}{1884100816354}a^{19}+\frac{47336599819}{1884100816354}a^{18}+\frac{6042847213}{1884100816354}a^{17}-\frac{310367824919}{1884100816354}a^{16}-\frac{26545729257}{1884100816354}a^{15}+\frac{50988854925}{942050408177}a^{14}-\frac{47570127795}{1884100816354}a^{13}+\frac{1433539499059}{1884100816354}a^{12}+\frac{2857469957}{81917426798}a^{11}-\frac{13859707585}{81917426798}a^{10}+\frac{238862941507}{1884100816354}a^{9}+\frac{303943113372}{942050408177}a^{8}+\frac{488258288511}{1884100816354}a^{7}-\frac{1594454018822}{942050408177}a^{6}+\frac{3261934713}{81917426798}a^{5}-\frac{540039960485}{81917426798}a^{4}-\frac{547362790763}{942050408177}a^{3}+\frac{6518861329529}{1884100816354}a^{2}-\frac{1164547856175}{1884100816354}a+\frac{4215286137703}{1884100816354}$, $\frac{289829875}{1884100816354}a^{19}+\frac{6171580788}{942050408177}a^{18}+\frac{3224398675}{942050408177}a^{17}-\frac{42287221599}{942050408177}a^{16}-\frac{45377925357}{1884100816354}a^{15}+\frac{26484302974}{942050408177}a^{14}-\frac{25964169715}{1884100816354}a^{13}+\frac{361421751587}{1884100816354}a^{12}+\frac{5912204645}{40958713399}a^{11}-\frac{11275835923}{81917426798}a^{10}+\frac{30026175701}{942050408177}a^{9}+\frac{323673739073}{1884100816354}a^{8}+\frac{58209376819}{1884100816354}a^{7}-\frac{467430988411}{942050408177}a^{6}+\frac{279609220}{40958713399}a^{5}-\frac{134337162047}{81917426798}a^{4}-\frac{1184139939506}{942050408177}a^{3}+\frac{3724090432937}{1884100816354}a^{2}-\frac{1604919393705}{1884100816354}a+\frac{715679747462}{942050408177}$, $\frac{6884034407}{1884100816354}a^{19}-\frac{7381162927}{942050408177}a^{18}-\frac{65919280693}{1884100816354}a^{17}+\frac{89713771385}{1884100816354}a^{16}+\frac{68842219891}{942050408177}a^{15}+\frac{13295012245}{1884100816354}a^{14}+\frac{228939640641}{1884100816354}a^{13}-\frac{459701498267}{1884100816354}a^{12}-\frac{25105543015}{81917426798}a^{11}-\frac{2649032981}{81917426798}a^{10}-\frac{90225844317}{942050408177}a^{9}-\frac{161673191134}{942050408177}a^{8}-\frac{452717844426}{942050408177}a^{7}+\frac{435203207459}{942050408177}a^{6}-\frac{37706260371}{81917426798}a^{5}+\frac{100450953426}{40958713399}a^{4}+\frac{6546158321403}{1884100816354}a^{3}-\frac{591927361243}{1884100816354}a^{2}+\frac{545971388653}{1884100816354}a-\frac{981004107081}{1884100816354}$, $\frac{4772796598}{942050408177}a^{19}-\frac{32316271295}{1884100816354}a^{18}-\frac{23862540954}{942050408177}a^{17}+\frac{214152625165}{1884100816354}a^{16}-\frac{95348393841}{1884100816354}a^{15}-\frac{85358367237}{1884100816354}a^{14}+\frac{434464177175}{1884100816354}a^{13}-\frac{949164947233}{1884100816354}a^{12}+\frac{7760460661}{40958713399}a^{11}+\frac{4505372546}{40958713399}a^{10}-\frac{358981142115}{1884100816354}a^{9}-\frac{284726010058}{942050408177}a^{8}-\frac{231098193858}{942050408177}a^{7}+\frac{1137825708806}{942050408177}a^{6}-\frac{168565998169}{81917426798}a^{5}+\frac{371544795533}{81917426798}a^{4}-\frac{1952508777891}{1884100816354}a^{3}-\frac{4663431914765}{1884100816354}a^{2}+\frac{6114776473173}{1884100816354}a-\frac{102095241915}{942050408177}$, $\frac{8608086465}{942050408177}a^{19}+\frac{11556108977}{1884100816354}a^{18}-\frac{64505514076}{942050408177}a^{17}-\frac{84215884291}{1884100816354}a^{16}+\frac{122116550157}{1884100816354}a^{15}+\frac{72664637123}{1884100816354}a^{14}+\frac{603169344179}{1884100816354}a^{13}+\frac{376188991153}{1884100816354}a^{12}-\frac{11591352708}{40958713399}a^{11}-\frac{6126728198}{40958713399}a^{10}-\frac{310642171575}{1884100816354}a^{9}-\frac{81778965022}{942050408177}a^{8}-\frac{799273436853}{942050408177}a^{7}-\frac{579807018145}{942050408177}a^{6}-\frac{153845933015}{81917426798}a^{5}-\frac{104553102033}{81917426798}a^{4}+\frac{6818788095797}{1884100816354}a^{3}+\frac{4876179692503}{1884100816354}a^{2}+\frac{4277425212307}{1884100816354}a+\frac{1704061676468}{942050408177}$
| 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: | \( 840640.634638 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{10}\cdot(2\pi)^{5}\cdot 840640.634638 \cdot 1}{2\cdot\sqrt{658618684118710741619590144}}\cr\approx \mathstrut & 0.164233844131 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^20 - 9*x^18 + 18*x^16 + 26*x^14 - 80*x^12 + 22*x^10 - 93*x^8 - 105*x^6 + 780*x^4 - 210*x^2 - 241)
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 - 9*x^18 + 18*x^16 + 26*x^14 - 80*x^12 + 22*x^10 - 93*x^8 - 105*x^6 + 780*x^4 - 210*x^2 - 241, 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 - 9*x^18 + 18*x^16 + 26*x^14 - 80*x^12 + 22*x^10 - 93*x^8 - 105*x^6 + 780*x^4 - 210*x^2 - 241);
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 - 9*x^18 + 18*x^16 + 26*x^14 - 80*x^12 + 22*x^10 - 93*x^8 - 105*x^6 + 780*x^4 - 210*x^2 - 241);
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
$C_2^{10}.C_2\wr C_5$ (as 20T846):
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 163840 |
| The 649 conjugacy class representatives for $C_2^{10}.C_2\wr C_5$ |
| Character table for $C_2^{10}.C_2\wr C_5$ |
Intermediate fields
| \(\Q(\zeta_{11})^+\), 10.6.51660490321.1 |
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 |
| Minimal sibling: | 20.6.658618684118710741619590144.3 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/padicField/3.10.0.1}{10} }{,}\,{\href{/padicField/3.5.0.1}{5} }^{2}$ | ${\href{/padicField/5.10.0.1}{10} }^{2}$ | ${\href{/padicField/7.10.0.1}{10} }^{2}$ | R | $20$ | $20$ | ${\href{/padicField/19.10.0.1}{10} }^{2}$ | ${\href{/padicField/23.4.0.1}{4} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }^{6}$ | ${\href{/padicField/29.5.0.1}{5} }^{4}$ | ${\href{/padicField/31.10.0.1}{10} }^{2}$ | $20$ | ${\href{/padicField/41.5.0.1}{5} }^{4}$ | ${\href{/padicField/43.4.0.1}{4} }{,}\,{\href{/padicField/43.2.0.1}{2} }^{5}{,}\,{\href{/padicField/43.1.0.1}{1} }^{6}$ | ${\href{/padicField/47.10.0.1}{10} }{,}\,{\href{/padicField/47.5.0.1}{5} }^{2}$ | ${\href{/padicField/53.10.0.1}{10} }^{2}$ | ${\href{/padicField/59.10.0.1}{10} }{,}\,{\href{/padicField/59.5.0.1}{5} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.5.0.1 | $x^{5} + x^{2} + 1$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ |
| 2.5.0.1 | $x^{5} + x^{2} + 1$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | |
| 2.10.10.9 | $x^{10} + 10 x^{9} + 22 x^{8} - 8 x^{7} + 72 x^{6} + 1328 x^{5} + 4496 x^{4} + 3680 x^{3} - 3696 x^{2} - 96 x - 32$ | $2$ | $5$ | $10$ | $C_2 \times (C_2^4 : C_5)$ | $[2, 2, 2, 2, 2]^{5}$ | |
|
\(11\)
| 11.10.8.5 | $x^{10} + 35 x^{9} + 500 x^{8} + 3710 x^{7} + 14985 x^{6} + 31389 x^{5} + 30355 x^{4} + 19790 x^{3} + 37110 x^{2} + 111495 x + 148840$ | $5$ | $2$ | $8$ | $C_{10}$ | $[\ ]_{5}^{2}$ |
| 11.10.8.5 | $x^{10} + 35 x^{9} + 500 x^{8} + 3710 x^{7} + 14985 x^{6} + 31389 x^{5} + 30355 x^{4} + 19790 x^{3} + 37110 x^{2} + 111495 x + 148840$ | $5$ | $2$ | $8$ | $C_{10}$ | $[\ ]_{5}^{2}$ | |
|
\(241\)
| $\Q_{241}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{241}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| 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 $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |