Normalized defining polynomial
\( x^{22} - 6 x^{21} - 84 x^{20} + 446 x^{19} + 3054 x^{18} - 13258 x^{17} - 47156 x^{16} + 87393 x^{15} + 214333 x^{14} + 4592960 x^{13} - 8169779 x^{12} - 97523140 x^{11} + 286905876 x^{10} + 443480439 x^{9} - 2833653296 x^{8} + 2392585056 x^{7} + 7429384462 x^{6} - 17535751907 x^{5} + 7148761681 x^{4} + 16729868412 x^{3} - 21211987463 x^{2} + 6184844508 x + 940596021 \)
Invariants
| Degree: | $22$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[14, 4]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(8454115376424649903675045731962308211426203045057=97^{5}\cdot 74843^{8}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $167.48$ | magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
| |
| Ramified primes: | $97, 74843$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| 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}{4603447815395421679477102307642764315797233718739215194085360570730617748306620658556620009535701872398432457} a^{21} + \frac{23871447099871272358005145600492213063743004424222432740729757203723793494749469408824979542092000793900906}{1534482605131807226492367435880921438599077906246405064695120190243539249435540219518873336511900624132810819} a^{20} - \frac{119163168292484882346111021279691935601686442485931694740430197224621162413734527479483382817461506694594444}{1534482605131807226492367435880921438599077906246405064695120190243539249435540219518873336511900624132810819} a^{19} - \frac{158176048554000673885654218090808886834469976072505285204080578025134477370196102112073893284096733577790257}{4603447815395421679477102307642764315797233718739215194085360570730617748306620658556620009535701872398432457} a^{18} + \frac{410888549771484046593854934029764214126442519934495590078713930148629341134948827651870312046856570155158322}{1534482605131807226492367435880921438599077906246405064695120190243539249435540219518873336511900624132810819} a^{17} - \frac{130507366584316448573443519880248783334778431829369024209603571864459129779620690208280660619006858469988103}{4603447815395421679477102307642764315797233718739215194085360570730617748306620658556620009535701872398432457} a^{16} - \frac{902485266179914891449335460985455006266958616385363289608966336327746614068399332742046271586895837482791511}{4603447815395421679477102307642764315797233718739215194085360570730617748306620658556620009535701872398432457} a^{15} + \frac{7034433721733528179206381245112058652658050492477655655191243973151661854834900530694550089814736486221996}{1534482605131807226492367435880921438599077906246405064695120190243539249435540219518873336511900624132810819} a^{14} + \frac{611584287051120804135894383164856334325170764243450573908907147732399457821483188383265571133659182126283959}{4603447815395421679477102307642764315797233718739215194085360570730617748306620658556620009535701872398432457} a^{13} + \frac{87544274856917147014904298871073586002421021769210458253408366226564452487671830313411988305456365795661889}{4603447815395421679477102307642764315797233718739215194085360570730617748306620658556620009535701872398432457} a^{12} - \frac{2196937005439826746001015773326163610033598160316655204682257218147876847919125264555341053973556428769184433}{4603447815395421679477102307642764315797233718739215194085360570730617748306620658556620009535701872398432457} a^{11} - \frac{682031577662458731913381819455014124142441170736872734029823741462498970215091114037778506243905926278049002}{4603447815395421679477102307642764315797233718739215194085360570730617748306620658556620009535701872398432457} a^{10} - \frac{630583256363379108168295829940467613061199483610828416907459197686818823245931600045764585429363807581885730}{1534482605131807226492367435880921438599077906246405064695120190243539249435540219518873336511900624132810819} a^{9} - \frac{275468545799463938697410071295848479113029850547140785222491582725323273596790567213788179420893448277099559}{1534482605131807226492367435880921438599077906246405064695120190243539249435540219518873336511900624132810819} a^{8} - \frac{1456715574846176320524529315680801102987063881234906991078201957722253388795234825170622518002341523399089709}{4603447815395421679477102307642764315797233718739215194085360570730617748306620658556620009535701872398432457} a^{7} + \frac{592174221606590414695141864782920612146973441135578509741102442843630605879761758581560415279337498010893331}{1534482605131807226492367435880921438599077906246405064695120190243539249435540219518873336511900624132810819} a^{6} + \frac{314400706978196108886128952958862026868229759968272952484997214650846422761193846416533537215122844750185260}{4603447815395421679477102307642764315797233718739215194085360570730617748306620658556620009535701872398432457} a^{5} + \frac{1546681714048523327156699627102555991618455262939126088445789119696474560955032984745472045938405663595174992}{4603447815395421679477102307642764315797233718739215194085360570730617748306620658556620009535701872398432457} a^{4} + \frac{988128525001367044462699833920098447477175436300155481304942783514689450276476920386928360305566550239526138}{4603447815395421679477102307642764315797233718739215194085360570730617748306620658556620009535701872398432457} a^{3} + \frac{666418268633179330924756995767624675542029121215060190449113343961756865369223825675976868437849277544428869}{1534482605131807226492367435880921438599077906246405064695120190243539249435540219518873336511900624132810819} a^{2} - \frac{1623504607626533272343842960100753043827155326479925994833669118355265974655502097501800640746621978116665459}{4603447815395421679477102307642764315797233718739215194085360570730617748306620658556620009535701872398432457} a + \frac{668740256640813089665591013037773177548789553798451248607947699721282432072999685069730040579394712871047815}{1534482605131807226492367435880921438599077906246405064695120190243539249435540219518873336511900624132810819}$
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
| Rank: | $17$ | magma: UnitRank(K);
sage: UK.rank()
gp: K.fu
| |
| Torsion generator: | \( -1 \) (order $2$) | magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
sage: UK.torsion_generator()
gp: K.tu[2]
| |
| Fundamental units: | Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH) | magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
| |
| Regulator: | \( 50471948791100000 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A non-solvable group of order 1351680 |
| The 112 conjugacy class representatives for t22n42 are not computed |
| Character table for t22n42 is not computed |
Intermediate fields
| 11.11.31376518243389673201.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Frobenius cycle types
| $p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 | 53 | 59 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/LocalNumberField/2.11.0.1}{11} }^{2}$ | ${\href{/LocalNumberField/3.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/5.12.0.1}{12} }{,}\,{\href{/LocalNumberField/5.6.0.1}{6} }{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }$ | ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }$ | ${\href{/LocalNumberField/11.11.0.1}{11} }^{2}$ | ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }$ | $22$ | ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ | $22$ | ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/31.11.0.1}{11} }^{2}$ | ${\href{/LocalNumberField/37.5.0.1}{5} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }$ | $22$ | ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/53.11.0.1}{11} }^{2}$ | ${\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$ |
Cycle lengths which are repeated in a cycle type are indicated by exponents.
Local algebras for ramified primes
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
| $97$ | 97.2.0.1 | $x^{2} - x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 97.10.5.1 | $x^{10} - 18818 x^{6} + 88529281 x^{2} - 214683506425$ | $2$ | $5$ | $5$ | $C_{10}$ | $[\ ]_{2}^{5}$ | |
| 97.10.0.1 | $x^{10} - 2 x + 57$ | $1$ | $10$ | $0$ | $C_{10}$ | $[\ ]^{10}$ | |
| 74843 | Data not computed | ||||||