Normalized defining polynomial
\( x^{22} - 242454 x^{20} - 66254755944 x^{18} + 9988149125214624 x^{16} + 2412251307077297217168 x^{14} - 97126283753436220637656848 x^{12} - 36043439335840553783024429840880 x^{10} - 968324930877205693808500678923025440 x^{8} + 411693823788000055227773052463075223008512 x^{6} - 14674638475171337576102371060253129696887988224 x^{4} + 164949721485379801300637463025582558350584505753600 x^{2} - 571411117069770362003780702945732790998630818613760000 \)
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: | \(1275054393023762909963506622668395857607199180560057153446822299847849103680658423408180099481492815135308821561344=2^{44}\cdot 3^{28}\cdot 7^{4}\cdot 13^{10}\cdot 23^{4}\cdot 137^{16}\cdot 2161^{10}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $153{,}679.15$ | 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: | $2, 3, 7, 13, 23, 137, 2161$ | 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 \(a\))
$1$, $a$, $a^{2}$, $\frac{1}{2} a^{3}$, $\frac{1}{56186} a^{4} - \frac{8855}{28093} a^{2}$, $\frac{1}{56186} a^{5} + \frac{10383}{56186} a^{3}$, $\frac{1}{3156866596} a^{6} - \frac{8855}{1578433298} a^{4} - \frac{6976}{28093} a^{2}$, $\frac{1}{3156866596} a^{7} - \frac{8855}{1578433298} a^{5} - \frac{6976}{28093} a^{3}$, $\frac{1}{177371706562856} a^{8} - \frac{8855}{88685853281428} a^{6} - \frac{3488}{789216649} a^{4} - \frac{6505}{56186} a^{2}$, $\frac{1}{177371706562856} a^{9} - \frac{8855}{88685853281428} a^{7} - \frac{3488}{789216649} a^{5} - \frac{6505}{56186} a^{3}$, $\frac{1}{9965806704940627216} a^{10} - \frac{8855}{4982903352470313608} a^{8} - \frac{1744}{22171463320357} a^{6} - \frac{1}{112372} a^{5} - \frac{6505}{3156866596} a^{4} + \frac{8855}{56186} a^{3} + \frac{23263}{56186} a^{2}$, $\frac{1}{49829033524703136080} a^{11} + \frac{47331}{24914516762351568040} a^{9} - \frac{52779}{443429266407140} a^{7} + \frac{95673}{15784332980} a^{5} + \frac{44971}{280930} a^{3} + \frac{1}{5} a$, $\frac{1}{1399847038809485201895440} a^{12} - \frac{45803}{1399847038809485201895440} a^{10} - \frac{61283}{24914516762351568040} a^{8} - \frac{2478}{110857316601785} a^{6} - \frac{1}{112372} a^{5} - \frac{69913}{15784332980} a^{4} + \frac{8855}{56186} a^{3} + \frac{35447}{280930} a^{2}$, $\frac{1}{2799694077618970403790880} a^{13} - \frac{1771}{279969407761897040379088} a^{11} - \frac{872}{3114314595293946005} a^{9} - \frac{62691}{886858532814280} a^{7} - \frac{25517}{3156866596} a^{5} - \frac{55981}{280930} a^{3} - \frac{1}{2} a^{2} - \frac{2}{5} a$, $\frac{1}{78651805722549735553697191840} a^{14} - \frac{1771}{7865180572254973555369719184} a^{12} - \frac{872}{87490439925592825118465} a^{10} - \frac{62691}{24914516762351568040} a^{8} + \frac{644}{22171463320357} a^{6} + \frac{3093}{607089730} a^{4} + \frac{2608}{140465} a^{2}$, $\frac{1}{78651805722549735553697191840} a^{15} + \frac{10383}{78651805722549735553697191840} a^{13} + \frac{2643}{699923519404742600947720} a^{11} - \frac{2792}{3114314595293946005} a^{9} + \frac{138441}{886858532814280} a^{7} - \frac{10011}{3946083245} a^{5} + \frac{64911}{280930} a^{3} - \frac{1}{2} a^{2} - \frac{1}{5} a$, $\frac{1}{4419130356327179441820030420722240} a^{16} - \frac{1771}{441913035632717944182003042072224} a^{14} - \frac{436}{2457868928829679236053037245} a^{12} - \frac{62691}{1399847038809485201895440} a^{10} + \frac{322}{622862919058789201} a^{8} + \frac{3093}{34109943569780} a^{6} - \frac{1}{112372} a^{5} + \frac{1304}{3946083245} a^{4} + \frac{8855}{56186} a^{3} - \frac{1107}{56186} a^{2}$, $\frac{1}{8838260712654358883640060841444480} a^{17} - \frac{1771}{883826071265435888364006084144448} a^{15} - \frac{218}{2457868928829679236053037245} a^{13} + \frac{5397}{699923519404742600947720} a^{11} + \frac{1801}{6228629190587892010} a^{9} - \frac{437}{5247683626120} a^{7} + \frac{16213}{6313733192} a^{5} - \frac{30477}{561860} a^{3} - \frac{1}{5} a$, $\frac{1}{399752145702964235630141369042106640390400} a^{18} - \frac{21415721}{199876072851482117815070684521053320195200} a^{16} + \frac{1945217}{444674992105422431333140561085175400} a^{14} - \frac{5665953}{31657351803326268560363119715600} a^{12} + \frac{10748611}{563438433120817793762914600} a^{10} + \frac{7491833}{8022474397477204908880} a^{8} - \frac{2807811}{21966803658938320} a^{6} - \frac{1}{112372} a^{5} - \frac{7610273}{5082555219560} a^{4} + \frac{8855}{56186} a^{3} - \frac{43299709}{113074325} a^{2} - \frac{6}{161}$, $\frac{1}{2398512874217785413780848214252639842342400} a^{19} + \frac{7938003}{399752145702964235630141369042106640390400} a^{17} + \frac{217531}{3557399936843379450665124488681403200} a^{15} + \frac{236109}{3957168975415783570045389964450} a^{13} + \frac{668049}{2253753732483271175051658400} a^{11} + \frac{27619181}{16044948794954409817760} a^{9} - \frac{6100817}{43933607317876640} a^{7} - \frac{67116863}{10165110439120} a^{5} - \frac{13532514}{113074325} a^{3} + \frac{629}{2415} a$, $\frac{1}{1481726785021316720511144273429268991459011383271438752370766341987216106458801064217720960000} a^{20} + \frac{1105981020431405486519287531258795958492118764413}{35279209167174207631217720795934975987119318649319970294542055761600383487114311052802880000} a^{18} + \frac{17519756046910411398492530050607939948639356047795733}{2197651231358518611562702858110070408196305401214179618248268165834929381071798823280000} a^{16} + \frac{1578192503452362997096668650496622944118426212248497}{850301263102257645631475138518983689344051899519367975872168436603783930807380000} a^{14} - \frac{73684326925120451369263093311292823058924589222530737}{1392299081718002765779655301031333417927113066525146011110232018074753882360000} a^{12} + \frac{1520914696857475671352502163065576638314420331541797849}{49560356021713692584617353113990439537504469673055423454605489555218520000} a^{10} - \frac{159706542106577114321177376396451293427353417476356063}{117610213746517382941461937407872517560731545635461795831952181576000} a^{8} - \frac{25300600177307441628930851833032007809275328558415483}{697743291605959865099621123932845178281253607870654586741372000} a^{6} - \frac{1}{112372} a^{5} + \frac{3602797485492995619883757740774376074952498333277551}{862392682348162879023369211108398368169578631527741423750} a^{4} + \frac{8855}{56186} a^{3} - \frac{101124194115562669927144903802221379858265124931901643}{532825721228479843994382734639897085357327467089516875} a^{2} + \frac{56018734779821855860352097937794227802560747473}{196689569358881224291311748543094541674480252475}$, $\frac{1}{133355410651918504846002984608634209231311024494429487713368970778849449581292095779594886400000} a^{21} - \frac{2463333941452828713452371872567667664730334726165709}{22225901775319750807667164101439034871885170749071581285561495129808241596882015963265814400000} a^{19} + \frac{2948605110388917378726185493262487641558733382162383833}{197788610822266675040643257229906336737667486109276165642344134925143644296461894095200000} a^{17} - \frac{3169202938322226277372826145626877442580153515377639497}{880061807310836663228576768367148118471093716002545855027694331884916368385638300000} a^{15} - \frac{1963003987359752921039670343963582133726458266723852791}{17900988193517178417166996727545715373348596569609020142845840232389692773200000} a^{13} + \frac{44222057150239265296305328033905224308252230910756612549}{4460432041954232332615561780259139558375402270574988110914494059969666800000} a^{11} - \frac{904832214656389875995206864375460489827412850171797851}{814224556706658804979351874362194352343526085168581663451976641680000} a^{9} + \frac{2564652553842079473860549920462934805144504732089797}{390042833816996197881775783564944509598216302536390762774680000} a^{7} + \frac{1334213358409218713687600171746381787906937344111257477}{155230682822669318224206457999511706270524153674993456275000} a^{5} - \frac{83342843064781911049632023524608499250692791368461299977}{671360408747884603432922245646270327550232608532791262500} a^{3} + \frac{337703813554360557482024050556937277345260150149}{8851030621149655093109028684439254375351611361375} a$
Class group and class number
Not computed
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: | Not computed | magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
| |
| Regulator: | Not computed | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A non-solvable group of order 20437401600 |
| The 200 conjugacy class representatives for t22n49 are not computed |
| Character table for t22n49 is not computed |
Intermediate fields
| 11.7.63019333158425674204677255696384.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Frobenius cycle types
| $p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 | 53 | 59 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | R | ${\href{/LocalNumberField/5.6.0.1}{6} }{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{2}$ | R | ${\href{/LocalNumberField/11.11.0.1}{11} }^{2}$ | R | ${\href{/LocalNumberField/17.11.0.1}{11} }^{2}$ | ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ | R | ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/31.11.0.1}{11} }^{2}$ | ${\href{/LocalNumberField/37.12.0.1}{12} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{4}$ | ${\href{/LocalNumberField/41.7.0.1}{7} }^{2}{,}\,{\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }$ | ${\href{/LocalNumberField/43.12.0.1}{12} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }$ | $18{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/53.10.0.1}{10} }{,}\,{\href{/LocalNumberField/53.8.0.1}{8} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/59.12.0.1}{12} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }$ |
In the table, R denotes a ramified prime. 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 |
|---|---|---|---|---|---|---|---|
| $2$ | 2.2.3.4 | $x^{2} + 10$ | $2$ | $1$ | $3$ | $C_2$ | $[3]$ |
| 2.8.21.18 | $x^{8} + 8 x^{6} + 56 x^{2} + 40$ | $8$ | $1$ | $21$ | $C_2 \wr C_2\wr C_2$ | $[2, 2, 3, 7/2, 7/2, 15/4]^{2}$ | |
| 2.12.20.78 | $x^{12} + 2 x^{11} + 2 x^{10} + 2 x^{9} + 4 x^{3} + 2 x^{2} + 2$ | $12$ | $1$ | $20$ | 12T149 | $[4/3, 4/3, 2, 2, 7/3, 7/3]_{3}^{2}$ | |
| $3$ | 3.2.1.1 | $x^{2} - 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 3.2.1.1 | $x^{2} - 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 3.9.13.2 | $x^{9} + 6 x^{5} + 3 x^{3} + 3$ | $9$ | $1$ | $13$ | $C_3^2 : D_{6} $ | $[3/2, 3/2, 5/3]_{2}^{2}$ | |
| 3.9.13.2 | $x^{9} + 6 x^{5} + 3 x^{3} + 3$ | $9$ | $1$ | $13$ | $C_3^2 : D_{6} $ | $[3/2, 3/2, 5/3]_{2}^{2}$ | |
| $7$ | 7.7.0.1 | $x^{7} - x + 2$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ |
| 7.7.0.1 | $x^{7} - x + 2$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | |
| 7.8.4.1 | $x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
| 13 | Data not computed | ||||||
| $23$ | 23.4.2.1 | $x^{4} + 299 x^{2} + 25921$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| 23.4.2.1 | $x^{4} + 299 x^{2} + 25921$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 23.7.0.1 | $x^{7} - x + 8$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | |
| 23.7.0.1 | $x^{7} - x + 8$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | |
| 137 | Data not computed | ||||||
| 2161 | Data not computed | ||||||