Normalized defining polynomial
\( x^{22} - x^{21} + x^{20} - 3 x^{19} + 3 x^{18} + x^{17} + 5 x^{16} - 4 x^{15} - 18 x^{14} + 11 x^{13} + 16 x^{12} + 32 x^{11} - 45 x^{10} - 81 x^{9} + 100 x^{8} + 75 x^{7} - 101 x^{6} - 41 x^{5} + 53 x^{4} + \cdots + 1 \)
Invariants
Degree: | $22$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[2, 10]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(289863980721044983135295513\) \(\medspace = 359753\cdot 28385393161^{2}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(15.95\) | 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: | $359753^{1/2}28385393161^{1/2}\approx 101053106.56209058$ | ||
Ramified primes: | \(359753\), \(28385393161\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{359753}) \) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $\frac{1}{1905641144321}a^{21}-\frac{174163645302}{1905641144321}a^{20}+\frac{200880859921}{1905641144321}a^{19}-\frac{625037984800}{1905641144321}a^{18}-\frac{282703934710}{1905641144321}a^{17}+\frac{455664307019}{1905641144321}a^{16}+\frac{618485916718}{1905641144321}a^{15}-\frac{793283426287}{1905641144321}a^{14}-\frac{200498667426}{1905641144321}a^{13}+\frac{37721721529}{1905641144321}a^{12}+\frac{805073550203}{1905641144321}a^{11}+\frac{139691662963}{1905641144321}a^{10}+\frac{327071997433}{1905641144321}a^{9}-\frac{484157436258}{1905641144321}a^{8}-\frac{509817804171}{1905641144321}a^{7}+\frac{558788108214}{1905641144321}a^{6}-\frac{363903847855}{1905641144321}a^{5}+\frac{714674888219}{1905641144321}a^{4}-\frac{885861087370}{1905641144321}a^{3}-\frac{441615939835}{1905641144321}a^{2}+\frac{636477542155}{1905641144321}a+\frac{888963507629}{1905641144321}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
Rank: | $11$ | 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: | $\frac{3768728854142}{1905641144321}a^{21}-\frac{6167589910592}{1905641144321}a^{20}+\frac{7519723568573}{1905641144321}a^{19}-\frac{15853623460267}{1905641144321}a^{18}+\frac{20764568361318}{1905641144321}a^{17}-\frac{8342847494564}{1905641144321}a^{16}+\frac{22528363768782}{1905641144321}a^{15}-\frac{27425551748298}{1905641144321}a^{14}-\frac{53682052462685}{1905641144321}a^{13}+\frac{78719359542282}{1905641144321}a^{12}+\frac{8411459498969}{1905641144321}a^{11}+\frac{116536800016243}{1905641144321}a^{10}-\frac{243581091462724}{1905641144321}a^{9}-\frac{156189312652512}{1905641144321}a^{8}+\frac{483988318079050}{1905641144321}a^{7}-\frac{28995558804042}{1905641144321}a^{6}-\frac{365367224828456}{1905641144321}a^{5}+\frac{84292659131978}{1905641144321}a^{4}+\frac{145118905435797}{1905641144321}a^{3}-\frac{40150306614134}{1905641144321}a^{2}-\frac{29162737047133}{1905641144321}a+\frac{7551444814109}{1905641144321}$, $\frac{406625973316}{1905641144321}a^{21}-\frac{1639035342013}{1905641144321}a^{20}+\frac{3372365619074}{1905641144321}a^{19}-\frac{6638292849980}{1905641144321}a^{18}+\frac{11387600589727}{1905641144321}a^{17}-\frac{15786576426026}{1905641144321}a^{16}+\frac{20534792714157}{1905641144321}a^{15}-\frac{27416374215739}{1905641144321}a^{14}+\frac{25059381252713}{1905641144321}a^{13}-\frac{11255157773246}{1905641144321}a^{12}-\frac{99457537426}{1905641144321}a^{11}+\frac{18378451547820}{1905641144321}a^{10}-\frac{61703637503141}{1905641144321}a^{9}+\frac{76077166843427}{1905641144321}a^{8}-\frac{17862540717404}{1905641144321}a^{7}-\frac{53177234679716}{1905641144321}a^{6}+\frac{53896994246448}{1905641144321}a^{5}-\frac{5216908155023}{1905641144321}a^{4}-\frac{26500061837285}{1905641144321}a^{3}+\frac{12422527740978}{1905641144321}a^{2}+\frac{6355370944415}{1905641144321}a-\frac{4329194007635}{1905641144321}$, $\frac{6284591426259}{1905641144321}a^{21}-\frac{11164413916710}{1905641144321}a^{20}+\frac{15131418430283}{1905641144321}a^{19}-\frac{31260652963459}{1905641144321}a^{18}+\frac{43820563416380}{1905641144321}a^{17}-\frac{29300497204341}{1905641144321}a^{16}+\frac{56672121927949}{1905641144321}a^{15}-\frac{71411078729053}{1905641144321}a^{14}-\frac{55200948864055}{1905641144321}a^{13}+\frac{106784658211370}{1905641144321}a^{12}+\frac{17556930652995}{1905641144321}a^{11}+\frac{193530156420163}{1905641144321}a^{10}-\frac{433522799321522}{1905641144321}a^{9}-\frac{168342906579420}{1905641144321}a^{8}+\frac{733785619199345}{1905641144321}a^{7}-\frac{93207241106888}{1905641144321}a^{6}-\frac{526335478335158}{1905641144321}a^{5}+\frac{132121161584644}{1905641144321}a^{4}+\frac{207175755463998}{1905641144321}a^{3}-\frac{54884744322167}{1905641144321}a^{2}-\frac{42990108545067}{1905641144321}a+\frac{6006552618379}{1905641144321}$, $\frac{5891020371580}{1905641144321}a^{21}-\frac{10825794494890}{1905641144321}a^{20}+\frac{14846969712485}{1905641144321}a^{19}-\frac{29400556230104}{1905641144321}a^{18}+\frac{41591726554795}{1905641144321}a^{17}-\frac{27483369870222}{1905641144321}a^{16}+\frac{49849032895764}{1905641144321}a^{15}-\frac{62673799259405}{1905641144321}a^{14}-\frac{55404879890189}{1905641144321}a^{13}+\frac{116896237052082}{1905641144321}a^{12}-\frac{4970905377666}{1905641144321}a^{11}+\frac{184854050894373}{1905641144321}a^{10}-\frac{417769972685268}{1905641144321}a^{9}-\frac{127479977275503}{1905641144321}a^{8}+\frac{723942153091989}{1905641144321}a^{7}-\frac{179482537630422}{1905641144321}a^{6}-\frac{484915096609374}{1905641144321}a^{5}+\frac{195533996809617}{1905641144321}a^{4}+\frac{178091181674148}{1905641144321}a^{3}-\frac{78577543144891}{1905641144321}a^{2}-\frac{35995250209310}{1905641144321}a+\frac{11469328502926}{1905641144321}$, $\frac{9526465526330}{1905641144321}a^{21}-\frac{17546459795690}{1905641144321}a^{20}+\frac{24003905182041}{1905641144321}a^{19}-\frac{48686618355714}{1905641144321}a^{18}+\frac{68981182535746}{1905641144321}a^{17}-\frac{47569713524065}{1905641144321}a^{16}+\frac{86887127083780}{1905641144321}a^{15}-\frac{110528576331401}{1905641144321}a^{14}-\frac{81025658356265}{1905641144321}a^{13}+\frac{173082622263375}{1905641144321}a^{12}+\frac{9375539485445}{1905641144321}a^{11}+\frac{297882896553063}{1905641144321}a^{10}-\frac{678088344925179}{1905641144321}a^{9}-\frac{213331738620451}{1905641144321}a^{8}+\frac{11\!\cdots\!11}{1905641144321}a^{7}-\frac{229203896715090}{1905641144321}a^{6}-\frac{772502394331918}{1905641144321}a^{5}+\frac{257178079553635}{1905641144321}a^{4}+\frac{285082848094511}{1905641144321}a^{3}-\frac{99000283342910}{1905641144321}a^{2}-\frac{54904207061339}{1905641144321}a+\frac{11783776983131}{1905641144321}$, $\frac{3936676684659}{1905641144321}a^{21}-\frac{7262891328540}{1905641144321}a^{20}+\frac{9942012189116}{1905641144321}a^{19}-\frac{19997709028491}{1905641144321}a^{18}+\frac{28360363508386}{1905641144321}a^{17}-\frac{19214324946537}{1905641144321}a^{16}+\frac{34909115694617}{1905641144321}a^{15}-\frac{44210962511161}{1905641144321}a^{14}-\frac{35273417612984}{1905641144321}a^{13}+\frac{75243980284280}{1905641144321}a^{12}+\frac{184101158424}{1905641144321}a^{11}+\frac{125663422689401}{1905641144321}a^{10}-\frac{283398496497223}{1905641144321}a^{9}-\frac{85003096390380}{1905641144321}a^{8}+\frac{474052120025688}{1905641144321}a^{7}-\frac{101915066650664}{1905641144321}a^{6}-\frac{317408811047291}{1905641144321}a^{5}+\frac{107062390409884}{1905641144321}a^{4}+\frac{118473443937935}{1905641144321}a^{3}-\frac{39167871776301}{1905641144321}a^{2}-\frac{24025656045513}{1905641144321}a+\frac{5204348086216}{1905641144321}$, $\frac{984427042237}{1905641144321}a^{21}-\frac{1274051499886}{1905641144321}a^{20}+\frac{1377633092563}{1905641144321}a^{19}-\frac{3261388781916}{1905641144321}a^{18}+\frac{3642926272391}{1905641144321}a^{17}+\frac{541245158866}{1905641144321}a^{16}+\frac{3435781425506}{1905641144321}a^{15}-\frac{2679379055668}{1905641144321}a^{14}-\frac{20133772613411}{1905641144321}a^{13}+\frac{21513761957244}{1905641144321}a^{12}+\frac{3183086950823}{1905641144321}a^{11}+\frac{36684070726735}{1905641144321}a^{10}-\frac{59672605070899}{1905641144321}a^{9}-\frac{58898936243864}{1905641144321}a^{8}+\frac{117327376118215}{1905641144321}a^{7}+\frac{29841647140782}{1905641144321}a^{6}-\frac{95187189449181}{1905641144321}a^{5}-\frac{16416214333067}{1905641144321}a^{4}+\frac{50596320012521}{1905641144321}a^{3}+\frac{4736436928456}{1905641144321}a^{2}-\frac{14555175317569}{1905641144321}a-\frac{1494090380620}{1905641144321}$, $\frac{7766446366187}{1905641144321}a^{21}-\frac{12414489410708}{1905641144321}a^{20}+\frac{15973453255490}{1905641144321}a^{19}-\frac{33572570958261}{1905641144321}a^{18}+\frac{44065453703040}{1905641144321}a^{17}-\frac{20873117508260}{1905641144321}a^{16}+\frac{53177976149911}{1905641144321}a^{15}-\frac{61454346359489}{1905641144321}a^{14}-\frac{99518425399502}{1905641144321}a^{13}+\frac{143470909864584}{1905641144321}a^{12}+\frac{22364036802146}{1905641144321}a^{11}+\frac{243280318807470}{1905641144321}a^{10}-\frac{484332929906417}{1905641144321}a^{9}-\frac{310199063499038}{1905641144321}a^{8}+\frac{931909533473963}{1905641144321}a^{7}-\frac{43415449537389}{1905641144321}a^{6}-\frac{689110842623022}{1905641144321}a^{5}+\frac{150764708120052}{1905641144321}a^{4}+\frac{260735712524480}{1905641144321}a^{3}-\frac{66458238366602}{1905641144321}a^{2}-\frac{52237029824272}{1905641144321}a+\frac{6793646422111}{1905641144321}$, $\frac{6579428896915}{1905641144321}a^{21}-\frac{12169217738586}{1905641144321}a^{20}+\frac{16862997364065}{1905641144321}a^{19}-\frac{33800179683670}{1905641144321}a^{18}+\frac{48427196017968}{1905641144321}a^{17}-\frac{34041390130445}{1905641144321}a^{16}+\frac{61252533062103}{1905641144321}a^{15}-\frac{78295726976823}{1905641144321}a^{14}-\frac{52112106324230}{1905641144321}a^{13}+\frac{118125958609346}{1905641144321}a^{12}+\frac{7432220371545}{1905641144321}a^{11}+\frac{201350286374259}{1905641144321}a^{10}-\frac{468293774224837}{1905641144321}a^{9}-\frac{138243892222159}{1905641144321}a^{8}+\frac{786271531921571}{1905641144321}a^{7}-\frac{166991456839498}{1905641144321}a^{6}-\frac{537233488523989}{1905641144321}a^{5}+\frac{185336998511112}{1905641144321}a^{4}+\frac{198594042392744}{1905641144321}a^{3}-\frac{67917970702851}{1905641144321}a^{2}-\frac{38859021887116}{1905641144321}a+\frac{6466476572487}{1905641144321}$, $\frac{9135135203093}{1905641144321}a^{21}-\frac{15830166387517}{1905641144321}a^{20}+\frac{21787241316325}{1905641144321}a^{19}-\frac{44422740002040}{1905641144321}a^{18}+\frac{61932812366723}{1905641144321}a^{17}-\frac{40415638549028}{1905641144321}a^{16}+\frac{79863384234985}{1905641144321}a^{15}-\frac{97820105192479}{1905641144321}a^{14}-\frac{83687189164091}{1905641144321}a^{13}+\frac{156011704957735}{1905641144321}a^{12}+\frac{20797950699008}{1905641144321}a^{11}+\frac{283286113259978}{1905641144321}a^{10}-\frac{614081899842696}{1905641144321}a^{9}-\frac{250894837660172}{1905641144321}a^{8}+\frac{10\!\cdots\!96}{1905641144321}a^{7}-\frac{138545049894604}{1905641144321}a^{6}-\frac{753784430649425}{1905641144321}a^{5}+\frac{200935950358378}{1905641144321}a^{4}+\frac{291215636398257}{1905641144321}a^{3}-\frac{83325330630170}{1905641144321}a^{2}-\frac{61212365991980}{1905641144321}a+\frac{8093815508820}{1905641144321}$, $\frac{17792269672330}{1905641144321}a^{21}-\frac{29142162458155}{1905641144321}a^{20}+\frac{36219174303942}{1905641144321}a^{19}-\frac{75365607276549}{1905641144321}a^{18}+\frac{99944138452140}{1905641144321}a^{17}-\frac{42875910306232}{1905641144321}a^{16}+\frac{110820360579157}{1905641144321}a^{15}-\frac{135145351315999}{1905641144321}a^{14}-\frac{241087876788255}{1905641144321}a^{13}+\frac{362259065530917}{1905641144321}a^{12}+\frac{45092380847369}{1905641144321}a^{11}+\frac{536158914675151}{1905641144321}a^{10}-\frac{11\!\cdots\!42}{1905641144321}a^{9}-\frac{717772160459828}{1905641144321}a^{8}+\frac{22\!\cdots\!58}{1905641144321}a^{7}-\frac{156681348498175}{1905641144321}a^{6}-\frac{17\!\cdots\!19}{1905641144321}a^{5}+\frac{420528365857497}{1905641144321}a^{4}+\frac{682913474934971}{1905641144321}a^{3}-\frac{189110116179019}{1905641144321}a^{2}-\frac{146152594630065}{1905641144321}a+\frac{25905183581521}{1905641144321}$ (assuming GRH) | 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: | \( 15832.2441917 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{10}\cdot 15832.2441917 \cdot 1}{2\cdot\sqrt{289863980721044983135295513}}\cr\approx \mathstrut & 0.178350377482 \end{aligned}\] (assuming GRH)
Galois group
$C_2^{10}.(C_2\times S_{11})$ (as 22T53):
A non-solvable group of order 81749606400 |
The 752 conjugacy class representatives for $C_2^{10}.(C_2\times S_{11})$ are not computed |
Character table for $C_2^{10}.(C_2\times S_{11})$ is not computed |
Intermediate fields
11.3.28385393161.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 22 sibling: | data not computed |
Degree 44 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 | ${\href{/padicField/2.11.0.1}{11} }^{2}$ | $22$ | ${\href{/padicField/5.6.0.1}{6} }{,}\,{\href{/padicField/5.5.0.1}{5} }^{2}{,}\,{\href{/padicField/5.3.0.1}{3} }^{2}$ | ${\href{/padicField/7.9.0.1}{9} }^{2}{,}\,{\href{/padicField/7.2.0.1}{2} }^{2}$ | ${\href{/padicField/11.10.0.1}{10} }{,}\,{\href{/padicField/11.4.0.1}{4} }^{3}$ | ${\href{/padicField/13.8.0.1}{8} }^{2}{,}\,{\href{/padicField/13.3.0.1}{3} }^{2}$ | ${\href{/padicField/17.11.0.1}{11} }^{2}$ | ${\href{/padicField/19.10.0.1}{10} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | ${\href{/padicField/23.8.0.1}{8} }^{2}{,}\,{\href{/padicField/23.6.0.1}{6} }$ | $16{,}\,{\href{/padicField/29.3.0.1}{3} }^{2}$ | ${\href{/padicField/31.7.0.1}{7} }^{2}{,}\,{\href{/padicField/31.3.0.1}{3} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }$ | $16{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.4.0.1}{4} }^{2}{,}\,{\href{/padicField/41.3.0.1}{3} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.11.0.1}{11} }^{2}$ | $22$ | ${\href{/padicField/53.12.0.1}{12} }{,}\,{\href{/padicField/53.8.0.1}{8} }{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.14.0.1}{14} }{,}\,{\href{/padicField/59.4.0.1}{4} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(359753\) | 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 $14$ | $1$ | $14$ | $0$ | $C_{14}$ | $[\ ]^{14}$ | ||
\(28385393161\) | $\Q_{28385393161}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{28385393161}$ | $x$ | $1$ | $1$ | $0$ | 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 $14$ | $1$ | $14$ | $0$ | $C_{14}$ | $[\ ]^{14}$ |