Normalized defining polynomial
\( x^{22} - 4 x^{21} + 41 x^{20} - 62 x^{19} + 712 x^{18} - 691 x^{17} + 8218 x^{16} - 3289 x^{15} + 62044 x^{14} - 12972 x^{13} + 331973 x^{12} + 1202 x^{11} + 1201112 x^{10} + 23541 x^{9} + 2960817 x^{8} + 250587 x^{7} + 4248123 x^{6} - 781164 x^{5} + 3132885 x^{4} - 746848 x^{3} + 1635703 x^{2} - 490413 x + 335241 \)
Invariants
| Degree: | $22$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[0, 11]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(-174398723209634833558490883294218000698177947=-\,3^{11}\cdot 74843^{8}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $102.56$ | 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: | $3, 74843$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is not Galois over $\Q$. | |||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{3} a^{15} - \frac{1}{3} a^{14} - \frac{1}{3} a^{13} - \frac{1}{3} a^{12} + \frac{1}{3} a^{10} - \frac{1}{3} a^{9} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{3} a^{16} + \frac{1}{3} a^{14} + \frac{1}{3} a^{13} - \frac{1}{3} a^{12} + \frac{1}{3} a^{11} - \frac{1}{3} a^{9} - \frac{1}{3} a^{6} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a$, $\frac{1}{9} a^{17} + \frac{1}{9} a^{15} + \frac{1}{9} a^{14} - \frac{4}{9} a^{13} + \frac{1}{9} a^{12} - \frac{1}{9} a^{10} + \frac{1}{3} a^{9} + \frac{1}{3} a^{8} + \frac{2}{9} a^{7} + \frac{4}{9} a^{5} - \frac{1}{9} a^{4} - \frac{2}{9} a^{2} + \frac{1}{3} a$, $\frac{1}{27} a^{18} + \frac{1}{27} a^{17} - \frac{2}{27} a^{16} - \frac{1}{27} a^{15} - \frac{1}{9} a^{14} + \frac{2}{9} a^{13} - \frac{11}{27} a^{12} + \frac{5}{27} a^{11} - \frac{1}{27} a^{10} + \frac{4}{9} a^{9} + \frac{5}{27} a^{8} + \frac{2}{27} a^{7} - \frac{11}{27} a^{6} - \frac{4}{9} a^{5} - \frac{7}{27} a^{4} + \frac{10}{27} a^{3} + \frac{13}{27} a^{2} - \frac{4}{9} a + \frac{1}{3}$, $\frac{1}{27} a^{19} + \frac{1}{27} a^{16} + \frac{1}{27} a^{15} + \frac{4}{9} a^{14} - \frac{2}{27} a^{13} - \frac{8}{27} a^{12} - \frac{2}{9} a^{11} + \frac{10}{27} a^{10} + \frac{2}{27} a^{9} + \frac{2}{9} a^{8} - \frac{7}{27} a^{7} - \frac{1}{27} a^{6} - \frac{10}{27} a^{5} - \frac{13}{27} a^{4} + \frac{1}{9} a^{3} - \frac{4}{27} a^{2} + \frac{1}{9} a - \frac{1}{3}$, $\frac{1}{293445423} a^{20} - \frac{27355}{10868349} a^{19} + \frac{2954365}{293445423} a^{18} - \frac{14056396}{293445423} a^{17} - \frac{33523546}{293445423} a^{16} + \frac{1401650}{293445423} a^{15} - \frac{99793667}{293445423} a^{14} - \frac{49034552}{293445423} a^{13} - \frac{71628227}{293445423} a^{12} - \frac{702754}{3622783} a^{11} - \frac{12134150}{293445423} a^{10} - \frac{95408}{3622783} a^{9} + \frac{95382190}{293445423} a^{8} - \frac{112425911}{293445423} a^{7} - \frac{43229354}{97815141} a^{6} + \frac{78699326}{293445423} a^{5} - \frac{37151581}{293445423} a^{4} - \frac{30516749}{97815141} a^{3} - \frac{63263621}{293445423} a^{2} - \frac{11555338}{97815141} a + \frac{13556944}{32605047}$, $\frac{1}{4595727019055596840753456665892009680012130642298362047669} a^{21} - \frac{1124811694536991969138814760665372996749814784554}{4595727019055596840753456665892009680012130642298362047669} a^{20} - \frac{24700588349653952075864758257416067173351553674504087084}{4595727019055596840753456665892009680012130642298362047669} a^{19} + \frac{1780077192790431320689365961810352676208021921972418557}{170212111816873957064942839477481840000449283048087483247} a^{18} - \frac{142567312907439677229831464984676937334832893986070040716}{4595727019055596840753456665892009680012130642298362047669} a^{17} + \frac{115782760124719607832742058008545390015821916975319184605}{4595727019055596840753456665892009680012130642298362047669} a^{16} - \frac{190812585852337045347257271127196048712753909511485445552}{1531909006351865613584485555297336560004043547432787349223} a^{15} + \frac{2000876124945060298335576626697840999662649341644449809078}{4595727019055596840753456665892009680012130642298362047669} a^{14} + \frac{1111228684735369661415216498203106668012412689906505578297}{4595727019055596840753456665892009680012130642298362047669} a^{13} - \frac{1121098634204242185796449650899531368953671368789833703656}{4595727019055596840753456665892009680012130642298362047669} a^{12} + \frac{1791203610350517216723927898703472484387440410900643988831}{4595727019055596840753456665892009680012130642298362047669} a^{11} - \frac{1797238496010390478119036676166875993874777944151757848426}{4595727019055596840753456665892009680012130642298362047669} a^{10} + \frac{616236731474599027371244161994027366296534189391587743263}{4595727019055596840753456665892009680012130642298362047669} a^{9} + \frac{26603389934576139205444770813971882777442209139254915438}{58173759734880972667765274251797590886229501801245089211} a^{8} + \frac{1567044114590355723043837727078887362796071597916407800764}{4595727019055596840753456665892009680012130642298362047669} a^{7} + \frac{6385223161027348213272043582967214354349803861569327105}{58173759734880972667765274251797590886229501801245089211} a^{6} - \frac{4420204783845070104074668331619431928239630692484127350}{23812057093552315237064542310321293678819329752841254133} a^{5} + \frac{2218018930953982345295886498773544283567917890572339872167}{4595727019055596840753456665892009680012130642298362047669} a^{4} + \frac{736161600849391998467574560815005246619132213163409488440}{4595727019055596840753456665892009680012130642298362047669} a^{3} - \frac{479130008576378934972608733015955734188242026807755850524}{4595727019055596840753456665892009680012130642298362047669} a^{2} - \frac{460152598570793315111911077062768604990346421288350153276}{1531909006351865613584485555297336560004043547432787349223} a + \frac{1315464416736899087516580642723683791053011373700564786}{2645784121505812804118282478924588186535481083649028237}$
Class group and class number
$C_{2}\times C_{2}\times C_{2}\times C_{3580}$, which has order $28640$ (assuming GRH)
Unit group
| Rank: | $10$ | magma: UnitRank(K);
sage: UK.rank()
gp: K.fu
| |
| Torsion generator: | \( \frac{244150480953480274008917699828905627488621173}{140951399918411307327772190173257829685451170866227} a^{21} - \frac{941793195357776211005953198601852804610660691}{140951399918411307327772190173257829685451170866227} a^{20} + \frac{9775830001318244013526705131076126200844388918}{140951399918411307327772190173257829685451170866227} a^{19} - \frac{4434765590905397595483227821760664197498931395}{46983799972803769109257396724419276561817056955409} a^{18} + \frac{167760184255851131715991863727201315031339947136}{140951399918411307327772190173257829685451170866227} a^{17} - \frac{137329838156907194310437544023144038384879492621}{140951399918411307327772190173257829685451170866227} a^{16} + \frac{70979820266376335058729509123313910950924829452}{5220422219200418789917488524935475173535228550601} a^{15} - \frac{438211946621070971674090974884194852919853959909}{140951399918411307327772190173257829685451170866227} a^{14} + \frac{14290365624026369091062048112602914081053218954179}{140951399918411307327772190173257829685451170866227} a^{13} - \frac{543287038141193063158278763061372264064816483226}{140951399918411307327772190173257829685451170866227} a^{12} + \frac{75239174110337692979740664031660222350074311155396}{140951399918411307327772190173257829685451170866227} a^{11} + \frac{14389418227120276724400568956271845641751306491651}{140951399918411307327772190173257829685451170866227} a^{10} + \frac{265827703981257208875958532591072402490616694577491}{140951399918411307327772190173257829685451170866227} a^{9} + \frac{691096166349763769653625827533695548704586639693}{1784194935676092497819901141433643413739888238813} a^{8} + \frac{631906029484707167207266362512836818020467451673311}{140951399918411307327772190173257829685451170866227} a^{7} + \frac{2427634991855678357449602019300511080789536443350}{1784194935676092497819901141433643413739888238813} a^{6} + \frac{4342869295413691894710402716782974549126078606854}{730318134292286566465140881726724506142234045939} a^{5} + \frac{3022687128782794943355671762558562997407435587332}{140951399918411307327772190173257829685451170866227} a^{4} + \frac{502228101976131696773832488845893519040600711809647}{140951399918411307327772190173257829685451170866227} a^{3} + \frac{144378447741498728470396802329855307990082262260185}{140951399918411307327772190173257829685451170866227} a^{2} + \frac{76189383639496925397009945554183440113631410889147}{46983799972803769109257396724419276561817056955409} a + \frac{50699474853675010847476954466395793384355859985}{81146459365809618496126764636302722904692671771} \) (order $6$) | 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: | \( 3049992559.69 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A non-solvable group of order 1320 |
| The 16 conjugacy class representatives for t22n13 |
| Character table for t22n13 |
Intermediate fields
| \(\Q(\sqrt{-3}) \), 11.11.31376518243389673201.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Degree 24 sibling: | data not computed |
| Arithmetically equvalently siblings: | data not computed |
Frobenius cycle types
| $p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 | 53 | 59 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | $22$ | R | ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/7.5.0.1}{5} }^{4}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{2}$ | $22$ | ${\href{/LocalNumberField/13.5.0.1}{5} }^{4}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ | $22$ | ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ | $22$ | ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }$ | ${\href{/LocalNumberField/31.11.0.1}{11} }^{2}$ | ${\href{/LocalNumberField/37.5.0.1}{5} }^{4}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ | $22$ | ${\href{/LocalNumberField/43.5.0.1}{5} }^{4}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ | $22$ | ${\href{/LocalNumberField/59.2.0.1}{2} }^{11}$ |
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 |
|---|---|---|---|---|---|---|---|
| $3$ | 3.2.1.2 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 3.2.1.2 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 3.6.3.2 | $x^{6} - 9 x^{2} + 27$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
| 3.6.3.2 | $x^{6} - 9 x^{2} + 27$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
| 3.6.3.2 | $x^{6} - 9 x^{2} + 27$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
| 74843 | Data not computed | ||||||