Normalized defining polynomial
\( x^{22} + 330 x^{20} - 44 x^{19} + 42933 x^{18} + 22044 x^{17} + 2904550 x^{16} + 4754706 x^{15} + 112890888 x^{14} + 312942256 x^{13} + 2797111350 x^{12} + 8683357308 x^{11} + 46886008312 x^{10} + 118200277272 x^{9} + 429913692021 x^{8} + 903013471261 x^{7} + 1909585780575 x^{6} + 2829530153934 x^{5} + 3125769821658 x^{4} + 1357934108949 x^{3} + 2797610392467 x^{2} - 3135184653051 x + 14798920078539 \)
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: | \(-8922302982217185011636920905359755379521162753565903207=-\,11^{41}\cdot 13^{11}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $314.59$ | 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: | $11, 13$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(1573=11^{2}\cdot 13\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{1573}(1,·)$, $\chi_{1573}(1286,·)$, $\chi_{1573}(1288,·)$, $\chi_{1573}(714,·)$, $\chi_{1573}(716,·)$, $\chi_{1573}(142,·)$, $\chi_{1573}(144,·)$, $\chi_{1573}(1429,·)$, $\chi_{1573}(1431,·)$, $\chi_{1573}(857,·)$, $\chi_{1573}(859,·)$, $\chi_{1573}(285,·)$, $\chi_{1573}(287,·)$, $\chi_{1573}(1572,·)$, $\chi_{1573}(1000,·)$, $\chi_{1573}(1002,·)$, $\chi_{1573}(428,·)$, $\chi_{1573}(430,·)$, $\chi_{1573}(1143,·)$, $\chi_{1573}(1145,·)$, $\chi_{1573}(571,·)$, $\chi_{1573}(573,·)$$\rbrace$ | ||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $\frac{1}{3} a^{3} - \frac{1}{3} a$, $\frac{1}{3} a^{4} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{5} - \frac{1}{3} a$, $\frac{1}{9} a^{6} + \frac{1}{9} a^{4} - \frac{2}{9} a^{2}$, $\frac{1}{9} a^{7} + \frac{1}{9} a^{5} + \frac{1}{9} a^{3} - \frac{1}{3} a$, $\frac{1}{27} a^{8} - \frac{1}{27} a^{7} + \frac{1}{27} a^{6} + \frac{2}{27} a^{5} + \frac{4}{27} a^{4} - \frac{1}{27} a^{3} + \frac{4}{9} a^{2} + \frac{1}{3} a$, $\frac{1}{81} a^{9} - \frac{1}{27} a^{7} + \frac{1}{27} a^{5} - \frac{10}{81} a^{3} + \frac{1}{9} a$, $\frac{1}{81} a^{10} - \frac{1}{27} a^{7} - \frac{1}{27} a^{6} + \frac{2}{27} a^{5} - \frac{7}{81} a^{4} - \frac{1}{27} a^{3} - \frac{2}{9} a^{2} + \frac{1}{3} a$, $\frac{1}{81} a^{11} + \frac{1}{27} a^{7} + \frac{8}{81} a^{5} - \frac{4}{27} a^{3}$, $\frac{1}{2187} a^{12} + \frac{1}{243} a^{11} - \frac{7}{2187} a^{10} - \frac{1}{729} a^{9} - \frac{10}{729} a^{8} + \frac{1}{27} a^{7} + \frac{68}{2187} a^{6} + \frac{8}{81} a^{5} + \frac{4}{2187} a^{4} + \frac{70}{729} a^{3} + \frac{41}{243} a^{2} - \frac{7}{81} a$, $\frac{1}{2187} a^{13} - \frac{7}{2187} a^{11} + \frac{2}{729} a^{10} - \frac{1}{729} a^{9} + \frac{1}{81} a^{8} + \frac{68}{2187} a^{7} - \frac{8}{243} a^{6} - \frac{77}{2187} a^{5} - \frac{86}{729} a^{4} + \frac{20}{243} a^{3} - \frac{31}{81} a^{2} - \frac{2}{9} a$, $\frac{1}{2187} a^{14} - \frac{4}{729} a^{11} + \frac{2}{2187} a^{10} + \frac{2}{729} a^{9} + \frac{20}{2187} a^{8} - \frac{8}{243} a^{7} - \frac{29}{729} a^{6} - \frac{68}{729} a^{5} - \frac{8}{2187} a^{4} - \frac{59}{729} a^{3} - \frac{37}{243} a^{2} + \frac{5}{81} a$, $\frac{1}{6561} a^{15} + \frac{1}{6561} a^{13} - \frac{32}{6561} a^{11} + \frac{4}{729} a^{10} - \frac{19}{6561} a^{9} + \frac{1}{81} a^{8} - \frac{19}{6561} a^{7} - \frac{10}{243} a^{6} - \frac{355}{6561} a^{5} - \frac{100}{729} a^{4} - \frac{106}{729} a^{3} - \frac{35}{81} a^{2} + \frac{11}{81} a$, $\frac{1}{6561} a^{16} + \frac{1}{6561} a^{14} + \frac{1}{6561} a^{12} + \frac{1}{729} a^{11} - \frac{7}{6561} a^{10} - \frac{2}{729} a^{9} - \frac{37}{6561} a^{8} - \frac{10}{243} a^{7} - \frac{55}{6561} a^{6} + \frac{53}{729} a^{5} - \frac{274}{2187} a^{4} - \frac{31}{729} a^{3} - \frac{56}{243} a^{2} + \frac{31}{81} a$, $\frac{1}{59049} a^{17} - \frac{2}{59049} a^{16} + \frac{4}{59049} a^{15} - \frac{2}{59049} a^{14} - \frac{2}{59049} a^{13} - \frac{8}{59049} a^{12} + \frac{272}{59049} a^{11} - \frac{151}{59049} a^{10} + \frac{5}{59049} a^{9} - \frac{880}{59049} a^{8} + \frac{1721}{59049} a^{7} + \frac{1133}{59049} a^{6} - \frac{1549}{19683} a^{5} + \frac{458}{6561} a^{4} + \frac{350}{2187} a^{3} - \frac{16}{729} a^{2} - \frac{76}{243} a$, $\frac{1}{59049} a^{18} - \frac{1}{19683} a^{15} - \frac{2}{19683} a^{14} + \frac{2}{19683} a^{13} + \frac{13}{59049} a^{12} - \frac{79}{19683} a^{11} - \frac{8}{2187} a^{10} - \frac{17}{19683} a^{9} + \frac{230}{19683} a^{8} + \frac{736}{19683} a^{7} - \frac{923}{59049} a^{6} - \frac{2567}{19683} a^{5} + \frac{202}{6561} a^{4} - \frac{254}{2187} a^{3} - \frac{35}{729} a^{2} - \frac{50}{243} a$, $\frac{1}{177147} a^{19} + \frac{1}{177147} a^{18} - \frac{1}{59049} a^{16} - \frac{1}{19683} a^{15} - \frac{8}{177147} a^{13} - \frac{8}{177147} a^{12} + \frac{317}{59049} a^{11} + \frac{325}{59049} a^{10} + \frac{89}{19683} a^{9} + \frac{7}{19683} a^{8} - \frac{551}{177147} a^{7} - \frac{2927}{177147} a^{6} + \frac{2134}{59049} a^{5} + \frac{1444}{19683} a^{4} + \frac{709}{6561} a^{3} + \frac{268}{2187} a^{2} - \frac{137}{729} a$, $\frac{1}{6949653957} a^{20} - \frac{11102}{6949653957} a^{19} - \frac{2926}{772183773} a^{18} - \frac{8051}{2316551319} a^{17} + \frac{52967}{2316551319} a^{16} - \frac{64156}{2316551319} a^{15} + \frac{217789}{6949653957} a^{14} - \frac{459332}{6949653957} a^{13} - \frac{28529}{2316551319} a^{12} + \frac{1730720}{2316551319} a^{11} + \frac{14010373}{2316551319} a^{10} + \frac{2402053}{2316551319} a^{9} + \frac{25440400}{6949653957} a^{8} + \frac{269139370}{6949653957} a^{7} + \frac{86448056}{2316551319} a^{6} + \frac{42313925}{772183773} a^{5} - \frac{13260178}{257394591} a^{4} + \frac{10497737}{85798197} a^{3} - \frac{2956912}{28599399} a^{2} - \frac{2723417}{9533133} a + \frac{4940}{13077}$, $\frac{1}{1254298479631474445080665500536405716646053663645964782298524322165952625235884209985962916783447303} a^{21} - \frac{90210102983906519095617858901955468375932258298618696743147796333474508053115267477180073}{1254298479631474445080665500536405716646053663645964782298524322165952625235884209985962916783447303} a^{20} - \frac{1859339949040112775263036175659453456244425817056246929513679487237253321197556009065163024730}{1254298479631474445080665500536405716646053663645964782298524322165952625235884209985962916783447303} a^{19} - \frac{886443011310693307260937113860369811875681208487346677378234383811377824313102484400988284465}{418099493210491481693555166845468572215351221215321594099508107388650875078628069995320972261149101} a^{18} + \frac{1134942947323854037819513993007048873655041494700446618854751460368142789299319239942768539322}{139366497736830493897851722281822857405117073738440531366502702462883625026209356665106990753716367} a^{17} + \frac{3949445390492798151999037984383552098616602876031992621625262847557413489514623104200158313843}{418099493210491481693555166845468572215351221215321594099508107388650875078628069995320972261149101} a^{16} + \frac{16294579035564136108808520278723072258759333911169965385370403405491612807772572886704094838635}{1254298479631474445080665500536405716646053663645964782298524322165952625235884209985962916783447303} a^{15} + \frac{231384611408395308867827342383625866902256789218868990412780596404238055403019538662623654603839}{1254298479631474445080665500536405716646053663645964782298524322165952625235884209985962916783447303} a^{14} + \frac{116279307013414969528093766422265637535597834647361674218497750080125847852238524847612678327168}{1254298479631474445080665500536405716646053663645964782298524322165952625235884209985962916783447303} a^{13} - \frac{6291900163506415556463946178733142302403584781075385930875731974894412040769880994636332505969}{46455499245610164632617240760607619135039024579480177122167567487627875008736452221702330251238789} a^{12} + \frac{680110024104164330692569053174026291979628684241581430437032922739402961158823808699982027217813}{139366497736830493897851722281822857405117073738440531366502702462883625026209356665106990753716367} a^{11} + \frac{824986873802323859352806914604737410783340932750972557796876103752873508349582918517828454111367}{418099493210491481693555166845468572215351221215321594099508107388650875078628069995320972261149101} a^{10} + \frac{6088907349014673264353599287293276067256661157530779531211931340782729502014552792942883925355058}{1254298479631474445080665500536405716646053663645964782298524322165952625235884209985962916783447303} a^{9} + \frac{452964728967809356612370371502805521753592626686793115205054417158635471625232415825781859456523}{1254298479631474445080665500536405716646053663645964782298524322165952625235884209985962916783447303} a^{8} - \frac{30043670331451058312464670344695846076923725136754307495792660128002581876085174099553104266753585}{1254298479631474445080665500536405716646053663645964782298524322165952625235884209985962916783447303} a^{7} + \frac{17302602776898523916948418023848916301111185923567371085382375558474010266818319237449634039213297}{418099493210491481693555166845468572215351221215321594099508107388650875078628069995320972261149101} a^{6} + \frac{4541881794608799446680894560050247015405620295818418613570438616583519934897446888481689908699321}{139366497736830493897851722281822857405117073738440531366502702462883625026209356665106990753716367} a^{5} + \frac{6676079854706856743468501695971720787787185320266310321079740654615372216066170409456431824730870}{46455499245610164632617240760607619135039024579480177122167567487627875008736452221702330251238789} a^{4} - \frac{2054489119087337990459886413484197643299700403350640251899477948429700782689015193057078691489985}{15485166415203388210872413586869206378346341526493392374055855829209291669578817407234110083746263} a^{3} - \frac{1166455224676249219313997172395500363766277564129486285603400453451332649086041815707493686389231}{5161722138401129403624137862289735459448780508831130791351951943069763889859605802411370027915421} a^{2} - \frac{208226636545657126655086753806641130074246382938353820962122164400547389154667248293663403436168}{1720574046133709801208045954096578486482926836277043597117317314356587963286535267470456675971807} a - \frac{365326787055904085183832060134654073101195559432320229283337702602359238036033418494642473679}{2360183876726625241711997193548118637150791270613228528281642406524812020969184180343561969783}$
Class group and class number
$C_{38227153810}$, which has order $38227153810$ (assuming GRH)
Unit group
| Rank: | $10$ | 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: | \( 285114946276.13544 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A cyclic group of order 22 |
| The 22 conjugacy class representatives for $C_{22}$ |
| Character table for $C_{22}$ is not computed |
Intermediate fields
| \(\Q(\sqrt{-143}) \), 11.11.672749994932560009201.1 |
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 | ${\href{/LocalNumberField/2.11.0.1}{11} }^{2}$ | ${\href{/LocalNumberField/3.1.0.1}{1} }^{22}$ | $22$ | ${\href{/LocalNumberField/7.11.0.1}{11} }^{2}$ | R | R | $22$ | ${\href{/LocalNumberField/19.11.0.1}{11} }^{2}$ | ${\href{/LocalNumberField/23.11.0.1}{11} }^{2}$ | $22$ | $22$ | $22$ | ${\href{/LocalNumberField/41.11.0.1}{11} }^{2}$ | $22$ | $22$ | ${\href{/LocalNumberField/53.11.0.1}{11} }^{2}$ | $22$ |
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 |
|---|---|---|---|---|---|---|---|
| 11 | Data not computed | ||||||
| 13 | Data not computed | ||||||