Normalized defining polynomial
\( x^{22} - 9 x^{21} - 112 x^{20} + 1045 x^{19} + 5483 x^{18} - 55170 x^{17} - 113261 x^{16} + 1745278 x^{15} - 5270 x^{14} - 34466274 x^{13} + 58267605 x^{12} + 400502344 x^{11} - 1424328742 x^{10} - 1847606785 x^{9} + 17240521227 x^{8} - 17428198991 x^{7} - 90564489526 x^{6} + 285808707218 x^{5} - 50921455947 x^{4} - 1357787819668 x^{3} + 3508052324155 x^{2} - 4172241594415 x + 2276865982021 \)
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: | \(-2706331951477505060212136736411776523762281466154952826611=-\,11^{11}\cdot 199^{20}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $407.91$ | 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: | $11, 199$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(2189=11\cdot 199\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{2189}(1792,·)$, $\chi_{2189}(1,·)$, $\chi_{2189}(1013,·)$, $\chi_{2189}(461,·)$, $\chi_{2189}(320,·)$, $\chi_{2189}(1297,·)$, $\chi_{2189}(659,·)$, $\chi_{2189}(615,·)$, $\chi_{2189}(857,·)$, $\chi_{2189}(859,·)$, $\chi_{2189}(1308,·)$, $\chi_{2189}(1695,·)$, $\chi_{2189}(736,·)$, $\chi_{2189}(1057,·)$, $\chi_{2189}(1319,·)$, $\chi_{2189}(1706,·)$, $\chi_{2189}(1255,·)$, $\chi_{2189}(1134,·)$, $\chi_{2189}(1717,·)$, $\chi_{2189}(1979,·)$, $\chi_{2189}(188,·)$, $\chi_{2189}(2111,·)$$\rbrace$ | ||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{703} a^{18} - \frac{100}{703} a^{17} - \frac{43}{703} a^{16} - \frac{233}{703} a^{15} - \frac{16}{703} a^{14} - \frac{153}{703} a^{13} + \frac{117}{703} a^{12} + \frac{205}{703} a^{11} + \frac{265}{703} a^{10} + \frac{288}{703} a^{9} - \frac{9}{19} a^{8} - \frac{265}{703} a^{7} + \frac{347}{703} a^{6} - \frac{179}{703} a^{5} + \frac{322}{703} a^{4} - \frac{335}{703} a^{3} - \frac{317}{703} a^{2} + \frac{281}{703} a + \frac{3}{19}$, $\frac{1}{703} a^{19} - \frac{201}{703} a^{17} - \frac{315}{703} a^{16} - \frac{117}{703} a^{15} - \frac{347}{703} a^{14} + \frac{283}{703} a^{13} - \frac{46}{703} a^{12} - \frac{325}{703} a^{11} + \frac{2}{19} a^{10} + \frac{347}{703} a^{9} + \frac{179}{703} a^{8} - \frac{142}{703} a^{7} + \frac{2}{19} a^{6} - \frac{3}{703} a^{5} + \frac{230}{703} a^{4} - \frac{73}{703} a^{3} + \frac{216}{703} a^{2} + \frac{91}{703} a - \frac{4}{19}$, $\frac{1}{2626156363540903} a^{20} - \frac{321252877327}{2626156363540903} a^{19} - \frac{10176819635}{70977199014619} a^{18} + \frac{454827856043398}{2626156363540903} a^{17} + \frac{307824448790471}{2626156363540903} a^{16} - \frac{111233098388279}{2626156363540903} a^{15} + \frac{1282102136738447}{2626156363540903} a^{14} - \frac{602539465206743}{2626156363540903} a^{13} - \frac{117109448953850}{2626156363540903} a^{12} + \frac{8015567659987}{2626156363540903} a^{11} + \frac{933792528405671}{2626156363540903} a^{10} + \frac{8474239181998}{70977199014619} a^{9} + \frac{950935809903226}{2626156363540903} a^{8} + \frac{1112984094077295}{2626156363540903} a^{7} - \frac{315210415000504}{2626156363540903} a^{6} - \frac{592466508622591}{2626156363540903} a^{5} - \frac{583657301249132}{2626156363540903} a^{4} - \frac{172847426791504}{2626156363540903} a^{3} - \frac{385409431893082}{2626156363540903} a^{2} - \frac{884193517199425}{2626156363540903} a - \frac{30724908589001}{70977199014619}$, $\frac{1}{992011648228913197741186039129553884416607572789429576252122524481838738790840353236751733845089} a^{21} - \frac{115195649379298916913085888716291006717998417400007644021571097929361987849231731}{992011648228913197741186039129553884416607572789429576252122524481838738790840353236751733845089} a^{20} - \frac{7294812693598408982761941798394492040490549468505865306853160439069041984561961213469386541}{52211139380469115670588738901555467600874082778391030329059080235886249410044229117723775465531} a^{19} + \frac{248155719170598022498281565452872952709203378934335208232120697765389105279064654974207111934}{992011648228913197741186039129553884416607572789429576252122524481838738790840353236751733845089} a^{18} - \frac{486112813119036410735618316249436684660875141579459214879846690859468047380439440765245266306420}{992011648228913197741186039129553884416607572789429576252122524481838738790840353236751733845089} a^{17} - \frac{294429818562438469176293998639767013191190891505660433774647832236449746118194094720295261377656}{992011648228913197741186039129553884416607572789429576252122524481838738790840353236751733845089} a^{16} - \frac{3059597545212919884998140746364072087551672856155004291411023106472009789483268209869617193578}{52211139380469115670588738901555467600874082778391030329059080235886249410044229117723775465531} a^{15} - \frac{300975459533551642406844833215732173107340765794805654809883167782939232226481840819779883631800}{992011648228913197741186039129553884416607572789429576252122524481838738790840353236751733845089} a^{14} + \frac{310101856617557812802026959525250449189273730699626025778423488982052279444312938216207684838347}{992011648228913197741186039129553884416607572789429576252122524481838738790840353236751733845089} a^{13} - \frac{436039569539559031145578206966458052861255964636672736762486086296560782030462401073398098414814}{992011648228913197741186039129553884416607572789429576252122524481838738790840353236751733845089} a^{12} + \frac{352490523617567468598359268539412008779324927211866459987911810430740100246005380548804862835313}{992011648228913197741186039129553884416607572789429576252122524481838738790840353236751733845089} a^{11} - \frac{413611603412822273430917272979193035213365232000107637441247535343456894589817406959561055067321}{992011648228913197741186039129553884416607572789429576252122524481838738790840353236751733845089} a^{10} + \frac{20119051305162931713009093447449100739682018498902976730576975570803663125455087154165836782637}{992011648228913197741186039129553884416607572789429576252122524481838738790840353236751733845089} a^{9} - \frac{143528122606145497844590397359843526037107119586704636774029938693174321412363861212814339762932}{992011648228913197741186039129553884416607572789429576252122524481838738790840353236751733845089} a^{8} - \frac{112120401810874710491947472789332243008742199721683661770559558133445191171150972751238891745500}{992011648228913197741186039129553884416607572789429576252122524481838738790840353236751733845089} a^{7} - \frac{365669457441365248526352933684053718811906587237340778370990301310397673118001214829978918080822}{992011648228913197741186039129553884416607572789429576252122524481838738790840353236751733845089} a^{6} + \frac{1205620838889749919077096934437336515849716404347223825883864318646230366370055584769155558710}{4448482727483915684938053987128044324738150550625244736556603248797483133591212346353146788543} a^{5} - \frac{99857424721772469228459080402040398328154154644860194129534035968245709132200841378253942332300}{992011648228913197741186039129553884416607572789429576252122524481838738790840353236751733845089} a^{4} + \frac{268703096377378256378971790793163667134030322037030601815306830717718868854273601137649681565734}{992011648228913197741186039129553884416607572789429576252122524481838738790840353236751733845089} a^{3} - \frac{18294932206181675216046291033930795358304344723527166343168674122390895193985191951346351470312}{52211139380469115670588738901555467600874082778391030329059080235886249410044229117723775465531} a^{2} + \frac{9392740489272884119035853162194927963249561531584338381092615760047367012835188843788963288068}{52211139380469115670588738901555467600874082778391030329059080235886249410044229117723775465531} a + \frac{6984927203723933689290883316283410722601374023984536696965019839050183215419339932101951081876}{26811125627808464803815838895393348227475880345660258817624933094103749697049739276668965779597}$
Class group and class number
$C_{372188993}$, which has order $372188993$ (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: | \( 117135822355.96071 \) (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
| \(\Q(\sqrt{-11}) \), 11.11.97393677359695041798001.1 |
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 | $22$ | ${\href{/LocalNumberField/3.11.0.1}{11} }^{2}$ | ${\href{/LocalNumberField/5.11.0.1}{11} }^{2}$ | $22$ | R | $22$ | $22$ | ${\href{/LocalNumberField/19.2.0.1}{2} }^{11}$ | ${\href{/LocalNumberField/23.11.0.1}{11} }^{2}$ | $22$ | ${\href{/LocalNumberField/31.11.0.1}{11} }^{2}$ | ${\href{/LocalNumberField/37.1.0.1}{1} }^{22}$ | $22$ | ${\href{/LocalNumberField/43.2.0.1}{2} }^{11}$ | ${\href{/LocalNumberField/47.11.0.1}{11} }^{2}$ | ${\href{/LocalNumberField/53.11.0.1}{11} }^{2}$ | ${\href{/LocalNumberField/59.11.0.1}{11} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
| 11 | Data not computed | ||||||
| $199$ | 199.11.10.1 | $x^{11} - 199$ | $11$ | $1$ | $10$ | $C_{11}$ | $[\ ]_{11}$ |
| 199.11.10.1 | $x^{11} - 199$ | $11$ | $1$ | $10$ | $C_{11}$ | $[\ ]_{11}$ | |