Normalized defining polynomial
\( x^{22} - x^{21} + 31 x^{20} - 96 x^{19} + 743 x^{18} - 2148 x^{17} + 9669 x^{16} - 25959 x^{15} + 87182 x^{14} - 197946 x^{13} + 491216 x^{12} - 903191 x^{11} + 1776770 x^{10} - 2627379 x^{9} + 3793565 x^{8} - 3601844 x^{7} + 3006023 x^{6} - 1017019 x^{5} + 371579 x^{4} + 110469 x^{3} + 58034 x^{2} + 7047 x + 841 \)
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: | \(-588613008557716517601287069955881168578347=-\,3^{11}\cdot 67^{20}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $79.18$ | 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: | $3, 67$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(201=3\cdot 67\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{201}(64,·)$, $\chi_{201}(1,·)$, $\chi_{201}(131,·)$, $\chi_{201}(68,·)$, $\chi_{201}(193,·)$, $\chi_{201}(76,·)$, $\chi_{201}(14,·)$, $\chi_{201}(143,·)$, $\chi_{201}(82,·)$, $\chi_{201}(148,·)$, $\chi_{201}(149,·)$, $\chi_{201}(22,·)$, $\chi_{201}(89,·)$, $\chi_{201}(25,·)$, $\chi_{201}(91,·)$, $\chi_{201}(92,·)$, $\chi_{201}(158,·)$, $\chi_{201}(40,·)$, $\chi_{201}(196,·)$, $\chi_{201}(107,·)$, $\chi_{201}(59,·)$, $\chi_{201}(62,·)$$\rbrace$ | ||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{37} a^{17} - \frac{16}{37} a^{16} - \frac{16}{37} a^{15} + \frac{2}{37} a^{14} - \frac{18}{37} a^{13} + \frac{2}{37} a^{12} + \frac{8}{37} a^{11} - \frac{13}{37} a^{10} - \frac{1}{37} a^{9} + \frac{14}{37} a^{8} - \frac{5}{37} a^{7} + \frac{12}{37} a^{6} + \frac{1}{37} a^{5} - \frac{7}{37} a^{4} + \frac{6}{37} a^{3} - \frac{6}{37} a^{2} - \frac{13}{37} a + \frac{12}{37}$, $\frac{1}{37} a^{18} - \frac{13}{37} a^{16} + \frac{5}{37} a^{15} + \frac{14}{37} a^{14} + \frac{10}{37} a^{13} + \frac{3}{37} a^{12} + \frac{4}{37} a^{11} + \frac{13}{37} a^{10} - \frac{2}{37} a^{9} - \frac{3}{37} a^{8} + \frac{6}{37} a^{7} + \frac{8}{37} a^{6} + \frac{9}{37} a^{5} + \frac{5}{37} a^{4} + \frac{16}{37} a^{3} + \frac{2}{37} a^{2} - \frac{11}{37} a + \frac{7}{37}$, $\frac{1}{1073} a^{19} - \frac{9}{1073} a^{18} + \frac{10}{1073} a^{17} - \frac{209}{1073} a^{16} + \frac{267}{1073} a^{15} + \frac{189}{1073} a^{14} - \frac{57}{1073} a^{13} - \frac{421}{1073} a^{12} + \frac{383}{1073} a^{11} + \frac{137}{1073} a^{10} - \frac{45}{1073} a^{9} - \frac{274}{1073} a^{8} + \frac{24}{1073} a^{7} + \frac{139}{1073} a^{6} + \frac{95}{1073} a^{5} - \frac{449}{1073} a^{4} + \frac{440}{1073} a^{3} + \frac{55}{1073} a^{2} - \frac{526}{1073} a + \frac{15}{37}$, $\frac{1}{256447} a^{20} - \frac{48}{256447} a^{19} - \frac{161}{256447} a^{18} - \frac{2890}{256447} a^{17} + \frac{101218}{256447} a^{16} + \frac{87129}{256447} a^{15} + \frac{107296}{256447} a^{14} - \frac{45874}{256447} a^{13} - \frac{100938}{256447} a^{12} + \frac{20580}{256447} a^{11} - \frac{76815}{256447} a^{10} - \frac{33812}{256447} a^{9} + \frac{27414}{256447} a^{8} - \frac{10715}{256447} a^{7} + \frac{85328}{256447} a^{6} + \frac{44653}{256447} a^{5} - \frac{213}{8843} a^{4} + \frac{99214}{256447} a^{3} + \frac{54024}{256447} a^{2} + \frac{65058}{256447} a + \frac{598}{8843}$, $\frac{1}{363792406196870434044052044865158080654898895057} a^{21} + \frac{134061252255875920885634782192791050877138}{363792406196870434044052044865158080654898895057} a^{20} + \frac{21785071667451461986331537619890701651822143}{363792406196870434044052044865158080654898895057} a^{19} + \frac{3159980038539151862407229395586978195676763113}{363792406196870434044052044865158080654898895057} a^{18} - \frac{4395792072926501584288205228669343204787161817}{363792406196870434044052044865158080654898895057} a^{17} - \frac{41512880152371300308436009330954931120288229915}{363792406196870434044052044865158080654898895057} a^{16} - \frac{119214428507423729702000770867744722383167617228}{363792406196870434044052044865158080654898895057} a^{15} + \frac{112265822864258099607205632631456328023645196351}{363792406196870434044052044865158080654898895057} a^{14} + \frac{12956976985842996660684590137137928541276599238}{363792406196870434044052044865158080654898895057} a^{13} + \frac{107211772771037775663479035498529742650867567722}{363792406196870434044052044865158080654898895057} a^{12} + \frac{88144571670214693651417636439914781224894371600}{363792406196870434044052044865158080654898895057} a^{11} - \frac{142471254313888483665640725737244698378781196508}{363792406196870434044052044865158080654898895057} a^{10} - \frac{118606216891000930266177471968854092457921939503}{363792406196870434044052044865158080654898895057} a^{9} + \frac{39727990696448031484281925403360647965659075233}{363792406196870434044052044865158080654898895057} a^{8} - \frac{116748566213724200517485618146205276818219484317}{363792406196870434044052044865158080654898895057} a^{7} - \frac{130611312869622848878471695101571054277167315542}{363792406196870434044052044865158080654898895057} a^{6} - \frac{29811241497550517175565617423173726996814188209}{363792406196870434044052044865158080654898895057} a^{5} + \frac{67945975438484460055561267139630812523725371315}{363792406196870434044052044865158080654898895057} a^{4} - \frac{180535609569440502092563416020754253926563221849}{363792406196870434044052044865158080654898895057} a^{3} - \frac{115967448446663892613668166178937840723137075876}{363792406196870434044052044865158080654898895057} a^{2} + \frac{127312365898362793425393607794965841248700972553}{363792406196870434044052044865158080654898895057} a - \frac{6060152356385674477946480778757246534141697522}{12544565730926566691174208443626140712237892933}$
Class group and class number
$C_{23}\times C_{253}$, which has order $5819$ (assuming GRH)
Unit group
| Rank: | $10$ | magma: UnitRank(K);
sage: UK.rank()
gp: K.fu
| |
| Torsion generator: | \( \frac{189562659348677593313094899504554636424432}{1522143958982721481355866296506937575961920063} a^{21} - \frac{190298252352550244935323093566692439557982}{1522143958982721481355866296506937575961920063} a^{20} + \frac{5877475457995903306552345827281942102085092}{1522143958982721481355866296506937575961920063} a^{19} - \frac{18224463567626360078910365224878059642772460}{1522143958982721481355866296506937575961920063} a^{18} + \frac{140928151887084131998420888115071548213999121}{1522143958982721481355866296506937575961920063} a^{17} - \frac{407859612601517245750452833004569644103041810}{1522143958982721481355866296506937575961920063} a^{16} + \frac{1835000703828235306102410701122902140297387584}{1522143958982721481355866296506937575961920063} a^{15} - \frac{4931093548469852764491129273437932007531379707}{1522143958982721481355866296506937575961920063} a^{14} + \frac{16555552082333303784452984237276503097097944282}{1522143958982721481355866296506937575961920063} a^{13} - \frac{37627313273329278013657610709713049263100105644}{1522143958982721481355866296506937575961920063} a^{12} + \frac{93373838131794482109836400461043391319332408392}{1522143958982721481355866296506937575961920063} a^{11} - \frac{171921881044738922350922673005605013723738565686}{1522143958982721481355866296506937575961920063} a^{10} + \frac{338274312774060386582181185955116452257710506330}{1522143958982721481355866296506937575961920063} a^{9} - \frac{501258372292487864922888761778215105488904100115}{1522143958982721481355866296506937575961920063} a^{8} + \frac{724546972558380209123132748447249357741306976244}{1522143958982721481355866296506937575961920063} a^{7} - \frac{692195010731279944480633608169850489141697101396}{1522143958982721481355866296506937575961920063} a^{6} + \frac{582113517945816890899598565959449517511378178626}{1522143958982721481355866296506937575961920063} a^{5} - \frac{208351225414817690375148872175006431794387046394}{1522143958982721481355866296506937575961920063} a^{4} + \frac{83263501286189775559602364978163150665518364756}{1522143958982721481355866296506937575961920063} a^{3} + \frac{11117000723249250924969035390453585019704686040}{1522143958982721481355866296506937575961920063} a^{2} + \frac{12700672592593462857816922694417042225097219142}{1522143958982721481355866296506937575961920063} a + \frac{53159414187634214444563715304032454967089810}{52487722723542120046754010224377157791790347} \) (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: | \( 338444542.043 \) (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
| \(\Q(\sqrt{-3}) \), 11.11.1822837804551761449.1 |
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 | $22$ | R | $22$ | ${\href{/LocalNumberField/7.11.0.1}{11} }^{2}$ | $22$ | ${\href{/LocalNumberField/13.11.0.1}{11} }^{2}$ | $22$ | ${\href{/LocalNumberField/19.11.0.1}{11} }^{2}$ | $22$ | ${\href{/LocalNumberField/29.2.0.1}{2} }^{11}$ | ${\href{/LocalNumberField/31.11.0.1}{11} }^{2}$ | ${\href{/LocalNumberField/37.1.0.1}{1} }^{22}$ | $22$ | ${\href{/LocalNumberField/43.11.0.1}{11} }^{2}$ | $22$ | $22$ | $22$ |
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 |
|---|---|---|---|---|---|---|---|
| 3 | Data not computed | ||||||
| $67$ | 67.11.10.1 | $x^{11} - 67$ | $11$ | $1$ | $10$ | $C_{11}$ | $[\ ]_{11}$ |
| 67.11.10.1 | $x^{11} - 67$ | $11$ | $1$ | $10$ | $C_{11}$ | $[\ ]_{11}$ | |