Normalized defining polynomial
\(x^{44} + 41 x^{42} + 784 x^{40} + 9282 x^{38} + 76180 x^{36} + 459888 x^{34} + 2114697 x^{32} + 7568133 x^{30} + 21358299 x^{28} + 47872465 x^{26} + 85431991 x^{24} + 121194085 x^{22} + 135920335 x^{20} + 119357605 x^{18} + 80900650 x^{16} + 41459620 x^{14} + 15683335 x^{12} + 3901015 x^{10} + 2054320 x^{8} - 6489250 x^{6} + 32853580 x^{4} - 164244624 x^{2} + 821223649\)
Invariants
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $\frac{1}{28657} a^{23} + \frac{23}{28657} a^{21} + \frac{230}{28657} a^{19} + \frac{1311}{28657} a^{17} + \frac{4692}{28657} a^{15} + \frac{10948}{28657} a^{13} - \frac{11913}{28657} a^{11} - \frac{12212}{28657} a^{9} + \frac{9867}{28657} a^{7} + \frac{3289}{28657} a^{5} + \frac{506}{28657} a^{3} + \frac{23}{28657} a$, $\frac{1}{1836311903} a^{24} - \frac{701408709}{1836311903} a^{22} - \frac{740496650}{1836311903} a^{20} + \frac{310528563}{1836311903} a^{18} + \frac{797213775}{1836311903} a^{16} + \frac{821062655}{1836311903} a^{14} + \frac{397002165}{1836311903} a^{12} + \frac{316332411}{1836311903} a^{10} + \frac{8807566}{1836311903} a^{8} - \frac{913753813}{1836311903} a^{6} - \frac{328466028}{1836311903} a^{4} - \frac{400109011}{1836311903} a^{2} + \frac{15127}{64079}$, $\frac{1}{1836311903} a^{25} + \frac{25}{1836311903} a^{23} + \frac{701409008}{1836311903} a^{21} + \frac{39089919}{1836311903} a^{19} + \frac{351800646}{1836311903} a^{17} - \frac{676399496}{1836311903} a^{15} - \frac{36556349}{1836311903} a^{13} - \frac{346757081}{1836311903} a^{11} + \frac{800375453}{1836311903} a^{9} + \frac{662974061}{1836311903} a^{7} + \frac{197109930}{1836311903} a^{5} + \frac{104513114}{1836311903} a^{3} + \frac{39088194}{1836311903} a$, $\frac{1}{1836311903} a^{26} - \frac{126492297}{1836311903} a^{22} + \frac{188387139}{1836311903} a^{20} - \frac{66165817}{1836311903} a^{18} - \frac{407312938}{1836311903} a^{16} - \frac{363691791}{1836311903} a^{14} + \frac{746060212}{1836311903} a^{12} + \frac{237312790}{1836311903} a^{10} + \frac{442784911}{1836311903} a^{8} - \frac{831099484}{1836311903} a^{6} - \frac{865395701}{1836311903} a^{4} + \frac{860253954}{1836311903} a^{2} + \frac{6299}{64079}$, $\frac{1}{1836311903} a^{27} - \frac{351}{1836311903} a^{23} - \frac{574921909}{1836311903} a^{21} - \frac{354008685}{1836311903} a^{19} + \frac{155556998}{1836311903} a^{17} + \frac{7774172}{1836311903} a^{15} - \frac{835601745}{1836311903} a^{13} - \frac{885479448}{1836311903} a^{11} + \frac{61450782}{1836311903} a^{9} + \frac{409149561}{1836311903} a^{7} + \frac{160124615}{1836311903} a^{5} + \frac{594262025}{1836311903} a^{3} - \frac{582798605}{1836311903} a$, $\frac{1}{1836311903} a^{28} - \frac{703583766}{1836311903} a^{22} + \frac{487957391}{1836311903} a^{20} + \frac{808680334}{1836311903} a^{18} + \frac{710399941}{1836311903} a^{16} + \frac{892733292}{1836311903} a^{14} + \frac{738887742}{1836311903} a^{12} + \frac{915412863}{1836311903} a^{10} - \frac{172018579}{1836311903} a^{8} + \frac{787119277}{1836311903} a^{6} - \frac{845975817}{1836311903} a^{4} + \frac{374955065}{1836311903} a^{2} - \frac{8980}{64079}$, $\frac{1}{1836311903} a^{29} + \frac{3654}{1836311903} a^{23} + \frac{143660924}{1836311903} a^{21} - \frac{797972433}{1836311903} a^{19} - \frac{551379648}{1836311903} a^{17} + \frac{436106338}{1836311903} a^{15} + \frac{285528817}{1836311903} a^{13} + \frac{6003695}{1836311903} a^{11} - \frac{278197482}{1836311903} a^{9} - \frac{11112826}{1836311903} a^{7} - \frac{499949217}{1836311903} a^{5} + \frac{145680403}{1836311903} a^{3} - \frac{601636327}{1836311903} a$, $\frac{1}{1836311903} a^{30} - \frac{400332978}{1836311903} a^{22} + \frac{89353548}{1836311903} a^{20} - \frac{381992796}{1836311903} a^{18} - \frac{192349354}{1836311903} a^{16} + \frac{656236949}{1836311903} a^{14} + \frac{46496155}{1836311903} a^{12} + \frac{719671614}{1836311903} a^{10} + \frac{859655264}{1836311903} a^{8} - \frac{58556169}{1836311903} a^{6} - \frac{587437847}{1836311903} a^{4} - \frac{307584921}{1836311903} a^{2} + \frac{26119}{64079}$, $\frac{1}{1836311903} a^{31} - \frac{31465}{1836311903} a^{23} + \frac{114728832}{1836311903} a^{21} - \frac{128239956}{1836311903} a^{19} - \frac{582270069}{1836311903} a^{17} + \frac{323859176}{1836311903} a^{15} - \frac{729051982}{1836311903} a^{13} + \frac{829759336}{1836311903} a^{11} + \frac{639864294}{1836311903} a^{9} - \frac{190430751}{1836311903} a^{7} - \frac{631396041}{1836311903} a^{5} + \frac{250671327}{1836311903} a^{3} + \frac{773867467}{1836311903} a$, $\frac{1}{1836311903} a^{32} - \frac{913849599}{1836311903} a^{22} - \frac{729906942}{1836311903} a^{20} - \frac{816671137}{1836311903} a^{18} + \frac{634694571}{1836311903} a^{16} + \frac{771536189}{1836311903} a^{14} + \frac{73004952}{1836311903} a^{12} - \frac{607649754}{1836311903} a^{10} - \frac{343463914}{1836311903} a^{8} - \frac{759656815}{1836311903} a^{6} - \frac{169509609}{1836311903} a^{4} - \frac{738068583}{1836311903} a^{2} - \frac{7757}{64079}$, $\frac{1}{1836311903} a^{33} - \frac{18980}{1836311903} a^{23} + \frac{88766362}{1836311903} a^{21} + \frac{24814291}{1836311903} a^{19} - \frac{445036579}{1836311903} a^{17} + \frac{676507032}{1836311903} a^{15} + \frac{463374220}{1836311903} a^{13} + \frac{421458986}{1836311903} a^{11} - \frac{775548611}{1836311903} a^{9} - \frac{284382872}{1836311903} a^{7} - \frac{623188929}{1836311903} a^{5} + \frac{745936978}{1836311903} a^{3} + \frac{596380955}{1836311903} a$, $\frac{1}{1836311903} a^{34} + \frac{612766292}{1836311903} a^{22} + \frac{529702853}{1836311903} a^{20} + \frac{662192434}{1836311903} a^{18} + \frac{583875812}{1836311903} a^{16} - \frac{546554641}{1836311903} a^{14} - \frac{701499226}{1836311903} a^{12} + \frac{310001262}{1836311903} a^{10} - \frac{221163365}{1836311903} a^{8} + \frac{295364166}{1836311903} a^{6} + \frac{739636223}{1836311903} a^{4} - \frac{322928920}{1836311903} a^{2} - \frac{27539}{64079}$, $\frac{1}{1836311903} a^{35} - \frac{21185}{1836311903} a^{23} - \frac{710225797}{1836311903} a^{21} - \frac{719222648}{1836311903} a^{19} - \frac{312204924}{1836311903} a^{17} - \frac{80956627}{1836311903} a^{15} + \frac{384896140}{1836311903} a^{13} - \frac{728911565}{1836311903} a^{11} + \frac{168501034}{1836311903} a^{9} + \frac{896425186}{1836311903} a^{7} - \frac{284218039}{1836311903} a^{5} - \frac{56680675}{1836311903} a^{3} - \frac{192801870}{1836311903} a$, $\frac{1}{1836311903} a^{36} - \frac{617806886}{1836311903} a^{22} - \frac{528165569}{1836311903} a^{20} + \frac{566165685}{1836311903} a^{18} + \frac{332294857}{1836311903} a^{16} - \frac{785414804}{1836311903} a^{14} - \frac{546561780}{1836311903} a^{12} - \frac{867817881}{1836311903} a^{10} + \frac{180896790}{1836311903} a^{8} + \frac{241334982}{1836311903} a^{6} - \frac{823683388}{1836311903} a^{4} - \frac{86455657}{1836311903} a^{2} + \frac{6416}{64079}$, $\frac{1}{1836311903} a^{37} - \frac{21247}{1836311903} a^{23} + \frac{826720807}{1836311903} a^{21} - \frac{575465779}{1836311903} a^{19} + \frac{435718363}{1836311903} a^{17} + \frac{164620450}{1836311903} a^{15} - \frac{166124757}{1836311903} a^{13} - \frac{610028064}{1836311903} a^{11} - \frac{648029154}{1836311903} a^{9} - \frac{623282965}{1836311903} a^{7} + \frac{112318565}{1836311903} a^{5} + \frac{340054167}{1836311903} a^{3} - \frac{297562215}{1836311903} a$, $\frac{1}{1836311903} a^{38} - \frac{333026471}{1836311903} a^{22} - \frac{2787075}{13210877} a^{20} + \frac{367428945}{1836311903} a^{18} + \frac{424704603}{1836311903} a^{16} - \frac{10972472}{1836311903} a^{14} + \frac{314401212}{1836311903} a^{12} - \frac{434857617}{1836311903} a^{10} - \frac{792742269}{1836311903} a^{8} + \frac{910804173}{1836311903} a^{6} - \frac{592411349}{1836311903} a^{4} + \frac{710391958}{1836311903} a^{2} - \frac{16895}{64079}$, $\frac{1}{1836311903} a^{39} - \frac{7908}{1836311903} a^{23} - \frac{526792}{13210877} a^{21} - \frac{163401491}{1836311903} a^{19} - \frac{30192218}{1836311903} a^{17} - \frac{189304329}{1836311903} a^{15} - \frac{713810422}{1836311903} a^{13} + \frac{585023747}{1836311903} a^{11} - \frac{184568480}{1836311903} a^{9} - \frac{193341076}{1836311903} a^{7} + \frac{263748170}{1836311903} a^{5} + \frac{277089760}{1836311903} a^{3} - \frac{169980678}{1836311903} a$, $\frac{1}{1836311903} a^{40} + \frac{684964103}{1836311903} a^{22} - \frac{12251024}{1836311903} a^{20} + \frac{480669675}{1836311903} a^{18} + \frac{118465372}{1836311903} a^{16} + \frac{887088213}{1836311903} a^{14} - \frac{15209563}{1836311903} a^{12} + \frac{315325822}{1836311903} a^{10} - \frac{322961462}{1836311903} a^{8} + \frac{185933271}{1836311903} a^{6} - \frac{687228822}{1836311903} a^{4} - \frac{266630797}{1836311903} a^{2} - \frac{11177}{64079}$, $\frac{1}{1836311903} a^{41} + \frac{23672}{1836311903} a^{23} + \frac{760926190}{1836311903} a^{21} + \frac{867194203}{1836311903} a^{19} + \frac{118080898}{1836311903} a^{17} + \frac{692416211}{1836311903} a^{15} + \frac{754763701}{1836311903} a^{13} - \frac{559416607}{1836311903} a^{11} - \frac{231136255}{1836311903} a^{9} - \frac{493496366}{1836311903} a^{7} - \frac{301601400}{1836311903} a^{5} + \frac{216460784}{1836311903} a^{3} + \frac{452877925}{1836311903} a$, $\frac{1}{1836311903} a^{42} + \frac{575658712}{1836311903} a^{22} + \frac{470466965}{1836311903} a^{20} + \frac{42485271}{1836311903} a^{18} + \frac{825361542}{1836311903} a^{16} + \frac{84775893}{1836311903} a^{14} - \frac{150346933}{1836311903} a^{12} + \frac{27970987}{1836311903} a^{10} + \frac{353358224}{1836311903} a^{8} + \frac{160754499}{1836311903} a^{6} + \frac{719678298}{1836311903} a^{4} + \frac{136590643}{1836311903} a^{2} - \frac{12892}{64079}$, $\frac{1}{1836311903} a^{43} - \frac{27024}{1836311903} a^{23} + \frac{83878358}{1836311903} a^{21} - \frac{150776993}{1836311903} a^{19} + \frac{825553779}{1836311903} a^{17} + \frac{182111894}{1836311903} a^{15} - \frac{535333565}{1836311903} a^{13} - \frac{452813750}{1836311903} a^{11} - \frac{610710331}{1836311903} a^{9} - \frac{417686634}{1836311903} a^{7} + \frac{526864587}{1836311903} a^{5} + \frac{813200804}{1836311903} a^{3} - \frac{756034651}{1836311903} a$
Class group and class number
not computed
Unit group
Rank: | $21$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
| |
Torsion generator: | \( -1 \) (order $2$) ![]() | sage: UK.torsion_generator()
gp: K.tu[2]
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
| |
Fundamental units: | not computed ![]() | sage: UK.fundamental_units()
gp: K.fu
magma: [K!f(g): g in Generators(UK)];
| |
Regulator: | not computed | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
|
Class number formula
Galois group
$C_2\times C_{22}$ (as 44T2):
An abelian group of order 44 |
The 44 conjugacy class representatives for $C_2\times C_{22}$ |
Character table for $C_2\times C_{22}$ is not computed |
Intermediate fields
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 | R | $22^{2}$ | R | ${\href{/LocalNumberField/7.11.0.1}{11} }^{4}$ | $22^{2}$ | $22^{2}$ | $22^{2}$ | $22^{2}$ | R | ${\href{/LocalNumberField/29.11.0.1}{11} }^{4}$ | $22^{2}$ | $22^{2}$ | ${\href{/LocalNumberField/41.11.0.1}{11} }^{4}$ | ${\href{/LocalNumberField/43.11.0.1}{11} }^{4}$ | ${\href{/LocalNumberField/47.2.0.1}{2} }^{22}$ | $22^{2}$ | $22^{2}$ |
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 |
---|---|---|---|---|---|---|---|
2 | Data not computed | ||||||
5 | Data not computed | ||||||
23 | Data not computed |