Normalized defining polynomial
\( x^{44} + 41 x^{42} + 784 x^{40} + 9282 x^{38} + 76180 x^{36} + 459888 x^{34} + 2114697 x^{32} + \cdots + 821223649 \)
Invariants
Degree: | $44$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[0, 22]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: |
\(653\!\cdots\!000\)
\(\medspace = 2^{44}\cdot 5^{22}\cdot 23^{42}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(89.20\) | sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
magma: Abs(Discriminant(OK))^(1/Degree(K));
oscar: (1.0 * dK)^(1/degree(K))
| |
Galois root discriminant: | $2\cdot 5^{1/2}23^{21/22}\approx 89.19615099241642$ | ||
Ramified primes: |
\(2\), \(5\), \(23\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q\) | ||
$\card{ \Gal(K/\Q) }$: | $44$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
This field is Galois and abelian over $\Q$. | |||
Conductor: | \(460=2^{2}\cdot 5\cdot 23\) | ||
Dirichlet character group: | $\lbrace$$\chi_{460}(1,·)$, $\chi_{460}(259,·)$, $\chi_{460}(261,·)$, $\chi_{460}(51,·)$, $\chi_{460}(129,·)$, $\chi_{460}(11,·)$, $\chi_{460}(141,·)$, $\chi_{460}(399,·)$, $\chi_{460}(149,·)$, $\chi_{460}(279,·)$, $\chi_{460}(411,·)$, $\chi_{460}(389,·)$, $\chi_{460}(291,·)$, $\chi_{460}(39,·)$, $\chi_{460}(41,·)$, $\chi_{460}(171,·)$, $\chi_{460}(301,·)$, $\chi_{460}(431,·)$, $\chi_{460}(91,·)$, $\chi_{460}(179,·)$, $\chi_{460}(309,·)$, $\chi_{460}(439,·)$, $\chi_{460}(111,·)$, $\chi_{460}(441,·)$, $\chi_{460}(59,·)$, $\chi_{460}(189,·)$, $\chi_{460}(191,·)$, $\chi_{460}(451,·)$, $\chi_{460}(329,·)$, $\chi_{460}(81,·)$, $\chi_{460}(139,·)$, $\chi_{460}(121,·)$, $\chi_{460}(429,·)$, $\chi_{460}(219,·)$, $\chi_{460}(101,·)$, $\chi_{460}(229,·)$, $\chi_{460}(361,·)$, $\chi_{460}(109,·)$, $\chi_{460}(239,·)$, $\chi_{460}(89,·)$, $\chi_{460}(119,·)$, $\chi_{460}(249,·)$, $\chi_{460}(251,·)$, $\chi_{460}(381,·)$$\rbrace$ | ||
This is a CM field. | |||
Reflex fields: | unavailable$^{2097152}$ |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $\frac{1}{28657}a^{23}+\frac{23}{28657}a^{21}+\frac{230}{28657}a^{19}+\frac{1311}{28657}a^{17}+\frac{4692}{28657}a^{15}+\frac{10948}{28657}a^{13}-\frac{11913}{28657}a^{11}-\frac{12212}{28657}a^{9}+\frac{9867}{28657}a^{7}+\frac{3289}{28657}a^{5}+\frac{506}{28657}a^{3}+\frac{23}{28657}a$, $\frac{1}{1836311903}a^{24}-\frac{701408709}{1836311903}a^{22}-\frac{740496650}{1836311903}a^{20}+\frac{310528563}{1836311903}a^{18}+\frac{797213775}{1836311903}a^{16}+\frac{821062655}{1836311903}a^{14}+\frac{397002165}{1836311903}a^{12}+\frac{316332411}{1836311903}a^{10}+\frac{8807566}{1836311903}a^{8}-\frac{913753813}{1836311903}a^{6}-\frac{328466028}{1836311903}a^{4}-\frac{400109011}{1836311903}a^{2}+\frac{15127}{64079}$, $\frac{1}{1836311903}a^{25}+\frac{25}{1836311903}a^{23}+\frac{701409008}{1836311903}a^{21}+\frac{39089919}{1836311903}a^{19}+\frac{351800646}{1836311903}a^{17}-\frac{676399496}{1836311903}a^{15}-\frac{36556349}{1836311903}a^{13}-\frac{346757081}{1836311903}a^{11}+\frac{800375453}{1836311903}a^{9}+\frac{662974061}{1836311903}a^{7}+\frac{197109930}{1836311903}a^{5}+\frac{104513114}{1836311903}a^{3}+\frac{39088194}{1836311903}a$, $\frac{1}{1836311903}a^{26}-\frac{126492297}{1836311903}a^{22}+\frac{188387139}{1836311903}a^{20}-\frac{66165817}{1836311903}a^{18}-\frac{407312938}{1836311903}a^{16}-\frac{363691791}{1836311903}a^{14}+\frac{746060212}{1836311903}a^{12}+\frac{237312790}{1836311903}a^{10}+\frac{442784911}{1836311903}a^{8}-\frac{831099484}{1836311903}a^{6}-\frac{865395701}{1836311903}a^{4}+\frac{860253954}{1836311903}a^{2}+\frac{6299}{64079}$, $\frac{1}{1836311903}a^{27}-\frac{351}{1836311903}a^{23}-\frac{574921909}{1836311903}a^{21}-\frac{354008685}{1836311903}a^{19}+\frac{155556998}{1836311903}a^{17}+\frac{7774172}{1836311903}a^{15}-\frac{835601745}{1836311903}a^{13}-\frac{885479448}{1836311903}a^{11}+\frac{61450782}{1836311903}a^{9}+\frac{409149561}{1836311903}a^{7}+\frac{160124615}{1836311903}a^{5}+\frac{594262025}{1836311903}a^{3}-\frac{582798605}{1836311903}a$, $\frac{1}{1836311903}a^{28}-\frac{703583766}{1836311903}a^{22}+\frac{487957391}{1836311903}a^{20}+\frac{808680334}{1836311903}a^{18}+\frac{710399941}{1836311903}a^{16}+\frac{892733292}{1836311903}a^{14}+\frac{738887742}{1836311903}a^{12}+\frac{915412863}{1836311903}a^{10}-\frac{172018579}{1836311903}a^{8}+\frac{787119277}{1836311903}a^{6}-\frac{845975817}{1836311903}a^{4}+\frac{374955065}{1836311903}a^{2}-\frac{8980}{64079}$, $\frac{1}{1836311903}a^{29}+\frac{3654}{1836311903}a^{23}+\frac{143660924}{1836311903}a^{21}-\frac{797972433}{1836311903}a^{19}-\frac{551379648}{1836311903}a^{17}+\frac{436106338}{1836311903}a^{15}+\frac{285528817}{1836311903}a^{13}+\frac{6003695}{1836311903}a^{11}-\frac{278197482}{1836311903}a^{9}-\frac{11112826}{1836311903}a^{7}-\frac{499949217}{1836311903}a^{5}+\frac{145680403}{1836311903}a^{3}-\frac{601636327}{1836311903}a$, $\frac{1}{1836311903}a^{30}-\frac{400332978}{1836311903}a^{22}+\frac{89353548}{1836311903}a^{20}-\frac{381992796}{1836311903}a^{18}-\frac{192349354}{1836311903}a^{16}+\frac{656236949}{1836311903}a^{14}+\frac{46496155}{1836311903}a^{12}+\frac{719671614}{1836311903}a^{10}+\frac{859655264}{1836311903}a^{8}-\frac{58556169}{1836311903}a^{6}-\frac{587437847}{1836311903}a^{4}-\frac{307584921}{1836311903}a^{2}+\frac{26119}{64079}$, $\frac{1}{1836311903}a^{31}-\frac{31465}{1836311903}a^{23}+\frac{114728832}{1836311903}a^{21}-\frac{128239956}{1836311903}a^{19}-\frac{582270069}{1836311903}a^{17}+\frac{323859176}{1836311903}a^{15}-\frac{729051982}{1836311903}a^{13}+\frac{829759336}{1836311903}a^{11}+\frac{639864294}{1836311903}a^{9}-\frac{190430751}{1836311903}a^{7}-\frac{631396041}{1836311903}a^{5}+\frac{250671327}{1836311903}a^{3}+\frac{773867467}{1836311903}a$, $\frac{1}{1836311903}a^{32}-\frac{913849599}{1836311903}a^{22}-\frac{729906942}{1836311903}a^{20}-\frac{816671137}{1836311903}a^{18}+\frac{634694571}{1836311903}a^{16}+\frac{771536189}{1836311903}a^{14}+\frac{73004952}{1836311903}a^{12}-\frac{607649754}{1836311903}a^{10}-\frac{343463914}{1836311903}a^{8}-\frac{759656815}{1836311903}a^{6}-\frac{169509609}{1836311903}a^{4}-\frac{738068583}{1836311903}a^{2}-\frac{7757}{64079}$, $\frac{1}{1836311903}a^{33}-\frac{18980}{1836311903}a^{23}+\frac{88766362}{1836311903}a^{21}+\frac{24814291}{1836311903}a^{19}-\frac{445036579}{1836311903}a^{17}+\frac{676507032}{1836311903}a^{15}+\frac{463374220}{1836311903}a^{13}+\frac{421458986}{1836311903}a^{11}-\frac{775548611}{1836311903}a^{9}-\frac{284382872}{1836311903}a^{7}-\frac{623188929}{1836311903}a^{5}+\frac{745936978}{1836311903}a^{3}+\frac{596380955}{1836311903}a$, $\frac{1}{1836311903}a^{34}+\frac{612766292}{1836311903}a^{22}+\frac{529702853}{1836311903}a^{20}+\frac{662192434}{1836311903}a^{18}+\frac{583875812}{1836311903}a^{16}-\frac{546554641}{1836311903}a^{14}-\frac{701499226}{1836311903}a^{12}+\frac{310001262}{1836311903}a^{10}-\frac{221163365}{1836311903}a^{8}+\frac{295364166}{1836311903}a^{6}+\frac{739636223}{1836311903}a^{4}-\frac{322928920}{1836311903}a^{2}-\frac{27539}{64079}$, $\frac{1}{1836311903}a^{35}-\frac{21185}{1836311903}a^{23}-\frac{710225797}{1836311903}a^{21}-\frac{719222648}{1836311903}a^{19}-\frac{312204924}{1836311903}a^{17}-\frac{80956627}{1836311903}a^{15}+\frac{384896140}{1836311903}a^{13}-\frac{728911565}{1836311903}a^{11}+\frac{168501034}{1836311903}a^{9}+\frac{896425186}{1836311903}a^{7}-\frac{284218039}{1836311903}a^{5}-\frac{56680675}{1836311903}a^{3}-\frac{192801870}{1836311903}a$, $\frac{1}{1836311903}a^{36}-\frac{617806886}{1836311903}a^{22}-\frac{528165569}{1836311903}a^{20}+\frac{566165685}{1836311903}a^{18}+\frac{332294857}{1836311903}a^{16}-\frac{785414804}{1836311903}a^{14}-\frac{546561780}{1836311903}a^{12}-\frac{867817881}{1836311903}a^{10}+\frac{180896790}{1836311903}a^{8}+\frac{241334982}{1836311903}a^{6}-\frac{823683388}{1836311903}a^{4}-\frac{86455657}{1836311903}a^{2}+\frac{6416}{64079}$, $\frac{1}{1836311903}a^{37}-\frac{21247}{1836311903}a^{23}+\frac{826720807}{1836311903}a^{21}-\frac{575465779}{1836311903}a^{19}+\frac{435718363}{1836311903}a^{17}+\frac{164620450}{1836311903}a^{15}-\frac{166124757}{1836311903}a^{13}-\frac{610028064}{1836311903}a^{11}-\frac{648029154}{1836311903}a^{9}-\frac{623282965}{1836311903}a^{7}+\frac{112318565}{1836311903}a^{5}+\frac{340054167}{1836311903}a^{3}-\frac{297562215}{1836311903}a$, $\frac{1}{1836311903}a^{38}-\frac{333026471}{1836311903}a^{22}-\frac{2787075}{13210877}a^{20}+\frac{367428945}{1836311903}a^{18}+\frac{424704603}{1836311903}a^{16}-\frac{10972472}{1836311903}a^{14}+\frac{314401212}{1836311903}a^{12}-\frac{434857617}{1836311903}a^{10}-\frac{792742269}{1836311903}a^{8}+\frac{910804173}{1836311903}a^{6}-\frac{592411349}{1836311903}a^{4}+\frac{710391958}{1836311903}a^{2}-\frac{16895}{64079}$, $\frac{1}{1836311903}a^{39}-\frac{7908}{1836311903}a^{23}-\frac{526792}{13210877}a^{21}-\frac{163401491}{1836311903}a^{19}-\frac{30192218}{1836311903}a^{17}-\frac{189304329}{1836311903}a^{15}-\frac{713810422}{1836311903}a^{13}+\frac{585023747}{1836311903}a^{11}-\frac{184568480}{1836311903}a^{9}-\frac{193341076}{1836311903}a^{7}+\frac{263748170}{1836311903}a^{5}+\frac{277089760}{1836311903}a^{3}-\frac{169980678}{1836311903}a$, $\frac{1}{1836311903}a^{40}+\frac{684964103}{1836311903}a^{22}-\frac{12251024}{1836311903}a^{20}+\frac{480669675}{1836311903}a^{18}+\frac{118465372}{1836311903}a^{16}+\frac{887088213}{1836311903}a^{14}-\frac{15209563}{1836311903}a^{12}+\frac{315325822}{1836311903}a^{10}-\frac{322961462}{1836311903}a^{8}+\frac{185933271}{1836311903}a^{6}-\frac{687228822}{1836311903}a^{4}-\frac{266630797}{1836311903}a^{2}-\frac{11177}{64079}$, $\frac{1}{1836311903}a^{41}+\frac{23672}{1836311903}a^{23}+\frac{760926190}{1836311903}a^{21}+\frac{867194203}{1836311903}a^{19}+\frac{118080898}{1836311903}a^{17}+\frac{692416211}{1836311903}a^{15}+\frac{754763701}{1836311903}a^{13}-\frac{559416607}{1836311903}a^{11}-\frac{231136255}{1836311903}a^{9}-\frac{493496366}{1836311903}a^{7}-\frac{301601400}{1836311903}a^{5}+\frac{216460784}{1836311903}a^{3}+\frac{452877925}{1836311903}a$, $\frac{1}{1836311903}a^{42}+\frac{575658712}{1836311903}a^{22}+\frac{470466965}{1836311903}a^{20}+\frac{42485271}{1836311903}a^{18}+\frac{825361542}{1836311903}a^{16}+\frac{84775893}{1836311903}a^{14}-\frac{150346933}{1836311903}a^{12}+\frac{27970987}{1836311903}a^{10}+\frac{353358224}{1836311903}a^{8}+\frac{160754499}{1836311903}a^{6}+\frac{719678298}{1836311903}a^{4}+\frac{136590643}{1836311903}a^{2}-\frac{12892}{64079}$, $\frac{1}{1836311903}a^{43}-\frac{27024}{1836311903}a^{23}+\frac{83878358}{1836311903}a^{21}-\frac{150776993}{1836311903}a^{19}+\frac{825553779}{1836311903}a^{17}+\frac{182111894}{1836311903}a^{15}-\frac{535333565}{1836311903}a^{13}-\frac{452813750}{1836311903}a^{11}-\frac{610710331}{1836311903}a^{9}-\frac{417686634}{1836311903}a^{7}+\frac{526864587}{1836311903}a^{5}+\frac{813200804}{1836311903}a^{3}-\frac{756034651}{1836311903}a$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
not computed
Relative class number: data not computed
Unit group
Rank: | $21$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
Torsion generator: |
\( -1 \)
(order $2$)
| sage: UK.torsion_generator()
gp: K.tu[2]
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
oscar: torsion_units_generator(OK)
| |
Fundamental units: | not computed | sage: UK.fundamental_units()
gp: K.fu
magma: [K|fUK(g): g in Generators(UK)];
oscar: [K(fUK(a)) for a in gens(UK)]
| |
Regulator: | not computed | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr = \mathstrut &\frac{2^{0}\cdot(2\pi)^{22}\cdot R \cdot h}{2\cdot\sqrt{65347506006797164323533696798768413693852562329201668296707932160000000000000000000000}}\cr\mathstrut & \text{
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 $C_2\times C_{22}$ |
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 | ${\href{/padicField/7.11.0.1}{11} }^{4}$ | $22^{2}$ | $22^{2}$ | $22^{2}$ | $22^{2}$ | R | ${\href{/padicField/29.11.0.1}{11} }^{4}$ | $22^{2}$ | $22^{2}$ | ${\href{/padicField/41.11.0.1}{11} }^{4}$ | ${\href{/padicField/43.11.0.1}{11} }^{4}$ | ${\href{/padicField/47.2.0.1}{2} }^{22}$ | $22^{2}$ | $22^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\)
| Deg $44$ | $2$ | $22$ | $44$ | |||
\(5\)
| Deg $44$ | $2$ | $22$ | $22$ | |||
\(23\)
| 23.22.21.1 | $x^{22} + 506$ | $22$ | $1$ | $21$ | 22T1 | $[\ ]_{22}$ |
23.22.21.1 | $x^{22} + 506$ | $22$ | $1$ | $21$ | 22T1 | $[\ ]_{22}$ |