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Properties

Label 44.0.653...000.1
Degree $44$
Signature $[0, 22]$
Discriminant $6.535\times 10^{85}$
Root discriminant $89.20$
Ramified primes $2, 5, 23$
Class number not computed
Class group not computed
Galois group $C_2\times C_{22}$ (as 44T2)

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Show commands for: SageMath / Pari/GP / Magma

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(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)
 
gp: K = bnfinit(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, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![821223649, 0, -164244624, 0, 32853580, 0, -6489250, 0, 2054320, 0, 3901015, 0, 15683335, 0, 41459620, 0, 80900650, 0, 119357605, 0, 135920335, 0, 121194085, 0, 85431991, 0, 47872465, 0, 21358299, 0, 7568133, 0, 2114697, 0, 459888, 0, 76180, 0, 9282, 0, 784, 0, 41, 0, 1]);
 

\(x^{44} + 41 x^{42} + 784 x^{40} + 9282 x^{38} + 76180 x^{36} + 459888 x^{34} + 2114697 x^{32} + 7568133 x^{30} + 21358299 x^{28} + 47872465 x^{26} + 85431991 x^{24} + 121194085 x^{22} + 135920335 x^{20} + 119357605 x^{18} + 80900650 x^{16} + 41459620 x^{14} + 15683335 x^{12} + 3901015 x^{10} + 2054320 x^{8} - 6489250 x^{6} + 32853580 x^{4} - 164244624 x^{2} + 821223649\)  Toggle raw display

sage: K.defining_polynomial()
 
gp: K.pol
 
magma: DefiningPolynomial(K);
 

Invariants

Degree:  $44$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[0, 22]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(653\!\cdots\!000\)\(\medspace = 2^{44}\cdot 5^{22}\cdot 23^{42}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $89.20$
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
Ramified primes:  $2, 5, 23$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Gal(K/\Q)|$:  $44$
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.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $\frac{1}{28657} a^{23} + \frac{23}{28657} a^{21} + \frac{230}{28657} a^{19} + \frac{1311}{28657} a^{17} + \frac{4692}{28657} a^{15} + \frac{10948}{28657} a^{13} - \frac{11913}{28657} a^{11} - \frac{12212}{28657} a^{9} + \frac{9867}{28657} a^{7} + \frac{3289}{28657} a^{5} + \frac{506}{28657} a^{3} + \frac{23}{28657} a$, $\frac{1}{1836311903} a^{24} - \frac{701408709}{1836311903} a^{22} - \frac{740496650}{1836311903} a^{20} + \frac{310528563}{1836311903} a^{18} + \frac{797213775}{1836311903} a^{16} + \frac{821062655}{1836311903} a^{14} + \frac{397002165}{1836311903} a^{12} + \frac{316332411}{1836311903} a^{10} + \frac{8807566}{1836311903} a^{8} - \frac{913753813}{1836311903} a^{6} - \frac{328466028}{1836311903} a^{4} - \frac{400109011}{1836311903} a^{2} + \frac{15127}{64079}$, $\frac{1}{1836311903} a^{25} + \frac{25}{1836311903} a^{23} + \frac{701409008}{1836311903} a^{21} + \frac{39089919}{1836311903} a^{19} + \frac{351800646}{1836311903} a^{17} - \frac{676399496}{1836311903} a^{15} - \frac{36556349}{1836311903} a^{13} - \frac{346757081}{1836311903} a^{11} + \frac{800375453}{1836311903} a^{9} + \frac{662974061}{1836311903} a^{7} + \frac{197109930}{1836311903} a^{5} + \frac{104513114}{1836311903} a^{3} + \frac{39088194}{1836311903} a$, $\frac{1}{1836311903} a^{26} - \frac{126492297}{1836311903} a^{22} + \frac{188387139}{1836311903} a^{20} - \frac{66165817}{1836311903} a^{18} - \frac{407312938}{1836311903} a^{16} - \frac{363691791}{1836311903} a^{14} + \frac{746060212}{1836311903} a^{12} + \frac{237312790}{1836311903} a^{10} + \frac{442784911}{1836311903} a^{8} - \frac{831099484}{1836311903} a^{6} - \frac{865395701}{1836311903} a^{4} + \frac{860253954}{1836311903} a^{2} + \frac{6299}{64079}$, $\frac{1}{1836311903} a^{27} - \frac{351}{1836311903} a^{23} - \frac{574921909}{1836311903} a^{21} - \frac{354008685}{1836311903} a^{19} + \frac{155556998}{1836311903} a^{17} + \frac{7774172}{1836311903} a^{15} - \frac{835601745}{1836311903} a^{13} - \frac{885479448}{1836311903} a^{11} + \frac{61450782}{1836311903} a^{9} + \frac{409149561}{1836311903} a^{7} + \frac{160124615}{1836311903} a^{5} + \frac{594262025}{1836311903} a^{3} - \frac{582798605}{1836311903} a$, $\frac{1}{1836311903} a^{28} - \frac{703583766}{1836311903} a^{22} + \frac{487957391}{1836311903} a^{20} + \frac{808680334}{1836311903} a^{18} + \frac{710399941}{1836311903} a^{16} + \frac{892733292}{1836311903} a^{14} + \frac{738887742}{1836311903} a^{12} + \frac{915412863}{1836311903} a^{10} - \frac{172018579}{1836311903} a^{8} + \frac{787119277}{1836311903} a^{6} - \frac{845975817}{1836311903} a^{4} + \frac{374955065}{1836311903} a^{2} - \frac{8980}{64079}$, $\frac{1}{1836311903} a^{29} + \frac{3654}{1836311903} a^{23} + \frac{143660924}{1836311903} a^{21} - \frac{797972433}{1836311903} a^{19} - \frac{551379648}{1836311903} a^{17} + \frac{436106338}{1836311903} a^{15} + \frac{285528817}{1836311903} a^{13} + \frac{6003695}{1836311903} a^{11} - \frac{278197482}{1836311903} a^{9} - \frac{11112826}{1836311903} a^{7} - \frac{499949217}{1836311903} a^{5} + \frac{145680403}{1836311903} a^{3} - \frac{601636327}{1836311903} a$, $\frac{1}{1836311903} a^{30} - \frac{400332978}{1836311903} a^{22} + \frac{89353548}{1836311903} a^{20} - \frac{381992796}{1836311903} a^{18} - \frac{192349354}{1836311903} a^{16} + \frac{656236949}{1836311903} a^{14} + \frac{46496155}{1836311903} a^{12} + \frac{719671614}{1836311903} a^{10} + \frac{859655264}{1836311903} a^{8} - \frac{58556169}{1836311903} a^{6} - \frac{587437847}{1836311903} a^{4} - \frac{307584921}{1836311903} a^{2} + \frac{26119}{64079}$, $\frac{1}{1836311903} a^{31} - \frac{31465}{1836311903} a^{23} + \frac{114728832}{1836311903} a^{21} - \frac{128239956}{1836311903} a^{19} - \frac{582270069}{1836311903} a^{17} + \frac{323859176}{1836311903} a^{15} - \frac{729051982}{1836311903} a^{13} + \frac{829759336}{1836311903} a^{11} + \frac{639864294}{1836311903} a^{9} - \frac{190430751}{1836311903} a^{7} - \frac{631396041}{1836311903} a^{5} + \frac{250671327}{1836311903} a^{3} + \frac{773867467}{1836311903} a$, $\frac{1}{1836311903} a^{32} - \frac{913849599}{1836311903} a^{22} - \frac{729906942}{1836311903} a^{20} - \frac{816671137}{1836311903} a^{18} + \frac{634694571}{1836311903} a^{16} + \frac{771536189}{1836311903} a^{14} + \frac{73004952}{1836311903} a^{12} - \frac{607649754}{1836311903} a^{10} - \frac{343463914}{1836311903} a^{8} - \frac{759656815}{1836311903} a^{6} - \frac{169509609}{1836311903} a^{4} - \frac{738068583}{1836311903} a^{2} - \frac{7757}{64079}$, $\frac{1}{1836311903} a^{33} - \frac{18980}{1836311903} a^{23} + \frac{88766362}{1836311903} a^{21} + \frac{24814291}{1836311903} a^{19} - \frac{445036579}{1836311903} a^{17} + \frac{676507032}{1836311903} a^{15} + \frac{463374220}{1836311903} a^{13} + \frac{421458986}{1836311903} a^{11} - \frac{775548611}{1836311903} a^{9} - \frac{284382872}{1836311903} a^{7} - \frac{623188929}{1836311903} a^{5} + \frac{745936978}{1836311903} a^{3} + \frac{596380955}{1836311903} a$, $\frac{1}{1836311903} a^{34} + \frac{612766292}{1836311903} a^{22} + \frac{529702853}{1836311903} a^{20} + \frac{662192434}{1836311903} a^{18} + \frac{583875812}{1836311903} a^{16} - \frac{546554641}{1836311903} a^{14} - \frac{701499226}{1836311903} a^{12} + \frac{310001262}{1836311903} a^{10} - \frac{221163365}{1836311903} a^{8} + \frac{295364166}{1836311903} a^{6} + \frac{739636223}{1836311903} a^{4} - \frac{322928920}{1836311903} a^{2} - \frac{27539}{64079}$, $\frac{1}{1836311903} a^{35} - \frac{21185}{1836311903} a^{23} - \frac{710225797}{1836311903} a^{21} - \frac{719222648}{1836311903} a^{19} - \frac{312204924}{1836311903} a^{17} - \frac{80956627}{1836311903} a^{15} + \frac{384896140}{1836311903} a^{13} - \frac{728911565}{1836311903} a^{11} + \frac{168501034}{1836311903} a^{9} + \frac{896425186}{1836311903} a^{7} - \frac{284218039}{1836311903} a^{5} - \frac{56680675}{1836311903} a^{3} - \frac{192801870}{1836311903} a$, $\frac{1}{1836311903} a^{36} - \frac{617806886}{1836311903} a^{22} - \frac{528165569}{1836311903} a^{20} + \frac{566165685}{1836311903} a^{18} + \frac{332294857}{1836311903} a^{16} - \frac{785414804}{1836311903} a^{14} - \frac{546561780}{1836311903} a^{12} - \frac{867817881}{1836311903} a^{10} + \frac{180896790}{1836311903} a^{8} + \frac{241334982}{1836311903} a^{6} - \frac{823683388}{1836311903} a^{4} - \frac{86455657}{1836311903} a^{2} + \frac{6416}{64079}$, $\frac{1}{1836311903} a^{37} - \frac{21247}{1836311903} a^{23} + \frac{826720807}{1836311903} a^{21} - \frac{575465779}{1836311903} a^{19} + \frac{435718363}{1836311903} a^{17} + \frac{164620450}{1836311903} a^{15} - \frac{166124757}{1836311903} a^{13} - \frac{610028064}{1836311903} a^{11} - \frac{648029154}{1836311903} a^{9} - \frac{623282965}{1836311903} a^{7} + \frac{112318565}{1836311903} a^{5} + \frac{340054167}{1836311903} a^{3} - \frac{297562215}{1836311903} a$, $\frac{1}{1836311903} a^{38} - \frac{333026471}{1836311903} a^{22} - \frac{2787075}{13210877} a^{20} + \frac{367428945}{1836311903} a^{18} + \frac{424704603}{1836311903} a^{16} - \frac{10972472}{1836311903} a^{14} + \frac{314401212}{1836311903} a^{12} - \frac{434857617}{1836311903} a^{10} - \frac{792742269}{1836311903} a^{8} + \frac{910804173}{1836311903} a^{6} - \frac{592411349}{1836311903} a^{4} + \frac{710391958}{1836311903} a^{2} - \frac{16895}{64079}$, $\frac{1}{1836311903} a^{39} - \frac{7908}{1836311903} a^{23} - \frac{526792}{13210877} a^{21} - \frac{163401491}{1836311903} a^{19} - \frac{30192218}{1836311903} a^{17} - \frac{189304329}{1836311903} a^{15} - \frac{713810422}{1836311903} a^{13} + \frac{585023747}{1836311903} a^{11} - \frac{184568480}{1836311903} a^{9} - \frac{193341076}{1836311903} a^{7} + \frac{263748170}{1836311903} a^{5} + \frac{277089760}{1836311903} a^{3} - \frac{169980678}{1836311903} a$, $\frac{1}{1836311903} a^{40} + \frac{684964103}{1836311903} a^{22} - \frac{12251024}{1836311903} a^{20} + \frac{480669675}{1836311903} a^{18} + \frac{118465372}{1836311903} a^{16} + \frac{887088213}{1836311903} a^{14} - \frac{15209563}{1836311903} a^{12} + \frac{315325822}{1836311903} a^{10} - \frac{322961462}{1836311903} a^{8} + \frac{185933271}{1836311903} a^{6} - \frac{687228822}{1836311903} a^{4} - \frac{266630797}{1836311903} a^{2} - \frac{11177}{64079}$, $\frac{1}{1836311903} a^{41} + \frac{23672}{1836311903} a^{23} + \frac{760926190}{1836311903} a^{21} + \frac{867194203}{1836311903} a^{19} + \frac{118080898}{1836311903} a^{17} + \frac{692416211}{1836311903} a^{15} + \frac{754763701}{1836311903} a^{13} - \frac{559416607}{1836311903} a^{11} - \frac{231136255}{1836311903} a^{9} - \frac{493496366}{1836311903} a^{7} - \frac{301601400}{1836311903} a^{5} + \frac{216460784}{1836311903} a^{3} + \frac{452877925}{1836311903} a$, $\frac{1}{1836311903} a^{42} + \frac{575658712}{1836311903} a^{22} + \frac{470466965}{1836311903} a^{20} + \frac{42485271}{1836311903} a^{18} + \frac{825361542}{1836311903} a^{16} + \frac{84775893}{1836311903} a^{14} - \frac{150346933}{1836311903} a^{12} + \frac{27970987}{1836311903} a^{10} + \frac{353358224}{1836311903} a^{8} + \frac{160754499}{1836311903} a^{6} + \frac{719678298}{1836311903} a^{4} + \frac{136590643}{1836311903} a^{2} - \frac{12892}{64079}$, $\frac{1}{1836311903} a^{43} - \frac{27024}{1836311903} a^{23} + \frac{83878358}{1836311903} a^{21} - \frac{150776993}{1836311903} a^{19} + \frac{825553779}{1836311903} a^{17} + \frac{182111894}{1836311903} a^{15} - \frac{535333565}{1836311903} a^{13} - \frac{452813750}{1836311903} a^{11} - \frac{610710331}{1836311903} a^{9} - \frac{417686634}{1836311903} a^{7} + \frac{526864587}{1836311903} a^{5} + \frac{813200804}{1836311903} a^{3} - \frac{756034651}{1836311903} a$  Toggle raw display

sage: K.integral_basis()
 
gp: K.zk
 
magma: IntegralBasis(K);
 

Class group and class number

not computed

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $21$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( -1 \) (order $2$)  Toggle raw display
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
Fundamental units:  not computed  Toggle raw display
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

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) $ not computed

Galois group

$C_2\times C_{22}$ (as 44T2):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
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

\(\Q(\sqrt{-5}) \), \(\Q(\sqrt{-115}) \), \(\Q(\sqrt{23}) \), \(\Q(\sqrt{-5}, \sqrt{23})\), \(\Q(\zeta_{23})^+\), 22.0.351468714257323283030813737164800000000000.1, 22.0.1927323443393334271838358868310546875.1, \(\Q(\zeta_{92})^+\)

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.

sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
gp: idealfactors = idealprimedec(K, p); \\ get the data
 
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
 
magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
magma: idealfactors := Factorization(p*Integers(K)); // get the data
 
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
2Data not computed
5Data not computed
23Data not computed