LMFDB - Number field 44.0.65347506006797164323533696798768413693852562329201668296707932160000000000000000000000.1

Show commands: Magma / Oscar / PariGP / SageMath

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(y^44 + 41*y^42 + 784*y^40 + 9282*y^38 + 76180*y^36 + 459888*y^34 + 2114697*y^32 + 7568133*y^30 + 21358299*y^28 + 47872465*y^26 + 85431991*y^24 + 121194085*y^22 + 135920335*y^20 + 119357605*y^18 + 80900650*y^16 + 41459620*y^14 + 15683335*y^12 + 3901015*y^10 + 2054320*y^8 - 6489250*y^6 + 32853580*y^4 - 164244624*y^2 + 821223649, 1)

magma: R<x> := PolynomialRing(Rationals()); 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);

oscar: Qx, x = PolynomialRing(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)

$ x^{44} + 41 x^{42} + 784 x^{40} + 9282 x^{38} + 76180 x^{36} + 459888 x^{34} + 2114697 x^{32} + \cdots + 821223649 $ 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

sage: K.defining_polynomial()

gp: K.pol

magma: DefiningPolynomial(K);

oscar: defining_polynomial(K)

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:	$653\!\cdots\!000$ 65347506006797164323533696798768413693852562329201668296707932160000000000000000000000 $\medspace = 2^{44}\cdot 5^{22}\cdot 23^{42}$	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:	$2$, $5$, $23$ 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 $a$)

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

sage: K.integral_basis()

gp: K.zk

magma: IntegralBasis(K);

oscar: basis(OK)

Monogenic:		Not computed
Index:		$1$
Inessential primes:		None

Class group and class number

not computed

sage: K.class_group().invariants()

gp: K.clgp

magma: ClassGroup(K);

oscar: class_group(K)

Unit group

sage: UK = K.unit_group()

magma: UK, fUK := UnitGroup(K);

oscar: UK, fUK = unit_group(OK)

Rank:	$21$	sage: UK.rank() gp: K.fu magma: UnitRank(K); oscar: rank(UK)
Torsion generator:	$ -1 $ -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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr $ not computed \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula

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)

DK = K.disc(); r1,r2 = K.signature(); RK = K.regulator(); RR = RK.parent()

hK = K.class_number(); wK = K.unit_group().torsion_generator().order();

2^r1 * (2*RR(pi))^r2 * RK * hK / (wK * RR(sqrt(abs(DK))))

# self-contained Pari/GP code snippet to compute the analytic class number formula

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);

[polcoeff (lfunrootres (lfuncreate (K))[1][1][2], -1), 2^K.r1 * (2*Pi)^K.r2 * K.reg * K.no / (K.tu[1] * sqrt (abs (K.disc)))]

/* self-contained Magma code snippet to compute the analytic class number formula */

Qx<x> := PolynomialRing(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);

OK := Integers(K); DK := Discriminant(OK);

UK, fUK := UnitGroup(OK); clK, fclK := ClassGroup(OK);

r1,r2 := Signature(K); RK := Regulator(K); RR := Parent(RK);

hK := #clK; wK := #TorsionSubgroup(UK);

2^r1 * (2*Pi(RR))^r2 * RK * hK / (wK * Sqrt(RR!Abs(DK)));

# self-contained Oscar code snippet to compute the analytic class number formula

Qx, x = PolynomialRing(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);

OK = ring_of_integers(K); DK = discriminant(OK);

UK, fUK = unit_group(OK); clK, fclK = class_group(OK);

r1,r2 = signature(K); RK = regulator(K); RR = parent(RK);

hK = order(clK); wK = torsion_units_order(K);

2^r1 * (2*pi)^r2 * RK * hK / (wK * sqrt(RR(abs(DK))))

Galois group

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

sage: K.galois_group(type='pari')

gp: polgalois(K.pol)

magma: G = GaloisGroup(K);

oscar: G, Gtx = galois_group(K); G, transitive_group_identification(G)

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

$\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.

sage: K.subfields()[1:-1]

gp: L = nfsubfields(K); L[2..length(b)]

magma: L := Subfields(K); L[2..#L];

oscar: subfields(K)[2:end-1]

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.

# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Sage:

p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]

\\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Pari:

p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])

// to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7 in Magma:

p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];

# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Oscar:

p = 7; pfac = factor(ideal(ring_of_integers(K), p)); [(e, valuation(norm(pr),p)) for (pr,e) in pfac]

Local algebras for ramified primes

$p$	Label	Polynomial	$e$	$f$	$c$	Galois group	Slope content
$2$ 2		Deg $44$	$2$	$22$	$44$
$5$ 5		Deg $44$	$2$	$22$	$22$
$23$ 23	23.22.21.1	$x^{22} + 506$	$22$	$1$	$21$	22T1	$[\ ]_{22}$
$23$ 23	23.22.21.1	$x^{22} + 506$	$22$	$1$	$21$	22T1	$[\ ]_{22}$

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