Number field 44.44.666454163935483494165986073535521413339908119119439689887653437787720225729135378569.1: \(\Q(\zeta_{89})^+\)

• Label: 44.44.666...569.1
• Degree: $44$
• Signature: $[44, 0]$
• Discriminant: $6.665\times 10^{83}$
• Root discriminant: \(80.37\)
• Ramified prime: $89$
• Class number: $1$ (GRH)
• Class group: trivial (GRH)
• Galois group: $C_{44}$ (as 44T1)

Normalized defining polynomial

x^44 - x^43 - 43*x^42 + 42*x^41 + 861*x^40 - 820*x^39 - 10660*x^38 + 9880*x^37 + 91390*x^36 - 82251*x^35 - 575757*x^34 + 501942*x^33 + 2760681*x^32 - 2324784*x^31 - 10295472*x^30 + 8347680*x^29 + 30260340*x^28 - 23535820*x^27 - 70607460*x^26 + 52451256*x^25 + 131128140*x^24 - 92561040*x^23 - 193536720*x^22 + 129024480*x^21 + 225792840*x^20 - 141120525*x^19 - 206253075*x^18 + 119759850*x^17 + 145422675*x^16 - 77558760*x^15 - 77558760*x^14 + 37442160*x^13 + 30421755*x^12 - 13037895*x^11 - 8436285*x^10 + 3124550*x^9 + 1562275*x^8 - 480700*x^7 - 177100*x^6 + 42504*x^5 + 10626*x^4 - 1771*x^3 - 253*x^2 + 22*x + 1

Invariants

Degree:		$44$
Signature:		$[44, 0]$
Discriminant:		666454163935483494165986073535521413339908119119439689887653437787720225729135378569
Root discriminant:		\(80.37\)
Galois root discriminant:		$89^{43/44}\approx 80.36846772511805$
Ramified primes:		\(89\)
Discriminant root field:		\(\Q(\sqrt{89}) \)
$\card{ \Gal(K/\Q) }$:		$44$
This field is Galois and abelian over $\Q$.
Conductor:		\(89\)
Dirichlet character group:		$\lbrace$$\chi_{89}(1,·)$, $\chi_{89}(2,·)$, $\chi_{89}(4,·)$, $\chi_{89}(5,·)$, $\chi_{89}(8,·)$, $\chi_{89}(9,·)$, $\chi_{89}(10,·)$, $\chi_{89}(11,·)$, $\chi_{89}(16,·)$, $\chi_{89}(17,·)$, $\chi_{89}(18,·)$, $\chi_{89}(20,·)$, $\chi_{89}(21,·)$, $\chi_{89}(22,·)$, $\chi_{89}(25,·)$, $\chi_{89}(32,·)$, $\chi_{89}(34,·)$, $\chi_{89}(36,·)$, $\chi_{89}(39,·)$, $\chi_{89}(40,·)$, $\chi_{89}(42,·)$, $\chi_{89}(44,·)$, $\chi_{89}(45,·)$, $\chi_{89}(47,·)$, $\chi_{89}(49,·)$, $\chi_{89}(50,·)$, $\chi_{89}(53,·)$, $\chi_{89}(55,·)$, $\chi_{89}(57,·)$, $\chi_{89}(64,·)$, $\chi_{89}(67,·)$, $\chi_{89}(68,·)$, $\chi_{89}(69,·)$, $\chi_{89}(71,·)$, $\chi_{89}(72,·)$, $\chi_{89}(73,·)$, $\chi_{89}(78,·)$, $\chi_{89}(79,·)$, $\chi_{89}(80,·)$, $\chi_{89}(81,·)$, $\chi_{89}(84,·)$, $\chi_{89}(85,·)$, $\chi_{89}(87,·)$, $\chi_{89}(88,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $a^{33}$, $a^{34}$, $a^{35}$, $a^{36}$, $a^{37}$, $a^{38}$, $a^{39}$, $a^{40}$, $a^{41}$, $a^{42}$, $a^{43}$

Monogenic:		Yes
Index:		$1$
Inessential primes:		None

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

Unit group

Rank:		$43$
Torsion generator:		\( -1 \)  (order $2$)
Fundamental units:		$a^{12}-12a^{10}+54a^{8}-112a^{6}+105a^{4}-36a^{2}+2$, $a^{24}-24a^{22}+252a^{20}-1520a^{18}+5814a^{16}-14688a^{14}+24752a^{12}-27456a^{10}+19305a^{8}-8008a^{6}+1716a^{4}-144a^{2}+2$, $a^{41}-41a^{39}+779a^{37}-9102a^{35}+73185a^{33}-429352a^{31}+1901416a^{29}-6487184a^{27}+17250012a^{25}-35937525a^{23}+58659315a^{21}-74657310a^{19}+73370115a^{17}-54826020a^{15}+30458900a^{13}-12183560a^{11}+3350479a^{9}-591261a^{7}+59983a^{5}-2870a^{3}+41a$, $a$, $a^{2}-2$, $a^{10}-10a^{8}+36a^{6}-56a^{4}+35a^{2}-6$, $a^{36}-36a^{34}+594a^{32}-5952a^{30}+40455a^{28}-197316a^{26}+712530a^{24}-1937520a^{22}+3996135a^{20}-6249100a^{18}+7354710a^{16}-6418656a^{14}+4056234a^{12}-1790712a^{10}+523260a^{8}-93024a^{6}+8721a^{4}-324a^{2}+2$, $a^{41}-41a^{39}+779a^{37}-9102a^{35}+73185a^{33}-429352a^{31}+1901416a^{29}-6487184a^{27}+17250012a^{25}-35937525a^{23}+58659315a^{21}-74657310a^{19}+73370115a^{17}-54826020a^{15}+30458900a^{13}-12183560a^{11}+3350479a^{9}-591261a^{7}+59983a^{5}-2870a^{3}+41a-1$, $a^{6}-6a^{4}+9a^{2}-2$, $a^{3}-3a$, $a^{43}-42a^{41}+820a^{39}-9880a^{37}+82251a^{35}-501942a^{33}+2324784a^{31}-8347680a^{29}+23535820a^{27}-52451256a^{25}+92561040a^{23}-129024480a^{21}+141120525a^{19}-119759850a^{17}+77558760a^{15}-37442160a^{13}+13037895a^{11}-3124550a^{9}+480700a^{7}-42504a^{5}+1771a^{3}-22a$, $a^{43}-43a^{41}+860a^{39}-10621a^{37}+90687a^{35}-567987a^{33}+2701776a^{31}-9970840a^{29}+28915436a^{27}-66335412a^{25}+120609840a^{23}-173376645a^{21}+195747825a^{19}-171655785a^{17}+115000920a^{15}-57500460a^{13}+20764055a^{11}-5167525a^{9}+826804a^{7}-76153a^{5}+3311a^{3}-43a$, $a^{22}-22a^{20}+209a^{18}-1122a^{16}+3740a^{14}-8008a^{12}+11011a^{10}-9438a^{8}+4719a^{6}-1210a^{4}+121a^{2}-2$, $a^{32}-32a^{30}-a^{29}+464a^{28}+29a^{27}-4031a^{26}-377a^{25}+23374a^{24}+2899a^{23}-95381a^{22}-14651a^{21}+281359a^{20}+51129a^{19}-606671a^{18}-125971a^{17}+955604a^{16}+220133a^{15}-1087371a^{14}-270181a^{13}+873885a^{12}+227239a^{11}-478193a^{10}-125477a^{9}+168452a^{8}+42483a^{7}-35029a^{6}-7904a^{5}+3703a^{4}+658a^{3}-138a^{2}-12a$, $a^{18}-18a^{16}+135a^{14}-546a^{12}+1287a^{10}-1782a^{8}+1386a^{6}-540a^{4}+81a^{2}-2$, $a^{41}-41a^{39}-a^{38}+779a^{37}+38a^{36}-9102a^{35}-666a^{34}+73185a^{33}+7140a^{32}-429351a^{31}-52359a^{30}+1901385a^{29}+278226a^{28}-6486749a^{27}-1107162a^{26}+17246358a^{25}+3362579a^{24}-35917051a^{23}-7871151a^{22}+58578608a^{21}+14241118a^{20}-74427311a^{19}-19851390a^{18}+72890746a^{17}+21122262a^{16}-54095411a^{15}-16878859a^{14}+29653457a^{13}+9878442a^{12}-11555920a^{11}-4077306a^{10}+3018146a^{9}+1120459a^{8}-479282a^{7}-187090a^{6}+38563a^{5}+16296a^{4}-1000a^{3}-580a^{2}-10a+5$, $a^{43}-a^{42}-43a^{41}+41a^{40}+861a^{39}-779a^{38}-10659a^{37}+9101a^{36}+91353a^{35}-73149a^{34}-575127a^{33}+428757a^{32}+2754135a^{31}-1895433a^{30}-10249065a^{29}+6446295a^{28}+30022569a^{27}-17049070a^{26}-69697614a^{25}+35204871a^{24}+128478091a^{23}-56643665a^{22}-187603089a^{21}+70443595a^{20}+215547857a^{19}-66682819a^{18}-192650785a^{17}+46836785a^{16}+131655123a^{15}-23393581a^{14}-67098557a^{13}+7684051a^{12}+24599548a^{11}-1374569a^{10}-6149703a^{9}+33474a^{8}+965130a^{7}+29470a^{6}-83146a^{5}-3430a^{4}+3156a^{3}+49a^{2}-33a+2$, $a^{4}-4a^{2}+2$, $a^{38}-38a^{36}-a^{35}+665a^{34}+35a^{33}-7106a^{32}-560a^{31}+51832a^{30}+5425a^{29}-273296a^{28}-35525a^{27}+1076103a^{26}+166257a^{25}-3223350a^{24}-573300a^{23}+7413705a^{22}+1480050a^{21}-13123110a^{20}-2877876a^{19}+17809935a^{18}+4206144a^{17}-18349629a^{16}-4576416a^{15}+14115084a^{14}+3640875a^{13}-7904352a^{12}-2059239a^{11}+3104970a^{10}+794067a^{9}-809424a^{8}-196308a^{7}+128205a^{6}+28386a^{5}-10494a^{4}-2070a^{3}+297a^{2}+54a+1$, $a^{27}-27a^{25}+324a^{23}-2277a^{21}+10395a^{19}-32319a^{17}+69768a^{15}-104652a^{13}+107406a^{11}-72930a^{9}+30888a^{7}-7371a^{5}+819a^{3}-27a-1$, $a^{42}-a^{41}-42a^{40}+41a^{39}+819a^{38}-779a^{37}-9842a^{36}+9101a^{35}+81585a^{34}-73150a^{33}-494802a^{32}+428792a^{31}+2272424a^{30}-1895992a^{29}-8069423a^{28}+6451688a^{27}+22428224a^{26}-17084132a^{25}-49085050a^{24}+35367125a^{23}+84669740a^{22}-57193939a^{21}-114704955a^{20}+71830794a^{19}+121049760a^{18}-69291271a^{17}-98191829a^{16}+50474564a^{15}+60023584a^{14}-27099700a^{13}-26873106a^{12}+10369592a^{11}+8455019a^{10}-2700128a^{9}-1757766a^{8}+448742a^{7}+219241a^{6}-43434a^{5}-13909a^{4}+2129a^{3}+327a^{2}-41a-2$, $a^{31}-31a^{29}+434a^{27}-3627a^{25}+20150a^{23}-78430a^{21}+219604a^{19}-447051a^{17}+660858a^{15}-700910a^{13}+520676a^{11}-260338a^{9}+82212a^{7}-14756a^{5}+1240a^{3}-31a-1$, $a^{43}-43a^{41}+861a^{39}-10660a^{37}-a^{36}+91389a^{35}+36a^{34}-575721a^{33}-595a^{32}+2760087a^{31}+5984a^{30}-10289520a^{29}-40919a^{28}+30219885a^{27}+201347a^{26}-70410145a^{25}-735904a^{24}+130415635a^{23}+2032900a^{22}-191599476a^{21}-4277471a^{20}+221798477a^{19}+6855542a^{18}-200011309a^{17}-8309022a^{16}+138088467a^{15}+7501488a^{14}-71179544a^{13}-4919851a^{12}+26417728a^{11}+2253979a^{10}-6692322a^{9}-678248a^{8}+1066274a^{7}+121100a^{6}-93688a^{5}-10690a^{4}+3660a^{3}+325a^{2}-40a$, $a^{32}-32a^{30}+464a^{28}-4032a^{26}+23400a^{24}-95680a^{22}+283360a^{20}-615296a^{18}+980628a^{16}-1136960a^{14}+940576a^{12}-537472a^{10}+201552a^{8}-45696a^{6}+5440a^{4}-256a^{2}+3$, $a^{42}-42a^{40}-a^{39}+819a^{38}+39a^{37}-9841a^{36}-702a^{35}+81549a^{34}+7735a^{33}-494208a^{32}-58344a^{31}+2266472a^{30}+319176a^{29}-8028968a^{28}-1308944a^{27}+22230908a^{26}+4102136a^{25}-48372520a^{24}-9924500a^{23}+82732220a^{22}+18599020a^{21}-110708820a^{20}-26935160a^{19}+114800660a^{18}+29903339a^{17}-90837120a^{16}-25090623a^{15}+53604945a^{14}+15565081a^{13}-22816990a^{12}-6908903a^{11}+6664735a^{10}+2088020a^{9}-1235365a^{8}-396748a^{7}+127140a^{6}+41041a^{5}-5650a^{4}-1672a^{3}+75a^{2}+11a+1$, $a^{41}-41a^{39}-a^{38}+779a^{37}+38a^{36}-9102a^{35}-666a^{34}+73185a^{33}+7140a^{32}-429351a^{31}-52359a^{30}+1901385a^{29}+278226a^{28}-6486749a^{27}-1107162a^{26}+17246358a^{25}+3362579a^{24}-35917051a^{23}-7871151a^{22}+58578609a^{21}+14241118a^{20}-74427332a^{19}-19851390a^{18}+72890935a^{17}+21122262a^{16}-54096363a^{15}-16878859a^{14}+29656397a^{13}+9878442a^{12}-11561653a^{11}-4077306a^{10}+3025153a^{9}+1120459a^{8}-484430a^{7}-187090a^{6}+40642a^{5}+16296a^{4}-1385a^{3}-580a^{2}+11a+5$, $a^{41}-41a^{39}+779a^{37}-9102a^{35}-a^{34}+73185a^{33}+34a^{32}-429352a^{31}-527a^{30}+1901416a^{29}+4930a^{28}-6487184a^{27}-31059a^{26}+17250012a^{25}+139230a^{24}-35937525a^{23}-457470a^{22}+58659315a^{21}+1118260a^{20}-74657310a^{19}-2042975a^{18}+73370115a^{17}+2778446a^{16}-54826020a^{15}-2778446a^{14}+30458900a^{13}+1998724a^{12}-12183560a^{11}-999362a^{10}+3350479a^{9}+329460a^{8}-591260a^{7}-65892a^{6}+59976a^{5}+6936a^{4}-2856a^{3}-289a^{2}+34a+1$, $a^{29}-29a^{27}+377a^{25}-2900a^{23}+14674a^{21}-51359a^{19}+127281a^{17}-224808a^{15}+281010a^{13}-243542a^{11}+140998a^{9}-51272a^{7}+10556a^{5}-1015a^{3}+29a-1$, $a^{41}-41a^{39}+779a^{37}-9102a^{35}+73185a^{33}-429352a^{31}+1901416a^{29}-6487184a^{27}+17250012a^{25}-a^{24}-35937525a^{23}+24a^{22}+58659315a^{21}-252a^{20}-74657310a^{19}+1520a^{18}+73370116a^{17}-5814a^{16}-54826037a^{15}+14688a^{14}+30459019a^{13}-24752a^{12}-12184002a^{11}+27456a^{10}+3351414a^{9}-19305a^{8}-592382a^{7}+8008a^{6}+60690a^{5}-1716a^{4}-3060a^{3}+144a^{2}+51a-3$, $a^{26}-26a^{24}+299a^{22}-2002a^{20}+8645a^{18}-25194a^{16}+50388a^{14}-a^{13}-68952a^{12}+13a^{11}+63206a^{10}-65a^{9}-37180a^{8}+156a^{7}+13013a^{6}-182a^{5}-2366a^{4}+91a^{3}+169a^{2}-13a-1$, $a^{32}-a^{31}-31a^{30}+30a^{29}+435a^{28}-406a^{27}-3655a^{26}+3277a^{25}+20500a^{24}-17574a^{23}-81006a^{22}+66033a^{21}+232001a^{20}-178640a^{19}-488015a^{18}+352089a^{17}+755820a^{16}-505818a^{15}-855950a^{14}+524552a^{13}+697034a^{12}-384540a^{11}-396474a^{10}+192270a^{9}+150280a^{8}-61828a^{7}-35140a^{6}+11571a^{5}+4425a^{4}-1044a^{3}-226a^{2}+28a+1$, $a^{41}-41a^{39}+779a^{37}-9102a^{35}-a^{34}+73185a^{33}+34a^{32}-429352a^{31}-527a^{30}+1901416a^{29}+4929a^{28}-6487183a^{27}-31031a^{26}+17249985a^{25}+138880a^{24}-35937201a^{23}-454894a^{22}+58657039a^{21}+1105862a^{20}-74646936a^{19}-2001991a^{18}+73337985a^{17}+2683314a^{16}-54757204a^{15}-2622607a^{14}+30357188a^{13}+1820105a^{12}-12081887a^{11}-859299a^{10}+3284556a^{9}+257312a^{8}-565520a^{7}-43162a^{6}+54684a^{5}+3122a^{4}-2422a^{3}-42a^{2}+28a-1$, $a^{41}-41a^{39}+779a^{37}-9102a^{35}+73185a^{33}-429352a^{31}+1901416a^{29}-6487184a^{27}+17250012a^{25}-a^{24}-35937525a^{23}+24a^{22}+58659315a^{21}-252a^{20}-74657310a^{19}+1520a^{18}+73370116a^{17}-5814a^{16}-54826037a^{15}+14688a^{14}+30459019a^{13}-24752a^{12}-12184002a^{11}+27456a^{10}+3351414a^{9}-19305a^{8}-592383a^{7}+8008a^{6}+60697a^{5}-1716a^{4}-3074a^{3}+144a^{2}+58a-3$, $a^{30}-30a^{28}+405a^{26}-a^{25}-3250a^{24}+25a^{23}+17250a^{22}-275a^{21}-63755a^{20}+1750a^{19}+168225a^{18}-7125a^{17}-319600a^{16}+19379a^{15}+435250a^{14}-35685a^{13}-417625a^{12}+44110a^{11}+273131a^{10}-35475a^{9}-115060a^{8}+17425a^{7}+28335a^{6}-4628a^{5}-3425a^{4}+515a^{3}+150a^{2}-15a-1$, $a^{14}-14a^{12}+77a^{10}-210a^{8}+294a^{6}-196a^{4}+49a^{2}-1$, $a^{17}-17a^{15}+119a^{13}-442a^{11}+935a^{9}-1122a^{7}+714a^{5}-204a^{3}+17a$, $a^{42}-a^{41}-41a^{40}+41a^{39}+779a^{38}-779a^{37}-9102a^{36}+9101a^{35}+73186a^{34}-73151a^{33}-429385a^{32}+428825a^{31}+1901911a^{30}-1896486a^{29}-6491650a^{28}+6456124a^{27}+17277040a^{26}-17110756a^{25}-36053381a^{24}+35479756a^{23}+59021404a^{22}-57539053a^{21}-75494211a^{20}+72605689a^{19}+74806420a^{18}-70566457a^{17}-56648880a^{16}+51997051a^{15}+32150109a^{14}-28390678a^{13}-13308930a^{12}+11119704a^{11}+3872847a^{10}-2982046a^{9}-753797a^{8}+510926a^{7}+91861a^{6}-50065a^{5}-6391a^{4}+2280a^{3}+210a^{2}-26a-2$, $a^{37}-37a^{35}+629a^{33}-6512a^{31}+45880a^{29}-232841a^{27}-a^{26}+878787a^{25}+26a^{24}-2510820a^{23}-299a^{22}+5476185a^{21}+2002a^{20}-9126975a^{19}-8645a^{18}+11560835a^{17}+25194a^{16}-10994920a^{15}-50388a^{14}+7696444a^{13}+68952a^{12}-3848222a^{11}-63206a^{10}+1314610a^{9}+37180a^{8}-286824a^{7}-13013a^{6}+35853a^{5}+2366a^{4}-2109a^{3}-169a^{2}+37a+1$, $a^{43}-43a^{41}-a^{40}+861a^{39}+40a^{38}-10660a^{37}-741a^{36}+91389a^{35}+8436a^{34}-575721a^{33}-66045a^{32}+2760087a^{31}+376992a^{30}-10289520a^{29}-1623159a^{28}+30219885a^{27}+5379587a^{26}-70410144a^{25}-13883779a^{24}+130415610a^{23}+28045900a^{22}-191599201a^{21}-44337491a^{20}+221796727a^{19}+54575942a^{18}-200004184a^{17}-51768673a^{16}+138069087a^{15}+37217504a^{14}-71143844a^{13}-19777955a^{12}+26373528a^{11}+7484347a^{10}-6656572a^{9}-1904694a^{8}+1048399a^{7}+297340a^{6}-88683a^{5}-24346a^{4}+3010a^{3}+805a^{2}-14a-6$, $a^{42}-a^{41}-42a^{40}+41a^{39}+819a^{38}-780a^{37}-9841a^{36}+9139a^{35}+81549a^{34}-73814a^{33}-494208a^{32}+435863a^{31}+2266473a^{30}-1947265a^{29}-8028999a^{28}+6719591a^{27}+22231341a^{26}-18125173a^{25}-48376120a^{24}+38428220a^{23}+82752045a^{22}-64057345a^{21}-110784950a^{20}+83566430a^{19}+115009640a^{18}-84491005a^{17}-91250560a^{16}+65179310a^{15}+54191495a^{14}-37489469a^{13}-23402965a^{12}+15553577a^{11}+7062990a^{10}-4437785a^{9}-1409045a^{8}+811249a^{7}+171105a^{6}-84858a^{5}-11000a^{4}+4133a^{3}+276a^{2}-59a-2$, $a^{39}-39a^{37}+702a^{35}-7735a^{33}+58344a^{31}-319176a^{29}-a^{28}+1308945a^{27}+28a^{26}-4102164a^{25}-350a^{24}+9924849a^{23}+2576a^{22}-18601572a^{21}-12397a^{20}+26947305a^{19}+40964a^{18}-29942783a^{17}-94963a^{16}+25179771a^{15}+155056a^{14}-15705433a^{13}-176462a^{12}+7060508a^{11}+136488a^{10}-2196689a^{9}-68728a^{8}+445467a^{7}+21055a^{6}-53339a^{5}-3515a^{4}+3080a^{3}+251a^{2}-44a-2$, $a^{43}-43a^{41}-a^{40}+860a^{39}+40a^{38}-10621a^{37}-741a^{36}+90687a^{35}+8436a^{34}-567986a^{33}-66044a^{32}+2701743a^{31}+376959a^{30}-9970345a^{29}-1622665a^{28}+28910970a^{27}+5375150a^{26}-66308385a^{25}-13857129a^{24}+120494011a^{23}+27932971a^{22}-173014878a^{21}-43990398a^{20}+194913155a^{19}+53792630a^{18}-170229417a^{17}-50469566a^{16}+113207794a^{15}+35649018a^{14}-55869921a^{13}-18427657a^{12}+19722638a^{11}+6684391a^{10}-4722015a^{9}-1596805a^{8}+708608a^{7}+227236a^{6}-59071a^{5}-16230a^{4}+2245a^{3}+405a^{2}-30a-1$, $a^{7}-7a^{5}+14a^{3}-7a$ (assuming GRH)
Regulator:		\( 9928065388501928000000000000 \) (assuming GRH)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{44}\cdot(2\pi)^{0}\cdot 9928065388501928000000000000 \cdot 1}{2\cdot\sqrt{666454163935483494165986073535521413339908119119439689887653437787720225729135378569}}\cr\approx \mathstrut & 0.106971798982240 \end{aligned}\] (assuming GRH)

Galois group

$C_{44}$ (as 44T1):

A cyclic group of order 44

The 44 conjugacy class representatives for $C_{44}$

Character table for $C_{44}$

Intermediate fields

\(\Q(\sqrt{89}) \), 4.4.704969.1, 11.11.31181719929966183601.1, 22.22.86534669543385676516186776267386878120889.1

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	${\href{/padicField/2.11.0.1}{11} }^{4}$	$44$	$22^{2}$	$44$	${\href{/padicField/11.11.0.1}{11} }^{4}$	$44$	$22^{2}$	$44$	$44$	$44$	$44$	${\href{/padicField/37.4.0.1}{4} }^{11}$	$44$	$44$	$22^{2}$	$22^{2}$	$44$

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
\(89\)		Deg $44$	$44$	$1$	$43$