Number field 44.44.482179487665033966874817964307376476160282778171317425736173928285756467369658548224.1: \(\Q(\zeta_{184})^+\)

• Label: 44.44.482...224.1
• Degree: $44$
• Signature: $[44, 0]$
• Discriminant: $4.822\times 10^{83}$
• Root discriminant: \(79.78\)
• Ramified primes: $2,23$
• Class number: $1$ (GRH)
• Class group: trivial (GRH)
• Galois group: $C_2\times C_{22}$ (as 44T2)

Normalized defining polynomial

x^44 - 44*x^42 + 901*x^40 - 11400*x^38 + 99790*x^36 - 641208*x^34 + 3131721*x^32 - 11878176*x^30 + 35442612*x^28 - 83778736*x^26 + 157236844*x^24 - 233880352*x^22 + 274130056*x^20 - 250699168*x^18 + 176290339*x^16 - 93382192*x^14 + 36217051*x^12 - 9883588*x^10 + 1792219*x^8 - 197912*x^6 + 11506*x^4 - 264*x^2 + 1

Invariants

Degree:		$44$
Signature:		$[44, 0]$
Discriminant:		482179487665033966874817964307376476160282778171317425736173928285756467369658548224
Root discriminant:		\(79.78\)
Galois root discriminant:		$2^{2}23^{21/22}\approx 79.77946278015138$
Ramified primes:		\(2\), \(23\)
Discriminant root field:		\(\Q\)
$\card{ \Gal(K/\Q) }$:		$44$
This field is Galois and abelian over $\Q$.
Conductor:		\(184=2^{3}\cdot 23\)
Dirichlet character group:		$\lbrace$$\chi_{184}(1,·)$, $\chi_{184}(43,·)$, $\chi_{184}(133,·)$, $\chi_{184}(7,·)$, $\chi_{184}(9,·)$, $\chi_{184}(11,·)$, $\chi_{184}(13,·)$, $\chi_{184}(15,·)$, $\chi_{184}(19,·)$, $\chi_{184}(25,·)$, $\chi_{184}(155,·)$, $\chi_{184}(29,·)$, $\chi_{184}(159,·)$, $\chi_{184}(91,·)$, $\chi_{184}(165,·)$, $\chi_{184}(177,·)$, $\chi_{184}(41,·)$, $\chi_{184}(171,·)$, $\chi_{184}(173,·)$, $\chi_{184}(175,·)$, $\chi_{184}(49,·)$, $\chi_{184}(51,·)$, $\chi_{184}(183,·)$, $\chi_{184}(63,·)$, $\chi_{184}(67,·)$, $\chi_{184}(73,·)$, $\chi_{184}(79,·)$, $\chi_{184}(77,·)$, $\chi_{184}(141,·)$, $\chi_{184}(81,·)$, $\chi_{184}(83,·)$, $\chi_{184}(85,·)$, $\chi_{184}(143,·)$, $\chi_{184}(135,·)$, $\chi_{184}(93,·)$, $\chi_{184}(99,·)$, $\chi_{184}(101,·)$, $\chi_{184}(103,·)$, $\chi_{184}(105,·)$, $\chi_{184}(107,·)$, $\chi_{184}(111,·)$, $\chi_{184}(117,·)$, $\chi_{184}(169,·)$, $\chi_{184}(121,·)$$\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:		$a^{16}-16a^{14}+104a^{12}-352a^{10}+660a^{8}-672a^{6}+336a^{4}-64a^{2}+2$, $a^{36}-36a^{34}+594a^{32}-5952a^{30}+40455a^{28}-197316a^{26}+712530a^{24}-1937520a^{22}+3996135a^{20}-6249100a^{18}+7354710a^{16}-6418656a^{14}+4056234a^{12}-1790712a^{10}+523260a^{8}-93024a^{6}+8721a^{4}-324a^{2}+2$, $a^{20}-20a^{18}+170a^{16}-800a^{14}+2275a^{12}-4004a^{10}+4290a^{8}-2640a^{6}+825a^{4}-100a^{2}+1$, $a^{32}-32a^{30}+464a^{28}-4032a^{26}+23400a^{24}-95680a^{22}+283360a^{20}-615296a^{18}+980628a^{16}-1136960a^{14}+940576a^{12}-537472a^{10}+201552a^{8}-45696a^{6}+5440a^{4}-256a^{2}+2$, $a^{20}-20a^{18}+170a^{16}-800a^{14}+2275a^{12}-4004a^{10}+4290a^{8}-2640a^{6}+825a^{4}-100a^{2}+2$, $a^{8}-8a^{6}+20a^{4}-16a^{2}+2$, $a^{28}-28a^{26}+350a^{24}-2576a^{22}+12397a^{20}-40964a^{18}+94962a^{16}-155040a^{14}+176358a^{12}-136136a^{10}+68068a^{8}-20384a^{6}+3185a^{4}-196a^{2}+1$, $a^{32}-32a^{30}+464a^{28}-4032a^{26}+23400a^{24}-95680a^{22}+283360a^{20}-615296a^{18}+980628a^{16}-1136960a^{14}+940576a^{12}-537472a^{10}+201552a^{8}-45696a^{6}+5440a^{4}-256a^{2}+3$, $a^{24}-24a^{22}+252a^{20}-1520a^{18}+5814a^{16}-14688a^{14}+24752a^{12}-27456a^{10}+19305a^{8}-8008a^{6}+1716a^{4}-144a^{2}+3$, $a^{40}-40a^{38}+739a^{36}-8364a^{34}+64856a^{32}-365056a^{30}+1541784a^{28}-4980896a^{26}+12434996a^{24}-24072928a^{22}+36051740a^{20}-41431856a^{18}+36015793a^{16}-23157008a^{14}+10650263a^{12}-3330964a^{10}+654368a^{8}-70720a^{6}+3340a^{4}-48a^{2}-1$, $a^{21}-21a^{19}+189a^{17}-952a^{15}+2940a^{13}-5733a^{11}+7007a^{9}-5148a^{7}+2079a^{5}-385a^{3}+21a$, $a^{29}-29a^{27}+377a^{25}-2900a^{23}+14674a^{21}-51359a^{19}+127281a^{17}-224808a^{15}+281010a^{13}-243542a^{11}+140998a^{9}-51272a^{7}+10556a^{5}-1015a^{3}+29a$, $a^{37}-37a^{35}+629a^{33}-6512a^{31}+45880a^{29}-232841a^{27}+878787a^{25}-2510820a^{23}+5476185a^{21}-9126975a^{19}+11560835a^{17}-10994920a^{15}+7696444a^{13}-3848222a^{11}+1314610a^{9}-286824a^{7}+35853a^{5}-2109a^{3}+37a$, $a^{27}-27a^{25}+324a^{23}-2277a^{21}+10395a^{19}-32319a^{17}+69768a^{15}-104652a^{13}+107406a^{11}-72930a^{9}+30888a^{7}-7371a^{5}+819a^{3}-27a$, $a^{41}-41a^{39}+779a^{37}-9102a^{35}+73185a^{33}-429352a^{31}+1901416a^{29}-6487184a^{27}+17250012a^{25}-35937525a^{23}+58659315a^{21}-74657310a^{19}+73370115a^{17}-54826020a^{15}+30458900a^{13}-12183560a^{11}+3350479a^{9}-591261a^{7}+59983a^{5}-2870a^{3}+41a$, $a^{3}-3a$, $a^{7}-7a^{5}+14a^{3}-7a$, $a$, $a^{33}-33a^{31}+495a^{29}-4466a^{27}+27027a^{25}-115830a^{23}+361790a^{21}-834900a^{19}+1427679a^{17}-1797818a^{15}+1641486a^{13}-1058148a^{11}+461890a^{9}-127908a^{7}+20196a^{5}-1496a^{3}+33a$, $a^{11}-11a^{9}+44a^{7}-77a^{5}+55a^{3}-11a$, $a^{15}-15a^{13}+90a^{11}-275a^{9}+450a^{7}-378a^{5}+140a^{3}-15a$, $a^{25}-25a^{23}+275a^{21}-1750a^{19}+7125a^{17}-19380a^{15}+35700a^{13}-44200a^{11}+35750a^{9}-17875a^{7}+5005a^{5}-650a^{3}+25a+1$, $a^{29}-29a^{27}+377a^{25}-2900a^{23}+14674a^{21}-51359a^{19}+127281a^{17}-224808a^{15}+281010a^{13}-243542a^{11}+140998a^{9}-51272a^{7}+10556a^{5}-1015a^{3}+29a+1$, $a^{10}-10a^{8}+35a^{6}-a^{5}-50a^{4}+5a^{3}+25a^{2}-5a-1$, $a^{21}-21a^{19}+189a^{17}-952a^{15}+2940a^{13}-5733a^{11}+7007a^{9}-5148a^{7}+2079a^{5}-385a^{3}+21a+1$, $a^{34}-34a^{32}+527a^{30}-4930a^{28}+31059a^{26}-139230a^{24}+457470a^{22}-1118260a^{20}+2042975a^{18}+a^{17}-2778446a^{16}-17a^{15}+2778446a^{14}+119a^{13}-1998724a^{12}-442a^{11}+999362a^{10}+935a^{9}-329460a^{8}-1122a^{7}+65892a^{6}+714a^{5}-6936a^{4}-204a^{3}+289a^{2}+17a-1$, $a^{17}-17a^{15}+119a^{13}-442a^{11}+935a^{9}-1122a^{7}+714a^{5}-204a^{3}+17a-1$, $a^{42}-42a^{40}+819a^{38}-9842a^{36}+81585a^{34}-494802a^{32}+2272424a^{30}-8069424a^{28}+22428252a^{26}-a^{25}-49085400a^{24}+25a^{23}+84672315a^{22}-275a^{21}-114717330a^{20}+1750a^{19}+121090515a^{18}-7125a^{17}-98285670a^{16}+19380a^{15}+60174900a^{14}-35700a^{13}-27041560a^{12}+44200a^{11}+8580495a^{10}-35750a^{9}-1817046a^{8}+17875a^{7}+235543a^{6}-5005a^{5}-16170a^{4}+650a^{3}+441a^{2}-25a-3$, $a^{23}-23a^{21}+230a^{19}-1311a^{17}+4692a^{15}-10948a^{13}+16744a^{11}-16445a^{9}+9867a^{7}-a^{6}-3289a^{5}+6a^{4}+506a^{3}-9a^{2}-23a+2$, $a^{31}+a^{30}-31a^{29}-30a^{28}+434a^{27}+405a^{26}-3627a^{25}-3250a^{24}+20150a^{23}+17250a^{22}-78430a^{21}-63756a^{20}+219604a^{19}+168245a^{18}-447051a^{17}-319770a^{16}+660858a^{15}+436050a^{14}-700910a^{13}-419900a^{12}+520676a^{11}+277134a^{10}-260338a^{9}-119340a^{8}+82212a^{7}+30940a^{6}-14756a^{5}-4200a^{4}+1240a^{3}+225a^{2}-31a-3$, $a^{39}-39a^{37}+702a^{35}-7735a^{33}+58344a^{31}-319176a^{29}+1308944a^{27}-4102137a^{25}+9924525a^{23}-18599295a^{21}+26936910a^{19}-29910465a^{17}+25110020a^{15}-15600900a^{13}+6953544a^{11}-2124694a^{9}+415701a^{7}-46683a^{5}+2470a^{3}-39a-1$, $a^{43}-43a^{41}+860a^{39}-10621a^{37}+90687a^{35}-567987a^{33}-a^{32}+2701776a^{31}+32a^{30}-9970840a^{29}-464a^{28}+28915436a^{27}+4032a^{26}-66335412a^{25}-23400a^{24}+120609840a^{23}+95680a^{22}-173376645a^{21}-283360a^{20}+195747825a^{19}+615296a^{18}-171655785a^{17}-980628a^{16}+115000920a^{15}+1136960a^{14}-57500460a^{13}-940576a^{12}+20764055a^{11}+537472a^{10}-5167525a^{9}-201552a^{8}+826804a^{7}+45696a^{6}-76153a^{5}-5440a^{4}+3311a^{3}+256a^{2}-43a-2$, $a^{19}-19a^{17}+152a^{15}-665a^{13}+1729a^{11}-2717a^{9}+2508a^{7}+a^{6}-1254a^{5}-6a^{4}+285a^{3}+9a^{2}-19a-2$, $a^{38}-38a^{36}+665a^{34}-7106a^{32}+51832a^{30}-273296a^{28}+a^{27}+1076103a^{26}-27a^{25}-3223350a^{24}+324a^{23}+7413705a^{22}-2277a^{21}-13123110a^{20}+10395a^{19}+17809935a^{18}-32319a^{17}-18349630a^{16}+69768a^{15}+14115100a^{14}-104652a^{13}-7904456a^{12}+107405a^{11}+3105322a^{10}-72919a^{9}-810084a^{8}+30844a^{7}+128877a^{6}-7294a^{5}-10830a^{4}+764a^{3}+361a^{2}-16a-3$, $a^{15}-15a^{13}+90a^{11}+a^{10}-275a^{9}-10a^{8}+450a^{7}+35a^{6}-377a^{5}-50a^{4}+135a^{3}+25a^{2}-10a-1$, $a^{40}+a^{39}-40a^{38}-39a^{37}+739a^{36}+702a^{35}-8364a^{34}-7735a^{33}+64857a^{32}+58344a^{31}-365088a^{30}-319176a^{29}+1542248a^{28}+1308944a^{27}-4984928a^{26}-4102137a^{25}+12458396a^{24}+9924525a^{23}-24168608a^{22}-18599295a^{21}+36335099a^{20}+26936910a^{19}-42047132a^{18}-29910465a^{17}+36996251a^{16}+25110020a^{15}-24293168a^{14}-15600900a^{13}+11588564a^{12}+6953544a^{11}-3864432a^{10}-2124694a^{9}+851631a^{8}+415701a^{7}-113784a^{6}-46683a^{5}+7975a^{4}+2470a^{3}-220a^{2}-39a+1$, $a^{2}+a-1$, $a^{41}-41a^{39}+779a^{37}-9102a^{35}+a^{34}+73185a^{33}-34a^{32}-429352a^{31}+527a^{30}+1901416a^{29}-4930a^{28}-6487184a^{27}+31059a^{26}+17250012a^{25}-139231a^{24}-35937525a^{23}+457494a^{22}+58659315a^{21}-1118512a^{20}-74657310a^{19}+2044495a^{18}+73370114a^{17}-2784260a^{16}-54826003a^{15}+2793134a^{14}+30458781a^{13}-2023476a^{12}-12183118a^{11}+1026818a^{10}+3349544a^{9}-348765a^{8}-590138a^{7}+73900a^{6}+59262a^{5}-8652a^{4}-2652a^{3}+433a^{2}+17a-3$, $a^{12}-12a^{10}+54a^{8}-a^{7}-112a^{6}+7a^{5}+105a^{4}-14a^{3}-36a^{2}+7a+2$, $a^{15}-15a^{13}+90a^{11}-275a^{9}+450a^{7}-378a^{5}+140a^{3}-15a-1$, $a^{5}-5a^{3}+5a-1$, $a^{37}-37a^{35}+629a^{33}-6512a^{31}+45880a^{29}-232841a^{27}+878787a^{25}-2510820a^{23}+5476185a^{21}-9126975a^{19}+11560835a^{17}-10994920a^{15}+7696444a^{13}-3848222a^{11}+1314610a^{9}-286824a^{7}+35853a^{5}-2109a^{3}+37a+1$, $a^{23}-23a^{21}+230a^{19}-1311a^{17}+4692a^{15}-10948a^{13}+16744a^{11}-16445a^{9}+9867a^{7}-3289a^{5}+506a^{3}-23a-1$ (assuming GRH)
Regulator:		\( 8626989567142591000000000000 \) (assuming GRH)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{44}\cdot(2\pi)^{0}\cdot 8626989567142591000000000000 \cdot 1}{2\cdot\sqrt{482179487665033966874817964307376476160282778171317425736173928285756467369658548224}}\cr\approx \mathstrut & 0.109281015182506 \end{aligned}\] (assuming GRH)

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

\(\Q(\sqrt{2}) \), \(\Q(\sqrt{46}) \), \(\Q(\sqrt{23}) \), \(\Q(\sqrt{2}, \sqrt{23})\), \(\Q(\zeta_{23})^+\), 22.22.14741666340843480753092741810452692992.1, 22.22.339058325839400057321133061640411938816.1, \(\Q(\zeta_{92})^+\)

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	$22^{2}$	$22^{2}$	${\href{/padicField/7.11.0.1}{11} }^{4}$	$22^{2}$	$22^{2}$	$22^{2}$	$22^{2}$	R	$22^{2}$	$22^{2}$	$22^{2}$	${\href{/padicField/41.11.0.1}{11} }^{4}$	$22^{2}$	${\href{/padicField/47.2.0.1}{2} }^{22}$	$22^{2}$	$22^{2}$

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 $44$	$4$	$11$	$88$
\(23\)	23.22.21.1	$x^{22} + 506$	$22$	$1$	$21$	22T1	$[\ ]_{22}$
	23.22.21.1	$x^{22} + 506$	$22$	$1$	$21$	22T1	$[\ ]_{22}$