Number field 44.44.181375764426442332776050749828434842919201892356144879261148162186145782470703125.1: \(\Q(\zeta_{115})^+\)

• Label: 44.44.181...125.1
• Degree: $44$
• Signature: $[44, 0]$
• Discriminant: $1.814\times 10^{80}$
• Root discriminant: \(66.69\)
• Ramified primes: $5,23$
• Class number: $1$ (GRH)
• Class group: trivial (GRH)
• Galois group: $C_{44}$ (as 44T1)

Normalized defining polynomial

x^44 - x^43 - 44*x^42 + 43*x^41 + 902*x^40 - 859*x^39 - 11441*x^38 + 10582*x^37 + 100567*x^36 - 89985*x^35 - 650236*x^34 + 560251*x^33 + 3203650*x^32 - 2643399*x^31 - 12294569*x^30 + 9651170*x^29 + 37253225*x^28 - 27602055*x^27 - 89808680*x^26 + 62206625*x^25 + 172779011*x^24 - 110572386*x^23 - 265002643*x^22 + 154430258*x^21 + 322457859*x^20 - 168027624*x^19 - 308473797*x^18 + 140446403*x^17 + 228768475*x^16 - 88323383*x^15 - 128871012*x^14 + 40552321*x^13 + 53558102*x^12 - 13016729*x^11 - 15742859*x^10 + 2742874*x^9 + 3073662*x^8 - 347233*x^7 - 361455*x^6 + 24089*x^5 + 21786*x^4 - 986*x^3 - 504*x^2 + 24*x + 1

Invariants

Degree:		$44$
Signature:		$[44, 0]$
Discriminant:		181375764426442332776050749828434842919201892356144879261148162186145782470703125
Root discriminant:		\(66.69\)
Galois root discriminant:		$5^{3/4}23^{21/22}\approx 66.68967783806693$
Ramified primes:		\(5\), \(23\)
Discriminant root field:		\(\Q(\sqrt{5}) \)
$\card{ \Gal(K/\Q) }$:		$44$
This field is Galois and abelian over $\Q$.
Conductor:		\(115=5\cdot 23\)
Dirichlet character group:		$\lbrace$$\chi_{115}(1,·)$, $\chi_{115}(4,·)$, $\chi_{115}(6,·)$, $\chi_{115}(7,·)$, $\chi_{115}(9,·)$, $\chi_{115}(16,·)$, $\chi_{115}(17,·)$, $\chi_{115}(22,·)$, $\chi_{115}(24,·)$, $\chi_{115}(26,·)$, $\chi_{115}(28,·)$, $\chi_{115}(29,·)$, $\chi_{115}(31,·)$, $\chi_{115}(33,·)$, $\chi_{115}(36,·)$, $\chi_{115}(37,·)$, $\chi_{115}(38,·)$, $\chi_{115}(39,·)$, $\chi_{115}(41,·)$, $\chi_{115}(42,·)$, $\chi_{115}(43,·)$, $\chi_{115}(49,·)$, $\chi_{115}(53,·)$, $\chi_{115}(54,·)$, $\chi_{115}(57,·)$, $\chi_{115}(59,·)$, $\chi_{115}(63,·)$, $\chi_{115}(64,·)$, $\chi_{115}(67,·)$, $\chi_{115}(68,·)$, $\chi_{115}(71,·)$, $\chi_{115}(81,·)$, $\chi_{115}(83,·)$, $\chi_{115}(88,·)$, $\chi_{115}(94,·)$, $\chi_{115}(96,·)$, $\chi_{115}(97,·)$, $\chi_{115}(101,·)$, $\chi_{115}(102,·)$, $\chi_{115}(103,·)$, $\chi_{115}(104,·)$, $\chi_{115}(107,·)$, $\chi_{115}(112,·)$, $\chi_{115}(113,·)$$\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:		$a^{23}-23a^{21}+230a^{19}-1311a^{17}+4692a^{15}-10948a^{13}+16744a^{11}-16445a^{9}+9867a^{7}-3289a^{5}+506a^{3}-23a$, $a^{5}-5a^{3}+5a$, $a^{10}-10a^{8}+35a^{6}-50a^{4}+25a^{2}-2$, $a^{20}-20a^{18}+170a^{16}-800a^{14}+2275a^{12}-4004a^{10}+4290a^{8}-2640a^{6}+825a^{4}-100a^{2}+2$, $a^{37}-37a^{35}+629a^{33}+a^{32}-6512a^{31}-32a^{30}+45880a^{29}+464a^{28}-232841a^{27}-4032a^{26}+878787a^{25}+23400a^{24}-2510820a^{23}-95680a^{22}+5476185a^{21}+283360a^{20}-9126975a^{19}-615296a^{18}+11560835a^{17}+980628a^{16}-10994920a^{15}-1136959a^{14}+7696444a^{13}+940562a^{12}-3848222a^{11}-537395a^{10}+1314611a^{9}+201342a^{8}-286833a^{7}-45402a^{6}+35880a^{5}+5244a^{4}-2139a^{3}-207a^{2}+46a$, $a^{30}-30a^{28}+405a^{26}-3250a^{24}+17250a^{22}-63756a^{20}+168245a^{18}-319770a^{16}+436050a^{14}-419900a^{12}+277134a^{10}-119340a^{8}+30940a^{6}-4200a^{4}+225a^{2}-2$, $a^{40}-40a^{38}+740a^{36}-8400a^{34}+65450a^{32}-371008a^{30}+1582240a^{28}-5178240a^{26}+13147875a^{24}-26013000a^{22}+40060020a^{20}-47720400a^{18}+43459650a^{16}-29716000a^{14}+14858000a^{12}-5230016a^{10}+1225785a^{8}-175560a^{6}+13300a^{4}-400a^{2}+2$, $a^{40}-40a^{38}+740a^{36}-8400a^{34}+65450a^{32}-371007a^{30}+1582210a^{28}-5177835a^{26}+13144625a^{24}-25995750a^{22}+39996265a^{20}-47552175a^{18}+43140050a^{16}-29280750a^{14}+14440375a^{12}-4956885a^{10}+1110725a^{8}-147225a^{6}+9875a^{4}-250a^{2}+1$, $a^{20}-20a^{18}+170a^{16}-800a^{14}+2275a^{12}-4003a^{10}+4280a^{8}-2605a^{6}+775a^{4}-75a^{2}+1$, $a^{42}-43a^{40}+859a^{38}+a^{37}-10582a^{36}-38a^{35}+89985a^{34}+664a^{33}-560251a^{32}-7072a^{31}+2643400a^{30}+51305a^{29}-9651200a^{28}-268365a^{27}+27602460a^{26}+1045016a^{25}-62209875a^{24}-3083771a^{23}+110589635a^{22}+6953683a^{21}-154493991a^{20}-11992704a^{19}+168195639a^{18}+15727497a^{17}-140764862a^{16}-15481884a^{15}+88754741a^{14}+11195637a^{13}-40961273a^{12}-5752397a^{11}+13277119a^{10}+1994564a^{9}-2845769a^{8}-429362a^{7}+368306a^{6}+49381a^{5}-25000a^{4}-2165a^{3}+705a^{2}+5a-2$, $a^{10}-10a^{8}+35a^{6}-50a^{4}+25a^{2}-1$, $a^{27}-27a^{25}+324a^{23}-2277a^{21}+10395a^{19}-32319a^{17}+69768a^{15}-104652a^{13}+107406a^{11}-72930a^{9}+30888a^{7}-7371a^{5}+819a^{3}-27a$, $a^{43}-43a^{41}+859a^{39}+a^{38}-10582a^{37}-38a^{36}+89985a^{35}+664a^{34}-560251a^{33}-7072a^{32}+2643399a^{31}+51305a^{30}-9651170a^{29}-268365a^{28}+27602055a^{27}+1045016a^{26}-62206624a^{25}-3083771a^{24}+110572361a^{23}+6953683a^{22}-154429982a^{21}-11992704a^{20}+168025853a^{19}+15727497a^{18}-140439089a^{17}-15481885a^{16}+88303051a^{15}+11195652a^{14}-40513681a^{13}-5752487a^{12}+12966796a^{11}+1994839a^{10}-2700117a^{9}-429812a^{8}+324210a^{7}+49759a^{6}-17005a^{5}-2305a^{4}-49a^{3}+21a^{2}+22a-1$, $a^{7}-7a^{5}+14a^{3}-7a$, $a^{39}-39a^{37}+702a^{35}-7735a^{33}+58344a^{31}+a^{30}-319176a^{29}-30a^{28}+1308944a^{27}+405a^{26}-4102137a^{25}-3250a^{24}+9924525a^{23}+17250a^{22}-18599295a^{21}-63756a^{20}+26936910a^{19}+168245a^{18}-29910465a^{17}-319769a^{16}+25110020a^{15}+436034a^{14}-15600900a^{13}-419796a^{12}+6953544a^{11}+276782a^{10}-2124694a^{9}-118680a^{8}+415702a^{7}+30268a^{6}-46690a^{5}-3864a^{4}+2484a^{3}+161a^{2}-46a$, $a^{23}-23a^{21}+230a^{19}-1311a^{17}+4692a^{15}-10948a^{13}+16744a^{11}-16445a^{9}+9867a^{7}-3290a^{5}+511a^{3}-28a$, $a^{42}-42a^{40}+819a^{38}-9842a^{36}+81585a^{34}-494802a^{32}+2272424a^{30}-8069424a^{28}+22428252a^{26}-49085400a^{24}+84672315a^{22}-114717330a^{20}+121090515a^{18}-98285670a^{16}+60174900a^{14}-27041560a^{12}+8580495a^{10}-1817046a^{8}+235543a^{6}-16170a^{4}+441a^{2}-2$, $a^{30}-30a^{28}+405a^{26}-3250a^{24}-a^{23}+17250a^{22}+23a^{21}-63756a^{20}-230a^{19}+168245a^{18}+1311a^{17}-319770a^{16}-4692a^{15}+436050a^{14}+10948a^{13}-419900a^{12}-16744a^{11}+277134a^{10}+16445a^{9}-119340a^{8}-9867a^{7}+30940a^{6}+3289a^{5}-4200a^{4}-506a^{3}+225a^{2}+23a-2$, $a^{2}-2$, $a^{23}-23a^{21}+230a^{19}-1311a^{17}+4692a^{15}-10948a^{13}+16744a^{11}-a^{10}-16445a^{9}+10a^{8}+9867a^{7}-35a^{6}-3289a^{5}+50a^{4}+506a^{3}-25a^{2}-23a+2$, $a^{39}-a^{38}-40a^{37}+38a^{36}+740a^{35}-664a^{34}-8400a^{33}+7071a^{32}+65449a^{31}-51273a^{30}-370975a^{29}+267901a^{28}+1581747a^{27}-1040984a^{26}-5173831a^{25}+3060371a^{24}+13121574a^{23}-6858003a^{22}-25902623a^{21}+11709345a^{20}+39725071a^{19}-15112221a^{18}-46976512a^{17}+14501427a^{16}+42249522a^{15}-10059493a^{14}-28287083a^{13}+4814200a^{12}+13657152a^{11}-1461447a^{10}-4535177a^{9}+232750a^{8}+963294a^{7}-6962a^{6}-116279a^{5}-2165a^{4}+6490a^{3}+116a^{2}-120a-3$, $a^{25}-26a^{23}+298a^{21}-1980a^{19}+8436a^{17}-24072a^{15}+46648a^{13}-60944a^{11}+52195a^{9}-27742a^{7}+8294a^{5}-1156a^{3}+48a-1$, $a^{23}-23a^{21}-a^{20}+230a^{19}+20a^{18}-1311a^{17}-170a^{16}+4692a^{15}+800a^{14}-10948a^{13}-2275a^{12}+16744a^{11}+4004a^{10}-16445a^{9}-4290a^{8}+9867a^{7}+2640a^{6}-3289a^{5}-825a^{4}+506a^{3}+100a^{2}-23a-2$, $a^{43}+a^{42}-43a^{41}-43a^{40}+859a^{39}+859a^{38}-10582a^{37}-10582a^{36}+89985a^{35}+89985a^{34}-560252a^{33}-560252a^{32}+2643432a^{31}+2643432a^{30}-9651664a^{29}-9651663a^{28}+27606493a^{27}+27606464a^{26}-62233303a^{25}-62232926a^{24}+110685664a^{23}+110682763a^{22}-154779902a^{21}-154765204a^{20}+168823061a^{19}+168771452a^{18}-141784783a^{17}-141656021a^{16}+89980199a^{15}+89749898a^{14}-42040562a^{13}-41746315a^{12}+13963756a^{11}+13699389a^{10}-3153954a^{9}-2992010a^{8}+461933a^{7}+397851a^{6}-42748a^{5}-27855a^{4}+2481a^{3}+781a^{2}-68a-3$, $a^{9}-9a^{7}+27a^{5}-30a^{3}+9a$, $a^{16}-16a^{14}+104a^{12}-352a^{10}+660a^{8}-672a^{6}+336a^{4}-64a^{2}+2$, $a^{36}-36a^{34}+a^{33}+594a^{32}-33a^{31}-5952a^{30}+495a^{29}+40455a^{28}-4466a^{27}-197316a^{26}+27027a^{25}+712530a^{24}-115830a^{23}-1937520a^{22}+361790a^{21}+3996135a^{20}-834900a^{19}-6249100a^{18}+1427679a^{17}+7354710a^{16}-1797818a^{15}-6418656a^{14}+1641487a^{13}+4056234a^{12}-1058161a^{11}-1790711a^{10}+461955a^{9}+523250a^{8}-128064a^{7}-92989a^{6}+20378a^{5}+8671a^{4}-1587a^{3}-299a^{2}+46a$, $a^{13}-13a^{11}+65a^{9}-156a^{7}+182a^{5}-91a^{3}+13a$, $a^{41}-41a^{39}+779a^{37}-9102a^{35}+73185a^{33}-429352a^{31}+1901416a^{29}+a^{28}-6487184a^{27}-28a^{26}+17250012a^{25}+350a^{24}-35937525a^{23}-2576a^{22}+58659315a^{21}+12397a^{20}-74657310a^{19}-40963a^{18}+73370115a^{17}+94944a^{16}-54826020a^{15}-154905a^{14}+30458900a^{13}+175812a^{12}-12183560a^{11}-134849a^{10}+3350479a^{9}+66286a^{8}-591261a^{7}-18998a^{6}+59984a^{5}+2645a^{4}-2875a^{3}-115a^{2}+46a$, $a^{40}-40a^{38}+740a^{36}-8400a^{34}+65450a^{32}-371008a^{30}+1582240a^{28}-5178240a^{26}+13147875a^{24}-26013000a^{22}+40060020a^{20}-47720400a^{18}+43459650a^{16}-29716000a^{14}+14857999a^{12}-5230004a^{10}+1225731a^{8}-175448a^{6}+13195a^{4}-364a^{2}$, $a^{28}-28a^{26}+350a^{24}-2576a^{22}+12397a^{20}-40964a^{18}+94962a^{16}-155040a^{14}+176358a^{12}-136136a^{10}+68068a^{8}-20384a^{6}+3185a^{4}-196a^{2}+2$, $a^{39}-39a^{37}+702a^{35}-7735a^{33}+58344a^{31}+a^{30}-319176a^{29}-30a^{28}+1308944a^{27}+405a^{26}-4102137a^{25}-3250a^{24}+9924525a^{23}+17250a^{22}-18599295a^{21}-63756a^{20}+26936910a^{19}+168245a^{18}-29910465a^{17}-319769a^{16}+25110020a^{15}+436034a^{14}-15600900a^{13}-419796a^{12}+6953544a^{11}+276783a^{10}-2124694a^{9}-118690a^{8}+415702a^{7}+30303a^{6}-46690a^{5}-3914a^{4}+2484a^{3}+186a^{2}-46a-2$, $a^{31}-31a^{29}+434a^{27}-3627a^{25}+20150a^{23}-78430a^{21}+219604a^{19}-447051a^{17}+660858a^{15}-700910a^{13}+520676a^{11}-260338a^{9}+82212a^{7}-14756a^{5}+1240a^{3}-31a$, $a^{26}-26a^{24}+299a^{22}-2002a^{20}+8645a^{18}-25194a^{16}+50388a^{14}-68952a^{12}+63206a^{10}-37180a^{8}+13013a^{6}-2366a^{4}+169a^{2}-2$, $a^{14}-14a^{12}+77a^{10}-210a^{8}+294a^{6}-196a^{4}+49a^{2}-1$, $a^{21}-21a^{19}+189a^{17}-952a^{15}+2940a^{13}-5733a^{11}+7007a^{9}-5148a^{7}+2079a^{5}-385a^{3}+21a+1$, $a^{18}-18a^{16}+135a^{14}-546a^{12}+1287a^{10}-1782a^{8}+1386a^{6}-541a^{4}+85a^{2}-4$, $a^{6}-5a^{4}+6a^{2}-1$, $a^{42}-43a^{40}+859a^{38}+a^{37}-10582a^{36}-38a^{35}+89985a^{34}+664a^{33}-560251a^{32}-7072a^{31}+2643399a^{30}+51305a^{29}-9651170a^{28}-268365a^{27}+27602055a^{26}+1045016a^{25}-62206625a^{24}-3083771a^{23}+110572385a^{22}+6953682a^{21}-154430235a^{20}-11992683a^{19}+168027394a^{18}+15727308a^{17}-140445092a^{16}-15480933a^{15}+88318691a^{14}+11192712a^{13}-40541373a^{12}-5746754a^{11}+12999985a^{10}+1987832a^{9}-2726429a^{8}-424664a^{7}+337366a^{6}+47680a^{5}-20800a^{4}-1920a^{3}+480a^{2}-a-1$, $a^{3}-3a$, $a^{26}-26a^{24}+299a^{22}-2002a^{20}+8645a^{18}-25194a^{16}+50388a^{14}-68953a^{12}+63218a^{10}-37234a^{8}+13125a^{6}-2471a^{4}+205a^{2}-4$, $a^{32}-32a^{30}+464a^{28}-4032a^{26}+23400a^{24}-95680a^{22}+283360a^{20}-615296a^{18}+980628a^{16}-1136960a^{14}+940576a^{12}-537472a^{10}+201552a^{8}-45696a^{6}+5440a^{4}-256a^{2}+2$, $a^{13}-13a^{11}+65a^{9}-156a^{7}+182a^{5}-91a^{3}+13a+1$ (assuming GRH)
Regulator:		\( 183282479706502550000000000 \) (assuming GRH)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{44}\cdot(2\pi)^{0}\cdot 183282479706502550000000000 \cdot 1}{2\cdot\sqrt{181375764426442332776050749828434842919201892356144879261148162186145782470703125}}\cr\approx \mathstrut & 0.119707438429760 \end{aligned}\] (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 $C_{44}$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.66125.1, \(\Q(\zeta_{23})^+\), 22.22.83796671451884098775580820361328125.1

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	$44$	$44$	R	$44$	$22^{2}$	$44$	$44$	${\href{/padicField/19.11.0.1}{11} }^{4}$	R	$22^{2}$	${\href{/padicField/31.11.0.1}{11} }^{4}$	$44$	${\href{/padicField/41.11.0.1}{11} }^{4}$	$44$	${\href{/padicField/47.4.0.1}{4} }^{11}$	$44$	$22^{2}$

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
\(5\)		Deg $44$	$4$	$11$	$33$
\(23\)		Deg $44$	$22$	$2$	$42$