Number field 41.41.57959375186337998161464929843210464026538099255933595673241672975683189751201.1: \(\Q(\zeta_{83})^+\)

• Label: 41.41.579...201.1
• Degree: $41$
• Signature: $[41, 0]$
• Discriminant: $5.796\times 10^{76}$
• Root discriminant: \(74.52\)
• Ramified prime: $83$
• Class number: $1$ (GRH)
• Class group: trivial (GRH)
• Galois group: $C_{41}$ (as 41T1)

Normalized defining polynomial

x^41 - x^40 - 40*x^39 + 39*x^38 + 741*x^37 - 703*x^36 - 8436*x^35 + 7770*x^34 + 66045*x^33 - 58905*x^32 - 376992*x^31 + 324632*x^30 + 1623160*x^29 - 1344904*x^28 - 5379616*x^27 + 4272048*x^26 + 13884156*x^25 - 10518300*x^24 - 28048800*x^23 + 20160075*x^22 + 44352165*x^21 - 30045015*x^20 - 54627300*x^19 + 34597290*x^18 + 51895935*x^17 - 30421755*x^16 - 37442160*x^15 + 20058300*x^14 + 20058300*x^13 - 9657700*x^12 - 7726160*x^11 + 3268760*x^10 + 2042975*x^9 - 735471*x^8 - 346104*x^7 + 100947*x^6 + 33649*x^5 - 7315*x^4 - 1540*x^3 + 210*x^2 + 21*x - 1

Invariants

Degree:		$41$
Signature:		$[41, 0]$
Discriminant:		\(57959375186337998161464929843210464026538099255933595673241672975683189751201\) \(\medspace = 83^{40}\)
Root discriminant:		\(74.52\)
Galois root discriminant:		$83^{40/41}\approx 74.51973633924607$
Ramified primes:		\(83\)
Discriminant root field:		\(\Q\)
$\card{ \Gal(K/\Q) }$:		$41$
This field is Galois and abelian over $\Q$.
Conductor:		\(83\)
Dirichlet character group:		$\lbrace$$\chi_{83}(1,·)$, $\chi_{83}(3,·)$, $\chi_{83}(4,·)$, $\chi_{83}(7,·)$, $\chi_{83}(9,·)$, $\chi_{83}(10,·)$, $\chi_{83}(11,·)$, $\chi_{83}(12,·)$, $\chi_{83}(16,·)$, $\chi_{83}(17,·)$, $\chi_{83}(21,·)$, $\chi_{83}(23,·)$, $\chi_{83}(25,·)$, $\chi_{83}(26,·)$, $\chi_{83}(27,·)$, $\chi_{83}(28,·)$, $\chi_{83}(29,·)$, $\chi_{83}(30,·)$, $\chi_{83}(31,·)$, $\chi_{83}(33,·)$, $\chi_{83}(36,·)$, $\chi_{83}(37,·)$, $\chi_{83}(38,·)$, $\chi_{83}(40,·)$, $\chi_{83}(41,·)$, $\chi_{83}(44,·)$, $\chi_{83}(48,·)$, $\chi_{83}(49,·)$, $\chi_{83}(51,·)$, $\chi_{83}(59,·)$, $\chi_{83}(61,·)$, $\chi_{83}(63,·)$, $\chi_{83}(64,·)$, $\chi_{83}(65,·)$, $\chi_{83}(68,·)$, $\chi_{83}(69,·)$, $\chi_{83}(70,·)$, $\chi_{83}(75,·)$, $\chi_{83}(77,·)$, $\chi_{83}(78,·)$, $\chi_{83}(81,·)$$\rbrace$
This is not a CM field.

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

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

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

Class group and class number

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

Unit group

Rank:		$40$
Torsion generator:		\( -1 \)  (order $2$)
Fundamental units:		$a^{10}-10a^{8}+35a^{6}-50a^{4}+25a^{2}-2$, $a^{30}-30a^{28}+405a^{26}-3250a^{24}+17250a^{22}-63756a^{20}+168245a^{18}-319770a^{16}+436050a^{14}-419900a^{12}+277134a^{10}-119340a^{8}+30940a^{6}-4200a^{4}+225a^{2}-2$, $a^{23}-23a^{21}+230a^{19}-1311a^{17}+4692a^{15}-10948a^{13}+16744a^{11}-16445a^{9}+9867a^{7}-3289a^{5}+506a^{3}-23a$, $a^{40}-40a^{38}+740a^{36}-8400a^{34}+65450a^{32}-371008a^{30}+1582240a^{28}-5178240a^{26}+13147875a^{24}-26013000a^{22}+40060020a^{20}-47720400a^{18}+43459650a^{16}-29716000a^{14}+14858000a^{12}-5230016a^{10}+1225785a^{8}-175560a^{6}+13300a^{4}-400a^{2}+3$, $a^{37}-37a^{35}+629a^{33}-6512a^{31}+45880a^{29}-232841a^{27}+878787a^{25}-2510820a^{23}+5476185a^{21}-9126975a^{19}+11560835a^{17}-10994920a^{15}+7696444a^{13}-3848222a^{11}+1314610a^{9}-286824a^{7}+35853a^{5}-2109a^{3}+37a$, $a^{5}-5a^{3}+5a$, $a^{39}-39a^{37}+702a^{35}-7735a^{33}+58344a^{31}-319176a^{29}+1308944a^{27}-4102137a^{25}+9924525a^{23}-18599295a^{21}+26936910a^{19}-29910465a^{17}+25110020a^{15}-15600900a^{13}+6953544a^{11}-2124694a^{9}+415701a^{7}-46683a^{5}+2470a^{3}-39a$, $a^{36}-36a^{34}+594a^{32}-a^{31}-5952a^{30}+31a^{29}+40455a^{28}-434a^{27}-197315a^{26}+3627a^{25}+712504a^{24}-20150a^{23}-1937221a^{22}+78429a^{21}+3994133a^{20}-219583a^{19}-6240455a^{18}+446862a^{17}+7329516a^{16}-659906a^{15}-6368268a^{14}+697970a^{13}+3987282a^{12}-514943a^{11}-1727506a^{10}+253331a^{9}+486080a^{8}-77064a^{7}-80011a^{6}+12676a^{5}+6355a^{4}-850a^{3}-155a^{2}+5a+1$, $a^{35}-35a^{33}+560a^{31}-5425a^{29}+35525a^{27}-166257a^{25}+573300a^{23}-1480050a^{21}+2877875a^{19}-4206125a^{17}+4576264a^{15}-3640210a^{13}+2057510a^{11}-791350a^{9}+193800a^{7}-27132a^{5}+1785a^{3}-35a$, $a^{22}-22a^{20}+209a^{18}-1122a^{16}+3740a^{14}-8008a^{12}+11011a^{10}-9438a^{8}+4719a^{6}-1210a^{4}+121a^{2}-2$, $a^{37}-37a^{35}+629a^{33}-6512a^{31}+45880a^{29}-232841a^{27}+878787a^{25}-2510820a^{23}+5476185a^{21}-9126975a^{19}+11560835a^{17}-10994920a^{15}+7696444a^{13}-3848222a^{11}+1314611a^{9}-286833a^{7}+35880a^{5}-2139a^{3}+46a-1$, $a^{37}-37a^{35}+629a^{33}-6512a^{31}+45880a^{29}-232841a^{27}+878787a^{25}-2510819a^{23}+5476162a^{21}-9126745a^{19}+11559524a^{17}-10990228a^{15}+7685496a^{13}-3831478a^{11}+1298165a^{9}-276957a^{7}+32564a^{5}-1603a^{3}+14a-1$, $a^{13}-13a^{11}+65a^{9}-156a^{7}+182a^{5}-91a^{3}+13a$, $a^{32}-32a^{30}+464a^{28}-4032a^{26}+23400a^{24}-95680a^{22}+283360a^{20}-615296a^{18}+980628a^{16}-1136960a^{14}+940576a^{12}-537472a^{10}+201552a^{8}-45696a^{6}+5440a^{4}-256a^{2}+2$, $a^{26}-26a^{24}+299a^{22}-2002a^{20}+8645a^{18}-25194a^{16}+50388a^{14}-68952a^{12}+63206a^{10}-37180a^{8}+13013a^{6}-2366a^{4}+169a^{2}-2$, $a^{40}-a^{39}-40a^{38}+38a^{37}+741a^{36}-665a^{35}-8435a^{34}+7105a^{33}+66010a^{32}-51800a^{31}-376432a^{30}+272832a^{29}+1617736a^{28}-1072072a^{27}-5344120a^{26}+3199976a^{25}+13718276a^{24}-7318324a^{23}-27478399a^{22}+12841751a^{21}+42886766a^{20}-17203264a^{19}-51800554a^{18}+17394026a^{17}+47815781a^{16}-13027729a^{15}-33086029a^{14}+7030571a^{13}+16688271a^{12}-2627129a^{11}-5895889a^{10}+641631a^{9}+1377102a^{8}-93840a^{7}-194787a^{6}+7107a^{5}+14422a^{4}-208a^{3}-418a^{2}+2a+3$, $a^{37}-37a^{35}+629a^{33}-6512a^{31}+45880a^{29}-a^{28}-232841a^{27}+28a^{26}+878787a^{25}-350a^{24}-2510820a^{23}+2576a^{22}+5476185a^{21}-12397a^{20}-9126975a^{19}+40963a^{18}+11560835a^{17}-94944a^{16}-10994920a^{15}+154905a^{14}+7696444a^{13}-175812a^{12}-3848222a^{11}+134849a^{10}+1314611a^{9}-66286a^{8}-286833a^{7}+18998a^{6}+35880a^{5}-2645a^{4}-2139a^{3}+115a^{2}+46a-1$, $a^{22}-21a^{20}+190a^{18}-969a^{16}+3060a^{14}-6188a^{12}+8008a^{10}-6435a^{8}+3003a^{6}-715a^{4}+66a^{2}-1$, $a^{11}-11a^{9}+44a^{7}-77a^{5}+55a^{3}-11a$, $a^{40}-a^{39}-39a^{38}+38a^{37}+702a^{36}-666a^{35}-7734a^{34}+7140a^{33}+58311a^{32}-52359a^{31}-318681a^{30}+278225a^{29}+1304479a^{28}-1107134a^{27}-4075137a^{26}+3362230a^{25}+9809020a^{24}-7868600a^{23}-18239805a^{22}+14228995a^{21}+26112635a^{20}-19812155a^{19}-28516415a^{18}+21034235a^{17}+23386645a^{16}-16742230a^{15}-14074895a^{14}+9734726a^{13}+6019105a^{12}-3979208a^{11}-1751255a^{10}+1080265a^{9}+327470a^{8}-178477a^{7}-36509a^{6}+15538a^{5}+2145a^{4}-575a^{3}-45a^{2}+14a+1$, $a^{9}-9a^{7}+27a^{5}-30a^{3}+9a$, $a^{40}-a^{39}-40a^{38}+39a^{37}+740a^{36}-703a^{35}-8399a^{34}+7770a^{33}+65416a^{32}-58905a^{31}-370480a^{30}+324631a^{29}+1577280a^{28}-1344874a^{27}-5146775a^{26}+4271643a^{25}+13005369a^{24}-10515050a^{23}-25537981a^{22}+20142825a^{21}+38876003a^{20}-29981259a^{19}-45500555a^{18}+34429045a^{17}+40336411a^{16}-30101985a^{15}-26451932a^{14}+19622251a^{13}+12372804a^{12}-9237814a^{11}-3894682a^{10}+2991703a^{9}+744810a^{8}-616341a^{7}-69146a^{6}+70301a^{5}+1078a^{4}-3311a^{3}+77a^{2}+34a$, $a^{23}-23a^{21}+230a^{19}-1311a^{17}+4692a^{15}-10948a^{13}+16744a^{11}-16445a^{9}+9867a^{7}-3289a^{5}+506a^{3}-23a-1$, $a^{37}-37a^{35}+629a^{33}-6512a^{31}+45880a^{29}-232841a^{27}+878787a^{25}-2510820a^{23}+5476185a^{21}-9126975a^{19}+11560835a^{17}-10994920a^{15}+7696444a^{13}-3848222a^{11}+1314610a^{9}-286824a^{7}+35853a^{5}-2109a^{3}+37a-1$, $a^{38}-38a^{36}-a^{35}+665a^{34}+35a^{33}-7105a^{32}-560a^{31}+51800a^{30}+5424a^{29}-272832a^{28}-35496a^{27}+1072071a^{26}+165880a^{25}-3199950a^{24}-570400a^{23}+7318025a^{22}+1465376a^{21}-12839750a^{20}-2826517a^{19}+17194639a^{18}+4078863a^{17}-17369001a^{16}-4351608a^{15}+12978124a^{14}+3359864a^{13}-6963776a^{12}-1815684a^{11}+2567498a^{10}+653004a^{9}-607872a^{8}-144880a^{7}+82509a^{6}+17648a^{5}-5054a^{4}-965a^{3}+41a^{2}+15a+3$, $a^{36}-36a^{34}+595a^{32}-5984a^{30}+40919a^{28}-201348a^{26}+735930a^{24}-2033200a^{22}+4279495a^{20}-6864396a^{18}-a^{17}+8335338a^{16}+17a^{15}-7555616a^{14}-119a^{13}+4996810a^{12}+442a^{11}-2328184a^{10}-935a^{9}+724812a^{8}+1122a^{7}-138720a^{6}-714a^{5}+14161a^{4}+204a^{3}-579a^{2}-17a+2$, $a^{37}-37a^{35}+629a^{33}-6512a^{31}+45880a^{29}-a^{28}-232840a^{27}+28a^{26}+878760a^{25}-350a^{24}-2510496a^{23}+2576a^{22}+5473908a^{21}-12397a^{20}-9116579a^{19}+40963a^{18}+11528497a^{17}-94944a^{16}-10925000a^{15}+154905a^{14}+7591127a^{13}-175812a^{12}-3739087a^{11}+134849a^{10}+1238964a^{9}-66286a^{8}-253437a^{7}+18998a^{6}+27255a^{5}-2645a^{4}-1035a^{3}+115a^{2}-1$, $a^{25}-25a^{23}+275a^{21}-1750a^{19}+7125a^{17}-19380a^{15}+35700a^{13}-44200a^{11}+35750a^{9}-17875a^{7}+5005a^{5}-650a^{3}+25a$, $a^{37}-a^{36}-37a^{35}+36a^{34}+629a^{33}-594a^{32}-6512a^{31}+5952a^{30}+45880a^{29}-40456a^{28}-232840a^{27}+197344a^{26}+878760a^{25}-712880a^{24}-2510496a^{23}+1940096a^{22}+5473908a^{21}-4008532a^{20}-9116579a^{19}+6290063a^{18}+11528497a^{17}-7449654a^{16}-10925000a^{15}+6573561a^{14}+7591127a^{13}-4232046a^{12}-3739087a^{11}+1925560a^{10}+1238964a^{9}-589536a^{8}-253437a^{7}+111987a^{6}+27255a^{5}-11316a^{4}-1035a^{3}+414a^{2}-1$, $a^{9}-9a^{7}+27a^{5}-30a^{3}+9a-1$, $a^{40}-a^{39}-39a^{38}+39a^{37}+702a^{36}-703a^{35}-7734a^{34}+7770a^{33}+58310a^{32}-58905a^{31}-318648a^{30}+324631a^{29}+1303984a^{28}-1344874a^{27}-4070672a^{26}+4271643a^{25}+9782019a^{24}-10515050a^{23}-18124275a^{22}+20142825a^{21}+25752871a^{20}-29981259a^{19}-27690411a^{18}+34429045a^{17}+21985659a^{16}-30101986a^{15}-12333092a^{14}+19622266a^{13}+4460340a^{12}-9237904a^{11}-778349a^{10}+2991978a^{9}-74712a^{8}-616791a^{7}+64450a^{6}+70679a^{5}-10962a^{4}-3451a^{3}+559a^{2}+48a-4$, $a^{29}-29a^{27}+378a^{25}-2925a^{23}+14949a^{21}-53109a^{19}+134406a^{17}-244188a^{15}+316710a^{13}-287742a^{11}+176748a^{9}-69147a^{7}+15561a^{5}-a^{4}-1665a^{3}+4a^{2}+54a-3$, $a^{15}-15a^{13}+90a^{11}-275a^{9}+450a^{7}-378a^{5}+140a^{3}-15a$, $a^{34}-34a^{32}+527a^{30}-4930a^{28}+31059a^{26}-139230a^{24}+457470a^{22}-1118260a^{20}-a^{19}+2042975a^{18}+19a^{17}-2778446a^{16}-152a^{15}+2778446a^{14}+665a^{13}-1998724a^{12}-1729a^{11}+999362a^{10}+2717a^{9}-329460a^{8}-2508a^{7}+65892a^{6}+1254a^{5}-6936a^{4}-285a^{3}+289a^{2}+19a-2$, $a^{37}-37a^{35}+629a^{33}-6512a^{31}+45880a^{29}-a^{28}-232841a^{27}+28a^{26}+878787a^{25}-350a^{24}-2510820a^{23}+2576a^{22}+5476185a^{21}-12397a^{20}-9126974a^{19}+40963a^{18}+11560816a^{17}-94944a^{16}-10994768a^{15}+154905a^{14}+7695779a^{13}-175812a^{12}-3846493a^{11}+134849a^{10}+1311894a^{9}-66286a^{8}-284325a^{7}+18998a^{6}+34626a^{5}-2645a^{4}-1854a^{3}+115a^{2}+27a-1$, $a^{37}-37a^{35}+629a^{33}-a^{32}-6512a^{31}+32a^{30}+45880a^{29}-464a^{28}-232841a^{27}+4032a^{26}+878787a^{25}-23400a^{24}-2510819a^{23}+95680a^{22}+5476162a^{21}-283360a^{20}-9126745a^{19}+615296a^{18}+11559524a^{17}-980628a^{16}-10990228a^{15}+1136959a^{14}+7685496a^{13}-940562a^{12}-3831478a^{11}+537395a^{10}+1298166a^{9}-201342a^{8}-276966a^{7}+45402a^{6}+32591a^{5}-5244a^{4}-1633a^{3}+207a^{2}+23a-1$, $a^{35}-35a^{33}+560a^{31}-5425a^{29}+35525a^{27}-166257a^{25}-a^{24}+573300a^{23}+24a^{22}-1480050a^{21}-252a^{20}+2877875a^{19}+1520a^{18}-4206125a^{17}-5814a^{16}+4576264a^{15}+14688a^{14}-3640210a^{13}-24752a^{12}+2057510a^{11}+27456a^{10}-791350a^{9}-19305a^{8}+193800a^{7}+8008a^{6}-27132a^{5}-1716a^{4}+1785a^{3}+144a^{2}-35a-3$, $a^{24}-24a^{22}+252a^{20}-1520a^{18}+5814a^{16}-14688a^{14}+24753a^{12}-27468a^{10}+19359a^{8}-8120a^{6}+1821a^{4}-180a^{2}+5$, $a^{27}-27a^{25}+324a^{23}-2276a^{21}+10374a^{19}-32130a^{17}+68817a^{15}-101727a^{13}+101763a^{11}-66197a^{9}+26181a^{7}-5643a^{5}+545a^{3}-15a$, $a^{20}-20a^{18}+170a^{16}-800a^{14}+2275a^{12}-4004a^{10}+4290a^{8}-2640a^{6}+825a^{4}-100a^{2}+2$ (assuming GRH)
Regulator:		\( 29837391421490950000000000 \) (assuming GRH)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{41}\cdot(2\pi)^{0}\cdot 29837391421490950000000000 \cdot 1}{2\cdot\sqrt{57959375186337998161464929843210464026538099255933595673241672975683189751201}}\cr\approx \mathstrut & 0.136269552790778 \end{aligned}\] (assuming GRH)

Galois group

$C_{41}$ (as 41T1):

A cyclic group of order 41

The 41 conjugacy class representatives for $C_{41}$

Character table for $C_{41}$

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q$.

Frobenius cycle types

$p$	$2$	$3$	$5$	$7$	$11$	$13$	$17$	$19$	$23$	$29$	$31$	$37$	$41$	$43$	$47$	$53$	$59$
Cycle type	$41$	$41$	$41$	$41$	$41$	$41$	$41$	$41$	$41$	$41$	$41$	$41$	$41$	$41$	$41$	$41$	$41$

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
\(83\)		Deg $41$	$41$	$1$	$40$