Number field 42.42.98118980687896783910098639727548084722605054105289047332129555342401833439729.1: \(\Q(\zeta_{129})^+\)

• Label: 42.42.981...729.1
• Degree: $42$
• Signature: $[42, 0]$
• Discriminant: $9.812\times 10^{76}$
• Root discriminant: \(68.10\)
• Ramified primes: $3,43$
• Class number: $1$ (GRH)
• Class group: trivial (GRH)
• Galois group: $C_{42}$ (as 42T1)

Normalized defining polynomial

x^42 - x^41 - 42*x^40 + 42*x^39 + 818*x^38 - 818*x^37 - 9803*x^36 + 9803*x^35 + 80884*x^34 - 80884*x^33 - 487103*x^32 + 487103*x^31 + 2214673*x^30 - 2214673*x^29 - 7756167*x^28 + 7756167*x^27 + 21159269*x^26 - 21159269*x^25 - 45176143*x^24 + 45176143*x^23 + 75433697*x^22 - 75433697*x^21 - 97942948*x^20 + 97942948*x^19 + 97804877*x^18 - 97804877*x^17 - 73850908*x^16 + 73850908*x^15 + 41150012*x^14 - 41150012*x^13 - 16350448*x^12 + 16350448*x^11 + 4413607*x^10 - 4413607*x^9 - 753918*x^8 + 753918*x^7 + 72886*x^6 - 72886*x^5 - 3267*x^4 + 3267*x^3 + 44*x^2 - 44*x + 1

Invariants

Degree:		$42$
Signature:		$[42, 0]$
Discriminant:		\(98118980687896783910098639727548084722605054105289047332129555342401833439729\) \(\medspace = 3^{21}\cdot 43^{41}\)
Root discriminant:		\(68.10\)
Galois root discriminant:		$3^{1/2}43^{41/42}\approx 68.09841078009121$
Ramified primes:		\(3\), \(43\)
Discriminant root field:		\(\Q(\sqrt{129}) \)
$\card{ \Gal(K/\Q) }$:		$42$
This field is Galois and abelian over $\Q$.
Conductor:		\(129=3\cdot 43\)
Dirichlet character group:		$\lbrace$$\chi_{129}(128,·)$, $\chi_{129}(1,·)$, $\chi_{129}(2,·)$, $\chi_{129}(4,·)$, $\chi_{129}(5,·)$, $\chi_{129}(8,·)$, $\chi_{129}(10,·)$, $\chi_{129}(13,·)$, $\chi_{129}(16,·)$, $\chi_{129}(20,·)$, $\chi_{129}(25,·)$, $\chi_{129}(26,·)$, $\chi_{129}(29,·)$, $\chi_{129}(31,·)$, $\chi_{129}(32,·)$, $\chi_{129}(40,·)$, $\chi_{129}(49,·)$, $\chi_{129}(50,·)$, $\chi_{129}(52,·)$, $\chi_{129}(58,·)$, $\chi_{129}(62,·)$, $\chi_{129}(64,·)$, $\chi_{129}(65,·)$, $\chi_{129}(67,·)$, $\chi_{129}(71,·)$, $\chi_{129}(77,·)$, $\chi_{129}(79,·)$, $\chi_{129}(80,·)$, $\chi_{129}(89,·)$, $\chi_{129}(97,·)$, $\chi_{129}(98,·)$, $\chi_{129}(100,·)$, $\chi_{129}(103,·)$, $\chi_{129}(104,·)$, $\chi_{129}(109,·)$, $\chi_{129}(113,·)$, $\chi_{129}(116,·)$, $\chi_{129}(119,·)$, $\chi_{129}(121,·)$, $\chi_{129}(124,·)$, $\chi_{129}(125,·)$, $\chi_{129}(127,·)$$\rbrace$
This is not a CM field.

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

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

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

Class group and class number

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

Unit group

Rank:		$41$
Torsion generator:		\( -1 \)  (order $2$)
Fundamental units:		$a^{12}-12a^{10}+54a^{8}-112a^{6}+105a^{4}-36a^{2}+2$, $a^{24}-24a^{22}+252a^{20}-1520a^{18}+5814a^{16}-14688a^{14}+24752a^{12}-27456a^{10}+19305a^{8}-8008a^{6}+1716a^{4}-144a^{2}+2$, $a^{38}-38a^{36}+665a^{34}-7106a^{32}+51832a^{30}-273296a^{28}+1076103a^{26}-3223350a^{24}+7413705a^{22}-13123110a^{20}+17809935a^{18}-18349630a^{16}+14115100a^{14}-7904456a^{12}+3105322a^{10}-810084a^{8}+128877a^{6}+a^{5}-10830a^{4}-5a^{3}+361a^{2}+5a-2$, $a^{33}-33a^{31}+495a^{29}-4466a^{27}+27027a^{25}-115830a^{23}+361790a^{21}-834900a^{19}+1427679a^{17}-1797818a^{15}+1641486a^{13}-1058148a^{11}+461890a^{9}-127908a^{7}+20196a^{5}-1496a^{3}+33a$, $a^{23}-23a^{21}+a^{20}+230a^{19}-20a^{18}-1311a^{17}+170a^{16}+4692a^{15}-800a^{14}-10948a^{13}+2275a^{12}+16744a^{11}-4004a^{10}-16445a^{9}+4290a^{8}+9867a^{7}-2640a^{6}-3289a^{5}+825a^{4}+506a^{3}-100a^{2}-23a+2$, $a^{6}-6a^{4}+9a^{2}-2$, $a^{38}-38a^{36}+a^{35}+665a^{34}-36a^{33}-7106a^{32}+593a^{31}+51831a^{30}-5920a^{29}-273266a^{28}+39991a^{27}+1075698a^{26}-193284a^{25}-3220100a^{24}+689131a^{23}+7396455a^{22}-1841863a^{21}-13059353a^{20}+3713005a^{19}+17641669a^{18}-5635115a^{17}-18029672a^{16}+6378773a^{15}+13678115a^{14}-5292629a^{13}-7481735a^{12}+3132312a^{11}+2822897a^{10}-1269410a^{9}-684671a^{8}+331125a^{7}+93903a^{6}-50238a^{5}-5245a^{4}+3641a^{3}-61a^{2}-68a+5$, $a^{38}-38a^{36}+665a^{34}-a^{33}-7106a^{32}+33a^{31}+51832a^{30}-495a^{29}-273296a^{28}+4466a^{27}+1076103a^{26}-27027a^{25}-3223350a^{24}+115831a^{23}+7413705a^{22}-361813a^{21}-13123109a^{20}+835130a^{19}+17809915a^{18}-1428990a^{17}-18349460a^{16}+1802509a^{15}+14114300a^{14}-1652419a^{13}-7902181a^{12}+1074802a^{11}+3101318a^{10}-478060a^{9}-805794a^{8}+137325a^{7}+126237a^{6}-23106a^{5}-10005a^{4}+1857a^{3}+261a^{2}-36a-1$, $a^{41}-41a^{39}+779a^{37}-9102a^{35}+73184a^{33}-429319a^{31}+1900921a^{29}-6482718a^{27}+17222985a^{25}-35821695a^{23}+58297525a^{21}-73822410a^{19}+71942436a^{17}-53028202a^{15}+28817414a^{13}-11125412a^{11}+2888589a^{9}-463353a^{7}-a^{6}+39787a^{5}+6a^{4}-1374a^{3}-8a^{2}+8a$, $a^{35}-35a^{33}+560a^{31}-5425a^{29}+35525a^{27}-166257a^{25}+573300a^{23}-1480050a^{21}+2877875a^{19}-4206125a^{17}+4576264a^{15}-3640210a^{13}+2057510a^{11}-791350a^{9}+a^{8}+193800a^{7}-8a^{6}-27132a^{5}+20a^{4}+1785a^{3}-16a^{2}-35a+1$, $a^{39}-a^{38}-39a^{37}+38a^{36}+702a^{35}-665a^{34}-7734a^{33}+7106a^{32}+58311a^{31}-51832a^{30}-318682a^{29}+273296a^{28}+1304507a^{27}-1076103a^{26}-4075487a^{25}+3223351a^{24}+9811594a^{23}-7413729a^{22}-18252156a^{21}+13123361a^{20}+26153139a^{19}-17811435a^{18}-28608756a^{17}+18355274a^{16}+23532319a^{15}-14128989a^{14}-14229491a^{13}+7926947a^{12}+6122284a^{11}-3128851a^{10}-1787631a^{9}+825309a^{8}+329639a^{7}-134539a^{6}-34106a^{5}+11917a^{4}+1598a^{3}-454a^{2}-23a+5$, $a^{33}-33a^{31}+495a^{29}-4466a^{27}+27027a^{25}-115830a^{23}+361790a^{21}-834900a^{19}+1427679a^{17}-1797818a^{15}+1641486a^{13}-1058148a^{11}+461890a^{9}-127908a^{7}+20196a^{5}-1496a^{3}+33a+1$, $a^{35}-35a^{33}+560a^{31}-5425a^{29}+35525a^{27}+a^{26}-166257a^{25}-26a^{24}+573300a^{23}+299a^{22}-1480050a^{21}-2002a^{20}+2877875a^{19}+8644a^{18}-4206124a^{17}-25176a^{16}+4576247a^{15}+50253a^{14}-3640091a^{13}-68406a^{12}+2057068a^{11}+61919a^{10}-790416a^{9}-35397a^{8}+192687a^{7}+11619a^{6}-26445a^{5}-1806a^{4}+1611a^{3}+72a^{2}-27a+1$, $a^{38}-38a^{36}+665a^{34}-7106a^{32}+51832a^{30}-273296a^{28}+1076103a^{26}-3223350a^{24}+7413705a^{22}-13123110a^{20}+17809935a^{18}-18349630a^{16}+14115100a^{14}-7904456a^{12}+3105322a^{10}-810084a^{8}+128877a^{6}+a^{5}-10830a^{4}-5a^{3}+361a^{2}+5a-3$, $a^{24}-24a^{22}+252a^{20}-1520a^{18}+5814a^{16}-14688a^{14}+24752a^{12}-27456a^{10}+19305a^{8}-8008a^{6}+1716a^{4}-144a^{2}+3$, $a^{41}-41a^{39}+779a^{37}-a^{36}-9101a^{35}+36a^{34}+73150a^{33}-593a^{32}-428792a^{31}+5919a^{30}+1895991a^{29}-39961a^{28}-6451660a^{27}+192880a^{26}+17083782a^{25}-685906a^{24}-35364549a^{23}+1824889a^{22}+57181541a^{21}-3651021a^{20}-71789809a^{19}+5474204a^{18}+69196121a^{17}-6079506a^{16}-50318589a^{15}+4896034a^{14}+26920521a^{13}-2764711a^{12}-10228164a^{11}+1039324a^{10}+2625976a^{9}-239601a^{8}-424279a^{7}+29505a^{6}+38780a^{5}-1532a^{4}-1669a^{3}+33a^{2}+21a-2$, $a^{27}-27a^{25}+324a^{23}-2277a^{21}+10395a^{19}-32319a^{17}+69768a^{15}-104652a^{13}+107406a^{11}-72930a^{9}+30888a^{7}-7371a^{5}+819a^{3}-27a+1$, $a^{32}-32a^{30}+464a^{28}-4032a^{26}+23400a^{24}-95680a^{22}+283360a^{20}-615296a^{18}+980628a^{16}-1136960a^{14}+940576a^{12}+a^{11}-537472a^{10}-11a^{9}+201552a^{8}+44a^{7}-45696a^{6}-77a^{5}+5440a^{4}+55a^{3}-256a^{2}-11a+1$, $a^{38}-38a^{36}+665a^{34}-a^{33}-7106a^{32}+33a^{31}+51832a^{30}-495a^{29}-273296a^{28}+4466a^{27}+1076103a^{26}-27027a^{25}-3223350a^{24}+115830a^{23}+7413705a^{22}-361790a^{21}-13123110a^{20}+834900a^{19}+17809935a^{18}-1427679a^{17}-18349630a^{16}+1797818a^{15}+14115100a^{14}-1641486a^{13}-7904456a^{12}+1058148a^{11}+3105322a^{10}-461890a^{9}-810084a^{8}+127908a^{7}+128877a^{6}-20195a^{5}-10830a^{4}+1491a^{3}+361a^{2}-28a-3$, $a^{38}-38a^{36}+665a^{34}-7106a^{32}+51832a^{30}-273296a^{28}+1076103a^{26}-3223351a^{24}+7413729a^{22}-13123362a^{20}+17811455a^{18}-18355444a^{16}+14129788a^{14}-7929208a^{12}+3132778a^{10}-829389a^{8}+136885a^{6}+a^{5}-12546a^{4}-5a^{3}+505a^{2}+5a-5$, $a^{37}-37a^{35}+629a^{33}-6512a^{31}+45880a^{29}-232841a^{27}+878787a^{25}-2510820a^{23}+5476185a^{21}-9126975a^{19}+11560835a^{17}-10994920a^{15}+7696444a^{13}-3848222a^{11}+1314610a^{9}-286824a^{7}+35853a^{5}-2109a^{3}+37a$, $a^{30}-30a^{28}+405a^{26}-3250a^{24}+17250a^{22}-63756a^{20}+168245a^{18}-319770a^{16}+436050a^{14}+a^{13}-419900a^{12}-13a^{11}+277134a^{10}+65a^{9}-119340a^{8}-156a^{7}+30940a^{6}+182a^{5}-4200a^{4}-91a^{3}+225a^{2}+13a-2$, $a^{17}-17a^{15}+119a^{13}-442a^{11}+935a^{9}-1122a^{7}+714a^{5}-204a^{3}+17a$, $a^{28}-28a^{26}+350a^{24}-2576a^{22}+12397a^{20}-40964a^{18}+94962a^{16}-155040a^{14}+176358a^{12}-136136a^{10}+68068a^{8}-20384a^{6}+3185a^{4}-196a^{2}+2$, $a^{14}-14a^{12}+77a^{10}-210a^{8}+294a^{6}-196a^{4}+49a^{2}-2$, $a^{7}-7a^{5}+14a^{3}-7a$, $a^{34}-34a^{32}+527a^{30}-4930a^{28}+31059a^{26}-139230a^{24}+457470a^{22}-1118260a^{20}+2042975a^{18}-2778446a^{16}+2778446a^{14}-1998724a^{12}+999362a^{10}-329460a^{8}+65892a^{6}-6936a^{4}+289a^{2}-2$, $a^{40}-40a^{38}+740a^{36}-8400a^{34}+65450a^{32}-371008a^{30}+1582240a^{28}-5178240a^{26}+13147875a^{24}-26013000a^{22}+40060020a^{20}-47720400a^{18}+43459650a^{16}-29716000a^{14}+14858000a^{12}-5230016a^{10}+1225785a^{8}-175560a^{6}+13300a^{4}+a^{3}-400a^{2}-3a+2$, $a^{38}-38a^{36}+665a^{34}-7106a^{32}+51832a^{30}-273296a^{28}+1076103a^{26}-3223350a^{24}+7413705a^{22}-13123110a^{20}+a^{19}+17809935a^{18}-19a^{17}-18349630a^{16}+152a^{15}+14115100a^{14}-665a^{13}-7904456a^{12}+1729a^{11}+3105322a^{10}-2717a^{9}-810084a^{8}+2508a^{7}+128877a^{6}-1254a^{5}-10830a^{4}+285a^{3}+361a^{2}-19a-1$, $a^{41}-42a^{39}+2a^{38}+818a^{37}-77a^{36}-9803a^{35}+1367a^{34}+80883a^{33}-14839a^{32}-487070a^{31}+110110a^{30}+2214177a^{29}-591483a^{28}-7751672a^{27}+2376145a^{26}+21131865a^{25}-7271837a^{24}-45057412a^{23}+17109793a^{22}+75057210a^{21}-31015752a^{20}-97056458a^{19}+43138548a^{18}+96248587a^{17}-45562838a^{16}-71823437a^{15}+35918433a^{14}+39215888a^{13}-20594278a^{12}-15030195a^{11}+8271494a^{10}+3791282a^{9}-2202452a^{8}-561913a^{7}+357097a^{6}+37213a^{5}-30422a^{4}+175a^{3}+969a^{2}-75a$, $a^{25}-25a^{23}+275a^{21}-1750a^{19}+7125a^{17}-19380a^{15}+35700a^{13}-44200a^{11}+35749a^{9}-17866a^{7}+4978a^{5}-620a^{3}+16a$, $a^{21}-21a^{19}+189a^{17}-952a^{15}+2939a^{13}-5720a^{11}+6942a^{9}-4992a^{7}+1897a^{5}-294a^{3}+8a$, $a^{15}-15a^{13}+89a^{11}-264a^{9}+406a^{7}-301a^{5}+85a^{3}-4a$, $a^{41}-42a^{39}+a^{38}+818a^{37}-39a^{36}-9803a^{35}+701a^{34}+80884a^{33}-7699a^{32}-487103a^{31}+57751a^{30}+2214673a^{29}-313257a^{28}-7756167a^{27}+1268983a^{26}+21159269a^{25}-3909257a^{24}-45176143a^{23}+9238618a^{22}+75433697a^{21}-16774382a^{20}-97942948a^{19}+23285638a^{18}+97804877a^{17}-24434762a^{16}-73850908a^{15}+19024888a^{14}+41150012a^{13}-10691112a^{12}-16350448a^{11}+4166888a^{10}+4413607a^{9}-1063128a^{8}-753918a^{7}+162657a^{6}+72886a^{5}-12903a^{4}-3267a^{3}+397a^{2}+45a-3$, $a^{40}-40a^{38}+740a^{36}-8400a^{34}+65450a^{32}-371008a^{30}+a^{29}+1582240a^{28}-29a^{27}-5178240a^{26}+377a^{25}+13147875a^{24}-2900a^{23}-26013000a^{22}+14674a^{21}+40060020a^{20}-51359a^{19}-47720400a^{18}+127281a^{17}+43459650a^{16}-224808a^{15}-29715999a^{14}+281010a^{13}+14857986a^{12}-243542a^{11}-5229939a^{10}+140998a^{9}+1225575a^{8}-51272a^{7}-175266a^{6}+10556a^{5}+13104a^{4}-1015a^{3}-351a^{2}+29a$, $a^{41}-41a^{39}+779a^{37}-9102a^{35}+73185a^{33}-429352a^{31}+1901416a^{29}-6487184a^{27}+17250012a^{25}-a^{24}-35937525a^{23}+24a^{22}+58659315a^{21}-252a^{20}-74657310a^{19}+1520a^{18}+73370115a^{17}-5814a^{16}-54826020a^{15}+14688a^{14}+30458900a^{13}-24752a^{12}-12183560a^{11}+27456a^{10}+3350479a^{9}-19305a^{8}-591261a^{7}+8008a^{6}+59983a^{5}-1716a^{4}-2870a^{3}+144a^{2}+41a-2$, $a^{41}-41a^{39}-a^{38}+779a^{37}+38a^{36}-9102a^{35}-664a^{34}+73185a^{33}+7071a^{32}-429352a^{31}-51273a^{30}+1901416a^{29}+267902a^{28}-6487183a^{27}-1041012a^{26}+17249984a^{25}+3060720a^{24}-35937176a^{23}-6860555a^{22}+58656763a^{21}+11721491a^{20}-74645165a^{19}-15151685a^{18}+73330671a^{17}+14590744a^{16}-54736872a^{15}-10200628a^{14}+30318548a^{13}+4967963a^{12}-12031955a^{11}-1573702a^{10}+3241810a^{9}+284934a^{8}-542541a^{7}-21021a^{6}+47676a^{5}-377a^{4}-1437a^{3}+52a^{2}-12a+1$, $a^{31}-31a^{29}+434a^{27}-3627a^{25}+20150a^{23}-78430a^{21}+219604a^{19}-447051a^{17}+660858a^{15}-700910a^{13}+a^{12}+520676a^{11}-12a^{10}-260338a^{9}+54a^{8}+82212a^{7}-112a^{6}-14756a^{5}+105a^{4}+1240a^{3}-36a^{2}-31a+2$, $a^{36}-36a^{34}+594a^{32}-a^{31}-5952a^{30}+31a^{29}+40455a^{28}-434a^{27}-197316a^{26}+3627a^{25}+712530a^{24}-20150a^{23}-1937520a^{22}+78430a^{21}+3996135a^{20}-219604a^{19}-6249100a^{18}+447051a^{17}+7354710a^{16}-660858a^{15}-6418656a^{14}+700910a^{13}+4056233a^{12}-520676a^{11}-1790700a^{10}+260338a^{9}+523206a^{8}-82212a^{7}-92912a^{6}+14756a^{5}+8616a^{4}-1240a^{3}-288a^{2}+31a$, $a^{41}-41a^{39}+779a^{37}-9102a^{35}+73185a^{33}-429352a^{31}+1901416a^{29}-6487184a^{27}+17250012a^{25}-35937525a^{23}-a^{22}+58659314a^{21}+22a^{20}-74657289a^{19}-209a^{18}+73369926a^{17}+1122a^{16}-54825068a^{15}-3740a^{14}+30455960a^{13}+8008a^{12}-12177827a^{11}-11011a^{10}+3343472a^{9}+9438a^{8}-586113a^{7}-4719a^{6}+57904a^{5}+1210a^{4}-2485a^{3}-120a^{2}+20a$, $a^{23}-23a^{21}+230a^{19}-1311a^{17}+4692a^{15}-10948a^{13}+16744a^{11}-16445a^{9}+9867a^{7}-3289a^{5}+506a^{3}-23a$ (assuming GRH)
Regulator:		\( 16535537450237775000000000 \) (assuming GRH)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{42}\cdot(2\pi)^{0}\cdot 16535537450237775000000000 \cdot 1}{2\cdot\sqrt{98118980687896783910098639727548084722605054105289047332129555342401833439729}}\cr\approx \mathstrut & 0.116083801936201 \end{aligned}\] (assuming GRH)

Galois group

$C_{42}$ (as 42T1):

A cyclic group of order 42

The 42 conjugacy class representatives for $C_{42}$

Character table for $C_{42}$

Intermediate fields

\(\Q(\sqrt{129}) \), 3.3.1849.1, 6.6.3969227961.1, 7.7.6321363049.1, 14.14.3757843639805369947326441.1, \(\Q(\zeta_{43})^+\)

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	${\href{/padicField/2.7.0.1}{7} }^{6}$	R	$21^{2}$	${\href{/padicField/7.6.0.1}{6} }^{7}$	${\href{/padicField/11.14.0.1}{14} }^{3}$	$21^{2}$	$42$	$42$	$42$	$21^{2}$	$21^{2}$	${\href{/padicField/37.6.0.1}{6} }^{7}$	${\href{/padicField/41.14.0.1}{14} }^{3}$	R	${\href{/padicField/47.14.0.1}{14} }^{3}$	$42$	${\href{/padicField/59.14.0.1}{14} }^{3}$

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
\(3\)		Deg $42$	$2$	$21$	$21$
\(43\)		Deg $42$	$42$	$1$	$41$