Number field 42.42.1236219045653317330764639083558080790009887216550178193020957667462329030021.1: \(\Q(\zeta_{147})^+\)

• Label: 42.42.123...021.1
• Degree: $42$
• Signature: $[42, 0]$
• Discriminant: $1.236\times 10^{75}$
• Root discriminant: \(61.36\)
• Ramified primes: $3,7$
• Class number: $1$ (GRH)
• Class group: trivial (GRH)
• Galois group: $C_{42}$ (as 42T1)

Normalized defining polynomial

x^42 - 42*x^40 + 819*x^38 - 9842*x^36 - x^35 + 81585*x^34 + 35*x^33 - 494802*x^32 - 560*x^31 + 2272424*x^30 + 5425*x^29 - 8069424*x^28 - 35525*x^27 + 22428252*x^26 + 166257*x^25 - 49085400*x^24 - 573300*x^23 + 84672315*x^22 + 1480051*x^21 - 114717330*x^20 - 2877896*x^19 + 121090515*x^18 + 4206314*x^17 - 98285670*x^16 - 4577216*x^15 + 60174899*x^14 + 3643150*x^13 - 27041546*x^12 - 2063243*x^11 + 8580418*x^10 + 798357*x^9 - 1816836*x^8 - 198948*x^7 + 235249*x^6 + 29211*x^5 - 15974*x^4 - 2170*x^3 + 392*x^2 + 56*x + 1

Invariants

Degree:		$42$
Signature:		$[42, 0]$
Discriminant:		\(1236219045653317330764639083558080790009887216550178193020957667462329030021\) \(\medspace = 3^{21}\cdot 7^{77}\)
Root discriminant:		\(61.36\)
Galois root discriminant:		$3^{1/2}7^{11/6}\approx 61.363063610071144$
Ramified primes:		\(3\), \(7\)
Discriminant root field:		\(\Q(\sqrt{21}) \)
$\card{ \Gal(K/\Q) }$:		$42$
This field is Galois and abelian over $\Q$.
Conductor:		\(147=3\cdot 7^{2}\)
Dirichlet character group:		$\lbrace$$\chi_{147}(1,·)$, $\chi_{147}(130,·)$, $\chi_{147}(131,·)$, $\chi_{147}(4,·)$, $\chi_{147}(5,·)$, $\chi_{147}(142,·)$, $\chi_{147}(143,·)$, $\chi_{147}(16,·)$, $\chi_{147}(17,·)$, $\chi_{147}(146,·)$, $\chi_{147}(20,·)$, $\chi_{147}(22,·)$, $\chi_{147}(25,·)$, $\chi_{147}(26,·)$, $\chi_{147}(37,·)$, $\chi_{147}(38,·)$, $\chi_{147}(41,·)$, $\chi_{147}(43,·)$, $\chi_{147}(46,·)$, $\chi_{147}(47,·)$, $\chi_{147}(58,·)$, $\chi_{147}(59,·)$, $\chi_{147}(62,·)$, $\chi_{147}(64,·)$, $\chi_{147}(67,·)$, $\chi_{147}(68,·)$, $\chi_{147}(79,·)$, $\chi_{147}(80,·)$, $\chi_{147}(83,·)$, $\chi_{147}(85,·)$, $\chi_{147}(88,·)$, $\chi_{147}(89,·)$, $\chi_{147}(100,·)$, $\chi_{147}(101,·)$, $\chi_{147}(104,·)$, $\chi_{147}(106,·)$, $\chi_{147}(109,·)$, $\chi_{147}(110,·)$, $\chi_{147}(121,·)$, $\chi_{147}(122,·)$, $\chi_{147}(125,·)$, $\chi_{147}(127,·)$$\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:		$a^{21}-21a^{19}+189a^{17}-952a^{15}+2940a^{13}-5733a^{11}+7007a^{9}-5148a^{7}+2079a^{5}-385a^{3}+21a$, $a^{21}-21a^{19}+189a^{17}-952a^{15}+2940a^{13}-5733a^{11}+7007a^{9}-5148a^{7}+2079a^{5}-385a^{3}+21a+1$, $a^{24}-24a^{22}+252a^{20}-1520a^{18}+5814a^{16}-14688a^{14}+24752a^{12}-27456a^{10}+19305a^{8}-8008a^{6}+1716a^{4}-144a^{2}+2$, $a^{39}-a^{38}-39a^{37}+38a^{36}+702a^{35}-665a^{34}-7735a^{33}+7105a^{32}+58344a^{31}-51800a^{30}-319176a^{29}+272832a^{28}+1308944a^{27}-1072071a^{26}-4102137a^{25}+3199951a^{24}+9924525a^{23}-7318049a^{22}-18599295a^{21}+12840002a^{20}+26936910a^{19}-17196158a^{18}-29910466a^{17}+17374798a^{16}+25110037a^{15}-12992693a^{14}-15601019a^{13}+6988086a^{12}+6953985a^{11}-2594019a^{10}-2125618a^{9}+626055a^{8}+416779a^{7}-89803a^{6}-47320a^{5}+6566a^{4}+2620a^{3}-168a^{2}-48a$, $a^{39}-39a^{37}+702a^{35}-7735a^{33}+58344a^{31}-319176a^{29}+1308944a^{27}-4102137a^{25}+9924525a^{23}-18599295a^{21}+26936910a^{19}-29910465a^{17}+25110020a^{15}-15600900a^{13}+6953544a^{11}-2124694a^{9}+415701a^{7}-46683a^{5}+2470a^{3}-39a$, $a^{38}-38a^{36}+665a^{34}-7106a^{32}+51832a^{30}-273296a^{28}+1076103a^{26}-3223350a^{24}+7413705a^{22}-13123110a^{20}+17809935a^{18}-18349630a^{16}+14115100a^{14}-7904456a^{12}+a^{11}+3105322a^{10}-11a^{9}-810084a^{8}+44a^{7}+128877a^{6}-77a^{5}-10830a^{4}+55a^{3}+361a^{2}-11a-2$, $a^{32}-32a^{30}+464a^{28}-4032a^{26}+23400a^{24}-95680a^{22}+283360a^{20}-615296a^{18}+a^{17}+980628a^{16}-17a^{15}-1136960a^{14}+119a^{13}+940576a^{12}-442a^{11}-537472a^{10}+935a^{9}+201552a^{8}-1122a^{7}-45696a^{6}+714a^{5}+5440a^{4}-204a^{3}-256a^{2}+17a+2$, $a^{15}-15a^{13}+90a^{11}-275a^{9}+450a^{7}-378a^{5}+140a^{3}-15a$, $a^{30}-30a^{28}+405a^{26}-3250a^{24}+17250a^{22}-63756a^{20}+168245a^{18}-319770a^{16}+436050a^{14}-419900a^{12}+277134a^{10}-119340a^{8}+30940a^{6}-4200a^{4}+225a^{2}-2$, $a^{27}-27a^{25}+324a^{23}-2277a^{21}+10395a^{19}-32319a^{17}+69768a^{15}-104652a^{13}+107406a^{11}-72930a^{9}+30888a^{7}-7371a^{5}+819a^{3}-27a$, $a^{40}-40a^{38}-a^{37}+740a^{36}+37a^{35}-8400a^{34}-630a^{33}+65450a^{32}+6545a^{31}-371008a^{30}-46375a^{29}+1582240a^{28}+237307a^{27}-5178240a^{26}-905814a^{25}+13147875a^{24}+2626651a^{23}-26013000a^{22}-5837998a^{21}+40060020a^{20}+9962106a^{19}-47720400a^{18}-12989844a^{17}+43459649a^{16}+12797582a^{15}-29715984a^{14}-9349543a^{13}+14857895a^{12}+4924843a^{11}-5229652a^{10}-1795662a^{9}+1225071a^{8}+427107a^{7}-174776a^{6}-60593a^{5}+12859a^{4}+4401a^{3}-299a^{2}-117a-4$, $a^{33}-33a^{31}+495a^{29}-4466a^{27}+27027a^{25}-115830a^{23}+361790a^{21}-834900a^{19}+1427679a^{17}-1797818a^{15}+1641486a^{13}-1058148a^{11}+461890a^{9}-127908a^{7}+20196a^{5}-1496a^{3}+33a$, $a^{30}-30a^{28}+405a^{26}-3250a^{24}+17250a^{22}-63756a^{20}+168245a^{18}-319770a^{16}+436050a^{14}-419900a^{12}+277134a^{10}-119340a^{8}+30940a^{6}-4200a^{4}+225a^{2}-1$, $a^{29}-29a^{27}+377a^{25}-2900a^{23}+14674a^{21}+a^{20}-51359a^{19}-20a^{18}+127281a^{17}+170a^{16}-224808a^{15}-800a^{14}+281010a^{13}+2275a^{12}-243542a^{11}-4004a^{10}+140998a^{9}+4290a^{8}-51272a^{7}-2640a^{6}+10556a^{5}+825a^{4}-1015a^{3}-100a^{2}+29a+2$, $a^{36}-36a^{34}+594a^{32}-5952a^{30}+40455a^{28}-197316a^{26}+712530a^{24}-1937520a^{22}+3996135a^{20}-6249100a^{18}+7354710a^{16}-6418656a^{14}+4056234a^{12}-1790712a^{10}+523260a^{8}-93024a^{6}+8721a^{4}-324a^{2}+2$, $a^{41}-41a^{39}+779a^{37}-a^{36}-9102a^{35}+36a^{34}+73185a^{33}-594a^{32}-429352a^{31}+5952a^{30}+1901417a^{29}-40455a^{28}-6487214a^{27}+197316a^{26}+17250416a^{25}-712530a^{24}-35940749a^{23}+1937520a^{22}+58676266a^{21}-3996134a^{20}-74719064a^{19}+6249080a^{18}+73529715a^{17}-7354540a^{16}-55120597a^{15}+6417856a^{14}+30844577a^{13}-4053959a^{12}-12534598a^{11}+1786708a^{10}+3564682a^{9}-518969a^{8}-673871a^{7}+90375a^{6}+78288a^{5}-7870a^{4}-4844a^{3}+199a^{2}+112a+4$, $a^{41}-40a^{39}+740a^{37}-8401a^{35}+65485a^{33}-a^{32}-371568a^{31}+31a^{30}+1587665a^{29}-434a^{28}-5213766a^{27}+3628a^{26}+13314159a^{25}-20175a^{24}-26586623a^{23}+78705a^{22}+41542325a^{21}-221354a^{20}-50608461a^{19}+454176a^{18}+47696971a^{17}-680238a^{16}-34358276a^{15}+736609a^{14}+18594750a^{13}-564863a^{12}-7383569a^{11}+296023a^{10}+2079967a^{9}-99930a^{8}-394857a^{7}+19571a^{6}+46257a^{5}-1779a^{4}-2819a^{3}+27a^{2}+60a+2$, $a^{24}-24a^{22}+252a^{20}-1520a^{18}+5814a^{16}-14688a^{14}+24753a^{12}-27468a^{10}+19359a^{8}-8120a^{6}+1821a^{4}-180a^{2}+5$, $a^{9}-9a^{7}+27a^{5}-30a^{3}+9a$, $a^{3}-3a$, $a^{39}-a^{38}-39a^{37}+38a^{36}+702a^{35}-665a^{34}-7735a^{33}+7106a^{32}+58345a^{31}-51832a^{30}-319207a^{29}+273296a^{28}+1309378a^{27}-1076103a^{26}-4105765a^{25}+3223350a^{24}+9944700a^{23}-7413705a^{22}-18678000a^{21}+13123110a^{20}+27158264a^{19}-17809934a^{18}-30364642a^{17}+18349612a^{16}+25790275a^{15}-14114965a^{14}-16337629a^{13}+7903910a^{12}+7518862a^{11}-3104034a^{10}-2421717a^{9}+808292a^{8}+516910a^{7}-127456a^{6}-67158a^{5}+10239a^{4}+4565a^{3}-251a^{2}-115a-4$, $a^{41}-41a^{39}+779a^{37}-a^{36}-9102a^{35}+35a^{34}+73185a^{33}-560a^{32}-429352a^{31}+5425a^{30}+1901416a^{29}-35525a^{28}-6487184a^{27}+166257a^{26}+17250012a^{25}-573300a^{24}-35937525a^{23}+1480051a^{22}+58659315a^{21}-2877896a^{20}-74657310a^{19}+4206314a^{18}+73370115a^{17}-4577216a^{16}-54826021a^{15}+3643150a^{14}+30458914a^{13}-2063243a^{12}-12183637a^{11}+798357a^{10}+3350689a^{9}-198948a^{8}-591555a^{7}+29210a^{6}+60179a^{5}-2164a^{4}-2919a^{3}+47a^{2}+44a+2$, $a^{37}-37a^{35}+629a^{33}-6512a^{31}+45880a^{29}-232841a^{27}+878787a^{25}-2510820a^{23}+5476185a^{21}-9126975a^{19}+11560835a^{17}-10994920a^{15}+7696444a^{13}+a^{12}-3848222a^{11}-12a^{10}+1314610a^{9}+54a^{8}-286824a^{7}-112a^{6}+35853a^{5}+105a^{4}-2109a^{3}-36a^{2}+37a+2$, $a^{33}-33a^{31}+495a^{29}-4466a^{27}+27027a^{25}-115830a^{23}+361790a^{21}-834900a^{19}+1427679a^{17}+a^{16}-1797818a^{15}-16a^{14}+1641486a^{13}+104a^{12}-1058148a^{11}-352a^{10}+461890a^{9}+660a^{8}-127908a^{7}-672a^{6}+20196a^{5}+336a^{4}-1496a^{3}-64a^{2}+33a+2$, $a^{8}-8a^{6}+20a^{4}-16a^{2}+2$, $a^{39}-a^{38}-39a^{37}+38a^{36}+702a^{35}-665a^{34}-7735a^{33}+7105a^{32}+58344a^{31}-51800a^{30}-319176a^{29}+272832a^{28}+1308944a^{27}-1072071a^{26}-4102137a^{25}+3199951a^{24}+9924525a^{23}-7318049a^{22}-18599295a^{21}+12840002a^{20}+26936910a^{19}-17196158a^{18}-29910466a^{17}+17374798a^{16}+25110037a^{15}-12992693a^{14}-15601019a^{13}+6988086a^{12}+6953985a^{11}-2594019a^{10}-2125618a^{9}+626055a^{8}+416779a^{7}-89803a^{6}-47320a^{5}+6565a^{4}+2620a^{3}-164a^{2}-48a-2$, $a^{32}-32a^{30}+464a^{28}-4032a^{26}+23400a^{24}-95680a^{22}+283360a^{20}-615296a^{18}+980628a^{16}-1136960a^{14}+940576a^{12}-537472a^{10}+201552a^{8}-45696a^{6}+5440a^{4}-256a^{2}+2$, $a^{10}-10a^{8}+35a^{6}-50a^{4}+25a^{2}-2$, $a^{29}-29a^{27}+377a^{25}-2900a^{23}+14674a^{21}-51359a^{19}+127281a^{17}-224808a^{15}+281010a^{13}-243542a^{11}+140998a^{9}-51272a^{7}+10556a^{5}-1015a^{3}+29a$, $a^{39}-a^{38}-39a^{37}+38a^{36}+702a^{35}-665a^{34}-7735a^{33}+7105a^{32}+58344a^{31}-51800a^{30}-319176a^{29}+272832a^{28}+1308944a^{27}-1072071a^{26}-4102137a^{25}+3199951a^{24}+9924525a^{23}-7318049a^{22}-18599296a^{21}+12840002a^{20}+26936931a^{19}-17196158a^{18}-29910655a^{17}+17374798a^{16}+25110989a^{15}-12992693a^{14}-15603959a^{13}+6988086a^{12}+6959718a^{11}-2594019a^{10}-2132625a^{9}+626055a^{8}+421927a^{7}-89803a^{6}-49399a^{5}+6565a^{4}+3005a^{3}-164a^{2}-69a-2$, $a^{35}-35a^{33}+560a^{31}-5425a^{29}+35525a^{27}-166257a^{25}+573300a^{23}-a^{22}-1480051a^{21}+22a^{20}+2877896a^{19}-209a^{18}-4206314a^{17}+1122a^{16}+4577216a^{15}-3739a^{14}-3643150a^{13}+7994a^{12}+2063243a^{11}-10934a^{10}-798357a^{9}+9228a^{8}+198948a^{7}-4425a^{6}-29211a^{5}+1014a^{4}+2170a^{3}-72a^{2}-56a-1$, $a^{41}-41a^{39}+779a^{37}-a^{36}-9103a^{35}+35a^{34}+73220a^{33}-560a^{32}-429912a^{31}+5425a^{30}+1906841a^{29}-35525a^{28}-6522709a^{27}+166257a^{26}+17416269a^{25}-573300a^{24}-36510825a^{23}+1480051a^{22}+60139365a^{21}-2877896a^{20}-77535185a^{19}+4206314a^{18}+77576240a^{17}-4577216a^{16}-59402285a^{15}+3643149a^{14}+34099124a^{13}-2063229a^{12}-14241147a^{11}+798280a^{10}+4142039a^{9}-198738a^{8}-785355a^{7}+28917a^{6}+87311a^{5}-1974a^{4}-4704a^{3}+7a^{2}+79a+2$, $a^{24}-24a^{22}+252a^{20}-1520a^{18}+5815a^{16}-14704a^{14}+24856a^{12}-27808a^{10}+19966a^{8}-8688a^{6}+2072a^{4}-224a^{2}+7$, $a^{40}-40a^{38}+740a^{36}-8400a^{34}+65450a^{32}-371008a^{30}+1582240a^{28}-5178240a^{26}+13147875a^{24}-26013000a^{22}+40060020a^{20}-47720400a^{18}+43459650a^{16}-29716000a^{14}+14858000a^{12}-5230016a^{10}+1225785a^{8}-175560a^{6}+13300a^{4}-400a^{2}+2$, $a^{41}-41a^{39}+779a^{37}-a^{36}-9102a^{35}+36a^{34}+73184a^{33}-594a^{32}-429319a^{31}+5952a^{30}+1900922a^{29}-40455a^{28}-6482748a^{27}+197316a^{26}+17223389a^{25}-712530a^{24}-35824919a^{23}+1937520a^{22}+58314476a^{21}-3996134a^{20}-73884164a^{19}+6249080a^{18}+72102037a^{17}-7354541a^{16}-53322796a^{15}+6417872a^{14}+29203210a^{13}-4054063a^{12}-11476892a^{11}+1787060a^{10}+3103727a^{9}-519629a^{8}-547085a^{7}+91047a^{6}+58806a^{5}-8206a^{4}-3552a^{3}+263a^{2}+96a+3$, $a^{38}-38a^{36}+665a^{34}-7106a^{32}+51832a^{30}-273296a^{28}+1076103a^{26}-3223350a^{24}+7413705a^{22}-13123110a^{20}+17809935a^{18}-18349630a^{16}+14115100a^{14}-7904456a^{12}+3105322a^{10}-810084a^{8}+128877a^{6}-10830a^{4}+361a^{2}-2$, $a^{35}-35a^{33}+560a^{31}-5425a^{29}+35525a^{27}-166257a^{25}+573300a^{23}-1480051a^{21}+2877896a^{19}-4206314a^{17}-a^{16}+4577216a^{15}+17a^{14}-3643150a^{13}-118a^{12}+2063243a^{11}+429a^{10}-798357a^{9}-870a^{8}+198948a^{7}+966a^{6}-29211a^{5}-532a^{4}+2170a^{3}+113a^{2}-56a-5$, $a^{5}-5a^{3}+5a$, $a^{23}-23a^{21}+230a^{19}-1311a^{17}+4692a^{15}-10948a^{13}+16744a^{11}-16445a^{9}+9867a^{7}-3289a^{5}+506a^{3}-23a$, $a^{40}-40a^{38}-a^{37}+740a^{36}+37a^{35}-8399a^{34}-629a^{33}+65415a^{32}+6512a^{31}-370448a^{30}-45880a^{29}+1576816a^{28}+232841a^{27}-5142745a^{26}-878787a^{25}+12982021a^{24}+2510820a^{23}-25442899a^{22}-5476185a^{21}+38596646a^{20}+9126976a^{19}-44902529a^{18}-11560854a^{17}+39405999a^{16}+10995073a^{15}-25414916a^{14}-7697124a^{13}+11567648a^{12}+3850041a^{11}-3478902a^{10}-1317601a^{9}+611953a^{8}+289773a^{7}-45373a^{6}-37459a^{5}-1246a^{4}+2509a^{3}+266a^{2}-67a-7$, $a^{40}-40a^{38}-a^{37}+740a^{36}+37a^{35}-8400a^{34}-629a^{33}+65450a^{32}+6512a^{31}-371007a^{30}-45880a^{29}+1582210a^{28}+232841a^{27}-5177836a^{26}-878787a^{25}+13144651a^{24}+2510820a^{23}-25996049a^{22}-5476185a^{21}+39998266a^{20}+9126976a^{19}-47560800a^{18}-11560854a^{17}+43165073a^{16}+10995072a^{15}-29330322a^{14}-7697109a^{13}+14506948a^{12}+3849951a^{11}-5015736a^{10}-1317326a^{9}+1142965a^{8}+289323a^{7}-156961a^{6}-37081a^{5}+11130a^{4}+2369a^{3}-280a^{2}-52a$ (assuming GRH)
Regulator:		\( 1966641158581684500000000 \) (assuming GRH)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{42}\cdot(2\pi)^{0}\cdot 1966641158581684500000000 \cdot 1}{2\cdot\sqrt{1236219045653317330764639083558080790009887216550178193020957667462329030021}}\cr\approx \mathstrut & 0.123000600330838 \end{aligned}\] (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

\(\Q(\sqrt{21}) \), \(\Q(\zeta_{7})^+\), \(\Q(\zeta_{21})^+\), 7.7.13841287201.1, 14.14.2932917071205091238064909.1, \(\Q(\zeta_{49})^+\)

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	$42$	R	$21^{2}$	R	$42$	${\href{/padicField/13.14.0.1}{14} }^{3}$	$21^{2}$	${\href{/padicField/19.6.0.1}{6} }^{7}$	$42$	${\href{/padicField/29.14.0.1}{14} }^{3}$	${\href{/padicField/31.6.0.1}{6} }^{7}$	$21^{2}$	${\href{/padicField/41.7.0.1}{7} }^{6}$	${\href{/padicField/43.7.0.1}{7} }^{6}$	$21^{2}$	$42$	$21^{2}$

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
\(3\)		Deg $42$	$2$	$21$	$21$
\(7\)		Deg $42$	$42$	$1$	$77$