Normalized defining polynomial

\( x^{42} - 42 x^{40} + 819 x^{38} - 9842 x^{36} + 81585 x^{34} - 494802 x^{32} + 2272424 x^{30} - 8069425 x^{28} + \cdots - 7 \) x^42 - 42*x^40 + 819*x^38 - 9842*x^36 + 81585*x^34 - 494802*x^32 + 2272424*x^30 - 8069425*x^28 + 22428280*x^26 - 49085750*x^24 + 84674891*x^22 - 114729727*x^20 + 121131479*x^18 - 98380632*x^16 + 60329941*x^14 - 27217932*x^12 + 8716708*x^10 - 1885324*x^8 + 256221*x^6 - 19551*x^4 + 686*x^2 - 7

Invariants

Degree:		$42$	sage: K.degree()   gp: poldegree(K.pol)   magma: Degree(K);   oscar: degree(K)
Signature:		$[42, 0]$	sage: K.signature()   gp: K.sign   magma: Signature(K);   oscar: signature(K)
Discriminant:		\(519767234928222794437622861788597020192717533652199079454480438860528408854528\) 519767234928222794437622861788597020192717533652199079454480438860528408854528 \(\medspace = 2^{42}\cdot 7^{77}\)	sage: K.disc()   gp: K.disc   magma: OK := Integers(K); Discriminant(OK);   oscar: OK = ring_of_integers(K); discriminant(OK)
Root discriminant:		\(70.86\)	sage: (K.disc().abs())^(1./K.degree())   gp: abs(K.disc)^(1/poldegree(K.pol))   magma: Abs(Discriminant(OK))^(1/Degree(K));   oscar: (1.0 * dK)^(1/degree(K))
Galois root discriminant:		$2\cdot 7^{11/6}\approx 70.8559625871494$
Ramified primes:		\(2\), \(7\) 2, 7	sage: K.disc().support()   gp: factor(abs(K.disc))[,1]~   magma: PrimeDivisors(Discriminant(OK));   oscar: prime_divisors(discriminant((OK)))
Discriminant root field:		\(\Q(\sqrt{7}) \)
$\card{ \Gal(K/\Q) }$:		$42$	sage: K.automorphisms()   magma: Automorphisms(K);   oscar: automorphisms(K)
This field is Galois and abelian over $\Q$.
Conductor:		\(196=2^{2}\cdot 7^{2}\)
Dirichlet character group:		$\lbrace$$\chi_{196}(1,·)$, $\chi_{196}(3,·)$, $\chi_{196}(65,·)$, $\chi_{196}(9,·)$, $\chi_{196}(139,·)$, $\chi_{196}(141,·)$, $\chi_{196}(143,·)$, $\chi_{196}(19,·)$, $\chi_{196}(149,·)$, $\chi_{196}(25,·)$, $\chi_{196}(27,·)$, $\chi_{196}(29,·)$, $\chi_{196}(31,·)$, $\chi_{196}(37,·)$, $\chi_{196}(167,·)$, $\chi_{196}(169,·)$, $\chi_{196}(171,·)$, $\chi_{196}(47,·)$, $\chi_{196}(177,·)$, $\chi_{196}(53,·)$, $\chi_{196}(137,·)$, $\chi_{196}(57,·)$, $\chi_{196}(159,·)$, $\chi_{196}(193,·)$, $\chi_{196}(195,·)$, $\chi_{196}(55,·)$, $\chi_{196}(81,·)$, $\chi_{196}(75,·)$, $\chi_{196}(83,·)$, $\chi_{196}(85,·)$, $\chi_{196}(87,·)$, $\chi_{196}(93,·)$, $\chi_{196}(165,·)$, $\chi_{196}(59,·)$, $\chi_{196}(131,·)$, $\chi_{196}(103,·)$, $\chi_{196}(109,·)$, $\chi_{196}(111,·)$, $\chi_{196}(113,·)$, $\chi_{196}(115,·)$, $\chi_{196}(187,·)$, $\chi_{196}(121,·)$$\rbrace$
This is not a CM field.

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

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

Class group and class number

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

Unit group

Rank:		$41$	sage: UK.rank()   gp: K.fu   magma: UnitRank(K);   oscar: rank(UK)
Torsion generator:		\( -1 \) -1  (order $2$)	sage: UK.torsion_generator()   gp: K.tu[2]   magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);   oscar: torsion_units_generator(OK)
Fundamental units:		$a^{14}-14a^{12}+77a^{10}-210a^{8}+294a^{6}-196a^{4}+49a^{2}-2$, $a^{14}-14a^{12}+77a^{10}-210a^{8}+294a^{6}-196a^{4}+49a^{2}-3$, $a^{12}-12a^{10}+54a^{8}-112a^{6}+105a^{4}-36a^{2}+2$, $a^{24}-24a^{22}+252a^{20}-1520a^{18}+5814a^{16}-14688a^{14}+24752a^{12}-27456a^{10}+19305a^{8}-8008a^{6}+1716a^{4}-144a^{2}+3$, $a^{26}-26a^{24}+299a^{22}-2002a^{20}+8645a^{18}-25194a^{16}+50388a^{14}-68952a^{12}+63206a^{10}-37180a^{8}+13013a^{6}-2366a^{4}+169a^{2}-3$, $a^{40}-39a^{38}+702a^{36}-7735a^{34}+58344a^{32}-319177a^{30}+1308974a^{28}-4102543a^{26}+9927801a^{24}-18616843a^{22}+27002646a^{20}-30087146a^{18}+25453862a^{16}-16083598a^{14}+7434389a^{12}-2454035a^{10}+562837a^{8}-86029a^{6}+7931a^{4}-349a^{2}+5$, $a^{16}-16a^{14}+104a^{12}-352a^{10}+660a^{8}-672a^{6}+336a^{4}-64a^{2}+3$, $a^{4}-4a^{2}+2$, $a^{24}-24a^{22}+252a^{20}-1520a^{18}+5814a^{16}-14688a^{14}+24752a^{12}-27456a^{10}+19305a^{8}-8008a^{6}+1716a^{4}-144a^{2}+2$, $a^{2}-2$, $a^{30}-30a^{28}+405a^{26}-3250a^{24}+17250a^{22}-63756a^{20}+168245a^{18}-319770a^{16}+436050a^{14}-419900a^{12}+277134a^{10}-119340a^{8}+30940a^{6}-4200a^{4}+225a^{2}-2$, $a^{40}-40a^{38}+740a^{36}-8400a^{34}+65450a^{32}-371009a^{30}+1582270a^{28}-5178646a^{26}+13151151a^{24}-26030549a^{22}+40125778a^{20}-47897290a^{18}+43804615a^{16}-30202454a^{14}+15346957a^{12}-5570720a^{10}+1383019a^{8}-220297a^{6}+20307a^{4}-895a^{2}+12$, $a^{38}-38a^{36}+665a^{34}-7106a^{32}+51832a^{30}-273296a^{28}+1076103a^{26}-3223350a^{24}+7413705a^{22}-13123110a^{20}+17809935a^{18}-18349630a^{16}+14115100a^{14}-7904456a^{12}+3105322a^{10}-810084a^{8}+128877a^{6}-10830a^{4}+361a^{2}-2$, $a^{10}-10a^{8}+35a^{6}-50a^{4}+25a^{2}-2$, $a^{18}-18a^{16}+135a^{14}-546a^{12}+1287a^{10}-1782a^{8}+1386a^{6}-540a^{4}+81a^{2}-2$, $a^{20}-20a^{18}+170a^{16}-800a^{14}+2275a^{12}-4004a^{10}+4290a^{8}-2640a^{6}+825a^{4}-100a^{2}+2$, $a^{16}-16a^{14}+104a^{12}-352a^{10}+660a^{8}-672a^{6}+336a^{4}-64a^{2}+2$, $a^{32}-32a^{30}+464a^{28}-4032a^{26}+23400a^{24}-95680a^{22}+283360a^{20}-615296a^{18}+980628a^{16}-1136960a^{14}+940576a^{12}-537472a^{10}+201552a^{8}-45696a^{6}+5440a^{4}-256a^{2}+2$, $a^{8}-8a^{6}+20a^{4}-16a^{2}+2$, $a^{40}-39a^{38}+702a^{36}-7735a^{34}+58343a^{32}-319144a^{30}+1308480a^{28}-4098105a^{26}+9901124a^{24}-18503591a^{22}+26653298a^{20}-29293648a^{18}+24123560a^{16}-14449117a^{14}+5987670a^{12}-1558478a^{10}+193052a^{8}+8442a^{6}-5277a^{4}+471a^{2}-10$, $a^{21}-21a^{19}+189a^{17}-952a^{15}+a^{14}+2940a^{13}-14a^{12}-5733a^{11}+77a^{10}+7007a^{9}-210a^{8}-5148a^{7}+294a^{6}+2079a^{5}-196a^{4}-385a^{3}+49a^{2}+21a-2$, $a^{35}-35a^{33}+560a^{31}-5425a^{29}+a^{28}+35525a^{27}-28a^{26}-166257a^{25}+350a^{24}+573300a^{23}-2576a^{22}-1480050a^{21}+12397a^{20}+2877875a^{19}-40964a^{18}-4206125a^{17}+94962a^{16}+4576264a^{15}-155041a^{14}-3640210a^{13}+176372a^{12}+2057510a^{11}-136213a^{10}-791350a^{9}+68278a^{8}+193800a^{7}-20678a^{6}-27132a^{5}+3381a^{4}+1785a^{3}-245a^{2}-35a+4$, $a^{19}-19a^{17}+a^{16}+152a^{15}-16a^{14}-665a^{13}+104a^{12}+1729a^{11}-352a^{10}-2717a^{9}+660a^{8}+2508a^{7}-672a^{6}-1254a^{5}+336a^{4}+285a^{3}-64a^{2}-19a+2$, $a^{28}-28a^{26}+350a^{24}-2576a^{22}+a^{21}+12397a^{20}-21a^{19}-40964a^{18}+189a^{17}+94962a^{16}-952a^{15}-155041a^{14}+2940a^{13}+176372a^{12}-5733a^{11}-136213a^{10}+7007a^{9}+68278a^{8}-5148a^{7}-20678a^{6}+2079a^{5}+3381a^{4}-385a^{3}-245a^{2}+21a+5$, $a^{30}-30a^{28}+405a^{26}-3250a^{24}-a^{23}+17250a^{22}+23a^{21}-63756a^{20}-230a^{19}+168245a^{18}+1311a^{17}-319770a^{16}-4692a^{15}+436050a^{14}+10948a^{13}-419900a^{12}-16744a^{11}+277134a^{10}+16445a^{9}-119340a^{8}-9867a^{7}+30940a^{6}+3289a^{5}-4200a^{4}-506a^{3}+225a^{2}+23a-2$, $a^{25}-25a^{23}+275a^{21}-1750a^{19}+7125a^{17}-19380a^{15}+35700a^{13}-44200a^{11}+a^{10}+35750a^{9}-10a^{8}-17875a^{7}+35a^{6}+5005a^{5}-50a^{4}-650a^{3}+25a^{2}+25a-2$, $a^{15}-15a^{13}+90a^{11}-a^{10}-275a^{9}+10a^{8}+450a^{7}-35a^{6}-377a^{5}+50a^{4}+135a^{3}-25a^{2}-10a+1$, $a^{33}-33a^{31}+495a^{29}-4466a^{27}+27027a^{25}-115830a^{23}+361790a^{21}-834900a^{19}+1427679a^{17}-1797818a^{15}+1641486a^{13}-1058148a^{11}+461890a^{9}-127908a^{7}+20196a^{5}-1496a^{3}-a^{2}+33a+2$, $a^{9}-9a^{7}+27a^{5}-30a^{3}+a^{2}+9a-2$, $a^{40}-40a^{38}+740a^{36}-8400a^{34}-a^{33}+65450a^{32}+33a^{31}-371008a^{30}-495a^{29}+1582240a^{28}+4466a^{27}-5178240a^{26}-27027a^{25}+13147875a^{24}+115830a^{23}-26013000a^{22}-361790a^{21}+40060020a^{20}+834900a^{19}-47720400a^{18}-1427679a^{17}+43459650a^{16}+1797818a^{15}-29716000a^{14}-1641486a^{13}+14858000a^{12}+1058148a^{11}-5230016a^{10}-461890a^{9}+1225785a^{8}+127908a^{7}-175560a^{6}-20196a^{5}+13300a^{4}+1496a^{3}-400a^{2}-33a+2$, $a^{41}-41a^{39}+779a^{37}-a^{36}-9102a^{35}+36a^{34}+73185a^{33}-594a^{32}-429352a^{31}+5952a^{30}+1901415a^{29}-40455a^{28}-6487156a^{27}+197316a^{26}+17249662a^{25}-712530a^{24}-35934949a^{23}+1937521a^{22}+58646918a^{21}-3996158a^{20}-74616346a^{19}+6249329a^{18}+73275153a^{17}-7356002a^{16}-54670979a^{15}+6423196a^{14}+30282527a^{13}-4066517a^{12}-12047334a^{11}+1805727a^{10}+3282136a^{9}-536988a^{8}-570427a^{7}+100383a^{6}+56420a^{5}-10756a^{4}-2534a^{3}+545a^{2}+24a-6$, $a^{40}-40a^{38}+740a^{36}-8400a^{34}+65450a^{32}-371008a^{30}-a^{29}+1582240a^{28}+29a^{27}-5178240a^{26}-377a^{25}+13147875a^{24}+2900a^{23}-26013000a^{22}-14674a^{21}+40060020a^{20}+51359a^{19}-47720400a^{18}-127281a^{17}+43459650a^{16}+224808a^{15}-29716000a^{14}-281010a^{13}+14858000a^{12}+243543a^{11}-5230016a^{10}-141009a^{9}+1225785a^{8}+51316a^{7}-175560a^{6}-10633a^{5}+13300a^{4}+1070a^{3}-400a^{2}-40a+1$, $a^{3}-a^{2}-2a+1$, $a^{39}+a^{38}-39a^{37}-38a^{36}+702a^{35}+665a^{34}-7735a^{33}-7107a^{32}+58343a^{31}+51864a^{30}-319145a^{29}-273760a^{28}+1308510a^{27}+1080135a^{26}-4098510a^{25}-3246751a^{24}+9904375a^{23}+7509409a^{22}-18520865a^{21}-13406722a^{20}+26717306a^{19}+18426751a^{18}-29463413a^{17}-19336072a^{16}+24449145a^{15}+15266748a^{14}-14899871a^{13}-8869784a^{12}+6432427a^{11}+3670250a^{10}-1863432a^{9}-1030941a^{8}+332411a^{7}+182581a^{6}-31290a^{5}-17987a^{4}+1081a^{3}+765a^{2}-2a-8$, $a^{41}-41a^{39}+779a^{37}-9102a^{35}+73185a^{33}-429352a^{31}+1901416a^{29}-6487184a^{27}+17250011a^{25}-35937500a^{23}+58659040a^{21}-74655560a^{19}+73362990a^{17}-a^{16}-54806640a^{15}+16a^{14}+30423200a^{13}-104a^{12}-12139360a^{11}+352a^{10}+3314729a^{9}-660a^{8}-573386a^{7}+672a^{6}+54978a^{5}-336a^{4}-2220a^{3}+64a^{2}+16a-1$, $a^{27}-27a^{25}+324a^{23}-2277a^{21}+10395a^{19}+a^{18}-32319a^{17}-18a^{16}+69768a^{15}+135a^{14}-104652a^{13}-546a^{12}+107406a^{11}+1287a^{10}-72929a^{9}-1782a^{8}+30879a^{7}+1386a^{6}-7344a^{5}-540a^{4}+789a^{3}+81a^{2}-18a-1$, $a^{37}-37a^{35}+629a^{33}-6512a^{31}+45880a^{29}-232841a^{27}+878787a^{25}-a^{24}-2510820a^{23}+24a^{22}+5476185a^{21}-252a^{20}-9126975a^{19}+1520a^{18}+11560835a^{17}-5814a^{16}-10994920a^{15}+14688a^{14}+7696443a^{13}-24752a^{12}-3848209a^{11}+27456a^{10}+1314545a^{9}-19305a^{8}-286668a^{7}+8008a^{6}+35671a^{5}-1716a^{4}-2018a^{3}+144a^{2}+24a-1$, $a^{33}+a^{32}-33a^{31}-32a^{30}+495a^{29}+464a^{28}-4466a^{27}-4032a^{26}+27027a^{25}+23400a^{24}-115830a^{23}-95680a^{22}+361790a^{21}+283360a^{20}-834900a^{19}-615296a^{18}+1427679a^{17}+980628a^{16}-1797818a^{15}-1136960a^{14}+1641486a^{13}+940576a^{12}-1058148a^{11}-537472a^{10}+461890a^{9}+201552a^{8}-127908a^{7}-45696a^{6}+20196a^{5}+5440a^{4}-1496a^{3}-256a^{2}+32a+1$, $a^{13}-13a^{11}+65a^{9}-156a^{7}+a^{6}+182a^{5}-6a^{4}-91a^{3}+9a^{2}+13a-2$, $a^{33}-33a^{31}+495a^{29}-4466a^{27}-a^{26}+27027a^{25}+26a^{24}-115830a^{23}-299a^{22}+361790a^{21}+2002a^{20}-834900a^{19}-8645a^{18}+1427679a^{17}+25194a^{16}-1797818a^{15}-50388a^{14}+1641486a^{13}+68952a^{12}-1058148a^{11}-63206a^{10}+461890a^{9}+37180a^{8}-127908a^{7}-13013a^{6}+20196a^{5}+2366a^{4}-1496a^{3}-169a^{2}+33a+2$, $a^{39}-39a^{37}+702a^{35}-7735a^{33}+58344a^{31}-319176a^{29}+1308944a^{27}+a^{26}-4102137a^{25}-26a^{24}+9924525a^{23}+299a^{22}-18599295a^{21}-2002a^{20}+26936910a^{19}+8645a^{18}-29910465a^{17}-25194a^{16}+25110020a^{15}+50388a^{14}-15600899a^{13}-68952a^{12}+6953531a^{11}+63206a^{10}-2124629a^{9}-37180a^{8}+415545a^{7}+13013a^{6}-46501a^{5}-2366a^{4}+2379a^{3}+169a^{2}-26a-1$ a^14 - 14*a^12 + 77*a^10 - 210*a^8 + 294*a^6 - 196*a^4 + 49*a^2 - 2, a^14 - 14*a^12 + 77*a^10 - 210*a^8 + 294*a^6 - 196*a^4 + 49*a^2 - 3, a^12 - 12*a^10 + 54*a^8 - 112*a^6 + 105*a^4 - 36*a^2 + 2, a^24 - 24*a^22 + 252*a^20 - 1520*a^18 + 5814*a^16 - 14688*a^14 + 24752*a^12 - 27456*a^10 + 19305*a^8 - 8008*a^6 + 1716*a^4 - 144*a^2 + 3, a^26 - 26*a^24 + 299*a^22 - 2002*a^20 + 8645*a^18 - 25194*a^16 + 50388*a^14 - 68952*a^12 + 63206*a^10 - 37180*a^8 + 13013*a^6 - 2366*a^4 + 169*a^2 - 3, a^40 - 39*a^38 + 702*a^36 - 7735*a^34 + 58344*a^32 - 319177*a^30 + 1308974*a^28 - 4102543*a^26 + 9927801*a^24 - 18616843*a^22 + 27002646*a^20 - 30087146*a^18 + 25453862*a^16 - 16083598*a^14 + 7434389*a^12 - 2454035*a^10 + 562837*a^8 - 86029*a^6 + 7931*a^4 - 349*a^2 + 5, a^16 - 16*a^14 + 104*a^12 - 352*a^10 + 660*a^8 - 672*a^6 + 336*a^4 - 64*a^2 + 3, a^4 - 4*a^2 + 2, a^24 - 24*a^22 + 252*a^20 - 1520*a^18 + 5814*a^16 - 14688*a^14 + 24752*a^12 - 27456*a^10 + 19305*a^8 - 8008*a^6 + 1716*a^4 - 144*a^2 + 2, a^2 - 2, a^30 - 30*a^28 + 405*a^26 - 3250*a^24 + 17250*a^22 - 63756*a^20 + 168245*a^18 - 319770*a^16 + 436050*a^14 - 419900*a^12 + 277134*a^10 - 119340*a^8 + 30940*a^6 - 4200*a^4 + 225*a^2 - 2, a^40 - 40*a^38 + 740*a^36 - 8400*a^34 + 65450*a^32 - 371009*a^30 + 1582270*a^28 - 5178646*a^26 + 13151151*a^24 - 26030549*a^22 + 40125778*a^20 - 47897290*a^18 + 43804615*a^16 - 30202454*a^14 + 15346957*a^12 - 5570720*a^10 + 1383019*a^8 - 220297*a^6 + 20307*a^4 - 895*a^2 + 12, a^38 - 38*a^36 + 665*a^34 - 7106*a^32 + 51832*a^30 - 273296*a^28 + 1076103*a^26 - 3223350*a^24 + 7413705*a^22 - 13123110*a^20 + 17809935*a^18 - 18349630*a^16 + 14115100*a^14 - 7904456*a^12 + 3105322*a^10 - 810084*a^8 + 128877*a^6 - 10830*a^4 + 361*a^2 - 2, a^10 - 10*a^8 + 35*a^6 - 50*a^4 + 25*a^2 - 2, a^18 - 18*a^16 + 135*a^14 - 546*a^12 + 1287*a^10 - 1782*a^8 + 1386*a^6 - 540*a^4 + 81*a^2 - 2, a^20 - 20*a^18 + 170*a^16 - 800*a^14 + 2275*a^12 - 4004*a^10 + 4290*a^8 - 2640*a^6 + 825*a^4 - 100*a^2 + 2, a^16 - 16*a^14 + 104*a^12 - 352*a^10 + 660*a^8 - 672*a^6 + 336*a^4 - 64*a^2 + 2, a^32 - 32*a^30 + 464*a^28 - 4032*a^26 + 23400*a^24 - 95680*a^22 + 283360*a^20 - 615296*a^18 + 980628*a^16 - 1136960*a^14 + 940576*a^12 - 537472*a^10 + 201552*a^8 - 45696*a^6 + 5440*a^4 - 256*a^2 + 2, a^8 - 8*a^6 + 20*a^4 - 16*a^2 + 2, a^40 - 39*a^38 + 702*a^36 - 7735*a^34 + 58343*a^32 - 319144*a^30 + 1308480*a^28 - 4098105*a^26 + 9901124*a^24 - 18503591*a^22 + 26653298*a^20 - 29293648*a^18 + 24123560*a^16 - 14449117*a^14 + 5987670*a^12 - 1558478*a^10 + 193052*a^8 + 8442*a^6 - 5277*a^4 + 471*a^2 - 10, a^21 - 21*a^19 + 189*a^17 - 952*a^15 + a^14 + 2940*a^13 - 14*a^12 - 5733*a^11 + 77*a^10 + 7007*a^9 - 210*a^8 - 5148*a^7 + 294*a^6 + 2079*a^5 - 196*a^4 - 385*a^3 + 49*a^2 + 21*a - 2, a^35 - 35*a^33 + 560*a^31 - 5425*a^29 + a^28 + 35525*a^27 - 28*a^26 - 166257*a^25 + 350*a^24 + 573300*a^23 - 2576*a^22 - 1480050*a^21 + 12397*a^20 + 2877875*a^19 - 40964*a^18 - 4206125*a^17 + 94962*a^16 + 4576264*a^15 - 155041*a^14 - 3640210*a^13 + 176372*a^12 + 2057510*a^11 - 136213*a^10 - 791350*a^9 + 68278*a^8 + 193800*a^7 - 20678*a^6 - 27132*a^5 + 3381*a^4 + 1785*a^3 - 245*a^2 - 35*a + 4, a^19 - 19*a^17 + a^16 + 152*a^15 - 16*a^14 - 665*a^13 + 104*a^12 + 1729*a^11 - 352*a^10 - 2717*a^9 + 660*a^8 + 2508*a^7 - 672*a^6 - 1254*a^5 + 336*a^4 + 285*a^3 - 64*a^2 - 19*a + 2, a^28 - 28*a^26 + 350*a^24 - 2576*a^22 + a^21 + 12397*a^20 - 21*a^19 - 40964*a^18 + 189*a^17 + 94962*a^16 - 952*a^15 - 155041*a^14 + 2940*a^13 + 176372*a^12 - 5733*a^11 - 136213*a^10 + 7007*a^9 + 68278*a^8 - 5148*a^7 - 20678*a^6 + 2079*a^5 + 3381*a^4 - 385*a^3 - 245*a^2 + 21*a + 5, a^30 - 30*a^28 + 405*a^26 - 3250*a^24 - a^23 + 17250*a^22 + 23*a^21 - 63756*a^20 - 230*a^19 + 168245*a^18 + 1311*a^17 - 319770*a^16 - 4692*a^15 + 436050*a^14 + 10948*a^13 - 419900*a^12 - 16744*a^11 + 277134*a^10 + 16445*a^9 - 119340*a^8 - 9867*a^7 + 30940*a^6 + 3289*a^5 - 4200*a^4 - 506*a^3 + 225*a^2 + 23*a - 2, a^25 - 25*a^23 + 275*a^21 - 1750*a^19 + 7125*a^17 - 19380*a^15 + 35700*a^13 - 44200*a^11 + a^10 + 35750*a^9 - 10*a^8 - 17875*a^7 + 35*a^6 + 5005*a^5 - 50*a^4 - 650*a^3 + 25*a^2 + 25*a - 2, a^15 - 15*a^13 + 90*a^11 - a^10 - 275*a^9 + 10*a^8 + 450*a^7 - 35*a^6 - 377*a^5 + 50*a^4 + 135*a^3 - 25*a^2 - 10*a + 1, a^33 - 33*a^31 + 495*a^29 - 4466*a^27 + 27027*a^25 - 115830*a^23 + 361790*a^21 - 834900*a^19 + 1427679*a^17 - 1797818*a^15 + 1641486*a^13 - 1058148*a^11 + 461890*a^9 - 127908*a^7 + 20196*a^5 - 1496*a^3 - a^2 + 33*a + 2, a^9 - 9*a^7 + 27*a^5 - 30*a^3 + a^2 + 9*a - 2, a^40 - 40*a^38 + 740*a^36 - 8400*a^34 - a^33 + 65450*a^32 + 33*a^31 - 371008*a^30 - 495*a^29 + 1582240*a^28 + 4466*a^27 - 5178240*a^26 - 27027*a^25 + 13147875*a^24 + 115830*a^23 - 26013000*a^22 - 361790*a^21 + 40060020*a^20 + 834900*a^19 - 47720400*a^18 - 1427679*a^17 + 43459650*a^16 + 1797818*a^15 - 29716000*a^14 - 1641486*a^13 + 14858000*a^12 + 1058148*a^11 - 5230016*a^10 - 461890*a^9 + 1225785*a^8 + 127908*a^7 - 175560*a^6 - 20196*a^5 + 13300*a^4 + 1496*a^3 - 400*a^2 - 33*a + 2, a^41 - 41*a^39 + 779*a^37 - a^36 - 9102*a^35 + 36*a^34 + 73185*a^33 - 594*a^32 - 429352*a^31 + 5952*a^30 + 1901415*a^29 - 40455*a^28 - 6487156*a^27 + 197316*a^26 + 17249662*a^25 - 712530*a^24 - 35934949*a^23 + 1937521*a^22 + 58646918*a^21 - 3996158*a^20 - 74616346*a^19 + 6249329*a^18 + 73275153*a^17 - 7356002*a^16 - 54670979*a^15 + 6423196*a^14 + 30282527*a^13 - 4066517*a^12 - 12047334*a^11 + 1805727*a^10 + 3282136*a^9 - 536988*a^8 - 570427*a^7 + 100383*a^6 + 56420*a^5 - 10756*a^4 - 2534*a^3 + 545*a^2 + 24*a - 6, a^40 - 40*a^38 + 740*a^36 - 8400*a^34 + 65450*a^32 - 371008*a^30 - a^29 + 1582240*a^28 + 29*a^27 - 5178240*a^26 - 377*a^25 + 13147875*a^24 + 2900*a^23 - 26013000*a^22 - 14674*a^21 + 40060020*a^20 + 51359*a^19 - 47720400*a^18 - 127281*a^17 + 43459650*a^16 + 224808*a^15 - 29716000*a^14 - 281010*a^13 + 14858000*a^12 + 243543*a^11 - 5230016*a^10 - 141009*a^9 + 1225785*a^8 + 51316*a^7 - 175560*a^6 - 10633*a^5 + 13300*a^4 + 1070*a^3 - 400*a^2 - 40*a + 1, a^3 - a^2 - 2*a + 1, a^39 + a^38 - 39*a^37 - 38*a^36 + 702*a^35 + 665*a^34 - 7735*a^33 - 7107*a^32 + 58343*a^31 + 51864*a^30 - 319145*a^29 - 273760*a^28 + 1308510*a^27 + 1080135*a^26 - 4098510*a^25 - 3246751*a^24 + 9904375*a^23 + 7509409*a^22 - 18520865*a^21 - 13406722*a^20 + 26717306*a^19 + 18426751*a^18 - 29463413*a^17 - 19336072*a^16 + 24449145*a^15 + 15266748*a^14 - 14899871*a^13 - 8869784*a^12 + 6432427*a^11 + 3670250*a^10 - 1863432*a^9 - 1030941*a^8 + 332411*a^7 + 182581*a^6 - 31290*a^5 - 17987*a^4 + 1081*a^3 + 765*a^2 - 2*a - 8, a^41 - 41*a^39 + 779*a^37 - 9102*a^35 + 73185*a^33 - 429352*a^31 + 1901416*a^29 - 6487184*a^27 + 17250011*a^25 - 35937500*a^23 + 58659040*a^21 - 74655560*a^19 + 73362990*a^17 - a^16 - 54806640*a^15 + 16*a^14 + 30423200*a^13 - 104*a^12 - 12139360*a^11 + 352*a^10 + 3314729*a^9 - 660*a^8 - 573386*a^7 + 672*a^6 + 54978*a^5 - 336*a^4 - 2220*a^3 + 64*a^2 + 16*a - 1, a^27 - 27*a^25 + 324*a^23 - 2277*a^21 + 10395*a^19 + a^18 - 32319*a^17 - 18*a^16 + 69768*a^15 + 135*a^14 - 104652*a^13 - 546*a^12 + 107406*a^11 + 1287*a^10 - 72929*a^9 - 1782*a^8 + 30879*a^7 + 1386*a^6 - 7344*a^5 - 540*a^4 + 789*a^3 + 81*a^2 - 18*a - 1, a^37 - 37*a^35 + 629*a^33 - 6512*a^31 + 45880*a^29 - 232841*a^27 + 878787*a^25 - a^24 - 2510820*a^23 + 24*a^22 + 5476185*a^21 - 252*a^20 - 9126975*a^19 + 1520*a^18 + 11560835*a^17 - 5814*a^16 - 10994920*a^15 + 14688*a^14 + 7696443*a^13 - 24752*a^12 - 3848209*a^11 + 27456*a^10 + 1314545*a^9 - 19305*a^8 - 286668*a^7 + 8008*a^6 + 35671*a^5 - 1716*a^4 - 2018*a^3 + 144*a^2 + 24*a - 1, a^33 + a^32 - 33*a^31 - 32*a^30 + 495*a^29 + 464*a^28 - 4466*a^27 - 4032*a^26 + 27027*a^25 + 23400*a^24 - 115830*a^23 - 95680*a^22 + 361790*a^21 + 283360*a^20 - 834900*a^19 - 615296*a^18 + 1427679*a^17 + 980628*a^16 - 1797818*a^15 - 1136960*a^14 + 1641486*a^13 + 940576*a^12 - 1058148*a^11 - 537472*a^10 + 461890*a^9 + 201552*a^8 - 127908*a^7 - 45696*a^6 + 20196*a^5 + 5440*a^4 - 1496*a^3 - 256*a^2 + 32*a + 1, a^13 - 13*a^11 + 65*a^9 - 156*a^7 + a^6 + 182*a^5 - 6*a^4 - 91*a^3 + 9*a^2 + 13*a - 2, a^33 - 33*a^31 + 495*a^29 - 4466*a^27 - a^26 + 27027*a^25 + 26*a^24 - 115830*a^23 - 299*a^22 + 361790*a^21 + 2002*a^20 - 834900*a^19 - 8645*a^18 + 1427679*a^17 + 25194*a^16 - 1797818*a^15 - 50388*a^14 + 1641486*a^13 + 68952*a^12 - 1058148*a^11 - 63206*a^10 + 461890*a^9 + 37180*a^8 - 127908*a^7 - 13013*a^6 + 20196*a^5 + 2366*a^4 - 1496*a^3 - 169*a^2 + 33*a + 2, a^39 - 39*a^37 + 702*a^35 - 7735*a^33 + 58344*a^31 - 319176*a^29 + 1308944*a^27 + a^26 - 4102137*a^25 - 26*a^24 + 9924525*a^23 + 299*a^22 - 18599295*a^21 - 2002*a^20 + 26936910*a^19 + 8645*a^18 - 29910465*a^17 - 25194*a^16 + 25110020*a^15 + 50388*a^14 - 15600899*a^13 - 68952*a^12 + 6953531*a^11 + 63206*a^10 - 2124629*a^9 - 37180*a^8 + 415545*a^7 + 13013*a^6 - 46501*a^5 - 2366*a^4 + 2379*a^3 + 169*a^2 - 26*a - 1 (assuming GRH)	sage: UK.fundamental_units()   gp: K.fu   magma: [K|fUK(g): g in Generators(UK)];   oscar: [K(fUK(a)) for a in gens(UK)]
Regulator:		\( 40972406918644810000000000 \) (assuming GRH)	sage: K.regulator()   gp: K.reg   magma: Regulator(K);   oscar: regulator(K)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{42}\cdot(2\pi)^{0}\cdot 40972406918644810000000000 \cdot 1}{2\cdot\sqrt{519767234928222794437622861788597020192717533652199079454480438860528408854528}}\cr\approx \mathstrut & 0.124973188408209 \end{aligned}\] (assuming GRH)

Galois group

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

Intermediate fields

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Frobenius cycle types

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