Number field 42.0.16778739246697564329550246936340186720321059686137149293129858772813957238199098503.1

• Label: 42.0.167...503.1
• Degree: $42$
• Signature: $[0, 21]$
• Discriminant: $-1.678\times 10^{82}$
• Root discriminant: \(90.73\)
• Ramified primes: $7,29$
• Class number: not computed
• Class group: not computed
• Galois group: $C_{42}$ (as 42T1)

Normalized defining polynomial

x^42 - x^41 + 13*x^40 - 18*x^39 + 139*x^38 - 250*x^37 + 1451*x^36 + 321*x^35 + 11819*x^34 + 7069*x^33 + 101147*x^32 + 34723*x^31 + 906407*x^30 - 143662*x^29 + 2718926*x^28 - 434521*x^27 + 7093735*x^26 - 1731307*x^25 + 17818760*x^24 - 17036040*x^23 + 50814721*x^22 - 44435389*x^21 + 124550915*x^20 - 97201554*x^19 + 258557330*x^18 - 191477895*x^17 + 222793332*x^16 - 183429282*x^15 + 205234539*x^14 - 104619060*x^13 + 150123312*x^12 + 66144604*x^11 + 21062459*x^10 + 5929237*x^9 + 1586736*x^8 + 394344*x^7 + 106707*x^6 + 17635*x^5 + 2858*x^4 + 449*x^3 + 67*x^2 + 9*x + 1

Invariants

Degree:		$42$
Signature:		$[0, 21]$
Discriminant:		-16778739246697564329550246936340186720321059686137149293129858772813957238199098503
Root discriminant:		\(90.73\)
Galois root discriminant:		$7^{5/6}29^{6/7}\approx 90.72612557219027$
Ramified primes:		\(7\), \(29\)
Discriminant root field:		\(\Q(\sqrt{-7}) \)
$\card{ \Gal(K/\Q) }$:		$42$
This field is Galois and abelian over $\Q$.
Conductor:		\(203=7\cdot 29\)
Dirichlet character group:		$\lbrace$$\chi_{203}(1,·)$, $\chi_{203}(132,·)$, $\chi_{203}(136,·)$, $\chi_{203}(139,·)$, $\chi_{203}(141,·)$, $\chi_{203}(16,·)$, $\chi_{203}(152,·)$, $\chi_{203}(146,·)$, $\chi_{203}(20,·)$, $\chi_{203}(23,·)$, $\chi_{203}(24,·)$, $\chi_{203}(25,·)$, $\chi_{203}(30,·)$, $\chi_{203}(36,·)$, $\chi_{203}(165,·)$, $\chi_{203}(169,·)$, $\chi_{203}(170,·)$, $\chi_{203}(45,·)$, $\chi_{203}(52,·)$, $\chi_{203}(53,·)$, $\chi_{203}(54,·)$, $\chi_{203}(59,·)$, $\chi_{203}(190,·)$, $\chi_{203}(181,·)$, $\chi_{203}(65,·)$, $\chi_{203}(194,·)$, $\chi_{203}(197,·)$, $\chi_{203}(198,·)$, $\chi_{203}(199,·)$, $\chi_{203}(74,·)$, $\chi_{203}(78,·)$, $\chi_{203}(81,·)$, $\chi_{203}(82,·)$, $\chi_{203}(83,·)$, $\chi_{203}(88,·)$, $\chi_{203}(94,·)$, $\chi_{203}(103,·)$, $\chi_{203}(107,·)$, $\chi_{203}(110,·)$, $\chi_{203}(111,·)$, $\chi_{203}(117,·)$, $\chi_{203}(123,·)$$\rbrace$
This is a CM field.
Reflex fields:		unavailable$^{1048576}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $a^{33}$, $a^{34}$, $a^{35}$, $\frac{1}{17}a^{36}+\frac{2}{17}a^{35}-\frac{7}{17}a^{34}+\frac{4}{17}a^{33}-\frac{3}{17}a^{31}+\frac{4}{17}a^{30}+\frac{5}{17}a^{29}-\frac{7}{17}a^{28}+\frac{1}{17}a^{27}-\frac{5}{17}a^{26}-\frac{7}{17}a^{24}-\frac{2}{17}a^{23}+\frac{8}{17}a^{22}-\frac{1}{17}a^{21}-\frac{5}{17}a^{20}+\frac{4}{17}a^{19}-\frac{7}{17}a^{18}-\frac{7}{17}a^{17}-\frac{8}{17}a^{16}+\frac{2}{17}a^{15}-\frac{3}{17}a^{14}-\frac{5}{17}a^{12}+\frac{1}{17}a^{11}-\frac{2}{17}a^{10}-\frac{4}{17}a^{9}+\frac{6}{17}a^{8}+\frac{7}{17}a^{7}+\frac{8}{17}a^{5}-\frac{5}{17}a^{4}+\frac{8}{17}a^{3}-\frac{1}{17}a^{2}+\frac{1}{17}a+\frac{8}{17}$, $\frac{1}{11\!\cdots\!51}a^{37}-\frac{89\!\cdots\!50}{11\!\cdots\!51}a^{36}+\frac{16\!\cdots\!28}{11\!\cdots\!51}a^{35}-\frac{17\!\cdots\!21}{11\!\cdots\!51}a^{34}-\frac{24\!\cdots\!23}{11\!\cdots\!51}a^{33}+\frac{50\!\cdots\!04}{11\!\cdots\!51}a^{32}+\frac{22\!\cdots\!06}{11\!\cdots\!51}a^{31}-\frac{29\!\cdots\!65}{11\!\cdots\!51}a^{30}-\frac{57\!\cdots\!07}{11\!\cdots\!51}a^{29}+\frac{37\!\cdots\!50}{11\!\cdots\!51}a^{28}-\frac{11\!\cdots\!28}{11\!\cdots\!51}a^{27}+\frac{17\!\cdots\!15}{11\!\cdots\!51}a^{26}-\frac{50\!\cdots\!65}{11\!\cdots\!51}a^{25}-\frac{15\!\cdots\!91}{11\!\cdots\!51}a^{24}-\frac{57\!\cdots\!08}{11\!\cdots\!51}a^{23}+\frac{22\!\cdots\!74}{11\!\cdots\!51}a^{22}+\frac{56\!\cdots\!66}{11\!\cdots\!51}a^{21}+\frac{46\!\cdots\!28}{11\!\cdots\!51}a^{20}+\frac{45\!\cdots\!62}{11\!\cdots\!51}a^{19}-\frac{47\!\cdots\!37}{11\!\cdots\!51}a^{18}+\frac{33\!\cdots\!75}{11\!\cdots\!51}a^{17}+\frac{31\!\cdots\!83}{11\!\cdots\!51}a^{16}+\frac{10\!\cdots\!69}{68\!\cdots\!03}a^{15}-\frac{20\!\cdots\!15}{11\!\cdots\!51}a^{14}+\frac{31\!\cdots\!67}{11\!\cdots\!51}a^{13}+\frac{19\!\cdots\!05}{11\!\cdots\!51}a^{12}+\frac{27\!\cdots\!53}{11\!\cdots\!51}a^{11}-\frac{20\!\cdots\!09}{11\!\cdots\!51}a^{10}-\frac{14\!\cdots\!90}{68\!\cdots\!03}a^{9}+\frac{41\!\cdots\!70}{11\!\cdots\!51}a^{8}+\frac{35\!\cdots\!71}{11\!\cdots\!51}a^{7}-\frac{46\!\cdots\!59}{11\!\cdots\!51}a^{6}-\frac{37\!\cdots\!28}{11\!\cdots\!51}a^{5}-\frac{28\!\cdots\!96}{11\!\cdots\!51}a^{4}+\frac{43\!\cdots\!40}{11\!\cdots\!51}a^{3}+\frac{29\!\cdots\!36}{11\!\cdots\!51}a^{2}-\frac{46\!\cdots\!60}{11\!\cdots\!51}a+\frac{47\!\cdots\!26}{11\!\cdots\!51}$, $\frac{1}{11\!\cdots\!51}a^{38}-\frac{49\!\cdots\!95}{11\!\cdots\!51}a^{36}-\frac{36\!\cdots\!39}{11\!\cdots\!51}a^{35}+\frac{30\!\cdots\!71}{11\!\cdots\!51}a^{34}-\frac{98\!\cdots\!14}{11\!\cdots\!51}a^{33}+\frac{19\!\cdots\!11}{11\!\cdots\!51}a^{32}-\frac{47\!\cdots\!56}{11\!\cdots\!51}a^{31}-\frac{17\!\cdots\!41}{11\!\cdots\!51}a^{30}+\frac{57\!\cdots\!28}{11\!\cdots\!51}a^{29}+\frac{30\!\cdots\!69}{11\!\cdots\!51}a^{28}+\frac{58\!\cdots\!85}{11\!\cdots\!51}a^{27}-\frac{48\!\cdots\!34}{11\!\cdots\!51}a^{26}-\frac{18\!\cdots\!31}{11\!\cdots\!51}a^{25}-\frac{18\!\cdots\!72}{11\!\cdots\!51}a^{24}-\frac{48\!\cdots\!93}{11\!\cdots\!51}a^{23}-\frac{23\!\cdots\!40}{11\!\cdots\!51}a^{22}+\frac{56\!\cdots\!51}{11\!\cdots\!51}a^{21}-\frac{36\!\cdots\!66}{11\!\cdots\!51}a^{20}+\frac{44\!\cdots\!00}{11\!\cdots\!51}a^{19}-\frac{39\!\cdots\!91}{11\!\cdots\!51}a^{18}+\frac{54\!\cdots\!92}{11\!\cdots\!51}a^{17}-\frac{18\!\cdots\!82}{11\!\cdots\!51}a^{16}-\frac{55\!\cdots\!44}{11\!\cdots\!51}a^{15}+\frac{18\!\cdots\!64}{68\!\cdots\!03}a^{14}+\frac{44\!\cdots\!86}{11\!\cdots\!51}a^{13}-\frac{74\!\cdots\!43}{11\!\cdots\!51}a^{12}-\frac{49\!\cdots\!87}{11\!\cdots\!51}a^{11}+\frac{34\!\cdots\!85}{11\!\cdots\!51}a^{10}+\frac{46\!\cdots\!30}{11\!\cdots\!51}a^{9}+\frac{50\!\cdots\!58}{11\!\cdots\!51}a^{8}-\frac{27\!\cdots\!19}{11\!\cdots\!51}a^{7}-\frac{46\!\cdots\!65}{11\!\cdots\!51}a^{6}+\frac{47\!\cdots\!42}{11\!\cdots\!51}a^{5}-\frac{17\!\cdots\!80}{11\!\cdots\!51}a^{4}-\frac{19\!\cdots\!64}{11\!\cdots\!51}a^{3}-\frac{47\!\cdots\!73}{11\!\cdots\!51}a^{2}+\frac{26\!\cdots\!31}{11\!\cdots\!51}a+\frac{32\!\cdots\!06}{11\!\cdots\!51}$, $\frac{1}{11\!\cdots\!51}a^{39}+\frac{25\!\cdots\!82}{11\!\cdots\!51}a^{36}+\frac{36\!\cdots\!77}{11\!\cdots\!51}a^{35}-\frac{56\!\cdots\!07}{11\!\cdots\!51}a^{34}-\frac{16\!\cdots\!90}{11\!\cdots\!51}a^{33}+\frac{43\!\cdots\!12}{11\!\cdots\!51}a^{32}+\frac{39\!\cdots\!80}{11\!\cdots\!51}a^{31}-\frac{13\!\cdots\!37}{11\!\cdots\!51}a^{30}-\frac{41\!\cdots\!83}{11\!\cdots\!51}a^{29}+\frac{52\!\cdots\!58}{11\!\cdots\!51}a^{28}+\frac{22\!\cdots\!88}{11\!\cdots\!51}a^{27}-\frac{33\!\cdots\!63}{11\!\cdots\!51}a^{26}-\frac{17\!\cdots\!06}{11\!\cdots\!51}a^{25}+\frac{30\!\cdots\!87}{11\!\cdots\!51}a^{24}-\frac{36\!\cdots\!36}{11\!\cdots\!51}a^{23}-\frac{44\!\cdots\!19}{11\!\cdots\!51}a^{22}+\frac{20\!\cdots\!33}{11\!\cdots\!51}a^{21}+\frac{28\!\cdots\!43}{11\!\cdots\!51}a^{20}+\frac{43\!\cdots\!74}{11\!\cdots\!51}a^{19}+\frac{33\!\cdots\!15}{11\!\cdots\!51}a^{18}-\frac{28\!\cdots\!17}{11\!\cdots\!51}a^{17}+\frac{57\!\cdots\!63}{11\!\cdots\!51}a^{16}+\frac{29\!\cdots\!12}{11\!\cdots\!51}a^{15}+\frac{12\!\cdots\!03}{11\!\cdots\!51}a^{14}+\frac{34\!\cdots\!59}{11\!\cdots\!51}a^{13}-\frac{46\!\cdots\!49}{11\!\cdots\!51}a^{12}-\frac{37\!\cdots\!57}{11\!\cdots\!51}a^{11}+\frac{38\!\cdots\!08}{11\!\cdots\!51}a^{10}+\frac{16\!\cdots\!74}{68\!\cdots\!03}a^{9}-\frac{41\!\cdots\!88}{11\!\cdots\!51}a^{8}-\frac{46\!\cdots\!50}{11\!\cdots\!51}a^{7}+\frac{26\!\cdots\!35}{11\!\cdots\!51}a^{6}-\frac{39\!\cdots\!80}{11\!\cdots\!51}a^{5}-\frac{49\!\cdots\!69}{11\!\cdots\!51}a^{4}+\frac{53\!\cdots\!45}{11\!\cdots\!51}a^{3}+\frac{15\!\cdots\!23}{11\!\cdots\!51}a^{2}-\frac{49\!\cdots\!67}{11\!\cdots\!51}a-\frac{32\!\cdots\!78}{11\!\cdots\!51}$, $\frac{1}{11\!\cdots\!51}a^{40}-\frac{60\!\cdots\!90}{11\!\cdots\!51}a^{36}-\frac{27\!\cdots\!62}{11\!\cdots\!51}a^{35}-\frac{45\!\cdots\!09}{11\!\cdots\!51}a^{34}-\frac{30\!\cdots\!51}{11\!\cdots\!51}a^{33}-\frac{23\!\cdots\!56}{11\!\cdots\!51}a^{32}+\frac{19\!\cdots\!50}{68\!\cdots\!03}a^{31}-\frac{38\!\cdots\!21}{11\!\cdots\!51}a^{30}+\frac{17\!\cdots\!87}{11\!\cdots\!51}a^{29}+\frac{29\!\cdots\!46}{11\!\cdots\!51}a^{28}+\frac{13\!\cdots\!95}{11\!\cdots\!51}a^{27}+\frac{20\!\cdots\!75}{68\!\cdots\!03}a^{26}+\frac{43\!\cdots\!72}{11\!\cdots\!51}a^{25}+\frac{11\!\cdots\!39}{11\!\cdots\!51}a^{24}-\frac{51\!\cdots\!14}{11\!\cdots\!51}a^{23}-\frac{79\!\cdots\!27}{11\!\cdots\!51}a^{22}-\frac{34\!\cdots\!68}{11\!\cdots\!51}a^{21}+\frac{13\!\cdots\!85}{11\!\cdots\!51}a^{20}-\frac{25\!\cdots\!97}{11\!\cdots\!51}a^{19}-\frac{24\!\cdots\!52}{11\!\cdots\!51}a^{18}+\frac{33\!\cdots\!39}{11\!\cdots\!51}a^{17}-\frac{37\!\cdots\!36}{11\!\cdots\!51}a^{16}-\frac{49\!\cdots\!57}{11\!\cdots\!51}a^{15}-\frac{22\!\cdots\!83}{11\!\cdots\!51}a^{14}+\frac{10\!\cdots\!92}{11\!\cdots\!51}a^{13}+\frac{12\!\cdots\!82}{68\!\cdots\!03}a^{12}+\frac{16\!\cdots\!53}{11\!\cdots\!51}a^{11}+\frac{87\!\cdots\!82}{68\!\cdots\!03}a^{10}+\frac{11\!\cdots\!85}{11\!\cdots\!51}a^{9}+\frac{25\!\cdots\!28}{11\!\cdots\!51}a^{8}+\frac{22\!\cdots\!80}{11\!\cdots\!51}a^{7}+\frac{27\!\cdots\!31}{11\!\cdots\!51}a^{6}-\frac{45\!\cdots\!37}{68\!\cdots\!03}a^{5}-\frac{29\!\cdots\!59}{11\!\cdots\!51}a^{4}-\frac{71\!\cdots\!02}{11\!\cdots\!51}a^{3}-\frac{31\!\cdots\!43}{11\!\cdots\!51}a^{2}-\frac{10\!\cdots\!05}{11\!\cdots\!51}a+\frac{58\!\cdots\!78}{11\!\cdots\!51}$, $\frac{1}{11\!\cdots\!51}a^{41}+\frac{17\!\cdots\!43}{11\!\cdots\!51}a^{36}-\frac{52\!\cdots\!47}{11\!\cdots\!51}a^{35}-\frac{40\!\cdots\!74}{11\!\cdots\!51}a^{34}+\frac{52\!\cdots\!81}{11\!\cdots\!51}a^{33}-\frac{33\!\cdots\!74}{11\!\cdots\!51}a^{32}+\frac{53\!\cdots\!27}{11\!\cdots\!51}a^{31}+\frac{26\!\cdots\!73}{11\!\cdots\!51}a^{30}+\frac{11\!\cdots\!58}{11\!\cdots\!51}a^{29}-\frac{49\!\cdots\!11}{11\!\cdots\!51}a^{28}+\frac{31\!\cdots\!40}{11\!\cdots\!51}a^{27}-\frac{59\!\cdots\!37}{11\!\cdots\!51}a^{26}+\frac{50\!\cdots\!00}{11\!\cdots\!51}a^{25}-\frac{50\!\cdots\!63}{11\!\cdots\!51}a^{24}+\frac{43\!\cdots\!80}{11\!\cdots\!51}a^{23}+\frac{49\!\cdots\!83}{11\!\cdots\!51}a^{22}-\frac{35\!\cdots\!99}{11\!\cdots\!51}a^{21}-\frac{53\!\cdots\!84}{11\!\cdots\!51}a^{20}+\frac{27\!\cdots\!91}{11\!\cdots\!51}a^{19}-\frac{42\!\cdots\!32}{11\!\cdots\!51}a^{18}+\frac{24\!\cdots\!35}{11\!\cdots\!51}a^{17}-\frac{10\!\cdots\!43}{11\!\cdots\!51}a^{16}+\frac{52\!\cdots\!12}{11\!\cdots\!51}a^{15}-\frac{30\!\cdots\!60}{11\!\cdots\!51}a^{14}+\frac{57\!\cdots\!07}{11\!\cdots\!51}a^{13}+\frac{21\!\cdots\!86}{11\!\cdots\!51}a^{12}-\frac{18\!\cdots\!77}{11\!\cdots\!51}a^{11}-\frac{29\!\cdots\!84}{11\!\cdots\!51}a^{10}-\frac{35\!\cdots\!35}{11\!\cdots\!51}a^{9}+\frac{26\!\cdots\!09}{11\!\cdots\!51}a^{8}-\frac{34\!\cdots\!27}{11\!\cdots\!51}a^{7}-\frac{43\!\cdots\!62}{11\!\cdots\!51}a^{6}-\frac{32\!\cdots\!22}{11\!\cdots\!51}a^{5}-\frac{26\!\cdots\!68}{11\!\cdots\!51}a^{4}-\frac{48\!\cdots\!52}{11\!\cdots\!51}a^{3}-\frac{15\!\cdots\!06}{11\!\cdots\!51}a^{2}+\frac{91\!\cdots\!99}{11\!\cdots\!51}a+\frac{31\!\cdots\!36}{11\!\cdots\!51}$

Monogenic:		Not computed
Index:		$1$
Inessential primes:		None

Class group and class number

not computed

Unit group

Rank:		$20$
Torsion generator:		\( \frac{1186100698624582900239519268600628928}{68812784185047783029870312204350653203} a^{41} - \frac{593050349312291450119759634300314464}{68812784185047783029870312204350653203} a^{40} + \frac{14445012079677956035059859664029088016}{68812784185047783029870312204350653203} a^{39} - \frac{13216551487513116603976761521165240859}{68812784185047783029870312204350653203} a^{38} + \frac{149194523591277891951556673714693395872}{68812784185047783029870312204350653203} a^{37} - \frac{206678046735333570366736232553659590704}{68812784185047783029870312204350653203} a^{36} + \frac{1519013748284961364988890051883926883184}{68812784185047783029870312204350653203} a^{35} + \frac{1342454187146844881838950183615090405616}{68812784185047783029870312204350653203} a^{34} + \frac{13645114240673384645587615294422456728592}{68812784185047783029870312204350653203} a^{33} + \frac{15332893174076929520382014087952987384960}{68812784185047783029870312204350653203} a^{32} + \frac{119670440938100233700292971106932990040763}{68812784185047783029870312204350653203} a^{31} + \frac{98975867226297783800344170395906053224000}{68812784185047783029870312204350653203} a^{30} + \frac{1057419956058974115916590829139136066670464}{68812784185047783029870312204350653203} a^{29} + \frac{358194023558255190607540268231133403237152}{68812784185047783029870312204350653203} a^{28} + \frac{2795627485651155650035544520898996377006720}{68812784185047783029870312204350653203} a^{27} + \frac{1190241079638306662998530370487496573430416}{68812784185047783029870312204350653203} a^{26} + \frac{7113524142397604779198672202817346720523040}{68812784185047783029870312204350653203} a^{25} + \frac{2434277806836715207885795474461170260281703}{68812784185047783029870312204350653203} a^{24} + \frac{17385222105796744832808365236469576900524896}{68812784185047783029870312204350653203} a^{23} - \frac{8678486245646583861159940991647593367748832}{68812784185047783029870312204350653203} a^{22} + \frac{43301464548334823254625474578828974696664656}{68812784185047783029870312204350653203} a^{21} - \frac{15319408818841659003311655364645148477759328}{68812784185047783029870312204350653203} a^{20} + \frac{101282902768538641735555424445091306775909520}{68812784185047783029870312204350653203} a^{19} - \frac{22332474976468556109191641243489096330596480}{68812784185047783029870312204350653203} a^{18} + \frac{199662557300016500661691621818281765635048966}{68812784185047783029870312204350653203} a^{17} - \frac{31440714183447289528059198000241584795594864}{68812784185047783029870312204350653203} a^{16} + \frac{48007647831825071103750365809467813164258592}{68812784185047783029870312204350653203} a^{15} - \frac{1484949910517465785715679112380544599459888}{68812784185047783029870312204350653203} a^{14} + \frac{41595672981393094624801081200177216917697936}{68812784185047783029870312204350653203} a^{13} + \frac{76995164978369564809491651286082015477473536}{68812784185047783029870312204350653203} a^{12} + \frac{30001814081864313097746114540830513478258288}{68812784185047783029870312204350653203} a^{11} + \frac{216061521390126675204192078224130140696762755}{68812784185047783029870312204350653203} a^{10} + \frac{2543526865447208945659207155440512415341440}{68812784185047783029870312204350653203} a^{9} + \frac{665696390737214484040151087709436798733088}{68812784185047783029870312204350653203} a^{8} + \frac{170307073666056230983573679570096426482224}{68812784185047783029870312204350653203} a^{7} + \frac{40465604484625582516021559127213356822112}{68812784185047783029870312204350653203} a^{6} + \frac{6660167226473215946077086321633352971600}{68812784185047783029870312204350653203} a^{5} + \frac{1072489195992042495295148172943954399968}{68812784185047783029870312204350653203} a^{4} - \frac{14199422276057202244392332908220744896433}{68812784185047783029870312204350653203} a^{3} + \frac{24399785800277133947784396382641509376}{68812784185047783029870312204350653203} a^{2} + \frac{3177055442744418482784426612323113200}{68812784185047783029870312204350653203} a + \frac{296525174656145725059879817150157232}{68812784185047783029870312204350653203} \)  (order $14$)
Fundamental units:		not computed
Regulator:		not computed

Class number formula

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

Galois group

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

A cyclic group of order 42

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

Character table for $C_{42}$ is not computed

Intermediate fields

\(\Q(\sqrt{-7}) \), \(\Q(\zeta_{7})^+\), \(\Q(\zeta_{7})\), 7.7.594823321.1, 14.0.291381688005381590432263.1, 21.21.142736986105602839685204351151303673689.1

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	$21^{2}$	$42$	$42$	R	$21^{2}$	${\href{/padicField/13.14.0.1}{14} }^{3}$	${\href{/padicField/17.6.0.1}{6} }^{7}$	$42$	$21^{2}$	R	$42$	$21^{2}$	${\href{/padicField/41.2.0.1}{2} }^{21}$	${\href{/padicField/43.7.0.1}{7} }^{6}$	$42$	$21^{2}$	${\href{/padicField/59.6.0.1}{6} }^{7}$

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
\(7\)		Deg $42$	$6$	$7$	$35$
\(29\)	29.7.6.2	$x^{7} + 29$	$7$	$1$	$6$	$C_7$	$[\ ]_{7}$
	29.7.6.2	$x^{7} + 29$	$7$	$1$	$6$	$C_7$	$[\ ]_{7}$
	29.7.6.2	$x^{7} + 29$	$7$	$1$	$6$	$C_7$	$[\ ]_{7}$
	29.7.6.2	$x^{7} + 29$	$7$	$1$	$6$	$C_7$	$[\ ]_{7}$
	29.7.6.2	$x^{7} + 29$	$7$	$1$	$6$	$C_7$	$[\ ]_{7}$
	29.7.6.2	$x^{7} + 29$	$7$	$1$	$6$	$C_7$	$[\ ]_{7}$