Number field 26.0.23382739460171668835485199054409361747964940454234567.1

• Label: 26.0.233...567.1
• Degree: $26$
• Signature: $[0, 13]$
• Discriminant: $-2.338\times 10^{52}$
• Root discriminant: \(103.32\)
• Ramified primes: $7,53$
• Class number: $1152931$ (GRH)
• Class group: [1152931] (GRH)
• Galois group: $C_{26}$ (as 26T1)

Normalized defining polynomial

x^26 - 11*x^25 + 32*x^24 + 72*x^23 - 470*x^22 - 300*x^21 + 5790*x^20 - 11543*x^19 + 2138*x^18 - 1493*x^17 + 126811*x^16 - 489500*x^15 + 1145956*x^14 - 2174910*x^13 + 4404928*x^12 - 9349303*x^11 + 21707936*x^10 - 45568357*x^9 + 94651964*x^8 - 167065424*x^7 + 281189852*x^6 - 383212608*x^5 + 505907604*x^4 - 494051417*x^3 + 493165779*x^2 - 275771917*x + 183561919

Invariants

Degree:		$26$
Signature:		$[0, 13]$
Discriminant:		\(-23382739460171668835485199054409361747964940454234567\) \(\medspace = -\,7^{13}\cdot 53^{24}\)
Root discriminant:		\(103.32\)
Galois root discriminant:		$7^{1/2}53^{12/13}\approx 103.32092467462952$
Ramified primes:		\(7\), \(53\)
Discriminant root field:		\(\Q(\sqrt{-7}) \)
$\card{ \Gal(K/\Q) }$:		$26$
This field is Galois and abelian over $\Q$.
Conductor:		\(371=7\cdot 53\)
Dirichlet character group:		$\lbrace$$\chi_{371}(1,·)$, $\chi_{371}(258,·)$, $\chi_{371}(195,·)$, $\chi_{371}(69,·)$, $\chi_{371}(134,·)$, $\chi_{371}(97,·)$, $\chi_{371}(328,·)$, $\chi_{371}(13,·)$, $\chi_{371}(15,·)$, $\chi_{371}(148,·)$, $\chi_{371}(342,·)$, $\chi_{371}(281,·)$, $\chi_{371}(153,·)$, $\chi_{371}(155,·)$, $\chi_{371}(160,·)$, $\chi_{371}(225,·)$, $\chi_{371}(99,·)$, $\chi_{371}(36,·)$, $\chi_{371}(293,·)$, $\chi_{371}(169,·)$, $\chi_{371}(365,·)$, $\chi_{371}(174,·)$, $\chi_{371}(307,·)$, $\chi_{371}(309,·)$, $\chi_{371}(183,·)$, $\chi_{371}(314,·)$$\rbrace$
This is a CM field.
Reflex fields:		unavailable$^{4096}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{23}a^{18}-\frac{6}{23}a^{17}+\frac{11}{23}a^{16}-\frac{11}{23}a^{15}-\frac{2}{23}a^{14}+\frac{2}{23}a^{13}+\frac{9}{23}a^{12}+\frac{4}{23}a^{11}-\frac{8}{23}a^{10}+\frac{7}{23}a^{9}+\frac{10}{23}a^{7}+\frac{3}{23}a^{6}+\frac{2}{23}a^{5}+\frac{8}{23}a^{4}-\frac{11}{23}a^{3}+\frac{1}{23}a^{2}+\frac{3}{23}a$, $\frac{1}{23}a^{19}-\frac{2}{23}a^{17}+\frac{9}{23}a^{16}+\frac{1}{23}a^{15}-\frac{10}{23}a^{14}-\frac{2}{23}a^{13}-\frac{11}{23}a^{12}-\frac{7}{23}a^{11}+\frac{5}{23}a^{10}-\frac{4}{23}a^{9}+\frac{10}{23}a^{8}-\frac{6}{23}a^{7}-\frac{3}{23}a^{6}-\frac{3}{23}a^{5}-\frac{9}{23}a^{4}+\frac{4}{23}a^{3}+\frac{9}{23}a^{2}-\frac{5}{23}a$, $\frac{1}{23}a^{20}-\frac{3}{23}a^{17}-\frac{9}{23}a^{15}-\frac{6}{23}a^{14}-\frac{7}{23}a^{13}+\frac{11}{23}a^{12}-\frac{10}{23}a^{11}+\frac{3}{23}a^{10}+\frac{1}{23}a^{9}-\frac{6}{23}a^{8}-\frac{6}{23}a^{7}+\frac{3}{23}a^{6}-\frac{5}{23}a^{5}-\frac{3}{23}a^{4}+\frac{10}{23}a^{3}-\frac{3}{23}a^{2}+\frac{6}{23}a$, $\frac{1}{23}a^{21}+\frac{5}{23}a^{17}+\frac{1}{23}a^{16}+\frac{7}{23}a^{15}+\frac{10}{23}a^{14}-\frac{6}{23}a^{13}-\frac{6}{23}a^{12}-\frac{8}{23}a^{11}-\frac{8}{23}a^{9}-\frac{6}{23}a^{8}+\frac{10}{23}a^{7}+\frac{4}{23}a^{6}+\frac{3}{23}a^{5}+\frac{11}{23}a^{4}+\frac{10}{23}a^{3}+\frac{9}{23}a^{2}+\frac{9}{23}a$, $\frac{1}{23}a^{22}+\frac{8}{23}a^{17}-\frac{2}{23}a^{16}-\frac{4}{23}a^{15}+\frac{4}{23}a^{14}+\frac{7}{23}a^{13}-\frac{7}{23}a^{12}+\frac{3}{23}a^{11}+\frac{9}{23}a^{10}+\frac{5}{23}a^{9}+\frac{10}{23}a^{8}+\frac{11}{23}a^{6}+\frac{1}{23}a^{5}-\frac{7}{23}a^{4}-\frac{5}{23}a^{3}+\frac{4}{23}a^{2}+\frac{8}{23}a$, $\frac{1}{23}a^{23}-\frac{1}{23}a$, $\frac{1}{10007415161}a^{24}-\frac{176153202}{10007415161}a^{23}+\frac{17782985}{10007415161}a^{22}+\frac{73398035}{10007415161}a^{21}-\frac{23241509}{10007415161}a^{20}+\frac{137024165}{10007415161}a^{19}-\frac{142825772}{10007415161}a^{18}-\frac{1470907843}{10007415161}a^{17}+\frac{35599056}{435105007}a^{16}-\frac{1987021286}{10007415161}a^{15}+\frac{140652819}{10007415161}a^{14}-\frac{1817617091}{10007415161}a^{13}+\frac{3134224801}{10007415161}a^{12}+\frac{1337899672}{10007415161}a^{11}-\frac{4196757732}{10007415161}a^{10}-\frac{3723761276}{10007415161}a^{9}+\frac{3673186742}{10007415161}a^{8}+\frac{80441129}{10007415161}a^{7}-\frac{1765275883}{10007415161}a^{6}-\frac{2270336062}{10007415161}a^{5}-\frac{1492477418}{10007415161}a^{4}-\frac{1889174221}{10007415161}a^{3}-\frac{4728644963}{10007415161}a^{2}+\frac{3488671863}{10007415161}a+\frac{211781356}{435105007}$, $\frac{1}{56\!\cdots\!91}a^{25}+\frac{16\!\cdots\!95}{56\!\cdots\!91}a^{24}-\frac{98\!\cdots\!03}{56\!\cdots\!91}a^{23}-\frac{73\!\cdots\!96}{56\!\cdots\!91}a^{22}+\frac{10\!\cdots\!28}{56\!\cdots\!91}a^{21}+\frac{31\!\cdots\!86}{56\!\cdots\!91}a^{20}+\frac{81\!\cdots\!50}{56\!\cdots\!91}a^{19}-\frac{40\!\cdots\!07}{56\!\cdots\!91}a^{18}+\frac{16\!\cdots\!78}{56\!\cdots\!91}a^{17}+\frac{18\!\cdots\!32}{56\!\cdots\!91}a^{16}+\frac{21\!\cdots\!92}{56\!\cdots\!91}a^{15}-\frac{12\!\cdots\!33}{56\!\cdots\!91}a^{14}+\frac{15\!\cdots\!70}{56\!\cdots\!91}a^{13}+\frac{23\!\cdots\!33}{56\!\cdots\!91}a^{12}-\frac{18\!\cdots\!62}{52\!\cdots\!13}a^{11}-\frac{12\!\cdots\!01}{56\!\cdots\!91}a^{10}+\frac{16\!\cdots\!76}{56\!\cdots\!91}a^{9}+\frac{11\!\cdots\!54}{56\!\cdots\!91}a^{8}-\frac{53\!\cdots\!89}{56\!\cdots\!91}a^{7}-\frac{22\!\cdots\!02}{56\!\cdots\!91}a^{6}-\frac{28\!\cdots\!52}{56\!\cdots\!91}a^{5}-\frac{24\!\cdots\!13}{56\!\cdots\!91}a^{4}-\frac{86\!\cdots\!52}{56\!\cdots\!91}a^{3}+\frac{26\!\cdots\!86}{56\!\cdots\!91}a^{2}+\frac{17\!\cdots\!72}{56\!\cdots\!91}a-\frac{69\!\cdots\!43}{24\!\cdots\!17}$

Monogenic:		No
Index:		Not computed
Inessential primes:		$23$

Class group and class number

$C_{1152931}$, which has order $1152931$ (assuming GRH)

Unit group

Rank:		$12$
Torsion generator:		\( -1 \)  (order $2$)
Fundamental units:		$\frac{47\!\cdots\!92}{56\!\cdots\!91}a^{25}+\frac{51\!\cdots\!78}{56\!\cdots\!91}a^{24}-\frac{91\!\cdots\!78}{56\!\cdots\!91}a^{23}+\frac{27\!\cdots\!95}{56\!\cdots\!91}a^{22}+\frac{87\!\cdots\!97}{56\!\cdots\!91}a^{21}-\frac{46\!\cdots\!64}{56\!\cdots\!91}a^{20}-\frac{55\!\cdots\!55}{56\!\cdots\!91}a^{19}+\frac{56\!\cdots\!32}{56\!\cdots\!91}a^{18}-\frac{59\!\cdots\!07}{56\!\cdots\!91}a^{17}-\frac{12\!\cdots\!14}{56\!\cdots\!91}a^{16}+\frac{37\!\cdots\!92}{56\!\cdots\!91}a^{15}+\frac{13\!\cdots\!13}{56\!\cdots\!91}a^{14}-\frac{33\!\cdots\!59}{56\!\cdots\!91}a^{13}+\frac{54\!\cdots\!90}{56\!\cdots\!91}a^{12}-\frac{91\!\cdots\!15}{52\!\cdots\!13}a^{11}+\frac{23\!\cdots\!50}{56\!\cdots\!91}a^{10}-\frac{49\!\cdots\!47}{56\!\cdots\!91}a^{9}+\frac{11\!\cdots\!55}{56\!\cdots\!91}a^{8}-\frac{23\!\cdots\!62}{56\!\cdots\!91}a^{7}+\frac{48\!\cdots\!19}{56\!\cdots\!91}a^{6}-\frac{76\!\cdots\!49}{56\!\cdots\!91}a^{5}+\frac{12\!\cdots\!25}{56\!\cdots\!91}a^{4}-\frac{16\!\cdots\!83}{68\!\cdots\!77}a^{3}+\frac{17\!\cdots\!11}{56\!\cdots\!91}a^{2}-\frac{11\!\cdots\!86}{56\!\cdots\!91}a+\frac{44\!\cdots\!76}{24\!\cdots\!17}$, $\frac{33\!\cdots\!16}{56\!\cdots\!91}a^{25}-\frac{47\!\cdots\!63}{56\!\cdots\!91}a^{24}+\frac{20\!\cdots\!68}{56\!\cdots\!91}a^{23}+\frac{56\!\cdots\!22}{56\!\cdots\!91}a^{22}-\frac{28\!\cdots\!80}{56\!\cdots\!91}a^{21}+\frac{30\!\cdots\!00}{56\!\cdots\!91}a^{20}+\frac{29\!\cdots\!66}{56\!\cdots\!91}a^{19}-\frac{97\!\cdots\!45}{56\!\cdots\!91}a^{18}+\frac{16\!\cdots\!28}{24\!\cdots\!17}a^{17}+\frac{16\!\cdots\!18}{56\!\cdots\!91}a^{16}+\frac{47\!\cdots\!38}{56\!\cdots\!91}a^{15}-\frac{32\!\cdots\!99}{56\!\cdots\!91}a^{14}+\frac{70\!\cdots\!03}{56\!\cdots\!91}a^{13}-\frac{11\!\cdots\!46}{56\!\cdots\!91}a^{12}+\frac{20\!\cdots\!69}{52\!\cdots\!13}a^{11}-\frac{52\!\cdots\!34}{56\!\cdots\!91}a^{10}+\frac{11\!\cdots\!74}{56\!\cdots\!91}a^{9}-\frac{25\!\cdots\!02}{56\!\cdots\!91}a^{8}+\frac{52\!\cdots\!45}{56\!\cdots\!91}a^{7}-\frac{96\!\cdots\!13}{56\!\cdots\!91}a^{6}+\frac{15\!\cdots\!30}{56\!\cdots\!91}a^{5}-\frac{21\!\cdots\!95}{56\!\cdots\!91}a^{4}+\frac{29\!\cdots\!41}{68\!\cdots\!77}a^{3}-\frac{25\!\cdots\!03}{56\!\cdots\!91}a^{2}+\frac{77\!\cdots\!20}{24\!\cdots\!17}a-\frac{22\!\cdots\!54}{10\!\cdots\!79}$, $\frac{24\!\cdots\!14}{56\!\cdots\!91}a^{25}-\frac{14\!\cdots\!40}{56\!\cdots\!91}a^{24}-\frac{53\!\cdots\!30}{56\!\cdots\!91}a^{23}+\frac{47\!\cdots\!25}{56\!\cdots\!91}a^{22}+\frac{18\!\cdots\!61}{56\!\cdots\!91}a^{21}-\frac{62\!\cdots\!83}{56\!\cdots\!91}a^{20}+\frac{43\!\cdots\!79}{56\!\cdots\!91}a^{19}+\frac{47\!\cdots\!99}{56\!\cdots\!91}a^{18}-\frac{72\!\cdots\!99}{56\!\cdots\!91}a^{17}-\frac{14\!\cdots\!70}{56\!\cdots\!91}a^{16}+\frac{28\!\cdots\!08}{56\!\cdots\!91}a^{15}+\frac{67\!\cdots\!85}{56\!\cdots\!91}a^{14}-\frac{20\!\cdots\!55}{56\!\cdots\!91}a^{13}+\frac{30\!\cdots\!90}{56\!\cdots\!91}a^{12}-\frac{41\!\cdots\!11}{52\!\cdots\!13}a^{11}+\frac{12\!\cdots\!77}{56\!\cdots\!91}a^{10}-\frac{21\!\cdots\!98}{56\!\cdots\!91}a^{9}+\frac{65\!\cdots\!84}{56\!\cdots\!91}a^{8}-\frac{12\!\cdots\!72}{56\!\cdots\!91}a^{7}+\frac{30\!\cdots\!43}{56\!\cdots\!91}a^{6}-\frac{46\!\cdots\!54}{56\!\cdots\!91}a^{5}+\frac{87\!\cdots\!46}{56\!\cdots\!91}a^{4}-\frac{10\!\cdots\!67}{68\!\cdots\!77}a^{3}+\frac{13\!\cdots\!26}{56\!\cdots\!91}a^{2}-\frac{74\!\cdots\!16}{56\!\cdots\!91}a+\frac{42\!\cdots\!97}{24\!\cdots\!17}$, $\frac{16\!\cdots\!26}{56\!\cdots\!91}a^{25}-\frac{20\!\cdots\!56}{56\!\cdots\!91}a^{24}+\frac{69\!\cdots\!66}{56\!\cdots\!91}a^{23}+\frac{97\!\cdots\!78}{56\!\cdots\!91}a^{22}-\frac{10\!\cdots\!24}{56\!\cdots\!91}a^{21}+\frac{15\!\cdots\!12}{56\!\cdots\!91}a^{20}+\frac{11\!\cdots\!38}{56\!\cdots\!91}a^{19}-\frac{29\!\cdots\!49}{56\!\cdots\!91}a^{18}+\frac{27\!\cdots\!47}{56\!\cdots\!91}a^{17}+\frac{45\!\cdots\!40}{56\!\cdots\!91}a^{16}+\frac{22\!\cdots\!28}{56\!\cdots\!91}a^{15}-\frac{11\!\cdots\!79}{56\!\cdots\!91}a^{14}+\frac{23\!\cdots\!97}{56\!\cdots\!91}a^{13}-\frac{36\!\cdots\!95}{56\!\cdots\!91}a^{12}+\frac{70\!\cdots\!87}{52\!\cdots\!13}a^{11}-\frac{18\!\cdots\!76}{56\!\cdots\!91}a^{10}+\frac{40\!\cdots\!62}{56\!\cdots\!91}a^{9}-\frac{87\!\cdots\!56}{56\!\cdots\!91}a^{8}+\frac{17\!\cdots\!19}{56\!\cdots\!91}a^{7}-\frac{31\!\cdots\!23}{56\!\cdots\!91}a^{6}+\frac{48\!\cdots\!21}{56\!\cdots\!91}a^{5}-\frac{68\!\cdots\!89}{56\!\cdots\!91}a^{4}+\frac{91\!\cdots\!67}{68\!\cdots\!77}a^{3}-\frac{82\!\cdots\!28}{56\!\cdots\!91}a^{2}+\frac{53\!\cdots\!23}{56\!\cdots\!91}a-\frac{18\!\cdots\!69}{24\!\cdots\!17}$, $\frac{19\!\cdots\!42}{56\!\cdots\!91}a^{25}-\frac{42\!\cdots\!16}{56\!\cdots\!91}a^{24}+\frac{12\!\cdots\!12}{24\!\cdots\!17}a^{23}-\frac{30\!\cdots\!88}{56\!\cdots\!91}a^{22}-\frac{33\!\cdots\!06}{56\!\cdots\!91}a^{21}+\frac{82\!\cdots\!17}{56\!\cdots\!91}a^{20}+\frac{30\!\cdots\!96}{56\!\cdots\!91}a^{19}-\frac{14\!\cdots\!17}{56\!\cdots\!91}a^{18}+\frac{98\!\cdots\!72}{56\!\cdots\!91}a^{17}+\frac{30\!\cdots\!68}{56\!\cdots\!91}a^{16}+\frac{32\!\cdots\!88}{56\!\cdots\!91}a^{15}-\frac{42\!\cdots\!38}{56\!\cdots\!91}a^{14}+\frac{96\!\cdots\!22}{56\!\cdots\!91}a^{13}-\frac{15\!\cdots\!52}{56\!\cdots\!91}a^{12}+\frac{26\!\cdots\!10}{52\!\cdots\!13}a^{11}-\frac{68\!\cdots\!63}{56\!\cdots\!91}a^{10}+\frac{14\!\cdots\!55}{56\!\cdots\!91}a^{9}-\frac{34\!\cdots\!32}{56\!\cdots\!91}a^{8}+\frac{68\!\cdots\!58}{56\!\cdots\!91}a^{7}-\frac{13\!\cdots\!26}{56\!\cdots\!91}a^{6}+\frac{20\!\cdots\!31}{56\!\cdots\!91}a^{5}-\frac{30\!\cdots\!38}{56\!\cdots\!91}a^{4}+\frac{41\!\cdots\!02}{68\!\cdots\!77}a^{3}-\frac{40\!\cdots\!23}{56\!\cdots\!91}a^{2}+\frac{25\!\cdots\!06}{56\!\cdots\!91}a-\frac{95\!\cdots\!70}{24\!\cdots\!17}$, $\frac{34\!\cdots\!72}{56\!\cdots\!91}a^{25}-\frac{38\!\cdots\!96}{56\!\cdots\!91}a^{24}+\frac{11\!\cdots\!54}{56\!\cdots\!91}a^{23}+\frac{28\!\cdots\!34}{56\!\cdots\!91}a^{22}-\frac{18\!\cdots\!10}{56\!\cdots\!91}a^{21}-\frac{11\!\cdots\!47}{56\!\cdots\!91}a^{20}+\frac{22\!\cdots\!04}{56\!\cdots\!91}a^{19}-\frac{40\!\cdots\!87}{56\!\cdots\!91}a^{18}-\frac{19\!\cdots\!34}{56\!\cdots\!91}a^{17}+\frac{45\!\cdots\!67}{56\!\cdots\!91}a^{16}+\frac{47\!\cdots\!60}{56\!\cdots\!91}a^{15}-\frac{18\!\cdots\!47}{56\!\cdots\!91}a^{14}+\frac{35\!\cdots\!23}{56\!\cdots\!91}a^{13}-\frac{56\!\cdots\!22}{56\!\cdots\!91}a^{12}+\frac{10\!\cdots\!01}{52\!\cdots\!13}a^{11}-\frac{27\!\cdots\!39}{56\!\cdots\!91}a^{10}+\frac{64\!\cdots\!56}{56\!\cdots\!91}a^{9}-\frac{13\!\cdots\!48}{56\!\cdots\!91}a^{8}+\frac{26\!\cdots\!77}{56\!\cdots\!91}a^{7}-\frac{45\!\cdots\!32}{56\!\cdots\!91}a^{6}+\frac{70\!\cdots\!10}{56\!\cdots\!91}a^{5}-\frac{86\!\cdots\!90}{56\!\cdots\!91}a^{4}+\frac{12\!\cdots\!87}{68\!\cdots\!77}a^{3}-\frac{84\!\cdots\!08}{56\!\cdots\!91}a^{2}+\frac{66\!\cdots\!47}{56\!\cdots\!91}a-\frac{95\!\cdots\!46}{24\!\cdots\!17}$, $\frac{29\!\cdots\!34}{56\!\cdots\!91}a^{25}-\frac{23\!\cdots\!22}{56\!\cdots\!91}a^{24}+\frac{31\!\cdots\!70}{56\!\cdots\!91}a^{23}+\frac{42\!\cdots\!62}{56\!\cdots\!91}a^{22}-\frac{51\!\cdots\!84}{56\!\cdots\!91}a^{21}-\frac{47\!\cdots\!45}{56\!\cdots\!91}a^{20}+\frac{10\!\cdots\!72}{56\!\cdots\!91}a^{19}+\frac{18\!\cdots\!44}{56\!\cdots\!91}a^{18}-\frac{55\!\cdots\!07}{56\!\cdots\!91}a^{17}-\frac{90\!\cdots\!23}{56\!\cdots\!91}a^{16}+\frac{36\!\cdots\!38}{56\!\cdots\!91}a^{15}-\frac{18\!\cdots\!96}{56\!\cdots\!91}a^{14}-\frac{14\!\cdots\!20}{56\!\cdots\!91}a^{13}-\frac{81\!\cdots\!57}{56\!\cdots\!91}a^{12}+\frac{13\!\cdots\!98}{52\!\cdots\!13}a^{11}-\frac{24\!\cdots\!55}{56\!\cdots\!91}a^{10}+\frac{94\!\cdots\!38}{56\!\cdots\!91}a^{9}-\frac{52\!\cdots\!09}{56\!\cdots\!91}a^{8}+\frac{12\!\cdots\!86}{56\!\cdots\!91}a^{7}+\frac{34\!\cdots\!65}{56\!\cdots\!91}a^{6}-\frac{44\!\cdots\!39}{56\!\cdots\!91}a^{5}+\frac{22\!\cdots\!91}{56\!\cdots\!91}a^{4}-\frac{19\!\cdots\!36}{68\!\cdots\!77}a^{3}+\frac{48\!\cdots\!24}{56\!\cdots\!91}a^{2}-\frac{17\!\cdots\!31}{56\!\cdots\!91}a+\frac{19\!\cdots\!35}{24\!\cdots\!17}$, $\frac{13\!\cdots\!30}{29\!\cdots\!99}a^{25}-\frac{14\!\cdots\!12}{29\!\cdots\!99}a^{24}+\frac{37\!\cdots\!72}{29\!\cdots\!99}a^{23}+\frac{11\!\cdots\!73}{29\!\cdots\!99}a^{22}-\frac{58\!\cdots\!51}{29\!\cdots\!99}a^{21}-\frac{64\!\cdots\!73}{29\!\cdots\!99}a^{20}+\frac{74\!\cdots\!19}{29\!\cdots\!99}a^{19}-\frac{12\!\cdots\!45}{29\!\cdots\!99}a^{18}-\frac{61\!\cdots\!95}{29\!\cdots\!99}a^{17}-\frac{58\!\cdots\!64}{29\!\cdots\!99}a^{16}+\frac{17\!\cdots\!70}{29\!\cdots\!99}a^{15}-\frac{59\!\cdots\!55}{29\!\cdots\!99}a^{14}+\frac{13\!\cdots\!47}{29\!\cdots\!99}a^{13}-\frac{25\!\cdots\!78}{29\!\cdots\!99}a^{12}+\frac{49\!\cdots\!45}{27\!\cdots\!57}a^{11}-\frac{11\!\cdots\!17}{29\!\cdots\!99}a^{10}+\frac{26\!\cdots\!92}{29\!\cdots\!99}a^{9}-\frac{54\!\cdots\!98}{29\!\cdots\!99}a^{8}+\frac{11\!\cdots\!54}{29\!\cdots\!99}a^{7}-\frac{19\!\cdots\!43}{29\!\cdots\!99}a^{6}+\frac{33\!\cdots\!68}{29\!\cdots\!99}a^{5}-\frac{43\!\cdots\!27}{29\!\cdots\!99}a^{4}+\frac{26\!\cdots\!01}{12\!\cdots\!13}a^{3}-\frac{54\!\cdots\!28}{29\!\cdots\!99}a^{2}+\frac{49\!\cdots\!91}{29\!\cdots\!99}a-\frac{63\!\cdots\!03}{12\!\cdots\!13}$, $\frac{30\!\cdots\!42}{56\!\cdots\!91}a^{25}-\frac{36\!\cdots\!76}{56\!\cdots\!91}a^{24}+\frac{11\!\cdots\!44}{56\!\cdots\!91}a^{23}+\frac{23\!\cdots\!33}{56\!\cdots\!91}a^{22}-\frac{18\!\cdots\!71}{56\!\cdots\!91}a^{21}-\frac{52\!\cdots\!20}{56\!\cdots\!91}a^{20}+\frac{22\!\cdots\!01}{56\!\cdots\!91}a^{19}-\frac{45\!\cdots\!36}{56\!\cdots\!91}a^{18}-\frac{22\!\cdots\!97}{56\!\cdots\!91}a^{17}+\frac{90\!\cdots\!26}{56\!\cdots\!91}a^{16}+\frac{45\!\cdots\!92}{56\!\cdots\!91}a^{15}-\frac{19\!\cdots\!44}{56\!\cdots\!91}a^{14}+\frac{34\!\cdots\!42}{56\!\cdots\!91}a^{13}-\frac{47\!\cdots\!28}{56\!\cdots\!91}a^{12}+\frac{98\!\cdots\!56}{52\!\cdots\!13}a^{11}-\frac{27\!\cdots\!10}{56\!\cdots\!91}a^{10}+\frac{62\!\cdots\!42}{56\!\cdots\!91}a^{9}-\frac{12\!\cdots\!50}{56\!\cdots\!91}a^{8}+\frac{24\!\cdots\!89}{56\!\cdots\!91}a^{7}-\frac{42\!\cdots\!47}{56\!\cdots\!91}a^{6}+\frac{65\!\cdots\!68}{56\!\cdots\!91}a^{5}-\frac{80\!\cdots\!63}{56\!\cdots\!91}a^{4}+\frac{10\!\cdots\!52}{68\!\cdots\!77}a^{3}-\frac{77\!\cdots\!28}{56\!\cdots\!91}a^{2}+\frac{56\!\cdots\!97}{56\!\cdots\!91}a-\frac{81\!\cdots\!42}{24\!\cdots\!17}$, $\frac{90\!\cdots\!32}{56\!\cdots\!91}a^{25}-\frac{15\!\cdots\!55}{56\!\cdots\!91}a^{24}+\frac{88\!\cdots\!16}{56\!\cdots\!91}a^{23}-\frac{51\!\cdots\!65}{56\!\cdots\!91}a^{22}-\frac{11\!\cdots\!57}{56\!\cdots\!91}a^{21}+\frac{21\!\cdots\!72}{56\!\cdots\!91}a^{20}+\frac{48\!\cdots\!71}{24\!\cdots\!17}a^{19}-\frac{45\!\cdots\!53}{56\!\cdots\!91}a^{18}+\frac{21\!\cdots\!89}{56\!\cdots\!91}a^{17}+\frac{10\!\cdots\!69}{56\!\cdots\!91}a^{16}+\frac{14\!\cdots\!26}{56\!\cdots\!91}a^{15}-\frac{14\!\cdots\!21}{56\!\cdots\!91}a^{14}+\frac{29\!\cdots\!73}{56\!\cdots\!91}a^{13}-\frac{44\!\cdots\!71}{56\!\cdots\!91}a^{12}+\frac{81\!\cdots\!59}{52\!\cdots\!13}a^{11}-\frac{21\!\cdots\!10}{56\!\cdots\!91}a^{10}+\frac{47\!\cdots\!45}{56\!\cdots\!91}a^{9}-\frac{10\!\cdots\!60}{56\!\cdots\!91}a^{8}+\frac{20\!\cdots\!58}{56\!\cdots\!91}a^{7}-\frac{39\!\cdots\!48}{56\!\cdots\!91}a^{6}+\frac{62\!\cdots\!97}{56\!\cdots\!91}a^{5}-\frac{89\!\cdots\!05}{56\!\cdots\!91}a^{4}+\frac{12\!\cdots\!29}{68\!\cdots\!77}a^{3}-\frac{11\!\cdots\!66}{56\!\cdots\!91}a^{2}+\frac{74\!\cdots\!85}{56\!\cdots\!91}a-\frac{23\!\cdots\!56}{24\!\cdots\!17}$, $\frac{10\!\cdots\!40}{56\!\cdots\!91}a^{25}-\frac{58\!\cdots\!27}{56\!\cdots\!91}a^{24}-\frac{24\!\cdots\!38}{56\!\cdots\!91}a^{23}+\frac{19\!\cdots\!08}{56\!\cdots\!91}a^{22}+\frac{16\!\cdots\!54}{56\!\cdots\!91}a^{21}-\frac{26\!\cdots\!23}{56\!\cdots\!91}a^{20}+\frac{61\!\cdots\!64}{56\!\cdots\!91}a^{19}+\frac{20\!\cdots\!45}{56\!\cdots\!91}a^{18}-\frac{19\!\cdots\!15}{56\!\cdots\!91}a^{17}-\frac{81\!\cdots\!04}{56\!\cdots\!91}a^{16}+\frac{93\!\cdots\!00}{56\!\cdots\!91}a^{15}+\frac{35\!\cdots\!55}{56\!\cdots\!91}a^{14}-\frac{72\!\cdots\!53}{56\!\cdots\!91}a^{13}+\frac{59\!\cdots\!37}{56\!\cdots\!91}a^{12}-\frac{82\!\cdots\!25}{52\!\cdots\!13}a^{11}+\frac{42\!\cdots\!38}{56\!\cdots\!91}a^{10}-\frac{72\!\cdots\!23}{56\!\cdots\!91}a^{9}+\frac{21\!\cdots\!82}{56\!\cdots\!91}a^{8}-\frac{36\!\cdots\!51}{56\!\cdots\!91}a^{7}+\frac{96\!\cdots\!47}{56\!\cdots\!91}a^{6}-\frac{13\!\cdots\!98}{56\!\cdots\!91}a^{5}+\frac{26\!\cdots\!43}{56\!\cdots\!91}a^{4}-\frac{26\!\cdots\!69}{68\!\cdots\!77}a^{3}+\frac{39\!\cdots\!92}{56\!\cdots\!91}a^{2}-\frac{16\!\cdots\!95}{56\!\cdots\!91}a+\frac{10\!\cdots\!24}{24\!\cdots\!17}$, $\frac{24\!\cdots\!34}{56\!\cdots\!91}a^{25}-\frac{37\!\cdots\!38}{56\!\cdots\!91}a^{24}+\frac{18\!\cdots\!98}{56\!\cdots\!91}a^{23}-\frac{27\!\cdots\!93}{56\!\cdots\!91}a^{22}-\frac{24\!\cdots\!09}{56\!\cdots\!91}a^{21}+\frac{36\!\cdots\!53}{56\!\cdots\!91}a^{20}+\frac{25\!\cdots\!41}{56\!\cdots\!91}a^{19}-\frac{91\!\cdots\!85}{56\!\cdots\!91}a^{18}+\frac{38\!\cdots\!65}{56\!\cdots\!91}a^{17}+\frac{18\!\cdots\!54}{56\!\cdots\!91}a^{16}+\frac{15\!\cdots\!60}{24\!\cdots\!17}a^{15}-\frac{28\!\cdots\!25}{56\!\cdots\!91}a^{14}+\frac{62\!\cdots\!63}{56\!\cdots\!91}a^{13}-\frac{97\!\cdots\!62}{56\!\cdots\!91}a^{12}+\frac{17\!\cdots\!95}{52\!\cdots\!13}a^{11}-\frac{45\!\cdots\!13}{56\!\cdots\!91}a^{10}+\frac{10\!\cdots\!08}{56\!\cdots\!91}a^{9}-\frac{22\!\cdots\!77}{56\!\cdots\!91}a^{8}+\frac{19\!\cdots\!52}{24\!\cdots\!17}a^{7}-\frac{84\!\cdots\!07}{56\!\cdots\!91}a^{6}+\frac{13\!\cdots\!82}{56\!\cdots\!91}a^{5}-\frac{18\!\cdots\!13}{56\!\cdots\!91}a^{4}+\frac{24\!\cdots\!19}{68\!\cdots\!77}a^{3}-\frac{22\!\cdots\!12}{56\!\cdots\!91}a^{2}+\frac{14\!\cdots\!20}{56\!\cdots\!91}a-\frac{48\!\cdots\!77}{24\!\cdots\!17}$ (assuming GRH)
Regulator:		\( 5382739421.971964 \) (assuming GRH)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{13}\cdot 5382739421.971964 \cdot 1152931}{2\cdot\sqrt{23382739460171668835485199054409361747964940454234567}}\cr\approx \mathstrut & 0.482688854072152 \end{aligned}\] (assuming GRH)

Galois group

$C_{26}$ (as 26T1):

A cyclic group of order 26

The 26 conjugacy class representatives for $C_{26}$

Character table for $C_{26}$ is not computed

Intermediate fields

\(\Q(\sqrt{-7}) \), 13.13.491258904256726154641.1

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	${\href{/padicField/2.13.0.1}{13} }^{2}$	$26$	$26$	R	${\href{/padicField/11.13.0.1}{13} }^{2}$	$26$	$26$	$26$	${\href{/padicField/23.1.0.1}{1} }^{26}$	${\href{/padicField/29.13.0.1}{13} }^{2}$	$26$	${\href{/padicField/37.13.0.1}{13} }^{2}$	$26$	${\href{/padicField/43.13.0.1}{13} }^{2}$	$26$	R	$26$

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 $26$	$2$	$13$	$13$
\(53\)	53.13.12.1	$x^{13} + 53$	$13$	$1$	$12$	$C_{13}$	$[\ ]_{13}$
	53.13.12.1	$x^{13} + 53$	$13$	$1$	$12$	$C_{13}$	$[\ ]_{13}$