Number field 26.0.275852727404510985186163978883510976421015681999.1

• Label: 26.0.275...999.1
• Degree: $26$
• Signature: $[0, 13]$
• Discriminant: $-2.759\times 10^{47}$
• Root discriminant: \(66.78\)
• Ramified prime: $79$
• Class number: $265$ (GRH)
• Class group: [265] (GRH)
• Galois group: $C_{26}$ (as 26T1)

Normalized defining polynomial

x^26 - x^25 + 2*x^24 + 48*x^23 - 41*x^22 + 75*x^21 + 827*x^20 - 597*x^19 + 990*x^18 + 6414*x^17 - 3755*x^16 + 5397*x^15 + 23537*x^14 - 10353*x^13 + 15676*x^12 + 35758*x^11 - 6997*x^10 + 39165*x^9 + 14468*x^8 + 8338*x^7 + 43270*x^6 - 25107*x^5 + 25775*x^4 - 10788*x^3 - 8746*x^2 + 10048*x + 4289

Invariants

Degree:		$26$
Signature:		$[0, 13]$
Discriminant:		\(-275852727404510985186163978883510976421015681999\) \(\medspace = -\,79^{25}\)
Root discriminant:		\(66.78\)
Galois root discriminant:		$79^{25/26}\approx 66.77923412581892$
Ramified primes:		\(79\)
Discriminant root field:		\(\Q(\sqrt{-79}) \)
$\card{ \Gal(K/\Q) }$:		$26$
This field is Galois and abelian over $\Q$.
Conductor:		\(79\)
Dirichlet character group:		$\lbrace$$\chi_{79}(64,·)$, $\chi_{79}(1,·)$, $\chi_{79}(67,·)$, $\chi_{79}(69,·)$, $\chi_{79}(65,·)$, $\chi_{79}(8,·)$, $\chi_{79}(10,·)$, $\chi_{79}(12,·)$, $\chi_{79}(14,·)$, $\chi_{79}(15,·)$, $\chi_{79}(17,·)$, $\chi_{79}(18,·)$, $\chi_{79}(21,·)$, $\chi_{79}(22,·)$, $\chi_{79}(27,·)$, $\chi_{79}(33,·)$, $\chi_{79}(38,·)$, $\chi_{79}(41,·)$, $\chi_{79}(71,·)$, $\chi_{79}(78,·)$, $\chi_{79}(46,·)$, $\chi_{79}(52,·)$, $\chi_{79}(57,·)$, $\chi_{79}(58,·)$, $\chi_{79}(61,·)$, $\chi_{79}(62,·)$$\rbrace$
This is a CM field.
Reflex fields:		unavailable$^{4096}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{23}a^{17}-\frac{7}{23}a^{15}-\frac{8}{23}a^{14}-\frac{2}{23}a^{13}-\frac{10}{23}a^{12}-\frac{2}{23}a^{11}-\frac{9}{23}a^{10}-\frac{1}{23}a^{9}+\frac{5}{23}a^{8}-\frac{2}{23}a^{7}+\frac{9}{23}a^{6}-\frac{4}{23}a^{5}+\frac{4}{23}a^{4}-\frac{4}{23}a^{2}-\frac{6}{23}a-\frac{10}{23}$, $\frac{1}{23}a^{18}-\frac{7}{23}a^{16}-\frac{8}{23}a^{15}-\frac{2}{23}a^{14}-\frac{10}{23}a^{13}-\frac{2}{23}a^{12}-\frac{9}{23}a^{11}-\frac{1}{23}a^{10}+\frac{5}{23}a^{9}-\frac{2}{23}a^{8}+\frac{9}{23}a^{7}-\frac{4}{23}a^{6}+\frac{4}{23}a^{5}-\frac{4}{23}a^{3}-\frac{6}{23}a^{2}-\frac{10}{23}a$, $\frac{1}{23}a^{19}-\frac{8}{23}a^{16}-\frac{5}{23}a^{15}+\frac{3}{23}a^{14}+\frac{7}{23}a^{13}-\frac{10}{23}a^{12}+\frac{8}{23}a^{11}+\frac{11}{23}a^{10}-\frac{9}{23}a^{9}-\frac{2}{23}a^{8}+\frac{5}{23}a^{7}-\frac{2}{23}a^{6}-\frac{5}{23}a^{5}+\frac{1}{23}a^{4}-\frac{6}{23}a^{3}+\frac{8}{23}a^{2}+\frac{4}{23}a-\frac{1}{23}$, $\frac{1}{23}a^{20}-\frac{5}{23}a^{16}-\frac{7}{23}a^{15}-\frac{11}{23}a^{14}-\frac{3}{23}a^{13}-\frac{3}{23}a^{12}-\frac{5}{23}a^{11}+\frac{11}{23}a^{10}-\frac{10}{23}a^{9}-\frac{1}{23}a^{8}+\frac{5}{23}a^{7}-\frac{2}{23}a^{6}-\frac{8}{23}a^{5}+\frac{3}{23}a^{4}+\frac{8}{23}a^{3}-\frac{5}{23}a^{2}-\frac{3}{23}a-\frac{11}{23}$, $\frac{1}{23}a^{21}-\frac{7}{23}a^{16}+\frac{3}{23}a^{14}+\frac{10}{23}a^{13}-\frac{9}{23}a^{12}+\frac{1}{23}a^{11}-\frac{9}{23}a^{10}-\frac{6}{23}a^{9}+\frac{7}{23}a^{8}+\frac{11}{23}a^{7}-\frac{9}{23}a^{6}+\frac{6}{23}a^{5}+\frac{5}{23}a^{4}-\frac{5}{23}a^{3}+\frac{5}{23}a-\frac{4}{23}$, $\frac{1}{23}a^{22}-\frac{1}{23}$, $\frac{1}{23}a^{23}-\frac{1}{23}a$, $\frac{1}{529}a^{24}-\frac{3}{529}a^{23}-\frac{8}{529}a^{22}-\frac{3}{529}a^{21}+\frac{1}{529}a^{20}+\frac{6}{529}a^{19}-\frac{6}{529}a^{18}+\frac{9}{529}a^{17}+\frac{217}{529}a^{16}+\frac{201}{529}a^{15}-\frac{246}{529}a^{14}-\frac{202}{529}a^{13}+\frac{70}{529}a^{12}+\frac{99}{529}a^{11}+\frac{190}{529}a^{10}+\frac{260}{529}a^{9}+\frac{2}{23}a^{8}+\frac{206}{529}a^{7}-\frac{250}{529}a^{6}-\frac{93}{529}a^{5}-\frac{177}{529}a^{4}-\frac{242}{529}a^{3}+\frac{19}{529}a^{2}+\frac{199}{529}a+\frac{235}{529}$, $\frac{1}{36\!\cdots\!69}a^{25}-\frac{17\!\cdots\!41}{36\!\cdots\!69}a^{24}+\frac{44\!\cdots\!44}{36\!\cdots\!69}a^{23}-\frac{71\!\cdots\!58}{36\!\cdots\!69}a^{22}-\frac{63\!\cdots\!58}{36\!\cdots\!69}a^{21}-\frac{42\!\cdots\!89}{36\!\cdots\!69}a^{20}+\frac{44\!\cdots\!04}{36\!\cdots\!69}a^{19}-\frac{43\!\cdots\!69}{36\!\cdots\!69}a^{18}+\frac{49\!\cdots\!92}{36\!\cdots\!69}a^{17}+\frac{87\!\cdots\!31}{36\!\cdots\!69}a^{16}+\frac{70\!\cdots\!78}{36\!\cdots\!69}a^{15}-\frac{73\!\cdots\!29}{36\!\cdots\!69}a^{14}-\frac{92\!\cdots\!82}{36\!\cdots\!69}a^{13}+\frac{16\!\cdots\!37}{36\!\cdots\!69}a^{12}+\frac{15\!\cdots\!90}{36\!\cdots\!69}a^{11}+\frac{92\!\cdots\!92}{36\!\cdots\!69}a^{10}-\frac{60\!\cdots\!67}{20\!\cdots\!49}a^{9}+\frac{11\!\cdots\!18}{36\!\cdots\!69}a^{8}-\frac{10\!\cdots\!03}{36\!\cdots\!69}a^{7}+\frac{91\!\cdots\!47}{36\!\cdots\!69}a^{6}-\frac{44\!\cdots\!67}{36\!\cdots\!69}a^{5}-\frac{11\!\cdots\!99}{36\!\cdots\!69}a^{4}+\frac{16\!\cdots\!25}{36\!\cdots\!69}a^{3}-\frac{21\!\cdots\!12}{36\!\cdots\!69}a^{2}+\frac{94\!\cdots\!59}{36\!\cdots\!69}a+\frac{20\!\cdots\!64}{84\!\cdots\!21}$

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

Class group and class number

$C_{265}$, which has order $265$ (assuming GRH)

Unit group

Rank:		$12$
Torsion generator:		\( -1 \)  (order $2$)
Fundamental units:		$\frac{36\!\cdots\!75}{36\!\cdots\!69}a^{25}-\frac{14\!\cdots\!83}{36\!\cdots\!69}a^{24}+\frac{63\!\cdots\!50}{36\!\cdots\!69}a^{23}+\frac{17\!\cdots\!03}{36\!\cdots\!69}a^{22}-\frac{41\!\cdots\!23}{36\!\cdots\!69}a^{21}+\frac{24\!\cdots\!09}{36\!\cdots\!69}a^{20}+\frac{31\!\cdots\!56}{36\!\cdots\!69}a^{19}-\frac{12\!\cdots\!81}{15\!\cdots\!03}a^{18}+\frac{33\!\cdots\!11}{36\!\cdots\!69}a^{17}+\frac{24\!\cdots\!90}{36\!\cdots\!69}a^{16}+\frac{12\!\cdots\!25}{36\!\cdots\!69}a^{15}+\frac{20\!\cdots\!17}{36\!\cdots\!69}a^{14}+\frac{89\!\cdots\!89}{36\!\cdots\!69}a^{13}+\frac{16\!\cdots\!96}{36\!\cdots\!69}a^{12}+\frac{68\!\cdots\!57}{36\!\cdots\!69}a^{11}+\frac{13\!\cdots\!50}{36\!\cdots\!69}a^{10}+\frac{28\!\cdots\!35}{20\!\cdots\!49}a^{9}+\frac{18\!\cdots\!44}{36\!\cdots\!69}a^{8}+\frac{10\!\cdots\!88}{36\!\cdots\!69}a^{7}+\frac{53\!\cdots\!86}{36\!\cdots\!69}a^{6}+\frac{18\!\cdots\!53}{36\!\cdots\!69}a^{5}-\frac{68\!\cdots\!31}{36\!\cdots\!69}a^{4}+\frac{29\!\cdots\!29}{36\!\cdots\!69}a^{3}-\frac{42\!\cdots\!95}{36\!\cdots\!69}a^{2}-\frac{72\!\cdots\!36}{36\!\cdots\!69}a+\frac{18\!\cdots\!12}{84\!\cdots\!21}$, $\frac{25\!\cdots\!72}{36\!\cdots\!69}a^{25}-\frac{38\!\cdots\!09}{36\!\cdots\!69}a^{24}+\frac{64\!\cdots\!85}{36\!\cdots\!69}a^{23}+\frac{12\!\cdots\!22}{36\!\cdots\!69}a^{22}-\frac{16\!\cdots\!09}{36\!\cdots\!69}a^{21}+\frac{25\!\cdots\!18}{36\!\cdots\!69}a^{20}+\frac{21\!\cdots\!72}{36\!\cdots\!69}a^{19}-\frac{26\!\cdots\!97}{36\!\cdots\!69}a^{18}+\frac{33\!\cdots\!40}{36\!\cdots\!69}a^{17}+\frac{17\!\cdots\!19}{36\!\cdots\!69}a^{16}-\frac{18\!\cdots\!89}{36\!\cdots\!69}a^{15}+\frac{18\!\cdots\!24}{36\!\cdots\!69}a^{14}+\frac{70\!\cdots\!71}{36\!\cdots\!69}a^{13}-\frac{60\!\cdots\!22}{36\!\cdots\!69}a^{12}+\frac{46\!\cdots\!73}{36\!\cdots\!69}a^{11}+\frac{13\!\cdots\!20}{36\!\cdots\!69}a^{10}-\frac{36\!\cdots\!59}{20\!\cdots\!49}a^{9}+\frac{78\!\cdots\!89}{36\!\cdots\!69}a^{8}+\frac{64\!\cdots\!13}{36\!\cdots\!69}a^{7}+\frac{26\!\cdots\!11}{36\!\cdots\!69}a^{6}+\frac{11\!\cdots\!27}{36\!\cdots\!69}a^{5}-\frac{12\!\cdots\!88}{36\!\cdots\!69}a^{4}+\frac{69\!\cdots\!82}{36\!\cdots\!69}a^{3}+\frac{19\!\cdots\!82}{36\!\cdots\!69}a^{2}-\frac{62\!\cdots\!68}{36\!\cdots\!69}a-\frac{12\!\cdots\!66}{84\!\cdots\!21}$, $\frac{83\!\cdots\!46}{15\!\cdots\!03}a^{25}-\frac{18\!\cdots\!83}{36\!\cdots\!69}a^{24}+\frac{35\!\cdots\!12}{36\!\cdots\!69}a^{23}+\frac{92\!\cdots\!53}{36\!\cdots\!69}a^{22}-\frac{77\!\cdots\!16}{36\!\cdots\!69}a^{21}+\frac{13\!\cdots\!46}{36\!\cdots\!69}a^{20}+\frac{16\!\cdots\!66}{36\!\cdots\!69}a^{19}-\frac{11\!\cdots\!02}{36\!\cdots\!69}a^{18}+\frac{17\!\cdots\!29}{36\!\cdots\!69}a^{17}+\frac{12\!\cdots\!74}{36\!\cdots\!69}a^{16}-\frac{74\!\cdots\!85}{36\!\cdots\!69}a^{15}+\frac{89\!\cdots\!21}{36\!\cdots\!69}a^{14}+\frac{47\!\cdots\!19}{36\!\cdots\!69}a^{13}-\frac{22\!\cdots\!00}{36\!\cdots\!69}a^{12}+\frac{25\!\cdots\!45}{36\!\cdots\!69}a^{11}+\frac{77\!\cdots\!71}{36\!\cdots\!69}a^{10}-\frac{11\!\cdots\!06}{20\!\cdots\!49}a^{9}+\frac{30\!\cdots\!21}{15\!\cdots\!03}a^{8}+\frac{38\!\cdots\!15}{36\!\cdots\!69}a^{7}+\frac{21\!\cdots\!30}{36\!\cdots\!69}a^{6}+\frac{88\!\cdots\!94}{36\!\cdots\!69}a^{5}-\frac{49\!\cdots\!96}{36\!\cdots\!69}a^{4}+\frac{34\!\cdots\!43}{36\!\cdots\!69}a^{3}+\frac{12\!\cdots\!32}{36\!\cdots\!69}a^{2}-\frac{40\!\cdots\!03}{36\!\cdots\!69}a+\frac{75\!\cdots\!08}{84\!\cdots\!21}$, $\frac{26\!\cdots\!88}{36\!\cdots\!69}a^{25}+\frac{56\!\cdots\!43}{36\!\cdots\!69}a^{24}-\frac{28\!\cdots\!76}{36\!\cdots\!69}a^{23}+\frac{17\!\cdots\!88}{36\!\cdots\!69}a^{22}+\frac{27\!\cdots\!57}{36\!\cdots\!69}a^{21}-\frac{10\!\cdots\!32}{36\!\cdots\!69}a^{20}+\frac{35\!\cdots\!69}{36\!\cdots\!69}a^{19}+\frac{49\!\cdots\!38}{36\!\cdots\!69}a^{18}-\frac{11\!\cdots\!94}{36\!\cdots\!69}a^{17}+\frac{24\!\cdots\!60}{36\!\cdots\!69}a^{16}+\frac{39\!\cdots\!02}{36\!\cdots\!69}a^{15}-\frac{35\!\cdots\!07}{36\!\cdots\!69}a^{14}+\frac{28\!\cdots\!96}{36\!\cdots\!69}a^{13}+\frac{14\!\cdots\!18}{36\!\cdots\!69}a^{12}+\frac{16\!\cdots\!16}{36\!\cdots\!69}a^{11}-\frac{29\!\cdots\!81}{36\!\cdots\!69}a^{10}+\frac{11\!\cdots\!30}{20\!\cdots\!49}a^{9}+\frac{11\!\cdots\!40}{36\!\cdots\!69}a^{8}+\frac{68\!\cdots\!04}{36\!\cdots\!69}a^{7}+\frac{64\!\cdots\!70}{36\!\cdots\!69}a^{6}+\frac{59\!\cdots\!65}{36\!\cdots\!69}a^{5}+\frac{21\!\cdots\!83}{36\!\cdots\!69}a^{4}-\frac{83\!\cdots\!58}{36\!\cdots\!69}a^{3}-\frac{10\!\cdots\!27}{36\!\cdots\!69}a^{2}+\frac{88\!\cdots\!01}{36\!\cdots\!69}a+\frac{92\!\cdots\!94}{84\!\cdots\!21}$, $\frac{58\!\cdots\!51}{36\!\cdots\!69}a^{25}-\frac{50\!\cdots\!41}{36\!\cdots\!69}a^{24}+\frac{93\!\cdots\!37}{36\!\cdots\!69}a^{23}+\frac{28\!\cdots\!70}{36\!\cdots\!69}a^{22}-\frac{20\!\cdots\!38}{36\!\cdots\!69}a^{21}+\frac{33\!\cdots\!95}{36\!\cdots\!69}a^{20}+\frac{50\!\cdots\!11}{36\!\cdots\!69}a^{19}-\frac{29\!\cdots\!92}{36\!\cdots\!69}a^{18}+\frac{40\!\cdots\!06}{36\!\cdots\!69}a^{17}+\frac{39\!\cdots\!74}{36\!\cdots\!69}a^{16}-\frac{18\!\cdots\!89}{36\!\cdots\!69}a^{15}+\frac{18\!\cdots\!44}{36\!\cdots\!69}a^{14}+\frac{15\!\cdots\!23}{36\!\cdots\!69}a^{13}-\frac{47\!\cdots\!38}{36\!\cdots\!69}a^{12}+\frac{46\!\cdots\!52}{36\!\cdots\!69}a^{11}+\frac{25\!\cdots\!97}{36\!\cdots\!69}a^{10}-\frac{12\!\cdots\!17}{20\!\cdots\!49}a^{9}+\frac{16\!\cdots\!39}{36\!\cdots\!69}a^{8}+\frac{14\!\cdots\!73}{36\!\cdots\!69}a^{7}+\frac{43\!\cdots\!48}{36\!\cdots\!69}a^{6}+\frac{23\!\cdots\!99}{36\!\cdots\!69}a^{5}-\frac{14\!\cdots\!08}{36\!\cdots\!69}a^{4}+\frac{12\!\cdots\!11}{36\!\cdots\!69}a^{3}-\frac{35\!\cdots\!29}{15\!\cdots\!03}a^{2}-\frac{13\!\cdots\!65}{36\!\cdots\!69}a+\frac{29\!\cdots\!75}{84\!\cdots\!21}$, $\frac{81\!\cdots\!09}{36\!\cdots\!69}a^{25}-\frac{31\!\cdots\!66}{36\!\cdots\!69}a^{24}+\frac{13\!\cdots\!07}{36\!\cdots\!69}a^{23}+\frac{16\!\cdots\!64}{15\!\cdots\!03}a^{22}-\frac{15\!\cdots\!76}{36\!\cdots\!69}a^{21}+\frac{32\!\cdots\!35}{36\!\cdots\!69}a^{20}+\frac{63\!\cdots\!34}{36\!\cdots\!69}a^{19}-\frac{25\!\cdots\!71}{36\!\cdots\!69}a^{18}+\frac{84\!\cdots\!77}{36\!\cdots\!69}a^{17}+\frac{49\!\cdots\!82}{36\!\cdots\!69}a^{16}-\frac{19\!\cdots\!26}{36\!\cdots\!69}a^{15}-\frac{29\!\cdots\!51}{36\!\cdots\!69}a^{14}+\frac{20\!\cdots\!35}{36\!\cdots\!69}a^{13}-\frac{69\!\cdots\!97}{36\!\cdots\!69}a^{12}-\frac{21\!\cdots\!52}{36\!\cdots\!69}a^{11}+\frac{32\!\cdots\!01}{36\!\cdots\!69}a^{10}-\frac{51\!\cdots\!79}{20\!\cdots\!49}a^{9}-\frac{41\!\cdots\!15}{36\!\cdots\!69}a^{8}-\frac{35\!\cdots\!65}{36\!\cdots\!69}a^{7}-\frac{48\!\cdots\!48}{36\!\cdots\!69}a^{6}-\frac{94\!\cdots\!12}{36\!\cdots\!69}a^{5}-\frac{99\!\cdots\!83}{36\!\cdots\!69}a^{4}+\frac{54\!\cdots\!63}{36\!\cdots\!69}a^{3}+\frac{42\!\cdots\!00}{36\!\cdots\!69}a^{2}-\frac{18\!\cdots\!37}{36\!\cdots\!69}a-\frac{59\!\cdots\!80}{84\!\cdots\!21}$, $\frac{62\!\cdots\!78}{36\!\cdots\!69}a^{25}-\frac{77\!\cdots\!77}{36\!\cdots\!69}a^{24}+\frac{82\!\cdots\!92}{36\!\cdots\!69}a^{23}+\frac{30\!\cdots\!69}{36\!\cdots\!69}a^{22}-\frac{34\!\cdots\!96}{36\!\cdots\!69}a^{21}+\frac{26\!\cdots\!83}{36\!\cdots\!69}a^{20}+\frac{53\!\cdots\!02}{36\!\cdots\!69}a^{19}-\frac{55\!\cdots\!17}{36\!\cdots\!69}a^{18}+\frac{27\!\cdots\!32}{36\!\cdots\!69}a^{17}+\frac{42\!\cdots\!61}{36\!\cdots\!69}a^{16}-\frac{41\!\cdots\!70}{36\!\cdots\!69}a^{15}+\frac{79\!\cdots\!12}{36\!\cdots\!69}a^{14}+\frac{16\!\cdots\!94}{36\!\cdots\!69}a^{13}-\frac{14\!\cdots\!10}{36\!\cdots\!69}a^{12}+\frac{13\!\cdots\!13}{36\!\cdots\!69}a^{11}+\frac{25\!\cdots\!82}{36\!\cdots\!69}a^{10}-\frac{12\!\cdots\!20}{20\!\cdots\!49}a^{9}+\frac{16\!\cdots\!75}{36\!\cdots\!69}a^{8}+\frac{56\!\cdots\!62}{36\!\cdots\!69}a^{7}-\frac{18\!\cdots\!70}{36\!\cdots\!69}a^{6}+\frac{32\!\cdots\!78}{36\!\cdots\!69}a^{5}-\frac{28\!\cdots\!86}{36\!\cdots\!69}a^{4}+\frac{15\!\cdots\!82}{36\!\cdots\!69}a^{3}+\frac{66\!\cdots\!87}{36\!\cdots\!69}a^{2}-\frac{14\!\cdots\!14}{36\!\cdots\!69}a+\frac{56\!\cdots\!26}{84\!\cdots\!21}$, $\frac{54\!\cdots\!19}{15\!\cdots\!03}a^{25}-\frac{14\!\cdots\!26}{36\!\cdots\!69}a^{24}+\frac{26\!\cdots\!66}{36\!\cdots\!69}a^{23}+\frac{60\!\cdots\!31}{36\!\cdots\!69}a^{22}-\frac{60\!\cdots\!43}{36\!\cdots\!69}a^{21}+\frac{10\!\cdots\!12}{36\!\cdots\!69}a^{20}+\frac{10\!\cdots\!58}{36\!\cdots\!69}a^{19}-\frac{91\!\cdots\!00}{36\!\cdots\!69}a^{18}+\frac{14\!\cdots\!77}{36\!\cdots\!69}a^{17}+\frac{86\!\cdots\!69}{36\!\cdots\!69}a^{16}-\frac{61\!\cdots\!30}{36\!\cdots\!69}a^{15}+\frac{84\!\cdots\!27}{36\!\cdots\!69}a^{14}+\frac{33\!\cdots\!61}{36\!\cdots\!69}a^{13}-\frac{19\!\cdots\!41}{36\!\cdots\!69}a^{12}+\frac{29\!\cdots\!82}{36\!\cdots\!69}a^{11}+\frac{59\!\cdots\!81}{36\!\cdots\!69}a^{10}-\frac{14\!\cdots\!54}{20\!\cdots\!49}a^{9}+\frac{32\!\cdots\!33}{15\!\cdots\!03}a^{8}+\frac{31\!\cdots\!82}{36\!\cdots\!69}a^{7}-\frac{17\!\cdots\!54}{36\!\cdots\!69}a^{6}+\frac{90\!\cdots\!99}{36\!\cdots\!69}a^{5}-\frac{46\!\cdots\!04}{36\!\cdots\!69}a^{4}+\frac{21\!\cdots\!15}{36\!\cdots\!69}a^{3}+\frac{49\!\cdots\!90}{36\!\cdots\!69}a^{2}-\frac{30\!\cdots\!08}{36\!\cdots\!69}a+\frac{36\!\cdots\!95}{84\!\cdots\!21}$, $\frac{94\!\cdots\!55}{36\!\cdots\!69}a^{25}-\frac{25\!\cdots\!28}{36\!\cdots\!69}a^{24}+\frac{39\!\cdots\!94}{36\!\cdots\!69}a^{23}+\frac{18\!\cdots\!29}{15\!\cdots\!03}a^{22}-\frac{11\!\cdots\!32}{36\!\cdots\!69}a^{21}+\frac{15\!\cdots\!90}{36\!\cdots\!69}a^{20}+\frac{69\!\cdots\!92}{36\!\cdots\!69}a^{19}-\frac{17\!\cdots\!14}{36\!\cdots\!69}a^{18}+\frac{23\!\cdots\!00}{36\!\cdots\!69}a^{17}+\frac{51\!\cdots\!66}{36\!\cdots\!69}a^{16}-\frac{12\!\cdots\!04}{36\!\cdots\!69}a^{15}+\frac{14\!\cdots\!36}{36\!\cdots\!69}a^{14}+\frac{18\!\cdots\!89}{36\!\cdots\!69}a^{13}-\frac{33\!\cdots\!24}{36\!\cdots\!69}a^{12}+\frac{43\!\cdots\!83}{36\!\cdots\!69}a^{11}+\frac{30\!\cdots\!86}{36\!\cdots\!69}a^{10}-\frac{95\!\cdots\!49}{20\!\cdots\!49}a^{9}+\frac{53\!\cdots\!69}{36\!\cdots\!69}a^{8}-\frac{33\!\cdots\!83}{36\!\cdots\!69}a^{7}+\frac{26\!\cdots\!31}{36\!\cdots\!69}a^{6}+\frac{13\!\cdots\!27}{36\!\cdots\!69}a^{5}-\frac{32\!\cdots\!35}{36\!\cdots\!69}a^{4}+\frac{17\!\cdots\!14}{36\!\cdots\!69}a^{3}+\frac{76\!\cdots\!87}{36\!\cdots\!69}a^{2}-\frac{10\!\cdots\!96}{36\!\cdots\!69}a+\frac{83\!\cdots\!34}{84\!\cdots\!21}$, $\frac{86\!\cdots\!06}{36\!\cdots\!69}a^{25}-\frac{15\!\cdots\!60}{36\!\cdots\!69}a^{24}+\frac{14\!\cdots\!54}{36\!\cdots\!69}a^{23}+\frac{42\!\cdots\!98}{36\!\cdots\!69}a^{22}-\frac{68\!\cdots\!52}{36\!\cdots\!69}a^{21}+\frac{54\!\cdots\!80}{36\!\cdots\!69}a^{20}+\frac{74\!\cdots\!69}{36\!\cdots\!69}a^{19}+\frac{98\!\cdots\!97}{36\!\cdots\!69}a^{18}+\frac{75\!\cdots\!54}{36\!\cdots\!69}a^{17}+\frac{59\!\cdots\!32}{36\!\cdots\!69}a^{16}+\frac{16\!\cdots\!06}{36\!\cdots\!69}a^{15}+\frac{44\!\cdots\!84}{36\!\cdots\!69}a^{14}+\frac{22\!\cdots\!07}{36\!\cdots\!69}a^{13}+\frac{94\!\cdots\!63}{36\!\cdots\!69}a^{12}+\frac{14\!\cdots\!84}{36\!\cdots\!69}a^{11}+\frac{36\!\cdots\!92}{36\!\cdots\!69}a^{10}+\frac{12\!\cdots\!57}{20\!\cdots\!49}a^{9}+\frac{39\!\cdots\!33}{36\!\cdots\!69}a^{8}+\frac{33\!\cdots\!32}{36\!\cdots\!69}a^{7}+\frac{25\!\cdots\!31}{36\!\cdots\!69}a^{6}+\frac{39\!\cdots\!58}{36\!\cdots\!69}a^{5}+\frac{32\!\cdots\!64}{36\!\cdots\!69}a^{4}+\frac{76\!\cdots\!69}{36\!\cdots\!69}a^{3}-\frac{13\!\cdots\!37}{36\!\cdots\!69}a^{2}-\frac{17\!\cdots\!59}{36\!\cdots\!69}a-\frac{36\!\cdots\!60}{84\!\cdots\!21}$, $\frac{38\!\cdots\!55}{36\!\cdots\!69}a^{25}-\frac{48\!\cdots\!86}{36\!\cdots\!69}a^{24}+\frac{70\!\cdots\!79}{36\!\cdots\!69}a^{23}+\frac{18\!\cdots\!66}{36\!\cdots\!69}a^{22}-\frac{21\!\cdots\!65}{36\!\cdots\!69}a^{21}+\frac{25\!\cdots\!25}{36\!\cdots\!69}a^{20}+\frac{32\!\cdots\!96}{36\!\cdots\!69}a^{19}-\frac{33\!\cdots\!50}{36\!\cdots\!69}a^{18}+\frac{13\!\cdots\!32}{15\!\cdots\!03}a^{17}+\frac{25\!\cdots\!54}{36\!\cdots\!69}a^{16}-\frac{24\!\cdots\!94}{36\!\cdots\!69}a^{15}+\frac{15\!\cdots\!66}{36\!\cdots\!69}a^{14}+\frac{96\!\cdots\!00}{36\!\cdots\!69}a^{13}-\frac{85\!\cdots\!95}{36\!\cdots\!69}a^{12}+\frac{44\!\cdots\!92}{36\!\cdots\!69}a^{11}+\frac{15\!\cdots\!30}{36\!\cdots\!69}a^{10}-\frac{70\!\cdots\!74}{20\!\cdots\!49}a^{9}+\frac{14\!\cdots\!13}{36\!\cdots\!69}a^{8}+\frac{45\!\cdots\!31}{36\!\cdots\!69}a^{7}-\frac{82\!\cdots\!45}{36\!\cdots\!69}a^{6}+\frac{21\!\cdots\!37}{36\!\cdots\!69}a^{5}-\frac{16\!\cdots\!81}{36\!\cdots\!69}a^{4}+\frac{82\!\cdots\!64}{36\!\cdots\!69}a^{3}+\frac{32\!\cdots\!00}{36\!\cdots\!69}a^{2}-\frac{87\!\cdots\!74}{36\!\cdots\!69}a+\frac{52\!\cdots\!73}{84\!\cdots\!21}$, $\frac{33\!\cdots\!08}{36\!\cdots\!69}a^{25}-\frac{17\!\cdots\!90}{36\!\cdots\!69}a^{24}+\frac{38\!\cdots\!95}{36\!\cdots\!69}a^{23}+\frac{16\!\cdots\!93}{36\!\cdots\!69}a^{22}-\frac{63\!\cdots\!43}{36\!\cdots\!69}a^{21}+\frac{12\!\cdots\!73}{36\!\cdots\!69}a^{20}+\frac{29\!\cdots\!82}{36\!\cdots\!69}a^{19}-\frac{75\!\cdots\!86}{36\!\cdots\!69}a^{18}+\frac{13\!\cdots\!36}{36\!\cdots\!69}a^{17}+\frac{23\!\cdots\!70}{36\!\cdots\!69}a^{16}-\frac{31\!\cdots\!79}{36\!\cdots\!69}a^{15}+\frac{52\!\cdots\!20}{36\!\cdots\!69}a^{14}+\frac{87\!\cdots\!46}{36\!\cdots\!69}a^{13}-\frac{13\!\cdots\!98}{36\!\cdots\!69}a^{12}+\frac{18\!\cdots\!62}{36\!\cdots\!69}a^{11}+\frac{13\!\cdots\!99}{36\!\cdots\!69}a^{10}+\frac{89\!\cdots\!57}{20\!\cdots\!49}a^{9}+\frac{11\!\cdots\!83}{36\!\cdots\!69}a^{8}+\frac{84\!\cdots\!79}{36\!\cdots\!69}a^{7}-\frac{74\!\cdots\!00}{36\!\cdots\!69}a^{6}+\frac{16\!\cdots\!05}{36\!\cdots\!69}a^{5}-\frac{45\!\cdots\!99}{36\!\cdots\!69}a^{4}+\frac{52\!\cdots\!86}{36\!\cdots\!69}a^{3}-\frac{18\!\cdots\!62}{36\!\cdots\!69}a^{2}-\frac{73\!\cdots\!98}{36\!\cdots\!69}a-\frac{47\!\cdots\!48}{84\!\cdots\!21}$ (assuming GRH)
Regulator:		\( 57529828940.82975 \) (assuming GRH)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{13}\cdot 57529828940.82975 \cdot 265}{2\cdot\sqrt{275852727404510985186163978883510976421015681999}}\cr\approx \mathstrut & 0.345230063779195 \end{aligned}\] (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 $C_{26}$

Intermediate fields

\(\Q(\sqrt{-79}) \), 13.13.59091511031674153381441.1

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	${\href{/padicField/2.13.0.1}{13} }^{2}$	$26$	${\href{/padicField/5.13.0.1}{13} }^{2}$	$26$	${\href{/padicField/11.13.0.1}{13} }^{2}$	${\href{/padicField/13.13.0.1}{13} }^{2}$	$26$	${\href{/padicField/19.13.0.1}{13} }^{2}$	${\href{/padicField/23.1.0.1}{1} }^{26}$	$26$	${\href{/padicField/31.13.0.1}{13} }^{2}$	$26$	$26$	$26$	$26$	$26$	$26$

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
\(79\)		Deg $26$	$26$	$1$	$25$