Number field 26.0.1040129483587052483222994327325975449420803964347689607283.1

• Label: 26.0.104...283.1
• Degree: $26$
• Signature: $[0, 13]$
• Discriminant: $-1.040\times 10^{57}$
• Root discriminant: \(155.94\)
• Ramified primes: $3,131$
• Class number: $1526201$ (GRH)
• Class group: [1526201] (GRH)
• Galois group: $C_{26}$ (as 26T1)

Normalized defining polynomial

x^26 - x^25 + 61*x^24 + 6*x^23 + 2428*x^22 + 745*x^21 + 53416*x^20 + 34671*x^19 + 835048*x^18 + 606289*x^17 + 7656274*x^16 + 7911027*x^15 + 50259437*x^14 + 53321755*x^13 + 222049643*x^12 + 235929538*x^11 + 711364146*x^10 + 658848433*x^9 + 1489046291*x^8 + 1147597018*x^7 + 2167670685*x^6 + 1324856027*x^5 + 1912484318*x^4 + 753653983*x^3 + 998748064*x^2 + 333252757*x + 245454889

Invariants

Degree:		$26$
Signature:		$[0, 13]$
Discriminant:		\(-1040129483587052483222994327325975449420803964347689607283\) \(\medspace = -\,3^{13}\cdot 131^{24}\)
Root discriminant:		\(155.94\)
Galois root discriminant:		$3^{1/2}131^{12/13}\approx 155.94264647177133$
Ramified primes:		\(3\), \(131\)
Discriminant root field:		\(\Q(\sqrt{-3}) \)
$\card{ \Gal(K/\Q) }$:		$26$
This field is Galois and abelian over $\Q$.
Conductor:		\(393=3\cdot 131\)
Dirichlet character group:		$\lbrace$$\chi_{393}(1,·)$, $\chi_{393}(322,·)$, $\chi_{393}(325,·)$, $\chi_{393}(211,·)$, $\chi_{393}(263,·)$, $\chi_{393}(194,·)$, $\chi_{393}(80,·)$, $\chi_{393}(193,·)$, $\chi_{393}(215,·)$, $\chi_{393}(346,·)$, $\chi_{393}(112,·)$, $\chi_{393}(230,·)$, $\chi_{393}(361,·)$, $\chi_{393}(170,·)$, $\chi_{393}(107,·)$, $\chi_{393}(301,·)$, $\chi_{393}(238,·)$, $\chi_{393}(176,·)$, $\chi_{393}(113,·)$, $\chi_{393}(307,·)$, $\chi_{393}(52,·)$, $\chi_{393}(374,·)$, $\chi_{393}(244,·)$, $\chi_{393}(314,·)$, $\chi_{393}(62,·)$, $\chi_{393}(191,·)$$\rbrace$
This is a CM field.
Reflex fields:		unavailable$^{4096}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $\frac{1}{4453}a^{21}-\frac{196}{4453}a^{20}+\frac{1765}{4453}a^{19}+\frac{1800}{4453}a^{18}-\frac{711}{4453}a^{17}+\frac{1638}{4453}a^{16}+\frac{790}{4453}a^{15}-\frac{1552}{4453}a^{14}+\frac{1980}{4453}a^{13}+\frac{246}{4453}a^{12}-\frac{12}{73}a^{11}+\frac{856}{4453}a^{10}+\frac{1682}{4453}a^{9}-\frac{1339}{4453}a^{8}+\frac{15}{4453}a^{7}-\frac{1475}{4453}a^{6}+\frac{675}{4453}a^{5}-\frac{841}{4453}a^{4}+\frac{847}{4453}a^{3}+\frac{1886}{4453}a^{2}+\frac{1326}{4453}a-\frac{1559}{4453}$, $\frac{1}{236009}a^{22}+\frac{18}{236009}a^{21}+\frac{35522}{236009}a^{20}-\frac{21260}{236009}a^{19}+\frac{77232}{236009}a^{18}+\frac{49869}{236009}a^{17}-\frac{107337}{236009}a^{16}+\frac{29465}{236009}a^{15}+\frac{26092}{236009}a^{14}-\frac{70317}{236009}a^{13}-\frac{54960}{236009}a^{12}+\frac{98029}{236009}a^{11}-\frac{37784}{236009}a^{10}+\frac{91429}{236009}a^{9}+\frac{83068}{236009}a^{8}+\frac{95248}{236009}a^{7}-\frac{52248}{236009}a^{6}-\frac{70135}{236009}a^{5}-\frac{103426}{236009}a^{4}-\frac{110754}{236009}a^{3}+\frac{21972}{236009}a^{2}-\frac{29505}{236009}a+\frac{22614}{236009}$, $\frac{1}{236009}a^{23}+\frac{6}{236009}a^{21}+\frac{100742}{236009}a^{20}-\frac{55619}{236009}a^{19}-\frac{19441}{236009}a^{18}-\frac{56385}{236009}a^{17}+\frac{15159}{236009}a^{16}+\frac{15122}{236009}a^{15}+\frac{31950}{236009}a^{14}-\frac{26804}{236009}a^{13}-\frac{17635}{236009}a^{12}-\frac{1880}{3869}a^{11}-\frac{87695}{236009}a^{10}-\frac{101285}{236009}a^{9}-\frac{63634}{236009}a^{8}+\frac{65498}{236009}a^{7}-\frac{87487}{236009}a^{6}+\frac{1156}{4453}a^{5}-\frac{41811}{236009}a^{4}+\frac{56982}{236009}a^{3}-\frac{6566}{236009}a^{2}-\frac{1221}{3233}a-\frac{60803}{236009}$, $\frac{1}{13\!\cdots\!51}a^{24}-\frac{3747429131818}{13\!\cdots\!51}a^{23}+\frac{275060234507}{15\!\cdots\!59}a^{22}+\frac{6900471573866}{18\!\cdots\!87}a^{21}-\frac{28\!\cdots\!47}{13\!\cdots\!51}a^{20}+\frac{51\!\cdots\!74}{13\!\cdots\!51}a^{19}+\frac{90\!\cdots\!78}{13\!\cdots\!51}a^{18}+\frac{58\!\cdots\!50}{13\!\cdots\!51}a^{17}+\frac{26\!\cdots\!22}{13\!\cdots\!51}a^{16}+\frac{68\!\cdots\!71}{13\!\cdots\!51}a^{15}+\frac{65\!\cdots\!17}{13\!\cdots\!51}a^{14}-\frac{45\!\cdots\!54}{13\!\cdots\!51}a^{13}-\frac{55\!\cdots\!64}{13\!\cdots\!51}a^{12}-\frac{16\!\cdots\!44}{13\!\cdots\!51}a^{11}-\frac{41\!\cdots\!55}{13\!\cdots\!51}a^{10}+\frac{20\!\cdots\!13}{13\!\cdots\!51}a^{9}+\frac{66\!\cdots\!73}{13\!\cdots\!51}a^{8}-\frac{91848836420757}{486155835752357}a^{7}-\frac{23\!\cdots\!31}{13\!\cdots\!51}a^{6}-\frac{52\!\cdots\!77}{13\!\cdots\!51}a^{5}-\frac{28\!\cdots\!08}{13\!\cdots\!51}a^{4}-\frac{42\!\cdots\!17}{13\!\cdots\!51}a^{3}-\frac{62\!\cdots\!10}{13\!\cdots\!51}a^{2}-\frac{18\!\cdots\!88}{13\!\cdots\!51}a+\frac{27\!\cdots\!56}{13\!\cdots\!51}$, $\frac{1}{68\!\cdots\!61}a^{25}-\frac{22\!\cdots\!22}{68\!\cdots\!61}a^{24}+\frac{57\!\cdots\!24}{68\!\cdots\!61}a^{23}-\frac{15\!\cdots\!79}{68\!\cdots\!61}a^{22}-\frac{39\!\cdots\!72}{68\!\cdots\!61}a^{21}-\frac{16\!\cdots\!42}{68\!\cdots\!61}a^{20}-\frac{11\!\cdots\!99}{68\!\cdots\!61}a^{19}-\frac{29\!\cdots\!83}{68\!\cdots\!61}a^{18}-\frac{15\!\cdots\!46}{68\!\cdots\!61}a^{17}+\frac{25\!\cdots\!71}{68\!\cdots\!61}a^{16}-\frac{19\!\cdots\!28}{68\!\cdots\!61}a^{15}-\frac{40\!\cdots\!35}{68\!\cdots\!61}a^{14}-\frac{77\!\cdots\!34}{68\!\cdots\!61}a^{13}-\frac{24\!\cdots\!91}{68\!\cdots\!61}a^{12}+\frac{19\!\cdots\!03}{68\!\cdots\!61}a^{11}-\frac{20\!\cdots\!74}{68\!\cdots\!61}a^{10}+\frac{87\!\cdots\!39}{68\!\cdots\!61}a^{9}-\frac{14\!\cdots\!01}{68\!\cdots\!61}a^{8}-\frac{65\!\cdots\!69}{68\!\cdots\!61}a^{7}+\frac{22\!\cdots\!30}{68\!\cdots\!61}a^{6}+\frac{20\!\cdots\!98}{68\!\cdots\!61}a^{5}-\frac{37\!\cdots\!94}{76\!\cdots\!49}a^{4}+\frac{20\!\cdots\!45}{68\!\cdots\!61}a^{3}-\frac{24\!\cdots\!93}{68\!\cdots\!61}a^{2}+\frac{28\!\cdots\!56}{68\!\cdots\!61}a+\frac{58\!\cdots\!04}{43\!\cdots\!83}$

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

Class group and class number

$C_{1526201}$, which has order $1526201$ (assuming GRH)

Unit group

Rank:		$12$
Torsion generator:		\( -\frac{329455162218607639739474271052372157327819327858696196457822372925527031}{102450172258413304951922364426581868481052708020451234373124097187654994475640829} a^{25} + \frac{382449285773061352463202330882995926698545269843782583663589300161797655}{102450172258413304951922364426581868481052708020451234373124097187654994475640829} a^{24} - \frac{19983683173891931514585932513630961718442961966576507509969433762436379456}{102450172258413304951922364426581868481052708020451234373124097187654994475640829} a^{23} + \frac{1165589220255354624096719616327954128879680767135675425801989544434727262}{102450172258413304951922364426581868481052708020451234373124097187654994475640829} a^{22} - \frac{789706791481582489415186542262774512693506680462926882306558212455972698286}{102450172258413304951922364426581868481052708020451234373124097187654994475640829} a^{21} - \frac{111130524336663289154019221199491539181803414385385831347841487128188179419}{102450172258413304951922364426581868481052708020451234373124097187654994475640829} a^{20} - \frac{17164617811514879173062612363501643693664044544466405840600079905064341630782}{102450172258413304951922364426581868481052708020451234373124097187654994475640829} a^{19} - \frac{8293242361556784603078760535909340166533007300370878053534082644817517067331}{102450172258413304951922364426581868481052708020451234373124097187654994475640829} a^{18} - \frac{264731642166341478063313211428140506319787881532295929419868501942888087570463}{102450172258413304951922364426581868481052708020451234373124097187654994475640829} a^{17} - \frac{146117688808442953374305561621791319039952308885812836168817518586509232242817}{102450172258413304951922364426581868481052708020451234373124097187654994475640829} a^{16} - \frac{2357344570311560407208212280110391701122999796421500163914884138364727283470060}{102450172258413304951922364426581868481052708020451234373124097187654994475640829} a^{15} - \frac{2048621336164522232391846032828886421377669210973551287407675503279044786635435}{102450172258413304951922364426581868481052708020451234373124097187654994475640829} a^{14} - \frac{14953225719560610482689392540751167214551206299215415566267820840249955322238547}{102450172258413304951922364426581868481052708020451234373124097187654994475640829} a^{13} - \frac{13199576623317117881518301943862238713561150199987085583277333506146560691852509}{102450172258413304951922364426581868481052708020451234373124097187654994475640829} a^{12} - \frac{62569136242436531205558597918618384898422325789182303432864959454153718918132351}{102450172258413304951922364426581868481052708020451234373124097187654994475640829} a^{11} - \frac{55278324283455900389816154739547882696947081801392358192013700611560220251340986}{102450172258413304951922364426581868481052708020451234373124097187654994475640829} a^{10} - \frac{188883231328563928240803940515919241680709000200648261795238142016401535877554456}{102450172258413304951922364426581868481052708020451234373124097187654994475640829} a^{9} - \frac{135575882010519618560984161506203462265935583971602061940371836934535266992865286}{102450172258413304951922364426581868481052708020451234373124097187654994475640829} a^{8} - \frac{353689201463328621977640836742151937080699613326922236440982519948358945493045567}{102450172258413304951922364426581868481052708020451234373124097187654994475640829} a^{7} - \frac{184154079468388344060491509726956554255348513236845838484147596851913195582986500}{102450172258413304951922364426581868481052708020451234373124097187654994475640829} a^{6} - \frac{462717645718179826089245748752953207559493593690151246774502142557073199958391887}{102450172258413304951922364426581868481052708020451234373124097187654994475640829} a^{5} - \frac{143408955154797596318343523597246568501916067770696204101598586294928690227442695}{102450172258413304951922364426581868481052708020451234373124097187654994475640829} a^{4} - \frac{310801080868168250222301057390399736570630474219810512857724677341784569046128288}{102450172258413304951922364426581868481052708020451234373124097187654994475640829} a^{3} + \frac{74586161094996602381070130838344093059409782503408382929567624990616373544514465}{102450172258413304951922364426581868481052708020451234373124097187654994475640829} a^{2} - \frac{129814807172290856042757032532144446524983494881260299501248507615593557319424538}{102450172258413304951922364426581868481052708020451234373124097187654994475640829} a + \frac{5028970955693085496145270277585125484195767844010597163009494336588022449286}{6539233564716493582174147215585745099958684369723063405446103094890853033487} \)  (order $6$)
Fundamental units:		$\frac{22\!\cdots\!67}{43\!\cdots\!83}a^{25}-\frac{59\!\cdots\!55}{43\!\cdots\!83}a^{24}+\frac{14\!\cdots\!12}{43\!\cdots\!83}a^{23}-\frac{19\!\cdots\!10}{43\!\cdots\!83}a^{22}+\frac{10\!\cdots\!43}{82\!\cdots\!11}a^{21}-\frac{65\!\cdots\!80}{43\!\cdots\!83}a^{20}+\frac{11\!\cdots\!69}{43\!\cdots\!83}a^{19}-\frac{93\!\cdots\!93}{43\!\cdots\!83}a^{18}+\frac{16\!\cdots\!50}{43\!\cdots\!83}a^{17}-\frac{12\!\cdots\!50}{43\!\cdots\!83}a^{16}+\frac{13\!\cdots\!38}{43\!\cdots\!83}a^{15}-\frac{38\!\cdots\!48}{43\!\cdots\!83}a^{14}+\frac{70\!\cdots\!60}{43\!\cdots\!83}a^{13}-\frac{16\!\cdots\!03}{43\!\cdots\!83}a^{12}+\frac{23\!\cdots\!72}{43\!\cdots\!83}a^{11}-\frac{29\!\cdots\!11}{43\!\cdots\!83}a^{10}+\frac{49\!\cdots\!16}{43\!\cdots\!83}a^{9}-\frac{22\!\cdots\!00}{43\!\cdots\!83}a^{8}+\frac{49\!\cdots\!42}{43\!\cdots\!83}a^{7}-\frac{50\!\cdots\!49}{43\!\cdots\!83}a^{6}+\frac{14\!\cdots\!72}{43\!\cdots\!83}a^{5}-\frac{92\!\cdots\!82}{43\!\cdots\!83}a^{4}-\frac{57\!\cdots\!20}{43\!\cdots\!83}a^{3}-\frac{56\!\cdots\!41}{43\!\cdots\!83}a^{2}-\frac{12\!\cdots\!83}{43\!\cdots\!83}a-\frac{34\!\cdots\!73}{43\!\cdots\!83}$, $\frac{74\!\cdots\!14}{43\!\cdots\!83}a^{25}-\frac{18\!\cdots\!20}{43\!\cdots\!83}a^{24}+\frac{46\!\cdots\!87}{43\!\cdots\!83}a^{23}-\frac{57\!\cdots\!76}{43\!\cdots\!83}a^{22}+\frac{17\!\cdots\!02}{43\!\cdots\!83}a^{21}-\frac{18\!\cdots\!75}{43\!\cdots\!83}a^{20}+\frac{38\!\cdots\!16}{43\!\cdots\!83}a^{19}-\frac{23\!\cdots\!58}{43\!\cdots\!83}a^{18}+\frac{56\!\cdots\!17}{43\!\cdots\!83}a^{17}-\frac{28\!\cdots\!56}{43\!\cdots\!83}a^{16}+\frac{47\!\cdots\!86}{43\!\cdots\!83}a^{15}+\frac{14\!\cdots\!58}{43\!\cdots\!83}a^{14}+\frac{25\!\cdots\!96}{43\!\cdots\!83}a^{13}+\frac{47\!\cdots\!08}{43\!\cdots\!83}a^{12}+\frac{95\!\cdots\!53}{43\!\cdots\!83}a^{11}+\frac{38\!\cdots\!75}{43\!\cdots\!83}a^{10}+\frac{24\!\cdots\!51}{43\!\cdots\!83}a^{9}+\frac{94\!\cdots\!43}{43\!\cdots\!83}a^{8}+\frac{39\!\cdots\!72}{43\!\cdots\!83}a^{7}+\frac{17\!\cdots\!83}{43\!\cdots\!83}a^{6}+\frac{43\!\cdots\!72}{43\!\cdots\!83}a^{5}+\frac{14\!\cdots\!08}{43\!\cdots\!83}a^{4}+\frac{23\!\cdots\!96}{43\!\cdots\!83}a^{3}+\frac{97\!\cdots\!51}{43\!\cdots\!83}a^{2}+\frac{79\!\cdots\!98}{43\!\cdots\!83}a+\frac{16\!\cdots\!93}{43\!\cdots\!83}$, $\frac{52\!\cdots\!25}{43\!\cdots\!83}a^{25}-\frac{14\!\cdots\!23}{43\!\cdots\!83}a^{24}+\frac{32\!\cdots\!70}{43\!\cdots\!83}a^{23}-\frac{49\!\cdots\!49}{43\!\cdots\!83}a^{22}+\frac{12\!\cdots\!92}{43\!\cdots\!83}a^{21}-\frac{16\!\cdots\!47}{43\!\cdots\!83}a^{20}+\frac{26\!\cdots\!72}{43\!\cdots\!83}a^{19}-\frac{23\!\cdots\!86}{43\!\cdots\!83}a^{18}+\frac{39\!\cdots\!61}{43\!\cdots\!83}a^{17}-\frac{30\!\cdots\!03}{43\!\cdots\!83}a^{16}+\frac{32\!\cdots\!75}{43\!\cdots\!83}a^{15}-\frac{82\!\cdots\!24}{43\!\cdots\!83}a^{14}+\frac{17\!\cdots\!52}{43\!\cdots\!83}a^{13}-\frac{12\!\cdots\!37}{43\!\cdots\!83}a^{12}+\frac{61\!\cdots\!76}{43\!\cdots\!83}a^{11}+\frac{98\!\cdots\!02}{43\!\cdots\!83}a^{10}+\frac{14\!\cdots\!13}{43\!\cdots\!83}a^{9}+\frac{20\!\cdots\!35}{43\!\cdots\!83}a^{8}+\frac{23\!\cdots\!49}{43\!\cdots\!83}a^{7}+\frac{90\!\cdots\!59}{82\!\cdots\!11}a^{6}+\frac{43\!\cdots\!69}{82\!\cdots\!11}a^{5}+\frac{55\!\cdots\!77}{43\!\cdots\!83}a^{4}+\frac{12\!\cdots\!33}{43\!\cdots\!83}a^{3}+\frac{11\!\cdots\!48}{43\!\cdots\!83}a^{2}+\frac{33\!\cdots\!95}{43\!\cdots\!83}a-\frac{89\!\cdots\!59}{43\!\cdots\!83}$, $\frac{20\!\cdots\!16}{43\!\cdots\!83}a^{25}+\frac{87\!\cdots\!82}{43\!\cdots\!83}a^{24}-\frac{42\!\cdots\!03}{43\!\cdots\!83}a^{23}+\frac{53\!\cdots\!98}{43\!\cdots\!83}a^{22}+\frac{37\!\cdots\!81}{43\!\cdots\!83}a^{21}+\frac{20\!\cdots\!72}{43\!\cdots\!83}a^{20}+\frac{10\!\cdots\!67}{43\!\cdots\!83}a^{19}+\frac{43\!\cdots\!03}{43\!\cdots\!83}a^{18}+\frac{35\!\cdots\!67}{43\!\cdots\!83}a^{17}+\frac{63\!\cdots\!39}{43\!\cdots\!83}a^{16}+\frac{50\!\cdots\!23}{43\!\cdots\!83}a^{15}+\frac{51\!\cdots\!46}{43\!\cdots\!83}a^{14}+\frac{69\!\cdots\!01}{43\!\cdots\!83}a^{13}+\frac{29\!\cdots\!71}{43\!\cdots\!83}a^{12}+\frac{40\!\cdots\!37}{43\!\cdots\!83}a^{11}+\frac{11\!\cdots\!89}{43\!\cdots\!83}a^{10}+\frac{16\!\cdots\!08}{43\!\cdots\!83}a^{9}+\frac{31\!\cdots\!86}{43\!\cdots\!83}a^{8}+\frac{36\!\cdots\!82}{43\!\cdots\!83}a^{7}+\frac{56\!\cdots\!19}{43\!\cdots\!83}a^{6}+\frac{56\!\cdots\!24}{43\!\cdots\!83}a^{5}+\frac{68\!\cdots\!47}{43\!\cdots\!83}a^{4}+\frac{46\!\cdots\!29}{43\!\cdots\!83}a^{3}+\frac{41\!\cdots\!00}{43\!\cdots\!83}a^{2}+\frac{16\!\cdots\!01}{43\!\cdots\!83}a+\frac{18\!\cdots\!11}{43\!\cdots\!83}$, $\frac{12\!\cdots\!34}{82\!\cdots\!11}a^{25}-\frac{14\!\cdots\!26}{43\!\cdots\!83}a^{24}+\frac{39\!\cdots\!00}{43\!\cdots\!83}a^{23}-\frac{41\!\cdots\!90}{43\!\cdots\!83}a^{22}+\frac{15\!\cdots\!28}{43\!\cdots\!83}a^{21}-\frac{12\!\cdots\!12}{43\!\cdots\!83}a^{20}+\frac{32\!\cdots\!14}{43\!\cdots\!83}a^{19}-\frac{14\!\cdots\!37}{43\!\cdots\!83}a^{18}+\frac{48\!\cdots\!98}{43\!\cdots\!83}a^{17}-\frac{15\!\cdots\!44}{43\!\cdots\!83}a^{16}+\frac{41\!\cdots\!68}{43\!\cdots\!83}a^{15}+\frac{66\!\cdots\!92}{43\!\cdots\!83}a^{14}+\frac{22\!\cdots\!98}{43\!\cdots\!83}a^{13}+\frac{72\!\cdots\!82}{43\!\cdots\!83}a^{12}+\frac{85\!\cdots\!32}{43\!\cdots\!83}a^{11}+\frac{43\!\cdots\!68}{43\!\cdots\!83}a^{10}+\frac{22\!\cdots\!25}{43\!\cdots\!83}a^{9}+\frac{10\!\cdots\!28}{43\!\cdots\!83}a^{8}+\frac{35\!\cdots\!34}{43\!\cdots\!83}a^{7}+\frac{17\!\cdots\!09}{43\!\cdots\!83}a^{6}+\frac{39\!\cdots\!42}{43\!\cdots\!83}a^{5}+\frac{14\!\cdots\!72}{43\!\cdots\!83}a^{4}+\frac{20\!\cdots\!22}{43\!\cdots\!83}a^{3}+\frac{96\!\cdots\!30}{43\!\cdots\!83}a^{2}+\frac{72\!\cdots\!36}{43\!\cdots\!83}a+\frac{26\!\cdots\!83}{43\!\cdots\!83}$, $\frac{11\!\cdots\!47}{43\!\cdots\!83}a^{25}-\frac{29\!\cdots\!63}{43\!\cdots\!83}a^{24}+\frac{68\!\cdots\!80}{43\!\cdots\!83}a^{23}+\frac{59\!\cdots\!32}{43\!\cdots\!83}a^{22}+\frac{27\!\cdots\!16}{43\!\cdots\!83}a^{21}+\frac{28\!\cdots\!26}{43\!\cdots\!83}a^{20}+\frac{60\!\cdots\!47}{43\!\cdots\!83}a^{19}+\frac{83\!\cdots\!44}{43\!\cdots\!83}a^{18}+\frac{96\!\cdots\!88}{43\!\cdots\!83}a^{17}+\frac{13\!\cdots\!96}{43\!\cdots\!83}a^{16}+\frac{88\!\cdots\!59}{43\!\cdots\!83}a^{15}+\frac{14\!\cdots\!56}{43\!\cdots\!83}a^{14}+\frac{60\!\cdots\!16}{43\!\cdots\!83}a^{13}+\frac{96\!\cdots\!21}{43\!\cdots\!83}a^{12}+\frac{51\!\cdots\!24}{82\!\cdots\!11}a^{11}+\frac{41\!\cdots\!75}{43\!\cdots\!83}a^{10}+\frac{88\!\cdots\!98}{43\!\cdots\!83}a^{9}+\frac{11\!\cdots\!38}{43\!\cdots\!83}a^{8}+\frac{18\!\cdots\!32}{43\!\cdots\!83}a^{7}+\frac{20\!\cdots\!97}{43\!\cdots\!83}a^{6}+\frac{25\!\cdots\!18}{43\!\cdots\!83}a^{5}+\frac{24\!\cdots\!60}{43\!\cdots\!83}a^{4}+\frac{19\!\cdots\!62}{43\!\cdots\!83}a^{3}+\frac{15\!\cdots\!93}{43\!\cdots\!83}a^{2}+\frac{69\!\cdots\!93}{43\!\cdots\!83}a+\frac{87\!\cdots\!67}{43\!\cdots\!83}$, $\frac{46\!\cdots\!13}{43\!\cdots\!83}a^{25}-\frac{59\!\cdots\!41}{43\!\cdots\!83}a^{24}+\frac{28\!\cdots\!27}{43\!\cdots\!83}a^{23}-\frac{39\!\cdots\!67}{43\!\cdots\!83}a^{22}+\frac{10\!\cdots\!23}{43\!\cdots\!83}a^{21}+\frac{71\!\cdots\!12}{43\!\cdots\!83}a^{20}+\frac{23\!\cdots\!10}{43\!\cdots\!83}a^{19}+\frac{10\!\cdots\!49}{43\!\cdots\!83}a^{18}+\frac{36\!\cdots\!38}{43\!\cdots\!83}a^{17}+\frac{19\!\cdots\!01}{43\!\cdots\!83}a^{16}+\frac{31\!\cdots\!18}{43\!\cdots\!83}a^{15}+\frac{29\!\cdots\!70}{43\!\cdots\!83}a^{14}+\frac{19\!\cdots\!18}{43\!\cdots\!83}a^{13}+\frac{19\!\cdots\!43}{43\!\cdots\!83}a^{12}+\frac{79\!\cdots\!47}{43\!\cdots\!83}a^{11}+\frac{84\!\cdots\!75}{43\!\cdots\!83}a^{10}+\frac{23\!\cdots\!12}{43\!\cdots\!83}a^{9}+\frac{22\!\cdots\!44}{43\!\cdots\!83}a^{8}+\frac{42\!\cdots\!71}{43\!\cdots\!83}a^{7}+\frac{38\!\cdots\!49}{43\!\cdots\!83}a^{6}+\frac{53\!\cdots\!03}{43\!\cdots\!83}a^{5}+\frac{42\!\cdots\!13}{43\!\cdots\!83}a^{4}+\frac{32\!\cdots\!01}{43\!\cdots\!83}a^{3}+\frac{26\!\cdots\!07}{43\!\cdots\!83}a^{2}+\frac{13\!\cdots\!89}{43\!\cdots\!83}a+\frac{10\!\cdots\!89}{43\!\cdots\!83}$, $\frac{50\!\cdots\!03}{43\!\cdots\!83}a^{25}-\frac{21\!\cdots\!30}{43\!\cdots\!83}a^{24}+\frac{30\!\cdots\!98}{43\!\cdots\!83}a^{23}+\frac{32\!\cdots\!04}{43\!\cdots\!83}a^{22}+\frac{12\!\cdots\!99}{43\!\cdots\!83}a^{21}+\frac{15\!\cdots\!97}{43\!\cdots\!83}a^{20}+\frac{27\!\cdots\!64}{43\!\cdots\!83}a^{19}+\frac{44\!\cdots\!68}{43\!\cdots\!83}a^{18}+\frac{44\!\cdots\!44}{43\!\cdots\!83}a^{17}+\frac{72\!\cdots\!63}{43\!\cdots\!83}a^{16}+\frac{42\!\cdots\!81}{43\!\cdots\!83}a^{15}+\frac{79\!\cdots\!12}{43\!\cdots\!83}a^{14}+\frac{29\!\cdots\!63}{43\!\cdots\!83}a^{13}+\frac{53\!\cdots\!83}{43\!\cdots\!83}a^{12}+\frac{14\!\cdots\!92}{43\!\cdots\!83}a^{11}+\frac{23\!\cdots\!69}{43\!\cdots\!83}a^{10}+\frac{48\!\cdots\!39}{43\!\cdots\!83}a^{9}+\frac{70\!\cdots\!25}{43\!\cdots\!83}a^{8}+\frac{10\!\cdots\!24}{43\!\cdots\!83}a^{7}+\frac{13\!\cdots\!01}{43\!\cdots\!83}a^{6}+\frac{15\!\cdots\!56}{43\!\cdots\!83}a^{5}+\frac{16\!\cdots\!55}{43\!\cdots\!83}a^{4}+\frac{13\!\cdots\!60}{43\!\cdots\!83}a^{3}+\frac{10\!\cdots\!88}{43\!\cdots\!83}a^{2}+\frac{44\!\cdots\!88}{43\!\cdots\!83}a+\frac{20\!\cdots\!82}{43\!\cdots\!83}$, $\frac{72\!\cdots\!04}{48\!\cdots\!47}a^{25}-\frac{18\!\cdots\!56}{48\!\cdots\!47}a^{24}+\frac{45\!\cdots\!97}{48\!\cdots\!47}a^{23}-\frac{59\!\cdots\!54}{48\!\cdots\!47}a^{22}+\frac{17\!\cdots\!40}{48\!\cdots\!47}a^{21}-\frac{19\!\cdots\!90}{48\!\cdots\!47}a^{20}+\frac{37\!\cdots\!93}{48\!\cdots\!47}a^{19}-\frac{25\!\cdots\!45}{48\!\cdots\!47}a^{18}+\frac{55\!\cdots\!35}{48\!\cdots\!47}a^{17}-\frac{31\!\cdots\!75}{48\!\cdots\!47}a^{16}+\frac{46\!\cdots\!19}{48\!\cdots\!47}a^{15}-\frac{23\!\cdots\!26}{48\!\cdots\!47}a^{14}+\frac{25\!\cdots\!23}{48\!\cdots\!47}a^{13}+\frac{34\!\cdots\!03}{48\!\cdots\!47}a^{12}+\frac{93\!\cdots\!01}{48\!\cdots\!47}a^{11}+\frac{33\!\cdots\!05}{48\!\cdots\!47}a^{10}+\frac{23\!\cdots\!89}{48\!\cdots\!47}a^{9}+\frac{83\!\cdots\!08}{48\!\cdots\!47}a^{8}+\frac{38\!\cdots\!50}{48\!\cdots\!47}a^{7}+\frac{16\!\cdots\!60}{48\!\cdots\!47}a^{6}+\frac{41\!\cdots\!80}{48\!\cdots\!47}a^{5}+\frac{12\!\cdots\!05}{48\!\cdots\!47}a^{4}+\frac{22\!\cdots\!61}{48\!\cdots\!47}a^{3}+\frac{88\!\cdots\!86}{48\!\cdots\!47}a^{2}+\frac{75\!\cdots\!27}{48\!\cdots\!47}a+\frac{13\!\cdots\!96}{48\!\cdots\!47}$, $\frac{20\!\cdots\!45}{43\!\cdots\!83}a^{25}-\frac{60\!\cdots\!09}{43\!\cdots\!83}a^{24}+\frac{12\!\cdots\!46}{43\!\cdots\!83}a^{23}-\frac{22\!\cdots\!95}{43\!\cdots\!83}a^{22}+\frac{49\!\cdots\!68}{43\!\cdots\!83}a^{21}-\frac{74\!\cdots\!07}{43\!\cdots\!83}a^{20}+\frac{10\!\cdots\!84}{43\!\cdots\!83}a^{19}-\frac{11\!\cdots\!60}{43\!\cdots\!83}a^{18}+\frac{15\!\cdots\!19}{43\!\cdots\!83}a^{17}-\frac{14\!\cdots\!07}{43\!\cdots\!83}a^{16}+\frac{12\!\cdots\!51}{43\!\cdots\!83}a^{15}-\frac{49\!\cdots\!68}{43\!\cdots\!83}a^{14}+\frac{66\!\cdots\!10}{43\!\cdots\!83}a^{13}-\frac{10\!\cdots\!67}{43\!\cdots\!83}a^{12}+\frac{24\!\cdots\!68}{43\!\cdots\!83}a^{11}+\frac{27\!\cdots\!96}{43\!\cdots\!83}a^{10}+\frac{58\!\cdots\!95}{43\!\cdots\!83}a^{9}+\frac{86\!\cdots\!71}{43\!\cdots\!83}a^{8}+\frac{97\!\cdots\!19}{43\!\cdots\!83}a^{7}+\frac{24\!\cdots\!52}{43\!\cdots\!83}a^{6}+\frac{10\!\cdots\!87}{43\!\cdots\!83}a^{5}+\frac{14\!\cdots\!57}{43\!\cdots\!83}a^{4}+\frac{72\!\cdots\!82}{43\!\cdots\!83}a^{3}+\frac{13\!\cdots\!44}{43\!\cdots\!83}a^{2}+\frac{17\!\cdots\!25}{43\!\cdots\!83}a-\frac{61\!\cdots\!35}{43\!\cdots\!83}$, $\frac{22\!\cdots\!87}{43\!\cdots\!83}a^{25}-\frac{44\!\cdots\!70}{43\!\cdots\!83}a^{24}+\frac{13\!\cdots\!98}{43\!\cdots\!83}a^{23}-\frac{10\!\cdots\!64}{43\!\cdots\!83}a^{22}+\frac{54\!\cdots\!96}{43\!\cdots\!83}a^{21}-\frac{30\!\cdots\!40}{43\!\cdots\!83}a^{20}+\frac{11\!\cdots\!23}{43\!\cdots\!83}a^{19}-\frac{16\!\cdots\!48}{43\!\cdots\!83}a^{18}+\frac{17\!\cdots\!46}{43\!\cdots\!83}a^{17}+\frac{25\!\cdots\!92}{43\!\cdots\!83}a^{16}+\frac{15\!\cdots\!06}{43\!\cdots\!83}a^{15}+\frac{77\!\cdots\!44}{43\!\cdots\!83}a^{14}+\frac{91\!\cdots\!80}{43\!\cdots\!83}a^{13}+\frac{65\!\cdots\!12}{43\!\cdots\!83}a^{12}+\frac{37\!\cdots\!02}{43\!\cdots\!83}a^{11}+\frac{33\!\cdots\!53}{43\!\cdots\!83}a^{10}+\frac{20\!\cdots\!91}{82\!\cdots\!11}a^{9}+\frac{97\!\cdots\!16}{43\!\cdots\!83}a^{8}+\frac{21\!\cdots\!40}{43\!\cdots\!83}a^{7}+\frac{18\!\cdots\!20}{43\!\cdots\!83}a^{6}+\frac{27\!\cdots\!50}{43\!\cdots\!83}a^{5}+\frac{21\!\cdots\!00}{43\!\cdots\!83}a^{4}+\frac{20\!\cdots\!84}{43\!\cdots\!83}a^{3}+\frac{13\!\cdots\!15}{43\!\cdots\!83}a^{2}+\frac{68\!\cdots\!21}{43\!\cdots\!83}a+\frac{19\!\cdots\!76}{43\!\cdots\!83}$, $\frac{56\!\cdots\!95}{43\!\cdots\!83}a^{25}-\frac{13\!\cdots\!76}{43\!\cdots\!83}a^{24}+\frac{35\!\cdots\!62}{43\!\cdots\!83}a^{23}-\frac{40\!\cdots\!99}{43\!\cdots\!83}a^{22}+\frac{13\!\cdots\!13}{43\!\cdots\!83}a^{21}-\frac{12\!\cdots\!69}{43\!\cdots\!83}a^{20}+\frac{29\!\cdots\!64}{43\!\cdots\!83}a^{19}-\frac{15\!\cdots\!81}{43\!\cdots\!83}a^{18}+\frac{43\!\cdots\!87}{43\!\cdots\!83}a^{17}-\frac{17\!\cdots\!81}{43\!\cdots\!83}a^{16}+\frac{36\!\cdots\!93}{43\!\cdots\!83}a^{15}+\frac{40\!\cdots\!32}{43\!\cdots\!83}a^{14}+\frac{20\!\cdots\!76}{43\!\cdots\!83}a^{13}+\frac{60\!\cdots\!83}{43\!\cdots\!83}a^{12}+\frac{76\!\cdots\!80}{43\!\cdots\!83}a^{11}+\frac{38\!\cdots\!82}{43\!\cdots\!83}a^{10}+\frac{19\!\cdots\!23}{43\!\cdots\!83}a^{9}+\frac{98\!\cdots\!17}{43\!\cdots\!83}a^{8}+\frac{33\!\cdots\!27}{43\!\cdots\!83}a^{7}+\frac{18\!\cdots\!09}{43\!\cdots\!83}a^{6}+\frac{38\!\cdots\!43}{43\!\cdots\!83}a^{5}+\frac{16\!\cdots\!55}{43\!\cdots\!83}a^{4}+\frac{22\!\cdots\!61}{43\!\cdots\!83}a^{3}+\frac{11\!\cdots\!08}{43\!\cdots\!83}a^{2}+\frac{75\!\cdots\!97}{43\!\cdots\!83}a+\frac{25\!\cdots\!92}{43\!\cdots\!83}$ (assuming GRH)
Regulator:		\( 1197545162478.713 \) (assuming GRH)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{13}\cdot 1197545162478.713 \cdot 1526201}{6\cdot\sqrt{1040129483587052483222994327325975449420803964347689607283}}\cr\approx \mathstrut & 0.224670974606927 \end{aligned}\] (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

\(\Q(\sqrt{-3}) \), 13.13.25542038069936263923006961.1

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	$26$	R	$26$	${\href{/padicField/7.13.0.1}{13} }^{2}$	$26$	${\href{/padicField/13.13.0.1}{13} }^{2}$	$26$	${\href{/padicField/19.13.0.1}{13} }^{2}$	$26$	$26$	${\href{/padicField/31.13.0.1}{13} }^{2}$	${\href{/padicField/37.13.0.1}{13} }^{2}$	$26$	${\href{/padicField/43.13.0.1}{13} }^{2}$	$26$	${\href{/padicField/53.2.0.1}{2} }^{13}$	$26$

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
\(3\)		Deg $26$	$2$	$13$	$13$
\(131\)		Deg $26$	$13$	$2$	$24$