Number field 26.0.80198818302747948116281352414518078412749016374883916055923.1

• Label: 26.0.801...923.1
• Degree: $26$
• Signature: $[0, 13]$
• Discriminant: $-8.020\times 10^{58}$
• Root discriminant: \(184.31\)
• Ramified primes: $3,157$
• Class number: $10628423$ (GRH)
• Class group: [10628423] (GRH)
• Galois group: $C_{26}$ (as 26T1)

Normalized defining polynomial

x^26 - x^25 + 73*x^24 - 186*x^23 + 3641*x^22 - 9330*x^21 + 100019*x^20 - 222719*x^19 + 1875616*x^18 - 3492740*x^17 + 23704391*x^16 - 32264444*x^15 + 210535014*x^14 - 201710657*x^13 + 1351138459*x^12 - 636617440*x^11 + 6088122667*x^10 - 338323732*x^9 + 19670105030*x^8 + 9111135848*x^7 + 40261020210*x^6 + 33026511660*x^5 + 66399395218*x^4 + 51808567752*x^3 + 39465012376*x^2 + 13351703742*x + 3769837201

Invariants

Degree:		$26$
Signature:		$[0, 13]$
Discriminant:		\(-80198818302747948116281352414518078412749016374883916055923\) \(\medspace = -\,3^{13}\cdot 157^{24}\)
Root discriminant:		\(184.31\)
Galois root discriminant:		$3^{1/2}157^{12/13}\approx 184.30831211371242$
Ramified primes:		\(3\), \(157\)
Discriminant root field:		\(\Q(\sqrt{-3}) \)
$\card{ \Gal(K/\Q) }$:		$26$
This field is Galois and abelian over $\Q$.
Conductor:		\(471=3\cdot 157\)
Dirichlet character group:		$\lbrace$$\chi_{471}(256,·)$, $\chi_{471}(1,·)$, $\chi_{471}(130,·)$, $\chi_{471}(67,·)$, $\chi_{471}(196,·)$, $\chi_{471}(389,·)$, $\chi_{471}(328,·)$, $\chi_{471}(265,·)$, $\chi_{471}(203,·)$, $\chi_{471}(14,·)$, $\chi_{471}(16,·)$, $\chi_{471}(467,·)$, $\chi_{471}(407,·)$, $\chi_{471}(413,·)$, $\chi_{471}(158,·)$, $\chi_{471}(415,·)$, $\chi_{471}(224,·)$, $\chi_{471}(353,·)$, $\chi_{471}(101,·)$, $\chi_{471}(422,·)$, $\chi_{471}(232,·)$, $\chi_{471}(173,·)$, $\chi_{471}(46,·)$, $\chi_{471}(310,·)$, $\chi_{471}(250,·)$, $\chi_{471}(287,·)$$\rbrace$
This is a CM field.
Reflex fields:		unavailable$^{4096}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{13}a^{12}+\frac{6}{13}a^{11}-\frac{3}{13}a^{10}-\frac{5}{13}a^{9}-\frac{4}{13}a^{8}+\frac{2}{13}a^{7}-\frac{1}{13}a^{6}-\frac{6}{13}a^{5}+\frac{3}{13}a^{4}+\frac{5}{13}a^{3}+\frac{4}{13}a^{2}-\frac{2}{13}a$, $\frac{1}{13}a^{13}-\frac{1}{13}a$, $\frac{1}{13}a^{14}-\frac{1}{13}a^{2}$, $\frac{1}{13}a^{15}-\frac{1}{13}a^{3}$, $\frac{1}{13}a^{16}-\frac{1}{13}a^{4}$, $\frac{1}{13}a^{17}-\frac{1}{13}a^{5}$, $\frac{1}{13}a^{18}-\frac{1}{13}a^{6}$, $\frac{1}{13}a^{19}-\frac{1}{13}a^{7}$, $\frac{1}{169}a^{20}-\frac{2}{169}a^{19}+\frac{6}{169}a^{18}+\frac{1}{169}a^{17}+\frac{4}{169}a^{16}-\frac{1}{169}a^{15}+\frac{1}{169}a^{14}-\frac{5}{169}a^{13}-\frac{4}{169}a^{12}+\frac{67}{169}a^{11}+\frac{12}{169}a^{10}+\frac{20}{169}a^{9}-\frac{50}{169}a^{8}+\frac{72}{169}a^{7}+\frac{37}{169}a^{6}-\frac{55}{169}a^{5}+\frac{75}{169}a^{4}-\frac{58}{169}a^{3}-\frac{30}{169}a^{2}-\frac{4}{13}a$, $\frac{1}{169}a^{21}+\frac{2}{169}a^{19}+\frac{6}{169}a^{17}-\frac{6}{169}a^{16}-\frac{1}{169}a^{15}-\frac{3}{169}a^{14}-\frac{1}{169}a^{13}-\frac{6}{169}a^{12}-\frac{75}{169}a^{11}+\frac{70}{169}a^{10}-\frac{23}{169}a^{9}+\frac{63}{169}a^{8}+\frac{51}{169}a^{7}-\frac{72}{169}a^{6}+\frac{17}{169}a^{5}+\frac{79}{169}a^{4}+\frac{36}{169}a^{3}-\frac{34}{169}a^{2}+\frac{1}{13}a$, $\frac{1}{169}a^{22}+\frac{4}{169}a^{19}-\frac{6}{169}a^{18}+\frac{5}{169}a^{17}+\frac{4}{169}a^{16}-\frac{1}{169}a^{15}-\frac{3}{169}a^{14}+\frac{4}{169}a^{13}-\frac{2}{169}a^{12}-\frac{12}{169}a^{11}-\frac{73}{169}a^{10}+\frac{36}{169}a^{9}+\frac{60}{169}a^{8}+\frac{83}{169}a^{7}+\frac{47}{169}a^{6}-\frac{45}{169}a^{5}+\frac{68}{169}a^{4}+\frac{69}{169}a^{3}-\frac{5}{169}a^{2}-\frac{2}{13}a$, $\frac{1}{2197}a^{23}-\frac{4}{2197}a^{22}+\frac{4}{2197}a^{20}+\frac{30}{2197}a^{19}+\frac{3}{2197}a^{18}+\frac{75}{2197}a^{17}+\frac{35}{2197}a^{16}+\frac{40}{2197}a^{15}-\frac{49}{2197}a^{14}-\frac{70}{2197}a^{13}+\frac{48}{2197}a^{12}+\frac{287}{2197}a^{11}+\frac{3}{2197}a^{10}+\frac{501}{2197}a^{9}-\frac{872}{2197}a^{8}-\frac{1078}{2197}a^{7}-\frac{935}{2197}a^{6}+\frac{183}{2197}a^{5}+\frac{915}{2197}a^{4}-\frac{736}{2197}a^{3}+\frac{943}{2197}a^{2}+\frac{56}{169}a+\frac{6}{13}$, $\frac{1}{67\!\cdots\!49}a^{24}-\frac{29\!\cdots\!43}{67\!\cdots\!49}a^{23}+\frac{12\!\cdots\!15}{67\!\cdots\!49}a^{22}+\frac{17\!\cdots\!65}{67\!\cdots\!49}a^{21}+\frac{13\!\cdots\!64}{67\!\cdots\!49}a^{20}-\frac{23\!\cdots\!99}{67\!\cdots\!49}a^{19}-\frac{10\!\cdots\!89}{67\!\cdots\!49}a^{18}-\frac{35\!\cdots\!41}{67\!\cdots\!49}a^{17}-\frac{23\!\cdots\!76}{67\!\cdots\!49}a^{16}+\frac{42\!\cdots\!07}{67\!\cdots\!49}a^{15}+\frac{16\!\cdots\!52}{67\!\cdots\!49}a^{14}-\frac{18\!\cdots\!68}{67\!\cdots\!49}a^{13}-\frac{14\!\cdots\!30}{67\!\cdots\!49}a^{12}-\frac{22\!\cdots\!84}{67\!\cdots\!49}a^{11}-\frac{33\!\cdots\!36}{67\!\cdots\!49}a^{10}-\frac{11\!\cdots\!93}{67\!\cdots\!49}a^{9}+\frac{22\!\cdots\!39}{67\!\cdots\!49}a^{8}+\frac{27\!\cdots\!50}{67\!\cdots\!49}a^{7}+\frac{18\!\cdots\!59}{67\!\cdots\!49}a^{6}-\frac{39\!\cdots\!78}{67\!\cdots\!49}a^{5}+\frac{21\!\cdots\!92}{67\!\cdots\!49}a^{4}+\frac{11\!\cdots\!74}{67\!\cdots\!49}a^{3}-\frac{24\!\cdots\!51}{67\!\cdots\!49}a^{2}-\frac{10\!\cdots\!83}{52\!\cdots\!73}a-\frac{39\!\cdots\!98}{40\!\cdots\!21}$, $\frac{1}{41\!\cdots\!57}a^{25}+\frac{18\!\cdots\!54}{41\!\cdots\!57}a^{24}-\frac{46\!\cdots\!85}{41\!\cdots\!57}a^{23}-\frac{31\!\cdots\!06}{41\!\cdots\!57}a^{22}-\frac{26\!\cdots\!49}{41\!\cdots\!57}a^{21}+\frac{21\!\cdots\!93}{41\!\cdots\!57}a^{20}-\frac{11\!\cdots\!36}{41\!\cdots\!57}a^{19}+\frac{67\!\cdots\!09}{41\!\cdots\!57}a^{18}-\frac{88\!\cdots\!83}{41\!\cdots\!57}a^{17}+\frac{34\!\cdots\!21}{41\!\cdots\!57}a^{16}-\frac{70\!\cdots\!75}{41\!\cdots\!57}a^{15}+\frac{51\!\cdots\!38}{41\!\cdots\!57}a^{14}-\frac{56\!\cdots\!10}{41\!\cdots\!57}a^{13}-\frac{10\!\cdots\!12}{41\!\cdots\!57}a^{12}-\frac{19\!\cdots\!95}{41\!\cdots\!57}a^{11}-\frac{18\!\cdots\!76}{41\!\cdots\!57}a^{10}+\frac{18\!\cdots\!24}{41\!\cdots\!57}a^{9}-\frac{14\!\cdots\!86}{41\!\cdots\!57}a^{8}-\frac{43\!\cdots\!23}{41\!\cdots\!57}a^{7}+\frac{15\!\cdots\!68}{41\!\cdots\!57}a^{6}+\frac{12\!\cdots\!80}{41\!\cdots\!57}a^{5}+\frac{17\!\cdots\!84}{41\!\cdots\!57}a^{4}-\frac{11\!\cdots\!88}{41\!\cdots\!57}a^{3}-\frac{28\!\cdots\!32}{41\!\cdots\!57}a^{2}-\frac{58\!\cdots\!23}{31\!\cdots\!89}a-\frac{18\!\cdots\!15}{51\!\cdots\!11}$

Monogenic:		No
Index:		Not computed
Inessential primes:		$13$

Class group and class number

$C_{10628423}$, which has order $10628423$ (assuming GRH)

Unit group

Rank:		$12$
Torsion generator:		\( \frac{1595166435030487796822572744524709773516707148720972693310890082629196326831}{20969720622023478298535688459272272371414636056883444980991705766313955755461214215993} a^{25} - \frac{1828668060278383088605707809013779282604274019871762003690830266501022535371}{20969720622023478298535688459272272371414636056883444980991705766313955755461214215993} a^{24} + \frac{117285136076104534611451482996966405330180446021720861288343569937052981088795}{20969720622023478298535688459272272371414636056883444980991705766313955755461214215993} a^{23} - \frac{314315016528160115552375288686099629054746160664002863641030907008063165119571}{20969720622023478298535688459272272371414636056883444980991705766313955755461214215993} a^{22} + \frac{5894856654340718818487587678141219829327040585528914365996078527293658829743414}{20969720622023478298535688459272272371414636056883444980991705766313955755461214215993} a^{21} - \frac{15843159396091234696696489533566063290952283622185679994068739412024953664075487}{20969720622023478298535688459272272371414636056883444980991705766313955755461214215993} a^{20} + \frac{163873764051713704479455200138375712682832116094910441789928128543172233080356149}{20969720622023478298535688459272272371414636056883444980991705766313955755461214215993} a^{19} - \frac{384081013807585464859694286534949132366598971739902108539551500074639259855805760}{20969720622023478298535688459272272371414636056883444980991705766313955755461214215993} a^{18} + \frac{3101915041880881388220837359283679219551728957748627280329166443369918667705832652}{20969720622023478298535688459272272371414636056883444980991705766313955755461214215993} a^{17} - \frac{6137840588474167608175428228366375784352848246320919934840864036594949009699949039}{20969720622023478298535688459272272371414636056883444980991705766313955755461214215993} a^{16} + \frac{39701367358340440680610320612786849236526164318103299921444294874936807860428599214}{20969720622023478298535688459272272371414636056883444980991705766313955755461214215993} a^{15} - \frac{58998395158916237377040223598463074105055516725701204713671373409274673174524705124}{20969720622023478298535688459272272371414636056883444980991705766313955755461214215993} a^{14} + \frac{356667542533403940973386307904898792260742535460460593701415116456754735100984915235}{20969720622023478298535688459272272371414636056883444980991705766313955755461214215993} a^{13} - \frac{389130231516892044499426976330902917215057431293988932121154977052810323631622709323}{20969720622023478298535688459272272371414636056883444980991705766313955755461214215993} a^{12} + \frac{2318270762493962743892327812567776290005672286884483278859291131036038133705522775108}{20969720622023478298535688459272272371414636056883444980991705766313955755461214215993} a^{11} - \frac{1447335113451296742455423346712754326571054870674728614054633808233949097959816267123}{20969720622023478298535688459272272371414636056883444980991705766313955755461214215993} a^{10} + \frac{10585186393150275950102854899666486361438258641769437451881255361261861117117623964446}{20969720622023478298535688459272272371414636056883444980991705766313955755461214215993} a^{9} - \frac{2357517392289224131850368756658763424311859075677712213132598595849663629920937601352}{20969720622023478298535688459272272371414636056883444980991705766313955755461214215993} a^{8} + \frac{34614145103068117599800484721088804260272611020306946004531942322577182199227990282720}{20969720622023478298535688459272272371414636056883444980991705766313955755461214215993} a^{7} + \frac{9221370199641477472723855526396637795123956483108874820828598305945378478488427209964}{20969720622023478298535688459272272371414636056883444980991705766313955755461214215993} a^{6} + \frac{71767494262709653881358133433873013806556657835086379970946383998196293390507675381037}{20969720622023478298535688459272272371414636056883444980991705766313955755461214215993} a^{5} + \frac{45478115605062328651025913186245171006992283780932630556041486330336078846538620705118}{20969720622023478298535688459272272371414636056883444980991705766313955755461214215993} a^{4} + \frac{115787434030000901613030916627652429278110821832223232606833432253416618687237258200282}{20969720622023478298535688459272272371414636056883444980991705766313955755461214215993} a^{3} + \frac{69581805261663446298795981545903988605589900485864423134697112186312342856312676913231}{20969720622023478298535688459272272371414636056883444980991705766313955755461214215993} a^{2} + \frac{5801742304128635320272679068615497005177982229903519200312523911012261210890870029746}{1613055432463344484502745266097867105493433542837188075460900443562611981189324170461} a + \frac{32440291609931479242994258666310149979998809613726330805939753462166426133230565}{26271688992709074813966762750172919843864452887460513615220124815756152073964139} \)  (order $6$)
Fundamental units:		$\frac{11\!\cdots\!18}{87\!\cdots\!59}a^{25}-\frac{38\!\cdots\!93}{87\!\cdots\!59}a^{24}+\frac{84\!\cdots\!99}{87\!\cdots\!59}a^{23}-\frac{40\!\cdots\!13}{87\!\cdots\!59}a^{22}+\frac{46\!\cdots\!15}{87\!\cdots\!59}a^{21}-\frac{20\!\cdots\!04}{87\!\cdots\!59}a^{20}+\frac{14\!\cdots\!21}{87\!\cdots\!59}a^{19}-\frac{52\!\cdots\!51}{87\!\cdots\!59}a^{18}+\frac{27\!\cdots\!02}{87\!\cdots\!59}a^{17}-\frac{91\!\cdots\!92}{87\!\cdots\!59}a^{16}+\frac{37\!\cdots\!40}{87\!\cdots\!59}a^{15}-\frac{10\!\cdots\!76}{87\!\cdots\!59}a^{14}+\frac{34\!\cdots\!44}{87\!\cdots\!59}a^{13}-\frac{81\!\cdots\!98}{87\!\cdots\!59}a^{12}+\frac{22\!\cdots\!65}{87\!\cdots\!59}a^{11}-\frac{44\!\cdots\!00}{87\!\cdots\!59}a^{10}+\frac{97\!\cdots\!25}{87\!\cdots\!59}a^{9}-\frac{16\!\cdots\!90}{87\!\cdots\!59}a^{8}+\frac{28\!\cdots\!36}{87\!\cdots\!59}a^{7}-\frac{39\!\cdots\!84}{87\!\cdots\!59}a^{6}+\frac{39\!\cdots\!72}{87\!\cdots\!59}a^{5}-\frac{50\!\cdots\!15}{87\!\cdots\!59}a^{4}+\frac{26\!\cdots\!28}{87\!\cdots\!59}a^{3}-\frac{65\!\cdots\!82}{87\!\cdots\!59}a^{2}-\frac{18\!\cdots\!27}{67\!\cdots\!43}a-\frac{92\!\cdots\!07}{51\!\cdots\!11}$, $\frac{21\!\cdots\!94}{87\!\cdots\!59}a^{25}-\frac{58\!\cdots\!22}{87\!\cdots\!59}a^{24}+\frac{15\!\cdots\!19}{87\!\cdots\!59}a^{23}-\frac{66\!\cdots\!85}{87\!\cdots\!59}a^{22}+\frac{85\!\cdots\!45}{87\!\cdots\!59}a^{21}-\frac{33\!\cdots\!19}{87\!\cdots\!59}a^{20}+\frac{25\!\cdots\!72}{87\!\cdots\!59}a^{19}-\frac{87\!\cdots\!73}{87\!\cdots\!59}a^{18}+\frac{50\!\cdots\!60}{87\!\cdots\!59}a^{17}-\frac{15\!\cdots\!71}{87\!\cdots\!59}a^{16}+\frac{67\!\cdots\!31}{87\!\cdots\!59}a^{15}-\frac{17\!\cdots\!55}{87\!\cdots\!59}a^{14}+\frac{63\!\cdots\!35}{87\!\cdots\!59}a^{13}-\frac{13\!\cdots\!96}{87\!\cdots\!59}a^{12}+\frac{41\!\cdots\!21}{87\!\cdots\!59}a^{11}-\frac{73\!\cdots\!15}{87\!\cdots\!59}a^{10}+\frac{18\!\cdots\!41}{87\!\cdots\!59}a^{9}-\frac{27\!\cdots\!16}{87\!\cdots\!59}a^{8}+\frac{54\!\cdots\!95}{87\!\cdots\!59}a^{7}-\frac{62\!\cdots\!36}{87\!\cdots\!59}a^{6}+\frac{79\!\cdots\!26}{87\!\cdots\!59}a^{5}-\frac{80\!\cdots\!70}{87\!\cdots\!59}a^{4}+\frac{66\!\cdots\!66}{87\!\cdots\!59}a^{3}-\frac{11\!\cdots\!28}{87\!\cdots\!59}a^{2}-\frac{32\!\cdots\!59}{67\!\cdots\!43}a-\frac{33\!\cdots\!83}{51\!\cdots\!11}$, $\frac{17\!\cdots\!34}{87\!\cdots\!59}a^{25}-\frac{10\!\cdots\!00}{87\!\cdots\!59}a^{24}+\frac{13\!\cdots\!68}{87\!\cdots\!59}a^{23}-\frac{29\!\cdots\!32}{87\!\cdots\!59}a^{22}+\frac{67\!\cdots\!89}{87\!\cdots\!59}a^{21}-\frac{16\!\cdots\!06}{87\!\cdots\!59}a^{20}+\frac{19\!\cdots\!70}{87\!\cdots\!59}a^{19}-\frac{42\!\cdots\!27}{87\!\cdots\!59}a^{18}+\frac{38\!\cdots\!17}{87\!\cdots\!59}a^{17}-\frac{73\!\cdots\!79}{87\!\cdots\!59}a^{16}+\frac{52\!\cdots\!62}{87\!\cdots\!59}a^{15}-\frac{81\!\cdots\!21}{87\!\cdots\!59}a^{14}+\frac{50\!\cdots\!48}{87\!\cdots\!59}a^{13}-\frac{62\!\cdots\!90}{87\!\cdots\!59}a^{12}+\frac{34\!\cdots\!62}{87\!\cdots\!59}a^{11}-\frac{32\!\cdots\!63}{87\!\cdots\!59}a^{10}+\frac{16\!\cdots\!31}{87\!\cdots\!59}a^{9}-\frac{11\!\cdots\!16}{87\!\cdots\!59}a^{8}+\frac{50\!\cdots\!68}{87\!\cdots\!59}a^{7}-\frac{23\!\cdots\!97}{87\!\cdots\!59}a^{6}+\frac{86\!\cdots\!86}{87\!\cdots\!59}a^{5}-\frac{19\!\cdots\!69}{87\!\cdots\!59}a^{4}+\frac{16\!\cdots\!59}{87\!\cdots\!59}a^{3}-\frac{63\!\cdots\!96}{87\!\cdots\!59}a^{2}-\frac{18\!\cdots\!78}{67\!\cdots\!43}a-\frac{12\!\cdots\!76}{51\!\cdots\!11}$, $\frac{80\!\cdots\!82}{87\!\cdots\!59}a^{25}+\frac{18\!\cdots\!46}{87\!\cdots\!59}a^{24}+\frac{57\!\cdots\!96}{87\!\cdots\!59}a^{23}+\frac{37\!\cdots\!22}{87\!\cdots\!59}a^{22}+\frac{24\!\cdots\!95}{87\!\cdots\!59}a^{21}+\frac{16\!\cdots\!59}{87\!\cdots\!59}a^{20}+\frac{58\!\cdots\!81}{87\!\cdots\!59}a^{19}+\frac{64\!\cdots\!40}{87\!\cdots\!59}a^{18}+\frac{98\!\cdots\!01}{87\!\cdots\!59}a^{17}+\frac{16\!\cdots\!76}{87\!\cdots\!59}a^{16}+\frac{10\!\cdots\!14}{87\!\cdots\!59}a^{15}+\frac{28\!\cdots\!79}{87\!\cdots\!59}a^{14}+\frac{92\!\cdots\!58}{87\!\cdots\!59}a^{13}+\frac{30\!\cdots\!48}{87\!\cdots\!59}a^{12}+\frac{57\!\cdots\!81}{87\!\cdots\!59}a^{11}+\frac{22\!\cdots\!92}{87\!\cdots\!59}a^{10}+\frac{29\!\cdots\!99}{87\!\cdots\!59}a^{9}+\frac{10\!\cdots\!70}{87\!\cdots\!59}a^{8}+\frac{10\!\cdots\!02}{87\!\cdots\!59}a^{7}+\frac{31\!\cdots\!31}{87\!\cdots\!59}a^{6}+\frac{28\!\cdots\!32}{87\!\cdots\!59}a^{5}+\frac{45\!\cdots\!42}{87\!\cdots\!59}a^{4}+\frac{22\!\cdots\!93}{87\!\cdots\!59}a^{3}+\frac{25\!\cdots\!08}{87\!\cdots\!59}a^{2}+\frac{65\!\cdots\!75}{67\!\cdots\!43}a+\frac{16\!\cdots\!54}{51\!\cdots\!11}$, $\frac{13\!\cdots\!42}{87\!\cdots\!59}a^{25}-\frac{44\!\cdots\!11}{87\!\cdots\!59}a^{24}+\frac{10\!\cdots\!10}{87\!\cdots\!59}a^{23}-\frac{47\!\cdots\!52}{87\!\cdots\!59}a^{22}+\frac{57\!\cdots\!84}{87\!\cdots\!59}a^{21}-\frac{24\!\cdots\!02}{87\!\cdots\!59}a^{20}+\frac{17\!\cdots\!48}{87\!\cdots\!59}a^{19}-\frac{61\!\cdots\!84}{87\!\cdots\!59}a^{18}+\frac{33\!\cdots\!51}{87\!\cdots\!59}a^{17}-\frac{10\!\cdots\!53}{87\!\cdots\!59}a^{16}+\frac{45\!\cdots\!15}{87\!\cdots\!59}a^{15}-\frac{11\!\cdots\!72}{87\!\cdots\!59}a^{14}+\frac{41\!\cdots\!61}{87\!\cdots\!59}a^{13}-\frac{92\!\cdots\!35}{87\!\cdots\!59}a^{12}+\frac{26\!\cdots\!61}{87\!\cdots\!59}a^{11}-\frac{50\!\cdots\!41}{87\!\cdots\!59}a^{10}+\frac{11\!\cdots\!01}{87\!\cdots\!59}a^{9}-\frac{18\!\cdots\!44}{87\!\cdots\!59}a^{8}+\frac{34\!\cdots\!93}{87\!\cdots\!59}a^{7}-\frac{43\!\cdots\!86}{87\!\cdots\!59}a^{6}+\frac{48\!\cdots\!10}{87\!\cdots\!59}a^{5}-\frac{54\!\cdots\!25}{87\!\cdots\!59}a^{4}+\frac{33\!\cdots\!00}{87\!\cdots\!59}a^{3}-\frac{74\!\cdots\!90}{87\!\cdots\!59}a^{2}-\frac{21\!\cdots\!77}{67\!\cdots\!43}a-\frac{19\!\cdots\!19}{51\!\cdots\!11}$, $\frac{24\!\cdots\!18}{87\!\cdots\!59}a^{25}-\frac{97\!\cdots\!95}{87\!\cdots\!59}a^{24}+\frac{18\!\cdots\!26}{87\!\cdots\!59}a^{23}-\frac{98\!\cdots\!56}{87\!\cdots\!59}a^{22}+\frac{10\!\cdots\!60}{87\!\cdots\!59}a^{21}-\frac{49\!\cdots\!64}{87\!\cdots\!59}a^{20}+\frac{31\!\cdots\!94}{87\!\cdots\!59}a^{19}-\frac{12\!\cdots\!31}{87\!\cdots\!59}a^{18}+\frac{64\!\cdots\!61}{87\!\cdots\!59}a^{17}-\frac{22\!\cdots\!97}{87\!\cdots\!59}a^{16}+\frac{87\!\cdots\!66}{87\!\cdots\!59}a^{15}-\frac{25\!\cdots\!24}{87\!\cdots\!59}a^{14}+\frac{81\!\cdots\!03}{87\!\cdots\!59}a^{13}-\frac{20\!\cdots\!22}{87\!\cdots\!59}a^{12}+\frac{53\!\cdots\!97}{87\!\cdots\!59}a^{11}-\frac{11\!\cdots\!69}{87\!\cdots\!59}a^{10}+\frac{23\!\cdots\!63}{87\!\cdots\!59}a^{9}-\frac{44\!\cdots\!44}{87\!\cdots\!59}a^{8}+\frac{67\!\cdots\!63}{87\!\cdots\!59}a^{7}-\frac{10\!\cdots\!52}{87\!\cdots\!59}a^{6}+\frac{88\!\cdots\!50}{87\!\cdots\!59}a^{5}-\frac{14\!\cdots\!65}{87\!\cdots\!59}a^{4}+\frac{27\!\cdots\!22}{87\!\cdots\!59}a^{3}-\frac{17\!\cdots\!86}{87\!\cdots\!59}a^{2}-\frac{48\!\cdots\!77}{67\!\cdots\!43}a+\frac{25\!\cdots\!65}{51\!\cdots\!11}$, $\frac{38\!\cdots\!72}{87\!\cdots\!59}a^{25}-\frac{11\!\cdots\!50}{87\!\cdots\!59}a^{24}+\frac{28\!\cdots\!08}{87\!\cdots\!59}a^{23}-\frac{12\!\cdots\!98}{87\!\cdots\!59}a^{22}+\frac{15\!\cdots\!22}{87\!\cdots\!59}a^{21}-\frac{63\!\cdots\!94}{87\!\cdots\!59}a^{20}+\frac{46\!\cdots\!18}{87\!\cdots\!59}a^{19}-\frac{16\!\cdots\!16}{87\!\cdots\!59}a^{18}+\frac{90\!\cdots\!10}{87\!\cdots\!59}a^{17}-\frac{27\!\cdots\!24}{87\!\cdots\!59}a^{16}+\frac{12\!\cdots\!69}{87\!\cdots\!59}a^{15}-\frac{30\!\cdots\!15}{87\!\cdots\!59}a^{14}+\frac{11\!\cdots\!73}{87\!\cdots\!59}a^{13}-\frac{24\!\cdots\!85}{87\!\cdots\!59}a^{12}+\frac{72\!\cdots\!99}{87\!\cdots\!59}a^{11}-\frac{13\!\cdots\!56}{87\!\cdots\!59}a^{10}+\frac{31\!\cdots\!64}{87\!\cdots\!59}a^{9}-\frac{47\!\cdots\!72}{87\!\cdots\!59}a^{8}+\frac{93\!\cdots\!89}{87\!\cdots\!59}a^{7}-\frac{11\!\cdots\!51}{87\!\cdots\!59}a^{6}+\frac{13\!\cdots\!68}{87\!\cdots\!59}a^{5}-\frac{14\!\cdots\!28}{87\!\cdots\!59}a^{4}+\frac{10\!\cdots\!65}{87\!\cdots\!59}a^{3}-\frac{19\!\cdots\!78}{87\!\cdots\!59}a^{2}-\frac{55\!\cdots\!47}{67\!\cdots\!43}a-\frac{51\!\cdots\!25}{51\!\cdots\!11}$, $\frac{15\!\cdots\!46}{67\!\cdots\!43}a^{25}-\frac{58\!\cdots\!47}{67\!\cdots\!43}a^{24}+\frac{12\!\cdots\!98}{67\!\cdots\!43}a^{23}-\frac{61\!\cdots\!50}{67\!\cdots\!43}a^{22}+\frac{67\!\cdots\!80}{67\!\cdots\!43}a^{21}-\frac{30\!\cdots\!52}{67\!\cdots\!43}a^{20}+\frac{20\!\cdots\!70}{67\!\cdots\!43}a^{19}-\frac{80\!\cdots\!30}{67\!\cdots\!43}a^{18}+\frac{41\!\cdots\!43}{67\!\cdots\!43}a^{17}-\frac{14\!\cdots\!07}{67\!\cdots\!43}a^{16}+\frac{57\!\cdots\!58}{67\!\cdots\!43}a^{15}-\frac{16\!\cdots\!01}{67\!\cdots\!43}a^{14}+\frac{53\!\cdots\!42}{67\!\cdots\!43}a^{13}-\frac{12\!\cdots\!92}{67\!\cdots\!43}a^{12}+\frac{35\!\cdots\!16}{67\!\cdots\!43}a^{11}-\frac{72\!\cdots\!11}{67\!\cdots\!43}a^{10}+\frac{15\!\cdots\!07}{67\!\cdots\!43}a^{9}-\frac{27\!\cdots\!44}{67\!\cdots\!43}a^{8}+\frac{45\!\cdots\!26}{67\!\cdots\!43}a^{7}-\frac{66\!\cdots\!32}{67\!\cdots\!43}a^{6}+\frac{61\!\cdots\!30}{67\!\cdots\!43}a^{5}-\frac{85\!\cdots\!99}{67\!\cdots\!43}a^{4}+\frac{26\!\cdots\!70}{67\!\cdots\!43}a^{3}-\frac{10\!\cdots\!32}{67\!\cdots\!43}a^{2}-\frac{30\!\cdots\!36}{51\!\cdots\!11}a-\frac{25\!\cdots\!35}{39\!\cdots\!47}$, $\frac{66\!\cdots\!34}{87\!\cdots\!59}a^{25}-\frac{35\!\cdots\!23}{87\!\cdots\!59}a^{24}+\frac{54\!\cdots\!41}{87\!\cdots\!59}a^{23}-\frac{33\!\cdots\!47}{87\!\cdots\!59}a^{22}+\frac{31\!\cdots\!69}{87\!\cdots\!59}a^{21}-\frac{17\!\cdots\!37}{87\!\cdots\!59}a^{20}+\frac{10\!\cdots\!28}{87\!\cdots\!59}a^{19}-\frac{46\!\cdots\!55}{87\!\cdots\!59}a^{18}+\frac{21\!\cdots\!80}{87\!\cdots\!59}a^{17}-\frac{82\!\cdots\!33}{87\!\cdots\!59}a^{16}+\frac{30\!\cdots\!62}{87\!\cdots\!59}a^{15}-\frac{98\!\cdots\!33}{87\!\cdots\!59}a^{14}+\frac{28\!\cdots\!29}{87\!\cdots\!59}a^{13}-\frac{81\!\cdots\!98}{87\!\cdots\!59}a^{12}+\frac{19\!\cdots\!31}{87\!\cdots\!59}a^{11}-\frac{47\!\cdots\!09}{87\!\cdots\!59}a^{10}+\frac{86\!\cdots\!87}{87\!\cdots\!59}a^{9}-\frac{18\!\cdots\!88}{87\!\cdots\!59}a^{8}+\frac{25\!\cdots\!93}{87\!\cdots\!59}a^{7}-\frac{47\!\cdots\!84}{87\!\cdots\!59}a^{6}+\frac{31\!\cdots\!70}{87\!\cdots\!59}a^{5}-\frac{60\!\cdots\!26}{87\!\cdots\!59}a^{4}-\frac{17\!\cdots\!81}{87\!\cdots\!59}a^{3}-\frac{69\!\cdots\!72}{87\!\cdots\!59}a^{2}-\frac{19\!\cdots\!45}{67\!\cdots\!43}a-\frac{82\!\cdots\!09}{51\!\cdots\!11}$, $\frac{63\!\cdots\!30}{87\!\cdots\!59}a^{25}-\frac{65\!\cdots\!70}{87\!\cdots\!59}a^{24}+\frac{47\!\cdots\!26}{87\!\cdots\!59}a^{23}-\frac{11\!\cdots\!22}{87\!\cdots\!59}a^{22}+\frac{23\!\cdots\!51}{87\!\cdots\!59}a^{21}-\frac{57\!\cdots\!81}{87\!\cdots\!59}a^{20}+\frac{63\!\cdots\!75}{87\!\cdots\!59}a^{19}-\frac{12\!\cdots\!39}{87\!\cdots\!59}a^{18}+\frac{11\!\cdots\!19}{87\!\cdots\!59}a^{17}-\frac{17\!\cdots\!74}{87\!\cdots\!59}a^{16}+\frac{13\!\cdots\!87}{87\!\cdots\!59}a^{15}-\frac{12\!\cdots\!84}{87\!\cdots\!59}a^{14}+\frac{10\!\cdots\!67}{87\!\cdots\!59}a^{13}-\frac{32\!\cdots\!11}{87\!\cdots\!59}a^{12}+\frac{62\!\cdots\!72}{87\!\cdots\!59}a^{11}+\frac{15\!\cdots\!18}{87\!\cdots\!59}a^{10}+\frac{25\!\cdots\!07}{87\!\cdots\!59}a^{9}+\frac{18\!\cdots\!82}{87\!\cdots\!59}a^{8}+\frac{77\!\cdots\!63}{87\!\cdots\!59}a^{7}+\frac{60\!\cdots\!26}{87\!\cdots\!59}a^{6}+\frac{16\!\cdots\!76}{87\!\cdots\!59}a^{5}+\frac{12\!\cdots\!78}{87\!\cdots\!59}a^{4}+\frac{74\!\cdots\!32}{87\!\cdots\!59}a^{3}+\frac{23\!\cdots\!78}{87\!\cdots\!59}a^{2}+\frac{41\!\cdots\!48}{67\!\cdots\!43}a+\frac{38\!\cdots\!50}{51\!\cdots\!11}$, $\frac{52\!\cdots\!92}{87\!\cdots\!59}a^{25}-\frac{10\!\cdots\!36}{87\!\cdots\!59}a^{24}+\frac{38\!\cdots\!93}{87\!\cdots\!59}a^{23}-\frac{13\!\cdots\!05}{87\!\cdots\!59}a^{22}+\frac{20\!\cdots\!71}{87\!\cdots\!59}a^{21}-\frac{67\!\cdots\!28}{87\!\cdots\!59}a^{20}+\frac{57\!\cdots\!67}{87\!\cdots\!59}a^{19}-\frac{16\!\cdots\!66}{87\!\cdots\!59}a^{18}+\frac{10\!\cdots\!23}{87\!\cdots\!59}a^{17}-\frac{27\!\cdots\!05}{87\!\cdots\!59}a^{16}+\frac{14\!\cdots\!07}{87\!\cdots\!59}a^{15}-\frac{28\!\cdots\!70}{87\!\cdots\!59}a^{14}+\frac{12\!\cdots\!93}{87\!\cdots\!59}a^{13}-\frac{21\!\cdots\!28}{87\!\cdots\!59}a^{12}+\frac{79\!\cdots\!88}{87\!\cdots\!59}a^{11}-\frac{10\!\cdots\!41}{87\!\cdots\!59}a^{10}+\frac{34\!\cdots\!76}{87\!\cdots\!59}a^{9}-\frac{33\!\cdots\!86}{87\!\cdots\!59}a^{8}+\frac{10\!\cdots\!57}{87\!\cdots\!59}a^{7}-\frac{62\!\cdots\!29}{87\!\cdots\!59}a^{6}+\frac{16\!\cdots\!66}{87\!\cdots\!59}a^{5}-\frac{77\!\cdots\!76}{87\!\cdots\!59}a^{4}+\frac{20\!\cdots\!48}{87\!\cdots\!59}a^{3}-\frac{15\!\cdots\!28}{87\!\cdots\!59}a^{2}-\frac{45\!\cdots\!90}{67\!\cdots\!43}a-\frac{61\!\cdots\!63}{51\!\cdots\!11}$, $\frac{67\!\cdots\!36}{87\!\cdots\!59}a^{25}-\frac{21\!\cdots\!78}{87\!\cdots\!59}a^{24}+\frac{52\!\cdots\!42}{87\!\cdots\!59}a^{23}-\frac{23\!\cdots\!74}{87\!\cdots\!59}a^{22}+\frac{28\!\cdots\!68}{87\!\cdots\!59}a^{21}-\frac{12\!\cdots\!48}{87\!\cdots\!59}a^{20}+\frac{88\!\cdots\!02}{87\!\cdots\!59}a^{19}-\frac{32\!\cdots\!91}{87\!\cdots\!59}a^{18}+\frac{18\!\cdots\!58}{87\!\cdots\!59}a^{17}-\frac{57\!\cdots\!86}{87\!\cdots\!59}a^{16}+\frac{25\!\cdots\!44}{87\!\cdots\!59}a^{15}-\frac{67\!\cdots\!72}{87\!\cdots\!59}a^{14}+\frac{24\!\cdots\!90}{87\!\cdots\!59}a^{13}-\frac{55\!\cdots\!10}{87\!\cdots\!59}a^{12}+\frac{16\!\cdots\!56}{87\!\cdots\!59}a^{11}-\frac{31\!\cdots\!74}{87\!\cdots\!59}a^{10}+\frac{73\!\cdots\!61}{87\!\cdots\!59}a^{9}-\frac{12\!\cdots\!08}{87\!\cdots\!59}a^{8}+\frac{22\!\cdots\!82}{87\!\cdots\!59}a^{7}-\frac{28\!\cdots\!42}{87\!\cdots\!59}a^{6}+\frac{31\!\cdots\!28}{87\!\cdots\!59}a^{5}-\frac{37\!\cdots\!56}{87\!\cdots\!59}a^{4}+\frac{24\!\cdots\!88}{87\!\cdots\!59}a^{3}-\frac{49\!\cdots\!80}{87\!\cdots\!59}a^{2}-\frac{14\!\cdots\!48}{67\!\cdots\!43}a-\frac{85\!\cdots\!87}{51\!\cdots\!11}$ (assuming GRH)
Regulator:		\( 5466968796671.363 \) (assuming GRH)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{13}\cdot 5466968796671.363 \cdot 10628423}{6\cdot\sqrt{80198818302747948116281352414518078412749016374883916055923}}\cr\approx \mathstrut & 0.813426804349975 \end{aligned}\] (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

\(\Q(\sqrt{-3}) \), 13.13.224282727500720205065439601.1

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	$26$	${\href{/padicField/7.13.0.1}{13} }^{2}$	$26$	${\href{/padicField/13.1.0.1}{1} }^{26}$	$26$	${\href{/padicField/19.13.0.1}{13} }^{2}$	$26$	$26$	${\href{/padicField/31.13.0.1}{13} }^{2}$	${\href{/padicField/37.13.0.1}{13} }^{2}$	$26$	${\href{/padicField/43.13.0.1}{13} }^{2}$	$26$	$26$	$26$

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 $26$	$2$	$13$	$13$
\(157\)	157.13.12.1	$x^{13} + 157$	$13$	$1$	$12$	$C_{13}$	$[\ ]_{13}$
	157.13.12.1	$x^{13} + 157$	$13$	$1$	$12$	$C_{13}$	$[\ ]_{13}$