Number field 26.0.384766437057818380952237905666104641217782272563.1

• Label: 26.0.384...563.1
• Degree: $26$
• Signature: $[0, 13]$
• Discriminant: $-3.848\times 10^{47}$
• Root discriminant: \(67.64\)
• Ramified primes: $3,53$
• Class number: $2809$ (GRH)
• Class group: [53, 53] (GRH)
• Galois group: $C_{26}$ (as 26T1)

Normalized defining polynomial

x^26 - x^25 + 25*x^24 - 14*x^23 + 405*x^22 - 192*x^21 + 3603*x^20 - 1111*x^19 + 22650*x^18 - 5414*x^17 + 87624*x^16 - 5096*x^15 + 234451*x^14 - 19756*x^13 + 367484*x^12 - 30562*x^11 + 404986*x^10 - 57146*x^9 + 232119*x^8 - 34742*x^7 + 91429*x^6 - 16226*x^5 + 7319*x^4 + 98*x^3 + 168*x^2 - 10*x + 1

Invariants

Degree:		$26$
Signature:		$[0, 13]$
Discriminant:		\(-384766437057818380952237905666104641217782272563\) \(\medspace = -\,3^{13}\cdot 53^{24}\)
Root discriminant:		\(67.64\)
Galois root discriminant:		$3^{1/2}53^{12/13}\approx 67.63942259918862$
Ramified primes:		\(3\), \(53\)
Discriminant root field:		\(\Q(\sqrt{-3}) \)
$\card{ \Gal(K/\Q) }$:		$26$
This field is Galois and abelian over $\Q$.
Conductor:		\(159=3\cdot 53\)
Dirichlet character group:		$\lbrace$$\chi_{159}(1,·)$, $\chi_{159}(130,·)$, $\chi_{159}(68,·)$, $\chi_{159}(134,·)$, $\chi_{159}(10,·)$, $\chi_{159}(13,·)$, $\chi_{159}(142,·)$, $\chi_{159}(77,·)$, $\chi_{159}(16,·)$, $\chi_{159}(148,·)$, $\chi_{159}(152,·)$, $\chi_{159}(89,·)$, $\chi_{159}(155,·)$, $\chi_{159}(28,·)$, $\chi_{159}(95,·)$, $\chi_{159}(97,·)$, $\chi_{159}(100,·)$, $\chi_{159}(107,·)$, $\chi_{159}(44,·)$, $\chi_{159}(46,·)$, $\chi_{159}(47,·)$, $\chi_{159}(49,·)$, $\chi_{159}(116,·)$, $\chi_{159}(119,·)$, $\chi_{159}(121,·)$, $\chi_{159}(122,·)$$\rbrace$
This is a CM field.
Reflex fields:		unavailable$^{4096}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $\frac{1}{23}a^{22}-\frac{3}{23}a^{21}-\frac{8}{23}a^{20}+\frac{5}{23}a^{19}+\frac{8}{23}a^{18}-\frac{2}{23}a^{17}-\frac{10}{23}a^{16}-\frac{10}{23}a^{14}-\frac{10}{23}a^{13}+\frac{4}{23}a^{12}-\frac{9}{23}a^{11}-\frac{7}{23}a^{10}-\frac{7}{23}a^{9}-\frac{9}{23}a^{8}-\frac{2}{23}a^{7}+\frac{1}{23}a^{6}+\frac{3}{23}a^{5}+\frac{10}{23}a^{4}+\frac{2}{23}a^{3}-\frac{2}{23}a^{2}-\frac{1}{23}a+\frac{4}{23}$, $\frac{1}{23}a^{23}+\frac{6}{23}a^{21}+\frac{4}{23}a^{20}-\frac{1}{23}a^{18}+\frac{7}{23}a^{17}-\frac{7}{23}a^{16}-\frac{10}{23}a^{15}+\frac{6}{23}a^{14}-\frac{3}{23}a^{13}+\frac{3}{23}a^{12}-\frac{11}{23}a^{11}-\frac{5}{23}a^{10}-\frac{7}{23}a^{9}-\frac{6}{23}a^{8}-\frac{5}{23}a^{7}+\frac{6}{23}a^{6}-\frac{4}{23}a^{5}+\frac{9}{23}a^{4}+\frac{4}{23}a^{3}-\frac{7}{23}a^{2}+\frac{1}{23}a-\frac{11}{23}$, $\frac{1}{435105007}a^{24}-\frac{6511450}{435105007}a^{23}+\frac{6511474}{435105007}a^{22}-\frac{21353795}{435105007}a^{21}-\frac{29492337}{435105007}a^{20}+\frac{7684633}{435105007}a^{19}+\frac{170977751}{435105007}a^{18}+\frac{23875413}{435105007}a^{17}-\frac{175808914}{435105007}a^{16}+\frac{163443991}{435105007}a^{15}-\frac{35916163}{435105007}a^{14}-\frac{533332}{435105007}a^{13}-\frac{15687192}{435105007}a^{12}+\frac{138182201}{435105007}a^{11}-\frac{189374544}{435105007}a^{10}-\frac{197759169}{435105007}a^{9}-\frac{53900542}{435105007}a^{8}-\frac{11497280}{435105007}a^{7}-\frac{7263133}{18917609}a^{6}+\frac{179207798}{435105007}a^{5}+\frac{25522905}{435105007}a^{4}+\frac{11250547}{435105007}a^{3}+\frac{131851964}{435105007}a^{2}+\frac{183386491}{435105007}a+\frac{16110395}{435105007}$, $\frac{1}{13\!\cdots\!97}a^{25}-\frac{40\!\cdots\!66}{13\!\cdots\!97}a^{24}-\frac{69\!\cdots\!77}{13\!\cdots\!97}a^{23}+\frac{29\!\cdots\!07}{13\!\cdots\!97}a^{22}-\frac{13\!\cdots\!50}{13\!\cdots\!97}a^{21}-\frac{61\!\cdots\!30}{13\!\cdots\!97}a^{20}-\frac{67\!\cdots\!88}{13\!\cdots\!97}a^{19}-\frac{62\!\cdots\!16}{13\!\cdots\!97}a^{18}-\frac{37\!\cdots\!57}{13\!\cdots\!97}a^{17}+\frac{43\!\cdots\!21}{13\!\cdots\!97}a^{16}-\frac{67\!\cdots\!32}{13\!\cdots\!97}a^{15}-\frac{51\!\cdots\!25}{13\!\cdots\!97}a^{14}+\frac{38\!\cdots\!96}{13\!\cdots\!97}a^{13}+\frac{63\!\cdots\!81}{13\!\cdots\!97}a^{12}-\frac{19\!\cdots\!40}{13\!\cdots\!97}a^{11}+\frac{65\!\cdots\!72}{13\!\cdots\!97}a^{10}-\frac{30\!\cdots\!93}{16\!\cdots\!59}a^{9}+\frac{68\!\cdots\!40}{13\!\cdots\!97}a^{8}-\frac{66\!\cdots\!44}{13\!\cdots\!97}a^{7}-\frac{52\!\cdots\!11}{13\!\cdots\!97}a^{6}+\frac{15\!\cdots\!90}{13\!\cdots\!97}a^{5}-\frac{43\!\cdots\!22}{13\!\cdots\!97}a^{4}+\frac{50\!\cdots\!07}{13\!\cdots\!97}a^{3}+\frac{45\!\cdots\!99}{13\!\cdots\!97}a^{2}+\frac{37\!\cdots\!16}{13\!\cdots\!97}a-\frac{32\!\cdots\!36}{13\!\cdots\!97}$

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

Class group and class number

$C_{53}\times C_{53}$, which has order $2809$ (assuming GRH)

Unit group

Rank:		$12$
Torsion generator:		\( -\frac{229987870023954017059236497189737894134743541980}{3137680228602444474088975878125518885635965438371} a^{25} + \frac{217136480836551540404588632151115363363948328202}{3137680228602444474088975878125518885635965438371} a^{24} - \frac{5734368197620153028358493469558145698063203294598}{3137680228602444474088975878125518885635965438371} a^{23} + \frac{2895769853953695503445872483572537182694702551756}{3137680228602444474088975878125518885635965438371} a^{22} - \frac{92903090782795538430449179718308485758584153051060}{3137680228602444474088975878125518885635965438371} a^{21} + \frac{38910900127714861412854367362364439244898433859460}{3137680228602444474088975878125518885635965438371} a^{20} - \frac{825175169192090249025432598882797587021843700983579}{3137680228602444474088975878125518885635965438371} a^{19} + \frac{208619354791426775899170622718184951223205376339973}{3137680228602444474088975878125518885635965438371} a^{18} - \frac{5186025624562473117779830913522560002009908665030605}{3137680228602444474088975878125518885635965438371} a^{17} + \frac{950284046221967557819071670013626109187374876093148}{3137680228602444474088975878125518885635965438371} a^{16} - \frac{20026979216450597144491354405047755726381392519412945}{3137680228602444474088975878125518885635965438371} a^{15} + \frac{26026733398959552629061003677239193487412451502180}{3137680228602444474088975878125518885635965438371} a^{14} - \frac{53640081111470508555505879268919239920346609308326833}{3137680228602444474088975878125518885635965438371} a^{13} + \frac{1493283954000585215753084023395936297151511603120853}{3137680228602444474088975878125518885635965438371} a^{12} - \frac{83693528184162061027106813090510590280448512245797904}{3137680228602444474088975878125518885635965438371} a^{11} + \frac{2190262566064004435436005680569916912366060961235047}{3137680228602444474088975878125518885635965438371} a^{10} - \frac{91866436702243517298271969914984914726430188109122519}{3137680228602444474088975878125518885635965438371} a^{9} + \frac{7761435268729161722573189589323550986192528410571847}{3137680228602444474088975878125518885635965438371} a^{8} - \frac{51689186418254482677926859052944291289597257812419914}{3137680228602444474088975878125518885635965438371} a^{7} + \frac{4755838253545939984437862860116971870622966564403172}{3137680228602444474088975878125518885635965438371} a^{6} - \frac{20044193060199090632308192233670562915612723850599729}{3137680228602444474088975878125518885635965438371} a^{5} + \frac{2415825135601289194505318683440073804435597153774499}{3137680228602444474088975878125518885635965438371} a^{4} - \frac{1267780444146117601025600216051378277239936331305850}{3137680228602444474088975878125518885635965438371} a^{3} - \frac{180775132780258128332551968251888391460806926920950}{3137680228602444474088975878125518885635965438371} a^{2} - \frac{28118311719579089033381263797378756317127766600308}{3137680228602444474088975878125518885635965438371} a + \frac{1646897395182689836224362555359664278287504292319}{3137680228602444474088975878125518885635965438371} \)  (order $6$)
Fundamental units:		$\frac{57\!\cdots\!23}{13\!\cdots\!97}a^{25}-\frac{44\!\cdots\!89}{13\!\cdots\!97}a^{24}+\frac{14\!\cdots\!19}{13\!\cdots\!97}a^{23}-\frac{21\!\cdots\!51}{59\!\cdots\!39}a^{22}+\frac{23\!\cdots\!37}{13\!\cdots\!97}a^{21}-\frac{58\!\cdots\!82}{13\!\cdots\!97}a^{20}+\frac{20\!\cdots\!56}{13\!\cdots\!97}a^{19}-\frac{18\!\cdots\!99}{13\!\cdots\!97}a^{18}+\frac{12\!\cdots\!74}{13\!\cdots\!97}a^{17}-\frac{23\!\cdots\!51}{13\!\cdots\!97}a^{16}+\frac{50\!\cdots\!48}{13\!\cdots\!97}a^{15}+\frac{81\!\cdots\!74}{13\!\cdots\!97}a^{14}+\frac{13\!\cdots\!94}{13\!\cdots\!97}a^{13}+\frac{18\!\cdots\!19}{13\!\cdots\!97}a^{12}+\frac{21\!\cdots\!25}{13\!\cdots\!97}a^{11}+\frac{34\!\cdots\!98}{16\!\cdots\!59}a^{10}+\frac{23\!\cdots\!95}{13\!\cdots\!97}a^{9}+\frac{17\!\cdots\!06}{13\!\cdots\!97}a^{8}+\frac{13\!\cdots\!65}{13\!\cdots\!97}a^{7}+\frac{80\!\cdots\!81}{13\!\cdots\!97}a^{6}+\frac{51\!\cdots\!05}{13\!\cdots\!97}a^{5}+\frac{13\!\cdots\!72}{13\!\cdots\!97}a^{4}+\frac{35\!\cdots\!02}{13\!\cdots\!97}a^{3}+\frac{54\!\cdots\!49}{13\!\cdots\!97}a^{2}+\frac{18\!\cdots\!78}{13\!\cdots\!97}a+\frac{44\!\cdots\!00}{13\!\cdots\!97}$, $\frac{17\!\cdots\!90}{13\!\cdots\!97}a^{25}-\frac{20\!\cdots\!98}{13\!\cdots\!97}a^{24}+\frac{43\!\cdots\!94}{13\!\cdots\!97}a^{23}-\frac{32\!\cdots\!02}{13\!\cdots\!97}a^{22}+\frac{71\!\cdots\!32}{13\!\cdots\!97}a^{21}-\frac{46\!\cdots\!70}{13\!\cdots\!97}a^{20}+\frac{63\!\cdots\!12}{13\!\cdots\!97}a^{19}-\frac{30\!\cdots\!52}{13\!\cdots\!97}a^{18}+\frac{39\!\cdots\!44}{13\!\cdots\!97}a^{17}-\frac{16\!\cdots\!71}{13\!\cdots\!97}a^{16}+\frac{15\!\cdots\!52}{13\!\cdots\!97}a^{15}-\frac{36\!\cdots\!54}{13\!\cdots\!97}a^{14}+\frac{40\!\cdots\!70}{13\!\cdots\!97}a^{13}-\frac{10\!\cdots\!74}{13\!\cdots\!97}a^{12}+\frac{62\!\cdots\!56}{13\!\cdots\!97}a^{11}-\frac{20\!\cdots\!04}{16\!\cdots\!59}a^{10}+\frac{67\!\cdots\!04}{13\!\cdots\!97}a^{9}-\frac{22\!\cdots\!66}{13\!\cdots\!97}a^{8}+\frac{37\!\cdots\!78}{13\!\cdots\!97}a^{7}-\frac{12\!\cdots\!36}{13\!\cdots\!97}a^{6}+\frac{14\!\cdots\!80}{13\!\cdots\!97}a^{5}-\frac{49\!\cdots\!86}{13\!\cdots\!97}a^{4}+\frac{79\!\cdots\!48}{13\!\cdots\!97}a^{3}+\frac{95\!\cdots\!44}{13\!\cdots\!97}a^{2}-\frac{66\!\cdots\!52}{13\!\cdots\!97}a+\frac{83\!\cdots\!64}{13\!\cdots\!97}$, $\frac{12\!\cdots\!78}{31\!\cdots\!71}a^{25}-\frac{15\!\cdots\!02}{31\!\cdots\!71}a^{24}+\frac{32\!\cdots\!64}{31\!\cdots\!71}a^{23}-\frac{24\!\cdots\!40}{31\!\cdots\!71}a^{22}+\frac{52\!\cdots\!00}{31\!\cdots\!71}a^{21}-\frac{34\!\cdots\!61}{31\!\cdots\!71}a^{20}+\frac{46\!\cdots\!07}{31\!\cdots\!71}a^{19}-\frac{23\!\cdots\!95}{31\!\cdots\!71}a^{18}+\frac{29\!\cdots\!72}{31\!\cdots\!71}a^{17}-\frac{12\!\cdots\!75}{31\!\cdots\!71}a^{16}+\frac{11\!\cdots\!00}{31\!\cdots\!71}a^{15}-\frac{28\!\cdots\!47}{31\!\cdots\!71}a^{14}+\frac{30\!\cdots\!27}{31\!\cdots\!71}a^{13}-\frac{82\!\cdots\!16}{31\!\cdots\!71}a^{12}+\frac{48\!\cdots\!13}{31\!\cdots\!71}a^{11}-\frac{12\!\cdots\!61}{31\!\cdots\!71}a^{10}+\frac{53\!\cdots\!33}{31\!\cdots\!71}a^{9}-\frac{16\!\cdots\!06}{31\!\cdots\!71}a^{8}+\frac{32\!\cdots\!88}{31\!\cdots\!71}a^{7}-\frac{98\!\cdots\!91}{31\!\cdots\!71}a^{6}+\frac{13\!\cdots\!81}{31\!\cdots\!71}a^{5}-\frac{41\!\cdots\!70}{31\!\cdots\!71}a^{4}+\frac{15\!\cdots\!10}{31\!\cdots\!71}a^{3}-\frac{10\!\cdots\!32}{31\!\cdots\!71}a^{2}+\frac{65\!\cdots\!81}{31\!\cdots\!71}a-\frac{22\!\cdots\!80}{31\!\cdots\!71}$, $\frac{18\!\cdots\!57}{13\!\cdots\!97}a^{25}-\frac{17\!\cdots\!57}{13\!\cdots\!97}a^{24}+\frac{45\!\cdots\!29}{13\!\cdots\!97}a^{23}-\frac{23\!\cdots\!97}{13\!\cdots\!97}a^{22}+\frac{74\!\cdots\!96}{13\!\cdots\!97}a^{21}-\frac{32\!\cdots\!25}{13\!\cdots\!97}a^{20}+\frac{65\!\cdots\!03}{13\!\cdots\!97}a^{19}-\frac{17\!\cdots\!18}{13\!\cdots\!97}a^{18}+\frac{41\!\cdots\!22}{13\!\cdots\!97}a^{17}-\frac{81\!\cdots\!81}{13\!\cdots\!97}a^{16}+\frac{15\!\cdots\!96}{13\!\cdots\!97}a^{15}-\frac{24\!\cdots\!77}{13\!\cdots\!97}a^{14}+\frac{42\!\cdots\!46}{13\!\cdots\!97}a^{13}-\frac{18\!\cdots\!39}{13\!\cdots\!97}a^{12}+\frac{65\!\cdots\!44}{13\!\cdots\!97}a^{11}-\frac{34\!\cdots\!60}{16\!\cdots\!59}a^{10}+\frac{72\!\cdots\!38}{13\!\cdots\!97}a^{9}-\frac{74\!\cdots\!81}{13\!\cdots\!97}a^{8}+\frac{40\!\cdots\!28}{13\!\cdots\!97}a^{7}-\frac{44\!\cdots\!16}{13\!\cdots\!97}a^{6}+\frac{15\!\cdots\!86}{13\!\cdots\!97}a^{5}-\frac{22\!\cdots\!56}{13\!\cdots\!97}a^{4}+\frac{98\!\cdots\!84}{13\!\cdots\!97}a^{3}+\frac{13\!\cdots\!52}{13\!\cdots\!97}a^{2}+\frac{96\!\cdots\!86}{13\!\cdots\!97}a+\frac{11\!\cdots\!10}{13\!\cdots\!97}$, $\frac{49\!\cdots\!18}{13\!\cdots\!97}a^{25}-\frac{51\!\cdots\!45}{13\!\cdots\!97}a^{24}+\frac{12\!\cdots\!09}{13\!\cdots\!97}a^{23}-\frac{74\!\cdots\!73}{13\!\cdots\!97}a^{22}+\frac{20\!\cdots\!50}{13\!\cdots\!97}a^{21}-\frac{10\!\cdots\!01}{13\!\cdots\!97}a^{20}+\frac{17\!\cdots\!56}{13\!\cdots\!97}a^{19}-\frac{61\!\cdots\!07}{13\!\cdots\!97}a^{18}+\frac{11\!\cdots\!61}{13\!\cdots\!97}a^{17}-\frac{31\!\cdots\!57}{13\!\cdots\!97}a^{16}+\frac{43\!\cdots\!93}{13\!\cdots\!97}a^{15}-\frac{41\!\cdots\!63}{13\!\cdots\!97}a^{14}+\frac{11\!\cdots\!90}{13\!\cdots\!97}a^{13}-\frac{14\!\cdots\!10}{13\!\cdots\!97}a^{12}+\frac{17\!\cdots\!54}{13\!\cdots\!97}a^{11}-\frac{26\!\cdots\!55}{16\!\cdots\!59}a^{10}+\frac{19\!\cdots\!55}{13\!\cdots\!97}a^{9}-\frac{35\!\cdots\!74}{13\!\cdots\!97}a^{8}+\frac{10\!\cdots\!46}{13\!\cdots\!97}a^{7}-\frac{20\!\cdots\!00}{13\!\cdots\!97}a^{6}+\frac{42\!\cdots\!13}{13\!\cdots\!97}a^{5}-\frac{89\!\cdots\!35}{13\!\cdots\!97}a^{4}+\frac{25\!\cdots\!22}{13\!\cdots\!97}a^{3}+\frac{33\!\cdots\!86}{13\!\cdots\!97}a^{2}+\frac{16\!\cdots\!67}{13\!\cdots\!97}a+\frac{28\!\cdots\!58}{13\!\cdots\!97}$, $\frac{18\!\cdots\!99}{16\!\cdots\!59}a^{25}-\frac{16\!\cdots\!98}{16\!\cdots\!59}a^{24}+\frac{46\!\cdots\!72}{16\!\cdots\!59}a^{23}-\frac{20\!\cdots\!91}{16\!\cdots\!59}a^{22}+\frac{74\!\cdots\!21}{16\!\cdots\!59}a^{21}-\frac{27\!\cdots\!65}{16\!\cdots\!59}a^{20}+\frac{66\!\cdots\!26}{16\!\cdots\!59}a^{19}-\frac{13\!\cdots\!24}{16\!\cdots\!59}a^{18}+\frac{41\!\cdots\!11}{16\!\cdots\!59}a^{17}-\frac{52\!\cdots\!54}{16\!\cdots\!59}a^{16}+\frac{16\!\cdots\!61}{16\!\cdots\!59}a^{15}+\frac{88\!\cdots\!45}{16\!\cdots\!59}a^{14}+\frac{43\!\cdots\!30}{16\!\cdots\!59}a^{13}+\frac{12\!\cdots\!13}{16\!\cdots\!59}a^{12}+\frac{67\!\cdots\!13}{16\!\cdots\!59}a^{11}+\frac{20\!\cdots\!84}{16\!\cdots\!59}a^{10}+\frac{74\!\cdots\!52}{16\!\cdots\!59}a^{9}-\frac{21\!\cdots\!54}{16\!\cdots\!59}a^{8}+\frac{42\!\cdots\!64}{16\!\cdots\!59}a^{7}-\frac{16\!\cdots\!53}{16\!\cdots\!59}a^{6}+\frac{16\!\cdots\!24}{16\!\cdots\!59}a^{5}-\frac{11\!\cdots\!58}{16\!\cdots\!59}a^{4}+\frac{10\!\cdots\!84}{16\!\cdots\!59}a^{3}+\frac{15\!\cdots\!21}{16\!\cdots\!59}a^{2}+\frac{34\!\cdots\!80}{16\!\cdots\!59}a+\frac{12\!\cdots\!58}{16\!\cdots\!59}$, $\frac{16\!\cdots\!08}{13\!\cdots\!97}a^{25}-\frac{13\!\cdots\!24}{13\!\cdots\!97}a^{24}+\frac{40\!\cdots\!62}{13\!\cdots\!97}a^{23}-\frac{16\!\cdots\!99}{13\!\cdots\!97}a^{22}+\frac{65\!\cdots\!09}{13\!\cdots\!97}a^{21}-\frac{21\!\cdots\!95}{13\!\cdots\!97}a^{20}+\frac{58\!\cdots\!32}{13\!\cdots\!97}a^{19}-\frac{93\!\cdots\!82}{13\!\cdots\!97}a^{18}+\frac{15\!\cdots\!65}{59\!\cdots\!39}a^{17}-\frac{33\!\cdots\!45}{13\!\cdots\!97}a^{16}+\frac{14\!\cdots\!24}{13\!\cdots\!97}a^{15}+\frac{12\!\cdots\!58}{13\!\cdots\!97}a^{14}+\frac{38\!\cdots\!32}{13\!\cdots\!97}a^{13}+\frac{24\!\cdots\!78}{13\!\cdots\!97}a^{12}+\frac{59\!\cdots\!89}{13\!\cdots\!97}a^{11}+\frac{46\!\cdots\!54}{16\!\cdots\!59}a^{10}+\frac{65\!\cdots\!35}{13\!\cdots\!97}a^{9}+\frac{42\!\cdots\!54}{13\!\cdots\!97}a^{8}+\frac{37\!\cdots\!12}{13\!\cdots\!97}a^{7}-\frac{19\!\cdots\!78}{13\!\cdots\!97}a^{6}+\frac{14\!\cdots\!39}{13\!\cdots\!97}a^{5}-\frac{52\!\cdots\!82}{13\!\cdots\!97}a^{4}+\frac{95\!\cdots\!46}{13\!\cdots\!97}a^{3}+\frac{14\!\cdots\!10}{13\!\cdots\!97}a^{2}+\frac{31\!\cdots\!92}{13\!\cdots\!97}a+\frac{11\!\cdots\!12}{13\!\cdots\!97}$, $\frac{10\!\cdots\!06}{13\!\cdots\!97}a^{25}-\frac{84\!\cdots\!14}{13\!\cdots\!97}a^{24}+\frac{26\!\cdots\!34}{13\!\cdots\!97}a^{23}-\frac{94\!\cdots\!70}{13\!\cdots\!97}a^{22}+\frac{42\!\cdots\!40}{13\!\cdots\!97}a^{21}-\frac{11\!\cdots\!62}{13\!\cdots\!97}a^{20}+\frac{37\!\cdots\!80}{13\!\cdots\!97}a^{19}-\frac{40\!\cdots\!70}{13\!\cdots\!97}a^{18}+\frac{23\!\cdots\!50}{13\!\cdots\!97}a^{17}-\frac{87\!\cdots\!22}{13\!\cdots\!97}a^{16}+\frac{91\!\cdots\!16}{13\!\cdots\!97}a^{15}+\frac{13\!\cdots\!22}{13\!\cdots\!97}a^{14}+\frac{24\!\cdots\!10}{13\!\cdots\!97}a^{13}+\frac{29\!\cdots\!08}{13\!\cdots\!97}a^{12}+\frac{37\!\cdots\!68}{13\!\cdots\!97}a^{11}+\frac{46\!\cdots\!18}{13\!\cdots\!97}a^{10}+\frac{41\!\cdots\!98}{13\!\cdots\!97}a^{9}+\frac{25\!\cdots\!92}{13\!\cdots\!97}a^{8}+\frac{22\!\cdots\!12}{13\!\cdots\!97}a^{7}+\frac{12\!\cdots\!12}{13\!\cdots\!97}a^{6}+\frac{82\!\cdots\!14}{13\!\cdots\!97}a^{5}+\frac{24\!\cdots\!50}{13\!\cdots\!97}a^{4}+\frac{16\!\cdots\!88}{13\!\cdots\!97}a^{3}+\frac{17\!\cdots\!31}{13\!\cdots\!97}a^{2}+\frac{22\!\cdots\!68}{13\!\cdots\!97}a-\frac{90\!\cdots\!34}{13\!\cdots\!97}$, $\frac{15\!\cdots\!88}{13\!\cdots\!97}a^{25}-\frac{13\!\cdots\!70}{13\!\cdots\!97}a^{24}+\frac{38\!\cdots\!70}{13\!\cdots\!97}a^{23}-\frac{15\!\cdots\!50}{13\!\cdots\!97}a^{22}+\frac{63\!\cdots\!50}{13\!\cdots\!97}a^{21}-\frac{19\!\cdots\!90}{13\!\cdots\!97}a^{20}+\frac{55\!\cdots\!80}{13\!\cdots\!97}a^{19}-\frac{82\!\cdots\!25}{13\!\cdots\!97}a^{18}+\frac{35\!\cdots\!95}{13\!\cdots\!97}a^{17}-\frac{27\!\cdots\!15}{13\!\cdots\!97}a^{16}+\frac{13\!\cdots\!45}{13\!\cdots\!97}a^{15}+\frac{14\!\cdots\!20}{13\!\cdots\!97}a^{14}+\frac{36\!\cdots\!15}{13\!\cdots\!97}a^{13}+\frac{28\!\cdots\!15}{13\!\cdots\!97}a^{12}+\frac{57\!\cdots\!90}{13\!\cdots\!97}a^{11}+\frac{53\!\cdots\!00}{16\!\cdots\!59}a^{10}+\frac{63\!\cdots\!05}{13\!\cdots\!97}a^{9}+\frac{11\!\cdots\!05}{13\!\cdots\!97}a^{8}+\frac{35\!\cdots\!45}{13\!\cdots\!97}a^{7}+\frac{21\!\cdots\!60}{13\!\cdots\!97}a^{6}+\frac{13\!\cdots\!55}{13\!\cdots\!97}a^{5}-\frac{35\!\cdots\!00}{13\!\cdots\!97}a^{4}+\frac{93\!\cdots\!30}{13\!\cdots\!97}a^{3}+\frac{13\!\cdots\!70}{13\!\cdots\!97}a^{2}+\frac{37\!\cdots\!25}{13\!\cdots\!97}a+\frac{11\!\cdots\!80}{13\!\cdots\!97}$, $\frac{21\!\cdots\!19}{13\!\cdots\!97}a^{25}-\frac{19\!\cdots\!72}{13\!\cdots\!97}a^{24}+\frac{53\!\cdots\!44}{13\!\cdots\!97}a^{23}-\frac{23\!\cdots\!83}{13\!\cdots\!97}a^{22}+\frac{87\!\cdots\!50}{13\!\cdots\!97}a^{21}-\frac{31\!\cdots\!56}{13\!\cdots\!97}a^{20}+\frac{77\!\cdots\!93}{13\!\cdots\!97}a^{19}-\frac{14\!\cdots\!11}{13\!\cdots\!97}a^{18}+\frac{48\!\cdots\!83}{13\!\cdots\!97}a^{17}-\frac{25\!\cdots\!03}{59\!\cdots\!39}a^{16}+\frac{18\!\cdots\!28}{13\!\cdots\!97}a^{15}+\frac{11\!\cdots\!73}{13\!\cdots\!97}a^{14}+\frac{50\!\cdots\!99}{13\!\cdots\!97}a^{13}+\frac{17\!\cdots\!40}{13\!\cdots\!97}a^{12}+\frac{79\!\cdots\!14}{13\!\cdots\!97}a^{11}+\frac{14\!\cdots\!27}{71\!\cdots\!33}a^{10}+\frac{87\!\cdots\!40}{13\!\cdots\!97}a^{9}-\frac{19\!\cdots\!27}{13\!\cdots\!97}a^{8}+\frac{49\!\cdots\!37}{13\!\cdots\!97}a^{7}-\frac{16\!\cdots\!05}{13\!\cdots\!97}a^{6}+\frac{19\!\cdots\!70}{13\!\cdots\!97}a^{5}-\frac{12\!\cdots\!08}{13\!\cdots\!97}a^{4}+\frac{12\!\cdots\!72}{13\!\cdots\!97}a^{3}+\frac{18\!\cdots\!63}{13\!\cdots\!97}a^{2}+\frac{46\!\cdots\!60}{13\!\cdots\!97}a+\frac{15\!\cdots\!26}{13\!\cdots\!97}$, $\frac{10\!\cdots\!39}{13\!\cdots\!97}a^{25}-\frac{12\!\cdots\!51}{13\!\cdots\!97}a^{24}+\frac{27\!\cdots\!38}{13\!\cdots\!97}a^{23}-\frac{18\!\cdots\!74}{13\!\cdots\!97}a^{22}+\frac{43\!\cdots\!09}{13\!\cdots\!97}a^{21}-\frac{25\!\cdots\!80}{13\!\cdots\!97}a^{20}+\frac{39\!\cdots\!18}{13\!\cdots\!97}a^{19}-\frac{16\!\cdots\!62}{13\!\cdots\!97}a^{18}+\frac{24\!\cdots\!98}{13\!\cdots\!97}a^{17}-\frac{86\!\cdots\!29}{13\!\cdots\!97}a^{16}+\frac{95\!\cdots\!84}{13\!\cdots\!97}a^{15}-\frac{16\!\cdots\!28}{13\!\cdots\!97}a^{14}+\frac{25\!\cdots\!74}{13\!\cdots\!97}a^{13}-\frac{50\!\cdots\!44}{13\!\cdots\!97}a^{12}+\frac{40\!\cdots\!76}{13\!\cdots\!97}a^{11}-\frac{78\!\cdots\!59}{13\!\cdots\!97}a^{10}+\frac{19\!\cdots\!53}{59\!\cdots\!39}a^{9}-\frac{11\!\cdots\!37}{13\!\cdots\!97}a^{8}+\frac{26\!\cdots\!59}{13\!\cdots\!97}a^{7}-\frac{65\!\cdots\!37}{13\!\cdots\!97}a^{6}+\frac{10\!\cdots\!94}{13\!\cdots\!97}a^{5}-\frac{28\!\cdots\!44}{13\!\cdots\!97}a^{4}+\frac{11\!\cdots\!45}{13\!\cdots\!97}a^{3}-\frac{69\!\cdots\!55}{13\!\cdots\!97}a^{2}+\frac{26\!\cdots\!28}{13\!\cdots\!97}a-\frac{70\!\cdots\!22}{59\!\cdots\!39}$, $\frac{12\!\cdots\!88}{13\!\cdots\!97}a^{25}-\frac{14\!\cdots\!56}{13\!\cdots\!97}a^{24}+\frac{30\!\cdots\!78}{13\!\cdots\!97}a^{23}-\frac{21\!\cdots\!56}{13\!\cdots\!97}a^{22}+\frac{49\!\cdots\!00}{13\!\cdots\!97}a^{21}-\frac{31\!\cdots\!77}{13\!\cdots\!97}a^{20}+\frac{43\!\cdots\!53}{13\!\cdots\!97}a^{19}-\frac{20\!\cdots\!74}{13\!\cdots\!97}a^{18}+\frac{27\!\cdots\!40}{13\!\cdots\!97}a^{17}-\frac{10\!\cdots\!53}{13\!\cdots\!97}a^{16}+\frac{46\!\cdots\!30}{59\!\cdots\!39}a^{15}-\frac{23\!\cdots\!46}{13\!\cdots\!97}a^{14}+\frac{28\!\cdots\!32}{13\!\cdots\!97}a^{13}-\frac{69\!\cdots\!80}{13\!\cdots\!97}a^{12}+\frac{19\!\cdots\!50}{59\!\cdots\!39}a^{11}-\frac{10\!\cdots\!93}{13\!\cdots\!97}a^{10}+\frac{49\!\cdots\!36}{13\!\cdots\!97}a^{9}-\frac{14\!\cdots\!53}{13\!\cdots\!97}a^{8}+\frac{28\!\cdots\!90}{13\!\cdots\!97}a^{7}-\frac{85\!\cdots\!91}{13\!\cdots\!97}a^{6}+\frac{11\!\cdots\!14}{13\!\cdots\!97}a^{5}-\frac{36\!\cdots\!33}{13\!\cdots\!97}a^{4}+\frac{11\!\cdots\!85}{13\!\cdots\!97}a^{3}-\frac{40\!\cdots\!86}{59\!\cdots\!39}a^{2}+\frac{57\!\cdots\!98}{13\!\cdots\!97}a-\frac{22\!\cdots\!68}{13\!\cdots\!97}$ (assuming GRH)
Regulator:		\( 5382739421.971964 \) (assuming GRH)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{13}\cdot 5382739421.971964 \cdot 2809}{6\cdot\sqrt{384766437057818380952237905666104641217782272563}}\cr\approx \mathstrut & 0.0966370230307511 \end{aligned}\] (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

\(\Q(\sqrt{-3}) \), 13.13.491258904256726154641.1

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	$26$	R	$26$	${\href{/padicField/7.13.0.1}{13} }^{2}$	$26$	${\href{/padicField/13.13.0.1}{13} }^{2}$	$26$	${\href{/padicField/19.13.0.1}{13} }^{2}$	${\href{/padicField/23.2.0.1}{2} }^{13}$	$26$	${\href{/padicField/31.13.0.1}{13} }^{2}$	${\href{/padicField/37.13.0.1}{13} }^{2}$	$26$	${\href{/padicField/43.13.0.1}{13} }^{2}$	$26$	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
\(3\)		Deg $26$	$2$	$13$	$13$
\(53\)		Deg $26$	$13$	$2$	$24$