Number field 28.0.792537323068373529244880273632877655015215903801344.1

• Label: 28.0.792...344.1
• Degree: $28$
• Signature: $[0, 14]$
• Discriminant: $7.925\times 10^{50}$
• Root discriminant: \(65.74\)
• Ramified primes: $2,43$
• Class number: $43$ (GRH)
• Class group: [43] (GRH)
• Galois group: $C_2\times C_{14}$ (as 28T2)

Normalized defining polynomial

x^28 - 3*x^26 + 18*x^24 + 304*x^22 - 431*x^20 + 627*x^18 + 12915*x^16 - 6007*x^14 + 76379*x^12 - 24573*x^10 + 152173*x^8 - 11756*x^6 + 110547*x^4 - 8906*x^2 + 6241

Invariants

Degree:		$28$
Signature:		$[0, 14]$
Discriminant:		\(792537323068373529244880273632877655015215903801344\) \(\medspace = 2^{28}\cdot 43^{26}\)
Root discriminant:		\(65.74\)
Galois root discriminant:		$2\cdot 43^{13/14}\approx 65.73885461515177$
Ramified primes:		\(2\), \(43\)
Discriminant root field:		\(\Q\)
$\card{ \Gal(K/\Q) }$:		$28$
This field is Galois and abelian over $\Q$.
Conductor:		\(172=2^{2}\cdot 43\)
Dirichlet character group:		$\lbrace$$\chi_{172}(1,·)$, $\chi_{172}(131,·)$, $\chi_{172}(97,·)$, $\chi_{172}(133,·)$, $\chi_{172}(65,·)$, $\chi_{172}(137,·)$, $\chi_{172}(11,·)$, $\chi_{172}(145,·)$, $\chi_{172}(75,·)$, $\chi_{172}(107,·)$, $\chi_{172}(21,·)$, $\chi_{172}(151,·)$, $\chi_{172}(127,·)$, $\chi_{172}(27,·)$, $\chi_{172}(161,·)$, $\chi_{172}(35,·)$, $\chi_{172}(39,·)$, $\chi_{172}(41,·)$, $\chi_{172}(171,·)$, $\chi_{172}(45,·)$, $\chi_{172}(47,·)$, $\chi_{172}(113,·)$, $\chi_{172}(51,·)$, $\chi_{172}(121,·)$, $\chi_{172}(87,·)$, $\chi_{172}(59,·)$, $\chi_{172}(125,·)$, $\chi_{172}(85,·)$$\rbrace$
This is a CM field.
Reflex fields:		unavailable$^{8192}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $\frac{1}{79}a^{23}-\frac{26}{79}a^{21}-\frac{10}{79}a^{19}-\frac{17}{79}a^{17}-\frac{21}{79}a^{15}-\frac{19}{79}a^{13}+\frac{33}{79}a^{11}-\frac{7}{79}a^{9}+\frac{29}{79}a^{7}-\frac{2}{79}a^{5}+\frac{2}{79}a^{3}+\frac{36}{79}a$, $\frac{1}{59824093}a^{24}+\frac{408957}{59824093}a^{22}+\frac{1439751}{8546299}a^{20}+\frac{1682122}{8546299}a^{18}-\frac{6127419}{59824093}a^{16}-\frac{23873345}{59824093}a^{14}-\frac{27757328}{59824093}a^{12}+\frac{14546737}{59824093}a^{10}+\frac{16919459}{59824093}a^{8}+\frac{25292559}{59824093}a^{6}-\frac{15757338}{59824093}a^{4}-\frac{28138816}{59824093}a^{2}+\frac{32748}{757267}$, $\frac{1}{59824093}a^{25}-\frac{348310}{59824093}a^{23}+\frac{4252457}{8546299}a^{21}+\frac{2763932}{8546299}a^{19}+\frac{6746120}{59824093}a^{17}-\frac{7970738}{59824093}a^{15}-\frac{13369255}{59824093}a^{13}-\frac{10443074}{59824093}a^{11}+\frac{22220328}{59824093}a^{9}+\frac{3331816}{59824093}a^{7}-\frac{14242804}{59824093}a^{5}-\frac{29653350}{59824093}a^{3}-\frac{24674520}{59824093}a$, $\frac{1}{64\!\cdots\!23}a^{26}-\frac{18\!\cdots\!35}{25\!\cdots\!73}a^{24}+\frac{81\!\cdots\!03}{64\!\cdots\!23}a^{22}+\frac{18\!\cdots\!62}{92\!\cdots\!89}a^{20}-\frac{75\!\cdots\!19}{64\!\cdots\!23}a^{18}-\frac{15\!\cdots\!04}{64\!\cdots\!23}a^{16}-\frac{27\!\cdots\!62}{64\!\cdots\!23}a^{14}+\frac{24\!\cdots\!62}{64\!\cdots\!23}a^{12}+\frac{11\!\cdots\!79}{64\!\cdots\!23}a^{10}+\frac{81\!\cdots\!70}{64\!\cdots\!23}a^{8}-\frac{84\!\cdots\!48}{64\!\cdots\!23}a^{6}-\frac{29\!\cdots\!07}{64\!\cdots\!23}a^{4}-\frac{23\!\cdots\!84}{64\!\cdots\!23}a^{2}+\frac{22\!\cdots\!27}{10\!\cdots\!03}$, $\frac{1}{64\!\cdots\!23}a^{27}-\frac{18\!\cdots\!35}{25\!\cdots\!73}a^{25}-\frac{45\!\cdots\!67}{64\!\cdots\!23}a^{23}+\frac{45\!\cdots\!55}{92\!\cdots\!89}a^{21}+\frac{96\!\cdots\!58}{64\!\cdots\!23}a^{19}-\frac{58\!\cdots\!60}{64\!\cdots\!23}a^{17}+\frac{15\!\cdots\!62}{64\!\cdots\!23}a^{15}-\frac{13\!\cdots\!77}{64\!\cdots\!23}a^{13}+\frac{56\!\cdots\!61}{64\!\cdots\!23}a^{11}+\frac{81\!\cdots\!37}{64\!\cdots\!23}a^{9}+\frac{12\!\cdots\!14}{64\!\cdots\!23}a^{7}-\frac{13\!\cdots\!67}{64\!\cdots\!23}a^{5}+\frac{25\!\cdots\!99}{64\!\cdots\!23}a^{3}-\frac{27\!\cdots\!99}{81\!\cdots\!37}a$

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

Class group and class number

$C_{43}$, which has order $43$ (assuming GRH)

Unit group

Rank:		$13$
Torsion generator:		\( \frac{34746942274768009}{733138626249524197283} a^{27} - \frac{90849002669170407}{733138626249524197283} a^{25} + \frac{604982509902124085}{733138626249524197283} a^{23} + \frac{1533783770301570770}{104734089464217742469} a^{21} - \frac{10551754272131332116}{733138626249524197283} a^{19} + \frac{21953131397543139958}{733138626249524197283} a^{17} + \frac{445553048340535517168}{733138626249524197283} a^{15} - \frac{27840992264299803552}{733138626249524197283} a^{13} + \frac{2840242315285543325081}{733138626249524197283} a^{11} - \frac{78138797732291591290}{733138626249524197283} a^{9} + \frac{6149681341485093545410}{733138626249524197283} a^{7} + \frac{917775515316363541979}{733138626249524197283} a^{5} + \frac{4884488566730410537874}{733138626249524197283} a^{3} + \frac{10450539251070167142}{9280235775310432877} a \)  (order $4$)
Fundamental units:		$\frac{76\!\cdots\!10}{92\!\cdots\!89}a^{26}-\frac{35\!\cdots\!36}{92\!\cdots\!89}a^{24}+\frac{18\!\cdots\!12}{92\!\cdots\!89}a^{22}+\frac{20\!\cdots\!24}{92\!\cdots\!89}a^{20}-\frac{67\!\cdots\!26}{92\!\cdots\!89}a^{18}+\frac{11\!\cdots\!50}{92\!\cdots\!89}a^{16}+\frac{68\!\cdots\!24}{92\!\cdots\!89}a^{14}-\frac{19\!\cdots\!27}{92\!\cdots\!89}a^{12}+\frac{79\!\cdots\!64}{92\!\cdots\!89}a^{10}-\frac{19\!\cdots\!26}{92\!\cdots\!89}a^{8}+\frac{12\!\cdots\!32}{92\!\cdots\!89}a^{6}-\frac{40\!\cdots\!46}{92\!\cdots\!89}a^{4}+\frac{40\!\cdots\!63}{92\!\cdots\!89}a^{2}-\frac{26\!\cdots\!28}{14\!\cdots\!29}$, $\frac{47\!\cdots\!34}{92\!\cdots\!89}a^{26}-\frac{21\!\cdots\!14}{92\!\cdots\!89}a^{24}+\frac{98\!\cdots\!36}{92\!\cdots\!89}a^{22}+\frac{13\!\cdots\!83}{92\!\cdots\!89}a^{20}-\frac{42\!\cdots\!07}{92\!\cdots\!89}a^{18}+\frac{36\!\cdots\!41}{92\!\cdots\!89}a^{16}+\frac{59\!\cdots\!46}{92\!\cdots\!89}a^{14}-\frac{11\!\cdots\!91}{92\!\cdots\!89}a^{12}+\frac{29\!\cdots\!35}{92\!\cdots\!89}a^{10}-\frac{62\!\cdots\!78}{92\!\cdots\!89}a^{8}+\frac{47\!\cdots\!75}{92\!\cdots\!89}a^{6}-\frac{11\!\cdots\!91}{92\!\cdots\!89}a^{4}+\frac{12\!\cdots\!59}{92\!\cdots\!89}a^{2}-\frac{12\!\cdots\!43}{14\!\cdots\!29}$, $\frac{23\!\cdots\!76}{92\!\cdots\!89}a^{26}+\frac{37\!\cdots\!87}{92\!\cdots\!89}a^{24}+\frac{15\!\cdots\!34}{92\!\cdots\!89}a^{22}+\frac{86\!\cdots\!31}{92\!\cdots\!89}a^{20}+\frac{11\!\cdots\!73}{92\!\cdots\!89}a^{18}-\frac{33\!\cdots\!69}{92\!\cdots\!89}a^{16}+\frac{35\!\cdots\!65}{92\!\cdots\!89}a^{14}+\frac{83\!\cdots\!00}{92\!\cdots\!89}a^{12}+\frac{69\!\cdots\!05}{92\!\cdots\!89}a^{10}+\frac{45\!\cdots\!87}{92\!\cdots\!89}a^{8}-\frac{10\!\cdots\!00}{92\!\cdots\!89}a^{6}+\frac{80\!\cdots\!91}{92\!\cdots\!89}a^{4}-\frac{72\!\cdots\!10}{92\!\cdots\!89}a^{2}+\frac{43\!\cdots\!65}{14\!\cdots\!29}$, $\frac{30\!\cdots\!91}{92\!\cdots\!89}a^{26}-\frac{34\!\cdots\!48}{92\!\cdots\!89}a^{24}+\frac{14\!\cdots\!97}{92\!\cdots\!89}a^{22}+\frac{40\!\cdots\!02}{92\!\cdots\!89}a^{20}-\frac{89\!\cdots\!63}{92\!\cdots\!89}a^{18}+\frac{15\!\cdots\!56}{92\!\cdots\!89}a^{16}+\frac{10\!\cdots\!24}{92\!\cdots\!89}a^{14}-\frac{34\!\cdots\!88}{92\!\cdots\!89}a^{12}+\frac{52\!\cdots\!50}{92\!\cdots\!89}a^{10}-\frac{24\!\cdots\!33}{92\!\cdots\!89}a^{8}+\frac{12\!\cdots\!12}{92\!\cdots\!89}a^{6}-\frac{46\!\cdots\!40}{92\!\cdots\!89}a^{4}+\frac{45\!\cdots\!65}{92\!\cdots\!89}a^{2}-\frac{13\!\cdots\!15}{34\!\cdots\!59}$, $\frac{21\!\cdots\!15}{92\!\cdots\!89}a^{26}-\frac{47\!\cdots\!16}{92\!\cdots\!89}a^{24}+\frac{29\!\cdots\!08}{92\!\cdots\!89}a^{22}+\frac{69\!\cdots\!48}{92\!\cdots\!89}a^{20}-\frac{46\!\cdots\!22}{92\!\cdots\!89}a^{18}-\frac{79\!\cdots\!63}{92\!\cdots\!89}a^{16}+\frac{30\!\cdots\!36}{92\!\cdots\!89}a^{14}+\frac{96\!\cdots\!05}{92\!\cdots\!89}a^{12}+\frac{92\!\cdots\!64}{92\!\cdots\!89}a^{10}+\frac{82\!\cdots\!71}{92\!\cdots\!89}a^{8}+\frac{51\!\cdots\!72}{92\!\cdots\!89}a^{6}+\frac{13\!\cdots\!47}{92\!\cdots\!89}a^{4}-\frac{83\!\cdots\!77}{92\!\cdots\!89}a^{2}-\frac{37\!\cdots\!32}{14\!\cdots\!29}$, $\frac{17\!\cdots\!16}{92\!\cdots\!89}a^{26}-\frac{66\!\cdots\!41}{92\!\cdots\!89}a^{24}+\frac{21\!\cdots\!74}{92\!\cdots\!89}a^{22}-\frac{55\!\cdots\!88}{92\!\cdots\!89}a^{20}-\frac{19\!\cdots\!18}{92\!\cdots\!89}a^{18}+\frac{29\!\cdots\!82}{92\!\cdots\!89}a^{16}-\frac{84\!\cdots\!56}{92\!\cdots\!89}a^{14}-\frac{81\!\cdots\!08}{92\!\cdots\!89}a^{12}+\frac{54\!\cdots\!16}{92\!\cdots\!89}a^{10}-\frac{43\!\cdots\!82}{92\!\cdots\!89}a^{8}+\frac{20\!\cdots\!66}{92\!\cdots\!89}a^{6}-\frac{79\!\cdots\!53}{92\!\cdots\!89}a^{4}+\frac{77\!\cdots\!42}{92\!\cdots\!89}a^{2}-\frac{10\!\cdots\!68}{14\!\cdots\!29}$, $\frac{14\!\cdots\!74}{64\!\cdots\!23}a^{27}+\frac{40\!\cdots\!28}{64\!\cdots\!23}a^{26}-\frac{42\!\cdots\!02}{64\!\cdots\!23}a^{25}-\frac{14\!\cdots\!92}{64\!\cdots\!23}a^{24}+\frac{25\!\cdots\!07}{64\!\cdots\!23}a^{23}+\frac{80\!\cdots\!20}{64\!\cdots\!23}a^{22}+\frac{63\!\cdots\!03}{92\!\cdots\!89}a^{21}+\frac{16\!\cdots\!78}{92\!\cdots\!89}a^{20}-\frac{59\!\cdots\!91}{64\!\cdots\!23}a^{19}-\frac{23\!\cdots\!02}{64\!\cdots\!23}a^{18}+\frac{83\!\cdots\!88}{64\!\cdots\!23}a^{17}+\frac{41\!\cdots\!61}{64\!\cdots\!23}a^{16}+\frac{18\!\cdots\!10}{64\!\cdots\!23}a^{15}+\frac{50\!\cdots\!20}{64\!\cdots\!23}a^{14}-\frac{75\!\cdots\!91}{64\!\cdots\!23}a^{13}-\frac{54\!\cdots\!38}{64\!\cdots\!23}a^{12}+\frac{15\!\cdots\!13}{92\!\cdots\!89}a^{11}+\frac{34\!\cdots\!38}{64\!\cdots\!23}a^{10}-\frac{26\!\cdots\!60}{64\!\cdots\!23}a^{9}-\frac{24\!\cdots\!82}{64\!\cdots\!23}a^{8}+\frac{21\!\cdots\!61}{64\!\cdots\!23}a^{7}+\frac{76\!\cdots\!50}{64\!\cdots\!23}a^{6}+\frac{22\!\cdots\!66}{64\!\cdots\!23}a^{5}-\frac{28\!\cdots\!66}{64\!\cdots\!23}a^{4}+\frac{14\!\cdots\!53}{64\!\cdots\!23}a^{3}+\frac{52\!\cdots\!78}{64\!\cdots\!23}a^{2}-\frac{92\!\cdots\!28}{81\!\cdots\!37}a-\frac{23\!\cdots\!53}{10\!\cdots\!03}$, $\frac{16\!\cdots\!69}{64\!\cdots\!23}a^{27}-\frac{72\!\cdots\!58}{64\!\cdots\!23}a^{26}-\frac{56\!\cdots\!06}{36\!\cdots\!39}a^{25}+\frac{31\!\cdots\!08}{92\!\cdots\!89}a^{24}+\frac{40\!\cdots\!68}{64\!\cdots\!23}a^{23}-\frac{12\!\cdots\!78}{64\!\cdots\!23}a^{22}+\frac{58\!\cdots\!26}{92\!\cdots\!89}a^{21}-\frac{31\!\cdots\!68}{92\!\cdots\!89}a^{20}-\frac{23\!\cdots\!77}{64\!\cdots\!23}a^{19}+\frac{32\!\cdots\!90}{64\!\cdots\!23}a^{18}+\frac{21\!\cdots\!97}{64\!\cdots\!23}a^{17}-\frac{40\!\cdots\!40}{64\!\cdots\!23}a^{16}+\frac{27\!\cdots\!76}{92\!\cdots\!89}a^{15}-\frac{13\!\cdots\!04}{92\!\cdots\!89}a^{14}-\frac{11\!\cdots\!16}{92\!\cdots\!89}a^{13}+\frac{65\!\cdots\!32}{92\!\cdots\!89}a^{12}+\frac{10\!\cdots\!10}{64\!\cdots\!23}a^{11}-\frac{53\!\cdots\!32}{64\!\cdots\!23}a^{10}-\frac{55\!\cdots\!47}{92\!\cdots\!89}a^{9}+\frac{25\!\cdots\!19}{92\!\cdots\!89}a^{8}+\frac{10\!\cdots\!79}{64\!\cdots\!23}a^{7}-\frac{10\!\cdots\!00}{64\!\cdots\!23}a^{6}-\frac{93\!\cdots\!51}{92\!\cdots\!89}a^{5}+\frac{24\!\cdots\!30}{92\!\cdots\!89}a^{4}-\frac{11\!\cdots\!20}{64\!\cdots\!23}a^{3}-\frac{66\!\cdots\!78}{64\!\cdots\!23}a^{2}-\frac{58\!\cdots\!52}{81\!\cdots\!37}a-\frac{20\!\cdots\!05}{10\!\cdots\!03}$, $\frac{16\!\cdots\!94}{64\!\cdots\!23}a^{27}+\frac{32\!\cdots\!27}{64\!\cdots\!23}a^{26}-\frac{78\!\cdots\!13}{64\!\cdots\!23}a^{25}-\frac{87\!\cdots\!00}{64\!\cdots\!23}a^{24}+\frac{41\!\cdots\!98}{64\!\cdots\!23}a^{23}+\frac{54\!\cdots\!03}{64\!\cdots\!23}a^{22}+\frac{63\!\cdots\!01}{92\!\cdots\!89}a^{21}+\frac{14\!\cdots\!86}{92\!\cdots\!89}a^{20}-\frac{15\!\cdots\!17}{64\!\cdots\!23}a^{19}-\frac{11\!\cdots\!93}{64\!\cdots\!23}a^{18}+\frac{33\!\cdots\!87}{64\!\cdots\!23}a^{17}+\frac{14\!\cdots\!70}{64\!\cdots\!23}a^{16}+\frac{19\!\cdots\!00}{64\!\cdots\!23}a^{15}+\frac{42\!\cdots\!58}{64\!\cdots\!23}a^{14}-\frac{48\!\cdots\!27}{64\!\cdots\!23}a^{13}-\frac{96\!\cdots\!18}{64\!\cdots\!23}a^{12}+\frac{18\!\cdots\!80}{64\!\cdots\!23}a^{11}+\frac{23\!\cdots\!82}{64\!\cdots\!23}a^{10}-\frac{23\!\cdots\!59}{64\!\cdots\!23}a^{9}+\frac{16\!\cdots\!37}{64\!\cdots\!23}a^{8}+\frac{49\!\cdots\!79}{64\!\cdots\!23}a^{7}+\frac{57\!\cdots\!22}{92\!\cdots\!89}a^{6}-\frac{29\!\cdots\!29}{64\!\cdots\!23}a^{5}+\frac{16\!\cdots\!85}{64\!\cdots\!23}a^{4}+\frac{49\!\cdots\!23}{92\!\cdots\!89}a^{3}+\frac{17\!\cdots\!43}{64\!\cdots\!23}a^{2}-\frac{96\!\cdots\!42}{81\!\cdots\!37}a-\frac{12\!\cdots\!42}{10\!\cdots\!03}$, $\frac{25\!\cdots\!47}{64\!\cdots\!23}a^{27}-\frac{94\!\cdots\!10}{92\!\cdots\!89}a^{26}-\frac{79\!\cdots\!37}{64\!\cdots\!23}a^{25}+\frac{30\!\cdots\!50}{64\!\cdots\!23}a^{24}+\frac{70\!\cdots\!07}{92\!\cdots\!89}a^{23}-\frac{10\!\cdots\!64}{64\!\cdots\!23}a^{22}+\frac{10\!\cdots\!72}{92\!\cdots\!89}a^{21}-\frac{28\!\cdots\!38}{92\!\cdots\!89}a^{20}-\frac{11\!\cdots\!42}{64\!\cdots\!23}a^{19}+\frac{97\!\cdots\!34}{92\!\cdots\!89}a^{18}+\frac{24\!\cdots\!64}{64\!\cdots\!23}a^{17}+\frac{55\!\cdots\!28}{64\!\cdots\!23}a^{16}+\frac{32\!\cdots\!65}{64\!\cdots\!23}a^{15}-\frac{10\!\cdots\!22}{64\!\cdots\!23}a^{14}-\frac{19\!\cdots\!72}{64\!\cdots\!23}a^{13}+\frac{19\!\cdots\!70}{64\!\cdots\!23}a^{12}+\frac{22\!\cdots\!07}{64\!\cdots\!23}a^{11}+\frac{70\!\cdots\!58}{64\!\cdots\!23}a^{10}-\frac{65\!\cdots\!51}{64\!\cdots\!23}a^{9}+\frac{36\!\cdots\!92}{64\!\cdots\!23}a^{8}+\frac{50\!\cdots\!70}{64\!\cdots\!23}a^{7}+\frac{16\!\cdots\!38}{64\!\cdots\!23}a^{6}+\frac{42\!\cdots\!55}{64\!\cdots\!23}a^{5}-\frac{73\!\cdots\!96}{64\!\cdots\!23}a^{4}+\frac{36\!\cdots\!67}{64\!\cdots\!23}a^{3}+\frac{27\!\cdots\!65}{64\!\cdots\!23}a^{2}+\frac{41\!\cdots\!96}{11\!\cdots\!91}a+\frac{15\!\cdots\!81}{10\!\cdots\!03}$, $\frac{12\!\cdots\!27}{64\!\cdots\!23}a^{27}-\frac{19\!\cdots\!60}{10\!\cdots\!69}a^{26}-\frac{23\!\cdots\!23}{64\!\cdots\!23}a^{25}+\frac{20\!\cdots\!77}{73\!\cdots\!83}a^{24}+\frac{19\!\cdots\!51}{64\!\cdots\!23}a^{23}-\frac{17\!\cdots\!54}{73\!\cdots\!83}a^{22}+\frac{58\!\cdots\!62}{92\!\cdots\!89}a^{21}-\frac{63\!\cdots\!09}{10\!\cdots\!69}a^{20}-\frac{65\!\cdots\!53}{64\!\cdots\!23}a^{19}-\frac{23\!\cdots\!45}{10\!\cdots\!69}a^{18}+\frac{51\!\cdots\!95}{64\!\cdots\!23}a^{17}+\frac{32\!\cdots\!67}{73\!\cdots\!83}a^{16}+\frac{16\!\cdots\!65}{64\!\cdots\!23}a^{15}-\frac{18\!\cdots\!11}{73\!\cdots\!83}a^{14}+\frac{12\!\cdots\!24}{64\!\cdots\!23}a^{13}-\frac{18\!\cdots\!70}{73\!\cdots\!83}a^{12}+\frac{10\!\cdots\!66}{64\!\cdots\!23}a^{11}-\frac{77\!\cdots\!67}{73\!\cdots\!83}a^{10}+\frac{57\!\cdots\!01}{64\!\cdots\!23}a^{9}-\frac{86\!\cdots\!53}{73\!\cdots\!83}a^{8}+\frac{22\!\cdots\!18}{64\!\cdots\!23}a^{7}-\frac{13\!\cdots\!83}{73\!\cdots\!83}a^{6}+\frac{13\!\cdots\!36}{64\!\cdots\!23}a^{5}-\frac{10\!\cdots\!51}{73\!\cdots\!83}a^{4}+\frac{14\!\cdots\!63}{64\!\cdots\!23}a^{3}-\frac{11\!\cdots\!72}{73\!\cdots\!83}a^{2}+\frac{12\!\cdots\!77}{81\!\cdots\!37}a-\frac{82\!\cdots\!54}{11\!\cdots\!63}$, $\frac{71\!\cdots\!13}{64\!\cdots\!23}a^{27}-\frac{86\!\cdots\!21}{64\!\cdots\!23}a^{26}-\frac{20\!\cdots\!59}{64\!\cdots\!23}a^{25}+\frac{61\!\cdots\!79}{64\!\cdots\!23}a^{24}+\frac{12\!\cdots\!16}{64\!\cdots\!23}a^{23}-\frac{36\!\cdots\!55}{92\!\cdots\!89}a^{22}+\frac{31\!\cdots\!97}{92\!\cdots\!89}a^{21}-\frac{29\!\cdots\!68}{92\!\cdots\!89}a^{20}-\frac{28\!\cdots\!99}{64\!\cdots\!23}a^{19}+\frac{14\!\cdots\!46}{64\!\cdots\!23}a^{18}+\frac{30\!\cdots\!54}{64\!\cdots\!23}a^{17}-\frac{19\!\cdots\!04}{64\!\cdots\!23}a^{16}+\frac{94\!\cdots\!99}{64\!\cdots\!23}a^{15}-\frac{10\!\cdots\!52}{64\!\cdots\!23}a^{14}-\frac{33\!\cdots\!29}{64\!\cdots\!23}a^{13}+\frac{53\!\cdots\!98}{64\!\cdots\!23}a^{12}+\frac{48\!\cdots\!85}{64\!\cdots\!23}a^{11}-\frac{80\!\cdots\!30}{64\!\cdots\!23}a^{10}-\frac{68\!\cdots\!68}{64\!\cdots\!23}a^{9}+\frac{23\!\cdots\!73}{64\!\cdots\!23}a^{8}+\frac{12\!\cdots\!09}{92\!\cdots\!89}a^{7}-\frac{14\!\cdots\!38}{64\!\cdots\!23}a^{6}+\frac{16\!\cdots\!92}{64\!\cdots\!23}a^{5}+\frac{33\!\cdots\!68}{64\!\cdots\!23}a^{4}+\frac{47\!\cdots\!52}{64\!\cdots\!23}a^{3}-\frac{19\!\cdots\!86}{64\!\cdots\!23}a^{2}+\frac{12\!\cdots\!11}{81\!\cdots\!37}a+\frac{21\!\cdots\!12}{14\!\cdots\!29}$, $\frac{15\!\cdots\!67}{92\!\cdots\!89}a^{27}-\frac{23\!\cdots\!33}{64\!\cdots\!23}a^{26}-\frac{29\!\cdots\!53}{64\!\cdots\!23}a^{25}+\frac{78\!\cdots\!71}{64\!\cdots\!23}a^{24}+\frac{18\!\cdots\!99}{64\!\cdots\!23}a^{23}-\frac{44\!\cdots\!06}{64\!\cdots\!23}a^{22}+\frac{47\!\cdots\!36}{92\!\cdots\!89}a^{21}-\frac{97\!\cdots\!23}{92\!\cdots\!89}a^{20}-\frac{55\!\cdots\!20}{92\!\cdots\!89}a^{19}+\frac{12\!\cdots\!23}{64\!\cdots\!23}a^{18}+\frac{54\!\cdots\!72}{64\!\cdots\!23}a^{17}-\frac{20\!\cdots\!35}{64\!\cdots\!23}a^{16}+\frac{13\!\cdots\!79}{64\!\cdots\!23}a^{15}-\frac{28\!\cdots\!53}{64\!\cdots\!23}a^{14}-\frac{31\!\cdots\!76}{64\!\cdots\!23}a^{13}+\frac{26\!\cdots\!45}{64\!\cdots\!23}a^{12}+\frac{80\!\cdots\!47}{64\!\cdots\!23}a^{11}-\frac{19\!\cdots\!06}{64\!\cdots\!23}a^{10}-\frac{12\!\cdots\!62}{64\!\cdots\!23}a^{9}+\frac{13\!\cdots\!44}{64\!\cdots\!23}a^{8}+\frac{15\!\cdots\!50}{64\!\cdots\!23}a^{7}-\frac{51\!\cdots\!69}{92\!\cdots\!89}a^{6}+\frac{10\!\cdots\!33}{64\!\cdots\!23}a^{5}+\frac{17\!\cdots\!70}{64\!\cdots\!23}a^{4}+\frac{11\!\cdots\!99}{64\!\cdots\!23}a^{3}-\frac{25\!\cdots\!65}{64\!\cdots\!23}a^{2}+\frac{57\!\cdots\!27}{81\!\cdots\!37}a+\frac{23\!\cdots\!98}{10\!\cdots\!03}$ (assuming GRH)
Regulator:		\( 2714544445552.786 \) (assuming GRH)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{14}\cdot 2714544445552.786 \cdot 43}{4\cdot\sqrt{792537323068373529244880273632877655015215903801344}}\cr\approx \mathstrut & 0.154922221619408 \end{aligned}\] (assuming GRH)

Galois group

$C_2\times C_{14}$ (as 28T2):

An abelian group of order 28

The 28 conjugacy class representatives for $C_2\times C_{14}$

Character table for $C_2\times C_{14}$

Intermediate fields

\(\Q(\sqrt{43}) \), \(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-43}) \), \(\Q(i, \sqrt{43})\), 7.7.6321363049.1, 14.14.28152039412241052225421312.1, 14.0.654698590982350051753984.1, 14.0.1718264124282290785243.1

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	R	${\href{/padicField/3.14.0.1}{14} }^{2}$	${\href{/padicField/5.14.0.1}{14} }^{2}$	${\href{/padicField/7.2.0.1}{2} }^{14}$	${\href{/padicField/11.14.0.1}{14} }^{2}$	${\href{/padicField/13.7.0.1}{7} }^{4}$	${\href{/padicField/17.7.0.1}{7} }^{4}$	${\href{/padicField/19.14.0.1}{14} }^{2}$	${\href{/padicField/23.14.0.1}{14} }^{2}$	${\href{/padicField/29.14.0.1}{14} }^{2}$	${\href{/padicField/31.14.0.1}{14} }^{2}$	${\href{/padicField/37.2.0.1}{2} }^{14}$	${\href{/padicField/41.7.0.1}{7} }^{4}$	R	${\href{/padicField/47.14.0.1}{14} }^{2}$	${\href{/padicField/53.7.0.1}{7} }^{4}$	${\href{/padicField/59.14.0.1}{14} }^{2}$

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
\(2\)		Deg $28$	$2$	$14$	$28$
\(43\)		Deg $28$	$14$	$2$	$26$