Number field 28.0.503553375386417026489977159144851919778199649.1

• Label: 28.0.503...649.1
• Degree: $28$
• Signature: $[0, 14]$
• Discriminant: $5.036\times 10^{44}$
• Root discriminant: \(39.49\)
• Ramified primes: $3,29$
• Class number: $192$ (GRH)
• Class group: [4, 4, 12] (GRH)
• Galois group: $C_2\times C_{14}$ (as 28T2)

Normalized defining polynomial

x^28 - x^27 + 14*x^26 - 11*x^25 + 115*x^24 - 79*x^23 + 617*x^22 - 353*x^21 + 2421*x^20 - 1188*x^19 + 7015*x^18 - 2803*x^17 + 15415*x^16 - 5107*x^15 + 25195*x^14 - 6334*x^13 + 30532*x^12 - 6088*x^11 + 26057*x^10 - 3239*x^9 + 15207*x^8 - 1590*x^7 + 5362*x^6 - 70*x^5 + 1050*x^4 - 84*x^3 + 77*x^2 + 7*x + 1

Invariants

Degree:		$28$
Signature:		$[0, 14]$
Discriminant:		\(503553375386417026489977159144851919778199649\) \(\medspace = 3^{14}\cdot 29^{26}\)
Root discriminant:		\(39.49\)
Galois root discriminant:		$3^{1/2}29^{13/14}\approx 39.49131773649383$
Ramified primes:		\(3\), \(29\)
Discriminant root field:		\(\Q\)
$\card{ \Gal(K/\Q) }$:		$28$
This field is Galois and abelian over $\Q$.
Conductor:		\(87=3\cdot 29\)
Dirichlet character group:		$\lbrace$$\chi_{87}(64,·)$, $\chi_{87}(1,·)$, $\chi_{87}(86,·)$, $\chi_{87}(67,·)$, $\chi_{87}(4,·)$, $\chi_{87}(5,·)$, $\chi_{87}(7,·)$, $\chi_{87}(74,·)$, $\chi_{87}(13,·)$, $\chi_{87}(16,·)$, $\chi_{87}(82,·)$, $\chi_{87}(83,·)$, $\chi_{87}(20,·)$, $\chi_{87}(22,·)$, $\chi_{87}(23,·)$, $\chi_{87}(25,·)$, $\chi_{87}(28,·)$, $\chi_{87}(80,·)$, $\chi_{87}(34,·)$, $\chi_{87}(35,·)$, $\chi_{87}(38,·)$, $\chi_{87}(71,·)$, $\chi_{87}(49,·)$, $\chi_{87}(52,·)$, $\chi_{87}(53,·)$, $\chi_{87}(59,·)$, $\chi_{87}(62,·)$, $\chi_{87}(65,·)$$\rbrace$
This is a CM field.
Reflex fields:		unavailable$^{8192}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $\frac{1}{17}a^{26}+\frac{6}{17}a^{25}+\frac{8}{17}a^{24}-\frac{5}{17}a^{23}+\frac{2}{17}a^{22}+\frac{5}{17}a^{21}-\frac{5}{17}a^{20}+\frac{1}{17}a^{19}-\frac{1}{17}a^{18}-\frac{2}{17}a^{17}-\frac{6}{17}a^{16}+\frac{5}{17}a^{15}-\frac{4}{17}a^{14}-\frac{3}{17}a^{13}+\frac{2}{17}a^{12}-\frac{5}{17}a^{11}+\frac{5}{17}a^{10}+\frac{1}{17}a^{9}+\frac{1}{17}a^{8}+\frac{1}{17}a^{7}+\frac{2}{17}a^{6}+\frac{8}{17}a^{5}+\frac{1}{17}a^{4}-\frac{5}{17}a^{3}-\frac{2}{17}a^{2}+\frac{6}{17}a-\frac{6}{17}$, $\frac{1}{38\!\cdots\!41}a^{27}+\frac{94\!\cdots\!35}{38\!\cdots\!41}a^{26}-\frac{17\!\cdots\!03}{38\!\cdots\!41}a^{25}+\frac{24\!\cdots\!81}{38\!\cdots\!41}a^{24}-\frac{30\!\cdots\!08}{38\!\cdots\!41}a^{23}+\frac{42\!\cdots\!72}{38\!\cdots\!41}a^{22}-\frac{10\!\cdots\!33}{38\!\cdots\!41}a^{21}-\frac{18\!\cdots\!33}{38\!\cdots\!41}a^{20}+\frac{68\!\cdots\!85}{38\!\cdots\!41}a^{19}-\frac{15\!\cdots\!18}{38\!\cdots\!41}a^{18}+\frac{15\!\cdots\!06}{38\!\cdots\!41}a^{17}-\frac{17\!\cdots\!48}{38\!\cdots\!41}a^{16}-\frac{15\!\cdots\!06}{38\!\cdots\!41}a^{15}-\frac{18\!\cdots\!06}{38\!\cdots\!41}a^{14}-\frac{15\!\cdots\!47}{38\!\cdots\!41}a^{13}-\frac{51\!\cdots\!69}{38\!\cdots\!41}a^{12}+\frac{17\!\cdots\!29}{38\!\cdots\!41}a^{11}-\frac{36\!\cdots\!14}{38\!\cdots\!41}a^{10}+\frac{14\!\cdots\!08}{38\!\cdots\!41}a^{9}+\frac{75\!\cdots\!25}{38\!\cdots\!41}a^{8}-\frac{29\!\cdots\!80}{38\!\cdots\!41}a^{7}-\frac{83\!\cdots\!01}{22\!\cdots\!73}a^{6}+\frac{81\!\cdots\!39}{38\!\cdots\!41}a^{5}+\frac{25\!\cdots\!40}{38\!\cdots\!41}a^{4}+\frac{91\!\cdots\!70}{38\!\cdots\!41}a^{3}-\frac{98\!\cdots\!51}{38\!\cdots\!41}a^{2}+\frac{14\!\cdots\!85}{38\!\cdots\!41}a-\frac{68\!\cdots\!70}{38\!\cdots\!41}$

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

Class group and class number

$C_{4}\times C_{4}\times C_{12}$, which has order $192$ (assuming GRH)

Unit group

Rank:		$13$
Torsion generator:		\( -\frac{459718160048198964470148224851155793007}{3895907347685506947339847625632831500841} a^{27} + \frac{491963445984988705273217591819646684565}{3895907347685506947339847625632831500841} a^{26} - \frac{6448629807164339806286652834135729651835}{3895907347685506947339847625632831500841} a^{25} + \frac{5489048472554384718748952494620363866639}{3895907347685506947339847625632831500841} a^{24} - \frac{52951072199703845948981282554345438827087}{3895907347685506947339847625632831500841} a^{23} + \frac{39819307479811215431616417681920535084871}{3895907347685506947339847625632831500841} a^{22} - \frac{283987576946209170566480179195940845846935}{3895907347685506947339847625632831500841} a^{21} + \frac{180713865062290080350196112311444834870074}{3895907347685506947339847625632831500841} a^{20} - \frac{1112673865127612497690665577819613726783806}{3895907347685506947339847625632831500841} a^{19} + \frac{36344069922386707864807676303648846199636}{229171020452088643961167507390166558873} a^{18} - \frac{3217951691774023788210898570050862866706706}{3895907347685506947339847625632831500841} a^{17} + \frac{87876390160312095317718859986505853860340}{229171020452088643961167507390166558873} a^{16} - \frac{7047879163309816292391973949049811453306036}{3895907347685506947339847625632831500841} a^{15} + \frac{2797655820669082469076957101458329888476089}{3895907347685506947339847625632831500841} a^{14} - \frac{11468990009854706273972141903057069947674868}{3895907347685506947339847625632831500841} a^{13} + \frac{3642273987431523561845329655519556538625641}{3895907347685506947339847625632831500841} a^{12} - \frac{13797351382373102873603284632167040793349857}{3895907347685506947339847625632831500841} a^{11} + \frac{3691710695278352959379085590040921104499799}{3895907347685506947339847625632831500841} a^{10} - \frac{11655967246241058805749943256068328625815037}{3895907347685506947339847625632831500841} a^{9} + \frac{2250968703208521239456502251110489534766072}{3895907347685506947339847625632831500841} a^{8} - \frac{392647765542212833797689295166405365735365}{229171020452088643961167507390166558873} a^{7} + \frac{1198090561014405273787508711780039774147370}{3895907347685506947339847625632831500841} a^{6} - \frac{2288207023706649952564153692852833900212139}{3895907347685506947339847625632831500841} a^{5} + \frac{197340796559132081521154403951415280347726}{3895907347685506947339847625632831500841} a^{4} - \frac{415651585504784233687805415050104789263488}{3895907347685506947339847625632831500841} a^{3} + \frac{85650487097946204886580171571688541211264}{3895907347685506947339847625632831500841} a^{2} - \frac{27260512845676607370643210950855091389682}{3895907347685506947339847625632831500841} a + \frac{1495905399110794953147810115657088139776}{3895907347685506947339847625632831500841} \)  (order $6$)
Fundamental units:		$\frac{47\!\cdots\!44}{38\!\cdots\!41}a^{27}-\frac{46\!\cdots\!05}{38\!\cdots\!41}a^{26}+\frac{66\!\cdots\!44}{38\!\cdots\!41}a^{25}-\frac{50\!\cdots\!52}{38\!\cdots\!41}a^{24}+\frac{55\!\cdots\!65}{38\!\cdots\!41}a^{23}-\frac{36\!\cdots\!18}{38\!\cdots\!41}a^{22}+\frac{29\!\cdots\!42}{38\!\cdots\!41}a^{21}-\frac{16\!\cdots\!68}{38\!\cdots\!41}a^{20}+\frac{11\!\cdots\!70}{38\!\cdots\!41}a^{19}-\frac{53\!\cdots\!66}{38\!\cdots\!41}a^{18}+\frac{33\!\cdots\!69}{38\!\cdots\!41}a^{17}-\frac{12\!\cdots\!58}{38\!\cdots\!41}a^{16}+\frac{75\!\cdots\!44}{38\!\cdots\!41}a^{15}-\frac{22\!\cdots\!60}{38\!\cdots\!41}a^{14}+\frac{12\!\cdots\!12}{38\!\cdots\!41}a^{13}-\frac{26\!\cdots\!88}{38\!\cdots\!41}a^{12}+\frac{15\!\cdots\!79}{38\!\cdots\!41}a^{11}-\frac{23\!\cdots\!44}{38\!\cdots\!41}a^{10}+\frac{13\!\cdots\!90}{38\!\cdots\!41}a^{9}-\frac{10\!\cdots\!91}{38\!\cdots\!41}a^{8}+\frac{78\!\cdots\!22}{38\!\cdots\!41}a^{7}-\frac{22\!\cdots\!70}{22\!\cdots\!73}a^{6}+\frac{28\!\cdots\!61}{38\!\cdots\!41}a^{5}+\frac{12\!\cdots\!06}{38\!\cdots\!41}a^{4}+\frac{59\!\cdots\!12}{38\!\cdots\!41}a^{3}+\frac{29\!\cdots\!18}{38\!\cdots\!41}a^{2}+\frac{47\!\cdots\!20}{38\!\cdots\!41}a+\frac{43\!\cdots\!02}{38\!\cdots\!41}$, $\frac{38\!\cdots\!71}{38\!\cdots\!41}a^{27}-\frac{41\!\cdots\!38}{38\!\cdots\!41}a^{26}+\frac{97\!\cdots\!79}{38\!\cdots\!41}a^{25}-\frac{33\!\cdots\!97}{22\!\cdots\!73}a^{24}+\frac{91\!\cdots\!76}{38\!\cdots\!41}a^{23}-\frac{44\!\cdots\!42}{38\!\cdots\!41}a^{22}+\frac{58\!\cdots\!91}{38\!\cdots\!41}a^{21}-\frac{13\!\cdots\!14}{22\!\cdots\!73}a^{20}+\frac{24\!\cdots\!37}{38\!\cdots\!41}a^{19}-\frac{89\!\cdots\!72}{38\!\cdots\!41}a^{18}+\frac{79\!\cdots\!33}{38\!\cdots\!41}a^{17}-\frac{24\!\cdots\!12}{38\!\cdots\!41}a^{16}+\frac{18\!\cdots\!40}{38\!\cdots\!41}a^{15}-\frac{52\!\cdots\!56}{38\!\cdots\!41}a^{14}+\frac{33\!\cdots\!41}{38\!\cdots\!41}a^{13}-\frac{81\!\cdots\!06}{38\!\cdots\!41}a^{12}+\frac{41\!\cdots\!41}{38\!\cdots\!41}a^{11}-\frac{92\!\cdots\!91}{38\!\cdots\!41}a^{10}+\frac{40\!\cdots\!53}{38\!\cdots\!41}a^{9}-\frac{71\!\cdots\!32}{38\!\cdots\!41}a^{8}+\frac{23\!\cdots\!30}{38\!\cdots\!41}a^{7}-\frac{36\!\cdots\!80}{38\!\cdots\!41}a^{6}+\frac{14\!\cdots\!46}{38\!\cdots\!41}a^{5}-\frac{98\!\cdots\!21}{38\!\cdots\!41}a^{4}+\frac{38\!\cdots\!23}{38\!\cdots\!41}a^{3}-\frac{12\!\cdots\!16}{38\!\cdots\!41}a^{2}+\frac{11\!\cdots\!18}{38\!\cdots\!41}a-\frac{19\!\cdots\!21}{38\!\cdots\!41}$, $\frac{61\!\cdots\!62}{38\!\cdots\!41}a^{27}-\frac{93\!\cdots\!24}{22\!\cdots\!73}a^{26}+\frac{97\!\cdots\!98}{38\!\cdots\!41}a^{25}-\frac{20\!\cdots\!76}{38\!\cdots\!41}a^{24}+\frac{83\!\cdots\!95}{38\!\cdots\!41}a^{23}-\frac{15\!\cdots\!74}{38\!\cdots\!41}a^{22}+\frac{47\!\cdots\!07}{38\!\cdots\!41}a^{21}-\frac{80\!\cdots\!75}{38\!\cdots\!41}a^{20}+\frac{19\!\cdots\!10}{38\!\cdots\!41}a^{19}-\frac{30\!\cdots\!56}{38\!\cdots\!41}a^{18}+\frac{57\!\cdots\!52}{38\!\cdots\!41}a^{17}-\frac{82\!\cdots\!19}{38\!\cdots\!41}a^{16}+\frac{76\!\cdots\!65}{22\!\cdots\!73}a^{15}-\frac{17\!\cdots\!15}{38\!\cdots\!41}a^{14}+\frac{22\!\cdots\!08}{38\!\cdots\!41}a^{13}-\frac{15\!\cdots\!01}{22\!\cdots\!73}a^{12}+\frac{28\!\cdots\!04}{38\!\cdots\!41}a^{11}-\frac{30\!\cdots\!56}{38\!\cdots\!41}a^{10}+\frac{26\!\cdots\!56}{38\!\cdots\!41}a^{9}-\frac{23\!\cdots\!03}{38\!\cdots\!41}a^{8}+\frac{16\!\cdots\!24}{38\!\cdots\!41}a^{7}-\frac{12\!\cdots\!49}{38\!\cdots\!41}a^{6}+\frac{74\!\cdots\!12}{38\!\cdots\!41}a^{5}-\frac{33\!\cdots\!08}{38\!\cdots\!41}a^{4}+\frac{17\!\cdots\!88}{38\!\cdots\!41}a^{3}-\frac{50\!\cdots\!25}{38\!\cdots\!41}a^{2}+\frac{35\!\cdots\!32}{38\!\cdots\!41}a-\frac{80\!\cdots\!67}{38\!\cdots\!41}$, $\frac{59\!\cdots\!26}{38\!\cdots\!41}a^{27}-\frac{37\!\cdots\!12}{38\!\cdots\!41}a^{26}+\frac{80\!\cdots\!95}{38\!\cdots\!41}a^{25}-\frac{34\!\cdots\!70}{38\!\cdots\!41}a^{24}+\frac{65\!\cdots\!15}{38\!\cdots\!41}a^{23}-\frac{21\!\cdots\!23}{38\!\cdots\!41}a^{22}+\frac{34\!\cdots\!23}{38\!\cdots\!41}a^{21}-\frac{75\!\cdots\!00}{38\!\cdots\!41}a^{20}+\frac{13\!\cdots\!74}{38\!\cdots\!41}a^{19}-\frac{18\!\cdots\!42}{38\!\cdots\!41}a^{18}+\frac{39\!\cdots\!87}{38\!\cdots\!41}a^{17}-\frac{16\!\cdots\!61}{38\!\cdots\!41}a^{16}+\frac{85\!\cdots\!17}{38\!\cdots\!41}a^{15}+\frac{24\!\cdots\!51}{38\!\cdots\!41}a^{14}+\frac{13\!\cdots\!70}{38\!\cdots\!41}a^{13}+\frac{15\!\cdots\!55}{38\!\cdots\!41}a^{12}+\frac{16\!\cdots\!68}{38\!\cdots\!41}a^{11}+\frac{27\!\cdots\!09}{38\!\cdots\!41}a^{10}+\frac{14\!\cdots\!33}{38\!\cdots\!41}a^{9}+\frac{34\!\cdots\!61}{38\!\cdots\!41}a^{8}+\frac{83\!\cdots\!04}{38\!\cdots\!41}a^{7}+\frac{20\!\cdots\!83}{38\!\cdots\!41}a^{6}+\frac{28\!\cdots\!32}{38\!\cdots\!41}a^{5}+\frac{96\!\cdots\!51}{38\!\cdots\!41}a^{4}+\frac{58\!\cdots\!54}{38\!\cdots\!41}a^{3}+\frac{12\!\cdots\!39}{38\!\cdots\!41}a^{2}+\frac{12\!\cdots\!32}{38\!\cdots\!41}a+\frac{53\!\cdots\!26}{38\!\cdots\!41}$, $\frac{23\!\cdots\!90}{38\!\cdots\!41}a^{27}-\frac{14\!\cdots\!29}{38\!\cdots\!41}a^{26}+\frac{18\!\cdots\!03}{22\!\cdots\!73}a^{25}-\frac{13\!\cdots\!26}{38\!\cdots\!41}a^{24}+\frac{25\!\cdots\!49}{38\!\cdots\!41}a^{23}-\frac{85\!\cdots\!41}{38\!\cdots\!41}a^{22}+\frac{13\!\cdots\!37}{38\!\cdots\!41}a^{21}-\frac{30\!\cdots\!87}{38\!\cdots\!41}a^{20}+\frac{53\!\cdots\!42}{38\!\cdots\!41}a^{19}-\frac{74\!\cdots\!95}{38\!\cdots\!41}a^{18}+\frac{15\!\cdots\!54}{38\!\cdots\!41}a^{17}-\frac{73\!\cdots\!64}{38\!\cdots\!41}a^{16}+\frac{34\!\cdots\!43}{38\!\cdots\!41}a^{15}+\frac{69\!\cdots\!42}{38\!\cdots\!41}a^{14}+\frac{55\!\cdots\!69}{38\!\cdots\!41}a^{13}+\frac{54\!\cdots\!61}{38\!\cdots\!41}a^{12}+\frac{68\!\cdots\!17}{38\!\cdots\!41}a^{11}+\frac{58\!\cdots\!35}{22\!\cdots\!73}a^{10}+\frac{58\!\cdots\!15}{38\!\cdots\!41}a^{9}+\frac{12\!\cdots\!53}{38\!\cdots\!41}a^{8}+\frac{35\!\cdots\!13}{38\!\cdots\!41}a^{7}+\frac{73\!\cdots\!43}{38\!\cdots\!41}a^{6}+\frac{13\!\cdots\!49}{38\!\cdots\!41}a^{5}+\frac{33\!\cdots\!87}{38\!\cdots\!41}a^{4}+\frac{32\!\cdots\!76}{38\!\cdots\!41}a^{3}+\frac{39\!\cdots\!10}{38\!\cdots\!41}a^{2}+\frac{30\!\cdots\!39}{38\!\cdots\!41}a+\frac{29\!\cdots\!43}{38\!\cdots\!41}$, $\frac{45\!\cdots\!07}{38\!\cdots\!41}a^{27}-\frac{49\!\cdots\!65}{38\!\cdots\!41}a^{26}+\frac{64\!\cdots\!35}{38\!\cdots\!41}a^{25}-\frac{54\!\cdots\!39}{38\!\cdots\!41}a^{24}+\frac{52\!\cdots\!87}{38\!\cdots\!41}a^{23}-\frac{39\!\cdots\!71}{38\!\cdots\!41}a^{22}+\frac{28\!\cdots\!35}{38\!\cdots\!41}a^{21}-\frac{18\!\cdots\!74}{38\!\cdots\!41}a^{20}+\frac{11\!\cdots\!06}{38\!\cdots\!41}a^{19}-\frac{36\!\cdots\!36}{22\!\cdots\!73}a^{18}+\frac{32\!\cdots\!06}{38\!\cdots\!41}a^{17}-\frac{87\!\cdots\!40}{22\!\cdots\!73}a^{16}+\frac{70\!\cdots\!36}{38\!\cdots\!41}a^{15}-\frac{27\!\cdots\!89}{38\!\cdots\!41}a^{14}+\frac{11\!\cdots\!68}{38\!\cdots\!41}a^{13}-\frac{36\!\cdots\!41}{38\!\cdots\!41}a^{12}+\frac{13\!\cdots\!57}{38\!\cdots\!41}a^{11}-\frac{36\!\cdots\!99}{38\!\cdots\!41}a^{10}+\frac{11\!\cdots\!37}{38\!\cdots\!41}a^{9}-\frac{22\!\cdots\!72}{38\!\cdots\!41}a^{8}+\frac{39\!\cdots\!65}{22\!\cdots\!73}a^{7}-\frac{11\!\cdots\!70}{38\!\cdots\!41}a^{6}+\frac{22\!\cdots\!39}{38\!\cdots\!41}a^{5}-\frac{19\!\cdots\!26}{38\!\cdots\!41}a^{4}+\frac{41\!\cdots\!88}{38\!\cdots\!41}a^{3}-\frac{81\!\cdots\!23}{38\!\cdots\!41}a^{2}+\frac{27\!\cdots\!82}{38\!\cdots\!41}a+\frac{24\!\cdots\!65}{38\!\cdots\!41}$, $\frac{69\!\cdots\!95}{22\!\cdots\!73}a^{27}-\frac{34\!\cdots\!93}{38\!\cdots\!41}a^{26}+\frac{18\!\cdots\!18}{38\!\cdots\!41}a^{25}-\frac{43\!\cdots\!31}{38\!\cdots\!41}a^{24}+\frac{15\!\cdots\!73}{38\!\cdots\!41}a^{23}-\frac{33\!\cdots\!72}{38\!\cdots\!41}a^{22}+\frac{81\!\cdots\!22}{38\!\cdots\!41}a^{21}-\frac{16\!\cdots\!52}{38\!\cdots\!41}a^{20}+\frac{31\!\cdots\!33}{38\!\cdots\!41}a^{19}-\frac{60\!\cdots\!35}{38\!\cdots\!41}a^{18}+\frac{87\!\cdots\!36}{38\!\cdots\!41}a^{17}-\frac{15\!\cdots\!88}{38\!\cdots\!41}a^{16}+\frac{17\!\cdots\!38}{38\!\cdots\!41}a^{15}-\frac{31\!\cdots\!89}{38\!\cdots\!41}a^{14}+\frac{26\!\cdots\!71}{38\!\cdots\!41}a^{13}-\frac{43\!\cdots\!00}{38\!\cdots\!41}a^{12}+\frac{26\!\cdots\!92}{38\!\cdots\!41}a^{11}-\frac{44\!\cdots\!03}{38\!\cdots\!41}a^{10}+\frac{17\!\cdots\!85}{38\!\cdots\!41}a^{9}-\frac{27\!\cdots\!98}{38\!\cdots\!41}a^{8}+\frac{55\!\cdots\!12}{38\!\cdots\!41}a^{7}-\frac{10\!\cdots\!95}{38\!\cdots\!41}a^{6}+\frac{21\!\cdots\!79}{38\!\cdots\!41}a^{5}-\frac{23\!\cdots\!29}{38\!\cdots\!41}a^{4}+\frac{54\!\cdots\!19}{38\!\cdots\!41}a^{3}-\frac{22\!\cdots\!30}{38\!\cdots\!41}a^{2}-\frac{41\!\cdots\!55}{38\!\cdots\!41}a-\frac{33\!\cdots\!43}{38\!\cdots\!41}$, $\frac{30\!\cdots\!65}{38\!\cdots\!41}a^{27}-\frac{23\!\cdots\!19}{38\!\cdots\!41}a^{26}+\frac{42\!\cdots\!08}{38\!\cdots\!41}a^{25}-\frac{23\!\cdots\!13}{38\!\cdots\!41}a^{24}+\frac{34\!\cdots\!93}{38\!\cdots\!41}a^{23}-\frac{15\!\cdots\!29}{38\!\cdots\!41}a^{22}+\frac{10\!\cdots\!09}{22\!\cdots\!73}a^{21}-\frac{63\!\cdots\!01}{38\!\cdots\!41}a^{20}+\frac{42\!\cdots\!91}{22\!\cdots\!73}a^{19}-\frac{19\!\cdots\!78}{38\!\cdots\!41}a^{18}+\frac{20\!\cdots\!41}{38\!\cdots\!41}a^{17}-\frac{35\!\cdots\!53}{38\!\cdots\!41}a^{16}+\frac{46\!\cdots\!75}{38\!\cdots\!41}a^{15}-\frac{46\!\cdots\!88}{38\!\cdots\!41}a^{14}+\frac{75\!\cdots\!73}{38\!\cdots\!41}a^{13}-\frac{16\!\cdots\!02}{38\!\cdots\!41}a^{12}+\frac{54\!\cdots\!33}{22\!\cdots\!73}a^{11}+\frac{28\!\cdots\!82}{38\!\cdots\!41}a^{10}+\frac{80\!\cdots\!53}{38\!\cdots\!41}a^{9}+\frac{81\!\cdots\!07}{38\!\cdots\!41}a^{8}+\frac{48\!\cdots\!17}{38\!\cdots\!41}a^{7}+\frac{55\!\cdots\!60}{38\!\cdots\!41}a^{6}+\frac{18\!\cdots\!93}{38\!\cdots\!41}a^{5}+\frac{32\!\cdots\!46}{38\!\cdots\!41}a^{4}+\frac{42\!\cdots\!80}{38\!\cdots\!41}a^{3}+\frac{35\!\cdots\!02}{38\!\cdots\!41}a^{2}+\frac{22\!\cdots\!31}{22\!\cdots\!73}a+\frac{35\!\cdots\!73}{38\!\cdots\!41}$, $\frac{96\!\cdots\!74}{38\!\cdots\!41}a^{27}-\frac{37\!\cdots\!11}{38\!\cdots\!41}a^{26}+\frac{12\!\cdots\!86}{38\!\cdots\!41}a^{25}-\frac{25\!\cdots\!46}{38\!\cdots\!41}a^{24}+\frac{10\!\cdots\!54}{38\!\cdots\!41}a^{23}-\frac{10\!\cdots\!48}{38\!\cdots\!41}a^{22}+\frac{55\!\cdots\!09}{38\!\cdots\!41}a^{21}+\frac{91\!\cdots\!23}{38\!\cdots\!41}a^{20}+\frac{21\!\cdots\!88}{38\!\cdots\!41}a^{19}+\frac{20\!\cdots\!97}{38\!\cdots\!41}a^{18}+\frac{61\!\cdots\!77}{38\!\cdots\!41}a^{17}+\frac{11\!\cdots\!07}{38\!\cdots\!41}a^{16}+\frac{13\!\cdots\!20}{38\!\cdots\!41}a^{15}+\frac{34\!\cdots\!91}{38\!\cdots\!41}a^{14}+\frac{21\!\cdots\!09}{38\!\cdots\!41}a^{13}+\frac{71\!\cdots\!01}{38\!\cdots\!41}a^{12}+\frac{26\!\cdots\!49}{38\!\cdots\!41}a^{11}+\frac{96\!\cdots\!86}{38\!\cdots\!41}a^{10}+\frac{22\!\cdots\!97}{38\!\cdots\!41}a^{9}+\frac{94\!\cdots\!32}{38\!\cdots\!41}a^{8}+\frac{14\!\cdots\!20}{38\!\cdots\!41}a^{7}+\frac{53\!\cdots\!85}{38\!\cdots\!41}a^{6}+\frac{49\!\cdots\!08}{38\!\cdots\!41}a^{5}+\frac{20\!\cdots\!86}{38\!\cdots\!41}a^{4}+\frac{13\!\cdots\!87}{38\!\cdots\!41}a^{3}+\frac{24\!\cdots\!71}{38\!\cdots\!41}a^{2}+\frac{24\!\cdots\!52}{38\!\cdots\!41}a-\frac{25\!\cdots\!20}{38\!\cdots\!41}$, $\frac{89\!\cdots\!00}{38\!\cdots\!41}a^{27}-\frac{95\!\cdots\!98}{38\!\cdots\!41}a^{26}+\frac{12\!\cdots\!32}{38\!\cdots\!41}a^{25}-\frac{63\!\cdots\!23}{22\!\cdots\!73}a^{24}+\frac{10\!\cdots\!62}{38\!\cdots\!41}a^{23}-\frac{77\!\cdots\!49}{38\!\cdots\!41}a^{22}+\frac{55\!\cdots\!57}{38\!\cdots\!41}a^{21}-\frac{20\!\cdots\!34}{22\!\cdots\!73}a^{20}+\frac{21\!\cdots\!58}{38\!\cdots\!41}a^{19}-\frac{12\!\cdots\!12}{38\!\cdots\!41}a^{18}+\frac{63\!\cdots\!18}{38\!\cdots\!41}a^{17}-\frac{29\!\cdots\!53}{38\!\cdots\!41}a^{16}+\frac{13\!\cdots\!17}{38\!\cdots\!41}a^{15}-\frac{54\!\cdots\!21}{38\!\cdots\!41}a^{14}+\frac{22\!\cdots\!97}{38\!\cdots\!41}a^{13}-\frac{71\!\cdots\!34}{38\!\cdots\!41}a^{12}+\frac{27\!\cdots\!96}{38\!\cdots\!41}a^{11}-\frac{72\!\cdots\!68}{38\!\cdots\!41}a^{10}+\frac{22\!\cdots\!77}{38\!\cdots\!41}a^{9}-\frac{43\!\cdots\!58}{38\!\cdots\!41}a^{8}+\frac{13\!\cdots\!93}{38\!\cdots\!41}a^{7}-\frac{23\!\cdots\!50}{38\!\cdots\!41}a^{6}+\frac{45\!\cdots\!30}{38\!\cdots\!41}a^{5}-\frac{37\!\cdots\!14}{38\!\cdots\!41}a^{4}+\frac{82\!\cdots\!17}{38\!\cdots\!41}a^{3}-\frac{13\!\cdots\!79}{38\!\cdots\!41}a^{2}+\frac{54\!\cdots\!97}{38\!\cdots\!41}a+\frac{47\!\cdots\!86}{38\!\cdots\!41}$, $\frac{47\!\cdots\!85}{38\!\cdots\!41}a^{27}-\frac{12\!\cdots\!09}{38\!\cdots\!41}a^{26}+\frac{79\!\cdots\!94}{38\!\cdots\!41}a^{25}-\frac{16\!\cdots\!33}{38\!\cdots\!41}a^{24}+\frac{70\!\cdots\!95}{38\!\cdots\!41}a^{23}-\frac{75\!\cdots\!11}{22\!\cdots\!73}a^{22}+\frac{41\!\cdots\!69}{38\!\cdots\!41}a^{21}-\frac{66\!\cdots\!78}{38\!\cdots\!41}a^{20}+\frac{17\!\cdots\!32}{38\!\cdots\!41}a^{19}-\frac{25\!\cdots\!56}{38\!\cdots\!41}a^{18}+\frac{54\!\cdots\!76}{38\!\cdots\!41}a^{17}-\frac{72\!\cdots\!63}{38\!\cdots\!41}a^{16}+\frac{12\!\cdots\!44}{38\!\cdots\!41}a^{15}-\frac{15\!\cdots\!21}{38\!\cdots\!41}a^{14}+\frac{23\!\cdots\!59}{38\!\cdots\!41}a^{13}-\frac{24\!\cdots\!01}{38\!\cdots\!41}a^{12}+\frac{31\!\cdots\!35}{38\!\cdots\!41}a^{11}-\frac{29\!\cdots\!52}{38\!\cdots\!41}a^{10}+\frac{30\!\cdots\!07}{38\!\cdots\!41}a^{9}-\frac{24\!\cdots\!92}{38\!\cdots\!41}a^{8}+\frac{20\!\cdots\!30}{38\!\cdots\!41}a^{7}-\frac{14\!\cdots\!78}{38\!\cdots\!41}a^{6}+\frac{96\!\cdots\!11}{38\!\cdots\!41}a^{5}-\frac{48\!\cdots\!55}{38\!\cdots\!41}a^{4}+\frac{22\!\cdots\!94}{38\!\cdots\!41}a^{3}-\frac{63\!\cdots\!40}{38\!\cdots\!41}a^{2}+\frac{22\!\cdots\!92}{38\!\cdots\!41}a-\frac{10\!\cdots\!50}{38\!\cdots\!41}$, $\frac{61\!\cdots\!30}{38\!\cdots\!41}a^{27}-\frac{68\!\cdots\!05}{38\!\cdots\!41}a^{26}+\frac{87\!\cdots\!10}{38\!\cdots\!41}a^{25}-\frac{78\!\cdots\!87}{38\!\cdots\!41}a^{24}+\frac{71\!\cdots\!37}{38\!\cdots\!41}a^{23}-\frac{57\!\cdots\!14}{38\!\cdots\!41}a^{22}+\frac{22\!\cdots\!94}{22\!\cdots\!73}a^{21}-\frac{26\!\cdots\!06}{38\!\cdots\!41}a^{20}+\frac{89\!\cdots\!32}{22\!\cdots\!73}a^{19}-\frac{91\!\cdots\!45}{38\!\cdots\!41}a^{18}+\frac{44\!\cdots\!33}{38\!\cdots\!41}a^{17}-\frac{22\!\cdots\!77}{38\!\cdots\!41}a^{16}+\frac{97\!\cdots\!39}{38\!\cdots\!41}a^{15}-\frac{43\!\cdots\!42}{38\!\cdots\!41}a^{14}+\frac{16\!\cdots\!80}{38\!\cdots\!41}a^{13}-\frac{58\!\cdots\!24}{38\!\cdots\!41}a^{12}+\frac{11\!\cdots\!19}{22\!\cdots\!73}a^{11}-\frac{61\!\cdots\!84}{38\!\cdots\!41}a^{10}+\frac{16\!\cdots\!23}{38\!\cdots\!41}a^{9}-\frac{41\!\cdots\!08}{38\!\cdots\!41}a^{8}+\frac{97\!\cdots\!56}{38\!\cdots\!41}a^{7}-\frac{22\!\cdots\!04}{38\!\cdots\!41}a^{6}+\frac{34\!\cdots\!43}{38\!\cdots\!41}a^{5}-\frac{53\!\cdots\!44}{38\!\cdots\!41}a^{4}+\frac{66\!\cdots\!41}{38\!\cdots\!41}a^{3}-\frac{14\!\cdots\!24}{38\!\cdots\!41}a^{2}+\frac{29\!\cdots\!54}{22\!\cdots\!73}a-\frac{24\!\cdots\!78}{38\!\cdots\!41}$, $\frac{23\!\cdots\!73}{38\!\cdots\!41}a^{27}-\frac{27\!\cdots\!63}{38\!\cdots\!41}a^{26}+\frac{32\!\cdots\!54}{38\!\cdots\!41}a^{25}-\frac{31\!\cdots\!34}{38\!\cdots\!41}a^{24}+\frac{26\!\cdots\!12}{38\!\cdots\!41}a^{23}-\frac{23\!\cdots\!39}{38\!\cdots\!41}a^{22}+\frac{14\!\cdots\!69}{38\!\cdots\!41}a^{21}-\frac{10\!\cdots\!18}{38\!\cdots\!41}a^{20}+\frac{56\!\cdots\!30}{38\!\cdots\!41}a^{19}-\frac{38\!\cdots\!97}{38\!\cdots\!41}a^{18}+\frac{16\!\cdots\!81}{38\!\cdots\!41}a^{17}-\frac{97\!\cdots\!25}{38\!\cdots\!41}a^{16}+\frac{35\!\cdots\!66}{38\!\cdots\!41}a^{15}-\frac{19\!\cdots\!09}{38\!\cdots\!41}a^{14}+\frac{56\!\cdots\!06}{38\!\cdots\!41}a^{13}-\frac{26\!\cdots\!96}{38\!\cdots\!41}a^{12}+\frac{67\!\cdots\!68}{38\!\cdots\!41}a^{11}-\frac{29\!\cdots\!56}{38\!\cdots\!41}a^{10}+\frac{55\!\cdots\!37}{38\!\cdots\!41}a^{9}-\frac{20\!\cdots\!93}{38\!\cdots\!41}a^{8}+\frac{30\!\cdots\!21}{38\!\cdots\!41}a^{7}-\frac{11\!\cdots\!35}{38\!\cdots\!41}a^{6}+\frac{91\!\cdots\!52}{38\!\cdots\!41}a^{5}-\frac{30\!\cdots\!69}{38\!\cdots\!41}a^{4}+\frac{12\!\cdots\!59}{38\!\cdots\!41}a^{3}-\frac{64\!\cdots\!62}{38\!\cdots\!41}a^{2}+\frac{36\!\cdots\!58}{38\!\cdots\!41}a+\frac{19\!\cdots\!48}{38\!\cdots\!41}$ (assuming GRH)
Regulator:		\( 487075979.1876791 \) (assuming GRH)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{14}\cdot 487075979.1876791 \cdot 192}{6\cdot\sqrt{503553375386417026489977159144851919778199649}}\cr\approx \mathstrut & 0.103810693395349 \end{aligned}\] (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

\(\Q(\sqrt{29}) \), \(\Q(\sqrt{-87}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-3}, \sqrt{29})\), 7.7.594823321.1, \(\Q(\zeta_{29})^+\), 14.0.22439994995240462987343.1, 14.0.773792930870360792667.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Frobenius cycle types

$p$	$2$	$3$	$5$	$7$	$11$	$13$	$17$	$19$	$23$	$29$	$31$	$37$	$41$	$43$	$47$	$53$	$59$
Cycle type	${\href{/padicField/2.14.0.1}{14} }^{2}$	R	${\href{/padicField/5.14.0.1}{14} }^{2}$	${\href{/padicField/7.7.0.1}{7} }^{4}$	${\href{/padicField/11.14.0.1}{14} }^{2}$	${\href{/padicField/13.7.0.1}{7} }^{4}$	${\href{/padicField/17.2.0.1}{2} }^{14}$	${\href{/padicField/19.14.0.1}{14} }^{2}$	${\href{/padicField/23.14.0.1}{14} }^{2}$	R	${\href{/padicField/31.14.0.1}{14} }^{2}$	${\href{/padicField/37.14.0.1}{14} }^{2}$	${\href{/padicField/41.2.0.1}{2} }^{14}$	${\href{/padicField/43.14.0.1}{14} }^{2}$	${\href{/padicField/47.14.0.1}{14} }^{2}$	${\href{/padicField/53.14.0.1}{14} }^{2}$	${\href{/padicField/59.2.0.1}{2} }^{14}$

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 $28$	$2$	$14$	$14$
\(29\)		Deg $28$	$14$	$2$	$26$