Number field 28.0.17501529797217428894629579082505064100647449493408203125.1

• Label: 28.0.175...125.1
• Degree: $28$
• Signature: $[0, 14]$
• Discriminant: $1.750\times 10^{55}$
• Root discriminant: \(93.97\)
• Ramified primes: $5,7$
• Class number: $43793$ (GRH)
• Class group: [43793] (GRH)
• Galois group: $C_{28}$ (as 28T1)

Normalized defining polynomial

x^28 + 21*x^26 - 21*x^25 + 350*x^24 + 875*x^23 + 5964*x^22 + 15191*x^21 + 74431*x^20 + 149541*x^19 + 491337*x^18 + 902447*x^17 + 2521988*x^16 + 4058887*x^15 + 10825963*x^14 + 16164904*x^13 + 40448583*x^12 + 53367559*x^11 + 103699603*x^10 + 109113760*x^9 + 196642418*x^8 + 95224249*x^7 + 208289837*x^6 + 114584232*x^5 + 234771859*x^4 + 151478789*x^3 + 168609280*x^2 + 76664532*x + 88529281

Invariants

Degree:		$28$
Signature:		$[0, 14]$
Discriminant:		\(17501529797217428894629579082505064100647449493408203125\) \(\medspace = 5^{21}\cdot 7^{48}\)
Root discriminant:		\(93.97\)
Galois root discriminant:		$5^{3/4}7^{12/7}\approx 93.96519090194225$
Ramified primes:		\(5\), \(7\)
Discriminant root field:		\(\Q(\sqrt{5}) \)
$\card{ \Gal(K/\Q) }$:		$28$
This field is Galois and abelian over $\Q$.
Conductor:		\(245=5\cdot 7^{2}\)
Dirichlet character group:		$\lbrace$$\chi_{245}(64,·)$, $\chi_{245}(1,·)$, $\chi_{245}(197,·)$, $\chi_{245}(134,·)$, $\chi_{245}(71,·)$, $\chi_{245}(8,·)$, $\chi_{245}(204,·)$, $\chi_{245}(141,·)$, $\chi_{245}(78,·)$, $\chi_{245}(211,·)$, $\chi_{245}(148,·)$, $\chi_{245}(22,·)$, $\chi_{245}(218,·)$, $\chi_{245}(92,·)$, $\chi_{245}(29,·)$, $\chi_{245}(162,·)$, $\chi_{245}(99,·)$, $\chi_{245}(36,·)$, $\chi_{245}(232,·)$, $\chi_{245}(169,·)$, $\chi_{245}(106,·)$, $\chi_{245}(43,·)$, $\chi_{245}(239,·)$, $\chi_{245}(176,·)$, $\chi_{245}(113,·)$, $\chi_{245}(183,·)$, $\chi_{245}(57,·)$, $\chi_{245}(127,·)$$\rbrace$
This is a CM field.
Reflex fields:		unavailable$^{8192}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $\frac{1}{31}a^{21}+\frac{1}{31}a^{20}-\frac{2}{31}a^{19}-\frac{14}{31}a^{18}+\frac{1}{31}a^{17}-\frac{13}{31}a^{16}+\frac{12}{31}a^{15}+\frac{7}{31}a^{14}+\frac{13}{31}a^{13}-\frac{9}{31}a^{12}+\frac{13}{31}a^{11}-\frac{13}{31}a^{10}+\frac{15}{31}a^{9}-\frac{8}{31}a^{8}-\frac{12}{31}a^{7}-\frac{15}{31}a^{6}-\frac{9}{31}a^{5}+\frac{8}{31}a^{3}+\frac{5}{31}a^{2}+\frac{6}{31}a+\frac{12}{31}$, $\frac{1}{31}a^{22}-\frac{3}{31}a^{20}-\frac{12}{31}a^{19}+\frac{15}{31}a^{18}-\frac{14}{31}a^{17}-\frac{6}{31}a^{16}-\frac{5}{31}a^{15}+\frac{6}{31}a^{14}+\frac{9}{31}a^{13}-\frac{9}{31}a^{12}+\frac{5}{31}a^{11}-\frac{3}{31}a^{10}+\frac{8}{31}a^{9}-\frac{4}{31}a^{8}-\frac{3}{31}a^{7}+\frac{6}{31}a^{6}+\frac{9}{31}a^{5}+\frac{8}{31}a^{4}-\frac{3}{31}a^{3}+\frac{1}{31}a^{2}+\frac{6}{31}a-\frac{12}{31}$, $\frac{1}{31}a^{23}-\frac{9}{31}a^{20}+\frac{9}{31}a^{19}+\frac{6}{31}a^{18}-\frac{3}{31}a^{17}-\frac{13}{31}a^{16}+\frac{11}{31}a^{15}-\frac{1}{31}a^{14}-\frac{1}{31}a^{13}+\frac{9}{31}a^{12}+\frac{5}{31}a^{11}+\frac{10}{31}a^{9}+\frac{4}{31}a^{8}+\frac{1}{31}a^{7}-\frac{5}{31}a^{6}+\frac{12}{31}a^{5}-\frac{3}{31}a^{4}-\frac{6}{31}a^{3}-\frac{10}{31}a^{2}+\frac{6}{31}a+\frac{5}{31}$, $\frac{1}{589}a^{24}-\frac{8}{589}a^{23}+\frac{2}{589}a^{22}+\frac{1}{589}a^{21}+\frac{271}{589}a^{20}-\frac{203}{589}a^{19}-\frac{68}{589}a^{18}-\frac{7}{589}a^{17}+\frac{35}{589}a^{16}+\frac{83}{589}a^{15}-\frac{10}{31}a^{14}+\frac{10}{589}a^{13}+\frac{135}{589}a^{12}+\frac{286}{589}a^{11}-\frac{2}{589}a^{10}-\frac{34}{589}a^{9}-\frac{119}{589}a^{8}+\frac{16}{589}a^{7}-\frac{241}{589}a^{6}+\frac{294}{589}a^{5}+\frac{127}{589}a^{4}-\frac{136}{589}a^{3}-\frac{48}{589}a^{2}-\frac{250}{589}a-\frac{192}{589}$, $\frac{1}{23\!\cdots\!93}a^{25}+\frac{11\!\cdots\!94}{24\!\cdots\!69}a^{24}+\frac{18\!\cdots\!66}{23\!\cdots\!93}a^{23}+\frac{23\!\cdots\!57}{23\!\cdots\!93}a^{22}-\frac{23\!\cdots\!98}{23\!\cdots\!93}a^{21}-\frac{63\!\cdots\!23}{23\!\cdots\!93}a^{20}-\frac{40\!\cdots\!05}{23\!\cdots\!93}a^{19}-\frac{13\!\cdots\!83}{23\!\cdots\!93}a^{18}-\frac{79\!\cdots\!87}{23\!\cdots\!93}a^{17}-\frac{52\!\cdots\!28}{75\!\cdots\!03}a^{16}-\frac{39\!\cdots\!34}{23\!\cdots\!93}a^{15}+\frac{15\!\cdots\!85}{23\!\cdots\!93}a^{14}-\frac{27\!\cdots\!04}{23\!\cdots\!93}a^{13}+\frac{27\!\cdots\!30}{23\!\cdots\!93}a^{12}-\frac{27\!\cdots\!94}{23\!\cdots\!93}a^{11}-\frac{73\!\cdots\!68}{23\!\cdots\!93}a^{10}-\frac{12\!\cdots\!90}{23\!\cdots\!93}a^{9}+\frac{53\!\cdots\!03}{23\!\cdots\!93}a^{8}-\frac{14\!\cdots\!38}{23\!\cdots\!93}a^{7}+\frac{77\!\cdots\!00}{23\!\cdots\!93}a^{6}-\frac{41\!\cdots\!85}{23\!\cdots\!93}a^{5}+\frac{72\!\cdots\!82}{23\!\cdots\!93}a^{4}+\frac{36\!\cdots\!95}{23\!\cdots\!93}a^{3}+\frac{69\!\cdots\!12}{23\!\cdots\!93}a^{2}-\frac{61\!\cdots\!91}{12\!\cdots\!47}a-\frac{12\!\cdots\!51}{24\!\cdots\!69}$, $\frac{1}{22\!\cdots\!21}a^{26}+\frac{13\!\cdots\!39}{22\!\cdots\!21}a^{24}-\frac{14\!\cdots\!81}{22\!\cdots\!21}a^{23}+\frac{27\!\cdots\!28}{22\!\cdots\!21}a^{22}+\frac{21\!\cdots\!97}{22\!\cdots\!21}a^{21}+\frac{90\!\cdots\!06}{22\!\cdots\!21}a^{20}-\frac{28\!\cdots\!62}{22\!\cdots\!21}a^{19}+\frac{46\!\cdots\!76}{22\!\cdots\!21}a^{18}-\frac{15\!\cdots\!75}{22\!\cdots\!21}a^{17}+\frac{55\!\cdots\!10}{22\!\cdots\!21}a^{16}-\frac{25\!\cdots\!93}{22\!\cdots\!21}a^{15}+\frac{52\!\cdots\!36}{22\!\cdots\!21}a^{14}+\frac{34\!\cdots\!14}{22\!\cdots\!21}a^{13}-\frac{93\!\cdots\!50}{22\!\cdots\!21}a^{12}+\frac{56\!\cdots\!08}{22\!\cdots\!21}a^{11}-\frac{51\!\cdots\!61}{22\!\cdots\!21}a^{10}-\frac{78\!\cdots\!66}{22\!\cdots\!21}a^{9}+\frac{46\!\cdots\!72}{22\!\cdots\!21}a^{8}+\frac{91\!\cdots\!52}{22\!\cdots\!21}a^{7}+\frac{49\!\cdots\!19}{22\!\cdots\!21}a^{6}+\frac{10\!\cdots\!88}{22\!\cdots\!21}a^{5}-\frac{50\!\cdots\!74}{22\!\cdots\!21}a^{4}-\frac{14\!\cdots\!55}{22\!\cdots\!21}a^{3}-\frac{30\!\cdots\!38}{22\!\cdots\!21}a^{2}-\frac{67\!\cdots\!03}{23\!\cdots\!93}a-\frac{80\!\cdots\!79}{24\!\cdots\!69}$, $\frac{1}{22\!\cdots\!37}a^{27}+\frac{21}{22\!\cdots\!37}a^{25}+\frac{17\!\cdots\!18}{22\!\cdots\!37}a^{24}+\frac{15\!\cdots\!33}{22\!\cdots\!37}a^{23}+\frac{12\!\cdots\!87}{22\!\cdots\!37}a^{22}+\frac{84\!\cdots\!80}{22\!\cdots\!37}a^{21}+\frac{42\!\cdots\!94}{22\!\cdots\!37}a^{20}-\frac{16\!\cdots\!33}{22\!\cdots\!37}a^{19}+\frac{38\!\cdots\!93}{22\!\cdots\!37}a^{18}+\frac{19\!\cdots\!80}{71\!\cdots\!27}a^{17}-\frac{76\!\cdots\!05}{22\!\cdots\!37}a^{16}+\frac{11\!\cdots\!72}{22\!\cdots\!37}a^{15}+\frac{57\!\cdots\!73}{22\!\cdots\!37}a^{14}-\frac{82\!\cdots\!97}{22\!\cdots\!37}a^{13}-\frac{41\!\cdots\!60}{22\!\cdots\!37}a^{12}+\frac{47\!\cdots\!53}{22\!\cdots\!37}a^{11}-\frac{73\!\cdots\!62}{22\!\cdots\!37}a^{10}+\frac{51\!\cdots\!52}{22\!\cdots\!37}a^{9}+\frac{83\!\cdots\!12}{22\!\cdots\!37}a^{8}-\frac{97\!\cdots\!38}{22\!\cdots\!37}a^{7}-\frac{99\!\cdots\!92}{22\!\cdots\!37}a^{6}+\frac{10\!\cdots\!15}{22\!\cdots\!37}a^{5}+\frac{68\!\cdots\!69}{22\!\cdots\!37}a^{4}-\frac{64\!\cdots\!58}{71\!\cdots\!27}a^{3}+\frac{11\!\cdots\!92}{22\!\cdots\!21}a^{2}-\frac{41\!\cdots\!91}{23\!\cdots\!93}a+\frac{18\!\cdots\!45}{24\!\cdots\!69}$

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

Class group and class number

$C_{43793}$, which has order $43793$ (assuming GRH)

Unit group

Rank:		$13$
Torsion generator:		\( -\frac{290106923447510490333593123640811}{119831413034218376091108432809396358256759} a^{27} - \frac{222843898368442712553926096345220}{119831413034218376091108432809396358256759} a^{26} - \frac{5982297509771884723917896991172275}{119831413034218376091108432809396358256759} a^{25} + \frac{1559907288579098987877482674416540}{119831413034218376091108432809396358256759} a^{24} - \frac{93910132837434566448767022434814795}{119831413034218376091108432809396358256759} a^{23} - \frac{331865355846985373095135521073090029}{119831413034218376091108432809396358256759} a^{22} - \frac{19350278508326442206765916032643270}{1235375392105344083413488998035014002647} a^{21} - \frac{5617504701046296008737507518194392065}{119831413034218376091108432809396358256759} a^{20} - \frac{23965821331333627669182833340145466640}{119831413034218376091108432809396358256759} a^{19} - \frac{57109768395072945050452905739803188145}{119831413034218376091108432809396358256759} a^{18} - \frac{162585524651539498303083618489133783348}{119831413034218376091108432809396358256759} a^{17} - \frac{339149564437032811060971999055108973475}{119831413034218376091108432809396358256759} a^{16} - \frac{824014432726582307249367169113520375575}{119831413034218376091108432809396358256759} a^{15} - \frac{1535779069756829357581526167433682954285}{119831413034218376091108432809396358256759} a^{14} - \frac{3472177280572165452682075426400310189240}{119831413034218376091108432809396358256759} a^{13} - \frac{6122875332542490493865343529440531814861}{119831413034218376091108432809396358256759} a^{12} - \frac{12834680587142962194248036414440410520305}{119831413034218376091108432809396358256759} a^{11} - \frac{668091132718253726419059177143185500060}{3865529452716721809390594606754721234089} a^{10} - \frac{32295699566789186643433762488879752899905}{119831413034218376091108432809396358256759} a^{9} - \frac{41796109594368492237215737700170257040545}{119831413034218376091108432809396358256759} a^{8} - \frac{52576199986282719317765056168875721954752}{119831413034218376091108432809396358256759} a^{7} - \frac{44035663530304765956261085250974439837400}{119831413034218376091108432809396358256759} a^{6} - \frac{32505985078694750902884819367238263331370}{119831413034218376091108432809396358256759} a^{5} - \frac{55957769272500218160826806013525601406495}{119831413034218376091108432809396358256759} a^{4} - \frac{429808924766367991366573198905138480855}{1235375392105344083413488998035014002647} a^{3} - \frac{180884074966070294093427641726119682189642}{119831413034218376091108432809396358256759} a^{2} - \frac{2491227650835413194318478312567300685}{12735828784591176117664835031288804151} a - \frac{2271466426394400605954881527221522895}{12735828784591176117664835031288804151} \)  (order $10$)
Fundamental units:		$\frac{69\!\cdots\!11}{11\!\cdots\!59}a^{27}-\frac{14\!\cdots\!71}{11\!\cdots\!59}a^{26}+\frac{13\!\cdots\!59}{11\!\cdots\!59}a^{25}-\frac{43\!\cdots\!32}{11\!\cdots\!59}a^{24}+\frac{25\!\cdots\!01}{11\!\cdots\!59}a^{23}+\frac{14\!\cdots\!63}{11\!\cdots\!59}a^{22}+\frac{26\!\cdots\!63}{12\!\cdots\!47}a^{21}+\frac{17\!\cdots\!81}{11\!\cdots\!59}a^{20}+\frac{26\!\cdots\!36}{11\!\cdots\!59}a^{19}-\frac{74\!\cdots\!79}{11\!\cdots\!59}a^{18}+\frac{89\!\cdots\!68}{11\!\cdots\!59}a^{17}-\frac{10\!\cdots\!38}{11\!\cdots\!59}a^{16}+\frac{28\!\cdots\!71}{11\!\cdots\!59}a^{15}-\frac{28\!\cdots\!71}{38\!\cdots\!89}a^{14}+\frac{94\!\cdots\!12}{11\!\cdots\!59}a^{13}-\frac{43\!\cdots\!49}{11\!\cdots\!59}a^{12}+\frac{60\!\cdots\!39}{38\!\cdots\!89}a^{11}-\frac{20\!\cdots\!44}{11\!\cdots\!59}a^{10}-\frac{14\!\cdots\!03}{11\!\cdots\!59}a^{9}-\frac{66\!\cdots\!47}{11\!\cdots\!59}a^{8}-\frac{29\!\cdots\!14}{11\!\cdots\!59}a^{7}-\frac{18\!\cdots\!11}{11\!\cdots\!59}a^{6}-\frac{69\!\cdots\!16}{11\!\cdots\!59}a^{5}-\frac{75\!\cdots\!79}{11\!\cdots\!59}a^{4}-\frac{39\!\cdots\!75}{12\!\cdots\!47}a^{3}-\frac{20\!\cdots\!78}{11\!\cdots\!59}a^{2}-\frac{97\!\cdots\!39}{12\!\cdots\!47}a-\frac{44\!\cdots\!33}{12\!\cdots\!51}$, $\frac{19\!\cdots\!89}{24\!\cdots\!69}a^{27}+\frac{84\!\cdots\!19}{24\!\cdots\!69}a^{26}+\frac{30\!\cdots\!90}{24\!\cdots\!69}a^{25}+\frac{12\!\cdots\!74}{24\!\cdots\!69}a^{24}+\frac{29\!\cdots\!56}{24\!\cdots\!69}a^{23}+\frac{47\!\cdots\!17}{24\!\cdots\!69}a^{22}+\frac{15\!\cdots\!30}{24\!\cdots\!69}a^{21}+\frac{68\!\cdots\!62}{24\!\cdots\!69}a^{20}+\frac{20\!\cdots\!26}{24\!\cdots\!69}a^{19}+\frac{72\!\cdots\!01}{24\!\cdots\!69}a^{18}+\frac{13\!\cdots\!00}{24\!\cdots\!69}a^{17}+\frac{39\!\cdots\!65}{24\!\cdots\!69}a^{16}+\frac{69\!\cdots\!26}{24\!\cdots\!69}a^{15}+\frac{17\!\cdots\!60}{24\!\cdots\!69}a^{14}+\frac{26\!\cdots\!25}{24\!\cdots\!69}a^{13}+\frac{69\!\cdots\!60}{24\!\cdots\!69}a^{12}+\frac{94\!\cdots\!11}{24\!\cdots\!69}a^{11}+\frac{23\!\cdots\!90}{24\!\cdots\!69}a^{10}+\frac{20\!\cdots\!99}{24\!\cdots\!69}a^{9}+\frac{36\!\cdots\!69}{24\!\cdots\!69}a^{8}+\frac{17\!\cdots\!73}{24\!\cdots\!69}a^{7}+\frac{38\!\cdots\!62}{24\!\cdots\!69}a^{6}-\frac{70\!\cdots\!48}{24\!\cdots\!69}a^{5}+\frac{44\!\cdots\!31}{24\!\cdots\!69}a^{4}+\frac{28\!\cdots\!33}{24\!\cdots\!69}a^{3}+\frac{32\!\cdots\!91}{24\!\cdots\!69}a^{2}+\frac{15\!\cdots\!48}{24\!\cdots\!69}a-\frac{13\!\cdots\!35}{24\!\cdots\!69}$, $\frac{38\!\cdots\!08}{11\!\cdots\!59}a^{27}+\frac{25\!\cdots\!52}{11\!\cdots\!59}a^{26}+\frac{68\!\cdots\!65}{11\!\cdots\!59}a^{25}-\frac{17\!\cdots\!64}{11\!\cdots\!59}a^{24}+\frac{10\!\cdots\!97}{11\!\cdots\!59}a^{23}+\frac{47\!\cdots\!31}{11\!\cdots\!59}a^{22}+\frac{22\!\cdots\!82}{12\!\cdots\!47}a^{21}+\frac{64\!\cdots\!79}{11\!\cdots\!59}a^{20}+\frac{27\!\cdots\!24}{11\!\cdots\!59}a^{19}+\frac{65\!\cdots\!07}{11\!\cdots\!59}a^{18}+\frac{16\!\cdots\!24}{11\!\cdots\!59}a^{17}+\frac{38\!\cdots\!85}{11\!\cdots\!59}a^{16}+\frac{94\!\cdots\!45}{11\!\cdots\!59}a^{15}+\frac{17\!\cdots\!31}{11\!\cdots\!59}a^{14}+\frac{39\!\cdots\!84}{11\!\cdots\!59}a^{13}+\frac{68\!\cdots\!15}{11\!\cdots\!59}a^{12}+\frac{14\!\cdots\!63}{11\!\cdots\!59}a^{11}+\frac{76\!\cdots\!96}{38\!\cdots\!89}a^{10}+\frac{36\!\cdots\!23}{11\!\cdots\!59}a^{9}+\frac{47\!\cdots\!47}{11\!\cdots\!59}a^{8}+\frac{82\!\cdots\!95}{11\!\cdots\!59}a^{7}+\frac{50\!\cdots\!40}{11\!\cdots\!59}a^{6}+\frac{37\!\cdots\!42}{11\!\cdots\!59}a^{5}+\frac{63\!\cdots\!17}{11\!\cdots\!59}a^{4}+\frac{49\!\cdots\!93}{12\!\cdots\!47}a^{3}+\frac{15\!\cdots\!68}{11\!\cdots\!59}a^{2}+\frac{28\!\cdots\!71}{12\!\cdots\!51}a+\frac{25\!\cdots\!57}{12\!\cdots\!51}$, $\frac{87\!\cdots\!64}{23\!\cdots\!93}a^{27}-\frac{10\!\cdots\!60}{23\!\cdots\!93}a^{26}+\frac{17\!\cdots\!08}{23\!\cdots\!93}a^{25}-\frac{40\!\cdots\!64}{23\!\cdots\!93}a^{24}+\frac{31\!\cdots\!52}{23\!\cdots\!93}a^{23}+\frac{42\!\cdots\!04}{23\!\cdots\!93}a^{22}+\frac{39\!\cdots\!63}{23\!\cdots\!93}a^{21}+\frac{64\!\cdots\!52}{23\!\cdots\!93}a^{20}+\frac{44\!\cdots\!12}{23\!\cdots\!93}a^{19}+\frac{45\!\cdots\!36}{24\!\cdots\!69}a^{18}+\frac{22\!\cdots\!92}{23\!\cdots\!93}a^{17}+\frac{68\!\cdots\!44}{75\!\cdots\!03}a^{16}+\frac{51\!\cdots\!08}{12\!\cdots\!47}a^{15}+\frac{56\!\cdots\!08}{23\!\cdots\!93}a^{14}+\frac{39\!\cdots\!24}{23\!\cdots\!93}a^{13}+\frac{15\!\cdots\!44}{23\!\cdots\!93}a^{12}+\frac{12\!\cdots\!28}{23\!\cdots\!93}a^{11}+\frac{17\!\cdots\!92}{23\!\cdots\!93}a^{10}+\frac{15\!\cdots\!24}{23\!\cdots\!93}a^{9}-\frac{21\!\cdots\!24}{23\!\cdots\!93}a^{8}+\frac{20\!\cdots\!56}{23\!\cdots\!93}a^{7}-\frac{12\!\cdots\!49}{23\!\cdots\!93}a^{6}+\frac{22\!\cdots\!48}{23\!\cdots\!93}a^{5}-\frac{17\!\cdots\!08}{23\!\cdots\!93}a^{4}+\frac{26\!\cdots\!60}{24\!\cdots\!69}a^{3}-\frac{19\!\cdots\!56}{24\!\cdots\!69}a^{2}+\frac{22\!\cdots\!57}{23\!\cdots\!93}a-\frac{18\!\cdots\!32}{24\!\cdots\!69}$, $\frac{86\!\cdots\!83}{23\!\cdots\!93}a^{27}-\frac{10\!\cdots\!84}{23\!\cdots\!93}a^{26}+\frac{17\!\cdots\!51}{23\!\cdots\!93}a^{25}-\frac{39\!\cdots\!58}{23\!\cdots\!93}a^{24}+\frac{30\!\cdots\!94}{23\!\cdots\!93}a^{23}+\frac{42\!\cdots\!38}{23\!\cdots\!93}a^{22}+\frac{39\!\cdots\!09}{23\!\cdots\!93}a^{21}+\frac{64\!\cdots\!19}{23\!\cdots\!93}a^{20}+\frac{44\!\cdots\!64}{23\!\cdots\!93}a^{19}+\frac{45\!\cdots\!67}{24\!\cdots\!69}a^{18}+\frac{22\!\cdots\!74}{23\!\cdots\!93}a^{17}+\frac{20\!\cdots\!79}{23\!\cdots\!93}a^{16}+\frac{51\!\cdots\!51}{12\!\cdots\!47}a^{15}+\frac{56\!\cdots\!26}{23\!\cdots\!93}a^{14}+\frac{38\!\cdots\!28}{23\!\cdots\!93}a^{13}+\frac{15\!\cdots\!68}{23\!\cdots\!93}a^{12}+\frac{13\!\cdots\!20}{23\!\cdots\!93}a^{11}+\frac{17\!\cdots\!24}{23\!\cdots\!93}a^{10}+\frac{15\!\cdots\!53}{23\!\cdots\!93}a^{9}-\frac{21\!\cdots\!53}{23\!\cdots\!93}a^{8}+\frac{20\!\cdots\!57}{23\!\cdots\!93}a^{7}-\frac{12\!\cdots\!50}{23\!\cdots\!93}a^{6}+\frac{22\!\cdots\!31}{23\!\cdots\!93}a^{5}-\frac{17\!\cdots\!26}{23\!\cdots\!93}a^{4}+\frac{26\!\cdots\!70}{24\!\cdots\!69}a^{3}-\frac{19\!\cdots\!32}{24\!\cdots\!69}a^{2}+\frac{39\!\cdots\!14}{75\!\cdots\!03}a-\frac{18\!\cdots\!29}{24\!\cdots\!69}$, $\frac{46\!\cdots\!36}{22\!\cdots\!37}a^{27}-\frac{57\!\cdots\!52}{22\!\cdots\!21}a^{26}+\frac{96\!\cdots\!37}{22\!\cdots\!37}a^{25}-\frac{11\!\cdots\!91}{11\!\cdots\!23}a^{24}+\frac{17\!\cdots\!31}{22\!\cdots\!37}a^{23}+\frac{20\!\cdots\!46}{22\!\cdots\!37}a^{22}+\frac{22\!\cdots\!28}{22\!\cdots\!37}a^{21}+\frac{37\!\cdots\!53}{22\!\cdots\!37}a^{20}+\frac{25\!\cdots\!82}{22\!\cdots\!37}a^{19}+\frac{28\!\cdots\!68}{22\!\cdots\!37}a^{18}+\frac{14\!\cdots\!82}{22\!\cdots\!37}a^{17}+\frac{15\!\cdots\!93}{22\!\cdots\!37}a^{16}+\frac{21\!\cdots\!00}{71\!\cdots\!27}a^{15}+\frac{51\!\cdots\!45}{22\!\cdots\!37}a^{14}+\frac{28\!\cdots\!20}{22\!\cdots\!37}a^{13}+\frac{16\!\cdots\!12}{22\!\cdots\!37}a^{12}+\frac{99\!\cdots\!37}{22\!\cdots\!37}a^{11}+\frac{29\!\cdots\!12}{22\!\cdots\!37}a^{10}+\frac{19\!\cdots\!31}{22\!\cdots\!37}a^{9}-\frac{44\!\cdots\!87}{22\!\cdots\!37}a^{8}+\frac{32\!\cdots\!79}{22\!\cdots\!37}a^{7}-\frac{60\!\cdots\!27}{22\!\cdots\!37}a^{6}+\frac{45\!\cdots\!93}{22\!\cdots\!37}a^{5}-\frac{30\!\cdots\!62}{11\!\cdots\!23}a^{4}+\frac{18\!\cdots\!73}{22\!\cdots\!37}a^{3}-\frac{18\!\cdots\!67}{75\!\cdots\!03}a^{2}-\frac{33\!\cdots\!23}{24\!\cdots\!69}a-\frac{59\!\cdots\!73}{24\!\cdots\!69}$, $\frac{48\!\cdots\!82}{22\!\cdots\!37}a^{27}+\frac{73\!\cdots\!52}{24\!\cdots\!69}a^{26}+\frac{10\!\cdots\!38}{22\!\cdots\!37}a^{25}-\frac{92\!\cdots\!55}{22\!\cdots\!37}a^{24}+\frac{17\!\cdots\!36}{22\!\cdots\!37}a^{23}+\frac{43\!\cdots\!34}{22\!\cdots\!37}a^{22}+\frac{30\!\cdots\!36}{22\!\cdots\!37}a^{21}+\frac{79\!\cdots\!82}{22\!\cdots\!37}a^{20}+\frac{38\!\cdots\!38}{22\!\cdots\!37}a^{19}+\frac{80\!\cdots\!26}{22\!\cdots\!37}a^{18}+\frac{26\!\cdots\!98}{22\!\cdots\!37}a^{17}+\frac{48\!\cdots\!54}{22\!\cdots\!37}a^{16}+\frac{13\!\cdots\!76}{22\!\cdots\!37}a^{15}+\frac{22\!\cdots\!61}{22\!\cdots\!37}a^{14}+\frac{58\!\cdots\!06}{22\!\cdots\!37}a^{13}+\frac{88\!\cdots\!28}{22\!\cdots\!37}a^{12}+\frac{22\!\cdots\!46}{22\!\cdots\!37}a^{11}+\frac{29\!\cdots\!42}{22\!\cdots\!37}a^{10}+\frac{58\!\cdots\!56}{22\!\cdots\!37}a^{9}+\frac{31\!\cdots\!24}{11\!\cdots\!23}a^{8}+\frac{10\!\cdots\!92}{22\!\cdots\!37}a^{7}+\frac{52\!\cdots\!90}{22\!\cdots\!37}a^{6}+\frac{11\!\cdots\!02}{22\!\cdots\!37}a^{5}+\frac{27\!\cdots\!00}{22\!\cdots\!37}a^{4}+\frac{13\!\cdots\!22}{22\!\cdots\!37}a^{3}+\frac{86\!\cdots\!30}{22\!\cdots\!21}a^{2}+\frac{99\!\cdots\!76}{23\!\cdots\!93}a+\frac{46\!\cdots\!52}{24\!\cdots\!69}$, $\frac{51\!\cdots\!28}{22\!\cdots\!37}a^{27}-\frac{14\!\cdots\!65}{22\!\cdots\!21}a^{26}+\frac{11\!\cdots\!55}{22\!\cdots\!37}a^{25}-\frac{38\!\cdots\!21}{22\!\cdots\!37}a^{24}+\frac{21\!\cdots\!60}{22\!\cdots\!37}a^{23}-\frac{17\!\cdots\!53}{22\!\cdots\!37}a^{22}+\frac{19\!\cdots\!67}{22\!\cdots\!37}a^{21}+\frac{43\!\cdots\!75}{22\!\cdots\!37}a^{20}+\frac{20\!\cdots\!11}{22\!\cdots\!37}a^{19}-\frac{12\!\cdots\!33}{22\!\cdots\!37}a^{18}+\frac{94\!\cdots\!21}{22\!\cdots\!37}a^{17}-\frac{65\!\cdots\!22}{22\!\cdots\!37}a^{16}+\frac{39\!\cdots\!50}{22\!\cdots\!37}a^{15}-\frac{17\!\cdots\!70}{71\!\cdots\!27}a^{14}+\frac{17\!\cdots\!89}{22\!\cdots\!37}a^{13}-\frac{27\!\cdots\!11}{22\!\cdots\!37}a^{12}+\frac{58\!\cdots\!19}{22\!\cdots\!37}a^{11}-\frac{12\!\cdots\!49}{22\!\cdots\!37}a^{10}+\frac{69\!\cdots\!33}{22\!\cdots\!37}a^{9}-\frac{32\!\cdots\!32}{22\!\cdots\!37}a^{8}+\frac{20\!\cdots\!74}{22\!\cdots\!37}a^{7}-\frac{11\!\cdots\!00}{22\!\cdots\!37}a^{6}+\frac{88\!\cdots\!91}{22\!\cdots\!37}a^{5}-\frac{81\!\cdots\!98}{22\!\cdots\!37}a^{4}-\frac{56\!\cdots\!66}{22\!\cdots\!37}a^{3}-\frac{75\!\cdots\!42}{23\!\cdots\!93}a^{2}+\frac{30\!\cdots\!00}{23\!\cdots\!93}a-\frac{79\!\cdots\!05}{24\!\cdots\!69}$, $\frac{40\!\cdots\!08}{22\!\cdots\!37}a^{27}+\frac{21\!\cdots\!78}{23\!\cdots\!93}a^{26}+\frac{81\!\cdots\!27}{22\!\cdots\!37}a^{25}-\frac{45\!\cdots\!81}{22\!\cdots\!37}a^{24}+\frac{13\!\cdots\!66}{22\!\cdots\!37}a^{23}+\frac{13\!\cdots\!78}{71\!\cdots\!27}a^{22}+\frac{24\!\cdots\!59}{22\!\cdots\!37}a^{21}+\frac{69\!\cdots\!00}{22\!\cdots\!37}a^{20}+\frac{30\!\cdots\!63}{22\!\cdots\!37}a^{19}+\frac{69\!\cdots\!86}{22\!\cdots\!37}a^{18}+\frac{20\!\cdots\!49}{22\!\cdots\!37}a^{17}+\frac{39\!\cdots\!42}{22\!\cdots\!37}a^{16}+\frac{10\!\cdots\!84}{22\!\cdots\!37}a^{15}+\frac{17\!\cdots\!18}{22\!\cdots\!37}a^{14}+\frac{42\!\cdots\!38}{22\!\cdots\!37}a^{13}+\frac{70\!\cdots\!59}{22\!\cdots\!37}a^{12}+\frac{15\!\cdots\!05}{22\!\cdots\!37}a^{11}+\frac{22\!\cdots\!14}{22\!\cdots\!37}a^{10}+\frac{38\!\cdots\!10}{22\!\cdots\!37}a^{9}+\frac{22\!\cdots\!65}{11\!\cdots\!23}a^{8}+\frac{21\!\cdots\!55}{71\!\cdots\!27}a^{7}+\frac{35\!\cdots\!58}{22\!\cdots\!37}a^{6}+\frac{41\!\cdots\!61}{22\!\cdots\!37}a^{5}+\frac{48\!\cdots\!22}{22\!\cdots\!37}a^{4}+\frac{79\!\cdots\!35}{22\!\cdots\!37}a^{3}+\frac{55\!\cdots\!49}{22\!\cdots\!21}a^{2}+\frac{26\!\cdots\!25}{24\!\cdots\!69}a-\frac{45\!\cdots\!60}{12\!\cdots\!51}$, $\frac{75\!\cdots\!24}{22\!\cdots\!37}a^{27}+\frac{87\!\cdots\!72}{22\!\cdots\!21}a^{26}+\frac{15\!\cdots\!42}{22\!\cdots\!37}a^{25}-\frac{57\!\cdots\!27}{22\!\cdots\!37}a^{24}+\frac{24\!\cdots\!94}{22\!\cdots\!37}a^{23}+\frac{81\!\cdots\!54}{22\!\cdots\!37}a^{22}+\frac{54\!\cdots\!98}{22\!\cdots\!37}a^{21}+\frac{14\!\cdots\!85}{22\!\cdots\!37}a^{20}+\frac{63\!\cdots\!89}{22\!\cdots\!37}a^{19}+\frac{13\!\cdots\!09}{22\!\cdots\!37}a^{18}+\frac{42\!\cdots\!75}{22\!\cdots\!37}a^{17}+\frac{68\!\cdots\!42}{22\!\cdots\!37}a^{16}+\frac{20\!\cdots\!96}{22\!\cdots\!37}a^{15}+\frac{31\!\cdots\!58}{22\!\cdots\!37}a^{14}+\frac{87\!\cdots\!16}{22\!\cdots\!37}a^{13}+\frac{11\!\cdots\!13}{22\!\cdots\!37}a^{12}+\frac{32\!\cdots\!11}{22\!\cdots\!37}a^{11}+\frac{35\!\cdots\!36}{22\!\cdots\!37}a^{10}+\frac{79\!\cdots\!19}{22\!\cdots\!37}a^{9}+\frac{40\!\cdots\!33}{22\!\cdots\!37}a^{8}+\frac{12\!\cdots\!78}{22\!\cdots\!37}a^{7}+\frac{38\!\cdots\!48}{22\!\cdots\!37}a^{6}+\frac{22\!\cdots\!34}{22\!\cdots\!37}a^{5}-\frac{17\!\cdots\!89}{22\!\cdots\!37}a^{4}+\frac{33\!\cdots\!39}{71\!\cdots\!27}a^{3}-\frac{18\!\cdots\!97}{23\!\cdots\!93}a^{2}+\frac{18\!\cdots\!77}{23\!\cdots\!93}a+\frac{58\!\cdots\!85}{24\!\cdots\!69}$, $\frac{76\!\cdots\!81}{22\!\cdots\!37}a^{27}-\frac{90\!\cdots\!11}{22\!\cdots\!21}a^{26}+\frac{31\!\cdots\!41}{22\!\cdots\!37}a^{25}-\frac{20\!\cdots\!28}{22\!\cdots\!37}a^{24}+\frac{77\!\cdots\!28}{22\!\cdots\!37}a^{23}-\frac{28\!\cdots\!65}{22\!\cdots\!37}a^{22}+\frac{24\!\cdots\!04}{22\!\cdots\!37}a^{21}-\frac{29\!\cdots\!81}{22\!\cdots\!37}a^{20}+\frac{67\!\cdots\!81}{22\!\cdots\!37}a^{19}-\frac{18\!\cdots\!29}{11\!\cdots\!23}a^{18}+\frac{74\!\cdots\!82}{22\!\cdots\!37}a^{17}-\frac{19\!\cdots\!32}{22\!\cdots\!37}a^{16}+\frac{17\!\cdots\!42}{22\!\cdots\!37}a^{15}-\frac{89\!\cdots\!89}{22\!\cdots\!37}a^{14}+\frac{32\!\cdots\!53}{22\!\cdots\!37}a^{13}-\frac{40\!\cdots\!34}{22\!\cdots\!37}a^{12}+\frac{18\!\cdots\!95}{22\!\cdots\!37}a^{11}-\frac{14\!\cdots\!79}{22\!\cdots\!37}a^{10}+\frac{84\!\cdots\!47}{22\!\cdots\!37}a^{9}-\frac{30\!\cdots\!26}{22\!\cdots\!37}a^{8}+\frac{28\!\cdots\!72}{22\!\cdots\!37}a^{7}-\frac{67\!\cdots\!21}{22\!\cdots\!37}a^{6}+\frac{13\!\cdots\!95}{22\!\cdots\!37}a^{5}-\frac{15\!\cdots\!08}{22\!\cdots\!37}a^{4}+\frac{41\!\cdots\!53}{71\!\cdots\!27}a^{3}-\frac{51\!\cdots\!27}{22\!\cdots\!21}a^{2}+\frac{46\!\cdots\!62}{23\!\cdots\!93}a+\frac{10\!\cdots\!06}{24\!\cdots\!69}$, $\frac{54\!\cdots\!15}{22\!\cdots\!37}a^{27}-\frac{13\!\cdots\!64}{73\!\cdots\!91}a^{26}+\frac{11\!\cdots\!68}{22\!\cdots\!37}a^{25}-\frac{19\!\cdots\!33}{22\!\cdots\!37}a^{24}+\frac{19\!\cdots\!50}{22\!\cdots\!37}a^{23}+\frac{34\!\cdots\!59}{22\!\cdots\!37}a^{22}+\frac{28\!\cdots\!87}{22\!\cdots\!37}a^{21}+\frac{60\!\cdots\!87}{22\!\cdots\!37}a^{20}+\frac{11\!\cdots\!81}{71\!\cdots\!27}a^{19}+\frac{52\!\cdots\!99}{22\!\cdots\!37}a^{18}+\frac{66\!\cdots\!17}{71\!\cdots\!27}a^{17}+\frac{30\!\cdots\!01}{22\!\cdots\!37}a^{16}+\frac{97\!\cdots\!16}{22\!\cdots\!37}a^{15}+\frac{12\!\cdots\!25}{22\!\cdots\!37}a^{14}+\frac{40\!\cdots\!80}{22\!\cdots\!37}a^{13}+\frac{42\!\cdots\!49}{22\!\cdots\!37}a^{12}+\frac{14\!\cdots\!11}{22\!\cdots\!37}a^{11}+\frac{11\!\cdots\!55}{22\!\cdots\!37}a^{10}+\frac{30\!\cdots\!63}{22\!\cdots\!37}a^{9}+\frac{21\!\cdots\!16}{37\!\cdots\!33}a^{8}+\frac{43\!\cdots\!16}{22\!\cdots\!37}a^{7}-\frac{52\!\cdots\!01}{22\!\cdots\!37}a^{6}+\frac{36\!\cdots\!21}{22\!\cdots\!37}a^{5}-\frac{94\!\cdots\!01}{22\!\cdots\!37}a^{4}+\frac{31\!\cdots\!43}{22\!\cdots\!37}a^{3}-\frac{49\!\cdots\!60}{22\!\cdots\!21}a^{2}+\frac{38\!\cdots\!29}{23\!\cdots\!93}a+\frac{42\!\cdots\!25}{24\!\cdots\!69}$, $\frac{56\!\cdots\!98}{22\!\cdots\!37}a^{27}-\frac{14\!\cdots\!45}{23\!\cdots\!93}a^{26}+\frac{12\!\cdots\!15}{22\!\cdots\!37}a^{25}-\frac{39\!\cdots\!28}{22\!\cdots\!37}a^{24}+\frac{24\!\cdots\!18}{22\!\cdots\!37}a^{23}+\frac{31\!\cdots\!50}{22\!\cdots\!37}a^{22}+\frac{24\!\cdots\!19}{22\!\cdots\!37}a^{21}+\frac{17\!\cdots\!40}{22\!\cdots\!37}a^{20}+\frac{26\!\cdots\!71}{22\!\cdots\!37}a^{19}+\frac{10\!\cdots\!80}{22\!\cdots\!37}a^{18}+\frac{14\!\cdots\!71}{22\!\cdots\!37}a^{17}+\frac{15\!\cdots\!41}{22\!\cdots\!37}a^{16}+\frac{63\!\cdots\!19}{22\!\cdots\!37}a^{15}-\frac{13\!\cdots\!04}{22\!\cdots\!37}a^{14}+\frac{28\!\cdots\!30}{22\!\cdots\!37}a^{13}-\frac{37\!\cdots\!19}{71\!\cdots\!27}a^{12}+\frac{10\!\cdots\!24}{22\!\cdots\!37}a^{11}-\frac{70\!\cdots\!54}{22\!\cdots\!37}a^{10}+\frac{19\!\cdots\!15}{22\!\cdots\!37}a^{9}-\frac{20\!\cdots\!45}{22\!\cdots\!37}a^{8}+\frac{43\!\cdots\!53}{22\!\cdots\!37}a^{7}-\frac{10\!\cdots\!63}{22\!\cdots\!37}a^{6}+\frac{13\!\cdots\!23}{22\!\cdots\!37}a^{5}-\frac{98\!\cdots\!80}{22\!\cdots\!37}a^{4}+\frac{30\!\cdots\!27}{22\!\cdots\!37}a^{3}-\frac{52\!\cdots\!19}{22\!\cdots\!21}a^{2}-\frac{10\!\cdots\!61}{24\!\cdots\!69}a-\frac{18\!\cdots\!41}{24\!\cdots\!69}$ (assuming GRH)
Regulator:		\( 1494941023922.2002 \) (assuming GRH)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{14}\cdot 1494941023922.2002 \cdot 43793}{10\cdot\sqrt{17501529797217428894629579082505064100647449493408203125}}\cr\approx \mathstrut & 0.233888521310681 \end{aligned}\] (assuming GRH)

Galois group

$C_{28}$ (as 28T1):

A cyclic group of order 28

The 28 conjugacy class representatives for $C_{28}$

Character table for $C_{28}$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\), 7.7.13841287201.1, 14.14.14967283701606751125078125.1

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	$28$	$28$	R	R	${\href{/padicField/11.7.0.1}{7} }^{4}$	$28$	$28$	${\href{/padicField/19.2.0.1}{2} }^{14}$	$28$	${\href{/padicField/29.14.0.1}{14} }^{2}$	${\href{/padicField/31.1.0.1}{1} }^{28}$	$28$	${\href{/padicField/41.7.0.1}{7} }^{4}$	$28$	$28$	$28$	${\href{/padicField/59.14.0.1}{14} }^{2}$

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
\(5\)		Deg $28$	$4$	$7$	$21$
\(7\)		Deg $28$	$7$	$4$	$48$