Number field 28.0.128401607345812283715591125636314028226206818103790283203125.1

• Label: 28.0.128...125.1
• Degree: $28$
• Signature: $[0, 14]$
• Discriminant: $1.284\times 10^{59}$
• Root discriminant: \(129.13\)
• Ramified primes: $5,71$
• Class number: $617792$ (GRH)
• Class group: [2, 2, 2, 2, 14, 2758] (GRH)
• Galois group: $C_{28}$ (as 28T1)

Normalized defining polynomial

x^28 - x^27 + 31*x^26 - 64*x^25 + 743*x^24 + 360*x^23 + 14247*x^22 + 17824*x^21 + 211772*x^20 + 351618*x^19 + 2950835*x^18 + 5926264*x^17 + 35579743*x^16 + 98555653*x^15 + 337486035*x^14 + 730923194*x^13 + 2193380028*x^12 + 3489008888*x^11 + 6547998985*x^10 + 8180911099*x^9 + 13300322412*x^8 - 1313161326*x^7 + 8126173929*x^6 - 251536057*x^5 + 8464009280*x^4 + 3624030723*x^3 + 1759162063*x^2 + 629981485*x + 352275361

Invariants

Degree:		$28$
Signature:		$[0, 14]$
Discriminant:		\(128401607345812283715591125636314028226206818103790283203125\) \(\medspace = 5^{21}\cdot 71^{24}\)
Root discriminant:		\(129.13\)
Galois root discriminant:		$5^{3/4}71^{6/7}\approx 129.12798485937523$
Ramified primes:		\(5\), \(71\)
Discriminant root field:		\(\Q(\sqrt{5}) \)
$\card{ \Gal(K/\Q) }$:		$28$
This field is Galois and abelian over $\Q$.
Conductor:		\(355=5\cdot 71\)
Dirichlet character group:		$\lbrace$$\chi_{355}(1,·)$, $\chi_{355}(258,·)$, $\chi_{355}(261,·)$, $\chi_{355}(321,·)$, $\chi_{355}(72,·)$, $\chi_{355}(329,·)$, $\chi_{355}(332,·)$, $\chi_{355}(143,·)$, $\chi_{355}(214,·)$, $\chi_{355}(91,·)$, $\chi_{355}(101,·)$, $\chi_{355}(32,·)$, $\chi_{355}(48,·)$, $\chi_{355}(162,·)$, $\chi_{355}(243,·)$, $\chi_{355}(37,·)$, $\chi_{355}(103,·)$, $\chi_{355}(233,·)$, $\chi_{355}(172,·)$, $\chi_{355}(174,·)$, $\chi_{355}(304,·)$, $\chi_{355}(108,·)$, $\chi_{355}(179,·)$, $\chi_{355}(116,·)$, $\chi_{355}(119,·)$, $\chi_{355}(314,·)$, $\chi_{355}(187,·)$, $\chi_{355}(316,·)$$\rbrace$
This is a CM field.
Reflex fields:		unavailable$^{8192}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{5}a^{16}+\frac{1}{5}a^{12}+\frac{1}{5}a^{8}+\frac{1}{5}a^{4}+\frac{1}{5}$, $\frac{1}{5}a^{17}+\frac{1}{5}a^{13}+\frac{1}{5}a^{9}+\frac{1}{5}a^{5}+\frac{1}{5}a$, $\frac{1}{25}a^{18}+\frac{2}{25}a^{17}+\frac{2}{25}a^{16}-\frac{4}{25}a^{14}-\frac{8}{25}a^{13}-\frac{8}{25}a^{12}-\frac{9}{25}a^{10}+\frac{7}{25}a^{9}+\frac{7}{25}a^{8}+\frac{11}{25}a^{6}-\frac{3}{25}a^{5}-\frac{3}{25}a^{4}+\frac{6}{25}a^{2}+\frac{12}{25}a+\frac{12}{25}$, $\frac{1}{25}a^{19}-\frac{2}{25}a^{17}+\frac{1}{25}a^{16}-\frac{4}{25}a^{15}+\frac{8}{25}a^{13}-\frac{4}{25}a^{12}-\frac{9}{25}a^{11}-\frac{7}{25}a^{9}-\frac{9}{25}a^{8}+\frac{11}{25}a^{7}+\frac{3}{25}a^{5}+\frac{11}{25}a^{4}+\frac{6}{25}a^{3}-\frac{12}{25}a+\frac{6}{25}$, $\frac{1}{25}a^{20}-\frac{1}{25}$, $\frac{1}{25}a^{21}-\frac{1}{25}a$, $\frac{1}{25}a^{22}-\frac{1}{25}a^{2}$, $\frac{1}{25}a^{23}-\frac{1}{25}a^{3}$, $\frac{1}{2125}a^{24}+\frac{18}{2125}a^{23}+\frac{12}{2125}a^{22}+\frac{3}{425}a^{21}+\frac{37}{2125}a^{20}-\frac{22}{2125}a^{19}-\frac{1}{85}a^{18}+\frac{7}{125}a^{17}+\frac{168}{2125}a^{16}-\frac{287}{2125}a^{15}-\frac{26}{85}a^{14}+\frac{899}{2125}a^{13}+\frac{78}{2125}a^{12}-\frac{727}{2125}a^{11}+\frac{29}{85}a^{10}-\frac{596}{2125}a^{9}-\frac{62}{2125}a^{8}+\frac{158}{2125}a^{7}+\frac{9}{85}a^{6}-\frac{991}{2125}a^{5}+\frac{897}{2125}a^{4}+\frac{1}{5}a^{3}-\frac{987}{2125}a^{2}+\frac{324}{2125}a+\frac{271}{2125}$, $\frac{1}{24\!\cdots\!25}a^{25}-\frac{61\!\cdots\!33}{24\!\cdots\!25}a^{24}+\frac{24\!\cdots\!54}{24\!\cdots\!25}a^{23}-\frac{25\!\cdots\!47}{24\!\cdots\!25}a^{22}-\frac{35\!\cdots\!48}{24\!\cdots\!25}a^{21}-\frac{74\!\cdots\!49}{24\!\cdots\!25}a^{20}-\frac{31\!\cdots\!33}{24\!\cdots\!25}a^{19}+\frac{33\!\cdots\!19}{24\!\cdots\!25}a^{18}-\frac{17\!\cdots\!91}{24\!\cdots\!25}a^{17}-\frac{22\!\cdots\!97}{48\!\cdots\!25}a^{16}+\frac{50\!\cdots\!82}{24\!\cdots\!25}a^{15}+\frac{15\!\cdots\!49}{24\!\cdots\!25}a^{14}+\frac{80\!\cdots\!14}{24\!\cdots\!25}a^{13}+\frac{15\!\cdots\!88}{48\!\cdots\!25}a^{12}+\frac{68\!\cdots\!22}{24\!\cdots\!25}a^{11}-\frac{10\!\cdots\!88}{14\!\cdots\!25}a^{10}-\frac{38\!\cdots\!06}{24\!\cdots\!25}a^{9}-\frac{18\!\cdots\!02}{48\!\cdots\!25}a^{8}-\frac{36\!\cdots\!13}{24\!\cdots\!25}a^{7}+\frac{16\!\cdots\!02}{14\!\cdots\!25}a^{6}-\frac{13\!\cdots\!02}{24\!\cdots\!25}a^{5}+\frac{11\!\cdots\!23}{24\!\cdots\!25}a^{4}-\frac{18\!\cdots\!02}{24\!\cdots\!25}a^{3}-\frac{10\!\cdots\!64}{24\!\cdots\!25}a^{2}+\frac{17\!\cdots\!02}{24\!\cdots\!25}a+\frac{61\!\cdots\!97}{17\!\cdots\!25}$, $\frac{1}{16\!\cdots\!25}a^{26}+\frac{8}{98\!\cdots\!25}a^{25}+\frac{16\!\cdots\!72}{16\!\cdots\!25}a^{24}+\frac{12\!\cdots\!14}{16\!\cdots\!25}a^{23}-\frac{77\!\cdots\!16}{16\!\cdots\!25}a^{22}-\frac{27\!\cdots\!36}{16\!\cdots\!25}a^{21}-\frac{30\!\cdots\!19}{16\!\cdots\!25}a^{20}-\frac{85\!\cdots\!73}{16\!\cdots\!25}a^{19}-\frac{35\!\cdots\!64}{33\!\cdots\!25}a^{18}-\frac{10\!\cdots\!64}{16\!\cdots\!25}a^{17}-\frac{16\!\cdots\!03}{16\!\cdots\!25}a^{16}+\frac{77\!\cdots\!92}{16\!\cdots\!25}a^{15}+\frac{58\!\cdots\!18}{19\!\cdots\!25}a^{14}-\frac{72\!\cdots\!94}{16\!\cdots\!25}a^{13}-\frac{65\!\cdots\!63}{16\!\cdots\!25}a^{12}-\frac{24\!\cdots\!54}{98\!\cdots\!25}a^{11}-\frac{74\!\cdots\!74}{33\!\cdots\!25}a^{10}+\frac{37\!\cdots\!01}{16\!\cdots\!25}a^{9}+\frac{72\!\cdots\!56}{98\!\cdots\!25}a^{8}+\frac{52\!\cdots\!97}{16\!\cdots\!25}a^{7}-\frac{13\!\cdots\!71}{16\!\cdots\!25}a^{6}-\frac{13\!\cdots\!93}{33\!\cdots\!25}a^{5}+\frac{70\!\cdots\!64}{33\!\cdots\!25}a^{4}-\frac{21\!\cdots\!27}{16\!\cdots\!25}a^{3}+\frac{79\!\cdots\!21}{16\!\cdots\!25}a^{2}-\frac{17\!\cdots\!04}{12\!\cdots\!25}a-\frac{33\!\cdots\!21}{89\!\cdots\!25}$, $\frac{1}{22\!\cdots\!25}a^{27}-\frac{1}{22\!\cdots\!25}a^{26}+\frac{752}{91\!\cdots\!25}a^{25}+\frac{11\!\cdots\!49}{91\!\cdots\!25}a^{24}-\frac{11\!\cdots\!54}{22\!\cdots\!25}a^{23}+\frac{34\!\cdots\!71}{22\!\cdots\!25}a^{22}+\frac{92\!\cdots\!49}{13\!\cdots\!25}a^{21}+\frac{12\!\cdots\!93}{91\!\cdots\!25}a^{20}+\frac{35\!\cdots\!91}{22\!\cdots\!25}a^{19}+\frac{33\!\cdots\!16}{22\!\cdots\!25}a^{18}-\frac{80\!\cdots\!58}{91\!\cdots\!25}a^{17}+\frac{14\!\cdots\!43}{22\!\cdots\!25}a^{16}-\frac{12\!\cdots\!39}{22\!\cdots\!25}a^{15}+\frac{14\!\cdots\!61}{22\!\cdots\!25}a^{14}+\frac{42\!\cdots\!32}{91\!\cdots\!25}a^{13}+\frac{82\!\cdots\!03}{22\!\cdots\!25}a^{12}-\frac{99\!\cdots\!94}{22\!\cdots\!25}a^{11}-\frac{68\!\cdots\!44}{22\!\cdots\!25}a^{10}-\frac{22\!\cdots\!28}{91\!\cdots\!25}a^{9}-\frac{68\!\cdots\!12}{22\!\cdots\!25}a^{8}+\frac{37\!\cdots\!57}{91\!\cdots\!25}a^{7}-\frac{76\!\cdots\!19}{13\!\cdots\!25}a^{6}-\frac{82\!\cdots\!23}{18\!\cdots\!05}a^{5}-\frac{79\!\cdots\!02}{22\!\cdots\!25}a^{4}-\frac{37\!\cdots\!97}{91\!\cdots\!25}a^{3}+\frac{84\!\cdots\!35}{26\!\cdots\!33}a^{2}-\frac{34\!\cdots\!82}{12\!\cdots\!25}a+\frac{21\!\cdots\!58}{52\!\cdots\!25}$

Monogenic:		No
Index:		Not computed
Inessential primes:		$5$

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{14}\times C_{2758}$, which has order $617792$ (assuming GRH)

Unit group

Rank:		$13$
Torsion generator:		\( \frac{61683217279394625257925418413465867558477}{54018810049828879849070284682371709765684808152825} a^{27} - \frac{61606839159964520934535332311373404125419}{54018810049828879849070284682371709765684808152825} a^{26} + \frac{1918570038320218778052658508025844668211973}{54018810049828879849070284682371709765684808152825} a^{25} - \frac{3958117852489581071053352992919126996205454}{54018810049828879849070284682371709765684808152825} a^{24} + \frac{46026234802450703738840596388663938448009949}{54018810049828879849070284682371709765684808152825} a^{23} + \frac{21664411894449482071909643469644782426192814}{54018810049828879849070284682371709765684808152825} a^{22} + \frac{883837017018517444925358933536421068954600223}{54018810049828879849070284682371709765684808152825} a^{21} + \frac{1098711744559909436946731401619285314487036428}{54018810049828879849070284682371709765684808152825} a^{20} + \frac{13149079768261280874094534076641863227747555623}{54018810049828879849070284682371709765684808152825} a^{19} + \frac{21731483333452303677405036073228551143514531312}{54018810049828879849070284682371709765684808152825} a^{18} + \frac{183214071821223509248291806780095529341879052793}{54018810049828879849070284682371709765684808152825} a^{17} + \frac{366662395864220141225963953473212456217465719392}{54018810049828879849070284682371709765684808152825} a^{16} + \frac{2210746808774628049423339614114553389710700798383}{54018810049828879849070284682371709765684808152825} a^{15} + \frac{6100325092333221704186924797738632811060403403252}{54018810049828879849070284682371709765684808152825} a^{14} + \frac{21000538985625351330418961371379022537794959914703}{54018810049828879849070284682371709765684808152825} a^{13} + \frac{45499289520348579983358992120382599544297880359682}{54018810049828879849070284682371709765684808152825} a^{12} + \frac{136714463332068932164388098471322617445831310731718}{54018810049828879849070284682371709765684808152825} a^{11} + \frac{217503143885822948274432698880314094567177584552992}{54018810049828879849070284682371709765684808152825} a^{10} + \frac{412044916414836212232803115228408913849166372391438}{54018810049828879849070284682371709765684808152825} a^{9} + \frac{510568157305674818232252200297616632258156603416647}{54018810049828879849070284682371709765684808152825} a^{8} + \frac{830391758331734045464872155072194976488039554368826}{54018810049828879849070284682371709765684808152825} a^{7} - \frac{82203466664390326344475853080003485127684149533774}{54018810049828879849070284682371709765684808152825} a^{6} + \frac{20289843460841370306127716932177867828721936711719}{2160752401993155193962811387294868390627392326113} a^{5} - \frac{139599337999904612212153557850647177221134070789934}{54018810049828879849070284682371709765684808152825} a^{4} + \frac{528414921875664512668987592424320022913567105275764}{54018810049828879849070284682371709765684808152825} a^{3} + \frac{1651452663727924142866204874571931986777323274659}{394297883575393283569856092572056275661932906225} a^{2} + \frac{1170327487021997778835993887438296582024085263}{575617348285245669445045390616140548411580885} a + \frac{15296026691934797349729291196096783799375983}{21007932419169549979746182139275202496773025} \)  (order $10$)
Fundamental units:		$\frac{60\!\cdots\!26}{54\!\cdots\!25}a^{27}-\frac{94\!\cdots\!19}{54\!\cdots\!25}a^{26}+\frac{19\!\cdots\!12}{54\!\cdots\!25}a^{25}-\frac{49\!\cdots\!26}{54\!\cdots\!25}a^{24}+\frac{46\!\cdots\!29}{54\!\cdots\!25}a^{23}-\frac{35\!\cdots\!46}{54\!\cdots\!25}a^{22}+\frac{84\!\cdots\!01}{54\!\cdots\!25}a^{21}+\frac{59\!\cdots\!47}{54\!\cdots\!25}a^{20}+\frac{24\!\cdots\!54}{10\!\cdots\!65}a^{19}+\frac{14\!\cdots\!51}{54\!\cdots\!25}a^{18}+\frac{16\!\cdots\!47}{54\!\cdots\!25}a^{17}+\frac{25\!\cdots\!62}{54\!\cdots\!25}a^{16}+\frac{38\!\cdots\!34}{10\!\cdots\!65}a^{15}+\frac{47\!\cdots\!21}{54\!\cdots\!25}a^{14}+\frac{17\!\cdots\!12}{54\!\cdots\!25}a^{13}+\frac{32\!\cdots\!77}{54\!\cdots\!25}a^{12}+\frac{21\!\cdots\!14}{10\!\cdots\!65}a^{11}+\frac{13\!\cdots\!41}{54\!\cdots\!25}a^{10}+\frac{27\!\cdots\!02}{54\!\cdots\!25}a^{9}+\frac{27\!\cdots\!17}{54\!\cdots\!25}a^{8}+\frac{52\!\cdots\!69}{54\!\cdots\!25}a^{7}-\frac{10\!\cdots\!89}{10\!\cdots\!65}a^{6}+\frac{10\!\cdots\!36}{10\!\cdots\!65}a^{5}-\frac{28\!\cdots\!42}{54\!\cdots\!25}a^{4}+\frac{46\!\cdots\!91}{54\!\cdots\!25}a^{3}-\frac{35\!\cdots\!92}{28\!\cdots\!25}a^{2}-\frac{62\!\cdots\!23}{21\!\cdots\!25}a+\frac{50\!\cdots\!44}{84\!\cdots\!21}$, $\frac{13\!\cdots\!11}{22\!\cdots\!25}a^{27}-\frac{17\!\cdots\!92}{22\!\cdots\!25}a^{26}+\frac{40\!\cdots\!09}{22\!\cdots\!25}a^{25}-\frac{96\!\cdots\!87}{22\!\cdots\!25}a^{24}+\frac{99\!\cdots\!32}{22\!\cdots\!25}a^{23}+\frac{15\!\cdots\!92}{22\!\cdots\!25}a^{22}+\frac{18\!\cdots\!29}{22\!\cdots\!25}a^{21}+\frac{17\!\cdots\!79}{22\!\cdots\!25}a^{20}+\frac{26\!\cdots\!99}{22\!\cdots\!25}a^{19}+\frac{36\!\cdots\!46}{22\!\cdots\!25}a^{18}+\frac{36\!\cdots\!89}{22\!\cdots\!25}a^{17}+\frac{64\!\cdots\!76}{22\!\cdots\!25}a^{16}+\frac{43\!\cdots\!04}{22\!\cdots\!25}a^{15}+\frac{11\!\cdots\!91}{22\!\cdots\!25}a^{14}+\frac{39\!\cdots\!19}{22\!\cdots\!25}a^{13}+\frac{47\!\cdots\!88}{13\!\cdots\!25}a^{12}+\frac{25\!\cdots\!34}{22\!\cdots\!25}a^{11}+\frac{36\!\cdots\!36}{22\!\cdots\!25}a^{10}+\frac{71\!\cdots\!74}{22\!\cdots\!25}a^{9}+\frac{80\!\cdots\!66}{22\!\cdots\!25}a^{8}+\frac{14\!\cdots\!03}{22\!\cdots\!25}a^{7}-\frac{69\!\cdots\!02}{22\!\cdots\!25}a^{6}+\frac{23\!\cdots\!39}{45\!\cdots\!25}a^{5}-\frac{30\!\cdots\!27}{22\!\cdots\!25}a^{4}+\frac{10\!\cdots\!12}{22\!\cdots\!25}a^{3}+\frac{10\!\cdots\!32}{16\!\cdots\!25}a^{2}+\frac{91\!\cdots\!14}{24\!\cdots\!25}a+\frac{30\!\cdots\!09}{89\!\cdots\!25}$, $\frac{59\!\cdots\!99}{16\!\cdots\!25}a^{27}+\frac{22\!\cdots\!41}{16\!\cdots\!25}a^{26}+\frac{34\!\cdots\!67}{33\!\cdots\!25}a^{25}-\frac{25\!\cdots\!62}{33\!\cdots\!25}a^{24}+\frac{38\!\cdots\!54}{16\!\cdots\!25}a^{23}+\frac{85\!\cdots\!64}{16\!\cdots\!25}a^{22}+\frac{62\!\cdots\!41}{12\!\cdots\!25}a^{21}+\frac{88\!\cdots\!86}{67\!\cdots\!25}a^{20}+\frac{13\!\cdots\!64}{16\!\cdots\!25}a^{19}+\frac{37\!\cdots\!64}{16\!\cdots\!25}a^{18}+\frac{79\!\cdots\!63}{67\!\cdots\!25}a^{17}+\frac{58\!\cdots\!72}{16\!\cdots\!25}a^{16}+\frac{25\!\cdots\!69}{16\!\cdots\!25}a^{15}+\frac{85\!\cdots\!94}{16\!\cdots\!25}a^{14}+\frac{10\!\cdots\!98}{67\!\cdots\!25}a^{13}+\frac{67\!\cdots\!62}{16\!\cdots\!25}a^{12}+\frac{17\!\cdots\!49}{16\!\cdots\!25}a^{11}+\frac{35\!\cdots\!99}{16\!\cdots\!25}a^{10}+\frac{23\!\cdots\!08}{67\!\cdots\!25}a^{9}+\frac{88\!\cdots\!02}{16\!\cdots\!25}a^{8}+\frac{24\!\cdots\!11}{33\!\cdots\!25}a^{7}+\frac{66\!\cdots\!63}{16\!\cdots\!25}a^{6}-\frac{20\!\cdots\!52}{33\!\cdots\!25}a^{5}+\frac{58\!\cdots\!52}{16\!\cdots\!25}a^{4}+\frac{36\!\cdots\!78}{24\!\cdots\!25}a^{3}+\frac{23\!\cdots\!89}{33\!\cdots\!25}a^{2}-\frac{27\!\cdots\!91}{12\!\cdots\!25}a+\frac{12\!\cdots\!78}{89\!\cdots\!25}$, $\frac{99\!\cdots\!43}{22\!\cdots\!25}a^{27}-\frac{12\!\cdots\!24}{22\!\cdots\!25}a^{26}+\frac{18\!\cdots\!67}{13\!\cdots\!25}a^{25}-\frac{70\!\cdots\!07}{22\!\cdots\!25}a^{24}+\frac{75\!\cdots\!59}{22\!\cdots\!25}a^{23}+\frac{21\!\cdots\!94}{22\!\cdots\!25}a^{22}+\frac{16\!\cdots\!17}{27\!\cdots\!25}a^{21}+\frac{14\!\cdots\!24}{22\!\cdots\!25}a^{20}+\frac{20\!\cdots\!06}{22\!\cdots\!25}a^{19}+\frac{30\!\cdots\!03}{22\!\cdots\!25}a^{18}+\frac{28\!\cdots\!14}{22\!\cdots\!25}a^{17}+\frac{53\!\cdots\!87}{22\!\cdots\!25}a^{16}+\frac{34\!\cdots\!01}{22\!\cdots\!25}a^{15}+\frac{91\!\cdots\!88}{22\!\cdots\!25}a^{14}+\frac{31\!\cdots\!94}{22\!\cdots\!25}a^{13}+\frac{65\!\cdots\!27}{22\!\cdots\!25}a^{12}+\frac{20\!\cdots\!96}{22\!\cdots\!25}a^{11}+\frac{30\!\cdots\!98}{22\!\cdots\!25}a^{10}+\frac{57\!\cdots\!49}{22\!\cdots\!25}a^{9}+\frac{68\!\cdots\!17}{22\!\cdots\!25}a^{8}+\frac{11\!\cdots\!48}{22\!\cdots\!25}a^{7}-\frac{40\!\cdots\!68}{22\!\cdots\!25}a^{6}+\frac{16\!\cdots\!88}{45\!\cdots\!25}a^{5}-\frac{90\!\cdots\!36}{22\!\cdots\!25}a^{4}+\frac{71\!\cdots\!52}{22\!\cdots\!25}a^{3}+\frac{22\!\cdots\!52}{16\!\cdots\!25}a^{2}+\frac{48\!\cdots\!38}{89\!\cdots\!25}a+\frac{36\!\cdots\!66}{89\!\cdots\!25}$, $\frac{14\!\cdots\!78}{16\!\cdots\!25}a^{27}-\frac{12\!\cdots\!83}{16\!\cdots\!25}a^{26}+\frac{89\!\cdots\!22}{33\!\cdots\!25}a^{25}-\frac{17\!\cdots\!42}{33\!\cdots\!25}a^{24}+\frac{61\!\cdots\!14}{98\!\cdots\!25}a^{23}+\frac{44\!\cdots\!84}{98\!\cdots\!25}a^{22}+\frac{14\!\cdots\!07}{12\!\cdots\!25}a^{21}+\frac{11\!\cdots\!32}{67\!\cdots\!25}a^{20}+\frac{30\!\cdots\!28}{16\!\cdots\!25}a^{19}+\frac{55\!\cdots\!08}{16\!\cdots\!25}a^{18}+\frac{84\!\cdots\!52}{33\!\cdots\!25}a^{17}+\frac{91\!\cdots\!54}{16\!\cdots\!25}a^{16}+\frac{51\!\cdots\!13}{16\!\cdots\!25}a^{15}+\frac{14\!\cdots\!18}{16\!\cdots\!25}a^{14}+\frac{57\!\cdots\!26}{19\!\cdots\!25}a^{13}+\frac{10\!\cdots\!59}{16\!\cdots\!25}a^{12}+\frac{31\!\cdots\!73}{16\!\cdots\!25}a^{11}+\frac{50\!\cdots\!53}{16\!\cdots\!25}a^{10}+\frac{17\!\cdots\!32}{33\!\cdots\!25}a^{9}+\frac{11\!\cdots\!14}{16\!\cdots\!25}a^{8}+\frac{33\!\cdots\!76}{33\!\cdots\!25}a^{7}-\frac{35\!\cdots\!79}{16\!\cdots\!25}a^{6}+\frac{27\!\cdots\!63}{13\!\cdots\!65}a^{5}+\frac{79\!\cdots\!04}{16\!\cdots\!25}a^{4}+\frac{49\!\cdots\!38}{24\!\cdots\!25}a^{3}+\frac{23\!\cdots\!84}{33\!\cdots\!25}a^{2}-\frac{23\!\cdots\!52}{12\!\cdots\!25}a+\frac{12\!\cdots\!96}{89\!\cdots\!25}$, $\frac{64\!\cdots\!92}{16\!\cdots\!25}a^{27}-\frac{76\!\cdots\!12}{16\!\cdots\!25}a^{26}+\frac{39\!\cdots\!58}{33\!\cdots\!25}a^{25}-\frac{88\!\cdots\!54}{33\!\cdots\!25}a^{24}+\frac{47\!\cdots\!67}{16\!\cdots\!25}a^{23}+\frac{17\!\cdots\!82}{16\!\cdots\!25}a^{22}+\frac{65\!\cdots\!23}{12\!\cdots\!25}a^{21}+\frac{20\!\cdots\!68}{33\!\cdots\!25}a^{20}+\frac{13\!\cdots\!77}{16\!\cdots\!25}a^{19}+\frac{20\!\cdots\!62}{16\!\cdots\!25}a^{18}+\frac{36\!\cdots\!34}{33\!\cdots\!25}a^{17}+\frac{34\!\cdots\!91}{16\!\cdots\!25}a^{16}+\frac{21\!\cdots\!42}{16\!\cdots\!25}a^{15}+\frac{58\!\cdots\!27}{16\!\cdots\!25}a^{14}+\frac{39\!\cdots\!64}{33\!\cdots\!25}a^{13}+\frac{41\!\cdots\!86}{16\!\cdots\!25}a^{12}+\frac{12\!\cdots\!32}{16\!\cdots\!25}a^{11}+\frac{18\!\cdots\!92}{16\!\cdots\!25}a^{10}+\frac{66\!\cdots\!94}{33\!\cdots\!25}a^{9}+\frac{40\!\cdots\!31}{16\!\cdots\!25}a^{8}+\frac{13\!\cdots\!96}{33\!\cdots\!25}a^{7}-\frac{32\!\cdots\!81}{16\!\cdots\!25}a^{6}+\frac{65\!\cdots\!66}{33\!\cdots\!25}a^{5}+\frac{29\!\cdots\!71}{16\!\cdots\!25}a^{4}+\frac{18\!\cdots\!22}{24\!\cdots\!25}a^{3}+\frac{51\!\cdots\!08}{33\!\cdots\!25}a^{2}-\frac{22\!\cdots\!13}{12\!\cdots\!25}a+\frac{25\!\cdots\!24}{89\!\cdots\!25}$, $\frac{51\!\cdots\!06}{18\!\cdots\!05}a^{27}-\frac{21\!\cdots\!29}{45\!\cdots\!25}a^{26}+\frac{40\!\cdots\!83}{45\!\cdots\!25}a^{25}-\frac{10\!\cdots\!98}{45\!\cdots\!25}a^{24}+\frac{20\!\cdots\!17}{91\!\cdots\!25}a^{23}-\frac{15\!\cdots\!68}{45\!\cdots\!25}a^{22}+\frac{17\!\cdots\!58}{45\!\cdots\!25}a^{21}+\frac{21\!\cdots\!21}{91\!\cdots\!25}a^{20}+\frac{25\!\cdots\!04}{45\!\cdots\!25}a^{19}+\frac{26\!\cdots\!08}{45\!\cdots\!25}a^{18}+\frac{34\!\cdots\!23}{45\!\cdots\!25}a^{17}+\frac{10\!\cdots\!63}{91\!\cdots\!25}a^{16}+\frac{39\!\cdots\!84}{45\!\cdots\!25}a^{15}+\frac{94\!\cdots\!43}{45\!\cdots\!25}a^{14}+\frac{33\!\cdots\!83}{45\!\cdots\!25}a^{13}+\frac{12\!\cdots\!23}{91\!\cdots\!25}a^{12}+\frac{20\!\cdots\!14}{45\!\cdots\!25}a^{11}+\frac{24\!\cdots\!78}{45\!\cdots\!25}a^{10}+\frac{47\!\cdots\!68}{45\!\cdots\!25}a^{9}+\frac{81\!\cdots\!58}{91\!\cdots\!25}a^{8}+\frac{83\!\cdots\!69}{45\!\cdots\!25}a^{7}-\frac{14\!\cdots\!58}{45\!\cdots\!25}a^{6}+\frac{16\!\cdots\!54}{91\!\cdots\!25}a^{5}-\frac{42\!\cdots\!62}{45\!\cdots\!25}a^{4}+\frac{72\!\cdots\!64}{45\!\cdots\!25}a^{3}-\frac{53\!\cdots\!86}{24\!\cdots\!25}a^{2}-\frac{30\!\cdots\!51}{48\!\cdots\!25}a-\frac{69\!\cdots\!11}{20\!\cdots\!25}$, $\frac{19\!\cdots\!89}{22\!\cdots\!25}a^{27}-\frac{31\!\cdots\!81}{22\!\cdots\!25}a^{26}+\frac{62\!\cdots\!08}{22\!\cdots\!25}a^{25}-\frac{16\!\cdots\!79}{22\!\cdots\!25}a^{24}+\frac{15\!\cdots\!51}{22\!\cdots\!25}a^{23}-\frac{16\!\cdots\!39}{22\!\cdots\!25}a^{22}+\frac{27\!\cdots\!14}{22\!\cdots\!25}a^{21}+\frac{18\!\cdots\!78}{22\!\cdots\!25}a^{20}+\frac{79\!\cdots\!14}{45\!\cdots\!25}a^{19}+\frac{26\!\cdots\!32}{13\!\cdots\!25}a^{18}+\frac{54\!\cdots\!38}{22\!\cdots\!25}a^{17}+\frac{82\!\cdots\!93}{22\!\cdots\!25}a^{16}+\frac{12\!\cdots\!19}{45\!\cdots\!25}a^{15}+\frac{15\!\cdots\!99}{22\!\cdots\!25}a^{14}+\frac{54\!\cdots\!23}{22\!\cdots\!25}a^{13}+\frac{10\!\cdots\!78}{22\!\cdots\!25}a^{12}+\frac{41\!\cdots\!72}{27\!\cdots\!25}a^{11}+\frac{44\!\cdots\!29}{22\!\cdots\!25}a^{10}+\frac{90\!\cdots\!33}{22\!\cdots\!25}a^{9}+\frac{94\!\cdots\!88}{22\!\cdots\!25}a^{8}+\frac{17\!\cdots\!56}{22\!\cdots\!25}a^{7}-\frac{34\!\cdots\!27}{45\!\cdots\!25}a^{6}+\frac{34\!\cdots\!97}{45\!\cdots\!25}a^{5}-\frac{55\!\cdots\!44}{13\!\cdots\!25}a^{4}-\frac{47\!\cdots\!56}{22\!\cdots\!25}a^{3}-\frac{11\!\cdots\!63}{12\!\cdots\!25}a^{2}-\frac{20\!\cdots\!87}{89\!\cdots\!25}a-\frac{54\!\cdots\!23}{17\!\cdots\!25}$, $\frac{22\!\cdots\!89}{45\!\cdots\!25}a^{27}-\frac{33\!\cdots\!11}{45\!\cdots\!25}a^{26}+\frac{69\!\cdots\!88}{45\!\cdots\!25}a^{25}-\frac{17\!\cdots\!59}{45\!\cdots\!25}a^{24}+\frac{17\!\cdots\!41}{45\!\cdots\!25}a^{23}-\frac{76\!\cdots\!79}{45\!\cdots\!25}a^{22}+\frac{31\!\cdots\!64}{45\!\cdots\!25}a^{21}+\frac{22\!\cdots\!68}{45\!\cdots\!25}a^{20}+\frac{35\!\cdots\!54}{36\!\cdots\!21}a^{19}+\frac{30\!\cdots\!87}{27\!\cdots\!25}a^{18}+\frac{61\!\cdots\!48}{45\!\cdots\!25}a^{17}+\frac{95\!\cdots\!93}{45\!\cdots\!25}a^{16}+\frac{57\!\cdots\!39}{36\!\cdots\!21}a^{15}+\frac{17\!\cdots\!09}{45\!\cdots\!25}a^{14}+\frac{63\!\cdots\!33}{45\!\cdots\!25}a^{13}+\frac{12\!\cdots\!28}{45\!\cdots\!25}a^{12}+\frac{18\!\cdots\!32}{21\!\cdots\!13}a^{11}+\frac{50\!\cdots\!39}{45\!\cdots\!25}a^{10}+\frac{10\!\cdots\!93}{45\!\cdots\!25}a^{9}+\frac{96\!\cdots\!38}{45\!\cdots\!25}a^{8}+\frac{19\!\cdots\!86}{45\!\cdots\!25}a^{7}-\frac{38\!\cdots\!29}{91\!\cdots\!25}a^{6}+\frac{39\!\cdots\!53}{91\!\cdots\!25}a^{5}-\frac{62\!\cdots\!04}{27\!\cdots\!25}a^{4}+\frac{27\!\cdots\!09}{45\!\cdots\!25}a^{3}-\frac{13\!\cdots\!63}{24\!\cdots\!25}a^{2}-\frac{22\!\cdots\!17}{17\!\cdots\!25}a-\frac{60\!\cdots\!09}{35\!\cdots\!25}$, $\frac{37\!\cdots\!73}{45\!\cdots\!25}a^{27}-\frac{58\!\cdots\!72}{45\!\cdots\!25}a^{26}+\frac{11\!\cdots\!36}{45\!\cdots\!25}a^{25}-\frac{30\!\cdots\!83}{45\!\cdots\!25}a^{24}+\frac{29\!\cdots\!82}{45\!\cdots\!25}a^{23}-\frac{20\!\cdots\!63}{45\!\cdots\!25}a^{22}+\frac{52\!\cdots\!73}{45\!\cdots\!25}a^{21}+\frac{36\!\cdots\!11}{45\!\cdots\!25}a^{20}+\frac{15\!\cdots\!84}{91\!\cdots\!25}a^{19}+\frac{51\!\cdots\!54}{27\!\cdots\!25}a^{18}+\frac{10\!\cdots\!51}{45\!\cdots\!25}a^{17}+\frac{16\!\cdots\!01}{45\!\cdots\!25}a^{16}+\frac{24\!\cdots\!39}{91\!\cdots\!25}a^{15}+\frac{29\!\cdots\!28}{45\!\cdots\!25}a^{14}+\frac{10\!\cdots\!71}{45\!\cdots\!25}a^{13}+\frac{20\!\cdots\!21}{45\!\cdots\!25}a^{12}+\frac{78\!\cdots\!82}{54\!\cdots\!25}a^{11}+\frac{84\!\cdots\!38}{45\!\cdots\!25}a^{10}+\frac{17\!\cdots\!41}{45\!\cdots\!25}a^{9}+\frac{16\!\cdots\!66}{45\!\cdots\!25}a^{8}+\frac{32\!\cdots\!47}{45\!\cdots\!25}a^{7}-\frac{65\!\cdots\!16}{91\!\cdots\!25}a^{6}+\frac{13\!\cdots\!07}{18\!\cdots\!05}a^{5}-\frac{10\!\cdots\!18}{27\!\cdots\!25}a^{4}+\frac{33\!\cdots\!88}{45\!\cdots\!25}a^{3}-\frac{22\!\cdots\!41}{24\!\cdots\!25}a^{2}-\frac{38\!\cdots\!14}{17\!\cdots\!25}a-\frac{10\!\cdots\!72}{35\!\cdots\!25}$, $\frac{14\!\cdots\!27}{22\!\cdots\!25}a^{27}-\frac{22\!\cdots\!58}{22\!\cdots\!25}a^{26}+\frac{46\!\cdots\!44}{22\!\cdots\!25}a^{25}-\frac{11\!\cdots\!97}{22\!\cdots\!25}a^{24}+\frac{11\!\cdots\!43}{22\!\cdots\!25}a^{23}-\frac{75\!\cdots\!77}{22\!\cdots\!25}a^{22}+\frac{20\!\cdots\!77}{22\!\cdots\!25}a^{21}+\frac{14\!\cdots\!79}{22\!\cdots\!25}a^{20}+\frac{58\!\cdots\!12}{45\!\cdots\!25}a^{19}+\frac{20\!\cdots\!26}{13\!\cdots\!25}a^{18}+\frac{40\!\cdots\!59}{22\!\cdots\!25}a^{17}+\frac{62\!\cdots\!99}{22\!\cdots\!25}a^{16}+\frac{94\!\cdots\!77}{45\!\cdots\!25}a^{15}+\frac{11\!\cdots\!82}{22\!\cdots\!25}a^{14}+\frac{41\!\cdots\!14}{22\!\cdots\!25}a^{13}+\frac{79\!\cdots\!29}{22\!\cdots\!25}a^{12}+\frac{30\!\cdots\!26}{27\!\cdots\!25}a^{11}+\frac{33\!\cdots\!72}{22\!\cdots\!25}a^{10}+\frac{67\!\cdots\!19}{22\!\cdots\!25}a^{9}+\frac{65\!\cdots\!84}{22\!\cdots\!25}a^{8}+\frac{12\!\cdots\!83}{22\!\cdots\!25}a^{7}-\frac{25\!\cdots\!06}{45\!\cdots\!25}a^{6}+\frac{25\!\cdots\!81}{45\!\cdots\!25}a^{5}-\frac{41\!\cdots\!42}{13\!\cdots\!25}a^{4}+\frac{14\!\cdots\!92}{22\!\cdots\!25}a^{3}-\frac{86\!\cdots\!59}{12\!\cdots\!25}a^{2}-\frac{15\!\cdots\!16}{89\!\cdots\!25}a-\frac{40\!\cdots\!74}{17\!\cdots\!25}$, $\frac{67\!\cdots\!76}{19\!\cdots\!25}a^{27}-\frac{13\!\cdots\!49}{98\!\cdots\!25}a^{26}+\frac{99\!\cdots\!56}{98\!\cdots\!25}a^{25}-\frac{10\!\cdots\!88}{98\!\cdots\!25}a^{24}+\frac{22\!\cdots\!99}{98\!\cdots\!25}a^{23}+\frac{38\!\cdots\!04}{98\!\cdots\!25}a^{22}+\frac{34\!\cdots\!67}{71\!\cdots\!25}a^{21}+\frac{10\!\cdots\!51}{98\!\cdots\!25}a^{20}+\frac{74\!\cdots\!97}{98\!\cdots\!25}a^{19}+\frac{37\!\cdots\!44}{19\!\cdots\!25}a^{18}+\frac{10\!\cdots\!01}{98\!\cdots\!25}a^{17}+\frac{29\!\cdots\!97}{98\!\cdots\!25}a^{16}+\frac{13\!\cdots\!87}{98\!\cdots\!25}a^{15}+\frac{88\!\cdots\!49}{19\!\cdots\!25}a^{14}+\frac{14\!\cdots\!96}{98\!\cdots\!25}a^{13}+\frac{34\!\cdots\!87}{98\!\cdots\!25}a^{12}+\frac{92\!\cdots\!27}{98\!\cdots\!25}a^{11}+\frac{35\!\cdots\!54}{19\!\cdots\!25}a^{10}+\frac{29\!\cdots\!66}{98\!\cdots\!25}a^{9}+\frac{43\!\cdots\!27}{98\!\cdots\!25}a^{8}+\frac{59\!\cdots\!37}{98\!\cdots\!25}a^{7}+\frac{24\!\cdots\!69}{98\!\cdots\!25}a^{6}-\frac{39\!\cdots\!44}{19\!\cdots\!25}a^{5}+\frac{57\!\cdots\!01}{19\!\cdots\!25}a^{4}+\frac{90\!\cdots\!34}{71\!\cdots\!25}a^{3}+\frac{44\!\cdots\!66}{98\!\cdots\!25}a^{2}+\frac{11\!\cdots\!33}{52\!\cdots\!25}a+\frac{65\!\cdots\!74}{52\!\cdots\!25}$, $\frac{10\!\cdots\!38}{12\!\cdots\!25}a^{27}-\frac{21\!\cdots\!78}{12\!\cdots\!25}a^{26}+\frac{27\!\cdots\!61}{97\!\cdots\!45}a^{25}-\frac{80\!\cdots\!36}{97\!\cdots\!45}a^{24}+\frac{84\!\cdots\!98}{12\!\cdots\!25}a^{23}-\frac{30\!\cdots\!17}{12\!\cdots\!25}a^{22}+\frac{10\!\cdots\!27}{89\!\cdots\!25}a^{21}+\frac{35\!\cdots\!83}{97\!\cdots\!45}a^{20}+\frac{19\!\cdots\!83}{12\!\cdots\!25}a^{19}+\frac{14\!\cdots\!93}{12\!\cdots\!25}a^{18}+\frac{52\!\cdots\!14}{24\!\cdots\!25}a^{17}+\frac{31\!\cdots\!04}{12\!\cdots\!25}a^{16}+\frac{29\!\cdots\!43}{12\!\cdots\!25}a^{15}+\frac{65\!\cdots\!53}{12\!\cdots\!25}a^{14}+\frac{46\!\cdots\!19}{24\!\cdots\!25}a^{13}+\frac{35\!\cdots\!84}{12\!\cdots\!25}a^{12}+\frac{12\!\cdots\!53}{12\!\cdots\!25}a^{11}+\frac{92\!\cdots\!63}{12\!\cdots\!25}a^{10}+\frac{31\!\cdots\!24}{24\!\cdots\!25}a^{9}-\frac{37\!\cdots\!36}{12\!\cdots\!25}a^{8}+\frac{48\!\cdots\!39}{57\!\cdots\!85}a^{7}-\frac{17\!\cdots\!49}{12\!\cdots\!25}a^{6}+\frac{21\!\cdots\!79}{24\!\cdots\!25}a^{5}+\frac{34\!\cdots\!94}{12\!\cdots\!25}a^{4}+\frac{10\!\cdots\!93}{71\!\cdots\!85}a^{3}+\frac{33\!\cdots\!84}{71\!\cdots\!85}a^{2}-\frac{57\!\cdots\!87}{71\!\cdots\!25}a-\frac{33\!\cdots\!73}{89\!\cdots\!25}$ (assuming GRH)
Regulator:		\( 18641852176931.93 \) (assuming GRH)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{14}\cdot 18641852176931.93 \cdot 617792}{10\cdot\sqrt{128401607345812283715591125636314028226206818103790283203125}}\cr\approx \mathstrut & 0.480357158220585 \end{aligned}\] (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 $C_{28}$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\), 7.7.128100283921.1, 14.14.1282006464112563369862578125.1

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	$28$	${\href{/padicField/11.7.0.1}{7} }^{4}$	$28$	${\href{/padicField/17.4.0.1}{4} }^{7}$	${\href{/padicField/19.14.0.1}{14} }^{2}$	$28$	${\href{/padicField/29.14.0.1}{14} }^{2}$	${\href{/padicField/31.7.0.1}{7} }^{4}$	$28$	${\href{/padicField/41.7.0.1}{7} }^{4}$	$28$	$28$	$28$	${\href{/padicField/59.14.0.1}{14} }^{2}$

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
\(5\)	5.4.3.2	$x^{4} + 5$	$4$	$1$	$3$	$C_4$	$[\ ]_{4}$
	5.4.3.2	$x^{4} + 5$	$4$	$1$	$3$	$C_4$	$[\ ]_{4}$
	5.4.3.2	$x^{4} + 5$	$4$	$1$	$3$	$C_4$	$[\ ]_{4}$
	5.4.3.2	$x^{4} + 5$	$4$	$1$	$3$	$C_4$	$[\ ]_{4}$
	5.4.3.2	$x^{4} + 5$	$4$	$1$	$3$	$C_4$	$[\ ]_{4}$
	5.4.3.2	$x^{4} + 5$	$4$	$1$	$3$	$C_4$	$[\ ]_{4}$
	5.4.3.2	$x^{4} + 5$	$4$	$1$	$3$	$C_4$	$[\ ]_{4}$
\(71\)	71.7.6.1	$x^{7} + 71$	$7$	$1$	$6$	$C_7$	$[\ ]_{7}$
	71.7.6.1	$x^{7} + 71$	$7$	$1$	$6$	$C_7$	$[\ ]_{7}$
	71.7.6.1	$x^{7} + 71$	$7$	$1$	$6$	$C_7$	$[\ ]_{7}$
	71.7.6.1	$x^{7} + 71$	$7$	$1$	$6$	$C_7$	$[\ ]_{7}$