Number field 30.0.8221408887534945302579972677458642036796803806187.1

• Label: 30.0.822...187.1
• Degree: $30$
• Signature: $[0, 15]$
• Discriminant: $-8.221\times 10^{48}$
• Root discriminant: \(42.71\)
• Ramified primes: $3,31$
• Class number: $755$ (GRH)
• Class group: [755] (GRH)
• Galois group: $C_{30}$ (as 30T1)

Normalized defining polynomial

x^30 - x^29 + 15*x^28 - 12*x^27 + 131*x^26 - 92*x^25 + 755*x^24 - 449*x^23 + 3202*x^22 - 1648*x^21 + 10173*x^20 - 4368*x^19 + 24856*x^18 - 8980*x^17 + 46255*x^16 - 13276*x^15 + 65515*x^14 - 15154*x^13 + 68312*x^12 - 11198*x^11 + 51310*x^10 - 6592*x^9 + 25719*x^8 - 1518*x^7 + 8148*x^6 - 588*x^5 + 1330*x^4 + 56*x^3 + 92*x^2 - 8*x + 1

Invariants

Degree:		$30$
Signature:		$[0, 15]$
Discriminant:		\(-8221408887534945302579972677458642036796803806187\) \(\medspace = -\,3^{15}\cdot 31^{28}\)
Root discriminant:		\(42.71\)
Galois root discriminant:		$3^{1/2}31^{14/15}\approx 42.70691575177761$
Ramified primes:		\(3\), \(31\)
Discriminant root field:		\(\Q(\sqrt{-3}) \)
$\card{ \Gal(K/\Q) }$:		$30$
This field is Galois and abelian over $\Q$.
Conductor:		\(93=3\cdot 31\)
Dirichlet character group:		$\lbrace$$\chi_{93}(64,·)$, $\chi_{93}(1,·)$, $\chi_{93}(2,·)$, $\chi_{93}(67,·)$, $\chi_{93}(4,·)$, $\chi_{93}(5,·)$, $\chi_{93}(70,·)$, $\chi_{93}(7,·)$, $\chi_{93}(8,·)$, $\chi_{93}(10,·)$, $\chi_{93}(76,·)$, $\chi_{93}(14,·)$, $\chi_{93}(16,·)$, $\chi_{93}(82,·)$, $\chi_{93}(19,·)$, $\chi_{93}(20,·)$, $\chi_{93}(25,·)$, $\chi_{93}(28,·)$, $\chi_{93}(32,·)$, $\chi_{93}(80,·)$, $\chi_{93}(35,·)$, $\chi_{93}(38,·)$, $\chi_{93}(40,·)$, $\chi_{93}(41,·)$, $\chi_{93}(71,·)$, $\chi_{93}(47,·)$, $\chi_{93}(49,·)$, $\chi_{93}(50,·)$, $\chi_{93}(56,·)$, $\chi_{93}(59,·)$$\rbrace$
This is a CM field.
Reflex fields:		unavailable$^{16384}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $\frac{1}{61}a^{26}+\frac{2}{61}a^{25}-\frac{26}{61}a^{24}-\frac{9}{61}a^{23}-\frac{16}{61}a^{22}-\frac{26}{61}a^{21}-\frac{13}{61}a^{20}-\frac{14}{61}a^{19}+\frac{2}{61}a^{18}-\frac{15}{61}a^{17}-\frac{5}{61}a^{16}-\frac{14}{61}a^{15}+\frac{8}{61}a^{14}-\frac{19}{61}a^{13}-\frac{18}{61}a^{12}+\frac{24}{61}a^{11}-\frac{21}{61}a^{10}+\frac{26}{61}a^{9}+\frac{28}{61}a^{8}-\frac{11}{61}a^{7}+\frac{7}{61}a^{6}-\frac{6}{61}a^{5}-\frac{17}{61}a^{4}+\frac{6}{61}a^{3}-\frac{25}{61}a^{2}+\frac{10}{61}a+\frac{19}{61}$, $\frac{1}{61}a^{27}-\frac{30}{61}a^{25}-\frac{18}{61}a^{24}+\frac{2}{61}a^{23}+\frac{6}{61}a^{22}-\frac{22}{61}a^{21}+\frac{12}{61}a^{20}+\frac{30}{61}a^{19}-\frac{19}{61}a^{18}+\frac{25}{61}a^{17}-\frac{4}{61}a^{16}-\frac{25}{61}a^{15}+\frac{26}{61}a^{14}+\frac{20}{61}a^{13}-\frac{1}{61}a^{12}-\frac{8}{61}a^{11}+\frac{7}{61}a^{10}-\frac{24}{61}a^{9}-\frac{6}{61}a^{8}+\frac{29}{61}a^{7}-\frac{20}{61}a^{6}-\frac{5}{61}a^{5}-\frac{21}{61}a^{4}+\frac{24}{61}a^{3}-\frac{1}{61}a^{2}-\frac{1}{61}a+\frac{23}{61}$, $\frac{1}{61}a^{28}-\frac{19}{61}a^{25}+\frac{15}{61}a^{24}-\frac{20}{61}a^{23}-\frac{14}{61}a^{22}+\frac{25}{61}a^{21}+\frac{6}{61}a^{20}-\frac{12}{61}a^{19}+\frac{24}{61}a^{18}-\frac{27}{61}a^{17}+\frac{8}{61}a^{16}-\frac{28}{61}a^{15}+\frac{16}{61}a^{14}-\frac{22}{61}a^{13}+\frac{1}{61}a^{12}-\frac{5}{61}a^{11}+\frac{17}{61}a^{10}-\frac{19}{61}a^{9}+\frac{15}{61}a^{8}+\frac{16}{61}a^{7}+\frac{22}{61}a^{6}-\frac{18}{61}a^{5}+\frac{2}{61}a^{4}-\frac{4}{61}a^{3}-\frac{19}{61}a^{2}+\frac{18}{61}a+\frac{21}{61}$, $\frac{1}{75\!\cdots\!79}a^{29}+\frac{38\!\cdots\!99}{75\!\cdots\!79}a^{28}+\frac{16\!\cdots\!59}{75\!\cdots\!79}a^{27}-\frac{28\!\cdots\!24}{75\!\cdots\!79}a^{26}+\frac{99\!\cdots\!01}{75\!\cdots\!79}a^{25}+\frac{13\!\cdots\!33}{75\!\cdots\!79}a^{24}-\frac{64\!\cdots\!51}{75\!\cdots\!79}a^{23}-\frac{22\!\cdots\!27}{75\!\cdots\!79}a^{22}-\frac{19\!\cdots\!94}{75\!\cdots\!79}a^{21}-\frac{59\!\cdots\!67}{75\!\cdots\!79}a^{20}+\frac{69\!\cdots\!89}{75\!\cdots\!79}a^{19}+\frac{31\!\cdots\!40}{75\!\cdots\!79}a^{18}+\frac{26\!\cdots\!40}{75\!\cdots\!79}a^{17}+\frac{23\!\cdots\!67}{75\!\cdots\!79}a^{16}-\frac{29\!\cdots\!07}{75\!\cdots\!79}a^{15}+\frac{17\!\cdots\!19}{75\!\cdots\!79}a^{14}-\frac{14\!\cdots\!73}{75\!\cdots\!79}a^{13}+\frac{31\!\cdots\!79}{75\!\cdots\!79}a^{12}+\frac{60\!\cdots\!39}{75\!\cdots\!79}a^{11}-\frac{28\!\cdots\!86}{75\!\cdots\!79}a^{10}-\frac{41\!\cdots\!53}{75\!\cdots\!79}a^{9}-\frac{28\!\cdots\!28}{75\!\cdots\!79}a^{8}+\frac{16\!\cdots\!43}{75\!\cdots\!79}a^{7}-\frac{11\!\cdots\!50}{75\!\cdots\!79}a^{6}-\frac{16\!\cdots\!13}{75\!\cdots\!79}a^{5}+\frac{88\!\cdots\!12}{75\!\cdots\!79}a^{4}+\frac{29\!\cdots\!38}{75\!\cdots\!79}a^{3}+\frac{19\!\cdots\!55}{75\!\cdots\!79}a^{2}-\frac{18\!\cdots\!76}{75\!\cdots\!79}a+\frac{16\!\cdots\!85}{75\!\cdots\!79}$

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

Class group and class number

$C_{755}$, which has order $755$ (assuming GRH)

Unit group

Rank:		$14$
Torsion generator:		\( \frac{86409833652564845976273692487227615687646}{759857395299017198901947312187494172568579} a^{29} - \frac{82574480569149263065235702464437698149667}{759857395299017198901947312187494172568579} a^{28} + \frac{1289242792797114731662210662196013738864235}{759857395299017198901947312187494172568579} a^{27} - \frac{975959730486929796303617740834842984509661}{759857395299017198901947312187494172568579} a^{26} + \frac{11227817719854210221781892368379742320571227}{759857395299017198901947312187494172568579} a^{25} - \frac{7405655364015080290185948193029535056503476}{759857395299017198901947312187494172568579} a^{24} + \frac{64488312366088500034207267118783999154367215}{759857395299017198901947312187494172568579} a^{23} - \frac{35579964932960055873303138038456490057169180}{759857395299017198901947312187494172568579} a^{22} + \frac{272681806318500060650710760296943459936259983}{759857395299017198901947312187494172568579} a^{21} - \frac{128527015465108970998968081323293025252202179}{759857395299017198901947312187494172568579} a^{20} + \frac{863133371743457917451303038012858927068139458}{759857395299017198901947312187494172568579} a^{19} - \frac{5450349727930869607186974483743214067927800}{12456678611459298342654873970286789714239} a^{18} + \frac{2100881444527627256797385340560824579584156001}{759857395299017198901947312187494172568579} a^{17} - \frac{664507312482231198495660546590816940924667853}{759857395299017198901947312187494172568579} a^{16} + \frac{3889695929589655881885223559300164749421735283}{759857395299017198901947312187494172568579} a^{15} - \frac{935727872405118979223393009665961877220478435}{759857395299017198901947312187494172568579} a^{14} + \frac{5477165077124753102710070952364176587977252712}{759857395299017198901947312187494172568579} a^{13} - \frac{1005766733569688630409898385879848726826205902}{759857395299017198901947312187494172568579} a^{12} + \frac{5660624444817725809786301395201312657642511780}{759857395299017198901947312187494172568579} a^{11} - \frac{642635714791618432704284052464634742269537202}{759857395299017198901947312187494172568579} a^{10} + \frac{4205057008370573931080003032469558772195969137}{759857395299017198901947312187494172568579} a^{9} - \frac{321722807628743239892897530953725927796698927}{759857395299017198901947312187494172568579} a^{8} + \frac{2064296366113094533493901476879156686464846528}{759857395299017198901947312187494172568579} a^{7} + \frac{1030802241192079374449019734668133815375224}{759857395299017198901947312187494172568579} a^{6} + \frac{636637846260630871684161521375070431432417986}{759857395299017198901947312187494172568579} a^{5} - \frac{8452321548655737296924077656642752742586170}{759857395299017198901947312187494172568579} a^{4} + \frac{95620100167841271437018175714447563656730360}{759857395299017198901947312187494172568579} a^{3} + \frac{15084490256522422850953561898836223518653628}{759857395299017198901947312187494172568579} a^{2} + \frac{6029203372858148817500324479651534387343827}{759857395299017198901947312187494172568579} a + \frac{247389804989818164360839378287490495109261}{759857395299017198901947312187494172568579} \)  (order $6$)
Fundamental units:		$\frac{43\!\cdots\!25}{75\!\cdots\!79}a^{29}-\frac{67\!\cdots\!05}{75\!\cdots\!79}a^{28}+\frac{67\!\cdots\!98}{75\!\cdots\!79}a^{27}-\frac{87\!\cdots\!60}{75\!\cdots\!79}a^{26}+\frac{60\!\cdots\!13}{75\!\cdots\!79}a^{25}-\frac{71\!\cdots\!75}{75\!\cdots\!79}a^{24}+\frac{35\!\cdots\!43}{75\!\cdots\!79}a^{23}-\frac{37\!\cdots\!11}{75\!\cdots\!79}a^{22}+\frac{15\!\cdots\!82}{75\!\cdots\!79}a^{21}-\frac{14\!\cdots\!33}{75\!\cdots\!79}a^{20}+\frac{48\!\cdots\!56}{75\!\cdots\!79}a^{19}-\frac{42\!\cdots\!81}{75\!\cdots\!79}a^{18}+\frac{12\!\cdots\!82}{75\!\cdots\!79}a^{17}-\frac{97\!\cdots\!77}{75\!\cdots\!79}a^{16}+\frac{22\!\cdots\!67}{75\!\cdots\!79}a^{15}-\frac{16\!\cdots\!58}{75\!\cdots\!79}a^{14}+\frac{32\!\cdots\!61}{75\!\cdots\!79}a^{13}-\frac{21\!\cdots\!59}{75\!\cdots\!79}a^{12}+\frac{34\!\cdots\!99}{75\!\cdots\!79}a^{11}-\frac{20\!\cdots\!23}{75\!\cdots\!79}a^{10}+\frac{26\!\cdots\!77}{75\!\cdots\!79}a^{9}-\frac{14\!\cdots\!40}{75\!\cdots\!79}a^{8}+\frac{13\!\cdots\!90}{75\!\cdots\!79}a^{7}-\frac{60\!\cdots\!60}{75\!\cdots\!79}a^{6}+\frac{42\!\cdots\!01}{75\!\cdots\!79}a^{5}-\frac{18\!\cdots\!90}{75\!\cdots\!79}a^{4}+\frac{77\!\cdots\!91}{75\!\cdots\!79}a^{3}-\frac{18\!\cdots\!67}{75\!\cdots\!79}a^{2}+\frac{17\!\cdots\!88}{75\!\cdots\!79}a-\frac{91\!\cdots\!92}{75\!\cdots\!79}$, $\frac{96\!\cdots\!52}{75\!\cdots\!79}a^{29}+\frac{15\!\cdots\!53}{75\!\cdots\!79}a^{28}+\frac{13\!\cdots\!20}{75\!\cdots\!79}a^{27}+\frac{49\!\cdots\!66}{75\!\cdots\!79}a^{26}+\frac{11\!\cdots\!81}{75\!\cdots\!79}a^{25}+\frac{54\!\cdots\!58}{75\!\cdots\!79}a^{24}+\frac{63\!\cdots\!28}{75\!\cdots\!79}a^{23}+\frac{37\!\cdots\!63}{75\!\cdots\!79}a^{22}+\frac{26\!\cdots\!05}{75\!\cdots\!79}a^{21}+\frac{18\!\cdots\!31}{75\!\cdots\!79}a^{20}+\frac{81\!\cdots\!54}{75\!\cdots\!79}a^{19}+\frac{63\!\cdots\!78}{75\!\cdots\!79}a^{18}+\frac{19\!\cdots\!61}{75\!\cdots\!79}a^{17}+\frac{16\!\cdots\!85}{75\!\cdots\!79}a^{16}+\frac{35\!\cdots\!42}{75\!\cdots\!79}a^{15}+\frac{32\!\cdots\!10}{75\!\cdots\!79}a^{14}+\frac{51\!\cdots\!91}{75\!\cdots\!79}a^{13}+\frac{47\!\cdots\!55}{75\!\cdots\!79}a^{12}+\frac{53\!\cdots\!79}{75\!\cdots\!79}a^{11}+\frac{51\!\cdots\!43}{75\!\cdots\!79}a^{10}+\frac{42\!\cdots\!90}{75\!\cdots\!79}a^{9}+\frac{36\!\cdots\!39}{75\!\cdots\!79}a^{8}+\frac{22\!\cdots\!20}{75\!\cdots\!79}a^{7}+\frac{17\!\cdots\!97}{75\!\cdots\!79}a^{6}+\frac{95\!\cdots\!42}{75\!\cdots\!79}a^{5}+\frac{44\!\cdots\!17}{75\!\cdots\!79}a^{4}+\frac{20\!\cdots\!13}{75\!\cdots\!79}a^{3}+\frac{55\!\cdots\!68}{75\!\cdots\!79}a^{2}+\frac{60\!\cdots\!46}{75\!\cdots\!79}a+\frac{77\!\cdots\!51}{75\!\cdots\!79}$, $\frac{51\!\cdots\!67}{75\!\cdots\!79}a^{29}-\frac{72\!\cdots\!45}{75\!\cdots\!79}a^{28}+\frac{79\!\cdots\!83}{75\!\cdots\!79}a^{27}-\frac{93\!\cdots\!20}{75\!\cdots\!79}a^{26}+\frac{70\!\cdots\!21}{75\!\cdots\!79}a^{25}-\frac{75\!\cdots\!03}{75\!\cdots\!79}a^{24}+\frac{40\!\cdots\!63}{75\!\cdots\!79}a^{23}-\frac{39\!\cdots\!99}{75\!\cdots\!79}a^{22}+\frac{17\!\cdots\!23}{75\!\cdots\!79}a^{21}-\frac{15\!\cdots\!85}{75\!\cdots\!79}a^{20}+\frac{56\!\cdots\!08}{75\!\cdots\!79}a^{19}-\frac{44\!\cdots\!84}{75\!\cdots\!79}a^{18}+\frac{13\!\cdots\!49}{75\!\cdots\!79}a^{17}-\frac{99\!\cdots\!82}{75\!\cdots\!79}a^{16}+\frac{26\!\cdots\!77}{75\!\cdots\!79}a^{15}-\frac{16\!\cdots\!05}{75\!\cdots\!79}a^{14}+\frac{37\!\cdots\!31}{75\!\cdots\!79}a^{13}-\frac{21\!\cdots\!73}{75\!\cdots\!79}a^{12}+\frac{39\!\cdots\!79}{75\!\cdots\!79}a^{11}-\frac{20\!\cdots\!74}{75\!\cdots\!79}a^{10}+\frac{29\!\cdots\!39}{75\!\cdots\!79}a^{9}-\frac{14\!\cdots\!70}{75\!\cdots\!79}a^{8}+\frac{15\!\cdots\!10}{75\!\cdots\!79}a^{7}-\frac{61\!\cdots\!57}{75\!\cdots\!79}a^{6}+\frac{46\!\cdots\!37}{75\!\cdots\!79}a^{5}-\frac{18\!\cdots\!38}{75\!\cdots\!79}a^{4}+\frac{77\!\cdots\!38}{75\!\cdots\!79}a^{3}-\frac{19\!\cdots\!06}{75\!\cdots\!79}a^{2}+\frac{18\!\cdots\!70}{75\!\cdots\!79}a-\frac{10\!\cdots\!93}{75\!\cdots\!79}$, $\frac{50\!\cdots\!51}{75\!\cdots\!79}a^{29}-\frac{57\!\cdots\!45}{75\!\cdots\!79}a^{28}+\frac{75\!\cdots\!42}{75\!\cdots\!79}a^{27}-\frac{70\!\cdots\!67}{75\!\cdots\!79}a^{26}+\frac{65\!\cdots\!22}{75\!\cdots\!79}a^{25}-\frac{55\!\cdots\!45}{75\!\cdots\!79}a^{24}+\frac{37\!\cdots\!50}{75\!\cdots\!79}a^{23}-\frac{27\!\cdots\!13}{75\!\cdots\!79}a^{22}+\frac{16\!\cdots\!71}{75\!\cdots\!79}a^{21}-\frac{10\!\cdots\!11}{75\!\cdots\!79}a^{20}+\frac{50\!\cdots\!69}{75\!\cdots\!79}a^{19}-\frac{28\!\cdots\!55}{75\!\cdots\!79}a^{18}+\frac{12\!\cdots\!13}{75\!\cdots\!79}a^{17}-\frac{61\!\cdots\!86}{75\!\cdots\!79}a^{16}+\frac{22\!\cdots\!98}{75\!\cdots\!79}a^{15}-\frac{97\!\cdots\!45}{75\!\cdots\!79}a^{14}+\frac{31\!\cdots\!37}{75\!\cdots\!79}a^{13}-\frac{12\!\cdots\!74}{75\!\cdots\!79}a^{12}+\frac{32\!\cdots\!78}{75\!\cdots\!79}a^{11}-\frac{10\!\cdots\!32}{75\!\cdots\!79}a^{10}+\frac{23\!\cdots\!66}{75\!\cdots\!79}a^{9}-\frac{70\!\cdots\!78}{75\!\cdots\!79}a^{8}+\frac{11\!\cdots\!04}{75\!\cdots\!79}a^{7}-\frac{27\!\cdots\!95}{75\!\cdots\!79}a^{6}+\frac{32\!\cdots\!87}{75\!\cdots\!79}a^{5}-\frac{10\!\cdots\!70}{75\!\cdots\!79}a^{4}+\frac{43\!\cdots\!67}{75\!\cdots\!79}a^{3}-\frac{11\!\cdots\!78}{75\!\cdots\!79}a^{2}+\frac{11\!\cdots\!34}{75\!\cdots\!79}a-\frac{14\!\cdots\!89}{75\!\cdots\!79}$, $\frac{12\!\cdots\!08}{75\!\cdots\!79}a^{29}-\frac{11\!\cdots\!30}{75\!\cdots\!79}a^{28}+\frac{18\!\cdots\!61}{75\!\cdots\!79}a^{27}-\frac{13\!\cdots\!65}{75\!\cdots\!79}a^{26}+\frac{16\!\cdots\!87}{75\!\cdots\!79}a^{25}-\frac{98\!\cdots\!16}{75\!\cdots\!79}a^{24}+\frac{93\!\cdots\!73}{75\!\cdots\!79}a^{23}-\frac{76\!\cdots\!90}{12\!\cdots\!39}a^{22}+\frac{39\!\cdots\!12}{75\!\cdots\!79}a^{21}-\frac{16\!\cdots\!82}{75\!\cdots\!79}a^{20}+\frac{12\!\cdots\!51}{75\!\cdots\!79}a^{19}-\frac{41\!\cdots\!88}{75\!\cdots\!79}a^{18}+\frac{30\!\cdots\!28}{75\!\cdots\!79}a^{17}-\frac{80\!\cdots\!30}{75\!\cdots\!79}a^{16}+\frac{57\!\cdots\!64}{75\!\cdots\!79}a^{15}-\frac{10\!\cdots\!02}{75\!\cdots\!79}a^{14}+\frac{80\!\cdots\!76}{75\!\cdots\!79}a^{13}-\frac{10\!\cdots\!84}{75\!\cdots\!79}a^{12}+\frac{84\!\cdots\!55}{75\!\cdots\!79}a^{11}-\frac{49\!\cdots\!78}{75\!\cdots\!79}a^{10}+\frac{63\!\cdots\!15}{75\!\cdots\!79}a^{9}-\frac{12\!\cdots\!43}{75\!\cdots\!79}a^{8}+\frac{31\!\cdots\!38}{75\!\cdots\!79}a^{7}+\frac{17\!\cdots\!99}{75\!\cdots\!79}a^{6}+\frac{10\!\cdots\!36}{75\!\cdots\!79}a^{5}+\frac{50\!\cdots\!84}{75\!\cdots\!79}a^{4}+\frac{15\!\cdots\!95}{75\!\cdots\!79}a^{3}+\frac{27\!\cdots\!92}{75\!\cdots\!79}a^{2}+\frac{11\!\cdots\!50}{75\!\cdots\!79}a+\frac{43\!\cdots\!93}{75\!\cdots\!79}$, $\frac{12\!\cdots\!86}{75\!\cdots\!79}a^{29}-\frac{10\!\cdots\!25}{75\!\cdots\!79}a^{28}+\frac{18\!\cdots\!81}{75\!\cdots\!79}a^{27}-\frac{11\!\cdots\!59}{75\!\cdots\!79}a^{26}+\frac{16\!\cdots\!21}{75\!\cdots\!79}a^{25}-\frac{85\!\cdots\!44}{75\!\cdots\!79}a^{24}+\frac{93\!\cdots\!89}{75\!\cdots\!79}a^{23}-\frac{38\!\cdots\!76}{75\!\cdots\!79}a^{22}+\frac{39\!\cdots\!45}{75\!\cdots\!79}a^{21}-\frac{13\!\cdots\!37}{75\!\cdots\!79}a^{20}+\frac{12\!\cdots\!58}{75\!\cdots\!79}a^{19}-\frac{31\!\cdots\!52}{75\!\cdots\!79}a^{18}+\frac{30\!\cdots\!07}{75\!\cdots\!79}a^{17}-\frac{54\!\cdots\!78}{75\!\cdots\!79}a^{16}+\frac{56\!\cdots\!13}{75\!\cdots\!79}a^{15}-\frac{58\!\cdots\!97}{75\!\cdots\!79}a^{14}+\frac{80\!\cdots\!12}{75\!\cdots\!79}a^{13}-\frac{38\!\cdots\!66}{75\!\cdots\!79}a^{12}+\frac{83\!\cdots\!16}{75\!\cdots\!79}a^{11}+\frac{16\!\cdots\!66}{75\!\cdots\!79}a^{10}+\frac{63\!\cdots\!79}{75\!\cdots\!79}a^{9}+\frac{33\!\cdots\!11}{75\!\cdots\!79}a^{8}+\frac{31\!\cdots\!00}{75\!\cdots\!79}a^{7}+\frac{37\!\cdots\!88}{75\!\cdots\!79}a^{6}+\frac{10\!\cdots\!46}{75\!\cdots\!79}a^{5}+\frac{93\!\cdots\!42}{75\!\cdots\!79}a^{4}+\frac{16\!\cdots\!48}{75\!\cdots\!79}a^{3}+\frac{32\!\cdots\!56}{75\!\cdots\!79}a^{2}+\frac{14\!\cdots\!94}{75\!\cdots\!79}a+\frac{49\!\cdots\!83}{75\!\cdots\!79}$, $\frac{11\!\cdots\!85}{75\!\cdots\!79}a^{29}-\frac{13\!\cdots\!69}{75\!\cdots\!79}a^{28}+\frac{17\!\cdots\!35}{75\!\cdots\!79}a^{27}-\frac{15\!\cdots\!35}{75\!\cdots\!79}a^{26}+\frac{15\!\cdots\!86}{75\!\cdots\!79}a^{25}-\frac{12\!\cdots\!13}{75\!\cdots\!79}a^{24}+\frac{87\!\cdots\!38}{75\!\cdots\!79}a^{23}-\frac{61\!\cdots\!37}{75\!\cdots\!79}a^{22}+\frac{36\!\cdots\!47}{75\!\cdots\!79}a^{21}-\frac{22\!\cdots\!89}{75\!\cdots\!79}a^{20}+\frac{11\!\cdots\!43}{75\!\cdots\!79}a^{19}-\frac{62\!\cdots\!68}{75\!\cdots\!79}a^{18}+\frac{27\!\cdots\!02}{75\!\cdots\!79}a^{17}-\frac{13\!\cdots\!35}{75\!\cdots\!79}a^{16}+\frac{51\!\cdots\!82}{75\!\cdots\!79}a^{15}-\frac{20\!\cdots\!92}{75\!\cdots\!79}a^{14}+\frac{70\!\cdots\!73}{75\!\cdots\!79}a^{13}-\frac{24\!\cdots\!39}{75\!\cdots\!79}a^{12}+\frac{71\!\cdots\!02}{75\!\cdots\!79}a^{11}-\frac{19\!\cdots\!19}{75\!\cdots\!79}a^{10}+\frac{51\!\cdots\!08}{75\!\cdots\!79}a^{9}-\frac{13\!\cdots\!52}{75\!\cdots\!79}a^{8}+\frac{23\!\cdots\!80}{75\!\cdots\!79}a^{7}-\frac{44\!\cdots\!98}{75\!\cdots\!79}a^{6}+\frac{66\!\cdots\!81}{75\!\cdots\!79}a^{5}-\frac{20\!\cdots\!66}{75\!\cdots\!79}a^{4}+\frac{83\!\cdots\!18}{75\!\cdots\!79}a^{3}-\frac{22\!\cdots\!97}{75\!\cdots\!79}a^{2}+\frac{21\!\cdots\!40}{75\!\cdots\!79}a-\frac{15\!\cdots\!12}{75\!\cdots\!79}$, $\frac{41\!\cdots\!18}{75\!\cdots\!79}a^{29}-\frac{25\!\cdots\!08}{75\!\cdots\!79}a^{28}+\frac{60\!\cdots\!74}{75\!\cdots\!79}a^{27}-\frac{25\!\cdots\!64}{75\!\cdots\!79}a^{26}+\frac{51\!\cdots\!77}{75\!\cdots\!79}a^{25}-\frac{16\!\cdots\!49}{75\!\cdots\!79}a^{24}+\frac{29\!\cdots\!18}{75\!\cdots\!79}a^{23}-\frac{60\!\cdots\!71}{75\!\cdots\!79}a^{22}+\frac{12\!\cdots\!76}{75\!\cdots\!79}a^{21}-\frac{14\!\cdots\!86}{75\!\cdots\!79}a^{20}+\frac{37\!\cdots\!87}{75\!\cdots\!79}a^{19}-\frac{94\!\cdots\!60}{75\!\cdots\!79}a^{18}+\frac{90\!\cdots\!66}{75\!\cdots\!79}a^{17}+\frac{49\!\cdots\!58}{75\!\cdots\!79}a^{16}+\frac{16\!\cdots\!77}{75\!\cdots\!79}a^{15}+\frac{23\!\cdots\!09}{75\!\cdots\!79}a^{14}+\frac{22\!\cdots\!79}{75\!\cdots\!79}a^{13}+\frac{49\!\cdots\!19}{75\!\cdots\!79}a^{12}+\frac{22\!\cdots\!54}{75\!\cdots\!79}a^{11}+\frac{71\!\cdots\!43}{75\!\cdots\!79}a^{10}+\frac{15\!\cdots\!79}{75\!\cdots\!79}a^{9}+\frac{60\!\cdots\!12}{75\!\cdots\!79}a^{8}+\frac{68\!\cdots\!50}{75\!\cdots\!79}a^{7}+\frac{37\!\cdots\!62}{75\!\cdots\!79}a^{6}+\frac{17\!\cdots\!82}{75\!\cdots\!79}a^{5}+\frac{10\!\cdots\!45}{75\!\cdots\!79}a^{4}+\frac{36\!\cdots\!66}{75\!\cdots\!79}a^{3}+\frac{25\!\cdots\!34}{75\!\cdots\!79}a^{2}-\frac{15\!\cdots\!95}{75\!\cdots\!79}a+\frac{17\!\cdots\!00}{75\!\cdots\!79}$, $\frac{57\!\cdots\!03}{75\!\cdots\!79}a^{29}-\frac{74\!\cdots\!77}{75\!\cdots\!79}a^{28}+\frac{87\!\cdots\!15}{75\!\cdots\!79}a^{27}-\frac{94\!\cdots\!24}{75\!\cdots\!79}a^{26}+\frac{77\!\cdots\!65}{75\!\cdots\!79}a^{25}-\frac{75\!\cdots\!62}{75\!\cdots\!79}a^{24}+\frac{45\!\cdots\!78}{75\!\cdots\!79}a^{23}-\frac{39\!\cdots\!96}{75\!\cdots\!79}a^{22}+\frac{19\!\cdots\!29}{75\!\cdots\!79}a^{21}-\frac{15\!\cdots\!16}{75\!\cdots\!79}a^{20}+\frac{61\!\cdots\!26}{75\!\cdots\!79}a^{19}-\frac{43\!\cdots\!96}{75\!\cdots\!79}a^{18}+\frac{15\!\cdots\!07}{75\!\cdots\!79}a^{17}-\frac{95\!\cdots\!47}{75\!\cdots\!79}a^{16}+\frac{28\!\cdots\!02}{75\!\cdots\!79}a^{15}-\frac{15\!\cdots\!13}{75\!\cdots\!79}a^{14}+\frac{40\!\cdots\!71}{75\!\cdots\!79}a^{13}-\frac{20\!\cdots\!02}{75\!\cdots\!79}a^{12}+\frac{42\!\cdots\!47}{75\!\cdots\!79}a^{11}-\frac{18\!\cdots\!92}{75\!\cdots\!79}a^{10}+\frac{31\!\cdots\!72}{75\!\cdots\!79}a^{9}-\frac{13\!\cdots\!17}{75\!\cdots\!79}a^{8}+\frac{15\!\cdots\!72}{75\!\cdots\!79}a^{7}-\frac{56\!\cdots\!92}{75\!\cdots\!79}a^{6}+\frac{47\!\cdots\!73}{75\!\cdots\!79}a^{5}-\frac{18\!\cdots\!64}{75\!\cdots\!79}a^{4}+\frac{74\!\cdots\!48}{75\!\cdots\!79}a^{3}-\frac{19\!\cdots\!84}{75\!\cdots\!79}a^{2}+\frac{17\!\cdots\!34}{75\!\cdots\!79}a-\frac{11\!\cdots\!18}{75\!\cdots\!79}$, $\frac{98\!\cdots\!13}{75\!\cdots\!79}a^{29}-\frac{10\!\cdots\!43}{75\!\cdots\!79}a^{28}+\frac{14\!\cdots\!03}{75\!\cdots\!79}a^{27}-\frac{12\!\cdots\!49}{75\!\cdots\!79}a^{26}+\frac{12\!\cdots\!32}{75\!\cdots\!79}a^{25}-\frac{96\!\cdots\!79}{75\!\cdots\!79}a^{24}+\frac{74\!\cdots\!84}{75\!\cdots\!79}a^{23}-\frac{47\!\cdots\!63}{75\!\cdots\!79}a^{22}+\frac{31\!\cdots\!83}{75\!\cdots\!79}a^{21}-\frac{17\!\cdots\!34}{75\!\cdots\!79}a^{20}+\frac{10\!\cdots\!77}{75\!\cdots\!79}a^{19}-\frac{76\!\cdots\!36}{12\!\cdots\!39}a^{18}+\frac{24\!\cdots\!18}{75\!\cdots\!79}a^{17}-\frac{97\!\cdots\!88}{75\!\cdots\!79}a^{16}+\frac{45\!\cdots\!87}{75\!\cdots\!79}a^{15}-\frac{14\!\cdots\!35}{75\!\cdots\!79}a^{14}+\frac{64\!\cdots\!12}{75\!\cdots\!79}a^{13}-\frac{17\!\cdots\!87}{75\!\cdots\!79}a^{12}+\frac{66\!\cdots\!65}{75\!\cdots\!79}a^{11}-\frac{13\!\cdots\!89}{75\!\cdots\!79}a^{10}+\frac{49\!\cdots\!11}{75\!\cdots\!79}a^{9}-\frac{77\!\cdots\!21}{75\!\cdots\!79}a^{8}+\frac{24\!\cdots\!02}{75\!\cdots\!79}a^{7}-\frac{19\!\cdots\!62}{75\!\cdots\!79}a^{6}+\frac{76\!\cdots\!14}{75\!\cdots\!79}a^{5}-\frac{64\!\cdots\!79}{75\!\cdots\!79}a^{4}+\frac{12\!\cdots\!54}{75\!\cdots\!79}a^{3}+\frac{95\!\cdots\!91}{75\!\cdots\!79}a^{2}+\frac{88\!\cdots\!27}{75\!\cdots\!79}a-\frac{76\!\cdots\!80}{75\!\cdots\!79}$, $\frac{49\!\cdots\!04}{75\!\cdots\!79}a^{29}-\frac{76\!\cdots\!06}{75\!\cdots\!79}a^{28}+\frac{77\!\cdots\!35}{75\!\cdots\!79}a^{27}-\frac{98\!\cdots\!79}{75\!\cdots\!79}a^{26}+\frac{67\!\cdots\!15}{75\!\cdots\!79}a^{25}-\frac{79\!\cdots\!54}{75\!\cdots\!79}a^{24}+\frac{39\!\cdots\!65}{75\!\cdots\!79}a^{23}-\frac{41\!\cdots\!44}{75\!\cdots\!79}a^{22}+\frac{16\!\cdots\!17}{75\!\cdots\!79}a^{21}-\frac{16\!\cdots\!51}{75\!\cdots\!79}a^{20}+\frac{53\!\cdots\!47}{75\!\cdots\!79}a^{19}-\frac{46\!\cdots\!85}{75\!\cdots\!79}a^{18}+\frac{13\!\cdots\!34}{75\!\cdots\!79}a^{17}-\frac{10\!\cdots\!03}{75\!\cdots\!79}a^{16}+\frac{24\!\cdots\!62}{75\!\cdots\!79}a^{15}-\frac{17\!\cdots\!45}{75\!\cdots\!79}a^{14}+\frac{33\!\cdots\!62}{75\!\cdots\!79}a^{13}-\frac{22\!\cdots\!98}{75\!\cdots\!79}a^{12}+\frac{34\!\cdots\!95}{75\!\cdots\!79}a^{11}-\frac{19\!\cdots\!52}{75\!\cdots\!79}a^{10}+\frac{24\!\cdots\!83}{75\!\cdots\!79}a^{9}-\frac{13\!\cdots\!13}{75\!\cdots\!79}a^{8}+\frac{11\!\cdots\!61}{75\!\cdots\!79}a^{7}-\frac{51\!\cdots\!14}{75\!\cdots\!79}a^{6}+\frac{26\!\cdots\!34}{75\!\cdots\!79}a^{5}-\frac{14\!\cdots\!53}{75\!\cdots\!79}a^{4}+\frac{22\!\cdots\!35}{75\!\cdots\!79}a^{3}-\frac{35\!\cdots\!38}{75\!\cdots\!79}a^{2}-\frac{80\!\cdots\!92}{75\!\cdots\!79}a-\frac{15\!\cdots\!01}{75\!\cdots\!79}$, $\frac{83\!\cdots\!80}{75\!\cdots\!79}a^{29}-\frac{83\!\cdots\!72}{75\!\cdots\!79}a^{28}+\frac{12\!\cdots\!96}{75\!\cdots\!79}a^{27}-\frac{99\!\cdots\!12}{75\!\cdots\!79}a^{26}+\frac{10\!\cdots\!16}{75\!\cdots\!79}a^{25}-\frac{76\!\cdots\!89}{75\!\cdots\!79}a^{24}+\frac{61\!\cdots\!94}{75\!\cdots\!79}a^{23}-\frac{37\!\cdots\!54}{75\!\cdots\!79}a^{22}+\frac{26\!\cdots\!66}{75\!\cdots\!79}a^{21}-\frac{13\!\cdots\!76}{75\!\cdots\!79}a^{20}+\frac{82\!\cdots\!82}{75\!\cdots\!79}a^{19}-\frac{36\!\cdots\!26}{75\!\cdots\!79}a^{18}+\frac{19\!\cdots\!04}{75\!\cdots\!79}a^{17}-\frac{76\!\cdots\!64}{75\!\cdots\!79}a^{16}+\frac{36\!\cdots\!23}{75\!\cdots\!79}a^{15}-\frac{11\!\cdots\!94}{75\!\cdots\!79}a^{14}+\frac{51\!\cdots\!66}{75\!\cdots\!79}a^{13}-\frac{13\!\cdots\!37}{75\!\cdots\!79}a^{12}+\frac{51\!\cdots\!80}{75\!\cdots\!79}a^{11}-\frac{11\!\cdots\!14}{75\!\cdots\!79}a^{10}+\frac{37\!\cdots\!48}{75\!\cdots\!79}a^{9}-\frac{12\!\cdots\!74}{12\!\cdots\!39}a^{8}+\frac{17\!\cdots\!32}{75\!\cdots\!79}a^{7}-\frac{30\!\cdots\!55}{75\!\cdots\!79}a^{6}+\frac{48\!\cdots\!62}{75\!\cdots\!79}a^{5}-\frac{13\!\cdots\!08}{75\!\cdots\!79}a^{4}+\frac{48\!\cdots\!62}{75\!\cdots\!79}a^{3}-\frac{15\!\cdots\!50}{75\!\cdots\!79}a^{2}+\frac{15\!\cdots\!40}{75\!\cdots\!79}a-\frac{13\!\cdots\!30}{75\!\cdots\!79}$, $\frac{71\!\cdots\!17}{75\!\cdots\!79}a^{29}-\frac{91\!\cdots\!31}{75\!\cdots\!79}a^{28}+\frac{10\!\cdots\!10}{75\!\cdots\!79}a^{27}-\frac{11\!\cdots\!80}{75\!\cdots\!79}a^{26}+\frac{95\!\cdots\!47}{75\!\cdots\!79}a^{25}-\frac{91\!\cdots\!77}{75\!\cdots\!79}a^{24}+\frac{55\!\cdots\!77}{75\!\cdots\!79}a^{23}-\frac{46\!\cdots\!41}{75\!\cdots\!79}a^{22}+\frac{23\!\cdots\!66}{75\!\cdots\!79}a^{21}-\frac{18\!\cdots\!18}{75\!\cdots\!79}a^{20}+\frac{75\!\cdots\!09}{75\!\cdots\!79}a^{19}-\frac{51\!\cdots\!11}{75\!\cdots\!79}a^{18}+\frac{18\!\cdots\!50}{75\!\cdots\!79}a^{17}-\frac{11\!\cdots\!80}{75\!\cdots\!79}a^{16}+\frac{34\!\cdots\!13}{75\!\cdots\!79}a^{15}-\frac{18\!\cdots\!15}{75\!\cdots\!79}a^{14}+\frac{48\!\cdots\!72}{75\!\cdots\!79}a^{13}-\frac{23\!\cdots\!75}{75\!\cdots\!79}a^{12}+\frac{50\!\cdots\!43}{75\!\cdots\!79}a^{11}-\frac{21\!\cdots\!02}{75\!\cdots\!79}a^{10}+\frac{37\!\cdots\!38}{75\!\cdots\!79}a^{9}-\frac{14\!\cdots\!78}{75\!\cdots\!79}a^{8}+\frac{18\!\cdots\!70}{75\!\cdots\!79}a^{7}-\frac{61\!\cdots\!87}{75\!\cdots\!79}a^{6}+\frac{54\!\cdots\!10}{75\!\cdots\!79}a^{5}-\frac{20\!\cdots\!12}{75\!\cdots\!79}a^{4}+\frac{83\!\cdots\!87}{75\!\cdots\!79}a^{3}-\frac{21\!\cdots\!02}{75\!\cdots\!79}a^{2}+\frac{20\!\cdots\!52}{75\!\cdots\!79}a-\frac{24\!\cdots\!33}{75\!\cdots\!79}$, $\frac{24\!\cdots\!70}{75\!\cdots\!79}a^{29}-\frac{27\!\cdots\!24}{75\!\cdots\!79}a^{28}+\frac{36\!\cdots\!45}{75\!\cdots\!79}a^{27}-\frac{33\!\cdots\!05}{75\!\cdots\!79}a^{26}+\frac{31\!\cdots\!11}{75\!\cdots\!79}a^{25}-\frac{25\!\cdots\!43}{75\!\cdots\!79}a^{24}+\frac{18\!\cdots\!58}{75\!\cdots\!79}a^{23}-\frac{12\!\cdots\!57}{75\!\cdots\!79}a^{22}+\frac{76\!\cdots\!62}{75\!\cdots\!79}a^{21}-\frac{47\!\cdots\!19}{75\!\cdots\!79}a^{20}+\frac{24\!\cdots\!13}{75\!\cdots\!79}a^{19}-\frac{13\!\cdots\!63}{75\!\cdots\!79}a^{18}+\frac{58\!\cdots\!07}{75\!\cdots\!79}a^{17}-\frac{27\!\cdots\!70}{75\!\cdots\!79}a^{16}+\frac{10\!\cdots\!82}{75\!\cdots\!79}a^{15}-\frac{42\!\cdots\!97}{75\!\cdots\!79}a^{14}+\frac{14\!\cdots\!48}{75\!\cdots\!79}a^{13}-\frac{52\!\cdots\!64}{75\!\cdots\!79}a^{12}+\frac{14\!\cdots\!97}{75\!\cdots\!79}a^{11}-\frac{42\!\cdots\!89}{75\!\cdots\!79}a^{10}+\frac{10\!\cdots\!78}{75\!\cdots\!79}a^{9}-\frac{28\!\cdots\!47}{75\!\cdots\!79}a^{8}+\frac{50\!\cdots\!80}{75\!\cdots\!79}a^{7}-\frac{10\!\cdots\!19}{75\!\cdots\!79}a^{6}+\frac{14\!\cdots\!66}{75\!\cdots\!79}a^{5}-\frac{43\!\cdots\!66}{75\!\cdots\!79}a^{4}+\frac{17\!\cdots\!73}{75\!\cdots\!79}a^{3}-\frac{49\!\cdots\!67}{75\!\cdots\!79}a^{2}+\frac{46\!\cdots\!30}{75\!\cdots\!79}a-\frac{72\!\cdots\!88}{75\!\cdots\!79}$ (assuming GRH)
Regulator:		\( 4316173757.895952 \) (assuming GRH)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{15}\cdot 4316173757.895952 \cdot 755}{6\cdot\sqrt{8221408887534945302579972677458642036796803806187}}\cr\approx \mathstrut & 0.177876887881771 \end{aligned}\] (assuming GRH)

Galois group

$C_{30}$ (as 30T1):

A cyclic group of order 30

The 30 conjugacy class representatives for $C_{30}$

Character table for $C_{30}$ is not computed

Intermediate fields

\(\Q(\sqrt{-3}) \), 3.3.961.1, 5.5.923521.1, 6.0.24935067.1, 10.0.207252522098163.1, \(\Q(\zeta_{31})^+\)

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	${\href{/padicField/2.10.0.1}{10} }^{3}$	R	${\href{/padicField/5.6.0.1}{6} }^{5}$	$15^{2}$	$30$	$15^{2}$	$30$	$15^{2}$	${\href{/padicField/23.10.0.1}{10} }^{3}$	${\href{/padicField/29.10.0.1}{10} }^{3}$	R	${\href{/padicField/37.3.0.1}{3} }^{10}$	$30$	$15^{2}$	${\href{/padicField/47.10.0.1}{10} }^{3}$	$30$	$30$

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 $30$	$2$	$15$	$15$
\(31\)	31.15.14.1	$x^{15} + 31$	$15$	$1$	$14$	$C_{15}$	$[\ ]_{15}$
	31.15.14.1	$x^{15} + 31$	$15$	$1$	$14$	$C_{15}$	$[\ ]_{15}$