Number field 30.0.2686026609350084707957118496751488894605397847668243387.1

• Label: 30.0.268...387.1
• Degree: $30$
• Signature: $[0, 15]$
• Discriminant: $-2.686\times 10^{54}$
• Root discriminant: \(65.21\)
• Ramified primes: $3,11,13$
• Class number: $17701$ (GRH)
• Class group: [17701] (GRH)
• Galois group: $C_{30}$ (as 30T1)

Normalized defining polynomial

x^30 - 2*x^29 + 37*x^28 - 26*x^27 + 779*x^26 - 272*x^25 + 10230*x^24 + 363*x^23 + 94154*x^22 + 8189*x^21 + 584563*x^20 + 57350*x^19 + 2635781*x^18 - 46599*x^17 + 8080224*x^16 - 1181499*x^15 + 18426044*x^14 - 4657959*x^13 + 28663378*x^12 - 9417878*x^11 + 31882791*x^10 - 10347153*x^9 + 19262569*x^8 - 4679058*x^7 + 7661349*x^6 - 1721389*x^5 + 1862037*x^4 - 311177*x^3 + 283622*x^2 - 48863*x + 17161

Invariants

Degree:		$30$
Signature:		$[0, 15]$
Discriminant:		\(-2686026609350084707957118496751488894605397847668243387\) \(\medspace = -\,3^{15}\cdot 11^{24}\cdot 13^{20}\)
Root discriminant:		\(65.21\)
Galois root discriminant:		$3^{1/2}11^{4/5}13^{2/3}\approx 65.2084199466292$
Ramified primes:		\(3\), \(11\), \(13\)
Discriminant root field:		\(\Q(\sqrt{-3}) \)
$\card{ \Gal(K/\Q) }$:		$30$
This field is Galois and abelian over $\Q$.
Conductor:		\(429=3\cdot 11\cdot 13\)
Dirichlet character group:		$\lbrace$$\chi_{429}(256,·)$, $\chi_{429}(1,·)$, $\chi_{429}(386,·)$, $\chi_{429}(196,·)$, $\chi_{429}(133,·)$, $\chi_{429}(328,·)$, $\chi_{429}(269,·)$, $\chi_{429}(14,·)$, $\chi_{429}(16,·)$, $\chi_{429}(146,·)$, $\chi_{429}(334,·)$, $\chi_{429}(313,·)$, $\chi_{429}(152,·)$, $\chi_{429}(412,·)$, $\chi_{429}(157,·)$, $\chi_{429}(287,·)$, $\chi_{429}(224,·)$, $\chi_{429}(289,·)$, $\chi_{429}(419,·)$, $\chi_{429}(100,·)$, $\chi_{429}(295,·)$, $\chi_{429}(92,·)$, $\chi_{429}(170,·)$, $\chi_{429}(235,·)$, $\chi_{429}(302,·)$, $\chi_{429}(367,·)$, $\chi_{429}(113,·)$, $\chi_{429}(53,·)$, $\chi_{429}(185,·)$, $\chi_{429}(191,·)$$\rbrace$
This is a CM field.
Reflex fields:		unavailable$^{16384}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $\frac{1}{131}a^{27}+\frac{49}{131}a^{26}-\frac{65}{131}a^{25}-\frac{52}{131}a^{24}+\frac{36}{131}a^{23}+\frac{52}{131}a^{22}-\frac{58}{131}a^{21}-\frac{35}{131}a^{20}-\frac{40}{131}a^{19}-\frac{18}{131}a^{18}-\frac{65}{131}a^{17}-\frac{17}{131}a^{16}+\frac{55}{131}a^{15}+\frac{30}{131}a^{14}-\frac{32}{131}a^{13}-\frac{24}{131}a^{12}-\frac{21}{131}a^{11}+\frac{53}{131}a^{10}+\frac{64}{131}a^{8}+\frac{50}{131}a^{6}-\frac{3}{131}a^{5}+\frac{11}{131}a^{4}+\frac{56}{131}a^{3}+\frac{3}{131}a^{2}+\frac{41}{131}a$, $\frac{1}{21\!\cdots\!61}a^{28}-\frac{112904653830712}{21\!\cdots\!61}a^{27}-\frac{45\!\cdots\!36}{21\!\cdots\!61}a^{26}+\frac{369613443805360}{21\!\cdots\!61}a^{25}-\frac{51\!\cdots\!78}{21\!\cdots\!61}a^{24}+\frac{31\!\cdots\!65}{21\!\cdots\!61}a^{23}-\frac{33\!\cdots\!89}{21\!\cdots\!61}a^{22}+\frac{96\!\cdots\!53}{21\!\cdots\!61}a^{21}+\frac{61\!\cdots\!29}{21\!\cdots\!61}a^{20}+\frac{61\!\cdots\!63}{21\!\cdots\!61}a^{19}-\frac{78\!\cdots\!66}{21\!\cdots\!61}a^{18}+\frac{60\!\cdots\!37}{21\!\cdots\!61}a^{17}+\frac{16\!\cdots\!41}{21\!\cdots\!61}a^{16}-\frac{82\!\cdots\!47}{21\!\cdots\!61}a^{15}-\frac{10\!\cdots\!72}{21\!\cdots\!61}a^{14}-\frac{81\!\cdots\!67}{21\!\cdots\!61}a^{13}-\frac{90\!\cdots\!54}{21\!\cdots\!61}a^{12}-\frac{1434701079829}{130972619851523}a^{11}+\frac{54\!\cdots\!44}{21\!\cdots\!61}a^{10}+\frac{81\!\cdots\!24}{21\!\cdots\!61}a^{9}-\frac{56\!\cdots\!27}{21\!\cdots\!61}a^{8}-\frac{12\!\cdots\!93}{21\!\cdots\!61}a^{7}-\frac{77\!\cdots\!47}{21\!\cdots\!61}a^{6}+\frac{91\!\cdots\!81}{21\!\cdots\!61}a^{5}+\frac{94\!\cdots\!22}{21\!\cdots\!61}a^{4}+\frac{63\!\cdots\!32}{21\!\cdots\!61}a^{3}+\frac{66\!\cdots\!94}{21\!\cdots\!61}a^{2}-\frac{13\!\cdots\!95}{21\!\cdots\!61}a-\frac{161293452768987}{16\!\cdots\!31}$, $\frac{1}{26\!\cdots\!77}a^{29}-\frac{71\!\cdots\!50}{26\!\cdots\!77}a^{28}-\frac{22\!\cdots\!58}{26\!\cdots\!77}a^{27}-\frac{55\!\cdots\!40}{26\!\cdots\!77}a^{26}-\frac{60\!\cdots\!35}{26\!\cdots\!77}a^{25}-\frac{12\!\cdots\!64}{26\!\cdots\!77}a^{24}-\frac{10\!\cdots\!87}{26\!\cdots\!77}a^{23}-\frac{44\!\cdots\!77}{26\!\cdots\!77}a^{22}+\frac{10\!\cdots\!91}{26\!\cdots\!77}a^{21}-\frac{21\!\cdots\!42}{26\!\cdots\!77}a^{20}+\frac{69\!\cdots\!48}{26\!\cdots\!77}a^{19}-\frac{19\!\cdots\!40}{26\!\cdots\!77}a^{18}-\frac{84\!\cdots\!97}{26\!\cdots\!77}a^{17}-\frac{24\!\cdots\!04}{26\!\cdots\!77}a^{16}-\frac{48\!\cdots\!32}{26\!\cdots\!77}a^{15}+\frac{11\!\cdots\!75}{26\!\cdots\!77}a^{14}-\frac{57\!\cdots\!15}{26\!\cdots\!77}a^{13}+\frac{28\!\cdots\!98}{26\!\cdots\!77}a^{12}-\frac{13\!\cdots\!06}{26\!\cdots\!77}a^{11}+\frac{11\!\cdots\!14}{26\!\cdots\!77}a^{10}+\frac{44\!\cdots\!54}{26\!\cdots\!77}a^{9}+\frac{42\!\cdots\!22}{26\!\cdots\!77}a^{8}+\frac{61\!\cdots\!72}{26\!\cdots\!77}a^{7}+\frac{15\!\cdots\!28}{26\!\cdots\!77}a^{6}+\frac{28\!\cdots\!23}{26\!\cdots\!77}a^{5}-\frac{10\!\cdots\!37}{26\!\cdots\!77}a^{4}+\frac{11\!\cdots\!10}{26\!\cdots\!77}a^{3}-\frac{99\!\cdots\!18}{26\!\cdots\!77}a^{2}+\frac{73\!\cdots\!93}{26\!\cdots\!77}a-\frac{87\!\cdots\!43}{20\!\cdots\!67}$

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

Class group and class number

$C_{17701}$, which has order $17701$ (assuming GRH)

Unit group

Rank:		$14$
Torsion generator:		\( \frac{3469122458023311461595367191847959320626322764470425323514812626445781025969}{165728357388117096358708417902056199219397598725769235047825802274527150318382367} a^{29} - \frac{6717066443293691446301436849747158627360652684498777761411300266473012638218}{165728357388117096358708417902056199219397598725769235047825802274527150318382367} a^{28} + \frac{127496555602963866516340802628302050935114043698739835586105762917069493036407}{165728357388117096358708417902056199219397598725769235047825802274527150318382367} a^{27} - \frac{81085677897372969129451125648779690512467885881331539250794945304626392385037}{165728357388117096358708417902056199219397598725769235047825802274527150318382367} a^{26} + \frac{2681133322690340585583010282880961570190950790818811170906257984561473990802987}{165728357388117096358708417902056199219397598725769235047825802274527150318382367} a^{25} - \frac{757313706220647413176210305713884785761819912176604468001212756261504458991873}{165728357388117096358708417902056199219397598725769235047825802274527150318382367} a^{24} + \frac{35104520672290508168789270573159229281081350384486311345738227361615550172550542}{165728357388117096358708417902056199219397598725769235047825802274527150318382367} a^{23} + \frac{3702492457429490495940369012265223737014794137619882453031659083764106913459625}{165728357388117096358708417902056199219397598725769235047825802274527150318382367} a^{22} + \frac{322488402278320821385273906436715071007399163780954346302940871171165980987806119}{165728357388117096358708417902056199219397598725769235047825802274527150318382367} a^{21} + \frac{49955388773718695571985169565141186965901947858544577430513856869488431341718528}{165728357388117096358708417902056199219397598725769235047825802274527150318382367} a^{20} + \frac{1991434262518618930741751527196882905063012765997933150836845506627192411062210073}{165728357388117096358708417902056199219397598725769235047825802274527150318382367} a^{19} + \frac{333079724283414038250016231715542441613311490073625094107191742548546610216430141}{165728357388117096358708417902056199219397598725769235047825802274527150318382367} a^{18} + \frac{8922344828177445710277149536695827164919643064618647316264361097332909047035322388}{165728357388117096358708417902056199219397598725769235047825802274527150318382367} a^{17} + \frac{448266012271420245451774575858960281997978647802417831080308337824300293240327409}{165728357388117096358708417902056199219397598725769235047825802274527150318382367} a^{16} + \frac{26982369611001864820597067818875530189863738500623048245811259050570407604745080689}{165728357388117096358708417902056199219397598725769235047825802274527150318382367} a^{15} - \frac{2064595410008024312695743126314305683104607400704825253400120383354323064253077132}{165728357388117096358708417902056199219397598725769235047825802274527150318382367} a^{14} + \frac{60537309715476302250388262324567821096018996140311918470333436998395685668022768485}{165728357388117096358708417902056199219397598725769235047825802274527150318382367} a^{13} - \frac{10938429748695818772061903184077300834979358530914694166681904468563750710023177225}{165728357388117096358708417902056199219397598725769235047825802274527150318382367} a^{12} + \frac{91375640112679675462478552035594053182742602520878184357471468653137179951613751604}{165728357388117096358708417902056199219397598725769235047825802274527150318382367} a^{11} - \frac{23049637983875015519581920089255823038909607582084165495049611475491202556588196396}{165728357388117096358708417902056199219397598725769235047825802274527150318382367} a^{10} + \frac{97878863874223792175809507936081596240968599417777101463812717676368403646050908184}{165728357388117096358708417902056199219397598725769235047825802274527150318382367} a^{9} - \frac{23217376241536011278096003100596087905644717602126131333637300243341510677058683768}{165728357388117096358708417902056199219397598725769235047825802274527150318382367} a^{8} + \frac{53018046023662368350231302835697973535430025201170214963865004555872585546122231152}{165728357388117096358708417902056199219397598725769235047825802274527150318382367} a^{7} - \frac{47506146646888735421919548121349127714174922281180257228608442924277681849250023}{1265101964794786995104644411466077856636622891036406374410883986828451529147957} a^{6} + \frac{19437645838432197819643092742255333328843596417730835818498837068962049583280108648}{165728357388117096358708417902056199219397598725769235047825802274527150318382367} a^{5} - \frac{2000513276729214975236456871489668936926175911434307057288741889777271471003651608}{165728357388117096358708417902056199219397598725769235047825802274527150318382367} a^{4} + \frac{3931756859920564360439330016304764968938973050907108351778091825651924565482350295}{165728357388117096358708417902056199219397598725769235047825802274527150318382367} a^{3} + \frac{395600850807166887886493291534688398606893585211437642886024696820628048956156045}{165728357388117096358708417902056199219397598725769235047825802274527150318382367} a^{2} + \frac{496877501828557068710526414635088866008807693008853229137969573567432090951869620}{165728357388117096358708417902056199219397598725769235047825802274527150318382367} a + \frac{782881538383575440242767988043826456442088839814023528148427013809070122603510}{1265101964794786995104644411466077856636622891036406374410883986828451529147957} \)  (order $6$)
Fundamental units:		$\frac{64\!\cdots\!78}{41\!\cdots\!81}a^{29}-\frac{57\!\cdots\!21}{41\!\cdots\!81}a^{28}+\frac{35\!\cdots\!50}{41\!\cdots\!81}a^{27}-\frac{18\!\cdots\!25}{41\!\cdots\!81}a^{26}+\frac{70\!\cdots\!73}{41\!\cdots\!81}a^{25}-\frac{36\!\cdots\!52}{41\!\cdots\!81}a^{24}+\frac{95\!\cdots\!17}{41\!\cdots\!81}a^{23}-\frac{45\!\cdots\!13}{41\!\cdots\!81}a^{22}+\frac{82\!\cdots\!11}{41\!\cdots\!81}a^{21}-\frac{41\!\cdots\!67}{41\!\cdots\!81}a^{20}+\frac{55\!\cdots\!97}{41\!\cdots\!81}a^{19}-\frac{25\!\cdots\!66}{41\!\cdots\!81}a^{18}+\frac{27\!\cdots\!25}{41\!\cdots\!81}a^{17}-\frac{11\!\cdots\!95}{41\!\cdots\!81}a^{16}+\frac{11\!\cdots\!64}{41\!\cdots\!81}a^{15}-\frac{36\!\cdots\!45}{41\!\cdots\!81}a^{14}+\frac{34\!\cdots\!57}{41\!\cdots\!81}a^{13}-\frac{87\!\cdots\!90}{41\!\cdots\!81}a^{12}+\frac{76\!\cdots\!18}{41\!\cdots\!81}a^{11}-\frac{14\!\cdots\!10}{41\!\cdots\!81}a^{10}+\frac{11\!\cdots\!01}{41\!\cdots\!81}a^{9}-\frac{16\!\cdots\!94}{41\!\cdots\!81}a^{8}+\frac{11\!\cdots\!19}{41\!\cdots\!81}a^{7}-\frac{99\!\cdots\!79}{41\!\cdots\!81}a^{6}+\frac{45\!\cdots\!88}{41\!\cdots\!81}a^{5}-\frac{30\!\cdots\!51}{41\!\cdots\!81}a^{4}+\frac{11\!\cdots\!06}{41\!\cdots\!81}a^{3}-\frac{59\!\cdots\!50}{41\!\cdots\!81}a^{2}+\frac{97\!\cdots\!09}{31\!\cdots\!51}a-\frac{20\!\cdots\!60}{31\!\cdots\!51}$, $\frac{83\!\cdots\!00}{41\!\cdots\!81}a^{29}+\frac{11\!\cdots\!28}{41\!\cdots\!81}a^{28}+\frac{23\!\cdots\!65}{41\!\cdots\!81}a^{27}+\frac{86\!\cdots\!19}{41\!\cdots\!81}a^{26}+\frac{50\!\cdots\!88}{41\!\cdots\!81}a^{25}+\frac{20\!\cdots\!77}{41\!\cdots\!81}a^{24}+\frac{62\!\cdots\!67}{41\!\cdots\!81}a^{23}+\frac{29\!\cdots\!83}{41\!\cdots\!81}a^{22}+\frac{60\!\cdots\!15}{41\!\cdots\!81}a^{21}+\frac{27\!\cdots\!71}{41\!\cdots\!81}a^{20}+\frac{33\!\cdots\!43}{41\!\cdots\!81}a^{19}+\frac{16\!\cdots\!30}{41\!\cdots\!81}a^{18}+\frac{13\!\cdots\!24}{41\!\cdots\!81}a^{17}+\frac{72\!\cdots\!82}{41\!\cdots\!81}a^{16}+\frac{19\!\cdots\!94}{41\!\cdots\!81}a^{15}+\frac{21\!\cdots\!27}{41\!\cdots\!81}a^{14}-\frac{17\!\cdots\!95}{41\!\cdots\!81}a^{13}+\frac{50\!\cdots\!19}{41\!\cdots\!81}a^{12}-\frac{19\!\cdots\!06}{41\!\cdots\!81}a^{11}+\frac{80\!\cdots\!79}{41\!\cdots\!81}a^{10}-\frac{43\!\cdots\!29}{41\!\cdots\!81}a^{9}+\frac{93\!\cdots\!87}{41\!\cdots\!81}a^{8}-\frac{57\!\cdots\!06}{41\!\cdots\!81}a^{7}+\frac{60\!\cdots\!78}{41\!\cdots\!81}a^{6}-\frac{24\!\cdots\!09}{41\!\cdots\!81}a^{5}+\frac{17\!\cdots\!80}{41\!\cdots\!81}a^{4}-\frac{63\!\cdots\!19}{41\!\cdots\!81}a^{3}+\frac{33\!\cdots\!63}{41\!\cdots\!81}a^{2}-\frac{54\!\cdots\!29}{31\!\cdots\!51}a+\frac{79\!\cdots\!59}{31\!\cdots\!51}$, $\frac{12\!\cdots\!71}{41\!\cdots\!81}a^{29}-\frac{60\!\cdots\!02}{41\!\cdots\!81}a^{28}+\frac{19\!\cdots\!75}{41\!\cdots\!81}a^{27}-\frac{22\!\cdots\!37}{41\!\cdots\!81}a^{26}+\frac{36\!\cdots\!73}{41\!\cdots\!81}a^{25}-\frac{46\!\cdots\!78}{41\!\cdots\!81}a^{24}+\frac{53\!\cdots\!97}{41\!\cdots\!81}a^{23}-\frac{60\!\cdots\!68}{41\!\cdots\!81}a^{22}+\frac{41\!\cdots\!41}{41\!\cdots\!81}a^{21}-\frac{54\!\cdots\!04}{41\!\cdots\!81}a^{20}+\frac{31\!\cdots\!35}{41\!\cdots\!81}a^{19}-\frac{33\!\cdots\!30}{41\!\cdots\!81}a^{18}+\frac{17\!\cdots\!41}{41\!\cdots\!81}a^{17}-\frac{15\!\cdots\!02}{41\!\cdots\!81}a^{16}+\frac{91\!\cdots\!93}{41\!\cdots\!81}a^{15}-\frac{47\!\cdots\!92}{41\!\cdots\!81}a^{14}+\frac{32\!\cdots\!56}{41\!\cdots\!81}a^{13}-\frac{11\!\cdots\!23}{41\!\cdots\!81}a^{12}+\frac{82\!\cdots\!78}{41\!\cdots\!81}a^{11}-\frac{17\!\cdots\!90}{41\!\cdots\!81}a^{10}+\frac{13\!\cdots\!08}{41\!\cdots\!81}a^{9}-\frac{20\!\cdots\!55}{41\!\cdots\!81}a^{8}+\frac{13\!\cdots\!83}{41\!\cdots\!81}a^{7}-\frac{12\!\cdots\!78}{41\!\cdots\!81}a^{6}+\frac{56\!\cdots\!96}{41\!\cdots\!81}a^{5}-\frac{38\!\cdots\!92}{41\!\cdots\!81}a^{4}+\frac{14\!\cdots\!36}{41\!\cdots\!81}a^{3}-\frac{74\!\cdots\!55}{41\!\cdots\!81}a^{2}+\frac{12\!\cdots\!86}{31\!\cdots\!51}a-\frac{30\!\cdots\!14}{31\!\cdots\!51}$, $\frac{11\!\cdots\!41}{41\!\cdots\!81}a^{29}-\frac{41\!\cdots\!66}{41\!\cdots\!81}a^{28}+\frac{46\!\cdots\!53}{41\!\cdots\!81}a^{27}-\frac{10\!\cdots\!03}{41\!\cdots\!81}a^{26}+\frac{95\!\cdots\!24}{41\!\cdots\!81}a^{25}-\frac{18\!\cdots\!14}{41\!\cdots\!81}a^{24}+\frac{12\!\cdots\!41}{41\!\cdots\!81}a^{23}-\frac{20\!\cdots\!89}{41\!\cdots\!81}a^{22}+\frac{11\!\cdots\!24}{41\!\cdots\!81}a^{21}-\frac{17\!\cdots\!38}{41\!\cdots\!81}a^{20}+\frac{72\!\cdots\!54}{41\!\cdots\!81}a^{19}-\frac{10\!\cdots\!29}{41\!\cdots\!81}a^{18}+\frac{33\!\cdots\!97}{41\!\cdots\!81}a^{17}-\frac{52\!\cdots\!49}{41\!\cdots\!81}a^{16}+\frac{11\!\cdots\!24}{41\!\cdots\!81}a^{15}-\frac{17\!\cdots\!35}{41\!\cdots\!81}a^{14}+\frac{29\!\cdots\!12}{41\!\cdots\!81}a^{13}-\frac{44\!\cdots\!37}{41\!\cdots\!81}a^{12}+\frac{56\!\cdots\!82}{41\!\cdots\!81}a^{11}-\frac{74\!\cdots\!99}{41\!\cdots\!81}a^{10}+\frac{75\!\cdots\!02}{41\!\cdots\!81}a^{9}-\frac{85\!\cdots\!68}{41\!\cdots\!81}a^{8}+\frac{63\!\cdots\!46}{41\!\cdots\!81}a^{7}-\frac{52\!\cdots\!52}{41\!\cdots\!81}a^{6}+\frac{25\!\cdots\!49}{41\!\cdots\!81}a^{5}-\frac{16\!\cdots\!70}{41\!\cdots\!81}a^{4}+\frac{60\!\cdots\!65}{41\!\cdots\!81}a^{3}-\frac{31\!\cdots\!55}{41\!\cdots\!81}a^{2}+\frac{52\!\cdots\!32}{31\!\cdots\!51}a-\frac{13\!\cdots\!48}{31\!\cdots\!51}$, $\frac{23\!\cdots\!66}{26\!\cdots\!77}a^{29}-\frac{57\!\cdots\!04}{26\!\cdots\!77}a^{28}+\frac{88\!\cdots\!48}{26\!\cdots\!77}a^{27}-\frac{10\!\cdots\!45}{26\!\cdots\!77}a^{26}+\frac{18\!\cdots\!04}{26\!\cdots\!77}a^{25}-\frac{14\!\cdots\!56}{26\!\cdots\!77}a^{24}+\frac{24\!\cdots\!54}{26\!\cdots\!77}a^{23}-\frac{10\!\cdots\!42}{26\!\cdots\!77}a^{22}+\frac{21\!\cdots\!02}{26\!\cdots\!77}a^{21}-\frac{80\!\cdots\!88}{26\!\cdots\!77}a^{20}+\frac{13\!\cdots\!80}{26\!\cdots\!77}a^{19}-\frac{47\!\cdots\!68}{26\!\cdots\!77}a^{18}+\frac{60\!\cdots\!60}{26\!\cdots\!77}a^{17}-\frac{28\!\cdots\!70}{26\!\cdots\!77}a^{16}+\frac{18\!\cdots\!19}{26\!\cdots\!77}a^{15}-\frac{10\!\cdots\!97}{26\!\cdots\!77}a^{14}+\frac{43\!\cdots\!03}{26\!\cdots\!77}a^{13}-\frac{28\!\cdots\!95}{26\!\cdots\!77}a^{12}+\frac{69\!\cdots\!79}{26\!\cdots\!77}a^{11}-\frac{47\!\cdots\!07}{26\!\cdots\!77}a^{10}+\frac{78\!\cdots\!15}{26\!\cdots\!77}a^{9}-\frac{48\!\cdots\!46}{26\!\cdots\!77}a^{8}+\frac{47\!\cdots\!70}{26\!\cdots\!77}a^{7}-\frac{20\!\cdots\!04}{26\!\cdots\!77}a^{6}+\frac{14\!\cdots\!39}{26\!\cdots\!77}a^{5}-\frac{48\!\cdots\!08}{26\!\cdots\!77}a^{4}+\frac{28\!\cdots\!01}{26\!\cdots\!77}a^{3}-\frac{53\!\cdots\!92}{26\!\cdots\!77}a^{2}+\frac{15\!\cdots\!01}{26\!\cdots\!77}a+\frac{49\!\cdots\!40}{20\!\cdots\!67}$, $\frac{50\!\cdots\!98}{26\!\cdots\!77}a^{29}-\frac{12\!\cdots\!19}{26\!\cdots\!77}a^{28}+\frac{19\!\cdots\!12}{26\!\cdots\!77}a^{27}-\frac{22\!\cdots\!46}{26\!\cdots\!77}a^{26}+\frac{40\!\cdots\!19}{26\!\cdots\!77}a^{25}-\frac{33\!\cdots\!40}{26\!\cdots\!77}a^{24}+\frac{52\!\cdots\!57}{26\!\cdots\!77}a^{23}-\frac{23\!\cdots\!78}{26\!\cdots\!77}a^{22}+\frac{47\!\cdots\!04}{26\!\cdots\!77}a^{21}-\frac{19\!\cdots\!34}{26\!\cdots\!77}a^{20}+\frac{29\!\cdots\!21}{26\!\cdots\!77}a^{19}-\frac{11\!\cdots\!68}{26\!\cdots\!77}a^{18}+\frac{13\!\cdots\!22}{26\!\cdots\!77}a^{17}-\frac{64\!\cdots\!52}{26\!\cdots\!77}a^{16}+\frac{41\!\cdots\!98}{26\!\cdots\!77}a^{15}-\frac{24\!\cdots\!10}{26\!\cdots\!77}a^{14}+\frac{98\!\cdots\!72}{26\!\cdots\!77}a^{13}-\frac{63\!\cdots\!33}{26\!\cdots\!77}a^{12}+\frac{15\!\cdots\!81}{26\!\cdots\!77}a^{11}-\frac{10\!\cdots\!77}{26\!\cdots\!77}a^{10}+\frac{18\!\cdots\!83}{26\!\cdots\!77}a^{9}-\frac{10\!\cdots\!85}{26\!\cdots\!77}a^{8}+\frac{11\!\cdots\!39}{26\!\cdots\!77}a^{7}-\frac{44\!\cdots\!17}{26\!\cdots\!77}a^{6}+\frac{35\!\cdots\!38}{26\!\cdots\!77}a^{5}-\frac{10\!\cdots\!23}{26\!\cdots\!77}a^{4}+\frac{68\!\cdots\!16}{26\!\cdots\!77}a^{3}-\frac{10\!\cdots\!15}{26\!\cdots\!77}a^{2}+\frac{29\!\cdots\!73}{26\!\cdots\!77}a+\frac{25\!\cdots\!33}{20\!\cdots\!67}$, $\frac{74\!\cdots\!94}{26\!\cdots\!77}a^{29}-\frac{65\!\cdots\!44}{26\!\cdots\!77}a^{28}+\frac{40\!\cdots\!13}{26\!\cdots\!77}a^{27}-\frac{21\!\cdots\!84}{26\!\cdots\!77}a^{26}+\frac{81\!\cdots\!62}{26\!\cdots\!77}a^{25}-\frac{41\!\cdots\!56}{26\!\cdots\!77}a^{24}+\frac{11\!\cdots\!82}{26\!\cdots\!77}a^{23}-\frac{51\!\cdots\!35}{26\!\cdots\!77}a^{22}+\frac{96\!\cdots\!97}{26\!\cdots\!77}a^{21}-\frac{46\!\cdots\!82}{26\!\cdots\!77}a^{20}+\frac{64\!\cdots\!63}{26\!\cdots\!77}a^{19}-\frac{28\!\cdots\!40}{26\!\cdots\!77}a^{18}+\frac{31\!\cdots\!58}{26\!\cdots\!77}a^{17}-\frac{13\!\cdots\!07}{26\!\cdots\!77}a^{16}+\frac{12\!\cdots\!23}{26\!\cdots\!77}a^{15}-\frac{41\!\cdots\!72}{26\!\cdots\!77}a^{14}+\frac{39\!\cdots\!13}{26\!\cdots\!77}a^{13}-\frac{99\!\cdots\!56}{26\!\cdots\!77}a^{12}+\frac{88\!\cdots\!57}{26\!\cdots\!77}a^{11}-\frac{16\!\cdots\!20}{26\!\cdots\!77}a^{10}+\frac{13\!\cdots\!48}{26\!\cdots\!77}a^{9}-\frac{18\!\cdots\!88}{26\!\cdots\!77}a^{8}+\frac{12\!\cdots\!85}{26\!\cdots\!77}a^{7}-\frac{11\!\cdots\!19}{26\!\cdots\!77}a^{6}+\frac{52\!\cdots\!55}{26\!\cdots\!77}a^{5}-\frac{35\!\cdots\!21}{26\!\cdots\!77}a^{4}+\frac{13\!\cdots\!98}{26\!\cdots\!77}a^{3}-\frac{67\!\cdots\!35}{26\!\cdots\!77}a^{2}+\frac{11\!\cdots\!52}{20\!\cdots\!67}a-\frac{25\!\cdots\!82}{20\!\cdots\!67}$, $\frac{17\!\cdots\!40}{26\!\cdots\!77}a^{29}-\frac{31\!\cdots\!54}{26\!\cdots\!77}a^{28}+\frac{64\!\cdots\!43}{26\!\cdots\!77}a^{27}-\frac{32\!\cdots\!53}{26\!\cdots\!77}a^{26}+\frac{13\!\cdots\!34}{26\!\cdots\!77}a^{25}-\frac{20\!\cdots\!19}{26\!\cdots\!77}a^{24}+\frac{17\!\cdots\!47}{26\!\cdots\!77}a^{23}+\frac{41\!\cdots\!46}{26\!\cdots\!77}a^{22}+\frac{16\!\cdots\!41}{26\!\cdots\!77}a^{21}+\frac{46\!\cdots\!94}{26\!\cdots\!77}a^{20}+\frac{10\!\cdots\!64}{26\!\cdots\!77}a^{19}+\frac{30\!\cdots\!67}{26\!\cdots\!77}a^{18}+\frac{44\!\cdots\!48}{26\!\cdots\!77}a^{17}+\frac{63\!\cdots\!27}{20\!\cdots\!67}a^{16}+\frac{13\!\cdots\!88}{26\!\cdots\!77}a^{15}+\frac{83\!\cdots\!43}{26\!\cdots\!77}a^{14}+\frac{29\!\cdots\!08}{26\!\cdots\!77}a^{13}-\frac{11\!\cdots\!27}{26\!\cdots\!77}a^{12}+\frac{43\!\cdots\!23}{26\!\cdots\!77}a^{11}-\frac{43\!\cdots\!03}{26\!\cdots\!77}a^{10}+\frac{43\!\cdots\!59}{26\!\cdots\!77}a^{9}-\frac{30\!\cdots\!16}{26\!\cdots\!77}a^{8}+\frac{20\!\cdots\!93}{26\!\cdots\!77}a^{7}+\frac{29\!\cdots\!56}{26\!\cdots\!77}a^{6}+\frac{53\!\cdots\!17}{26\!\cdots\!77}a^{5}+\frac{13\!\cdots\!40}{26\!\cdots\!77}a^{4}+\frac{67\!\cdots\!84}{26\!\cdots\!77}a^{3}+\frac{37\!\cdots\!00}{26\!\cdots\!77}a^{2}-\frac{83\!\cdots\!02}{26\!\cdots\!77}a+\frac{20\!\cdots\!66}{20\!\cdots\!67}$, $\frac{85\!\cdots\!03}{26\!\cdots\!77}a^{29}-\frac{17\!\cdots\!13}{26\!\cdots\!77}a^{28}+\frac{31\!\cdots\!93}{26\!\cdots\!77}a^{27}-\frac{21\!\cdots\!99}{26\!\cdots\!77}a^{26}+\frac{65\!\cdots\!90}{26\!\cdots\!77}a^{25}-\frac{22\!\cdots\!03}{26\!\cdots\!77}a^{24}+\frac{85\!\cdots\!35}{26\!\cdots\!77}a^{23}+\frac{46\!\cdots\!42}{26\!\cdots\!77}a^{22}+\frac{77\!\cdots\!63}{26\!\cdots\!77}a^{21}+\frac{81\!\cdots\!75}{26\!\cdots\!77}a^{20}+\frac{47\!\cdots\!93}{26\!\cdots\!77}a^{19}+\frac{58\!\cdots\!92}{26\!\cdots\!77}a^{18}+\frac{21\!\cdots\!18}{26\!\cdots\!77}a^{17}+\frac{20\!\cdots\!66}{26\!\cdots\!77}a^{16}+\frac{62\!\cdots\!77}{26\!\cdots\!77}a^{15}-\frac{66\!\cdots\!19}{26\!\cdots\!77}a^{14}+\frac{13\!\cdots\!68}{26\!\cdots\!77}a^{13}-\frac{26\!\cdots\!62}{26\!\cdots\!77}a^{12}+\frac{19\!\cdots\!83}{26\!\cdots\!77}a^{11}-\frac{47\!\cdots\!03}{26\!\cdots\!77}a^{10}+\frac{19\!\cdots\!91}{26\!\cdots\!77}a^{9}-\frac{33\!\cdots\!84}{26\!\cdots\!77}a^{8}+\frac{78\!\cdots\!28}{26\!\cdots\!77}a^{7}+\frac{16\!\cdots\!68}{26\!\cdots\!77}a^{6}+\frac{11\!\cdots\!40}{26\!\cdots\!77}a^{5}+\frac{90\!\cdots\!68}{26\!\cdots\!77}a^{4}-\frac{26\!\cdots\!14}{26\!\cdots\!77}a^{3}+\frac{31\!\cdots\!86}{26\!\cdots\!77}a^{2}-\frac{69\!\cdots\!66}{26\!\cdots\!77}a+\frac{19\!\cdots\!09}{20\!\cdots\!67}$, $\frac{17\!\cdots\!61}{26\!\cdots\!77}a^{29}-\frac{44\!\cdots\!67}{26\!\cdots\!77}a^{28}+\frac{67\!\cdots\!48}{26\!\cdots\!77}a^{27}-\frac{81\!\cdots\!59}{26\!\cdots\!77}a^{26}+\frac{14\!\cdots\!54}{26\!\cdots\!77}a^{25}-\frac{12\!\cdots\!46}{26\!\cdots\!77}a^{24}+\frac{18\!\cdots\!90}{26\!\cdots\!77}a^{23}-\frac{88\!\cdots\!40}{26\!\cdots\!77}a^{22}+\frac{16\!\cdots\!36}{26\!\cdots\!77}a^{21}-\frac{72\!\cdots\!46}{26\!\cdots\!77}a^{20}+\frac{10\!\cdots\!95}{26\!\cdots\!77}a^{19}-\frac{42\!\cdots\!03}{26\!\cdots\!77}a^{18}+\frac{46\!\cdots\!97}{26\!\cdots\!77}a^{17}-\frac{18\!\cdots\!58}{20\!\cdots\!67}a^{16}+\frac{14\!\cdots\!78}{26\!\cdots\!77}a^{15}-\frac{90\!\cdots\!30}{26\!\cdots\!77}a^{14}+\frac{33\!\cdots\!13}{26\!\cdots\!77}a^{13}-\frac{23\!\cdots\!80}{26\!\cdots\!77}a^{12}+\frac{53\!\cdots\!71}{26\!\cdots\!77}a^{11}-\frac{38\!\cdots\!40}{26\!\cdots\!77}a^{10}+\frac{61\!\cdots\!25}{26\!\cdots\!77}a^{9}-\frac{40\!\cdots\!87}{26\!\cdots\!77}a^{8}+\frac{37\!\cdots\!01}{26\!\cdots\!77}a^{7}-\frac{16\!\cdots\!99}{26\!\cdots\!77}a^{6}+\frac{11\!\cdots\!47}{26\!\cdots\!77}a^{5}-\frac{42\!\cdots\!16}{26\!\cdots\!77}a^{4}+\frac{22\!\cdots\!85}{26\!\cdots\!77}a^{3}-\frac{47\!\cdots\!68}{26\!\cdots\!77}a^{2}+\frac{10\!\cdots\!77}{26\!\cdots\!77}a+\frac{35\!\cdots\!84}{20\!\cdots\!67}$, $\frac{10\!\cdots\!18}{26\!\cdots\!77}a^{29}-\frac{21\!\cdots\!67}{26\!\cdots\!77}a^{28}+\frac{37\!\cdots\!96}{26\!\cdots\!77}a^{27}-\frac{29\!\cdots\!26}{26\!\cdots\!77}a^{26}+\frac{78\!\cdots\!61}{26\!\cdots\!77}a^{25}-\frac{33\!\cdots\!75}{26\!\cdots\!77}a^{24}+\frac{10\!\cdots\!04}{26\!\cdots\!77}a^{23}-\frac{35\!\cdots\!19}{26\!\cdots\!77}a^{22}+\frac{92\!\cdots\!59}{26\!\cdots\!77}a^{21}+\frac{12\!\cdots\!27}{26\!\cdots\!77}a^{20}+\frac{56\!\cdots\!28}{26\!\cdots\!77}a^{19}+\frac{18\!\cdots\!52}{26\!\cdots\!77}a^{18}+\frac{24\!\cdots\!44}{26\!\cdots\!77}a^{17}-\frac{20\!\cdots\!59}{26\!\cdots\!77}a^{16}+\frac{73\!\cdots\!89}{26\!\cdots\!77}a^{15}-\frac{14\!\cdots\!60}{26\!\cdots\!77}a^{14}+\frac{16\!\cdots\!46}{26\!\cdots\!77}a^{13}-\frac{46\!\cdots\!73}{26\!\cdots\!77}a^{12}+\frac{23\!\cdots\!16}{26\!\cdots\!77}a^{11}-\frac{75\!\cdots\!85}{26\!\cdots\!77}a^{10}+\frac{23\!\cdots\!22}{26\!\cdots\!77}a^{9}-\frac{58\!\cdots\!18}{26\!\cdots\!77}a^{8}+\frac{92\!\cdots\!41}{26\!\cdots\!77}a^{7}+\frac{13\!\cdots\!54}{26\!\cdots\!77}a^{6}+\frac{14\!\cdots\!62}{26\!\cdots\!77}a^{5}+\frac{90\!\cdots\!27}{26\!\cdots\!77}a^{4}-\frac{24\!\cdots\!87}{26\!\cdots\!77}a^{3}+\frac{39\!\cdots\!28}{26\!\cdots\!77}a^{2}-\frac{78\!\cdots\!22}{26\!\cdots\!77}a+\frac{21\!\cdots\!84}{20\!\cdots\!67}$, $\frac{27\!\cdots\!10}{26\!\cdots\!77}a^{29}-\frac{28\!\cdots\!68}{26\!\cdots\!77}a^{28}+\frac{95\!\cdots\!70}{26\!\cdots\!77}a^{27}+\frac{29\!\cdots\!05}{26\!\cdots\!77}a^{26}+\frac{20\!\cdots\!62}{26\!\cdots\!77}a^{25}+\frac{13\!\cdots\!80}{26\!\cdots\!77}a^{24}+\frac{26\!\cdots\!89}{26\!\cdots\!77}a^{23}+\frac{28\!\cdots\!99}{26\!\cdots\!77}a^{22}+\frac{24\!\cdots\!72}{26\!\cdots\!77}a^{21}+\frac{26\!\cdots\!11}{26\!\cdots\!77}a^{20}+\frac{14\!\cdots\!57}{26\!\cdots\!77}a^{19}+\frac{16\!\cdots\!31}{26\!\cdots\!77}a^{18}+\frac{63\!\cdots\!98}{26\!\cdots\!77}a^{17}+\frac{65\!\cdots\!10}{26\!\cdots\!77}a^{16}+\frac{17\!\cdots\!86}{26\!\cdots\!77}a^{15}+\frac{17\!\cdots\!23}{26\!\cdots\!77}a^{14}+\frac{33\!\cdots\!09}{26\!\cdots\!77}a^{13}+\frac{34\!\cdots\!27}{26\!\cdots\!77}a^{12}+\frac{35\!\cdots\!46}{26\!\cdots\!77}a^{11}+\frac{49\!\cdots\!39}{26\!\cdots\!77}a^{10}+\frac{17\!\cdots\!85}{26\!\cdots\!77}a^{9}+\frac{58\!\cdots\!37}{26\!\cdots\!77}a^{8}-\frac{20\!\cdots\!00}{26\!\cdots\!77}a^{7}+\frac{38\!\cdots\!87}{26\!\cdots\!77}a^{6}-\frac{11\!\cdots\!97}{26\!\cdots\!77}a^{5}+\frac{10\!\cdots\!72}{26\!\cdots\!77}a^{4}-\frac{41\!\cdots\!34}{26\!\cdots\!77}a^{3}+\frac{19\!\cdots\!49}{26\!\cdots\!77}a^{2}-\frac{30\!\cdots\!29}{20\!\cdots\!67}a+\frac{44\!\cdots\!09}{20\!\cdots\!67}$, $\frac{13\!\cdots\!17}{26\!\cdots\!77}a^{29}-\frac{69\!\cdots\!19}{26\!\cdots\!77}a^{28}+\frac{45\!\cdots\!75}{26\!\cdots\!77}a^{27}+\frac{41\!\cdots\!53}{26\!\cdots\!77}a^{26}+\frac{97\!\cdots\!65}{26\!\cdots\!77}a^{25}+\frac{12\!\cdots\!94}{26\!\cdots\!77}a^{24}+\frac{12\!\cdots\!37}{26\!\cdots\!77}a^{23}+\frac{21\!\cdots\!38}{26\!\cdots\!77}a^{22}+\frac{11\!\cdots\!22}{26\!\cdots\!77}a^{21}+\frac{19\!\cdots\!78}{26\!\cdots\!77}a^{20}+\frac{72\!\cdots\!01}{26\!\cdots\!77}a^{19}+\frac{11\!\cdots\!16}{26\!\cdots\!77}a^{18}+\frac{31\!\cdots\!52}{26\!\cdots\!77}a^{17}+\frac{48\!\cdots\!73}{26\!\cdots\!77}a^{16}+\frac{82\!\cdots\!93}{26\!\cdots\!77}a^{15}+\frac{13\!\cdots\!11}{26\!\cdots\!77}a^{14}+\frac{14\!\cdots\!37}{26\!\cdots\!77}a^{13}+\frac{27\!\cdots\!23}{26\!\cdots\!77}a^{12}+\frac{12\!\cdots\!63}{26\!\cdots\!77}a^{11}+\frac{38\!\cdots\!50}{26\!\cdots\!77}a^{10}+\frac{87\!\cdots\!33}{26\!\cdots\!77}a^{9}+\frac{42\!\cdots\!73}{26\!\cdots\!77}a^{8}-\frac{17\!\cdots\!26}{26\!\cdots\!77}a^{7}+\frac{23\!\cdots\!07}{26\!\cdots\!77}a^{6}-\frac{86\!\cdots\!37}{26\!\cdots\!77}a^{5}+\frac{75\!\cdots\!86}{26\!\cdots\!77}a^{4}-\frac{46\!\cdots\!99}{26\!\cdots\!77}a^{3}+\frac{14\!\cdots\!01}{26\!\cdots\!77}a^{2}-\frac{21\!\cdots\!26}{20\!\cdots\!67}a-\frac{24\!\cdots\!08}{20\!\cdots\!67}$, $\frac{20\!\cdots\!48}{26\!\cdots\!77}a^{29}-\frac{52\!\cdots\!08}{26\!\cdots\!77}a^{28}+\frac{77\!\cdots\!36}{26\!\cdots\!77}a^{27}-\frac{96\!\cdots\!09}{26\!\cdots\!77}a^{26}+\frac{16\!\cdots\!51}{26\!\cdots\!77}a^{25}-\frac{14\!\cdots\!68}{26\!\cdots\!77}a^{24}+\frac{20\!\cdots\!62}{26\!\cdots\!77}a^{23}-\frac{11\!\cdots\!09}{26\!\cdots\!77}a^{22}+\frac{18\!\cdots\!86}{26\!\cdots\!77}a^{21}-\frac{92\!\cdots\!20}{26\!\cdots\!77}a^{20}+\frac{11\!\cdots\!02}{26\!\cdots\!77}a^{19}-\frac{54\!\cdots\!10}{26\!\cdots\!77}a^{18}+\frac{52\!\cdots\!86}{26\!\cdots\!77}a^{17}-\frac{30\!\cdots\!96}{26\!\cdots\!77}a^{16}+\frac{16\!\cdots\!05}{26\!\cdots\!77}a^{15}-\frac{11\!\cdots\!67}{26\!\cdots\!77}a^{14}+\frac{37\!\cdots\!93}{26\!\cdots\!77}a^{13}-\frac{28\!\cdots\!49}{26\!\cdots\!77}a^{12}+\frac{60\!\cdots\!51}{26\!\cdots\!77}a^{11}-\frac{46\!\cdots\!07}{26\!\cdots\!77}a^{10}+\frac{69\!\cdots\!03}{26\!\cdots\!77}a^{9}-\frac{48\!\cdots\!24}{26\!\cdots\!77}a^{8}+\frac{42\!\cdots\!21}{26\!\cdots\!77}a^{7}-\frac{20\!\cdots\!62}{26\!\cdots\!77}a^{6}+\frac{12\!\cdots\!63}{26\!\cdots\!77}a^{5}-\frac{55\!\cdots\!65}{26\!\cdots\!77}a^{4}+\frac{24\!\cdots\!73}{26\!\cdots\!77}a^{3}-\frac{63\!\cdots\!44}{26\!\cdots\!77}a^{2}+\frac{11\!\cdots\!43}{26\!\cdots\!77}a-\frac{65\!\cdots\!08}{20\!\cdots\!67}$ (assuming GRH)
Regulator:		\( 85915831770.81862 \) (assuming GRH)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{15}\cdot 85915831770.81862 \cdot 17701}{6\cdot\sqrt{2686026609350084707957118496751488894605397847668243387}}\cr\approx \mathstrut & 0.145232101178119 \end{aligned}\] (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 $C_{30}$

Intermediate fields

\(\Q(\sqrt{-3}) \), 3.3.169.1, \(\Q(\zeta_{11})^+\), 6.0.771147.1, 10.0.52089208083.1, 15.15.432659002790862279847129.1

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	$30$	R	${\href{/padicField/5.10.0.1}{10} }^{3}$	$15^{2}$	R	R	$30$	$15^{2}$	${\href{/padicField/23.6.0.1}{6} }^{5}$	$30$	${\href{/padicField/31.5.0.1}{5} }^{6}$	$15^{2}$	$30$	${\href{/padicField/43.3.0.1}{3} }^{10}$	${\href{/padicField/47.10.0.1}{10} }^{3}$	${\href{/padicField/53.10.0.1}{10} }^{3}$	$30$

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 $30$	$2$	$15$	$15$
\(11\)		Deg $30$	$5$	$6$	$24$
\(13\)	13.15.10.1	$x^{15} + 65 x^{12} + 12 x^{11} + 33 x^{10} + 1690 x^{9} - 2340 x^{8} - 12822 x^{7} + 22234 x^{6} - 11805 x^{5} + 254085 x^{4} + 160029 x^{3} + 311358 x^{2} - 328605 x + 351148$	$3$	$5$	$10$	$C_{15}$	$[\ ]_{3}^{5}$
	13.15.10.1	$x^{15} + 65 x^{12} + 12 x^{11} + 33 x^{10} + 1690 x^{9} - 2340 x^{8} - 12822 x^{7} + 22234 x^{6} - 11805 x^{5} + 254085 x^{4} + 160029 x^{3} + 311358 x^{2} - 328605 x + 351148$	$3$	$5$	$10$	$C_{15}$	$[\ ]_{3}^{5}$