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: | $30$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[0, 15]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(-2686026609350084707957118496751488894605397847668243387=-\,3^{15}\cdot 11^{24}\cdot 13^{20}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $65.21$ | magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
| |
| Ramified primes: | $3, 11, 13$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| 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. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $\frac{1}{131} a^{27} + \frac{49}{131} a^{26} - \frac{65}{131} a^{25} - \frac{52}{131} a^{24} + \frac{36}{131} a^{23} + \frac{52}{131} a^{22} - \frac{58}{131} a^{21} - \frac{35}{131} a^{20} - \frac{40}{131} a^{19} - \frac{18}{131} a^{18} - \frac{65}{131} a^{17} - \frac{17}{131} a^{16} + \frac{55}{131} a^{15} + \frac{30}{131} a^{14} - \frac{32}{131} a^{13} - \frac{24}{131} a^{12} - \frac{21}{131} a^{11} + \frac{53}{131} a^{10} + \frac{64}{131} a^{8} + \frac{50}{131} a^{6} - \frac{3}{131} a^{5} + \frac{11}{131} a^{4} + \frac{56}{131} a^{3} + \frac{3}{131} a^{2} + \frac{41}{131} a$, $\frac{1}{210473000101397461} a^{28} - \frac{112904653830712}{210473000101397461} a^{27} - \frac{4594183061072836}{210473000101397461} a^{26} + \frac{369613443805360}{210473000101397461} a^{25} - \frac{51944919284565778}{210473000101397461} a^{24} + \frac{3162248301381065}{210473000101397461} a^{23} - \frac{33636060190435089}{210473000101397461} a^{22} + \frac{96579084743881953}{210473000101397461} a^{21} + \frac{61938317467544829}{210473000101397461} a^{20} + \frac{61859379501610163}{210473000101397461} a^{19} - \frac{78037293277244966}{210473000101397461} a^{18} + \frac{60704106366924137}{210473000101397461} a^{17} + \frac{16450418908788841}{210473000101397461} a^{16} - \frac{82876244700337247}{210473000101397461} a^{15} - \frac{102238288256164672}{210473000101397461} a^{14} - \frac{81410474071149467}{210473000101397461} a^{13} - \frac{90416821625932054}{210473000101397461} a^{12} - \frac{1434701079829}{130972619851523} a^{11} + \frac{54963839942588244}{210473000101397461} a^{10} + \frac{81290594394817924}{210473000101397461} a^{9} - \frac{56658039766186927}{210473000101397461} a^{8} - \frac{12003694762518993}{210473000101397461} a^{7} - \frac{77380043567695547}{210473000101397461} a^{6} + \frac{91582069377107281}{210473000101397461} a^{5} + \frac{94566548794182622}{210473000101397461} a^{4} + \frac{63349107622336732}{210473000101397461} a^{3} + \frac{66617390284176294}{210473000101397461} a^{2} - \frac{13643151895312695}{210473000101397461} a - \frac{161293452768987}{1606664122911431}$, $\frac{1}{266269805964531330356962610553930177350379512036381119606953250289223685724320381278170333137177} a^{29} - \frac{71965518525714196667703113545156059969714190069018481928707689011712163542750}{266269805964531330356962610553930177350379512036381119606953250289223685724320381278170333137177} a^{28} - \frac{221111594527188337135557422954571747690394975421778598607711945416831272208569192648923358558}{266269805964531330356962610553930177350379512036381119606953250289223685724320381278170333137177} a^{27} - \frac{55968464434249667240949167316246646912478156840468682355090700838299202303054175040795268789140}{266269805964531330356962610553930177350379512036381119606953250289223685724320381278170333137177} a^{26} - \frac{60609263416897624723970605516077358818453224745045686079631600542126128986571084655010562006435}{266269805964531330356962610553930177350379512036381119606953250289223685724320381278170333137177} a^{25} - \frac{126510397232877925890525270230138015364670307163303232852727535890363104109862388122436181632564}{266269805964531330356962610553930177350379512036381119606953250289223685724320381278170333137177} a^{24} - \frac{100345011865174692236525021391668477313347435189048911973235371339870654488960022076513882591487}{266269805964531330356962610553930177350379512036381119606953250289223685724320381278170333137177} a^{23} - \frac{44188955117992667240541606416512690810020373251340960512298008186267744125097207353569471026577}{266269805964531330356962610553930177350379512036381119606953250289223685724320381278170333137177} a^{22} + \frac{10237478705569409090672962958113279829802404217474342630332546129881590221079897924856268678791}{266269805964531330356962610553930177350379512036381119606953250289223685724320381278170333137177} a^{21} - \frac{21810039379169451507804630529288623418613972951567759758414987244753870050541203398965171780542}{266269805964531330356962610553930177350379512036381119606953250289223685724320381278170333137177} a^{20} + \frac{69451315866373466627436608002688805614172770143257372352910673266963558925774065433578045939948}{266269805964531330356962610553930177350379512036381119606953250289223685724320381278170333137177} a^{19} - \frac{1967605594044433922109462430725123418606225101724763212369044958651507932400281374589300899240}{266269805964531330356962610553930177350379512036381119606953250289223685724320381278170333137177} a^{18} - \frac{84821118223512811041416378521666699031254474675147297707607014260577002591000019861536265759897}{266269805964531330356962610553930177350379512036381119606953250289223685724320381278170333137177} a^{17} - \frac{24513470260705541081769052259609669206210811364921008850344022467960626572688349645418694026704}{266269805964531330356962610553930177350379512036381119606953250289223685724320381278170333137177} a^{16} - \frac{48146356881040930199623471377926668309279539013702112205586262057557155246294182987564524330332}{266269805964531330356962610553930177350379512036381119606953250289223685724320381278170333137177} a^{15} + \frac{114628902370261169308612665522117247088762249400612495159649737851211411952607647824713181755475}{266269805964531330356962610553930177350379512036381119606953250289223685724320381278170333137177} a^{14} - \frac{57330207513052841727123156614381262245792979078902213573464910569076089682661472238516924773915}{266269805964531330356962610553930177350379512036381119606953250289223685724320381278170333137177} a^{13} + \frac{28934623235121404638896223080287100397657707079505904765241419306758283990310812403374022774398}{266269805964531330356962610553930177350379512036381119606953250289223685724320381278170333137177} a^{12} - \frac{131175524989288697145243173928996197294967778589293807502816024062394733953374106690910568922506}{266269805964531330356962610553930177350379512036381119606953250289223685724320381278170333137177} a^{11} + \frac{11578567781238915911088371315348389506678234395929078290294037355133957837717491570839456276214}{266269805964531330356962610553930177350379512036381119606953250289223685724320381278170333137177} a^{10} + \frac{44825089514389111611140469833114648196287995929635394952933014609748905635512446404080033123554}{266269805964531330356962610553930177350379512036381119606953250289223685724320381278170333137177} a^{9} + \frac{42615470809783026017530505103722707859720494608036689525606558538278864759458646892711938234922}{266269805964531330356962610553930177350379512036381119606953250289223685724320381278170333137177} a^{8} + \frac{61503753519807173959274682742430853956828961584683550054538871143988449315278951647133654062172}{266269805964531330356962610553930177350379512036381119606953250289223685724320381278170333137177} a^{7} + \frac{15590609898776302148932970931271666004560912569865083784432129549446899982468766554696782094728}{266269805964531330356962610553930177350379512036381119606953250289223685724320381278170333137177} a^{6} + \frac{2864919800262071256657178892273206742521717355576327770956856777862781737160220639562051153423}{266269805964531330356962610553930177350379512036381119606953250289223685724320381278170333137177} a^{5} - \frac{108100708498923409416924615100248159586782460967481698400448202606322885167657144657099330738137}{266269805964531330356962610553930177350379512036381119606953250289223685724320381278170333137177} a^{4} + \frac{114355786591909572586495264880362085497340926088085911720935437895773598314706583369341639360010}{266269805964531330356962610553930177350379512036381119606953250289223685724320381278170333137177} a^{3} - \frac{99368082018730480982587294993106578515605786380402865198289237310678302207967235423756753272218}{266269805964531330356962610553930177350379512036381119606953250289223685724320381278170333137177} a^{2} + \frac{73484104441079833934146103270354871901193317917456273650039473921754444311473890671354439861993}{266269805964531330356962610553930177350379512036381119606953250289223685724320381278170333137177} a - \frac{877459077975417851315274144121510378157090529306011420809765372342318943968900413237733605743}{2032593938660544506541699317205573872903660397224283355778269086177280043697102147161605596467}$
Class group and class number
$C_{17701}$, which has order $17701$ (assuming GRH)
Unit group
| Rank: | $14$ | magma: UnitRank(K);
sage: UK.rank()
gp: K.fu
| |
| 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 $6$) | magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
sage: UK.torsion_generator()
gp: K.tu[2]
| |
| Fundamental units: | Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH) | magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
| |
| Regulator: | \( 85915831770.81862 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A cyclic group of order 30 |
| The 30 conjugacy class representatives for $C_{30}$ |
| Character table for $C_{30}$ is not computed |
Intermediate fields
| \(\Q(\sqrt{-3}) \), 3.3.169.1, \(\Q(\zeta_{11})^+\), 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 | ${\href{/LocalNumberField/5.10.0.1}{10} }^{3}$ | $15^{2}$ | R | R | $30$ | $15^{2}$ | ${\href{/LocalNumberField/23.6.0.1}{6} }^{5}$ | $30$ | ${\href{/LocalNumberField/31.5.0.1}{5} }^{6}$ | $15^{2}$ | $30$ | ${\href{/LocalNumberField/43.3.0.1}{3} }^{10}$ | ${\href{/LocalNumberField/47.10.0.1}{10} }^{3}$ | ${\href{/LocalNumberField/53.10.0.1}{10} }^{3}$ | $30$ |
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 | Data not computed | ||||||
| 11 | Data not computed | ||||||
| $13$ | 13.15.10.1 | $x^{15} + 79092 x^{6} - 228488 x^{3} + 80199288$ | $3$ | $5$ | $10$ | $C_{15}$ | $[\ ]_{3}^{5}$ |
| 13.15.10.1 | $x^{15} + 79092 x^{6} - 228488 x^{3} + 80199288$ | $3$ | $5$ | $10$ | $C_{15}$ | $[\ ]_{3}^{5}$ | |