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Properties

Label 36.0.550...744.1
Degree $36$
Signature $[0, 18]$
Discriminant $5.502\times 10^{60}$
Root discriminant \(48.67\)
Ramified primes $2,3,7,199$
Class number not computed
Class group not computed
Galois group $C_2\times A_4\times D_6$ (as 36T334)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^36 - 12*x^35 + 117*x^34 - 801*x^33 + 4695*x^32 - 22944*x^31 + 99156*x^30 - 377772*x^29 + 1300485*x^28 - 4063386*x^27 + 11677437*x^26 - 31091397*x^25 + 77290173*x^24 - 180440796*x^23 + 396698646*x^22 - 823211451*x^21 + 1614415197*x^20 - 3001125603*x^19 + 5316812738*x^18 - 9023396952*x^17 + 14711412132*x^16 - 22930637760*x^15 + 33755151003*x^14 - 46228515924*x^13 + 58054312566*x^12 - 66243228153*x^11 + 68283850818*x^10 - 62712669783*x^9 + 49993043427*x^8 - 33497730303*x^7 + 18223735458*x^6 - 7755245535*x^5 + 2488299969*x^4 - 581944761*x^3 + 94855845*x^2 - 9757218*x + 507547)
 
gp: K = bnfinit(y^36 - 12*y^35 + 117*y^34 - 801*y^33 + 4695*y^32 - 22944*y^31 + 99156*y^30 - 377772*y^29 + 1300485*y^28 - 4063386*y^27 + 11677437*y^26 - 31091397*y^25 + 77290173*y^24 - 180440796*y^23 + 396698646*y^22 - 823211451*y^21 + 1614415197*y^20 - 3001125603*y^19 + 5316812738*y^18 - 9023396952*y^17 + 14711412132*y^16 - 22930637760*y^15 + 33755151003*y^14 - 46228515924*y^13 + 58054312566*y^12 - 66243228153*y^11 + 68283850818*y^10 - 62712669783*y^9 + 49993043427*y^8 - 33497730303*y^7 + 18223735458*y^6 - 7755245535*y^5 + 2488299969*y^4 - 581944761*y^3 + 94855845*y^2 - 9757218*y + 507547, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^36 - 12*x^35 + 117*x^34 - 801*x^33 + 4695*x^32 - 22944*x^31 + 99156*x^30 - 377772*x^29 + 1300485*x^28 - 4063386*x^27 + 11677437*x^26 - 31091397*x^25 + 77290173*x^24 - 180440796*x^23 + 396698646*x^22 - 823211451*x^21 + 1614415197*x^20 - 3001125603*x^19 + 5316812738*x^18 - 9023396952*x^17 + 14711412132*x^16 - 22930637760*x^15 + 33755151003*x^14 - 46228515924*x^13 + 58054312566*x^12 - 66243228153*x^11 + 68283850818*x^10 - 62712669783*x^9 + 49993043427*x^8 - 33497730303*x^7 + 18223735458*x^6 - 7755245535*x^5 + 2488299969*x^4 - 581944761*x^3 + 94855845*x^2 - 9757218*x + 507547);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^36 - 12*x^35 + 117*x^34 - 801*x^33 + 4695*x^32 - 22944*x^31 + 99156*x^30 - 377772*x^29 + 1300485*x^28 - 4063386*x^27 + 11677437*x^26 - 31091397*x^25 + 77290173*x^24 - 180440796*x^23 + 396698646*x^22 - 823211451*x^21 + 1614415197*x^20 - 3001125603*x^19 + 5316812738*x^18 - 9023396952*x^17 + 14711412132*x^16 - 22930637760*x^15 + 33755151003*x^14 - 46228515924*x^13 + 58054312566*x^12 - 66243228153*x^11 + 68283850818*x^10 - 62712669783*x^9 + 49993043427*x^8 - 33497730303*x^7 + 18223735458*x^6 - 7755245535*x^5 + 2488299969*x^4 - 581944761*x^3 + 94855845*x^2 - 9757218*x + 507547)
 

\( x^{36} - 12 x^{35} + 117 x^{34} - 801 x^{33} + 4695 x^{32} - 22944 x^{31} + 99156 x^{30} - 377772 x^{29} + \cdots + 507547 \) Copy content Toggle raw display

sage: K.defining_polynomial()
 
gp: K.pol
 
magma: DefiningPolynomial(K);
 
oscar: defining_polynomial(K)
 

Invariants

Degree:  $36$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[0, 18]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(5502152549065142937178823334131582447169322335358004697759744\) \(\medspace = 2^{24}\cdot 3^{62}\cdot 7^{12}\cdot 199^{6}\) Copy content Toggle raw display
sage: K.disc()
 
gp: K.disc
 
magma: OK := Integers(K); Discriminant(OK);
 
oscar: OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(48.67\)
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(OK))^(1/Degree(K));
 
oscar: (1.0 * dK)^(1/degree(K))
 
Galois root discriminant:  $2^{2/3}3^{31/18}7^{1/2}199^{1/2}\approx 392.98054405162793$
Ramified primes:   \(2\), \(3\), \(7\), \(199\) Copy content Toggle raw display
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(OK));
 
oscar: prime_divisors(discriminant((OK)))
 
Discriminant root field:  \(\Q\)
$\card{ \Aut(K/\Q) }$:  $4$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is not Galois over $\Q$.
This is a CM field.
Reflex fields:  unavailable$^{131072}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $\frac{1}{19}a^{32}-\frac{6}{19}a^{31}-\frac{8}{19}a^{30}+\frac{1}{19}a^{29}-\frac{2}{19}a^{28}+\frac{6}{19}a^{27}+\frac{6}{19}a^{26}-\frac{8}{19}a^{25}+\frac{3}{19}a^{24}-\frac{9}{19}a^{23}-\frac{5}{19}a^{22}-\frac{3}{19}a^{21}-\frac{4}{19}a^{19}-\frac{2}{19}a^{17}-\frac{5}{19}a^{16}-\frac{9}{19}a^{15}-\frac{6}{19}a^{14}+\frac{9}{19}a^{13}+\frac{7}{19}a^{12}-\frac{8}{19}a^{11}-\frac{9}{19}a^{10}-\frac{6}{19}a^{9}-\frac{4}{19}a^{8}-\frac{8}{19}a^{7}+\frac{9}{19}a^{6}+\frac{7}{19}a^{5}-\frac{4}{19}a^{4}+\frac{6}{19}a^{3}-\frac{7}{19}a^{2}+\frac{1}{19}a$, $\frac{1}{19}a^{33}-\frac{6}{19}a^{31}-\frac{9}{19}a^{30}+\frac{4}{19}a^{29}-\frac{6}{19}a^{28}+\frac{4}{19}a^{27}+\frac{9}{19}a^{26}-\frac{7}{19}a^{25}+\frac{9}{19}a^{24}-\frac{2}{19}a^{23}+\frac{5}{19}a^{22}+\frac{1}{19}a^{21}-\frac{4}{19}a^{20}-\frac{5}{19}a^{19}-\frac{2}{19}a^{18}+\frac{2}{19}a^{17}-\frac{1}{19}a^{16}-\frac{3}{19}a^{15}-\frac{8}{19}a^{14}+\frac{4}{19}a^{13}-\frac{4}{19}a^{12}-\frac{3}{19}a^{10}-\frac{2}{19}a^{9}+\frac{6}{19}a^{8}-\frac{1}{19}a^{7}+\frac{4}{19}a^{6}+\frac{1}{19}a^{4}-\frac{9}{19}a^{3}-\frac{3}{19}a^{2}+\frac{6}{19}a$, $\frac{1}{22580911}a^{34}-\frac{506682}{22580911}a^{33}-\frac{391654}{22580911}a^{32}-\frac{18672}{318041}a^{31}-\frac{5559326}{22580911}a^{30}+\frac{4710209}{22580911}a^{29}-\frac{3340501}{22580911}a^{28}-\frac{10766535}{22580911}a^{27}+\frac{10982780}{22580911}a^{26}-\frac{5642065}{22580911}a^{25}+\frac{6396796}{22580911}a^{24}-\frac{3552531}{22580911}a^{23}+\frac{3727970}{22580911}a^{22}-\frac{2370906}{22580911}a^{21}-\frac{4762167}{22580911}a^{20}-\frac{10624905}{22580911}a^{19}+\frac{5253748}{22580911}a^{18}+\frac{970009}{22580911}a^{17}-\frac{4491874}{22580911}a^{16}+\frac{8525575}{22580911}a^{15}+\frac{548840}{22580911}a^{14}+\frac{2987891}{22580911}a^{13}+\frac{1688673}{22580911}a^{12}-\frac{4473412}{22580911}a^{11}-\frac{4578282}{22580911}a^{10}+\frac{1692854}{22580911}a^{9}-\frac{5293603}{22580911}a^{8}+\frac{6790640}{22580911}a^{7}+\frac{7185793}{22580911}a^{6}-\frac{1777266}{22580911}a^{5}+\frac{8387479}{22580911}a^{4}-\frac{6278858}{22580911}a^{3}+\frac{2629279}{22580911}a^{2}+\frac{8381605}{22580911}a+\frac{569819}{1188469}$, $\frac{1}{65\!\cdots\!37}a^{35}-\frac{13\!\cdots\!76}{65\!\cdots\!37}a^{34}-\frac{12\!\cdots\!93}{65\!\cdots\!37}a^{33}-\frac{11\!\cdots\!13}{65\!\cdots\!37}a^{32}-\frac{21\!\cdots\!88}{65\!\cdots\!37}a^{31}+\frac{12\!\cdots\!11}{65\!\cdots\!37}a^{30}-\frac{27\!\cdots\!86}{65\!\cdots\!37}a^{29}-\frac{41\!\cdots\!81}{65\!\cdots\!37}a^{28}-\frac{11\!\cdots\!92}{65\!\cdots\!37}a^{27}+\frac{24\!\cdots\!72}{65\!\cdots\!37}a^{26}+\frac{30\!\cdots\!16}{65\!\cdots\!37}a^{25}-\frac{65\!\cdots\!59}{65\!\cdots\!37}a^{24}-\frac{82\!\cdots\!26}{65\!\cdots\!37}a^{23}-\frac{63\!\cdots\!90}{65\!\cdots\!37}a^{22}-\frac{11\!\cdots\!15}{74\!\cdots\!77}a^{21}-\frac{28\!\cdots\!77}{65\!\cdots\!37}a^{20}-\frac{27\!\cdots\!31}{65\!\cdots\!37}a^{19}+\frac{22\!\cdots\!94}{65\!\cdots\!37}a^{18}-\frac{84\!\cdots\!57}{65\!\cdots\!37}a^{17}-\frac{21\!\cdots\!89}{65\!\cdots\!37}a^{16}-\frac{64\!\cdots\!03}{65\!\cdots\!37}a^{15}-\frac{24\!\cdots\!18}{65\!\cdots\!37}a^{14}-\frac{10\!\cdots\!40}{65\!\cdots\!37}a^{13}+\frac{21\!\cdots\!18}{92\!\cdots\!47}a^{12}+\frac{13\!\cdots\!61}{18\!\cdots\!17}a^{11}+\frac{20\!\cdots\!71}{65\!\cdots\!37}a^{10}+\frac{19\!\cdots\!97}{65\!\cdots\!37}a^{9}-\frac{30\!\cdots\!94}{65\!\cdots\!37}a^{8}-\frac{33\!\cdots\!27}{65\!\cdots\!37}a^{7}+\frac{20\!\cdots\!80}{65\!\cdots\!37}a^{6}+\frac{10\!\cdots\!00}{65\!\cdots\!37}a^{5}-\frac{56\!\cdots\!80}{34\!\cdots\!23}a^{4}-\frac{28\!\cdots\!15}{65\!\cdots\!37}a^{3}+\frac{26\!\cdots\!01}{65\!\cdots\!37}a^{2}+\frac{31\!\cdots\!63}{65\!\cdots\!37}a+\frac{15\!\cdots\!58}{34\!\cdots\!23}$ Copy content Toggle raw display

sage: K.integral_basis()
 
gp: K.zk
 
magma: IntegralBasis(K);
 
oscar: basis(OK)
 

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

Class group and class number

not computed

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 
oscar: class_group(K)
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, fUK := UnitGroup(K);
 
oscar: UK, fUK = unit_group(OK)
 
Rank:  $17$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
oscar: rank(UK)
 
Torsion generator:   \( \frac{7783312265655205090344162447766778752472757767811605063109051650535419662296796521829387493576983779148567390994947718277217206870994895898}{100722113648322369600779407518555677352633942708063557457425633101195464816660895133445753716959743785647947008478301629356103356140617805102657} a^{35} - \frac{88211474806731679429013690113024746207915445080629995187714748513434157045144872288930610504092007633188015965857074256712013301584667071335}{100722113648322369600779407518555677352633942708063557457425633101195464816660895133445753716959743785647947008478301629356103356140617805102657} a^{34} + \frac{851997152300305085772175740934500613587148363771697539075248553994907845098695233056122581892104435762643871892104248498090258219098136087985}{100722113648322369600779407518555677352633942708063557457425633101195464816660895133445753716959743785647947008478301629356103356140617805102657} a^{33} - \frac{298321391947553327852238491326523408034239000744126527169252383559405525126159904120098338663360239039715209219042400401027161276808582309520}{5301163876227493136883126711502930386980733826740187234601349110589234990350573428076092300892618093981470895183068506808215966112664095005403} a^{32} + \frac{32779813369748387616719101201455936101459384056747766511457413891350873403786924970714897436458555923982075678173890392207304419031215836603044}{100722113648322369600779407518555677352633942708063557457425633101195464816660895133445753716959743785647947008478301629356103356140617805102657} a^{31} - \frac{156829546346431867995473702252001212981752449283163872647798908481972114572912471296384446515058715465657704412429069338907672542007025605125358}{100722113648322369600779407518555677352633942708063557457425633101195464816660895133445753716959743785647947008478301629356103356140617805102657} a^{30} + \frac{667794218400111457373081510717421232436250045289915972609381167352845929235515412800695483899902865250843470721227373001953776489308708070059642}{100722113648322369600779407518555677352633942708063557457425633101195464816660895133445753716959743785647947008478301629356103356140617805102657} a^{29} - \frac{2497860145881251386380231900197134167158762874739788822938655569021656869570579676231441703854089475648266713810121793078010185955761956652077625}{100722113648322369600779407518555677352633942708063557457425633101195464816660895133445753716959743785647947008478301629356103356140617805102657} a^{28} + \frac{8468362940735839820384670306688046924727016526215532592195687415751764764287894308158196614467767021004461771059739419502173317992312143339790930}{100722113648322369600779407518555677352633942708063557457425633101195464816660895133445753716959743785647947008478301629356103356140617805102657} a^{27} - \frac{26023388521225560454941079589815980596194083386350742793208618520307993990246958886100622434284754211066293187656760376534053324110938777949940491}{100722113648322369600779407518555677352633942708063557457425633101195464816660895133445753716959743785647947008478301629356103356140617805102657} a^{26} + \frac{73682167085972862638158450488293797436361162140916436683800566290176749435381647182658761346347030531925212396402631469817220084533818461172510711}{100722113648322369600779407518555677352633942708063557457425633101195464816660895133445753716959743785647947008478301629356103356140617805102657} a^{25} - \frac{193302726430180938955683924648665686085783937542238256877178195533461872560510869979132662565704328756021792076621861830726992082171721349556977768}{100722113648322369600779407518555677352633942708063557457425633101195464816660895133445753716959743785647947008478301629356103356140617805102657} a^{24} + \frac{6674788146252765275441397372607180609718310916838398651767734756920993978400510426241115372746624970706954107772681404554167904223392861350302444}{1418621318990455909870132500261347568346956939550190950104586381706978377699449227231630334041686532192224605753215515906423990931557997254967} a^{23} - \frac{1091612362051142360159458816477442340030241914826587780200471007751935705103823881790184962722383210628917320414936096855391667339064496248727353754}{100722113648322369600779407518555677352633942708063557457425633101195464816660895133445753716959743785647947008478301629356103356140617805102657} a^{22} + \frac{2367476507810127928147964004817508864279752471776370228413329224369191225303732555659833150947560482833701519364787811617466932869960867271105284879}{100722113648322369600779407518555677352633942708063557457425633101195464816660895133445753716959743785647947008478301629356103356140617805102657} a^{21} - \frac{4846307509986056235463678601063384199904531490764136275972681729793024526424289288865149005944464293406914942676040507768486854410722538053798858693}{100722113648322369600779407518555677352633942708063557457425633101195464816660895133445753716959743785647947008478301629356103356140617805102657} a^{20} + \frac{9371823611663220647664829677945582601345553188638268379503938094501484705745387330227911484394200951486715081849312736106170140051720667692948356063}{100722113648322369600779407518555677352633942708063557457425633101195464816660895133445753716959743785647947008478301629356103356140617805102657} a^{19} - \frac{17186267900951241090337306358003940256954885108536556241239251967060324906081976641907757463182981061608220585997633328890796180136214074018695055104}{100722113648322369600779407518555677352633942708063557457425633101195464816660895133445753716959743785647947008478301629356103356140617805102657} a^{18} + \frac{30069465406543681873555272924903946320246539359693817003557340859499392745817062605074489763801667124192387339717513288068900174604935317561536593884}{100722113648322369600779407518555677352633942708063557457425633101195464816660895133445753716959743785647947008478301629356103356140617805102657} a^{17} - \frac{50448517183690173052404748110208199061636713271979647552333653180006034594993595787257461794241401122709914930181537954184089291154363156192815900870}{100722113648322369600779407518555677352633942708063557457425633101195464816660895133445753716959743785647947008478301629356103356140617805102657} a^{16} + \frac{81327795305570058571797676383570668137802650030681044985043198772998645641690528460240233426819054151478712434384357757021568074582059126413901013978}{100722113648322369600779407518555677352633942708063557457425633101195464816660895133445753716959743785647947008478301629356103356140617805102657} a^{15} - \frac{125018749356453499772966391206916264981274854142038816566161508729529734894883146271054781937559653120077871079441335157679274984996258277891688990035}{100722113648322369600779407518555677352633942708063557457425633101195464816660895133445753716959743785647947008478301629356103356140617805102657} a^{14} + \frac{180599498473430707957793721858724629143776608397419496562755302780406571506799942154731787041542133585661009523966936663109865553315310733192037308120}{100722113648322369600779407518555677352633942708063557457425633101195464816660895133445753716959743785647947008478301629356103356140617805102657} a^{13} - \frac{241266889181653050706087706992061972088592029852683610144655500165360696445102122509189551777960467418505471583945812760773784232434263926521991367183}{100722113648322369600779407518555677352633942708063557457425633101195464816660895133445753716959743785647947008478301629356103356140617805102657} a^{12} + \frac{293658240234707437802391834273175041334493625624004795501934128672256514710745774637416276287328873936192748644471717193173040700791796170306309594359}{100722113648322369600779407518555677352633942708063557457425633101195464816660895133445753716959743785647947008478301629356103356140617805102657} a^{11} - \frac{323303448058996919482337085602371396302378429528470445473018850004187643483934070389051258862698927643701211317105623001373416889730527047074205182366}{100722113648322369600779407518555677352633942708063557457425633101195464816660895133445753716959743785647947008478301629356103356140617805102657} a^{10} + \frac{320104421424786080368930787204341236573184684652832557073320019608106662905435507447958073510429804332580638347824015901516090842955401794287895072567}{100722113648322369600779407518555677352633942708063557457425633101195464816660895133445753716959743785647947008478301629356103356140617805102657} a^{9} - \frac{279214445894608518219683156160123211878731960488291087794580570698809724762265071905231376029513509548436433694595452343926957685469075927019648369709}{100722113648322369600779407518555677352633942708063557457425633101195464816660895133445753716959743785647947008478301629356103356140617805102657} a^{8} + \frac{207340620761897970754296737987362234579273976442084696436183332352269366966008595842312998828997326466722286148131435073458390809420415253974739577171}{100722113648322369600779407518555677352633942708063557457425633101195464816660895133445753716959743785647947008478301629356103356140617805102657} a^{7} - \frac{6642642242796908044615693341414234821950463743674298245532932605371941463102453613639310757245310311452950158881407168104993433051665345543754138758}{5301163876227493136883126711502930386980733826740187234601349110589234990350573428076092300892618093981470895183068506808215966112664095005403} a^{6} + \frac{60371351515964625599925854884810063686861786481111240310896689244676489821907738777587119543813438905673644830221161167646651867996720637098868300315}{100722113648322369600779407518555677352633942708063557457425633101195464816660895133445753716959743785647947008478301629356103356140617805102657} a^{5} - \frac{21679576817892804332851994906040895195871266752443316330833669057483504654523415076453695909996370919406870663252975585861312070403916595090306464222}{100722113648322369600779407518555677352633942708063557457425633101195464816660895133445753716959743785647947008478301629356103356140617805102657} a^{4} + \frac{296378524306425211971865604924542138225759236634658350456754279088218275622833236824584328649184073376997044420475927902661627636395257505406521577}{5301163876227493136883126711502930386980733826740187234601349110589234990350573428076092300892618093981470895183068506808215966112664095005403} a^{3} - \frac{1020422704527609558866027879138964222072970221748779800759450281799805895177535627087472940359549067419310707897098105492725469695193417689682265581}{100722113648322369600779407518555677352633942708063557457425633101195464816660895133445753716959743785647947008478301629356103356140617805102657} a^{2} + \frac{116727490943432331007291615500078275994558233764038393496146175368675571959807760274608664518788717735194025079148489225431827466783696762279520230}{100722113648322369600779407518555677352633942708063557457425633101195464816660895133445753716959743785647947008478301629356103356140617805102657} a - \frac{364446096993364198385879142121879790033961736419784957889810342156422997085834782659570528880513493771294243571275561294269372067542213149383188}{5301163876227493136883126711502930386980733826740187234601349110589234990350573428076092300892618093981470895183068506808215966112664095005403} \)  (order $18$) Copy content Toggle raw display
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
oscar: torsion_units_generator(OK)
 
Fundamental units:  not computed
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K|fUK(g): g in Generators(UK)];
 
oscar: [K(fUK(a)) for a in gens(UK)]
 
Regulator:  not computed
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr $ not computed \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^36 - 12*x^35 + 117*x^34 - 801*x^33 + 4695*x^32 - 22944*x^31 + 99156*x^30 - 377772*x^29 + 1300485*x^28 - 4063386*x^27 + 11677437*x^26 - 31091397*x^25 + 77290173*x^24 - 180440796*x^23 + 396698646*x^22 - 823211451*x^21 + 1614415197*x^20 - 3001125603*x^19 + 5316812738*x^18 - 9023396952*x^17 + 14711412132*x^16 - 22930637760*x^15 + 33755151003*x^14 - 46228515924*x^13 + 58054312566*x^12 - 66243228153*x^11 + 68283850818*x^10 - 62712669783*x^9 + 49993043427*x^8 - 33497730303*x^7 + 18223735458*x^6 - 7755245535*x^5 + 2488299969*x^4 - 581944761*x^3 + 94855845*x^2 - 9757218*x + 507547)
 
DK = K.disc(); r1,r2 = K.signature(); RK = K.regulator(); RR = RK.parent()
 
hK = K.class_number(); wK = K.unit_group().torsion_generator().order();
 
2^r1 * (2*RR(pi))^r2 * RK * hK / (wK * RR(sqrt(abs(DK))))
 
# self-contained Pari/GP code snippet to compute the analytic class number formula
 
K = bnfinit(x^36 - 12*x^35 + 117*x^34 - 801*x^33 + 4695*x^32 - 22944*x^31 + 99156*x^30 - 377772*x^29 + 1300485*x^28 - 4063386*x^27 + 11677437*x^26 - 31091397*x^25 + 77290173*x^24 - 180440796*x^23 + 396698646*x^22 - 823211451*x^21 + 1614415197*x^20 - 3001125603*x^19 + 5316812738*x^18 - 9023396952*x^17 + 14711412132*x^16 - 22930637760*x^15 + 33755151003*x^14 - 46228515924*x^13 + 58054312566*x^12 - 66243228153*x^11 + 68283850818*x^10 - 62712669783*x^9 + 49993043427*x^8 - 33497730303*x^7 + 18223735458*x^6 - 7755245535*x^5 + 2488299969*x^4 - 581944761*x^3 + 94855845*x^2 - 9757218*x + 507547, 1);
 
[polcoeff (lfunrootres (lfuncreate (K))[1][1][2], -1), 2^K.r1 * (2*Pi)^K.r2 * K.reg * K.no / (K.tu[1] * sqrt (abs (K.disc)))]
 
/* self-contained Magma code snippet to compute the analytic class number formula */
 
Qx<x> := PolynomialRing(QQ); K<a> := NumberField(x^36 - 12*x^35 + 117*x^34 - 801*x^33 + 4695*x^32 - 22944*x^31 + 99156*x^30 - 377772*x^29 + 1300485*x^28 - 4063386*x^27 + 11677437*x^26 - 31091397*x^25 + 77290173*x^24 - 180440796*x^23 + 396698646*x^22 - 823211451*x^21 + 1614415197*x^20 - 3001125603*x^19 + 5316812738*x^18 - 9023396952*x^17 + 14711412132*x^16 - 22930637760*x^15 + 33755151003*x^14 - 46228515924*x^13 + 58054312566*x^12 - 66243228153*x^11 + 68283850818*x^10 - 62712669783*x^9 + 49993043427*x^8 - 33497730303*x^7 + 18223735458*x^6 - 7755245535*x^5 + 2488299969*x^4 - 581944761*x^3 + 94855845*x^2 - 9757218*x + 507547);
 
OK := Integers(K); DK := Discriminant(OK);
 
UK, fUK := UnitGroup(OK); clK, fclK := ClassGroup(OK);
 
r1,r2 := Signature(K); RK := Regulator(K); RR := Parent(RK);
 
hK := #clK; wK := #TorsionSubgroup(UK);
 
2^r1 * (2*Pi(RR))^r2 * RK * hK / (wK * Sqrt(RR!Abs(DK)));
 
# self-contained Oscar code snippet to compute the analytic class number formula
 
Qx, x = PolynomialRing(QQ); K, a = NumberField(x^36 - 12*x^35 + 117*x^34 - 801*x^33 + 4695*x^32 - 22944*x^31 + 99156*x^30 - 377772*x^29 + 1300485*x^28 - 4063386*x^27 + 11677437*x^26 - 31091397*x^25 + 77290173*x^24 - 180440796*x^23 + 396698646*x^22 - 823211451*x^21 + 1614415197*x^20 - 3001125603*x^19 + 5316812738*x^18 - 9023396952*x^17 + 14711412132*x^16 - 22930637760*x^15 + 33755151003*x^14 - 46228515924*x^13 + 58054312566*x^12 - 66243228153*x^11 + 68283850818*x^10 - 62712669783*x^9 + 49993043427*x^8 - 33497730303*x^7 + 18223735458*x^6 - 7755245535*x^5 + 2488299969*x^4 - 581944761*x^3 + 94855845*x^2 - 9757218*x + 507547);
 
OK = ring_of_integers(K); DK = discriminant(OK);
 
UK, fUK = unit_group(OK); clK, fclK = class_group(OK);
 
r1,r2 = signature(K); RK = regulator(K); RR = parent(RK);
 
hK = order(clK); wK = torsion_units_order(K);
 
2^r1 * (2*pi)^r2 * RK * hK / (wK * sqrt(RR(abs(DK))))
 

Galois group

$C_2\times A_4\times D_6$ (as 36T334):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: G = GaloisGroup(K);
 
oscar: G, Gtx = galois_group(K); G, transitive_group_identification(G)
 
A solvable group of order 288
The 48 conjugacy class representatives for $C_2\times A_4\times D_6$
Character table for $C_2\times A_4\times D_6$

Intermediate fields

\(\Q(\sqrt{-3}) \), 3.3.756.1, \(\Q(\zeta_{9})^+\), 6.0.1305639.1, 6.6.3916917.1, 6.0.1714608.1, \(\Q(\zeta_{9})\), 9.9.314987206464.1, 12.0.15342238784889.1, 18.0.297650820707983690149888.2, 18.0.781888920006838520203839074304.1, 18.18.2345666760020515560611517222912.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

sage: K.subfields()[1:-1]
 
gp: L = nfsubfields(K); L[2..length(b)]
 
magma: L := Subfields(K); L[2..#L];
 
oscar: subfields(K)[2:end-1]
 

Sibling fields

Degree 36 siblings: deg 36, deg 36, deg 36, deg 36, deg 36, deg 36, deg 36, some data not computed
Minimal sibling: This field is its own minimal sibling

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R R ${\href{/padicField/5.6.0.1}{6} }^{6}$ R ${\href{/padicField/11.6.0.1}{6} }^{6}$ ${\href{/padicField/13.6.0.1}{6} }^{4}{,}\,{\href{/padicField/13.3.0.1}{3} }^{4}$ ${\href{/padicField/17.6.0.1}{6} }^{6}$ ${\href{/padicField/19.2.0.1}{2} }^{14}{,}\,{\href{/padicField/19.1.0.1}{1} }^{8}$ ${\href{/padicField/23.6.0.1}{6} }^{6}$ ${\href{/padicField/29.6.0.1}{6} }^{6}$ ${\href{/padicField/31.6.0.1}{6} }^{4}{,}\,{\href{/padicField/31.3.0.1}{3} }^{4}$ ${\href{/padicField/37.6.0.1}{6} }^{2}{,}\,{\href{/padicField/37.3.0.1}{3} }^{8}$ ${\href{/padicField/41.6.0.1}{6} }^{6}$ ${\href{/padicField/43.3.0.1}{3} }^{12}$ ${\href{/padicField/47.6.0.1}{6} }^{6}$ ${\href{/padicField/53.2.0.1}{2} }^{18}$ ${\href{/padicField/59.6.0.1}{6} }^{6}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Sage:
 
p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
\\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Pari:
 
p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
 
// to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7 in Magma:
 
p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];
 
# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Oscar:
 
p = 7; pfac = factor(ideal(ring_of_integers(K), p)); [(e, valuation(norm(pr),p)) for (pr,e) in pfac]
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
\(2\) Copy content Toggle raw display 2.18.12.1$x^{18} + 10 x^{16} + 12 x^{15} + 28 x^{14} + 100 x^{13} + 92 x^{12} + 224 x^{11} + 640 x^{10} + 352 x^{9} + 736 x^{8} + 1760 x^{7} + 688 x^{6} + 1024 x^{5} + 1760 x^{4} + 448 x^{3} + 448 x^{2} + 320 x + 64$$3$$6$$12$$S_3 \times C_3$$[\ ]_{3}^{6}$
2.18.12.1$x^{18} + 10 x^{16} + 12 x^{15} + 28 x^{14} + 100 x^{13} + 92 x^{12} + 224 x^{11} + 640 x^{10} + 352 x^{9} + 736 x^{8} + 1760 x^{7} + 688 x^{6} + 1024 x^{5} + 1760 x^{4} + 448 x^{3} + 448 x^{2} + 320 x + 64$$3$$6$$12$$S_3 \times C_3$$[\ ]_{3}^{6}$
\(3\) Copy content Toggle raw display Deg $36$$18$$2$$62$
\(7\) Copy content Toggle raw display 7.3.0.1$x^{3} + 6 x^{2} + 4$$1$$3$$0$$C_3$$[\ ]^{3}$
7.3.0.1$x^{3} + 6 x^{2} + 4$$1$$3$$0$$C_3$$[\ ]^{3}$
7.3.0.1$x^{3} + 6 x^{2} + 4$$1$$3$$0$$C_3$$[\ ]^{3}$
7.3.0.1$x^{3} + 6 x^{2} + 4$$1$$3$$0$$C_3$$[\ ]^{3}$
7.6.3.2$x^{6} + 12 x^{5} + 57 x^{4} + 176 x^{3} + 699 x^{2} + 420 x + 1787$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
7.6.3.2$x^{6} + 12 x^{5} + 57 x^{4} + 176 x^{3} + 699 x^{2} + 420 x + 1787$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
7.6.3.2$x^{6} + 12 x^{5} + 57 x^{4} + 176 x^{3} + 699 x^{2} + 420 x + 1787$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
7.6.3.2$x^{6} + 12 x^{5} + 57 x^{4} + 176 x^{3} + 699 x^{2} + 420 x + 1787$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
\(199\) Copy content Toggle raw display $\Q_{199}$$x + 196$$1$$1$$0$Trivial$[\ ]$
$\Q_{199}$$x + 196$$1$$1$$0$Trivial$[\ ]$
$\Q_{199}$$x + 196$$1$$1$$0$Trivial$[\ ]$
$\Q_{199}$$x + 196$$1$$1$$0$Trivial$[\ ]$
199.2.0.1$x^{2} + 193 x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
199.2.0.1$x^{2} + 193 x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
199.2.1.1$x^{2} + 597$$2$$1$$1$$C_2$$[\ ]_{2}$
199.2.0.1$x^{2} + 193 x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
199.2.0.1$x^{2} + 193 x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
199.2.0.1$x^{2} + 193 x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
199.2.0.1$x^{2} + 193 x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
199.2.0.1$x^{2} + 193 x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
199.2.0.1$x^{2} + 193 x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
199.2.0.1$x^{2} + 193 x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
199.2.0.1$x^{2} + 193 x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
199.2.1.1$x^{2} + 597$$2$$1$$1$$C_2$$[\ ]_{2}$
199.4.2.1$x^{4} + 386 x^{3} + 37653 x^{2} + 77972 x + 7450967$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
199.4.2.1$x^{4} + 386 x^{3} + 37653 x^{2} + 77972 x + 7450967$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$