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Properties

Label 36.0.72253773345...8125.1
Degree $36$
Signature $[0, 18]$
Discriminant $3^{88}\cdot 5^{27}$
Root discriminant $49.04$
Ramified primes $3, 5$
Class number $2053$ (GRH)
Class group $[2053]$ (GRH)
Galois group $C_{36}$ (as 36T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -9, 81, -699, 6021, -4752, 22041, -19917, 55485, 26, 121878, 26649, 256629, 40932, 255879, 36792, 197460, 11664, 133435, 1935, 79785, -153, 34695, -189, 13050, -18, 4374, -1, 1260, 0, 273, 0, 54, 0, 9, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^36 + 9*x^34 + 54*x^32 + 273*x^30 + 1260*x^28 - x^27 + 4374*x^26 - 18*x^25 + 13050*x^24 - 189*x^23 + 34695*x^22 - 153*x^21 + 79785*x^20 + 1935*x^19 + 133435*x^18 + 11664*x^17 + 197460*x^16 + 36792*x^15 + 255879*x^14 + 40932*x^13 + 256629*x^12 + 26649*x^11 + 121878*x^10 + 26*x^9 + 55485*x^8 - 19917*x^7 + 22041*x^6 - 4752*x^5 + 6021*x^4 - 699*x^3 + 81*x^2 - 9*x + 1)
 
gp: K = bnfinit(x^36 + 9*x^34 + 54*x^32 + 273*x^30 + 1260*x^28 - x^27 + 4374*x^26 - 18*x^25 + 13050*x^24 - 189*x^23 + 34695*x^22 - 153*x^21 + 79785*x^20 + 1935*x^19 + 133435*x^18 + 11664*x^17 + 197460*x^16 + 36792*x^15 + 255879*x^14 + 40932*x^13 + 256629*x^12 + 26649*x^11 + 121878*x^10 + 26*x^9 + 55485*x^8 - 19917*x^7 + 22041*x^6 - 4752*x^5 + 6021*x^4 - 699*x^3 + 81*x^2 - 9*x + 1, 1)
 

Normalized defining polynomial

\( x^{36} + 9 x^{34} + 54 x^{32} + 273 x^{30} + 1260 x^{28} - x^{27} + 4374 x^{26} - 18 x^{25} + 13050 x^{24} - 189 x^{23} + 34695 x^{22} - 153 x^{21} + 79785 x^{20} + 1935 x^{19} + 133435 x^{18} + 11664 x^{17} + 197460 x^{16} + 36792 x^{15} + 255879 x^{14} + 40932 x^{13} + 256629 x^{12} + 26649 x^{11} + 121878 x^{10} + 26 x^{9} + 55485 x^{8} - 19917 x^{7} + 22041 x^{6} - 4752 x^{5} + 6021 x^{4} - 699 x^{3} + 81 x^{2} - 9 x + 1 \)

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

Invariants

Degree:  $36$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 18]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(7225377334561374804949923918873673793376691639423370361328125=3^{88}\cdot 5^{27}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $49.04$
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:  $3, 5$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(135=3^{3}\cdot 5\)
Dirichlet character group:    $\lbrace$$\chi_{135}(1,·)$, $\chi_{135}(4,·)$, $\chi_{135}(133,·)$, $\chi_{135}(7,·)$, $\chi_{135}(13,·)$, $\chi_{135}(16,·)$, $\chi_{135}(19,·)$, $\chi_{135}(22,·)$, $\chi_{135}(28,·)$, $\chi_{135}(31,·)$, $\chi_{135}(34,·)$, $\chi_{135}(37,·)$, $\chi_{135}(43,·)$, $\chi_{135}(46,·)$, $\chi_{135}(49,·)$, $\chi_{135}(52,·)$, $\chi_{135}(58,·)$, $\chi_{135}(61,·)$, $\chi_{135}(64,·)$, $\chi_{135}(67,·)$, $\chi_{135}(73,·)$, $\chi_{135}(76,·)$, $\chi_{135}(79,·)$, $\chi_{135}(82,·)$, $\chi_{135}(88,·)$, $\chi_{135}(91,·)$, $\chi_{135}(94,·)$, $\chi_{135}(97,·)$, $\chi_{135}(103,·)$, $\chi_{135}(106,·)$, $\chi_{135}(109,·)$, $\chi_{135}(112,·)$, $\chi_{135}(118,·)$, $\chi_{135}(121,·)$, $\chi_{135}(124,·)$, $\chi_{135}(127,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $\frac{1}{271} a^{30} - \frac{132}{271} a^{29} + \frac{118}{271} a^{28} + \frac{72}{271} a^{27} - \frac{40}{271} a^{26} - \frac{128}{271} a^{25} - \frac{7}{271} a^{24} + \frac{96}{271} a^{23} + \frac{50}{271} a^{22} + \frac{43}{271} a^{21} + \frac{73}{271} a^{20} + \frac{18}{271} a^{19} - \frac{56}{271} a^{18} + \frac{20}{271} a^{17} + \frac{13}{271} a^{16} - \frac{45}{271} a^{15} - \frac{39}{271} a^{14} - \frac{78}{271} a^{13} - \frac{21}{271} a^{12} - \frac{128}{271} a^{11} + \frac{52}{271} a^{10} + \frac{76}{271} a^{9} + \frac{16}{271} a^{8} - \frac{48}{271} a^{7} + \frac{116}{271} a^{6} + \frac{36}{271} a^{5} + \frac{19}{271} a^{4} - \frac{65}{271} a^{3} - \frac{46}{271} a^{2} - \frac{45}{271} a + \frac{10}{271}$, $\frac{1}{271} a^{31} + \frac{38}{271} a^{29} - \frac{70}{271} a^{28} - \frac{21}{271} a^{27} + \frac{12}{271} a^{26} - \frac{101}{271} a^{25} - \frac{15}{271} a^{24} - \frac{15}{271} a^{23} - \frac{132}{271} a^{22} + \frac{58}{271} a^{21} - \frac{102}{271} a^{20} - \frac{119}{271} a^{19} - \frac{55}{271} a^{18} - \frac{57}{271} a^{17} + \frac{45}{271} a^{16} - \frac{17}{271} a^{15} - \frac{77}{271} a^{14} - \frac{19}{271} a^{13} + \frac{81}{271} a^{12} - \frac{42}{271} a^{11} - \frac{106}{271} a^{10} + \frac{21}{271} a^{9} - \frac{104}{271} a^{8} + \frac{13}{271} a^{7} - \frac{99}{271} a^{6} - \frac{107}{271} a^{5} + \frac{4}{271} a^{4} + \frac{46}{271} a^{3} + \frac{116}{271} a^{2} + \frac{32}{271} a - \frac{35}{271}$, $\frac{1}{271} a^{32} + \frac{68}{271} a^{29} + \frac{102}{271} a^{28} - \frac{14}{271} a^{27} + \frac{64}{271} a^{26} - \frac{29}{271} a^{25} - \frac{20}{271} a^{24} + \frac{14}{271} a^{23} + \frac{55}{271} a^{22} - \frac{110}{271} a^{21} + \frac{88}{271} a^{20} + \frac{74}{271} a^{19} - \frac{97}{271} a^{18} + \frac{98}{271} a^{17} + \frac{31}{271} a^{16} + \frac{7}{271} a^{15} + \frac{108}{271} a^{14} + \frac{64}{271} a^{13} - \frac{57}{271} a^{12} - \frac{120}{271} a^{11} - \frac{58}{271} a^{10} - \frac{11}{271} a^{9} - \frac{53}{271} a^{8} + \frac{99}{271} a^{7} + \frac{92}{271} a^{6} - \frac{9}{271} a^{5} - \frac{134}{271} a^{4} - \frac{124}{271} a^{3} - \frac{117}{271} a^{2} + \frac{49}{271} a - \frac{109}{271}$, $\frac{1}{318883648678381348159186580491} a^{33} - \frac{95219291383905279185744329}{318883648678381348159186580491} a^{32} + \frac{493776682547585729833299192}{318883648678381348159186580491} a^{31} + \frac{319718808092385137362199860}{318883648678381348159186580491} a^{30} - \frac{128522254508942917885330874072}{318883648678381348159186580491} a^{29} + \frac{24736519936449702571912175282}{318883648678381348159186580491} a^{28} - \frac{68308705307503698328009006598}{318883648678381348159186580491} a^{27} - \frac{137909923381810725710417722791}{318883648678381348159186580491} a^{26} - \frac{4523783377269194185963303390}{318883648678381348159186580491} a^{25} - \frac{91227237534201648749781953917}{318883648678381348159186580491} a^{24} - \frac{42077842859838734118075872875}{318883648678381348159186580491} a^{23} + \frac{4693090116142366869005418126}{318883648678381348159186580491} a^{22} - \frac{157213114725633897901484226327}{318883648678381348159186580491} a^{21} + \frac{114316059471640706143768887741}{318883648678381348159186580491} a^{20} + \frac{128444353976047190874604360242}{318883648678381348159186580491} a^{19} + \frac{2982590212873405254483767885}{318883648678381348159186580491} a^{18} + \frac{71156223358266077005862341046}{318883648678381348159186580491} a^{17} + \frac{47672736559876526387827383803}{318883648678381348159186580491} a^{16} + \frac{57875655923449250061842116943}{318883648678381348159186580491} a^{15} - \frac{138185899607103247083336812985}{318883648678381348159186580491} a^{14} + \frac{101121602811926583919050621646}{318883648678381348159186580491} a^{13} + \frac{65940287753460716978473324165}{318883648678381348159186580491} a^{12} - \frac{19732099473577754896550691993}{318883648678381348159186580491} a^{11} + \frac{105399351850892134517943515714}{318883648678381348159186580491} a^{10} + \frac{51243100930378626618459188718}{318883648678381348159186580491} a^{9} + \frac{94583834461568715848344406864}{318883648678381348159186580491} a^{8} - \frac{116499542025746305117267606521}{318883648678381348159186580491} a^{7} + \frac{121089124654152517524947573383}{318883648678381348159186580491} a^{6} - \frac{54049184118354492730647844914}{318883648678381348159186580491} a^{5} + \frac{53844162906849480054867758861}{318883648678381348159186580491} a^{4} - \frac{125712120177027462773257173244}{318883648678381348159186580491} a^{3} - \frac{101905550026748417025233369229}{318883648678381348159186580491} a^{2} - \frac{77299181097134821023007001883}{318883648678381348159186580491} a - \frac{142037195598842167128102359058}{318883648678381348159186580491}$, $\frac{1}{318883648678381348159186580491} a^{34} - \frac{516236388781524581091894442}{318883648678381348159186580491} a^{32} - \frac{544078367788428716623672595}{318883648678381348159186580491} a^{31} + \frac{60642223156409370308545279}{318883648678381348159186580491} a^{30} - \frac{105631203429691394955433453438}{318883648678381348159186580491} a^{29} - \frac{151429470201693255632521676286}{318883648678381348159186580491} a^{28} + \frac{91897535536286131731733227845}{318883648678381348159186580491} a^{27} + \frac{144261133800951200806430240750}{318883648678381348159186580491} a^{26} + \frac{136250747043634635731997349043}{318883648678381348159186580491} a^{25} - \frac{84271841095319609542202108439}{318883648678381348159186580491} a^{24} + \frac{71758417690506291728238235165}{318883648678381348159186580491} a^{23} + \frac{130322969198587604986332506193}{318883648678381348159186580491} a^{22} - \frac{62963769247786471545777250606}{318883648678381348159186580491} a^{21} + \frac{24014995863918776691854084746}{318883648678381348159186580491} a^{20} - \frac{51445773015472060015032444271}{318883648678381348159186580491} a^{19} - \frac{123179437128314624655481863990}{318883648678381348159186580491} a^{18} + \frac{118131682352882902144642938826}{318883648678381348159186580491} a^{17} + \frac{92034947263987297824913556292}{318883648678381348159186580491} a^{16} + \frac{148840570859302158576640271150}{318883648678381348159186580491} a^{15} + \frac{103185717867744612317675201360}{318883648678381348159186580491} a^{14} + \frac{69440409141463883646280280791}{318883648678381348159186580491} a^{13} + \frac{58354469809364763675133548313}{318883648678381348159186580491} a^{12} - \frac{125912921164347621720012002288}{318883648678381348159186580491} a^{11} + \frac{92930653455787467315594177466}{318883648678381348159186580491} a^{10} + \frac{137559048480590755453813413205}{318883648678381348159186580491} a^{9} - \frac{102119248700771081720427096896}{318883648678381348159186580491} a^{8} + \frac{39114761826296693000953129482}{318883648678381348159186580491} a^{7} - \frac{77800971007139760486789674144}{318883648678381348159186580491} a^{6} + \frac{84488958029597271950146444002}{318883648678381348159186580491} a^{5} - \frac{151628095497863135641227584962}{318883648678381348159186580491} a^{4} - \frac{49116496991808757347787566798}{318883648678381348159186580491} a^{3} + \frac{37841232805976693647353966380}{318883648678381348159186580491} a^{2} + \frac{69796298098214087363536974034}{318883648678381348159186580491} a - \frac{79720114379888609569652606524}{318883648678381348159186580491}$, $\frac{1}{318883648678381348159186580491} a^{35} + \frac{379626760802403774207976771}{318883648678381348159186580491} a^{32} + \frac{365124513806611730008776986}{318883648678381348159186580491} a^{31} + \frac{347614463071307414864992999}{318883648678381348159186580491} a^{30} - \frac{101439505694470900282938002165}{318883648678381348159186580491} a^{29} - \frac{33628006721856698094328600132}{318883648678381348159186580491} a^{28} - \frac{93435839653333913965096478159}{318883648678381348159186580491} a^{27} + \frac{159093992928802860512135308714}{318883648678381348159186580491} a^{26} - \frac{132843405459621549264537321335}{318883648678381348159186580491} a^{25} - \frac{104236421936236328386443263122}{318883648678381348159186580491} a^{24} - \frac{28324729748471780921983925825}{318883648678381348159186580491} a^{23} + \frac{35045781965479859881439245314}{318883648678381348159186580491} a^{22} - \frac{65425653609991062106821327399}{318883648678381348159186580491} a^{21} + \frac{16744307885957112704238385730}{318883648678381348159186580491} a^{20} + \frac{50453916831124304912252623074}{318883648678381348159186580491} a^{19} - \frac{922898403622726668255939824}{318883648678381348159186580491} a^{18} - \frac{125889934450142522588860452769}{318883648678381348159186580491} a^{17} + \frac{37230873322333954903296239221}{318883648678381348159186580491} a^{16} + \frac{129945681207494755967572120393}{318883648678381348159186580491} a^{15} + \frac{92794148912197250774860301526}{318883648678381348159186580491} a^{14} - \frac{110488943157587675659309975187}{318883648678381348159186580491} a^{13} + \frac{120046820594047395787906337030}{318883648678381348159186580491} a^{12} - \frac{100414356153474158071443896481}{318883648678381348159186580491} a^{11} - \frac{88489928115184085630072772417}{318883648678381348159186580491} a^{10} - \frac{107233056855764878804431378580}{318883648678381348159186580491} a^{9} - \frac{143997554826883708174799599907}{318883648678381348159186580491} a^{8} + \frac{59543019106100151605236876177}{318883648678381348159186580491} a^{7} - \frac{37121646469137986332149812219}{318883648678381348159186580491} a^{6} + \frac{90790117116266444216787667166}{318883648678381348159186580491} a^{5} + \frac{135130051805075002152142950600}{318883648678381348159186580491} a^{4} - \frac{59262250215330553955353674075}{318883648678381348159186580491} a^{3} + \frac{117617585107506866499831619121}{318883648678381348159186580491} a^{2} - \frac{114594201208248071632476259506}{318883648678381348159186580491} a + \frac{6481279775257297063998136815}{318883648678381348159186580491}$

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

Class group and class number

$C_{2053}$, which has order $2053$ (assuming GRH)

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

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $17$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -\frac{37078126722002202006861542313}{318883648678381348159186580491} a^{35} + \frac{436455875773191844058481}{318883648678381348159186580491} a^{34} - \frac{333658735026101126566580102517}{318883648678381348159186580491} a^{33} - \frac{2001832515382426292362511413692}{318883648678381348159186580491} a^{31} - \frac{1480182397289716505792610}{318883648678381348159186580491} a^{30} - \frac{10120050565304844621717574223454}{318883648678381348159186580491} a^{29} - \frac{13321641575607448552133490}{318883648678381348159186580491} a^{28} - \frac{46707036344534054552684917046940}{318883648678381348159186580491} a^{27} + \frac{36998196872548557315548741373}{318883648678381348159186580491} a^{26} - \frac{162127412195570221127548157861832}{318883648678381348159186580491} a^{25} + \frac{666462480433178568880181583177}{318883648678381348159186580491} a^{24} - \frac{483692109456341644974828709097850}{318883648678381348159186580491} a^{23} + \frac{7005114943784870297034956932647}{318883648678381348159186580491} a^{22} - \frac{1285904382591396784306293929827185}{318883648678381348159186580491} a^{21} + \frac{5658326226016319928539573401869}{318883648678381348159186580491} a^{20} - \frac{2956910950505734882506211986534510}{318883648678381348159186580491} a^{19} - \frac{71769767834304661674662912786445}{318883648678381348159186580491} a^{18} - \frac{4944438863173323636514226673501915}{318883648678381348159186580491} a^{17} - \frac{432442662214383914939411766708312}{318883648678381348159186580491} a^{16} - \frac{7316584299086287263572473621894800}{318883648678381348159186580491} a^{15} - \frac{1363704680150200757916450339527637}{318883648678381348159186580491} a^{14} - \frac{9480526120410716560175702963328627}{318883648678381348159186580491} a^{13} - \frac{1516401008627681873656285520684826}{318883648678381348159186580491} a^{12} - \frac{9506607910984621671769727368555617}{318883648678381348159186580491} a^{11} - \frac{987059800891079377668764051858217}{318883648678381348159186580491} a^{10} - \frac{4510895350525870026258225464270138}{318883648678381348159186580491} a^{9} - \frac{704698898219707051214017450308}{318883648678381348159186580491} a^{8} - \frac{2055347117445578322892423492107135}{318883648678381348159186580491} a^{7} + \frac{737751338309605317795904999397121}{318883648678381348159186580491} a^{6} - \frac{816438213882899195093317957903443}{318883648678381348159186580491} a^{5} + \frac{173912471666319478348596835458414}{318883648678381348159186580491} a^{4} - \frac{223007849754180287853732376056783}{318883648678381348159186580491} a^{3} + \frac{25889794991069670850219363349667}{318883648678381348159186580491} a^{2} - \frac{3000104427220881360006168622773}{318883648678381348159186580491} a + \frac{333343456175478416950846276587}{318883648678381348159186580491} \) (order $10$)
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:  \( 6550249244897.024 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{36}$ (as 36T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 36
The 36 conjugacy class representatives for $C_{36}$
Character table for $C_{36}$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{9})^+\), \(\Q(\zeta_{5})\), 6.6.820125.1, \(\Q(\zeta_{27})^+\), 12.0.84075626953125.1, 18.18.1923380668327365689220703125.1

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 $36$ R R $36$ ${\href{/LocalNumberField/11.9.0.1}{9} }^{4}$ $36$ ${\href{/LocalNumberField/17.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{6}$ $36$ $18^{2}$ ${\href{/LocalNumberField/31.9.0.1}{9} }^{4}$ ${\href{/LocalNumberField/37.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/41.9.0.1}{9} }^{4}$ $36$ $36$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{9}$ $18^{2}$

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

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

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
3Data not computed
5Data not computed