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Properties

Label 36.0.22878331820...2289.2
Degree $36$
Signature $[0, 18]$
Discriminant $3^{18}\cdot 7^{30}\cdot 13^{30}$
Root discriminant $74.32$
Ramified primes $3, 7, 13$
Class number $4032$ (GRH)
Class group $[2, 2, 2, 2, 6, 42]$ (GRH)
Galois group $C_6^2$ (as 36T4)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![92525161, 39832279, -85948561, -7847932, 49718214, 61314085, -11566116, -106491054, -31771765, 64137758, 89890193, -4484353, -45496518, -30121865, 9985635, 15676868, 2568686, -7374521, -3553956, 614630, 1603263, 112199, 218454, 41016, -33121, 11554, 29133, -8707, -423, 1964, -47, -288, 131, 22, -11, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^36 - x^35 - 11*x^34 + 22*x^33 + 131*x^32 - 288*x^31 - 47*x^30 + 1964*x^29 - 423*x^28 - 8707*x^27 + 29133*x^26 + 11554*x^25 - 33121*x^24 + 41016*x^23 + 218454*x^22 + 112199*x^21 + 1603263*x^20 + 614630*x^19 - 3553956*x^18 - 7374521*x^17 + 2568686*x^16 + 15676868*x^15 + 9985635*x^14 - 30121865*x^13 - 45496518*x^12 - 4484353*x^11 + 89890193*x^10 + 64137758*x^9 - 31771765*x^8 - 106491054*x^7 - 11566116*x^6 + 61314085*x^5 + 49718214*x^4 - 7847932*x^3 - 85948561*x^2 + 39832279*x + 92525161)
 
gp: K = bnfinit(x^36 - x^35 - 11*x^34 + 22*x^33 + 131*x^32 - 288*x^31 - 47*x^30 + 1964*x^29 - 423*x^28 - 8707*x^27 + 29133*x^26 + 11554*x^25 - 33121*x^24 + 41016*x^23 + 218454*x^22 + 112199*x^21 + 1603263*x^20 + 614630*x^19 - 3553956*x^18 - 7374521*x^17 + 2568686*x^16 + 15676868*x^15 + 9985635*x^14 - 30121865*x^13 - 45496518*x^12 - 4484353*x^11 + 89890193*x^10 + 64137758*x^9 - 31771765*x^8 - 106491054*x^7 - 11566116*x^6 + 61314085*x^5 + 49718214*x^4 - 7847932*x^3 - 85948561*x^2 + 39832279*x + 92525161, 1)
 

Normalized defining polynomial

\( x^{36} - x^{35} - 11 x^{34} + 22 x^{33} + 131 x^{32} - 288 x^{31} - 47 x^{30} + 1964 x^{29} - 423 x^{28} - 8707 x^{27} + 29133 x^{26} + 11554 x^{25} - 33121 x^{24} + 41016 x^{23} + 218454 x^{22} + 112199 x^{21} + 1603263 x^{20} + 614630 x^{19} - 3553956 x^{18} - 7374521 x^{17} + 2568686 x^{16} + 15676868 x^{15} + 9985635 x^{14} - 30121865 x^{13} - 45496518 x^{12} - 4484353 x^{11} + 89890193 x^{10} + 64137758 x^{9} - 31771765 x^{8} - 106491054 x^{7} - 11566116 x^{6} + 61314085 x^{5} + 49718214 x^{4} - 7847932 x^{3} - 85948561 x^{2} + 39832279 x + 92525161 \)

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:  \(22878331820822683097801634238807198405761132789439230168181892642289=3^{18}\cdot 7^{30}\cdot 13^{30}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $74.32$
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:  $3, 7, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(273=3\cdot 7\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{273}(256,·)$, $\chi_{273}(1,·)$, $\chi_{273}(107,·)$, $\chi_{273}(257,·)$, $\chi_{273}(10,·)$, $\chi_{273}(16,·)$, $\chi_{273}(17,·)$, $\chi_{273}(22,·)$, $\chi_{273}(29,·)$, $\chi_{273}(160,·)$, $\chi_{273}(38,·)$, $\chi_{273}(92,·)$, $\chi_{273}(170,·)$, $\chi_{273}(263,·)$, $\chi_{273}(172,·)$, $\chi_{273}(173,·)$, $\chi_{273}(181,·)$, $\chi_{273}(62,·)$, $\chi_{273}(53,·)$, $\chi_{273}(194,·)$, $\chi_{273}(199,·)$, $\chi_{273}(74,·)$, $\chi_{273}(79,·)$, $\chi_{273}(82,·)$, $\chi_{273}(211,·)$, $\chi_{273}(220,·)$, $\chi_{273}(272,·)$, $\chi_{273}(100,·)$, $\chi_{273}(101,·)$, $\chi_{273}(103,·)$, $\chi_{273}(235,·)$, $\chi_{273}(191,·)$, $\chi_{273}(113,·)$, $\chi_{273}(244,·)$, $\chi_{273}(251,·)$, $\chi_{273}(166,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $\frac{1}{2} a^{27} - \frac{1}{2} a^{24} - \frac{1}{2} a^{21} - \frac{1}{2} a^{15} - \frac{1}{2} a^{12} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{28} - \frac{1}{2} a^{25} - \frac{1}{2} a^{22} - \frac{1}{2} a^{16} - \frac{1}{2} a^{13} - \frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{29} - \frac{1}{2} a^{26} - \frac{1}{2} a^{23} - \frac{1}{2} a^{17} - \frac{1}{2} a^{14} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{131014} a^{30} - \frac{18385}{131014} a^{29} + \frac{7515}{65507} a^{28} - \frac{8317}{131014} a^{27} + \frac{23503}{131014} a^{26} + \frac{20412}{65507} a^{25} + \frac{207}{131014} a^{24} - \frac{15935}{131014} a^{23} + \frac{10201}{65507} a^{22} - \frac{23795}{65507} a^{21} - \frac{2733}{65507} a^{20} - \frac{29110}{65507} a^{19} - \frac{24423}{131014} a^{18} - \frac{39271}{131014} a^{17} + \frac{438}{5039} a^{16} - \frac{1931}{10078} a^{15} - \frac{64245}{131014} a^{14} - \frac{23418}{65507} a^{13} - \frac{25657}{65507} a^{12} - \frac{18554}{65507} a^{11} + \frac{28844}{65507} a^{10} + \frac{13778}{65507} a^{9} + \frac{13899}{65507} a^{8} + \frac{29810}{65507} a^{7} - \frac{9949}{131014} a^{6} + \frac{58157}{131014} a^{5} - \frac{22833}{65507} a^{4} - \frac{16085}{131014} a^{3} + \frac{50225}{131014} a^{2} + \frac{20495}{65507} a + \frac{20430}{65507}$, $\frac{1}{44151718} a^{31} - \frac{75}{22075859} a^{30} + \frac{9724417}{44151718} a^{29} + \frac{5002508}{22075859} a^{28} + \frac{370817}{1698143} a^{27} + \frac{8849173}{44151718} a^{26} - \frac{5402178}{22075859} a^{25} - \frac{9715384}{22075859} a^{24} + \frac{11563961}{44151718} a^{23} + \frac{8746565}{44151718} a^{22} + \frac{8595227}{22075859} a^{21} - \frac{9120011}{22075859} a^{20} + \frac{13565775}{44151718} a^{19} + \frac{10770610}{22075859} a^{18} + \frac{21513523}{44151718} a^{17} - \frac{406490}{1698143} a^{16} + \frac{10715881}{22075859} a^{15} + \frac{11239981}{44151718} a^{14} - \frac{20136157}{44151718} a^{13} + \frac{9017011}{22075859} a^{12} - \frac{5986335}{22075859} a^{11} + \frac{4024342}{22075859} a^{10} + \frac{4621874}{22075859} a^{9} + \frac{5861615}{22075859} a^{8} - \frac{12701779}{44151718} a^{7} + \frac{7383050}{22075859} a^{6} + \frac{3557291}{44151718} a^{5} + \frac{420490}{22075859} a^{4} - \frac{4086643}{22075859} a^{3} - \frac{8016863}{44151718} a^{2} - \frac{11274071}{44151718} a + \frac{5570836}{22075859}$, $\frac{1}{44151718} a^{32} + \frac{12}{22075859} a^{30} + \frac{8022957}{44151718} a^{29} + \frac{832135}{44151718} a^{28} + \frac{1408215}{44151718} a^{27} + \frac{12052345}{44151718} a^{26} - \frac{13564115}{44151718} a^{25} + \frac{11573467}{44151718} a^{24} + \frac{886116}{22075859} a^{23} + \frac{20830331}{44151718} a^{22} - \frac{3974681}{44151718} a^{21} + \frac{19302811}{44151718} a^{20} - \frac{3720595}{22075859} a^{19} + \frac{641667}{1698143} a^{18} + \frac{11249677}{44151718} a^{17} - \frac{14052297}{44151718} a^{16} - \frac{39773}{131014} a^{15} - \frac{1795756}{22075859} a^{14} + \frac{798619}{3396286} a^{13} + \frac{778225}{3396286} a^{12} - \frac{655624}{1698143} a^{11} + \frac{4411973}{22075859} a^{10} - \frac{10640282}{22075859} a^{9} + \frac{9337649}{44151718} a^{8} + \frac{6409991}{22075859} a^{7} + \frac{9682150}{22075859} a^{6} - \frac{13572865}{44151718} a^{5} - \frac{8249973}{44151718} a^{4} + \frac{2939975}{44151718} a^{3} + \frac{1614253}{22075859} a^{2} + \frac{13802121}{44151718} a - \frac{9662401}{44151718}$, $\frac{1}{126185610044} a^{33} + \frac{63}{126185610044} a^{32} - \frac{5}{63092805022} a^{31} + \frac{46305}{31546402511} a^{30} + \frac{15230248157}{63092805022} a^{29} - \frac{4212866173}{63092805022} a^{28} - \frac{23598079731}{126185610044} a^{27} + \frac{989967915}{63092805022} a^{26} - \frac{15759784175}{63092805022} a^{25} + \frac{8098971531}{31546402511} a^{24} - \frac{503548145}{9706585388} a^{23} - \frac{27128714261}{63092805022} a^{22} + \frac{12006503950}{31546402511} a^{21} + \frac{12298647273}{126185610044} a^{20} + \frac{440067725}{4853292694} a^{19} + \frac{5581421383}{31546402511} a^{18} + \frac{2250284249}{31546402511} a^{17} + \frac{12588762226}{31546402511} a^{16} - \frac{28007864805}{63092805022} a^{15} + \frac{9061600073}{126185610044} a^{14} - \frac{7929435255}{63092805022} a^{13} + \frac{53910122561}{126185610044} a^{12} - \frac{26490988649}{63092805022} a^{11} - \frac{1417301191}{31546402511} a^{10} + \frac{50593787363}{126185610044} a^{9} + \frac{5773740217}{126185610044} a^{8} + \frac{9403702240}{31546402511} a^{7} - \frac{159257532}{31546402511} a^{6} + \frac{19615874089}{63092805022} a^{5} + \frac{2284062807}{4853292694} a^{4} - \frac{15714120334}{31546402511} a^{3} + \frac{56258818223}{126185610044} a^{2} + \frac{8378769669}{63092805022} a - \frac{9266698555}{126185610044}$, $\frac{1}{47242801351821295895593952664956} a^{34} + \frac{21739700502637842227}{11810700337955323973898488166239} a^{33} + \frac{69924224325051958174643}{47242801351821295895593952664956} a^{32} - \frac{663264924598354648809}{908515410611947997992191397403} a^{31} - \frac{18262859278734622499967934}{11810700337955323973898488166239} a^{30} - \frac{2691938441085783835672072397691}{23621400675910647947796976332478} a^{29} - \frac{344242598052175003882855315337}{3634061642447791991968765589612} a^{28} + \frac{3651361347358449153478420715215}{47242801351821295895593952664956} a^{27} - \frac{4073336075640306694643851095997}{23621400675910647947796976332478} a^{26} - \frac{8829573361639063925262203350581}{23621400675910647947796976332478} a^{25} - \frac{2644895952917804983368985721505}{47242801351821295895593952664956} a^{24} - \frac{8472950612849378315384943948731}{47242801351821295895593952664956} a^{23} + \frac{2126827516444506092229909311690}{11810700337955323973898488166239} a^{22} - \frac{22525269778435049096390556996809}{47242801351821295895593952664956} a^{21} + \frac{2585646517253466815312265012657}{47242801351821295895593952664956} a^{20} - \frac{5102772085462575700876767856706}{11810700337955323973898488166239} a^{19} - \frac{9198492943915782564833888798583}{23621400675910647947796976332478} a^{18} - \frac{6808253094849806380097669000193}{23621400675910647947796976332478} a^{17} + \frac{596184447953290471942864925787}{1817030821223895995984382794806} a^{16} - \frac{10270515996732458181771096522629}{47242801351821295895593952664956} a^{15} + \frac{16172162507769860972970013967539}{47242801351821295895593952664956} a^{14} - \frac{20320455953368752795131303178675}{47242801351821295895593952664956} a^{13} - \frac{566318066824435561003988837571}{47242801351821295895593952664956} a^{12} + \frac{11162621206528529472922594907433}{23621400675910647947796976332478} a^{11} + \frac{1035241913048799549909195874195}{3634061642447791991968765589612} a^{10} + \frac{653255393556418357071102582597}{11810700337955323973898488166239} a^{9} - \frac{14297408443127750092688542753161}{47242801351821295895593952664956} a^{8} - \frac{9515611600286252829663202743091}{23621400675910647947796976332478} a^{7} + \frac{452401674955043625195637239255}{908515410611947997992191397403} a^{6} - \frac{2842004867160021294008571301805}{23621400675910647947796976332478} a^{5} - \frac{592778990389239204399739605715}{1817030821223895995984382794806} a^{4} + \frac{11269338243273035919475007529239}{47242801351821295895593952664956} a^{3} + \frac{21881633847229594499306360154097}{47242801351821295895593952664956} a^{2} + \frac{19904715134001979627745469898905}{47242801351821295895593952664956} a + \frac{9885390187835346650332930062863}{47242801351821295895593952664956}$, $\frac{1}{1096337033758214743745788851366130256648063339838076717695587793774726725572966062740141909135940850542330346619133068} a^{35} + \frac{283030390392539758635043036763766117015786883115370968462209462974381241091509968375}{274084258439553685936447212841532564162015834959519179423896948443681681393241515685035477283985212635582586654783267} a^{34} + \frac{1912888947643557561134779823604140646735881771403789937318621670512287791832172802606624164573980990339839}{548168516879107371872894425683065128324031669919038358847793896887363362786483031370070954567970425271165173309566534} a^{33} - \frac{7225496282444337267629008850027626616947301732261478328029376450152507243538944805743674449048406292387088711}{1096337033758214743745788851366130256648063339838076717695587793774726725572966062740141909135940850542330346619133068} a^{32} - \frac{2286098314112749535858535927911895872095642068769542786003385014358236436202705152258159255848150900999058557}{274084258439553685936447212841532564162015834959519179423896948443681681393241515685035477283985212635582586654783267} a^{31} - \frac{866849723485403687911738102390741918062968626433575458995214402569793145736709215211419642312226278271782621913}{274084258439553685936447212841532564162015834959519179423896948443681681393241515685035477283985212635582586654783267} a^{30} + \frac{195337388399009033339831587662459467388991376719441276343746412344441492313985520236783437631805591140583280272072683}{1096337033758214743745788851366130256648063339838076717695587793774726725572966062740141909135940850542330346619133068} a^{29} + \frac{86894600432391369299046273791653845256280725993691677113473654823826690032162542581683049729024748183572607622870273}{1096337033758214743745788851366130256648063339838076717695587793774726725572966062740141909135940850542330346619133068} a^{28} - \frac{43884433075388761586887419789422900661089089006346289215707844352850863392696704760142168141904583208259816583406657}{1096337033758214743745788851366130256648063339838076717695587793774726725572966062740141909135940850542330346619133068} a^{27} - \frac{84222477849648155529214221480973925964697287742314089585848796631040766688241866059589836047359730892953272851142231}{548168516879107371872894425683065128324031669919038358847793896887363362786483031370070954567970425271165173309566534} a^{26} - \frac{37931454528303120921357001676297620753910885309318607152885245484036266342257479597781190087932168017167035930548043}{84333617981401134134291450105086942819081795372159747515045214905748209659458927903087839164303142349410026663010236} a^{25} - \frac{1417890642044447816522490692117419958597025066515287812272386335553249499819741887587986012325376085312790608151131}{3571130403121220663667064662430391715466004364293409503894422781025168487208358510554208173081240555512476699085124} a^{24} + \frac{251700301392646063455761678950033929459797633140310324932591134236540898866809344678474909561770137641145091172514051}{1096337033758214743745788851366130256648063339838076717695587793774726725572966062740141909135940850542330346619133068} a^{23} + \frac{118251479103892123612778889945412340766134299750886252115507768596581290332266765907048364548857415381145343869436079}{1096337033758214743745788851366130256648063339838076717695587793774726725572966062740141909135940850542330346619133068} a^{22} + \frac{453745173109340006170730752022545184838125166054138234822775388599722043184846793476001348443973290112902147552629573}{1096337033758214743745788851366130256648063339838076717695587793774726725572966062740141909135940850542330346619133068} a^{21} - \frac{432795767871463625532207252768925256729875945861331588386518938246376033217732908782560958976126043828415660166476345}{1096337033758214743745788851366130256648063339838076717695587793774726725572966062740141909135940850542330346619133068} a^{20} + \frac{143330194310084934278093586028967441230378927542989560635460500230444883520909249990383944140878756154924096272208669}{548168516879107371872894425683065128324031669919038358847793896887363362786483031370070954567970425271165173309566534} a^{19} + \frac{118076058334017833059419706045876150503147750194676519924356795353801502151874580446490719952779731415049444769348734}{274084258439553685936447212841532564162015834959519179423896948443681681393241515685035477283985212635582586654783267} a^{18} + \frac{53892580745859024544237180522753771174439361234398110338077529099012939722581500523507782616925886913223355947674559}{274084258439553685936447212841532564162015834959519179423896948443681681393241515685035477283985212635582586654783267} a^{17} - \frac{391448160635628938161242175587606603601065556621636091339506096869133839422527819978202114040423213574500274169649037}{1096337033758214743745788851366130256648063339838076717695587793774726725572966062740141909135940850542330346619133068} a^{16} - \frac{376660333335041020484483556850098920683501619590372993889974895879809289432773477676112602447336698954764304165559653}{1096337033758214743745788851366130256648063339838076717695587793774726725572966062740141909135940850542330346619133068} a^{15} - \frac{221115827334219449635900427770462021164656537320056977069271152873034040594005867569524026932059128094389024021004673}{548168516879107371872894425683065128324031669919038358847793896887363362786483031370070954567970425271165173309566534} a^{14} + \frac{373112318586617864130264336053226468948748918364565839758867349818133092068649297006782996630130851972995957401171145}{1096337033758214743745788851366130256648063339838076717695587793774726725572966062740141909135940850542330346619133068} a^{13} + \frac{303395143123805483397051073274202363870981354954304324201133152188598224902160883709589656846061928308443470091919613}{1096337033758214743745788851366130256648063339838076717695587793774726725572966062740141909135940850542330346619133068} a^{12} - \frac{31517919016794633257500833693836350370062340203175347756019041300775087083708208516240569297415006896194094988449331}{84333617981401134134291450105086942819081795372159747515045214905748209659458927903087839164303142349410026663010236} a^{11} + \frac{29589363893627180905639169733744508834291956733130115026272517189903070209271478646274385731153262906728147247290008}{274084258439553685936447212841532564162015834959519179423896948443681681393241515685035477283985212635582586654783267} a^{10} - \frac{59559959110506359972526829659164793555148018479846650678751105553652365647467330573575409550687626225282852177904599}{274084258439553685936447212841532564162015834959519179423896948443681681393241515685035477283985212635582586654783267} a^{9} - \frac{162579520346814017714153478590922446714034568555146849862953042381250891354737654491548317373110094264385039190576151}{1096337033758214743745788851366130256648063339838076717695587793774726725572966062740141909135940850542330346619133068} a^{8} + \frac{184376431314011871913998071217527833305932814646956755075380503232324657069813030652846472064389243562491835282963029}{548168516879107371872894425683065128324031669919038358847793896887363362786483031370070954567970425271165173309566534} a^{7} - \frac{12687271638165491002385115076935702077342041065854930067474749478228205641853607353781124636009113547200755653011743}{274084258439553685936447212841532564162015834959519179423896948443681681393241515685035477283985212635582586654783267} a^{6} - \frac{77719053189082265274556293478381868382258421590704649497358472768257568569344138405172263519072928628822903164511655}{548168516879107371872894425683065128324031669919038358847793896887363362786483031370070954567970425271165173309566534} a^{5} - \frac{459378887037360979057309810788529815512668710753966639824349922230742161807718067612452664974986277252332922682137179}{1096337033758214743745788851366130256648063339838076717695587793774726725572966062740141909135940850542330346619133068} a^{4} - \frac{26914542472662712068541638036092331303693138138894370563656760350615190719005376846627292722656547056513302124623197}{84333617981401134134291450105086942819081795372159747515045214905748209659458927903087839164303142349410026663010236} a^{3} + \frac{126278285228325615253403427868469405135636948322255481985700072195349633269952045530820685327276901847154793932329686}{274084258439553685936447212841532564162015834959519179423896948443681681393241515685035477283985212635582586654783267} a^{2} + \frac{472478318630966892430504517557411813382426754351071422843468546606074495300375412931345394143448918785731818264132011}{1096337033758214743745788851366130256648063339838076717695587793774726725572966062740141909135940850542330346619133068} a + \frac{52087455367233722495560336071640896176580660530651654574898423851858905877720867972189885563489845868626440735217}{113976196460985002988438387708299226182354022230801197390122444513434528077031506678463656215400857733894411749572}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{6}\times C_{42}$, which has order $4032$ (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{2608650006977459295996119686779241354295842424205993835264309204205257546158208394949}{581130829693271687034512745660558977729524500005704534185441997633801668986197421713641961588} a^{35} - \frac{4410852424277364540071675465820136773794498763531859479005339370515684538205765268291}{581130829693271687034512745660558977729524500005704534185441997633801668986197421713641961588} a^{34} - \frac{12096186575538551920582867808147122612290280226624788203263406193039605640301566910319}{290565414846635843517256372830279488864762250002852267092720998816900834493098710856820980794} a^{33} + \frac{35714738953017707658946012753121102378048480223393293249242260843670158749474918042071}{290565414846635843517256372830279488864762250002852267092720998816900834493098710856820980794} a^{32} + \frac{70351929661903084939885240316065877189061839232208070586706512623370775134878008652969}{145282707423317921758628186415139744432381125001426133546360499408450417246549355428410490397} a^{31} - \frac{225680018858527682239241964310716419086893311615209641597353430606604298582181646032165}{145282707423317921758628186415139744432381125001426133546360499408450417246549355428410490397} a^{30} + \frac{618034597739250465285973090300885130717766161307926284483715011025923729886967614067343}{581130829693271687034512745660558977729524500005704534185441997633801668986197421713641961588} a^{29} + \frac{1046167388198251254269469527631896529419597546205724568405673063459277831175132439468344}{145282707423317921758628186415139744432381125001426133546360499408450417246549355428410490397} a^{28} - \frac{786982278912906935627812188898116705402165627913854592968243279748268074090992990372284}{145282707423317921758628186415139744432381125001426133546360499408450417246549355428410490397} a^{27} - \frac{9278921609172671000585182812435581242439159911958080322729830612485595738705777706990153}{290565414846635843517256372830279488864762250002852267092720998816900834493098710856820980794} a^{26} + \frac{84748784514002110157657015711638014674998898287286164593933973180253496199964834452961991}{581130829693271687034512745660558977729524500005704534185441997633801668986197421713641961588} a^{25} - \frac{26853616939719606471523649294002253887544788761531220992378306467312251853571460741048}{473233574668788018757746535554201121929580211731029750965343646281597450314493014424789871} a^{24} - \frac{2252331035722305413742475110292488491407671729569302385584516485423208659777519532299482}{145282707423317921758628186415139744432381125001426133546360499408450417246549355428410490397} a^{23} + \frac{55934701101918059239610603877052701198593660135376924028503511024566522015736745311454575}{581130829693271687034512745660558977729524500005704534185441997633801668986197421713641961588} a^{22} + \frac{278861080699080109446422246294872495611654574990975133793548484059685611668555461554479069}{290565414846635843517256372830279488864762250002852267092720998816900834493098710856820980794} a^{21} + \frac{4142664229226175614903875198213318450953828726201575948764665444149690262741779527645397}{22351185757433526424404336371559960681904788461757866699440076832069294961007593142832383138} a^{20} + \frac{2105115350614668421742244675167612312305989746677128431139160210007571261274679465855331975}{290565414846635843517256372830279488864762250002852267092720998816900834493098710856820980794} a^{19} - \frac{726369707554678172754027625827148203839971297079080126394221628692964283250278207832187247}{290565414846635843517256372830279488864762250002852267092720998816900834493098710856820980794} a^{18} - \frac{1353475897512214720222232277945556482888067394560904072874242179999469567506883240372342872}{145282707423317921758628186415139744432381125001426133546360499408450417246549355428410490397} a^{17} - \frac{15873301627672016403382907336572911290112930936375955318585443342219762756068490860901414643}{581130829693271687034512745660558977729524500005704534185441997633801668986197421713641961588} a^{16} + \frac{4201536719837518992180788721369730658366365794716977170074206392055491669709500422125172091}{145282707423317921758628186415139744432381125001426133546360499408450417246549355428410490397} a^{15} + \frac{24331125488802560559125246318101634309722356757372529705224665000986754780970330925873174349}{581130829693271687034512745660558977729524500005704534185441997633801668986197421713641961588} a^{14} + \frac{48310940316557377761483117153663681743535805873891882417850888069257194291533092718232805}{22351185757433526424404336371559960681904788461757866699440076832069294961007593142832383138} a^{13} - \frac{26734554168190142032752036032582803438248957334767118133631525844631227146210905428621233131}{145282707423317921758628186415139744432381125001426133546360499408450417246549355428410490397} a^{12} - \frac{65599457457507022461901252578546938550929974663167018985683499113576930940815470769019121697}{581130829693271687034512745660558977729524500005704534185441997633801668986197421713641961588} a^{11} + \frac{60029737715374955345635183579379790932403244034982883107087826628360385295216380533263244271}{581130829693271687034512745660558977729524500005704534185441997633801668986197421713641961588} a^{10} + \frac{60376061745330597594336769711250419076338539272863646301987334421537348583717460256242432776}{145282707423317921758628186415139744432381125001426133546360499408450417246549355428410490397} a^{9} - \frac{55731431360195511456436846118837246997482661128361552011312663598041624826420630927729921523}{290565414846635843517256372830279488864762250002852267092720998816900834493098710856820980794} a^{8} - \frac{35468098448173286312028038998394020821631302226491996509874823769403289020362694323882259590}{145282707423317921758628186415139744432381125001426133546360499408450417246549355428410490397} a^{7} - \frac{49275885607966632749337845016400604141702369125808162042765526730335740583581884319935323375}{145282707423317921758628186415139744432381125001426133546360499408450417246549355428410490397} a^{6} + \frac{252625135355099869946501508136555846459987358097348364312871404068828691542997577711861508189}{290565414846635843517256372830279488864762250002852267092720998816900834493098710856820980794} a^{5} + \frac{202645060066735186651124413931148880437899361085947537054472895951908073506927886830428182255}{581130829693271687034512745660558977729524500005704534185441997633801668986197421713641961588} a^{4} + \frac{10032707421257233533101095532297792397180425239657829319857844676933756991142429700709397341}{145282707423317921758628186415139744432381125001426133546360499408450417246549355428410490397} a^{3} - \frac{437014655943347325431056106626906380343718474465066471540600493341710508373474992026857348287}{581130829693271687034512745660558977729524500005704534185441997633801668986197421713641961588} a^{2} - \frac{19205025184262349807284988233309675763993892979572422465711980278296848045659044682931946910}{145282707423317921758628186415139744432381125001426133546360499408450417246549355428410490397} a + \frac{1417204262719448907788606272634727573323538158741694989841898545349391213042437577456955}{1161824813256758832747912276305227190035595616059770594627304128915131248622912628279051} \) (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:  \( 6548181113293621.0 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_6^2$ (as 36T4):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 36
The 36 conjugacy class representatives for $C_6^2$
Character table for $C_6^2$ is not computed

Intermediate fields

\(\Q(\sqrt{-3}) \), \(\Q(\sqrt{273}) \), \(\Q(\sqrt{-91}) \), 3.3.169.1, 3.3.8281.2, \(\Q(\zeta_{7})^+\), 3.3.8281.1, \(\Q(\sqrt{-3}, \sqrt{-91})\), 6.0.771147.1, 6.0.1851523947.1, 6.0.64827.1, 6.0.1851523947.2, 6.6.3438544473.1, 6.0.127353499.1, 6.6.168488679177.1, 6.0.6240321451.2, 6.6.996974433.1, 6.0.36924979.1, 6.6.168488679177.2, 6.0.6240321451.1, 9.9.567869252041.1, 12.0.11823588092798847729.3, 12.0.28388435010810033397329.6, 12.0.993958020055671489.2, 12.0.28388435010810033397329.3, 18.0.6347285018761982937208599123.3, 18.18.4783129918873486243975221249718233.1, 18.0.243008175525757569678159896851.1

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 ${\href{/LocalNumberField/2.6.0.1}{6} }^{6}$ R ${\href{/LocalNumberField/5.6.0.1}{6} }^{6}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{6}$ R ${\href{/LocalNumberField/17.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{12}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{12}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{12}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{6}$

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

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
$3$3.12.6.2$x^{12} + 108 x^{6} - 243 x^{2} + 2916$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
3.12.6.2$x^{12} + 108 x^{6} - 243 x^{2} + 2916$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
3.12.6.2$x^{12} + 108 x^{6} - 243 x^{2} + 2916$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
7Data not computed
$13$13.6.5.5$x^{6} + 104$$6$$1$$5$$C_6$$[\ ]_{6}$
13.6.5.5$x^{6} + 104$$6$$1$$5$$C_6$$[\ ]_{6}$
13.6.5.5$x^{6} + 104$$6$$1$$5$$C_6$$[\ ]_{6}$
13.6.5.5$x^{6} + 104$$6$$1$$5$$C_6$$[\ ]_{6}$
13.6.5.5$x^{6} + 104$$6$$1$$5$$C_6$$[\ ]_{6}$
13.6.5.5$x^{6} + 104$$6$$1$$5$$C_6$$[\ ]_{6}$