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Properties

Label 36.0.15570799512...3125.1
Degree $36$
Signature $[0, 18]$
Discriminant $3^{48}\cdot 5^{27}\cdot 13^{30}$
Root discriminant $122.65$
Ramified primes $3, 5, 13$
Class number Not computed
Class group Not computed
Galois group $C_3\times C_{12}$ (as 36T3)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![22848924331, -51149414124, 101791204893, -64890324348, 69893697486, -18291665040, 123111472419, -109764491523, 136411565292, -65694612305, 69063367278, -26596418232, 11620240143, 2242273917, 2592680742, -4935139658, 3135439767, -976832769, -26520393, 226149234, -43702461, -22878764, 14119596, 441060, -1894259, 121467, 130713, -36200, 6351, 5415, -385, -210, -15, 8, -3, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^36 - 3*x^35 - 3*x^34 + 8*x^33 - 15*x^32 - 210*x^31 - 385*x^30 + 5415*x^29 + 6351*x^28 - 36200*x^27 + 130713*x^26 + 121467*x^25 - 1894259*x^24 + 441060*x^23 + 14119596*x^22 - 22878764*x^21 - 43702461*x^20 + 226149234*x^19 - 26520393*x^18 - 976832769*x^17 + 3135439767*x^16 - 4935139658*x^15 + 2592680742*x^14 + 2242273917*x^13 + 11620240143*x^12 - 26596418232*x^11 + 69063367278*x^10 - 65694612305*x^9 + 136411565292*x^8 - 109764491523*x^7 + 123111472419*x^6 - 18291665040*x^5 + 69893697486*x^4 - 64890324348*x^3 + 101791204893*x^2 - 51149414124*x + 22848924331)
 
gp: K = bnfinit(x^36 - 3*x^35 - 3*x^34 + 8*x^33 - 15*x^32 - 210*x^31 - 385*x^30 + 5415*x^29 + 6351*x^28 - 36200*x^27 + 130713*x^26 + 121467*x^25 - 1894259*x^24 + 441060*x^23 + 14119596*x^22 - 22878764*x^21 - 43702461*x^20 + 226149234*x^19 - 26520393*x^18 - 976832769*x^17 + 3135439767*x^16 - 4935139658*x^15 + 2592680742*x^14 + 2242273917*x^13 + 11620240143*x^12 - 26596418232*x^11 + 69063367278*x^10 - 65694612305*x^9 + 136411565292*x^8 - 109764491523*x^7 + 123111472419*x^6 - 18291665040*x^5 + 69893697486*x^4 - 64890324348*x^3 + 101791204893*x^2 - 51149414124*x + 22848924331, 1)
 

Normalized defining polynomial

\( x^{36} - 3 x^{35} - 3 x^{34} + 8 x^{33} - 15 x^{32} - 210 x^{31} - 385 x^{30} + 5415 x^{29} + 6351 x^{28} - 36200 x^{27} + 130713 x^{26} + 121467 x^{25} - 1894259 x^{24} + 441060 x^{23} + 14119596 x^{22} - 22878764 x^{21} - 43702461 x^{20} + 226149234 x^{19} - 26520393 x^{18} - 976832769 x^{17} + 3135439767 x^{16} - 4935139658 x^{15} + 2592680742 x^{14} + 2242273917 x^{13} + 11620240143 x^{12} - 26596418232 x^{11} + 69063367278 x^{10} - 65694612305 x^{9} + 136411565292 x^{8} - 109764491523 x^{7} + 123111472419 x^{6} - 18291665040 x^{5} + 69893697486 x^{4} - 64890324348 x^{3} + 101791204893 x^{2} - 51149414124 x + 22848924331 \)

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:  \(1557079951247999724699376224646452411705566349268816543288528919219970703125=3^{48}\cdot 5^{27}\cdot 13^{30}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $122.65$
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:  $3, 5, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(585=3^{2}\cdot 5\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{585}(256,·)$, $\chi_{585}(1,·)$, $\chi_{585}(517,·)$, $\chi_{585}(391,·)$, $\chi_{585}(139,·)$, $\chi_{585}(142,·)$, $\chi_{585}(16,·)$, $\chi_{585}(529,·)$, $\chi_{585}(274,·)$, $\chi_{585}(532,·)$, $\chi_{585}(277,·)$, $\chi_{585}(406,·)$, $\chi_{585}(283,·)$, $\chi_{585}(289,·)$, $\chi_{585}(298,·)$, $\chi_{585}(43,·)$, $\chi_{585}(433,·)$, $\chi_{585}(94,·)$, $\chi_{585}(61,·)$, $\chi_{585}(322,·)$, $\chi_{585}(451,·)$, $\chi_{585}(196,·)$, $\chi_{585}(334,·)$, $\chi_{585}(79,·)$, $\chi_{585}(337,·)$, $\chi_{585}(82,·)$, $\chi_{585}(211,·)$, $\chi_{585}(469,·)$, $\chi_{585}(88,·)$, $\chi_{585}(478,·)$, $\chi_{585}(484,·)$, $\chi_{585}(103,·)$, $\chi_{585}(493,·)$, $\chi_{585}(238,·)$, $\chi_{585}(472,·)$, $\chi_{585}(127,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $\frac{1}{85801} a^{30} - \frac{9657}{85801} a^{29} + \frac{3193}{85801} a^{28} + \frac{23768}{85801} a^{27} + \frac{26874}{85801} a^{26} + \frac{16034}{85801} a^{25} - \frac{23452}{85801} a^{24} + \frac{19912}{85801} a^{23} + \frac{31540}{85801} a^{22} + \frac{8287}{85801} a^{21} + \frac{23782}{85801} a^{20} + \frac{1267}{85801} a^{19} + \frac{1823}{85801} a^{18} - \frac{38224}{85801} a^{17} - \frac{16463}{85801} a^{16} - \frac{4394}{85801} a^{15} + \frac{26138}{85801} a^{14} - \frac{8859}{85801} a^{13} - \frac{29001}{85801} a^{12} - \frac{35291}{85801} a^{11} + \frac{20413}{85801} a^{10} - \frac{28469}{85801} a^{9} + \frac{31181}{85801} a^{8} - \frac{36715}{85801} a^{7} + \frac{41457}{85801} a^{6} - \frac{908}{85801} a^{5} + \frac{30523}{85801} a^{4} - \frac{40668}{85801} a^{3} + \frac{33814}{85801} a^{2} - \frac{34910}{85801} a + \frac{22218}{85801}$, $\frac{1}{85801} a^{31} + \frac{11231}{85801} a^{29} - \frac{29791}{85801} a^{28} + \frac{36775}{85801} a^{27} - \frac{9773}{85801} a^{26} + \frac{31882}{85801} a^{25} - \frac{27213}{85801} a^{24} + \frac{41683}{85801} a^{23} - \frac{3483}{85801} a^{22} - \frac{992}{85801} a^{21} - \frac{25236}{85801} a^{20} - \frac{32301}{85801} a^{19} - \frac{22718}{85801} a^{18} - \frac{29729}{85801} a^{17} + \frac{1668}{85801} a^{16} - \frac{21026}{85801} a^{15} - \frac{20735}{85801} a^{14} - \frac{36767}{85801} a^{13} + \frac{42317}{85801} a^{12} + \frac{16798}{85801} a^{11} + \frac{14975}{85801} a^{10} + \frac{12452}{85801} a^{9} + \frac{2493}{85801} a^{8} + \frac{14434}{85801} a^{7} + \frac{1875}{85801} a^{6} + \frac{13669}{85801} a^{5} - \frac{6492}{85801} a^{4} + \frac{14115}{85801} a^{3} + \frac{34083}{85801} a^{2} + \frac{8477}{85801} a - \frac{29075}{85801}$, $\frac{1}{85801} a^{32} - \frac{24488}{85801} a^{29} + \frac{41010}{85801} a^{28} - \frac{21270}{85801} a^{27} - \frac{27895}{85801} a^{26} - \frac{8768}{85801} a^{25} + \frac{22025}{85801} a^{24} - \frac{37749}{85801} a^{23} - \frac{40204}{85801} a^{22} - \frac{2448}{85801} a^{21} - \frac{29430}{85801} a^{20} - \frac{9429}{85801} a^{19} + \frac{2597}{85801} a^{18} + \frac{33009}{85801} a^{17} - \frac{26228}{85801} a^{16} - \frac{7296}{85801} a^{15} + \frac{18377}{85801} a^{14} + \frac{8586}{85801} a^{13} + \frac{26433}{85801} a^{12} - \frac{32424}{85801} a^{11} + \frac{14321}{85801} a^{10} - \frac{42495}{85801} a^{9} - \frac{25496}{85801} a^{8} - \frac{11566}{85801} a^{7} - \frac{33672}{85801} a^{6} - \frac{19063}{85801} a^{5} - \frac{14703}{85801} a^{4} - \frac{28133}{85801} a^{3} - \frac{1331}{85801} a^{2} + \frac{20366}{85801} a - \frac{21050}{85801}$, $\frac{1}{130958838509} a^{33} + \frac{308074}{130958838509} a^{32} - \frac{280643}{130958838509} a^{31} - \frac{580611}{130958838509} a^{30} + \frac{8683404048}{130958838509} a^{29} - \frac{29018894463}{130958838509} a^{28} - \frac{13150704423}{130958838509} a^{27} - \frac{719425068}{130958838509} a^{26} - \frac{57754234545}{130958838509} a^{25} - \frac{15889417430}{130958838509} a^{24} - \frac{3005645065}{130958838509} a^{23} + \frac{5089064708}{130958838509} a^{22} - \frac{19090028461}{130958838509} a^{21} - \frac{33405532}{364787851} a^{20} + \frac{60891125847}{130958838509} a^{19} - \frac{31814272376}{130958838509} a^{18} + \frac{19322776631}{130958838509} a^{17} - \frac{45786891060}{130958838509} a^{16} - \frac{7677091479}{130958838509} a^{15} - \frac{12557529232}{130958838509} a^{14} + \frac{9746185317}{130958838509} a^{13} - \frac{23373537373}{130958838509} a^{12} - \frac{20260532886}{130958838509} a^{11} + \frac{49668747887}{130958838509} a^{10} - \frac{40114968223}{130958838509} a^{9} + \frac{11247185353}{130958838509} a^{8} + \frac{3668890658}{130958838509} a^{7} - \frac{50454072723}{130958838509} a^{6} + \frac{10091808042}{130958838509} a^{5} + \frac{25550065649}{130958838509} a^{4} - \frac{9003528}{120035599} a^{3} - \frac{46947685762}{130958838509} a^{2} - \frac{2810168968}{130958838509} a - \frac{15068380021}{130958838509}$, $\frac{1}{2583686924944061} a^{34} + \frac{8058}{2583686924944061} a^{33} + \frac{9360611074}{2583686924944061} a^{32} - \frac{4837373220}{2583686924944061} a^{31} - \frac{14881758070}{2583686924944061} a^{30} + \frac{989880035310074}{2583686924944061} a^{29} - \frac{1109997404081887}{2583686924944061} a^{28} - \frac{41047481526972}{2583686924944061} a^{27} - \frac{577293407992260}{2583686924944061} a^{26} + \frac{850733919353273}{2583686924944061} a^{25} + \frac{521836828377930}{2583686924944061} a^{24} - \frac{1085799634078441}{2583686924944061} a^{23} + \frac{1092689507867997}{2583686924944061} a^{22} + \frac{657062297568220}{2583686924944061} a^{21} + \frac{912966279010452}{2583686924944061} a^{20} - \frac{417435195189339}{2583686924944061} a^{19} - \frac{1172494200801104}{2583686924944061} a^{18} - \frac{628751228667485}{2583686924944061} a^{17} + \frac{79304041447618}{2583686924944061} a^{16} + \frac{1196407590668395}{2583686924944061} a^{15} - \frac{423320712347070}{2583686924944061} a^{14} + \frac{1187260785454600}{2583686924944061} a^{13} + \frac{1037680713244032}{2583686924944061} a^{12} - \frac{2133096427531}{2583686924944061} a^{11} - \frac{309123013007273}{2583686924944061} a^{10} + \frac{725584641417573}{2583686924944061} a^{9} + \frac{383743281258725}{2583686924944061} a^{8} - \frac{1189098238736131}{2583686924944061} a^{7} + \frac{1173869890282048}{2583686924944061} a^{6} + \frac{1076222570014557}{2583686924944061} a^{5} + \frac{235924063699396}{2583686924944061} a^{4} + \frac{1124809769204943}{2583686924944061} a^{3} - \frac{76837392106022}{2583686924944061} a^{2} - \frac{676821269251097}{2583686924944061} a - \frac{42937465063}{130958838509}$, $\frac{1}{1765674412151682001155170027558991078592380950319852820596447985817723060991645966157698585687286096824138649997307176857800822492459432038485701663338278220349840764039114753283061590614286515321431} a^{35} - \frac{139578949634129000989498983925743861619979680137466694821772679060255777071539760686824403624603996796150009894695188032531777467239725055508112337852747014688018698992901932768284523}{1765674412151682001155170027558991078592380950319852820596447985817723060991645966157698585687286096824138649997307176857800822492459432038485701663338278220349840764039114753283061590614286515321431} a^{34} + \frac{6257299179626803573155549813493528142336555058381751519691242265149527535021035554245985355634483964655040887824376932024441267094419990146391613617230647488170325570659892134456631447029}{1765674412151682001155170027558991078592380950319852820596447985817723060991645966157698585687286096824138649997307176857800822492459432038485701663338278220349840764039114753283061590614286515321431} a^{33} + \frac{4790233029542254599770070477074275861419825189453110873196426033912980665669210285427294397722090053971568992043925756697418302108141785791445149784602980834780765093945540423902432453195158805}{1765674412151682001155170027558991078592380950319852820596447985817723060991645966157698585687286096824138649997307176857800822492459432038485701663338278220349840764039114753283061590614286515321431} a^{32} - \frac{9879406171561949931539351358374652987522290681570970935848590316492652823270415788335275763327633280951559849630040251781790286516312399017521451167941167990623298572721261704227628985462673207}{1765674412151682001155170027558991078592380950319852820596447985817723060991645966157698585687286096824138649997307176857800822492459432038485701663338278220349840764039114753283061590614286515321431} a^{31} + \frac{1217545121471979320301211988605378732531257370565351601002599554036684753169150631924500537428266522880717804309378933464942412250287489188979931759696630972649479243735726446972180307222582130}{1765674412151682001155170027558991078592380950319852820596447985817723060991645966157698585687286096824138649997307176857800822492459432038485701663338278220349840764039114753283061590614286515321431} a^{30} - \frac{564950525805222579429105638082857817578412107293080917210129574606719854351992644625184302213226556400187754150130854685666473287694610622590383958095166642447369001165322568835418531826660902698040}{1765674412151682001155170027558991078592380950319852820596447985817723060991645966157698585687286096824138649997307176857800822492459432038485701663338278220349840764039114753283061590614286515321431} a^{29} + \frac{23378480010270977836919302707937007461706338765171340555073134854696754378021401542161704269305593421898324399310412389433405809279941961617960304747675635242980332925007423500700179556619376296718}{1765674412151682001155170027558991078592380950319852820596447985817723060991645966157698585687286096824138649997307176857800822492459432038485701663338278220349840764039114753283061590614286515321431} a^{28} - \frac{853525178555869061117619199267419015435279544549659105213933195881642775368335450090221550196333853181864835344082751187845527502185985767223957878622289767174911075469901802177534968347052502465177}{1765674412151682001155170027558991078592380950319852820596447985817723060991645966157698585687286096824138649997307176857800822492459432038485701663338278220349840764039114753283061590614286515321431} a^{27} + \frac{729814016528775964486912167616617185158487535484875463736052611657959830550568155026820820887244646186419009694288611809261259614111174315253058029102448958354032488435531161537433838612106706362553}{1765674412151682001155170027558991078592380950319852820596447985817723060991645966157698585687286096824138649997307176857800822492459432038485701663338278220349840764039114753283061590614286515321431} a^{26} + \frac{534574205202408600159216225926665872400855628724632755882452932455961760192400920972386892794752509244585856908393965950442376120173304798851417081514968783945760699944582143477446299528733652796912}{1765674412151682001155170027558991078592380950319852820596447985817723060991645966157698585687286096824138649997307176857800822492459432038485701663338278220349840764039114753283061590614286515321431} a^{25} + \frac{145066258533943438638616528100939769161258110214947509793664640736380270038681595411382083278621061061260674935114652412511740971337673802825257093215669168842560414653635683710726191625592991568741}{1765674412151682001155170027558991078592380950319852820596447985817723060991645966157698585687286096824138649997307176857800822492459432038485701663338278220349840764039114753283061590614286515321431} a^{24} - \frac{146960770102411311629608054954071983807012729588802612376175436886217790289031037990812741608998874683911400013142235897111700990000897763422375763682906459745144059299053166122413062559453866065786}{1765674412151682001155170027558991078592380950319852820596447985817723060991645966157698585687286096824138649997307176857800822492459432038485701663338278220349840764039114753283061590614286515321431} a^{23} + \frac{697225760396175547839168078913659419622023796413572907988263647239465351938960774375871476814162259002748795597380563306366797332291304018972606383413636852139924303784308369153441615622277821071223}{1765674412151682001155170027558991078592380950319852820596447985817723060991645966157698585687286096824138649997307176857800822492459432038485701663338278220349840764039114753283061590614286515321431} a^{22} - \frac{610921780207824654464133662762960908464024759852604656708887506359636064915676921467747978115916165366132792905478453269716053129130382163120082173242931327047369335489862996097666233117750018265762}{1765674412151682001155170027558991078592380950319852820596447985817723060991645966157698585687286096824138649997307176857800822492459432038485701663338278220349840764039114753283061590614286515321431} a^{21} + \frac{315997683306131748693694844565242490169702562826554738975401911587212322638300504133795016328341042794986650668404181742465190867340222365924808960438486884367605051149063717145426143373798735704382}{1765674412151682001155170027558991078592380950319852820596447985817723060991645966157698585687286096824138649997307176857800822492459432038485701663338278220349840764039114753283061590614286515321431} a^{20} + \frac{312620002546455838923054177200820514953151451426332783047515978939350262314885924084037929499662154954408548919065559630422620964026958280061515311800201857547971827700710192433876717254245783924976}{1765674412151682001155170027558991078592380950319852820596447985817723060991645966157698585687286096824138649997307176857800822492459432038485701663338278220349840764039114753283061590614286515321431} a^{19} + \frac{171498696322756497814104870380492921990372020011093899835413573893489582991129563805095299201042776650859627580079567475424342225387211125708899618163026882318811589061611483399053336051070189420250}{1765674412151682001155170027558991078592380950319852820596447985817723060991645966157698585687286096824138649997307176857800822492459432038485701663338278220349840764039114753283061590614286515321431} a^{18} + \frac{332810184872460140306193056616476735070395619630620420522091719560606798149246269750959831398908664646646162329994131066687045426633164482806900473169659960380082303224732412536317352365354517278661}{1765674412151682001155170027558991078592380950319852820596447985817723060991645966157698585687286096824138649997307176857800822492459432038485701663338278220349840764039114753283061590614286515321431} a^{17} + \frac{752241286068595859652213270133920593825763595520138942929375200150519310414945020877523328452255153193931422891335767794953453099752213522905771070464167491504912162571468563906641780789773145217417}{1765674412151682001155170027558991078592380950319852820596447985817723060991645966157698585687286096824138649997307176857800822492459432038485701663338278220349840764039114753283061590614286515321431} a^{16} + \frac{172579281095602287711846384045060877334121788119461512570243195293572148115753449186894484192145863386994866901343286687145950674787104692579757069346709313584226213434286524036822792662061959387665}{1765674412151682001155170027558991078592380950319852820596447985817723060991645966157698585687286096824138649997307176857800822492459432038485701663338278220349840764039114753283061590614286515321431} a^{15} + \frac{813880513828871865018153454149303663739488053722257435700368692280189651480959455488629954168318812262086681017457296820902576653423668098356089165411874205316181070927071914782007441162517513978369}{1765674412151682001155170027558991078592380950319852820596447985817723060991645966157698585687286096824138649997307176857800822492459432038485701663338278220349840764039114753283061590614286515321431} a^{14} - \frac{494897206753077246876624686918191870112908003450572909107099908631308052075881703615265211746636818445836122349933456307389217631392992464167588727709219380133755350160850773853671262637697588134097}{1765674412151682001155170027558991078592380950319852820596447985817723060991645966157698585687286096824138649997307176857800822492459432038485701663338278220349840764039114753283061590614286515321431} a^{13} + \frac{100481981963505076680369262099714108717807758433231028635634513304713383924612601714021001803285417989199589588890748401116838284626572421627129180860458993708371182350112901445781108769752438913882}{1765674412151682001155170027558991078592380950319852820596447985817723060991645966157698585687286096824138649997307176857800822492459432038485701663338278220349840764039114753283061590614286515321431} a^{12} + \frac{633433564261463089714190328834926167382700493910484249926123691902460678100072751155996931490671212329998501275498460004818570651802789521679070016069982084393650350164834225554566801615645407872048}{1765674412151682001155170027558991078592380950319852820596447985817723060991645966157698585687286096824138649997307176857800822492459432038485701663338278220349840764039114753283061590614286515321431} a^{11} + \frac{198635569107239032804202027084089638489923839045509143618014479706215238220948346102000837490859731353593438537283887438129414999040763991197302575084563851339496699454464723894425878072690817056759}{1765674412151682001155170027558991078592380950319852820596447985817723060991645966157698585687286096824138649997307176857800822492459432038485701663338278220349840764039114753283061590614286515321431} a^{10} + \frac{32690589611488077820978917699267756643539307086106593076613956346281310971580030523503929221353844801976169760728806187332989231622817550079870823321238465021878813141477203715749135304042946826814}{1765674412151682001155170027558991078592380950319852820596447985817723060991645966157698585687286096824138649997307176857800822492459432038485701663338278220349840764039114753283061590614286515321431} a^{9} - \frac{413824808214600285462616728031058391183075719622894785760173762455094572196877439806427764169914201762509966582409065273284418857462351622836398688698393701355446871499093781897359936223712747571514}{1765674412151682001155170027558991078592380950319852820596447985817723060991645966157698585687286096824138649997307176857800822492459432038485701663338278220349840764039114753283061590614286515321431} a^{8} + \frac{879972742838038105878531173175813287198912239549871515258943872955329632505658727641355279489500921002964383040538605235980976187338000969154293438553649442550685473128541904676007208302068958262799}{1765674412151682001155170027558991078592380950319852820596447985817723060991645966157698585687286096824138649997307176857800822492459432038485701663338278220349840764039114753283061590614286515321431} a^{7} + \frac{867265613393750982982144222752055022395872915661196499739858694050811083140431717198821130314830300884784490488244802956624203609676346995638163235774552414344350561678376222692523819550474719305641}{1765674412151682001155170027558991078592380950319852820596447985817723060991645966157698585687286096824138649997307176857800822492459432038485701663338278220349840764039114753283061590614286515321431} a^{6} - \frac{851091533494772683158887351141834386566144169808142905935799855238008114489564058925755537468369351547500478377068013629777252027493556726004879467755171958039910773612633528781319701457230402067237}{1765674412151682001155170027558991078592380950319852820596447985817723060991645966157698585687286096824138649997307176857800822492459432038485701663338278220349840764039114753283061590614286515321431} a^{5} - \frac{715890618795680620386633438936263779394437461819309384514285662257028208631996471849185503880821223332954619514852814230897849506020945851162822180271062538975437170499943003684074291704993093198244}{1765674412151682001155170027558991078592380950319852820596447985817723060991645966157698585687286096824138649997307176857800822492459432038485701663338278220349840764039114753283061590614286515321431} a^{4} + \frac{339048891914902049516177252210156212416118818189879521048978559721925268845782736087181338799354881574200189341009806179937623061217045788018999984283674038644793944017916613127188132321343803396359}{1765674412151682001155170027558991078592380950319852820596447985817723060991645966157698585687286096824138649997307176857800822492459432038485701663338278220349840764039114753283061590614286515321431} a^{3} - \frac{316179993893264401358615764472189318896900643328268964640845392509748754749874975055267532323356019699962083241671495394606252423641641389675983444366943762264720897107968528884480936441976056169254}{1765674412151682001155170027558991078592380950319852820596447985817723060991645966157698585687286096824138649997307176857800822492459432038485701663338278220349840764039114753283061590614286515321431} a^{2} + \frac{456434227483824626226192860224233554797199946996277687317915711941343237330510051694408783062118107555813508263777685963819691943911687970901549387385128992529128772959375494023331989815979019480944}{1765674412151682001155170027558991078592380950319852820596447985817723060991645966157698585687286096824138649997307176857800822492459432038485701663338278220349840764039114753283061590614286515321431} a + \frac{5802440313564941930905266544152049193691844465510097156305092811683241716523855415719481049889450034722065300668191140845053544637568499929721084606851905344847527219933573940949660200043357162}{89496396784007400332260632954482795812883620574780922530105326464479855085997565317943057716421820509105309442815508989700482664730063968700172419450467749016667888085514458577883399595229688039}$

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

Class group and class number

Not computed

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:  \( -1 \) (order $2$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Not computed
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  Not computed
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times C_{12}$ (as 36T3):

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 $C_3\times C_{12}$
Character table for $C_3\times C_{12}$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{9})^+\), 3.3.169.1, 3.3.13689.1, 3.3.13689.2, 4.0.21125.1, 6.6.820125.1, 6.6.3570125.1, 6.6.23423590125.1, 6.6.23423590125.2, 9.9.2565164201769.1, 12.0.405816992857986328125.1, 12.0.269254866892578125.1, 12.0.11590539133016947517578125.1, 12.0.11590539133016947517578125.2, 18.18.12851694105541388560018283203125.1

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 ${\href{/LocalNumberField/2.12.0.1}{12} }^{3}$ R R ${\href{/LocalNumberField/7.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{6}$ R ${\href{/LocalNumberField/17.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{12}$ ${\href{/LocalNumberField/23.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/37.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/43.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/47.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{9}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{12}$

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
$5$5.12.9.1$x^{12} - 10 x^{8} - 375 x^{4} - 2000$$4$$3$$9$$C_{12}$$[\ ]_{4}^{3}$
5.12.9.1$x^{12} - 10 x^{8} - 375 x^{4} - 2000$$4$$3$$9$$C_{12}$$[\ ]_{4}^{3}$
5.12.9.1$x^{12} - 10 x^{8} - 375 x^{4} - 2000$$4$$3$$9$$C_{12}$$[\ ]_{4}^{3}$
13Data not computed