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Properties

Label 27.27.1197336751...1009.3
Degree $27$
Signature $[27, 0]$
Discriminant $3^{36}\cdot 7^{18}\cdot 19^{24}$
Root discriminant $216.89$
Ramified primes $3, 7, 19$
Class number $3$ (GRH)
Class group $[3]$ (GRH)
Galois group $C_3\times C_9$ (as 27T2)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-23416482473, 56808287184, 184625797869, -328067407732, -588923699037, 646172244414, 941607757277, -557167152231, -812037279015, 211466391591, 393523655928, -22064613804, -110231138950, -7767041349, 18094018455, 2750706459, -1747356537, -368176113, 98166647, 25802124, -3087864, -1007250, 49233, 21600, -300, -234, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^27 - 234*x^25 - 300*x^24 + 21600*x^23 + 49233*x^22 - 1007250*x^21 - 3087864*x^20 + 25802124*x^19 + 98166647*x^18 - 368176113*x^17 - 1747356537*x^16 + 2750706459*x^15 + 18094018455*x^14 - 7767041349*x^13 - 110231138950*x^12 - 22064613804*x^11 + 393523655928*x^10 + 211466391591*x^9 - 812037279015*x^8 - 557167152231*x^7 + 941607757277*x^6 + 646172244414*x^5 - 588923699037*x^4 - 328067407732*x^3 + 184625797869*x^2 + 56808287184*x - 23416482473)
 
gp: K = bnfinit(x^27 - 234*x^25 - 300*x^24 + 21600*x^23 + 49233*x^22 - 1007250*x^21 - 3087864*x^20 + 25802124*x^19 + 98166647*x^18 - 368176113*x^17 - 1747356537*x^16 + 2750706459*x^15 + 18094018455*x^14 - 7767041349*x^13 - 110231138950*x^12 - 22064613804*x^11 + 393523655928*x^10 + 211466391591*x^9 - 812037279015*x^8 - 557167152231*x^7 + 941607757277*x^6 + 646172244414*x^5 - 588923699037*x^4 - 328067407732*x^3 + 184625797869*x^2 + 56808287184*x - 23416482473, 1)
 

Normalized defining polynomial

\( x^{27} - 234 x^{25} - 300 x^{24} + 21600 x^{23} + 49233 x^{22} - 1007250 x^{21} - 3087864 x^{20} + 25802124 x^{19} + 98166647 x^{18} - 368176113 x^{17} - 1747356537 x^{16} + 2750706459 x^{15} + 18094018455 x^{14} - 7767041349 x^{13} - 110231138950 x^{12} - 22064613804 x^{11} + 393523655928 x^{10} + 211466391591 x^{9} - 812037279015 x^{8} - 557167152231 x^{7} + 941607757277 x^{6} + 646172244414 x^{5} - 588923699037 x^{4} - 328067407732 x^{3} + 184625797869 x^{2} + 56808287184 x - 23416482473 \)

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

Invariants

Degree:  $27$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[27, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1197336751300349460100016353165918520894496565611371829528951009=3^{36}\cdot 7^{18}\cdot 19^{24}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $216.89$
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:  $3, 7, 19$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1197=3^{2}\cdot 7\cdot 19\)
Dirichlet character group:    $\lbrace$$\chi_{1197}(1024,·)$, $\chi_{1197}(1,·)$, $\chi_{1197}(898,·)$, $\chi_{1197}(4,·)$, $\chi_{1197}(967,·)$, $\chi_{1197}(64,·)$, $\chi_{1197}(1033,·)$, $\chi_{1197}(970,·)$, $\chi_{1197}(256,·)$, $\chi_{1197}(16,·)$, $\chi_{1197}(1108,·)$, $\chi_{1197}(277,·)$, $\chi_{1197}(121,·)$, $\chi_{1197}(1156,·)$, $\chi_{1197}(1051,·)$, $\chi_{1197}(541,·)$, $\chi_{1197}(928,·)$, $\chi_{1197}(289,·)$, $\chi_{1197}(739,·)$, $\chi_{1197}(484,·)$, $\chi_{1197}(613,·)$, $\chi_{1197}(232,·)$, $\chi_{1197}(841,·)$, $\chi_{1197}(562,·)$, $\chi_{1197}(823,·)$, $\chi_{1197}(505,·)$, $\chi_{1197}(58,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{11} a^{15} + \frac{5}{11} a^{14} + \frac{1}{11} a^{13} + \frac{5}{11} a^{12} - \frac{5}{11} a^{11} - \frac{5}{11} a^{10} + \frac{5}{11} a^{9} + \frac{4}{11} a^{8} + \frac{2}{11} a^{7} - \frac{3}{11} a^{6} + \frac{3}{11} a^{5} - \frac{3}{11} a^{4} - \frac{1}{11} a^{3} + \frac{3}{11} a^{2} - \frac{5}{11} a - \frac{3}{11}$, $\frac{1}{11} a^{16} - \frac{2}{11} a^{14} + \frac{3}{11} a^{12} - \frac{2}{11} a^{11} - \frac{3}{11} a^{10} + \frac{1}{11} a^{9} + \frac{4}{11} a^{8} - \frac{2}{11} a^{7} - \frac{4}{11} a^{6} + \frac{4}{11} a^{5} + \frac{3}{11} a^{4} - \frac{3}{11} a^{3} + \frac{2}{11} a^{2} + \frac{4}{11}$, $\frac{1}{11} a^{17} - \frac{1}{11} a^{14} + \frac{5}{11} a^{13} - \frac{3}{11} a^{12} - \frac{2}{11} a^{11} + \frac{2}{11} a^{10} + \frac{3}{11} a^{9} - \frac{5}{11} a^{8} - \frac{2}{11} a^{6} - \frac{2}{11} a^{5} + \frac{2}{11} a^{4} - \frac{5}{11} a^{2} + \frac{5}{11} a + \frac{5}{11}$, $\frac{1}{44} a^{18} - \frac{1}{44} a^{17} - \frac{9}{22} a^{13} - \frac{5}{44} a^{12} - \frac{3}{11} a^{11} + \frac{7}{44} a^{10} + \frac{2}{11} a^{9} - \frac{1}{22} a^{8} + \frac{19}{44} a^{6} + \frac{9}{22} a^{5} - \frac{5}{44} a^{4} - \frac{17}{44} a^{3} - \frac{9}{44} a^{2} - \frac{4}{11} a + \frac{3}{44}$, $\frac{1}{44} a^{19} - \frac{1}{44} a^{17} - \frac{9}{22} a^{14} + \frac{21}{44} a^{13} - \frac{17}{44} a^{12} - \frac{5}{44} a^{11} + \frac{15}{44} a^{10} + \frac{3}{22} a^{9} - \frac{1}{22} a^{8} + \frac{19}{44} a^{7} - \frac{7}{44} a^{6} + \frac{13}{44} a^{5} - \frac{1}{2} a^{4} + \frac{9}{22} a^{3} + \frac{19}{44} a^{2} - \frac{13}{44} a + \frac{3}{44}$, $\frac{1}{44} a^{20} - \frac{1}{44} a^{17} - \frac{1}{22} a^{15} + \frac{13}{44} a^{14} - \frac{19}{44} a^{13} - \frac{9}{22} a^{12} + \frac{1}{4} a^{11} + \frac{21}{44} a^{10} - \frac{1}{22} a^{9} - \frac{7}{44} a^{8} - \frac{19}{44} a^{7} - \frac{4}{11} a^{6} + \frac{9}{44} a^{4} - \frac{7}{22} a^{3} - \frac{9}{22} a^{2} - \frac{5}{44} a - \frac{1}{44}$, $\frac{1}{1364} a^{21} + \frac{3}{341} a^{20} - \frac{1}{341} a^{19} - \frac{1}{124} a^{18} + \frac{17}{682} a^{17} + \frac{15}{682} a^{16} + \frac{17}{1364} a^{15} - \frac{33}{124} a^{14} + \frac{85}{682} a^{13} - \frac{431}{1364} a^{12} - \frac{503}{1364} a^{11} + \frac{86}{341} a^{10} - \frac{17}{44} a^{9} + \frac{621}{1364} a^{8} + \frac{105}{341} a^{7} - \frac{337}{682} a^{6} - \frac{1}{44} a^{5} - \frac{10}{31} a^{4} + \frac{134}{341} a^{3} - \frac{615}{1364} a^{2} - \frac{621}{1364} a + \frac{339}{682}$, $\frac{1}{1364} a^{22} + \frac{7}{1364} a^{20} + \frac{3}{682} a^{19} + \frac{1}{124} a^{18} + \frac{25}{1364} a^{17} + \frac{29}{1364} a^{16} - \frac{9}{1364} a^{15} - \frac{19}{44} a^{14} - \frac{549}{1364} a^{13} - \frac{663}{1364} a^{12} + \frac{45}{341} a^{11} + \frac{261}{682} a^{10} + \frac{125}{1364} a^{9} - \frac{553}{1364} a^{8} - \frac{18}{341} a^{7} - \frac{623}{1364} a^{6} + \frac{87}{1364} a^{5} - \frac{65}{341} a^{4} + \frac{106}{341} a^{3} - \frac{557}{1364} a^{2} - \frac{213}{682} a + \frac{79}{1364}$, $\frac{1}{1364} a^{23} + \frac{15}{1364} a^{20} + \frac{2}{341} a^{19} + \frac{9}{1364} a^{18} - \frac{27}{682} a^{17} + \frac{29}{1364} a^{16} - \frac{13}{682} a^{15} + \frac{659}{1364} a^{14} + \frac{255}{1364} a^{13} + \frac{37}{124} a^{12} - \frac{111}{1364} a^{11} + \frac{269}{682} a^{10} + \frac{40}{341} a^{9} + \frac{193}{682} a^{8} - \frac{587}{1364} a^{7} + \frac{21}{44} a^{6} - \frac{34}{341} a^{5} - \frac{54}{341} a^{4} + \frac{5}{11} a^{3} - \frac{523}{1364} a^{2} - \frac{149}{341} a + \frac{617}{1364}$, $\frac{1}{21568955188} a^{24} + \frac{34737}{695772748} a^{23} - \frac{467717}{21568955188} a^{22} - \frac{277065}{1960814108} a^{21} - \frac{59457401}{5392238797} a^{20} - \frac{222363871}{21568955188} a^{19} - \frac{227729635}{21568955188} a^{18} + \frac{308480665}{21568955188} a^{17} - \frac{171622089}{5392238797} a^{16} + \frac{188868587}{5392238797} a^{15} + \frac{90259457}{21568955188} a^{14} - \frac{443316635}{1960814108} a^{13} - \frac{2025038005}{5392238797} a^{12} + \frac{6205368187}{21568955188} a^{11} - \frac{3246445681}{21568955188} a^{10} - \frac{10495872157}{21568955188} a^{9} - \frac{183598849}{10784477594} a^{8} - \frac{4176761647}{10784477594} a^{7} - \frac{526765095}{21568955188} a^{6} + \frac{6227782879}{21568955188} a^{5} + \frac{2565632477}{21568955188} a^{4} - \frac{2076951425}{5392238797} a^{3} + \frac{9992419403}{21568955188} a^{2} - \frac{3448056759}{21568955188} a + \frac{10098707979}{21568955188}$, $\frac{1}{21568955188} a^{25} + \frac{116759}{10784477594} a^{23} - \frac{3803883}{21568955188} a^{22} - \frac{3153013}{10784477594} a^{21} - \frac{232067651}{21568955188} a^{20} - \frac{25308691}{5392238797} a^{19} + \frac{1952669}{490203527} a^{18} - \frac{47608893}{1960814108} a^{17} - \frac{709931905}{21568955188} a^{16} + \frac{540962035}{21568955188} a^{15} - \frac{2235055739}{21568955188} a^{14} - \frac{9099009143}{21568955188} a^{13} + \frac{269258867}{695772748} a^{12} + \frac{5066359629}{10784477594} a^{11} - \frac{825985099}{1960814108} a^{10} - \frac{5382053751}{21568955188} a^{9} - \frac{8979685855}{21568955188} a^{8} + \frac{1475013414}{5392238797} a^{7} + \frac{2808886737}{21568955188} a^{6} + \frac{471976800}{5392238797} a^{5} + \frac{1339813756}{5392238797} a^{4} - \frac{6681943497}{21568955188} a^{3} + \frac{2024614058}{5392238797} a^{2} + \frac{2075988435}{5392238797} a + \frac{2766509741}{21568955188}$, $\frac{1}{66200155352233865256600407946205800375077766632679530542662656095491007761929743915932235689291940795746454555468492} a^{26} + \frac{1036285430369519887920461572357552710753615864843322490716521289541658231500411877597629693208290906520891}{66200155352233865256600407946205800375077766632679530542662656095491007761929743915932235689291940795746454555468492} a^{25} - \frac{1435868583650749469709891417895509520963254263480901818284008968516941924389141732176718590661176664148857}{66200155352233865256600407946205800375077766632679530542662656095491007761929743915932235689291940795746454555468492} a^{24} - \frac{282995988836108630892393675608027670112758832045715189485918714137493503804261590451784223346475131118881967483}{66200155352233865256600407946205800375077766632679530542662656095491007761929743915932235689291940795746454555468492} a^{23} + \frac{2563125930072613189684657956696338049122859451854133757986539467797602429274340802816037258841538629470656723085}{66200155352233865256600407946205800375077766632679530542662656095491007761929743915932235689291940795746454555468492} a^{22} - \frac{21250305031877752929600123636895545325049584418250479085043247525534462101094362160238118188456441548847655690557}{66200155352233865256600407946205800375077766632679530542662656095491007761929743915932235689291940795746454555468492} a^{21} + \frac{89399443222039373487901660815377880272086418915373404181485772580982800504387998423356982801208031526939928659157}{33100077676116932628300203973102900187538883316339765271331328047745503880964871957966117844645970397873227277734246} a^{20} + \frac{16851639578580359004743740647545693268736053399604850534449439329334543642620987409748300227663124099682242658558}{1504548985278042392195463816959222735797221968924534785060514911261159267316585088998459902029362290812419421715193} a^{19} + \frac{40567580625568570155465637535948888112868796353858889251836763916077544444179369939886009786645730059853754387076}{16550038838058466314150101986551450093769441658169882635665664023872751940482435978983058922322985198936613638867123} a^{18} + \frac{1000345044286506891016261166058253306328790432201043362974338589489737350298129399851582969224173086621894833933315}{66200155352233865256600407946205800375077766632679530542662656095491007761929743915932235689291940795746454555468492} a^{17} - \frac{12773177298001098631064642194734370533793273500787452440297757088956722448040944835448330244494775781374720151927}{33100077676116932628300203973102900187538883316339765271331328047745503880964871957966117844645970397873227277734246} a^{16} - \frac{740323221250828116749670928408064900956882858980281574939258994807858765424278169153976661032788145069650853572324}{16550038838058466314150101986551450093769441658169882635665664023872751940482435978983058922322985198936613638867123} a^{15} + \frac{24826629615010901766530964282216157858958765261729045154890501640524036281595008837080440307591702523920317725033269}{66200155352233865256600407946205800375077766632679530542662656095491007761929743915932235689291940795746454555468492} a^{14} - \frac{24726311645118776229030896434482738620903504215083950482875121677071067599017086111928862919687905580597089311176089}{66200155352233865256600407946205800375077766632679530542662656095491007761929743915932235689291940795746454555468492} a^{13} + \frac{15436444966368181732832911551358514625853505182549087213638807550587515255190265466396867199435188515217415349792395}{33100077676116932628300203973102900187538883316339765271331328047745503880964871957966117844645970397873227277734246} a^{12} + \frac{3678726091926720598219625481355040452543992322082936050161850268881334927391628553019249034215187406360795920142749}{33100077676116932628300203973102900187538883316339765271331328047745503880964871957966117844645970397873227277734246} a^{11} - \frac{589959390968201770349163486025863676870453594257902559676424963703382722795275890642789275890887955960557099609551}{3009097970556084784390927633918445471594443937849069570121029822522318534633170177996919804058724581624838843430386} a^{10} + \frac{13846441339359344471952154896624877203777389107030672331743474235239478878002971431412757091370825843543947332652255}{66200155352233865256600407946205800375077766632679530542662656095491007761929743915932235689291940795746454555468492} a^{9} + \frac{7590948621910544499177689935274504120398780919090343600316956374884387032839967088826852515994327868622944754982647}{16550038838058466314150101986551450093769441658169882635665664023872751940482435978983058922322985198936613638867123} a^{8} - \frac{2814438559306753263889338258175567704483438450349664516185322466059324861152762952215297951748046333090720074776675}{16550038838058466314150101986551450093769441658169882635665664023872751940482435978983058922322985198936613638867123} a^{7} - \frac{4724816987874055351036729218162182695328319345700839874682369864692812021844066865470032344108366397635941072382693}{16550038838058466314150101986551450093769441658169882635665664023872751940482435978983058922322985198936613638867123} a^{6} - \frac{609674323575633445437461346964927481626968986474246832535006275289985258853642807355306008793374550280384426076815}{1504548985278042392195463816959222735797221968924534785060514911261159267316585088998459902029362290812419421715193} a^{5} - \frac{7225048843268911623635988043445944179080684374464013077820195144032279504644528170795479422157019360592466859781475}{16550038838058466314150101986551450093769441658169882635665664023872751940482435978983058922322985198936613638867123} a^{4} - \frac{847011872371335654761026272612078223206799006131461502279758342290464101077151121348219824356860511967692614218330}{16550038838058466314150101986551450093769441658169882635665664023872751940482435978983058922322985198936613638867123} a^{3} - \frac{54395261872581846769706986883157316840685606126543481225888199663749429170935113823117168851468358782270006280230}{1504548985278042392195463816959222735797221968924534785060514911261159267316585088998459902029362290812419421715193} a^{2} - \frac{13683050219781935877098435427230166355763956637951742674244127623433083166055829706335799436458038415062334123979971}{66200155352233865256600407946205800375077766632679530542662656095491007761929743915932235689291940795746454555468492} a + \frac{5367856675301606890658241918165126951248046142209756234929028061682592722579404871689015170451619436619940719926229}{66200155352233865256600407946205800375077766632679530542662656095491007761929743915932235689291940795746454555468492}$

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

Class group and class number

$C_{3}$, which has order $3$ (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:  $26$
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:  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:  \( 46774338025873890000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times C_9$ (as 27T2):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 27
The 27 conjugacy class representatives for $C_3\times C_9$
Character table for $C_3\times C_9$ is not computed

Intermediate fields

3.3.1432809.4, 3.3.361.1, 3.3.3969.1, 3.3.1432809.1, 9.9.2941473244627851129.10, 9.9.1998099208210609.2, 9.9.1061871841310654257569.2, 9.9.9025761726072081.1

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 ${\href{/LocalNumberField/2.9.0.1}{9} }^{3}$ R ${\href{/LocalNumberField/5.9.0.1}{9} }^{3}$ R ${\href{/LocalNumberField/11.3.0.1}{3} }^{9}$ ${\href{/LocalNumberField/13.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/17.9.0.1}{9} }^{3}$ R ${\href{/LocalNumberField/23.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/29.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/31.1.0.1}{1} }^{27}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{9}$ ${\href{/LocalNumberField/41.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/43.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/47.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/53.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/59.9.0.1}{9} }^{3}$

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
$7$7.9.6.1$x^{9} + 42 x^{6} + 539 x^{3} + 2744$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
7.9.6.1$x^{9} + 42 x^{6} + 539 x^{3} + 2744$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
7.9.6.1$x^{9} + 42 x^{6} + 539 x^{3} + 2744$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
19Data not computed