↑

Properties

Label 27.27.1197336751...1009.8
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 $12$ (GRH)
Class group $[2, 6]$ (GRH)
Galois group $C_3\times C_9$ (as 27T2)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![353597867, -3498281892, -31775939244, 149958202254, 1371196307376, 2956555931049, 1456668702112, -2310077106372, -2397853291923, 457955338088, 1102440345990, 64602332640, -249129898160, -40489809264, 31852073331, 7124089977, -2427785484, -659712528, 111309958, 35915889, -2979873, -1176444, 42558, 22626, -249, -234, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^27 - 234*x^25 - 249*x^24 + 22626*x^23 + 42558*x^22 - 1176444*x^21 - 2979873*x^20 + 35915889*x^19 + 111309958*x^18 - 659712528*x^17 - 2427785484*x^16 + 7124089977*x^15 + 31852073331*x^14 - 40489809264*x^13 - 249129898160*x^12 + 64602332640*x^11 + 1102440345990*x^10 + 457955338088*x^9 - 2397853291923*x^8 - 2310077106372*x^7 + 1456668702112*x^6 + 2956555931049*x^5 + 1371196307376*x^4 + 149958202254*x^3 - 31775939244*x^2 - 3498281892*x + 353597867)
 
gp: K = bnfinit(x^27 - 234*x^25 - 249*x^24 + 22626*x^23 + 42558*x^22 - 1176444*x^21 - 2979873*x^20 + 35915889*x^19 + 111309958*x^18 - 659712528*x^17 - 2427785484*x^16 + 7124089977*x^15 + 31852073331*x^14 - 40489809264*x^13 - 249129898160*x^12 + 64602332640*x^11 + 1102440345990*x^10 + 457955338088*x^9 - 2397853291923*x^8 - 2310077106372*x^7 + 1456668702112*x^6 + 2956555931049*x^5 + 1371196307376*x^4 + 149958202254*x^3 - 31775939244*x^2 - 3498281892*x + 353597867, 1)
 

Normalized defining polynomial

\( x^{27} - 234 x^{25} - 249 x^{24} + 22626 x^{23} + 42558 x^{22} - 1176444 x^{21} - 2979873 x^{20} + 35915889 x^{19} + 111309958 x^{18} - 659712528 x^{17} - 2427785484 x^{16} + 7124089977 x^{15} + 31852073331 x^{14} - 40489809264 x^{13} - 249129898160 x^{12} + 64602332640 x^{11} + 1102440345990 x^{10} + 457955338088 x^{9} - 2397853291923 x^{8} - 2310077106372 x^{7} + 1456668702112 x^{6} + 2956555931049 x^{5} + 1371196307376 x^{4} + 149958202254 x^{3} - 31775939244 x^{2} - 3498281892 x + 353597867 \)

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$$\chi_{1197}(64,·)$, $\chi_{1197}(1,·)$, $\chi_{1197}(1030,·)$, $\chi_{1197}(961,·)$, $\chi_{1197}(520,·)$, $\chi_{1197}(457,·)$, $\chi_{1197}(652,·)$, $\chi_{1197}(655,·)$, $\chi_{1197}(403,·)$, $\chi_{1197}(1108,·)$, $\chi_{1197}(85,·)$, $\chi_{1197}(25,·)$, $\chi_{1197}(541,·)$, $\chi_{1197}(928,·)$, $\chi_{1197}(289,·)$, $\chi_{1197}(739,·)$, $\chi_{1197}(613,·)$, $\chi_{1197}(358,·)$, $\chi_{1197}(169,·)$, $\chi_{1197}(43,·)$, $\chi_{1197}(814,·)$, $\chi_{1197}(625,·)$, $\chi_{1197}(1075,·)$, $\chi_{1197}(505,·)$, $\chi_{1197}(634,·)$, $\chi_{1197}(571,·)$, $\chi_{1197}(499,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{7} a^{15} + \frac{1}{7} a^{14} - \frac{2}{7} a^{13} + \frac{2}{7} a^{12} + \frac{3}{7} a^{11} + \frac{1}{7} a^{10} + \frac{3}{7} a^{8} + \frac{1}{7} a^{7} - \frac{2}{7} a^{6} + \frac{1}{7} a^{5} - \frac{2}{7} a^{4} - \frac{1}{7} a^{3}$, $\frac{1}{7} a^{16} - \frac{3}{7} a^{14} - \frac{3}{7} a^{13} + \frac{1}{7} a^{12} - \frac{2}{7} a^{11} - \frac{1}{7} a^{10} + \frac{3}{7} a^{9} - \frac{2}{7} a^{8} - \frac{3}{7} a^{7} + \frac{3}{7} a^{6} - \frac{3}{7} a^{5} + \frac{1}{7} a^{4} + \frac{1}{7} a^{3}$, $\frac{1}{7} a^{17} + \frac{2}{7} a^{13} - \frac{3}{7} a^{12} + \frac{1}{7} a^{11} - \frac{1}{7} a^{10} - \frac{2}{7} a^{9} - \frac{1}{7} a^{8} - \frac{1}{7} a^{7} - \frac{2}{7} a^{6} - \frac{3}{7} a^{5} + \frac{2}{7} a^{4} - \frac{3}{7} a^{3}$, $\frac{1}{245} a^{18} + \frac{8}{245} a^{17} + \frac{8}{245} a^{16} - \frac{16}{245} a^{15} - \frac{23}{49} a^{14} - \frac{17}{35} a^{13} - \frac{12}{245} a^{12} + \frac{18}{49} a^{11} + \frac{113}{245} a^{10} + \frac{13}{35} a^{9} - \frac{17}{245} a^{8} - \frac{1}{245} a^{7} - \frac{61}{245} a^{6} + \frac{99}{245} a^{5} - \frac{16}{49} a^{4} - \frac{1}{7} a^{3} + \frac{12}{35} a^{2} + \frac{8}{35} a + \frac{2}{5}$, $\frac{1}{245} a^{19} + \frac{2}{35} a^{17} - \frac{2}{49} a^{16} + \frac{13}{245} a^{15} + \frac{101}{245} a^{14} - \frac{22}{49} a^{13} + \frac{46}{245} a^{12} + \frac{58}{245} a^{11} + \frac{27}{245} a^{10} + \frac{12}{49} a^{9} - \frac{15}{49} a^{8} - \frac{88}{245} a^{7} - \frac{78}{245} a^{6} - \frac{67}{245} a^{5} + \frac{16}{49} a^{4} - \frac{3}{35} a^{3} + \frac{17}{35} a^{2} - \frac{3}{7} a - \frac{1}{5}$, $\frac{1}{245} a^{20} - \frac{17}{245} a^{17} + \frac{6}{245} a^{16} + \frac{2}{49} a^{15} - \frac{22}{49} a^{14} + \frac{32}{245} a^{13} + \frac{121}{245} a^{12} - \frac{78}{245} a^{11} - \frac{87}{245} a^{10} - \frac{19}{245} a^{9} + \frac{23}{49} a^{8} - \frac{64}{245} a^{7} + \frac{52}{245} a^{6} - \frac{46}{245} a^{5} + \frac{12}{35} a^{4} - \frac{3}{35} a^{3} - \frac{8}{35} a^{2} - \frac{2}{5} a + \frac{2}{5}$, $\frac{1}{9065} a^{21} - \frac{17}{9065} a^{20} + \frac{4}{9065} a^{19} - \frac{1}{1295} a^{18} - \frac{164}{9065} a^{17} + \frac{53}{9065} a^{16} - \frac{563}{9065} a^{15} + \frac{1891}{9065} a^{14} + \frac{4142}{9065} a^{13} - \frac{41}{9065} a^{12} + \frac{4471}{9065} a^{11} - \frac{1397}{9065} a^{10} - \frac{4502}{9065} a^{9} - \frac{4099}{9065} a^{8} + \frac{1373}{9065} a^{7} + \frac{2943}{9065} a^{6} - \frac{4222}{9065} a^{5} + \frac{4196}{9065} a^{4} + \frac{556}{1295} a^{3} - \frac{113}{259} a^{2} - \frac{568}{1295} a - \frac{33}{185}$, $\frac{1}{9065} a^{22} + \frac{11}{9065} a^{20} - \frac{13}{9065} a^{19} + \frac{13}{9065} a^{18} + \frac{8}{1813} a^{17} + \frac{6}{1295} a^{16} - \frac{58}{9065} a^{15} - \frac{564}{1813} a^{14} - \frac{2924}{9065} a^{13} + \frac{6}{35} a^{12} + \frac{788}{1813} a^{11} - \frac{4423}{9065} a^{10} - \frac{1601}{9065} a^{9} + \frac{1213}{9065} a^{8} - \frac{3279}{9065} a^{7} - \frac{293}{9065} a^{6} - \frac{4197}{9065} a^{5} + \frac{3148}{9065} a^{4} + \frac{451}{1295} a^{3} + \frac{113}{1295} a^{2} - \frac{193}{1295} a - \frac{43}{185}$, $\frac{1}{9065} a^{23} - \frac{11}{9065} a^{20} + \frac{6}{9065} a^{19} + \frac{6}{9065} a^{18} - \frac{559}{9065} a^{17} - \frac{419}{9065} a^{16} - \frac{3}{259} a^{15} + \frac{1472}{9065} a^{14} + \frac{463}{1295} a^{13} + \frac{893}{1813} a^{12} + \frac{3487}{9065} a^{11} - \frac{1108}{9065} a^{10} + \frac{1044}{9065} a^{9} - \frac{29}{245} a^{8} - \frac{4111}{9065} a^{7} + \frac{345}{1813} a^{6} + \frac{456}{1295} a^{5} - \frac{1374}{9065} a^{4} - \frac{564}{1295} a^{3} + \frac{324}{1295} a^{2} + \frac{619}{1295} a - \frac{81}{185}$, $\frac{1}{3646314665} a^{24} - \frac{147811}{3646314665} a^{23} - \frac{2931}{520902095} a^{22} + \frac{214}{3646314665} a^{21} + \frac{740636}{729262933} a^{20} - \frac{5718653}{3646314665} a^{19} + \frac{52701}{104180419} a^{18} + \frac{129004437}{3646314665} a^{17} + \frac{16528104}{520902095} a^{16} + \frac{36374978}{729262933} a^{15} - \frac{7090173}{14882917} a^{14} - \frac{512437257}{3646314665} a^{13} + \frac{29362939}{520902095} a^{12} - \frac{945014362}{3646314665} a^{11} + \frac{853239663}{3646314665} a^{10} + \frac{650783046}{3646314665} a^{9} + \frac{2677833}{10630655} a^{8} - \frac{502650}{729262933} a^{7} + \frac{139882009}{3646314665} a^{6} + \frac{32249647}{104180419} a^{5} + \frac{31542788}{104180419} a^{4} + \frac{247852798}{520902095} a^{3} + \frac{4887319}{74414585} a^{2} - \frac{33088716}{74414585} a - \frac{26287292}{74414585}$, $\frac{1}{375570410495} a^{25} - \frac{4}{75114082099} a^{24} + \frac{193882}{53652915785} a^{23} + \frac{3593154}{75114082099} a^{22} - \frac{17363117}{375570410495} a^{21} + \frac{370304883}{375570410495} a^{20} - \frac{5538301}{53652915785} a^{19} + \frac{49957025}{75114082099} a^{18} - \frac{452181558}{7664702255} a^{17} + \frac{5526908766}{375570410495} a^{16} + \frac{281143614}{10730583157} a^{15} - \frac{124285031692}{375570410495} a^{14} - \frac{4468925129}{10730583157} a^{13} - \frac{69457937246}{375570410495} a^{12} - \frac{28407706359}{375570410495} a^{11} + \frac{33738516844}{375570410495} a^{10} - \frac{2493755007}{10730583157} a^{9} + \frac{122868039646}{375570410495} a^{8} + \frac{59840133494}{375570410495} a^{7} - \frac{8998024089}{53652915785} a^{6} + \frac{25629575439}{53652915785} a^{5} - \frac{22912475561}{53652915785} a^{4} + \frac{74545731}{218991493} a^{3} + \frac{2140174019}{7664702255} a^{2} + \frac{2306786262}{7664702255} a + \frac{1706899}{10630655}$, $\frac{1}{158977845631535694908133348236503730923407790816086380359767529735213385952037616981036094837564474518945521958765} a^{26} - \frac{82835426005037020968297367520567730888904682281439939462007496274817378135735065013387383604249560966}{158977845631535694908133348236503730923407790816086380359767529735213385952037616981036094837564474518945521958765} a^{25} - \frac{3966113390669871779434067848773684757686073362341075619209123093148871076715129380080599664480487103839}{31795569126307138981626669647300746184681558163217276071953505947042677190407523396207218967512894903789104391753} a^{24} + \frac{5637530834348238438196302311288618562803172826293655421951723477043979478534496287166143727404555691363138734}{158977845631535694908133348236503730923407790816086380359767529735213385952037616981036094837564474518945521958765} a^{23} + \frac{52604533958471436956975510801608054419052830937941294954810592839062710430126851329185863089330836298889389}{4542224160901019854518095663900106597811651166173896581707643706720382455772503342315316995358984986255586341679} a^{22} - \frac{1306395569811286739332137717655482086199493664093481016409672453881129035296127516760133395356355623461483440}{31795569126307138981626669647300746184681558163217276071953505947042677190407523396207218967512894903789104391753} a^{21} - \frac{178163780820808680829194470184256352231965482107067957139231618572323143054884416849069477305786631638268606268}{158977845631535694908133348236503730923407790816086380359767529735213385952037616981036094837564474518945521958765} a^{20} + \frac{260873321651974058057761787835299299590262443484878834126736459211400837897975324554027382034121136464281987207}{158977845631535694908133348236503730923407790816086380359767529735213385952037616981036094837564474518945521958765} a^{19} - \frac{50230401568373875942966517153540360246203905947008086937348437001815636034883132624135647058782319930761886909}{31795569126307138981626669647300746184681558163217276071953505947042677190407523396207218967512894903789104391753} a^{18} - \frac{9011746322820635965605553222118878963160947575626384234759779433715105573795949182425104133336276813986369601186}{158977845631535694908133348236503730923407790816086380359767529735213385952037616981036094837564474518945521958765} a^{17} + \frac{960432923555828934007674725057223550177804343096901811250703744202487478046642550794401139481502431167835296873}{31795569126307138981626669647300746184681558163217276071953505947042677190407523396207218967512894903789104391753} a^{16} + \frac{59580067669142377672132108173641873012785334259931218617202983556406975579947192260955109750613887272856008788}{4296698530582045808327928330716317051983994346380712982696419722573334755460476134622597157772012824836365458345} a^{15} - \frac{29404029516752052153229899047711996915226576528579216055968831879184472862357752555066583990357064343238677774527}{158977845631535694908133348236503730923407790816086380359767529735213385952037616981036094837564474518945521958765} a^{14} - \frac{48908957502267598788437752297580848305916970838980969787545764411384563139800177614917351370096178877082490242247}{158977845631535694908133348236503730923407790816086380359767529735213385952037616981036094837564474518945521958765} a^{13} + \frac{13652555448888104984542038049403184405629965055307195005530903101326734802128300930457563248008724011414326271752}{31795569126307138981626669647300746184681558163217276071953505947042677190407523396207218967512894903789104391753} a^{12} - \frac{32445452220777140295880486428454990144867685086127560773846654429944944272552562895011375402493360799478676427486}{158977845631535694908133348236503730923407790816086380359767529735213385952037616981036094837564474518945521958765} a^{11} + \frac{2577894721229679745888204839482710482853366784608255957036394274515864605494284116407245548457987523240066745504}{22711120804505099272590478319500532989058255830869482908538218533601912278862516711576584976794924931277931708395} a^{10} + \frac{58115880148577551550734192984736699394529418769220922650289491072126701703288206660968429236275699119033508273469}{158977845631535694908133348236503730923407790816086380359767529735213385952037616981036094837564474518945521958765} a^{9} - \frac{43876137393534272616199613283266071482872572137648307120656806850152673207142714994217834657245472569838470760758}{158977845631535694908133348236503730923407790816086380359767529735213385952037616981036094837564474518945521958765} a^{8} + \frac{20129604594922604504701922370758185233201139970720883341057953049569518951227103495422446988717448371461462259201}{158977845631535694908133348236503730923407790816086380359767529735213385952037616981036094837564474518945521958765} a^{7} - \frac{11018168429479939329407704881898473365356098517428436100473145372117090986807079507477719460905446719035182148122}{31795569126307138981626669647300746184681558163217276071953505947042677190407523396207218967512894903789104391753} a^{6} - \frac{1728244046632152962195130843621455658118420720441432995356641143842804040273101510963973722821965768848831845919}{22711120804505099272590478319500532989058255830869482908538218533601912278862516711576584976794924931277931708395} a^{5} - \frac{2237657724332789699073457192801971530502295516676638280408896977251850467705928486100751262245599135144906570205}{4542224160901019854518095663900106597811651166173896581707643706720382455772503342315316995358984986255586341679} a^{4} - \frac{7398923368714647854848976837552584171925146404312692222443312461315687112956195397731277333148664876667877793993}{22711120804505099272590478319500532989058255830869482908538218533601912278862516711576584976794924931277931708395} a^{3} - \frac{112841337975590945462253303632582727549989640100494639822700636802676730299063173555164545825799863737376773447}{648889165843002836359727951985729513973093023739128083101091958102911779396071906045045285051283569465083763097} a^{2} - \frac{1449500769939793866108860493744619706432488093493003483215427067622800293640034338520780768981261752764875864162}{3244445829215014181798639759928647569865465118695640415505459790514558896980359530225226425256417847325418815485} a - \frac{5354033725963856509881389354092677347419939315047762392137081483192591784299110815219747005817831133825258831}{31499474070048681376685822911928617183159855521316897237917085344801542689129704176943945876275901430343871995}$

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

Class group and class number

$C_{2}\times C_{6}$, which has order $12$ (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:  \( 22150688642117730000000 \) (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

3.3.1432809.2, 3.3.361.1, 3.3.3969.2, 3.3.1432809.3, 9.9.2941473244627851129.9, 9.9.9025761726072081.2, 9.9.1998099208210609.2, 9.9.1061871841310654257569.5

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 ${\href{/LocalNumberField/23.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/29.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{9}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{9}$ ${\href{/LocalNumberField/41.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/43.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/47.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/53.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/59.9.0.1}{9} }^{3}$

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$7.3.2.1$x^{3} + 14$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.1$x^{3} + 14$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.1$x^{3} + 14$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.1$x^{3} + 14$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.1$x^{3} + 14$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.1$x^{3} + 14$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.1$x^{3} + 14$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.1$x^{3} + 14$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.1$x^{3} + 14$$3$$1$$2$$C_3$$[\ ]_{3}$
19Data not computed