↑

Properties

Label 27.27.4652158179...1761.1
Degree $27$
Signature $[27, 0]$
Discriminant $3^{36}\cdot 127^{24}$
Root discriminant $320.78$
Ramified primes $3, 127$
Class number $4$ (GRH)
Class group $[2, 2]$ (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![-2784493637, 7536057522, 33768425463, -89489663845, -61199808147, 245913582318, -3315665497, -284780543145, 85583050641, 163446427076, -78264957723, -48846266253, 32268931643, 7122066258, -7219642587, -263176183, 926380305, -58476672, -69113725, 8585901, 2946153, -494100, -68025, 14055, 755, -192, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^27 - 3*x^26 - 192*x^25 + 755*x^24 + 14055*x^23 - 68025*x^22 - 494100*x^21 + 2946153*x^20 + 8585901*x^19 - 69113725*x^18 - 58476672*x^17 + 926380305*x^16 - 263176183*x^15 - 7219642587*x^14 + 7122066258*x^13 + 32268931643*x^12 - 48846266253*x^11 - 78264957723*x^10 + 163446427076*x^9 + 85583050641*x^8 - 284780543145*x^7 - 3315665497*x^6 + 245913582318*x^5 - 61199808147*x^4 - 89489663845*x^3 + 33768425463*x^2 + 7536057522*x - 2784493637)
 
gp: K = bnfinit(x^27 - 3*x^26 - 192*x^25 + 755*x^24 + 14055*x^23 - 68025*x^22 - 494100*x^21 + 2946153*x^20 + 8585901*x^19 - 69113725*x^18 - 58476672*x^17 + 926380305*x^16 - 263176183*x^15 - 7219642587*x^14 + 7122066258*x^13 + 32268931643*x^12 - 48846266253*x^11 - 78264957723*x^10 + 163446427076*x^9 + 85583050641*x^8 - 284780543145*x^7 - 3315665497*x^6 + 245913582318*x^5 - 61199808147*x^4 - 89489663845*x^3 + 33768425463*x^2 + 7536057522*x - 2784493637, 1)
 

Normalized defining polynomial

\( x^{27} - 3 x^{26} - 192 x^{25} + 755 x^{24} + 14055 x^{23} - 68025 x^{22} - 494100 x^{21} + 2946153 x^{20} + 8585901 x^{19} - 69113725 x^{18} - 58476672 x^{17} + 926380305 x^{16} - 263176183 x^{15} - 7219642587 x^{14} + 7122066258 x^{13} + 32268931643 x^{12} - 48846266253 x^{11} - 78264957723 x^{10} + 163446427076 x^{9} + 85583050641 x^{8} - 284780543145 x^{7} - 3315665497 x^{6} + 245913582318 x^{5} - 61199808147 x^{4} - 89489663845 x^{3} + 33768425463 x^{2} + 7536057522 x - 2784493637 \)

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:  \(46521581790658445094934566845164072323103073629781639266605395671761=3^{36}\cdot 127^{24}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $320.78$
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:  $3, 127$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1143=3^{2}\cdot 127\)
Dirichlet character group:    $\lbrace$$\chi_{1143}(1,·)$, $\chi_{1143}(322,·)$, $\chi_{1143}(763,·)$, $\chi_{1143}(226,·)$, $\chi_{1143}(781,·)$, $\chi_{1143}(400,·)$, $\chi_{1143}(19,·)$, $\chi_{1143}(22,·)$, $\chi_{1143}(988,·)$, $\chi_{1143}(799,·)$, $\chi_{1143}(865,·)$, $\chi_{1143}(418,·)$, $\chi_{1143}(1123,·)$, $\chi_{1143}(484,·)$, $\chi_{1143}(37,·)$, $\chi_{1143}(742,·)$, $\chi_{1143}(103,·)$, $\chi_{1143}(361,·)$, $\chi_{1143}(814,·)$, $\chi_{1143}(433,·)$, $\chi_{1143}(403,·)$, $\chi_{1143}(52,·)$, $\chi_{1143}(382,·)$, $\chi_{1143}(607,·)$, $\chi_{1143}(1084,·)$, $\chi_{1143}(784,·)$, $\chi_{1143}(703,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{7} - \frac{1}{2}$, $\frac{1}{2} a^{8} - \frac{1}{2} a$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{4}$, $\frac{1}{8} a^{12} - \frac{1}{4} a^{11} + \frac{1}{8} a^{10} - \frac{1}{4} a^{9} + \frac{1}{8} a^{8} - \frac{1}{4} a^{7} + \frac{1}{8} a^{6} - \frac{1}{2} a^{5} + \frac{3}{8} a^{4} - \frac{1}{2} a^{3} - \frac{1}{8} a^{2} - \frac{1}{2} a - \frac{1}{8}$, $\frac{1}{8} a^{13} + \frac{1}{8} a^{11} + \frac{1}{8} a^{9} + \frac{1}{8} a^{7} - \frac{1}{4} a^{6} + \frac{3}{8} a^{5} - \frac{1}{4} a^{4} - \frac{1}{8} a^{3} - \frac{1}{4} a^{2} - \frac{1}{8} a + \frac{1}{4}$, $\frac{1}{8} a^{14} - \frac{1}{4} a^{11} - \frac{1}{4} a^{9} - \frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{2} a^{4} - \frac{1}{4} a^{3} + \frac{1}{4} a - \frac{3}{8}$, $\frac{1}{8} a^{15} - \frac{1}{4} a^{8} - \frac{1}{4} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2} + \frac{1}{8} a + \frac{1}{4}$, $\frac{1}{8} a^{16} - \frac{1}{4} a^{9} - \frac{1}{4} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{3}{8} a^{2} - \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{8} a^{17} - \frac{1}{4} a^{10} - \frac{1}{4} a^{9} - \frac{1}{2} a^{4} + \frac{1}{8} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2}$, $\frac{1}{16} a^{18} - \frac{1}{16} a^{17} - \frac{1}{16} a^{14} - \frac{1}{16} a^{13} - \frac{1}{16} a^{11} - \frac{1}{16} a^{9} - \frac{1}{16} a^{7} - \frac{1}{4} a^{6} + \frac{3}{16} a^{5} - \frac{5}{16} a^{4} + \frac{1}{4} a^{3} - \frac{1}{4} a^{2} + \frac{3}{16} a - \frac{3}{16}$, $\frac{1}{16} a^{19} - \frac{1}{16} a^{17} - \frac{1}{16} a^{15} - \frac{1}{16} a^{13} - \frac{1}{16} a^{12} + \frac{3}{16} a^{11} - \frac{1}{16} a^{10} + \frac{3}{16} a^{9} - \frac{1}{16} a^{8} + \frac{3}{16} a^{7} + \frac{3}{16} a^{6} + \frac{1}{8} a^{5} + \frac{7}{16} a^{4} + \frac{1}{4} a^{3} - \frac{1}{16} a^{2} - \frac{1}{4} a + \frac{7}{16}$, $\frac{1}{16} a^{20} - \frac{1}{16} a^{17} - \frac{1}{16} a^{16} - \frac{1}{16} a^{12} - \frac{1}{4} a^{11} - \frac{1}{16} a^{10} - \frac{1}{4} a^{9} - \frac{1}{16} a^{8} - \frac{1}{4} a^{7} + \frac{1}{8} a^{6} - \frac{1}{4} a^{5} - \frac{1}{16} a^{4} - \frac{3}{16} a^{3} - \frac{1}{4} a - \frac{1}{16}$, $\frac{1}{128} a^{21} - \frac{1}{32} a^{20} + \frac{1}{128} a^{19} - \frac{3}{128} a^{18} - \frac{3}{64} a^{17} - \frac{1}{32} a^{16} - \frac{5}{128} a^{15} - \frac{3}{64} a^{14} + \frac{3}{64} a^{13} - \frac{3}{128} a^{12} - \frac{1}{64} a^{11} - \frac{15}{128} a^{10} - \frac{3}{64} a^{9} - \frac{3}{128} a^{8} + \frac{1}{128} a^{7} - \frac{23}{128} a^{6} - \frac{35}{128} a^{5} + \frac{3}{8} a^{4} - \frac{3}{16} a^{3} - \frac{51}{128} a^{2} + \frac{3}{128} a + \frac{23}{128}$, $\frac{1}{128} a^{22} + \frac{1}{128} a^{20} + \frac{1}{128} a^{19} - \frac{1}{64} a^{18} + \frac{1}{32} a^{17} - \frac{5}{128} a^{16} + \frac{3}{64} a^{15} - \frac{1}{64} a^{14} + \frac{5}{128} a^{13} + \frac{1}{64} a^{12} + \frac{25}{128} a^{11} + \frac{7}{64} a^{10} + \frac{21}{128} a^{9} + \frac{5}{128} a^{8} + \frac{29}{128} a^{7} + \frac{1}{128} a^{6} + \frac{5}{32} a^{5} - \frac{3}{16} a^{4} - \frac{3}{128} a^{3} - \frac{9}{128} a^{2} - \frac{13}{128} a + \frac{7}{32}$, $\frac{1}{128} a^{23} - \frac{3}{128} a^{20} - \frac{3}{128} a^{19} - \frac{1}{128} a^{18} + \frac{1}{128} a^{17} + \frac{1}{64} a^{16} + \frac{3}{128} a^{15} + \frac{3}{128} a^{14} + \frac{1}{32} a^{13} + \frac{1}{32} a^{12} + \frac{3}{16} a^{11} - \frac{5}{32} a^{10} + \frac{19}{128} a^{9} - \frac{3}{16} a^{8} - \frac{3}{16} a^{7} - \frac{5}{128} a^{6} + \frac{51}{128} a^{5} + \frac{29}{128} a^{4} + \frac{23}{128} a^{3} - \frac{5}{64} a^{2} + \frac{33}{128} a + \frac{25}{128}$, $\frac{1}{33536} a^{24} - \frac{15}{16768} a^{23} + \frac{29}{8384} a^{22} - \frac{35}{16768} a^{21} - \frac{297}{33536} a^{20} + \frac{357}{16768} a^{19} + \frac{31}{2096} a^{18} + \frac{307}{16768} a^{17} - \frac{97}{33536} a^{16} - \frac{11}{262} a^{15} - \frac{475}{8384} a^{14} - \frac{289}{16768} a^{13} - \frac{1839}{33536} a^{12} - \frac{2773}{16768} a^{11} + \frac{501}{2096} a^{10} - \frac{1255}{8384} a^{9} + \frac{1157}{33536} a^{8} - \frac{1037}{8384} a^{7} - \frac{2527}{16768} a^{6} - \frac{201}{8384} a^{5} - \frac{1103}{33536} a^{4} + \frac{879}{2096} a^{3} - \frac{5839}{16768} a^{2} + \frac{5183}{16768} a + \frac{1949}{33536}$, $\frac{1}{167680} a^{25} + \frac{1}{83840} a^{23} + \frac{133}{83840} a^{22} - \frac{39}{167680} a^{21} + \frac{1011}{83840} a^{20} + \frac{205}{8384} a^{19} + \frac{673}{83840} a^{18} - \frac{9711}{167680} a^{17} - \frac{4517}{83840} a^{16} - \frac{817}{41920} a^{15} - \frac{5}{4192} a^{14} - \frac{587}{33536} a^{13} + \frac{3309}{83840} a^{12} - \frac{433}{10480} a^{11} + \frac{4153}{83840} a^{10} + \frac{34743}{167680} a^{9} + \frac{823}{4192} a^{8} - \frac{3709}{20960} a^{7} - \frac{521}{8384} a^{6} - \frac{17363}{167680} a^{5} + \frac{5973}{20960} a^{4} - \frac{7899}{20960} a^{3} + \frac{4807}{41920} a^{2} - \frac{36579}{167680} a + \frac{33689}{83840}$, $\frac{1}{30236434889966550904949990030319242219135555472135599675724574622402181481373225234043429446371321275063923590400} a^{26} - \frac{117225026987964245369447893228063216245587790940888202023091623377276044668963008806045266224814983541971}{472444295155727357889843594223738159673993054252118744933196478475034085646456644281928585099551894922873806100} a^{25} - \frac{277193836576080975660545530668131323618292993919528629468898982881826380435000215528379839920874361681133313}{30236434889966550904949990030319242219135555472135599675724574622402181481373225234043429446371321275063923590400} a^{24} + \frac{31346831176844952314662518632526103963946840552283959685743426855646892840579217166032175001187028382821161219}{15118217444983275452474995015159621109567777736067799837862287311201090740686612617021714723185660637531961795200} a^{23} + \frac{77051046794136608973408221097801314213847082024170267987230317131487100252409136473882918058027519680134299697}{30236434889966550904949990030319242219135555472135599675724574622402181481373225234043429446371321275063923590400} a^{22} + \frac{130566619130571953045156937136391167162632107649885509449696763451922794025368113372443117654843383087502481}{3779554361245818863118748753789905277391944434016949959465571827800272685171653154255428680796415159382990448800} a^{21} + \frac{120072213360497657012568038382971000672096754662401701812410096990070624563737763902207416231932986256576886557}{30236434889966550904949990030319242219135555472135599675724574622402181481373225234043429446371321275063923590400} a^{20} - \frac{456775255113273991836962469075361412704652466311589844156998777983822998078851465518466767151838605320876573}{39889755791512600138456451227334092637381999303608970548449306889712640476745679728289484757745806431482748800} a^{19} - \frac{66779589631764470764251351281238035377524098202018683195572359258739309688085030789038201263613175037469883461}{6047286977993310180989998006063848443827111094427119935144914924480436296274645046808685889274264255012784718080} a^{18} + \frac{12599172631708408257464647556488563414453630823413692703916929538838688405481793028080121719297707962363140503}{3023643488996655090494999003031924221913555547213559967572457462240218148137322523404342944637132127506392359040} a^{17} + \frac{157059166397128800485087408626877582864353843319144862285518559890211506470051281058606317105597921811331676623}{30236434889966550904949990030319242219135555472135599675724574622402181481373225234043429446371321275063923590400} a^{16} - \frac{711533819162894258502814474559738205195954471535892711864387802352197539076902302268593755986152204143791980669}{15118217444983275452474995015159621109567777736067799837862287311201090740686612617021714723185660637531961795200} a^{15} + \frac{20912075527131794703288554543959791370390296371485180194530276744937108780729569976647041116703286575492375875}{1209457395598662036197999601212769688765422218885423987028982984896087259254929009361737177854852851002556943616} a^{14} - \frac{436405580838732436026117191147524908688140515771714628949599334083738907232930082088121254600082187488187906653}{7559108722491637726237497507579810554783888868033899918931143655600545370343306308510857361592830318765980897600} a^{13} + \frac{34913624910343819953953063927947256654951737227027198589711904337380349098204490116193509085510870454703270293}{1209457395598662036197999601212769688765422218885423987028982984896087259254929009361737177854852851002556943616} a^{12} + \frac{649111370056572702941401471244167482658276069964299035960080257954946251710926636782570099131998073380673473317}{7559108722491637726237497507579810554783888868033899918931143655600545370343306308510857361592830318765980897600} a^{11} + \frac{4544497831690699763370088723792664480759393807024261190641318776412679984817743519516121099931905286192884897559}{30236434889966550904949990030319242219135555472135599675724574622402181481373225234043429446371321275063923590400} a^{10} + \frac{850054451037867075593582592654822856834130318818349054480099101003677730680626036878529324147155086465870860779}{15118217444983275452474995015159621109567777736067799837862287311201090740686612617021714723185660637531961795200} a^{9} - \frac{650658408469574908211007012381570585579376440497224860379389477272995904532003350572838117736327966358234927427}{30236434889966550904949990030319242219135555472135599675724574622402181481373225234043429446371321275063923590400} a^{8} + \frac{1764312129573330052114355787308509022562536076013815785598294200958006121920158522426383059608148877559754849787}{7559108722491637726237497507579810554783888868033899918931143655600545370343306308510857361592830318765980897600} a^{7} + \frac{3212653070765424808532662920896558334802456138976661349262746075079060505536813126316221146177671576186685115137}{30236434889966550904949990030319242219135555472135599675724574622402181481373225234043429446371321275063923590400} a^{6} - \frac{3776733843262432306020964874282945029161666441076392438392436790776460140376512503457849589160356191465794790707}{15118217444983275452474995015159621109567777736067799837862287311201090740686612617021714723185660637531961795200} a^{5} - \frac{9117975471763121502860617154475960527683217315694142618652775288032090853120275288323814642841327896460754474433}{30236434889966550904949990030319242219135555472135599675724574622402181481373225234043429446371321275063923590400} a^{4} + \frac{1124003162992911228617853173692689367965794993672231938873125526947092632554080382383764145543192690462910526903}{15118217444983275452474995015159621109567777736067799837862287311201090740686612617021714723185660637531961795200} a^{3} - \frac{1404250855680790307793040060593201808204488587117008895586079710699965461745652273081639733978081028035407707841}{30236434889966550904949990030319242219135555472135599675724574622402181481373225234043429446371321275063923590400} a^{2} + \frac{2597095416318278274706219890366009266035284659862285927059381825016706347490137235440233979186726120960983275347}{15118217444983275452474995015159621109567777736067799837862287311201090740686612617021714723185660637531961795200} a - \frac{2884930969721074238952497112594614065996430093076835268673481925573217527996726053442575484923400238188740999157}{30236434889966550904949990030319242219135555472135599675724574622402181481373225234043429446371321275063923590400}$

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

Class group and class number

$C_{2}\times C_{2}$, which has order $4$ (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:  \( 9878199703760603000000000 \) (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

\(\Q(\zeta_{9})^+\), 3.3.16129.1, 3.3.1306449.2, 3.3.1306449.1, 9.9.2229858897655236849.1, 9.9.67675234241018881.1, 9.9.35965394160281315137521.4, 9.9.35965394160281315137521.3

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 ${\href{/LocalNumberField/2.3.0.1}{3} }^{9}$ R ${\href{/LocalNumberField/5.3.0.1}{3} }^{9}$ ${\href{/LocalNumberField/7.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/11.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/13.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/17.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{9}$ ${\href{/LocalNumberField/23.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/29.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/31.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/37.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/41.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/43.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{9}$ ${\href{/LocalNumberField/53.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/59.9.0.1}{9} }^{3}$

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
$127$127.9.8.1$x^{9} - 127$$9$$1$$8$$C_9$$[\ ]_{9}$
127.9.8.1$x^{9} - 127$$9$$1$$8$$C_9$$[\ ]_{9}$
127.9.8.1$x^{9} - 127$$9$$1$$8$$C_9$$[\ ]_{9}$