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Properties

Label 27.27.1187411143...7281.1
Degree $27$
Signature $[27, 0]$
Discriminant $3^{36}\cdot 109^{24}$
Root discriminant $280.03$
Ramified primes $3, 109$
Class number $1$ (GRH)
Class group Trivial (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![668804824, -8074445220, 28531403934, -4007013731, -141722795001, 180395569233, 128423804151, -318723893835, 38975332611, 196283403790, -83218822605, -50785958448, 35528125026, 4569640299, -6982699857, 275657276, 759872085, -94362513, -49261956, 8773131, 1939944, -425713, -45264, 11610, 572, -168, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^27 - 3*x^26 - 168*x^25 + 572*x^24 + 11610*x^23 - 45264*x^22 - 425713*x^21 + 1939944*x^20 + 8773131*x^19 - 49261956*x^18 - 94362513*x^17 + 759872085*x^16 + 275657276*x^15 - 6982699857*x^14 + 4569640299*x^13 + 35528125026*x^12 - 50785958448*x^11 - 83218822605*x^10 + 196283403790*x^9 + 38975332611*x^8 - 318723893835*x^7 + 128423804151*x^6 + 180395569233*x^5 - 141722795001*x^4 - 4007013731*x^3 + 28531403934*x^2 - 8074445220*x + 668804824)
 
gp: K = bnfinit(x^27 - 3*x^26 - 168*x^25 + 572*x^24 + 11610*x^23 - 45264*x^22 - 425713*x^21 + 1939944*x^20 + 8773131*x^19 - 49261956*x^18 - 94362513*x^17 + 759872085*x^16 + 275657276*x^15 - 6982699857*x^14 + 4569640299*x^13 + 35528125026*x^12 - 50785958448*x^11 - 83218822605*x^10 + 196283403790*x^9 + 38975332611*x^8 - 318723893835*x^7 + 128423804151*x^6 + 180395569233*x^5 - 141722795001*x^4 - 4007013731*x^3 + 28531403934*x^2 - 8074445220*x + 668804824, 1)
 

Normalized defining polynomial

\( x^{27} - 3 x^{26} - 168 x^{25} + 572 x^{24} + 11610 x^{23} - 45264 x^{22} - 425713 x^{21} + 1939944 x^{20} + 8773131 x^{19} - 49261956 x^{18} - 94362513 x^{17} + 759872085 x^{16} + 275657276 x^{15} - 6982699857 x^{14} + 4569640299 x^{13} + 35528125026 x^{12} - 50785958448 x^{11} - 83218822605 x^{10} + 196283403790 x^{9} + 38975332611 x^{8} - 318723893835 x^{7} + 128423804151 x^{6} + 180395569233 x^{5} - 141722795001 x^{4} - 4007013731 x^{3} + 28531403934 x^{2} - 8074445220 x + 668804824 \)

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:  \(1187411143908093381248555091925855631936617111126489321417190047281=3^{36}\cdot 109^{24}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $280.03$
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:  $3, 109$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(981=3^{2}\cdot 109\)
Dirichlet character group:    $\lbrace$$\chi_{981}(256,·)$, $\chi_{981}(1,·)$, $\chi_{981}(214,·)$, $\chi_{981}(838,·)$, $\chi_{981}(583,·)$, $\chi_{981}(136,·)$, $\chi_{981}(910,·)$, $\chi_{981}(655,·)$, $\chi_{981}(16,·)$, $\chi_{981}(790,·)$, $\chi_{981}(343,·)$, $\chi_{981}(154,·)$, $\chi_{981}(463,·)$, $\chi_{981}(541,·)$, $\chi_{981}(670,·)$, $\chi_{981}(481,·)$, $\chi_{981}(868,·)$, $\chi_{981}(808,·)$, $\chi_{981}(172,·)$, $\chi_{981}(175,·)$, $\chi_{981}(328,·)$, $\chi_{981}(499,·)$, $\chi_{981}(502,·)$, $\chi_{981}(184,·)$, $\chi_{981}(826,·)$, $\chi_{981}(829,·)$, $\chi_{981}(511,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{8} - \frac{1}{2} a$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{5}$, $\frac{1}{4} a^{13} - \frac{1}{4} a^{11} - \frac{1}{4} a^{9} - \frac{1}{4} a^{7} + \frac{1}{4} a^{5} - \frac{1}{2} a^{4} + \frac{1}{4} a^{3} - \frac{1}{2} a^{2} + \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{4} a^{14} - \frac{1}{4} a^{12} - \frac{1}{4} a^{10} - \frac{1}{4} a^{8} + \frac{1}{4} a^{6} - \frac{1}{2} a^{5} + \frac{1}{4} a^{4} - \frac{1}{2} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{15} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} + \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{4} a^{16} - \frac{1}{4} a^{2}$, $\frac{1}{4} a^{17} - \frac{1}{4} a^{3}$, $\frac{1}{4} a^{18} - \frac{1}{4} a^{4}$, $\frac{1}{4} a^{19} - \frac{1}{4} a^{5}$, $\frac{1}{16} a^{20} - \frac{1}{16} a^{19} - \frac{1}{8} a^{17} + \frac{1}{16} a^{16} + \frac{1}{16} a^{15} - \frac{1}{8} a^{14} + \frac{1}{16} a^{13} + \frac{1}{8} a^{12} + \frac{1}{16} a^{11} - \frac{1}{8} a^{10} + \frac{1}{16} a^{9} - \frac{1}{8} a^{8} - \frac{3}{16} a^{7} - \frac{3}{16} a^{6} - \frac{1}{8} a^{5} - \frac{1}{8} a^{4} + \frac{3}{16} a^{3} - \frac{3}{16} a^{2} + \frac{1}{4}$, $\frac{1}{16} a^{21} - \frac{1}{16} a^{19} - \frac{1}{8} a^{18} - \frac{1}{16} a^{17} - \frac{1}{8} a^{16} - \frac{1}{16} a^{15} - \frac{1}{16} a^{14} - \frac{1}{16} a^{13} + \frac{3}{16} a^{12} + \frac{3}{16} a^{11} - \frac{1}{16} a^{10} + \frac{3}{16} a^{9} + \frac{3}{16} a^{8} - \frac{1}{8} a^{7} - \frac{5}{16} a^{6} - \frac{1}{2} a^{5} - \frac{7}{16} a^{4} - \frac{1}{4} a^{3} - \frac{7}{16} a^{2} - \frac{1}{2} a - \frac{1}{4}$, $\frac{1}{16} a^{22} + \frac{1}{16} a^{19} - \frac{1}{16} a^{18} + \frac{1}{16} a^{14} + \frac{1}{16} a^{12} - \frac{1}{4} a^{11} - \frac{3}{16} a^{10} - \frac{1}{4} a^{7} - \frac{7}{16} a^{6} + \frac{7}{16} a^{5} - \frac{1}{8} a^{4} - \frac{1}{4} a^{3} - \frac{7}{16} a^{2} - \frac{1}{2} a - \frac{1}{4}$, $\frac{1}{16} a^{23} - \frac{1}{8} a^{17} - \frac{1}{16} a^{16} - \frac{1}{8} a^{14} - \frac{1}{8} a^{12} - \frac{1}{4} a^{11} - \frac{1}{8} a^{10} - \frac{1}{16} a^{9} + \frac{1}{8} a^{8} - \frac{1}{4} a^{7} + \frac{3}{8} a^{6} - \frac{1}{2} a^{5} - \frac{3}{8} a^{4} - \frac{3}{8} a^{3} + \frac{7}{16} a^{2} + \frac{1}{4} a - \frac{1}{4}$, $\frac{1}{237063968} a^{24} + \frac{3796025}{237063968} a^{23} - \frac{7117823}{237063968} a^{22} - \frac{2085593}{118531984} a^{21} - \frac{67601}{29632996} a^{20} + \frac{13209299}{237063968} a^{19} - \frac{1081491}{237063968} a^{18} - \frac{8205013}{237063968} a^{17} - \frac{16632773}{237063968} a^{16} + \frac{2619767}{29632996} a^{15} + \frac{17100189}{237063968} a^{14} + \frac{5508055}{59265992} a^{13} - \frac{6786031}{237063968} a^{12} + \frac{4544177}{59265992} a^{11} - \frac{1887317}{14816498} a^{10} + \frac{15984775}{237063968} a^{9} - \frac{11867587}{59265992} a^{8} - \frac{386185}{118531984} a^{7} - \frac{115631635}{237063968} a^{6} - \frac{29639283}{237063968} a^{5} - \frac{987857}{14816498} a^{4} + \frac{99590533}{237063968} a^{3} - \frac{47489749}{118531984} a^{2} - \frac{982455}{59265992} a + \frac{4061839}{29632996}$, $\frac{1}{196051901536} a^{25} + \frac{133}{98025950768} a^{24} - \frac{38385153}{24506487692} a^{23} + \frac{1178537765}{196051901536} a^{22} + \frac{205128749}{98025950768} a^{21} + \frac{3594286681}{196051901536} a^{20} + \frac{5654723369}{49012975384} a^{19} - \frac{8698664687}{98025950768} a^{18} - \frac{5251968247}{98025950768} a^{17} - \frac{23615330845}{196051901536} a^{16} + \frac{8681275991}{196051901536} a^{15} - \frac{19010128773}{196051901536} a^{14} + \frac{23950861623}{196051901536} a^{13} + \frac{46983724351}{196051901536} a^{12} + \frac{15837499295}{98025950768} a^{11} - \frac{15176864975}{196051901536} a^{10} + \frac{30184017239}{196051901536} a^{9} + \frac{6514075675}{98025950768} a^{8} + \frac{2674606057}{196051901536} a^{7} + \frac{33953305303}{98025950768} a^{6} - \frac{8911145957}{196051901536} a^{5} + \frac{51193079405}{196051901536} a^{4} + \frac{82870622129}{196051901536} a^{3} + \frac{5481791319}{49012975384} a^{2} - \frac{15491906505}{49012975384} a + \frac{3438944}{7408249}$, $\frac{1}{11434940886164358814378596437420670433795647361837538353092001848234020614853016955147921236559994346976} a^{26} + \frac{5014387388194250344733312450493302208541429214130349393345154829148422180545961513224589231}{2858735221541089703594649109355167608448911840459384588273000462058505153713254238786980309139998586744} a^{25} - \frac{10752032579509967595725624862117664341803468904169193774817636861003203930942343270405024325751}{11434940886164358814378596437420670433795647361837538353092001848234020614853016955147921236559994346976} a^{24} + \frac{27795473024877173317592640363650293528242623997788565736052871014458448184790593604618228919821205065}{2858735221541089703594649109355167608448911840459384588273000462058505153713254238786980309139998586744} a^{23} - \frac{181835531874403466192203602676431287181589190141192772388499136627716391671893194280029656732765152949}{11434940886164358814378596437420670433795647361837538353092001848234020614853016955147921236559994346976} a^{22} - \frac{262874444841163755074731486122308055326690291988342916435233545887940645879312242970780858065821270849}{11434940886164358814378596437420670433795647361837538353092001848234020614853016955147921236559994346976} a^{21} + \frac{333815062865026349983991547261149329941952198890717354890717746488384564338570668224228800474207771}{69725249305880236673040222179394331913388093669741087518853669806305003749103761921633666076585331384} a^{20} - \frac{1038572189325019175306690334803170666808446899435283884679621972531446063033880353755240540754471232671}{11434940886164358814378596437420670433795647361837538353092001848234020614853016955147921236559994346976} a^{19} + \frac{823293849769643642657629873904419787451136762871909539310022400685740227662498947292222924802212454877}{11434940886164358814378596437420670433795647361837538353092001848234020614853016955147921236559994346976} a^{18} + \frac{655184940874455890802012319662536086300075888098588037756287930107440355013544293947144651028532902353}{5717470443082179407189298218710335216897823680918769176546000924117010307426508477573960618279997173488} a^{17} - \frac{2257347160497617743705489775715718798376211042063474949653004095097611106500016161142493579348817790}{357341902692636212949331138669395951056113980057423073534125057757313144214156779848372538642499823343} a^{16} + \frac{1190120309947902607257923252587004708033228599511913016171740354145311615962800800498964116959667425107}{11434940886164358814378596437420670433795647361837538353092001848234020614853016955147921236559994346976} a^{15} + \frac{333564844188837264436505624752066581659187196747089542497326118684982413711057104325024592075402001805}{2858735221541089703594649109355167608448911840459384588273000462058505153713254238786980309139998586744} a^{14} + \frac{389238417945475462011376069887453244956305611567906784216418757044479935338549375986611376502662767395}{11434940886164358814378596437420670433795647361837538353092001848234020614853016955147921236559994346976} a^{13} - \frac{1320081069509609490241477570119229854536406906782885921625332441633912653756608894437114327089809602145}{11434940886164358814378596437420670433795647361837538353092001848234020614853016955147921236559994346976} a^{12} - \frac{2188182488648470494663460779875285897818265875454773255516288103074013743455023793165106061644453873845}{11434940886164358814378596437420670433795647361837538353092001848234020614853016955147921236559994346976} a^{11} - \frac{403253950821589154591354301845311898254949366157022666171754039390567191182466304112946060987304349249}{11434940886164358814378596437420670433795647361837538353092001848234020614853016955147921236559994346976} a^{10} - \frac{796450696444534738003865352013856603636299050809073671256617887636801626543618034320610268213189161729}{11434940886164358814378596437420670433795647361837538353092001848234020614853016955147921236559994346976} a^{9} - \frac{345707248420563254043260700425065993937228933977385951855394015425631981607555142415945751590388215979}{11434940886164358814378596437420670433795647361837538353092001848234020614853016955147921236559994346976} a^{8} - \frac{651332534012695520877198242526622584408117711798388403480058310935577655230964185252652472163212599777}{2858735221541089703594649109355167608448911840459384588273000462058505153713254238786980309139998586744} a^{7} - \frac{201050143908968923273571161563434784877887989566583069123821032112276852222553270023337419286661189017}{714683805385272425898662277338791902112227960114846147068250115514626288428313559696745077284999646686} a^{6} + \frac{15501220503237372748818210588942254959092878906380105869094807992102156884043752757139085711155733103}{5717470443082179407189298218710335216897823680918769176546000924117010307426508477573960618279997173488} a^{5} + \frac{4590396067376088557509188328782128795299737906916721592347935034941813996754374789655176789184189026459}{11434940886164358814378596437420670433795647361837538353092001848234020614853016955147921236559994346976} a^{4} - \frac{2325085475310991682830704539517443755873603369992080359968849706285221961944634790080808633908072853499}{11434940886164358814378596437420670433795647361837538353092001848234020614853016955147921236559994346976} a^{3} - \frac{100073551237489711260017977233489031769874860243709449526284233198239257963260063832688567411590212633}{2858735221541089703594649109355167608448911840459384588273000462058505153713254238786980309139998586744} a^{2} + \frac{1077423855999738237967334057888661705796451819465201424652941866083745382587373591001885482875884800175}{2858735221541089703594649109355167608448911840459384588273000462058505153713254238786980309139998586744} a - \frac{10560137873170383255055236643998036504629296620686269691691800142899595291115922450830751739065399}{432094199144662893530025560664324003695421983140777597985640940456243221540697436334186866556831709}$

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

Class group and class number

Trivial group, which has order $1$ (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:  \( 5431785827487784000000000 \) (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

\(\Q(\zeta_{9})^+\), 3.3.11881.1, 3.3.962361.1, 3.3.962361.2, 9.9.891279760005451881.2, 9.9.10589294828624773798161.4, 9.9.10589294828624773798161.3, 9.9.19925626416901921.1

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 ${\href{/LocalNumberField/2.3.0.1}{3} }^{9}$ R ${\href{/LocalNumberField/5.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/7.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/11.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/13.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{9}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{9}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{9}$ ${\href{/LocalNumberField/29.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/31.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/37.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{9}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{9}$ ${\href{/LocalNumberField/47.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/53.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/59.9.0.1}{9} }^{3}$

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
$109$109.9.8.1$x^{9} - 109$$9$$1$$8$$C_9$$[\ ]_{9}$
109.9.8.1$x^{9} - 109$$9$$1$$8$$C_9$$[\ ]_{9}$
109.9.8.1$x^{9} - 109$$9$$1$$8$$C_9$$[\ ]_{9}$