Normalized defining polynomial
\( x^{27} - 3 x^{26} - 210 x^{25} + 278 x^{24} + 19929 x^{23} + 5937 x^{22} - 1083004 x^{21} - 1964481 x^{20} + 35357121 x^{19} + 118971079 x^{18} - 651688491 x^{17} - 3622053234 x^{16} + 4445915527 x^{15} + 60695501586 x^{14} + 61934493837 x^{13} - 500874109445 x^{12} - 1506251267832 x^{11} + 632120142195 x^{10} + 10208524999752 x^{9} + 17212808525700 x^{8} - 6295882065903 x^{7} - 67953265942575 x^{6} - 122172061784082 x^{5} - 118152234479289 x^{4} - 68213531737236 x^{3} - 22934996039574 x^{2} - 3986229684933 x - 260162527969 \)
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$$\chi_{1197}(1024,·)$, $\chi_{1197}(1,·)$, $\chi_{1197}(130,·)$, $\chi_{1197}(4,·)$, $\chi_{1197}(709,·)$, $\chi_{1197}(961,·)$, $\chi_{1197}(520,·)$, $\chi_{1197}(457,·)$, $\chi_{1197}(631,·)$, $\chi_{1197}(898,·)$, $\chi_{1197}(256,·)$, $\chi_{1197}(16,·)$, $\chi_{1197}(142,·)$, $\chi_{1197}(883,·)$, $\chi_{1197}(634,·)$, $\chi_{1197}(64,·)$, $\chi_{1197}(1138,·)$, $\chi_{1197}(1075,·)$, $\chi_{1197}(1012,·)$, $\chi_{1197}(757,·)$, $\chi_{1197}(823,·)$, $\chi_{1197}(568,·)$, $\chi_{1197}(505,·)$, $\chi_{1197}(442,·)$, $\chi_{1197}(571,·)$, $\chi_{1197}(253,·)$, $\chi_{1197}(1087,·)$$\rbrace$ | ||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $\frac{1}{57229} a^{25} - \frac{21902}{57229} a^{24} + \frac{26172}{57229} a^{23} + \frac{21417}{57229} a^{22} - \frac{8818}{57229} a^{21} - \frac{18624}{57229} a^{20} + \frac{13190}{57229} a^{19} + \frac{1324}{57229} a^{18} + \frac{25689}{57229} a^{17} - \frac{9680}{57229} a^{16} + \frac{10218}{57229} a^{15} + \frac{9715}{57229} a^{14} + \frac{9112}{57229} a^{13} - \frac{9394}{57229} a^{12} + \frac{9549}{57229} a^{11} + \frac{8860}{57229} a^{10} + \frac{25552}{57229} a^{9} - \frac{9601}{57229} a^{8} - \frac{908}{57229} a^{7} - \frac{8282}{57229} a^{6} - \frac{21830}{57229} a^{5} + \frac{27284}{57229} a^{4} - \frac{24712}{57229} a^{3} - \frac{21622}{57229} a^{2} + \frac{4283}{57229} a + \frac{6285}{57229}$, $\frac{1}{68037332159497038748874446405529384083582793645219535056897093235205411686873135626319530384154146534907614337689136511414021267109} a^{26} + \frac{550855602369210764863600194284224846215587372678458790205164529582014534537022417986739067053113880406586093235325349821230144}{68037332159497038748874446405529384083582793645219535056897093235205411686873135626319530384154146534907614337689136511414021267109} a^{25} - \frac{11240495588786471273781490289730854262115619407877028710647334613632787805714357715945216690399974180775600755906746218069477015690}{68037332159497038748874446405529384083582793645219535056897093235205411686873135626319530384154146534907614337689136511414021267109} a^{24} - \frac{13727681716594081152657378276973233794986366967217853716038190784230975184727599668780509999341620496482852445513674666756711461423}{68037332159497038748874446405529384083582793645219535056897093235205411686873135626319530384154146534907614337689136511414021267109} a^{23} + \frac{7877945932847480672594895789635418438412202339508034995608661670162115212747634420721767852974261399726802985420947492383445136050}{68037332159497038748874446405529384083582793645219535056897093235205411686873135626319530384154146534907614337689136511414021267109} a^{22} + \frac{22273700252256400422760706776364170273278874219808565570512182948918920822745872606130609800454180736737074825124970987227657382774}{68037332159497038748874446405529384083582793645219535056897093235205411686873135626319530384154146534907614337689136511414021267109} a^{21} - \frac{27203141410843689562806439037155315482406363067860912547818888145310972188084448433483539747854721304084578573486466779663266957356}{68037332159497038748874446405529384083582793645219535056897093235205411686873135626319530384154146534907614337689136511414021267109} a^{20} - \frac{13091286763715130307526132604567362766086824874005999421771954728604752503073146539422332135532818818906903896608455532066927147523}{68037332159497038748874446405529384083582793645219535056897093235205411686873135626319530384154146534907614337689136511414021267109} a^{19} - \frac{135385563282425668226588733410488358404158996617517645964997223188707307489294055295465840657276036488250773030932280123437308613}{297106253971602789296394962469560629185951063952923733872913070896093500816039893564714106481022473951561634662397976032375638721} a^{18} - \frac{20103074488773697143790700044912108522513071322874848757357188949495320453756822417570060631759514192569694898581540522998594473030}{68037332159497038748874446405529384083582793645219535056897093235205411686873135626319530384154146534907614337689136511414021267109} a^{17} - \frac{10564420245471711943274776556385211316904481394393523817474054871461548963886078632230760110980205058581707905068256340535839026658}{68037332159497038748874446405529384083582793645219535056897093235205411686873135626319530384154146534907614337689136511414021267109} a^{16} + \frac{993137921923493855610207090552925316644457634566502224379891018312693357462844876609582187532346446505048074863693910589599324670}{68037332159497038748874446405529384083582793645219535056897093235205411686873135626319530384154146534907614337689136511414021267109} a^{15} - \frac{5198180423981847124900318882163574997139827688758531011752524639232935522092752947321320365376706031868733844846653515365141519053}{68037332159497038748874446405529384083582793645219535056897093235205411686873135626319530384154146534907614337689136511414021267109} a^{14} + \frac{20848725876755656521306990424332802869545458112239652892797111870585433633279393583600715017806318055045233871450847751518627664377}{68037332159497038748874446405529384083582793645219535056897093235205411686873135626319530384154146534907614337689136511414021267109} a^{13} - \frac{9164862300200889241079800011041902046004325201128472586418058782831522479003936776705663896536008190668507747690286424635153631351}{68037332159497038748874446405529384083582793645219535056897093235205411686873135626319530384154146534907614337689136511414021267109} a^{12} - \frac{20944399878229567135217742226701882906796275353399868756131270384968060581283434590687098473562965278896858948783273192488244681278}{68037332159497038748874446405529384083582793645219535056897093235205411686873135626319530384154146534907614337689136511414021267109} a^{11} - \frac{33199533471847254680146747632503959086161676461891925550979195147686943564238946303182756419607731627807777566368970484085558144983}{68037332159497038748874446405529384083582793645219535056897093235205411686873135626319530384154146534907614337689136511414021267109} a^{10} + \frac{294889698518600031177395386566455571666601193749195754253465177250118208461284158030254972972047033772656851432948436482001423243}{68037332159497038748874446405529384083582793645219535056897093235205411686873135626319530384154146534907614337689136511414021267109} a^{9} - \frac{6024276618890388027930104523394287545239895363865473033723964815318316340843866378387511500862365294344785155241402372477459818144}{68037332159497038748874446405529384083582793645219535056897093235205411686873135626319530384154146534907614337689136511414021267109} a^{8} - \frac{22830135651605502501906154350678514968587981659613430278853918116781668429511817165389478677617893818996279492405154204774646371138}{68037332159497038748874446405529384083582793645219535056897093235205411686873135626319530384154146534907614337689136511414021267109} a^{7} + \frac{12141498856474527131041500610327440344535865346637487478238921391785367373874242363252086143693640554801720977960828676102120891974}{68037332159497038748874446405529384083582793645219535056897093235205411686873135626319530384154146534907614337689136511414021267109} a^{6} + \frac{24462413204460887767173983870308061638854196214103508041184036830189170708288766873332073302180702922821631409076557028415741204842}{68037332159497038748874446405529384083582793645219535056897093235205411686873135626319530384154146534907614337689136511414021267109} a^{5} - \frac{3215690387231900458568408182721532285029585529634421958037459382353035971053742417153482256472161635511319167213622552863437128}{68037332159497038748874446405529384083582793645219535056897093235205411686873135626319530384154146534907614337689136511414021267109} a^{4} + \frac{26686213721360243850122939050324189732299112249602141508548388493855248644941193089787872578609709633263343193847330411101223196218}{68037332159497038748874446405529384083582793645219535056897093235205411686873135626319530384154146534907614337689136511414021267109} a^{3} - \frac{31473435257489262812549058672143432670173752861502531097432796471461342535079186374443186500461677620391151754187381349571238658594}{68037332159497038748874446405529384083582793645219535056897093235205411686873135626319530384154146534907614337689136511414021267109} a^{2} - \frac{1788743292274574047488481789733550108175012901344253058069881263610150628096974778858129504596900305060820478160062816388446579376}{68037332159497038748874446405529384083582793645219535056897093235205411686873135626319530384154146534907614337689136511414021267109} a - \frac{31271318754092111809942742984701331547350217939028047801053542731573853043135480877554717494382176467478072235364061858990364344429}{68037332159497038748874446405529384083582793645219535056897093235205411686873135626319530384154146534907614337689136511414021267109}$
Class group and class number
$C_{2}\times C_{6}$, which has order $12$ (assuming GRH)
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: | \( 6121016983986815000000 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$C_3\times C_9$ (as 27T2):
| 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, \(\Q(\zeta_{19})^+\), 9.9.1061871841310654257569.6, 9.9.1061871841310654257569.2 |
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.
Local algebras for ramified primes
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
| 3 | Data 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 | $x^{9} + 42 x^{6} + 539 x^{3} + 2744$ | $3$ | $3$ | $6$ | $C_3^2$ | $[\ ]_{3}^{3}$ | |
| 19 | Data not computed | ||||||