Normalized defining polynomial
\( x^{27} - 9 x^{26} - 99 x^{25} + 906 x^{24} + 4833 x^{23} - 39654 x^{22} - 156102 x^{21} + 968139 x^{20} + 3563946 x^{19} - 13945195 x^{18} - 56219139 x^{17} + 112278861 x^{16} + 582670932 x^{15} - 354069522 x^{14} - 3707506512 x^{13} - 1396052364 x^{12} + 12836504439 x^{11} + 14919397542 x^{10} - 16968716582 x^{9} - 39300644385 x^{8} - 9891055872 x^{7} + 24100026927 x^{6} + 17213661954 x^{5} - 1129546062 x^{4} - 4247511909 x^{3} - 1269306432 x^{2} - 91305144 x + 5272453 \)
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: | \(521955940438809922293374595854472560734963287885444907593441=3^{66}\cdot 37^{18}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $162.84$ | 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: | $3, 37$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(999=3^{3}\cdot 37\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{999}(1,·)$, $\chi_{999}(898,·)$, $\chi_{999}(454,·)$, $\chi_{999}(10,·)$, $\chi_{999}(322,·)$, $\chi_{999}(334,·)$, $\chi_{999}(655,·)$, $\chi_{999}(787,·)$, $\chi_{999}(343,·)$, $\chi_{999}(100,·)$, $\chi_{999}(667,·)$, $\chi_{999}(988,·)$, $\chi_{999}(211,·)$, $\chi_{999}(544,·)$, $\chi_{999}(676,·)$, $\chi_{999}(232,·)$, $\chi_{999}(556,·)$, $\chi_{999}(877,·)$, $\chi_{999}(112,·)$, $\chi_{999}(433,·)$, $\chi_{999}(778,·)$, $\chi_{999}(565,·)$, $\chi_{999}(121,·)$, $\chi_{999}(889,·)$, $\chi_{999}(223,·)$, $\chi_{999}(445,·)$, $\chi_{999}(766,·)$$\rbrace$ | ||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $\frac{1}{288791} a^{25} - \frac{91354}{288791} a^{24} - \frac{131271}{288791} a^{23} - \frac{14753}{288791} a^{22} - \frac{33994}{288791} a^{21} + \frac{131530}{288791} a^{20} + \frac{106461}{288791} a^{19} + \frac{50218}{288791} a^{18} + \frac{99949}{288791} a^{17} - \frac{124135}{288791} a^{16} - \frac{120694}{288791} a^{15} + \frac{139417}{288791} a^{14} + \frac{129536}{288791} a^{13} + \frac{4519}{288791} a^{12} + \frac{58151}{288791} a^{11} - \frac{104880}{288791} a^{10} + \frac{93476}{288791} a^{9} + \frac{75358}{288791} a^{8} + \frac{126961}{288791} a^{7} + \frac{31494}{288791} a^{6} - \frac{69432}{288791} a^{5} - \frac{141068}{288791} a^{4} - \frac{44058}{288791} a^{3} + \frac{71300}{288791} a^{2} + \frac{18013}{288791} a - \frac{55461}{288791}$, $\frac{1}{1202965337037689365256624085936213879159527327114969999791193427288882133735328012993316444836881498092817344829} a^{26} - \frac{375491818708454488609607460897623430288858688119027108780702963800935658808585894371416405306726009152758}{1202965337037689365256624085936213879159527327114969999791193427288882133735328012993316444836881498092817344829} a^{25} + \frac{510416844382757458322878657128840637853141135543230188224091981138002433104433025739255069463704057781625326037}{1202965337037689365256624085936213879159527327114969999791193427288882133735328012993316444836881498092817344829} a^{24} + \frac{340435577628462633946298792185478288594492243362809962496437476368647681413974340111184594995394324437376491763}{1202965337037689365256624085936213879159527327114969999791193427288882133735328012993316444836881498092817344829} a^{23} - \frac{31768391180867038531506585527842650835387113893680695885888111998804738045422952319349019816981184202572156186}{1202965337037689365256624085936213879159527327114969999791193427288882133735328012993316444836881498092817344829} a^{22} - \frac{412004672853333292374775107255187148603031668219070371231190343702753188068663750059598066913412951509351491986}{1202965337037689365256624085936213879159527327114969999791193427288882133735328012993316444836881498092817344829} a^{21} + \frac{490028161268684225258846063552030351768244600330962262355710350869710348970910139133152600081977205589691041807}{1202965337037689365256624085936213879159527327114969999791193427288882133735328012993316444836881498092817344829} a^{20} - \frac{574954576180203906098399221175033197451426586602530073271027114850989778819182008730247935996032801202969166917}{1202965337037689365256624085936213879159527327114969999791193427288882133735328012993316444836881498092817344829} a^{19} - \frac{254269161844294864481176806746292254462503528378306329774367224330614055089715229062539763917947778061328351914}{1202965337037689365256624085936213879159527327114969999791193427288882133735328012993316444836881498092817344829} a^{18} - \frac{340590176244214039008193906000289841202649252988634897943217071690851763585010497859118180374193222517092547304}{1202965337037689365256624085936213879159527327114969999791193427288882133735328012993316444836881498092817344829} a^{17} - \frac{411561040328865898275124894403476395907498298595585816579245430356893620024914190644511052946297242812971672170}{1202965337037689365256624085936213879159527327114969999791193427288882133735328012993316444836881498092817344829} a^{16} - \frac{232755622197127283780890085663611501995706418744148166789579616485501627617531832770677949075502173239252291857}{1202965337037689365256624085936213879159527327114969999791193427288882133735328012993316444836881498092817344829} a^{15} + \frac{409356982512732956285202264626385957481814312584939412112522269345955860451598008690061425764933378009795760464}{1202965337037689365256624085936213879159527327114969999791193427288882133735328012993316444836881498092817344829} a^{14} + \frac{118700974469349211377501290766516916871374777521818785245803869553166271416918029114953138984717146820505600104}{1202965337037689365256624085936213879159527327114969999791193427288882133735328012993316444836881498092817344829} a^{13} - \frac{468105681077787184804245588927332552414287151988418261446018049462983629499990038970194025502451455786420346819}{1202965337037689365256624085936213879159527327114969999791193427288882133735328012993316444836881498092817344829} a^{12} - \frac{264172792714728267065453786495837334967648562019452225274172793078247389939351575847357895630567009697183513620}{1202965337037689365256624085936213879159527327114969999791193427288882133735328012993316444836881498092817344829} a^{11} + \frac{328051566676038887593974351191466981678879846614600057494496737689027014250819168734198812310055755616132612369}{1202965337037689365256624085936213879159527327114969999791193427288882133735328012993316444836881498092817344829} a^{10} + \frac{600421869580192492435449932275528528776401357653221204948508080583547530998748684620719045154551119148756275707}{1202965337037689365256624085936213879159527327114969999791193427288882133735328012993316444836881498092817344829} a^{9} - \frac{275325216461502007298590300866286096076156767658252538471917923045998331072245800568688785709709966411997365503}{1202965337037689365256624085936213879159527327114969999791193427288882133735328012993316444836881498092817344829} a^{8} - \frac{213156127422749087328708190943593768684551214207508146955300239963143602319349993067163247446154370354433147552}{1202965337037689365256624085936213879159527327114969999791193427288882133735328012993316444836881498092817344829} a^{7} + \frac{540092195746613462425501873043202141424763013125639203477128448591629062273239596528261424811822360737863738433}{1202965337037689365256624085936213879159527327114969999791193427288882133735328012993316444836881498092817344829} a^{6} + \frac{110628410234354710318529328652362738637344044114913604745890769343940192394766614819216023473822717108593300061}{1202965337037689365256624085936213879159527327114969999791193427288882133735328012993316444836881498092817344829} a^{5} + \frac{42924301280987610729490180331216564502223651981638946418762721069673954058516678894540979737816867968063753194}{1202965337037689365256624085936213879159527327114969999791193427288882133735328012993316444836881498092817344829} a^{4} + \frac{142335681697988327612377395294459808051182565801608935987120996453342733451439319490392113457302391819625255078}{1202965337037689365256624085936213879159527327114969999791193427288882133735328012993316444836881498092817344829} a^{3} - \frac{318728371110690997100854836470274354379913126218856917412621647072911124941788580555261209037243644528906432360}{1202965337037689365256624085936213879159527327114969999791193427288882133735328012993316444836881498092817344829} a^{2} - \frac{31490538873443201921409858938351962443134402748358409570354896327144850039482231606115097481683758170975749380}{1202965337037689365256624085936213879159527327114969999791193427288882133735328012993316444836881498092817344829} a + \frac{329619732680116652207605410450341561826071451379960970082969654128309341405577880369783916025536777068888471494}{1202965337037689365256624085936213879159527327114969999791193427288882133735328012993316444836881498092817344829}$
Class group and class number
Trivial group, which has order $1$ (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: | \( 1223192070462029700000 \) (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
| 3.3.110889.1, \(\Q(\zeta_{9})^+\), 3.3.1369.1, 3.3.110889.2, 9.9.1363532208525369.2, \(\Q(\zeta_{27})^+\), 9.9.80515213381214514081.2, 9.9.80515213381214514081.3 |
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 | ${\href{/LocalNumberField/2.9.0.1}{9} }^{3}$ | R | ${\href{/LocalNumberField/5.9.0.1}{9} }^{3}$ | ${\href{/LocalNumberField/7.9.0.1}{9} }^{3}$ | ${\href{/LocalNumberField/11.9.0.1}{9} }^{3}$ | ${\href{/LocalNumberField/13.9.0.1}{9} }^{3}$ | ${\href{/LocalNumberField/17.3.0.1}{3} }^{9}$ | ${\href{/LocalNumberField/19.3.0.1}{3} }^{9}$ | ${\href{/LocalNumberField/23.9.0.1}{9} }^{3}$ | ${\href{/LocalNumberField/29.9.0.1}{9} }^{3}$ | ${\href{/LocalNumberField/31.9.0.1}{9} }^{3}$ | R | ${\href{/LocalNumberField/41.9.0.1}{9} }^{3}$ | ${\href{/LocalNumberField/43.9.0.1}{9} }^{3}$ | ${\href{/LocalNumberField/47.9.0.1}{9} }^{3}$ | ${\href{/LocalNumberField/53.3.0.1}{3} }^{9}$ | ${\href{/LocalNumberField/59.9.0.1}{9} }^{3}$ |
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 | ||||||
| $37$ | 37.9.6.1 | $x^{9} + 222 x^{6} + 15059 x^{3} + 405224$ | $3$ | $3$ | $6$ | $C_3^2$ | $[\ ]_{3}^{3}$ |
| 37.9.6.1 | $x^{9} + 222 x^{6} + 15059 x^{3} + 405224$ | $3$ | $3$ | $6$ | $C_3^2$ | $[\ ]_{3}^{3}$ | |
| 37.9.6.1 | $x^{9} + 222 x^{6} + 15059 x^{3} + 405224$ | $3$ | $3$ | $6$ | $C_3^2$ | $[\ ]_{3}^{3}$ | |