Normalized defining polynomial
\( x^{27} - 513 x^{25} + 116964 x^{23} - 15617943 x^{21} + 1354686795 x^{19} - 724356 x^{18} - 80025043581 x^{17} + 247729752 x^{16} + 3282297025608 x^{15} - 35301489660 x^{14} - 93545465229828 x^{13} + 2712723360984 x^{12} + 1824136571981646 x^{11} - 121491253381212 x^{10} - 23532675108670668 x^{9} + 3196154512028808 x^{8} + 189215858520799146 x^{7} - 47232061122203496 x^{6} - 849500283479758155 x^{5} + 349639932982545360 x^{4} + 1619550605024888319 x^{3} - 996473809000254276 x^{2} - 34450965446244615 x - 53756552511721 \)
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: | \(3463254175600113063839837232871102966170246194081956872095753304432278349849=3^{94}\cdot 19^{24}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $627.71$ | 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: | $3, 19$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(1539=3^{4}\cdot 19\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{1539}(1,·)$, $\chi_{1539}(514,·)$, $\chi_{1539}(1027,·)$, $\chi_{1539}(196,·)$, $\chi_{1539}(709,·)$, $\chi_{1539}(454,·)$, $\chi_{1539}(967,·)$, $\chi_{1539}(1480,·)$, $\chi_{1539}(499,·)$, $\chi_{1539}(334,·)$, $\chi_{1539}(847,·)$, $\chi_{1539}(1360,·)$, $\chi_{1539}(403,·)$, $\chi_{1539}(916,·)$, $\chi_{1539}(1429,·)$, $\chi_{1539}(1222,·)$, $\chi_{1539}(235,·)$, $\chi_{1539}(748,·)$, $\chi_{1539}(301,·)$, $\chi_{1539}(814,·)$, $\chi_{1539}(1327,·)$, $\chi_{1539}(1261,·)$, $\chi_{1539}(1012,·)$, $\chi_{1539}(1525,·)$, $\chi_{1539}(313,·)$, $\chi_{1539}(826,·)$, $\chi_{1539}(1339,·)$$\rbrace$ | ||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{19} a^{9}$, $\frac{1}{19} a^{10}$, $\frac{1}{19} a^{11}$, $\frac{1}{19} a^{12}$, $\frac{1}{19} a^{13}$, $\frac{1}{26353} a^{14} + \frac{14}{1387} a^{13} - \frac{14}{1387} a^{12} - \frac{27}{1387} a^{11} + \frac{3}{1387} a^{10} + \frac{10}{1387} a^{9} + \frac{25}{73} a^{8} - \frac{24}{73} a^{7} - \frac{8}{73} a^{6} - \frac{48}{1387} a^{5} + \frac{24}{73} a^{4} + \frac{28}{73} a^{3} - \frac{6}{73} a^{2} + \frac{15}{73} a + \frac{35}{73}$, $\frac{1}{26353} a^{15} - \frac{15}{1387} a^{13} - \frac{26}{1387} a^{12} + \frac{31}{1387} a^{11} + \frac{15}{1387} a^{10} + \frac{5}{1387} a^{9} - \frac{31}{73} a^{8} + \frac{25}{73} a^{7} + \frac{161}{1387} a^{6} - \frac{34}{73} a^{5} - \frac{5}{73} a^{4} - \frac{8}{73} a^{3} + \frac{5}{73} a^{2} - \frac{13}{73} a + \frac{34}{73}$, $\frac{1}{26353} a^{16} + \frac{22}{1387} a^{13} - \frac{17}{1387} a^{12} - \frac{15}{1387} a^{11} - \frac{16}{1387} a^{10} - \frac{2}{1387} a^{9} - \frac{4}{73} a^{8} + \frac{579}{1387} a^{7} + \frac{22}{73} a^{6} + \frac{5}{73} a^{5} - \frac{30}{73} a^{4} + \frac{28}{73} a^{3} + \frac{29}{73} a^{2} + \frac{2}{73} a - \frac{26}{73}$, $\frac{1}{26353} a^{17} - \frac{29}{1387} a^{13} - \frac{3}{1387} a^{12} + \frac{28}{1387} a^{11} - \frac{15}{1387} a^{10} - \frac{22}{1387} a^{9} + \frac{370}{1387} a^{8} - \frac{20}{73} a^{7} - \frac{9}{73} a^{6} + \frac{4}{73} a^{5} - \frac{3}{73} a^{4} + \frac{5}{73} a^{3} + \frac{28}{73} a^{2} - \frac{18}{73} a - \frac{30}{73}$, $\frac{1}{124282301535703} a^{18} - \frac{18}{6541173765037} a^{16} + \frac{135}{344272303423} a^{14} - \frac{546}{18119594917} a^{12} + \frac{24453}{18119594917} a^{10} - \frac{59592464567}{6541173765037} a^{9} - \frac{643302}{18119594917} a^{8} - \frac{152212425743}{344272303423} a^{7} + \frac{9506574}{18119594917} a^{6} + \frac{3647404304}{18119594917} a^{5} - \frac{70373340}{18119594917} a^{4} - \frac{6535666185}{18119594917} a^{3} + \frac{200564019}{18119594917} a^{2} - \frac{8045690038}{18119594917} a - \frac{5187712434}{18119594917}$, $\frac{1}{124282301535703} a^{19} - \frac{18}{6541173765037} a^{17} + \frac{135}{344272303423} a^{15} - \frac{546}{18119594917} a^{13} + \frac{24453}{18119594917} a^{11} - \frac{59592464567}{6541173765037} a^{10} - \frac{643302}{18119594917} a^{9} - \frac{152212425743}{344272303423} a^{8} + \frac{9506574}{18119594917} a^{7} + \frac{3647404304}{18119594917} a^{6} - \frac{70373340}{18119594917} a^{5} - \frac{6535666185}{18119594917} a^{4} + \frac{200564019}{18119594917} a^{3} - \frac{8045690038}{18119594917} a^{2} - \frac{5187712434}{18119594917} a$, $\frac{1}{124282301535703} a^{20} - \frac{189}{344272303423} a^{16} + \frac{1884}{18119594917} a^{14} - \frac{2223}{248213629} a^{12} - \frac{59592464567}{6541173765037} a^{11} + \frac{7719624}{18119594917} a^{10} + \frac{7255666407}{344272303423} a^{9} - \frac{210502710}{18119594917} a^{8} - \frac{117426603}{18119594917} a^{7} + \frac{3180874968}{18119594917} a^{6} + \frac{8744151427}{18119594917} a^{5} - \frac{5747523344}{18119594917} a^{4} + \frac{3586244400}{18119594917} a^{3} + \frac{9046397313}{18119594917} a^{2} + \frac{2552434388}{18119594917} a + \frac{1522649438}{18119594917}$, $\frac{1}{2361363729178357} a^{21} - \frac{189}{6541173765037} a^{17} + \frac{1884}{344272303423} a^{15} - \frac{117}{248213629} a^{13} - \frac{1436681678259}{124282301535703} a^{12} + \frac{406296}{18119594917} a^{11} + \frac{152212425743}{6541173765037} a^{10} - \frac{11079090}{18119594917} a^{9} + \frac{126719737816}{344272303423} a^{8} + \frac{167414472}{18119594917} a^{7} + \frac{5228532948}{18119594917} a^{6} - \frac{1256164119}{18119594917} a^{5} + \frac{2096075486}{18119594917} a^{4} + \frac{9046397313}{344272303423} a^{3} + \frac{134338652}{18119594917} a^{2} - \frac{2780849227}{18119594917} a$, $\frac{1}{2361363729178357} a^{22} - \frac{1518}{344272303423} a^{16} + \frac{16974}{18119594917} a^{14} - \frac{1436681678259}{124282301535703} a^{13} - \frac{1554390}{18119594917} a^{12} + \frac{152212425743}{6541173765037} a^{11} + \frac{76731633}{18119594917} a^{10} + \frac{7294808608}{344272303423} a^{9} - \frac{2142683010}{18119594917} a^{8} - \frac{7122799200}{18119594917} a^{7} - \frac{3357246719}{18119594917} a^{6} - \frac{542193841}{18119594917} a^{5} + \frac{27356030375}{344272303423} a^{4} - \frac{4567514168}{18119594917} a^{3} - \frac{7339253678}{18119594917} a^{2} + \frac{8680966157}{18119594917} a - \frac{2131775818}{18119594917}$, $\frac{1}{431114697897476785906897} a^{23} + \frac{2741785}{22690247257761936100363} a^{22} - \frac{23}{22690247257761936100363} a^{21} - \frac{2665516}{1194223539882207163177} a^{20} + \frac{230}{1194223539882207163177} a^{19} + \frac{677688}{1194223539882207163177} a^{18} - \frac{69}{3308098448427166657} a^{17} - \frac{69518893398}{62853870520116166483} a^{16} + \frac{4692}{3308098448427166657} a^{15} + \frac{419525196007447467}{22690247257761936100363} a^{14} + \frac{30435120140898526923}{1194223539882207163177} a^{13} - \frac{5823863364025039740}{1194223539882207163177} a^{12} + \frac{1187621134745427760}{62853870520116166483} a^{11} - \frac{1125680791944250048}{62853870520116166483} a^{10} - \frac{824561734848706666}{62853870520116166483} a^{9} - \frac{1054348927327971982}{3308098448427166657} a^{8} - \frac{1516310248339724094}{3308098448427166657} a^{7} - \frac{46111674893146387}{174110444654061403} a^{6} - \frac{23610920571654081473}{62853870520116166483} a^{5} + \frac{377928178452376074}{3308098448427166657} a^{4} - \frac{1156049659604642771}{3308098448427166657} a^{3} - \frac{10537352824271248}{174110444654061403} a^{2} + \frac{48904413372220047}{174110444654061403} a + \frac{10700467058169193}{174110444654061403}$, $\frac{1}{431114697897476785906897} a^{24} - \frac{24}{22690247257761936100363} a^{22} + \frac{4049120}{22690247257761936100363} a^{21} + \frac{252}{1194223539882207163177} a^{20} + \frac{76269}{62853870520116166483} a^{19} - \frac{80}{3308098448427166657} a^{18} - \frac{791367678}{62853870520116166483} a^{17} + \frac{306}{174110444654061403} a^{16} + \frac{414167668020603216}{22690247257761936100363} a^{15} - \frac{14688}{174110444654061403} a^{14} - \frac{6211952665766280549}{1194223539882207163177} a^{13} + \frac{18753654192772418312}{1194223539882207163177} a^{12} + \frac{881783937567870946}{62853870520116166483} a^{11} + \frac{1369447173163715797}{62853870520116166483} a^{10} - \frac{13989815793623331}{3308098448427166657} a^{9} + \frac{488168188738730338}{3308098448427166657} a^{8} + \frac{53816788506340594}{174110444654061403} a^{7} - \frac{13198136872781396230}{62853870520116166483} a^{6} - \frac{17335296669914316}{174110444654061403} a^{5} - \frac{361687371449882411}{3308098448427166657} a^{4} - \frac{353618389616369806}{3308098448427166657} a^{3} - \frac{46385711491447351}{174110444654061403} a^{2} - \frac{8206694275596875}{174110444654061403} a + \frac{20033083514743597}{174110444654061403}$, $\frac{1}{8191179260052058932231043} a^{25} + \frac{1281912}{22690247257761936100363} a^{22} - \frac{300}{22690247257761936100363} a^{21} + \frac{3366598}{1194223539882207163177} a^{20} + \frac{4000}{1194223539882207163177} a^{19} + \frac{3222513}{1194223539882207163177} a^{18} - \frac{1350}{3308098448427166657} a^{17} + \frac{3857824785176312461}{431114697897476785906897} a^{16} + \frac{97920}{3308098448427166657} a^{15} + \frac{286643562953361110}{22690247257761936100363} a^{14} + \frac{9416364860699797607}{1194223539882207163177} a^{13} - \frac{26981433545006697429}{1194223539882207163177} a^{12} + \frac{834558361256577479}{62853870520116166483} a^{11} - \frac{249149610981261959}{62853870520116166483} a^{10} + \frac{1210462755404708105}{62853870520116166483} a^{9} + \frac{220052382969141875}{3308098448427166657} a^{8} + \frac{279845693910321374578}{1194223539882207163177} a^{7} + \frac{59917632104491244}{174110444654061403} a^{6} - \frac{2291268965748639630}{62853870520116166483} a^{5} + \frac{874429915661256724}{3308098448427166657} a^{4} - \frac{654455896864535740}{3308098448427166657} a^{3} - \frac{68972247002240666}{174110444654061403} a^{2} - \frac{29915802170597161}{174110444654061403} a - \frac{2056159755823244}{174110444654061403}$, $\frac{1}{8191179260052058932231043} a^{26} - \frac{325}{22690247257761936100363} a^{22} - \frac{5773}{310825304900848439731} a^{21} + \frac{4550}{1194223539882207163177} a^{20} + \frac{3290170}{1194223539882207163177} a^{19} - \frac{1625}{3308098448427166657} a^{18} + \frac{3858161841865739104}{431114697897476785906897} a^{17} + \frac{6630}{174110444654061403} a^{16} + \frac{282975919213989071}{22690247257761936100363} a^{15} - \frac{331500}{174110444654061403} a^{14} - \frac{26123689688581152080}{1194223539882207163177} a^{13} + \frac{23826651442792936446}{1194223539882207163177} a^{12} - \frac{1293771234345697833}{62853870520116166483} a^{11} + \frac{1135851475652521043}{62853870520116166483} a^{10} + \frac{60826252974016324}{3308098448427166657} a^{9} - \frac{98138793399371206785}{1194223539882207163177} a^{8} + \frac{24576192591024701}{174110444654061403} a^{7} - \frac{28957806968710093858}{62853870520116166483} a^{6} + \frac{24636094227570066}{174110444654061403} a^{5} + \frac{1331761182646919100}{3308098448427166657} a^{4} + \frac{92543722949286339}{3308098448427166657} a^{3} + \frac{11074185389955839}{174110444654061403} a^{2} + \frac{76205769234882871}{174110444654061403} a - \frac{31044544430803128}{174110444654061403}$
Class group and class number
$C_{9}$, which has order $9$ (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: | \( 25942643830764960000000000000 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A cyclic group of order 27 |
| The 27 conjugacy class representatives for $C_{27}$ |
| Character table for $C_{27}$ is not computed |
Intermediate fields
| \(\Q(\zeta_{9})^+\), 9.9.1476349596018920529.2 |
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 | $27$ | R | $27$ | $27$ | $27$ | $27$ | ${\href{/LocalNumberField/17.9.0.1}{9} }^{3}$ | R | $27$ | $27$ | $27$ | ${\href{/LocalNumberField/37.9.0.1}{9} }^{3}$ | $27$ | $27$ | $27$ | ${\href{/LocalNumberField/53.9.0.1}{9} }^{3}$ | $27$ |
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 | ||||||
| $19$ | 19.9.8.9 | $x^{9} - 4864$ | $9$ | $1$ | $8$ | $C_9$ | $[\ ]_{9}$ |
| 19.9.8.9 | $x^{9} - 4864$ | $9$ | $1$ | $8$ | $C_9$ | $[\ ]_{9}$ | |
| 19.9.8.9 | $x^{9} - 4864$ | $9$ | $1$ | $8$ | $C_9$ | $[\ ]_{9}$ | |