Normalized defining polynomial
\( x^{27} - 513 x^{25} + 116964 x^{23} - 15617943 x^{21} + 1354686795 x^{19} - 3058506 x^{18} - 80025043581 x^{17} + 1046009052 x^{16} + 3282297025608 x^{15} - 149056289910 x^{14} - 93545465229828 x^{13} + 11454147789084 x^{12} + 1824136571981646 x^{11} - 512982190268262 x^{10} - 23529533875005090 x^{9} + 13495377620903508 x^{8} + 188678707563985308 x^{7} - 199431691508907396 x^{6} - 818882678941369389 x^{5} + 1476312521559444360 x^{4} + 973178953658903259 x^{3} - 4207490686444416426 x^{2} + 3649867447339870227 x - 1053420808747042331 \)
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$$\chi_{1539}(1498,·)$, $\chi_{1539}(1,·)$, $\chi_{1539}(514,·)$, $\chi_{1539}(1027,·)$, $\chi_{1539}(142,·)$, $\chi_{1539}(655,·)$, $\chi_{1539}(1168,·)$, $\chi_{1539}(334,·)$, $\chi_{1539}(985,·)$, $\chi_{1539}(472,·)$, $\chi_{1539}(25,·)$, $\chi_{1539}(538,·)$, $\chi_{1539}(1051,·)$, $\chi_{1539}(157,·)$, $\chi_{1539}(670,·)$, $\chi_{1539}(1183,·)$, $\chi_{1539}(1360,·)$, $\chi_{1539}(847,·)$, $\chi_{1539}(232,·)$, $\chi_{1539}(745,·)$, $\chi_{1539}(1258,·)$, $\chi_{1539}(235,·)$, $\chi_{1539}(748,·)$, $\chi_{1539}(1261,·)$, $\chi_{1539}(112,·)$, $\chi_{1539}(625,·)$, $\chi_{1539}(1138,·)$$\rbrace$ | ||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{19} a^{9}$, $\frac{1}{19} a^{10}$, $\frac{1}{19} a^{11}$, $\frac{1}{19} a^{12}$, $\frac{1}{19} a^{13}$, $\frac{1}{361} a^{14} - \frac{6}{19} a^{5}$, $\frac{1}{361} a^{15} - \frac{6}{19} a^{6}$, $\frac{1}{361} a^{16} - \frac{6}{19} a^{7}$, $\frac{1}{361} a^{17} - \frac{6}{19} a^{8}$, $\frac{1}{1940867449847} a^{18} - \frac{18}{102150918413} a^{16} + \frac{135}{5376364127} a^{14} - \frac{546}{282966533} a^{12} + \frac{24453}{282966533} a^{10} - \frac{82699159}{102150918413} a^{9} - \frac{643302}{282966533} a^{8} + \frac{744292431}{5376364127} a^{7} + \frac{9506574}{282966533} a^{6} + \frac{30854971}{282966533} a^{5} - \frac{70373340}{282966533} a^{4} - \frac{116890381}{282966533} a^{3} - \frac{82402514}{282966533} a^{2} + \frac{128638759}{282966533} a - \frac{48406937}{282966533}$, $\frac{1}{1940867449847} a^{19} - \frac{18}{102150918413} a^{17} + \frac{135}{5376364127} a^{15} - \frac{546}{282966533} a^{13} + \frac{24453}{282966533} a^{11} - \frac{82699159}{102150918413} a^{10} - \frac{643302}{282966533} a^{9} + \frac{744292431}{5376364127} a^{8} + \frac{9506574}{282966533} a^{7} + \frac{30854971}{282966533} a^{6} - \frac{70373340}{282966533} a^{5} - \frac{116890381}{282966533} a^{4} - \frac{82402514}{282966533} a^{3} + \frac{128638759}{282966533} a^{2} - \frac{48406937}{282966533} a$, $\frac{1}{1940867449847} a^{20} - \frac{189}{5376364127} a^{16} + \frac{1884}{282966533} a^{14} - \frac{162279}{282966533} a^{12} - \frac{82699159}{102150918413} a^{11} - \frac{136293677}{5376364127} a^{10} + \frac{104607168}{5376364127} a^{9} + \frac{72463823}{282966533} a^{8} + \frac{128691678}{282966533} a^{7} + \frac{68243105}{282966533} a^{6} - \frac{34252020}{282966533} a^{5} - \frac{97929489}{282966533} a^{4} + \frac{50409610}{282966533} a^{3} + \frac{66586575}{282966533} a^{2} + \frac{134642963}{282966533} a + \frac{139852993}{282966533}$, $\frac{1}{36876481547093} a^{21} - \frac{189}{102150918413} a^{17} + \frac{1884}{5376364127} a^{15} - \frac{8541}{282966533} a^{13} - \frac{43093612175}{1940867449847} a^{12} + \frac{406296}{282966533} a^{11} - \frac{744292431}{102150918413} a^{10} + \frac{72463823}{5376364127} a^{9} + \frac{411658211}{5376364127} a^{8} - \frac{115552061}{282966533} a^{7} - \frac{76267615}{282966533} a^{6} - \frac{124297987}{282966533} a^{5} - \frac{116490666}{282966533} a^{4} + \frac{2613285372}{5376364127} a^{3} + \frac{51765398}{282966533} a^{2} - \frac{22425267}{282966533} a$, $\frac{1}{36876481547093} a^{22} - \frac{1518}{5376364127} a^{16} + \frac{16974}{282966533} a^{14} - \frac{43093612175}{1940867449847} a^{13} - \frac{1554390}{282966533} a^{12} - \frac{744292431}{102150918413} a^{11} + \frac{43068362}{5376364127} a^{10} + \frac{61709942}{5376364127} a^{9} + \frac{121049254}{282966533} a^{8} - \frac{39365057}{282966533} a^{7} + \frac{57825287}{282966533} a^{6} + \frac{43795792}{282966533} a^{5} + \frac{2203835923}{5376364127} a^{4} - \frac{62224334}{282966533} a^{3} + \frac{53140477}{282966533} a^{2} + \frac{140401713}{282966533} a - \frac{87859505}{282966533}$, $\frac{1}{37069598407618264806701} a^{23} + \frac{25272312}{1951031495137803410879} a^{22} - \frac{23}{1951031495137803410879} a^{21} + \frac{25988369}{102685868165147547941} a^{20} + \frac{230}{102685868165147547941} a^{19} - \frac{8613139}{102685868165147547941} a^{18} - \frac{69}{284448388269106781} a^{17} - \frac{822073219281}{5404519377113028839} a^{16} + \frac{4692}{284448388269106781} a^{15} - \frac{1981182699280714093}{1951031495137803410879} a^{14} + \frac{1162066254473309431}{102685868165147547941} a^{13} + \frac{1287500573843059630}{102685868165147547941} a^{12} - \frac{79225923876281477}{5404519377113028839} a^{11} - \frac{127381734322297059}{5404519377113028839} a^{10} - \frac{104736741689492034}{5404519377113028839} a^{9} - \frac{101005160441570293}{284448388269106781} a^{8} + \frac{130977930411053687}{284448388269106781} a^{7} - \frac{6798191378156250}{14970967803637199} a^{6} + \frac{1121865304318629519}{5404519377113028839} a^{5} + \frac{129115705079703484}{284448388269106781} a^{4} - \frac{46236828540155847}{284448388269106781} a^{3} + \frac{4100785975326130}{14970967803637199} a^{2} - \frac{4590874543049515}{14970967803637199} a - \frac{1585028391689613}{14970967803637199}$, $\frac{1}{37069598407618264806701} a^{24} - \frac{24}{1951031495137803410879} a^{22} + \frac{4009101}{1951031495137803410879} a^{21} + \frac{252}{102685868165147547941} a^{20} + \frac{21623285}{102685868165147547941} a^{19} - \frac{80}{284448388269106781} a^{18} - \frac{1146939219}{5404519377113028839} a^{17} + \frac{306}{14970967803637199} a^{16} - \frac{2043387771734780941}{1951031495137803410879} a^{15} - \frac{14688}{14970967803637199} a^{14} - \frac{1775538088666980813}{102685868165147547941} a^{13} - \frac{637233791144923098}{102685868165147547941} a^{12} + \frac{127522011346242688}{5404519377113028839} a^{11} - \frac{42051716782719137}{5404519377113028839} a^{10} + \frac{34622117108846}{284448388269106781} a^{9} + \frac{86386948540389932}{284448388269106781} a^{8} + \frac{5811621086694539}{14970967803637199} a^{7} - \frac{218740169377318190}{5404519377113028839} a^{6} + \frac{5705068471521190}{14970967803637199} a^{5} + \frac{134329184925109123}{284448388269106781} a^{4} + \frac{76548090428153658}{284448388269106781} a^{3} - \frac{5621784451285971}{14970967803637199} a^{2} + \frac{7091995027571932}{14970967803637199} a - \frac{1546400925485805}{14970967803637199}$, $\frac{1}{704322369744747031327319} a^{25} - \frac{3078636}{1951031495137803410879} a^{22} - \frac{300}{1951031495137803410879} a^{21} - \frac{12812102}{102685868165147547941} a^{20} + \frac{4000}{102685868165147547941} a^{19} + \frac{21932723}{102685868165147547941} a^{18} - \frac{1350}{284448388269106781} a^{17} + \frac{19575539808247419454}{37069598407618264806701} a^{16} + \frac{97920}{284448388269106781} a^{15} - \frac{2187513812771771958}{1951031495137803410879} a^{14} + \frac{2366012439099271985}{102685868165147547941} a^{13} + \frac{117761531207777445}{102685868165147547941} a^{12} - \frac{4712330022480664}{5404519377113028839} a^{11} + \frac{68493650848155938}{5404519377113028839} a^{10} - \frac{92055729029207492}{5404519377113028839} a^{9} - \frac{29724938714452249}{284448388269106781} a^{8} + \frac{23639867801637293114}{102685868165147547941} a^{7} + \frac{1520968829230888}{14970967803637199} a^{6} - \frac{1189356496527399486}{5404519377113028839} a^{5} - \frac{121567714814064026}{284448388269106781} a^{4} - \frac{107812587176822637}{284448388269106781} a^{3} - \frac{2752427335252940}{14970967803637199} a^{2} - \frac{5505792405811545}{14970967803637199} a - \frac{1551476988294976}{14970967803637199}$, $\frac{1}{704322369744747031327319} a^{26} - \frac{325}{1951031495137803410879} a^{22} - \frac{1577780}{1951031495137803410879} a^{21} + \frac{4550}{102685868165147547941} a^{20} - \frac{15764722}{102685868165147547941} a^{19} - \frac{1625}{284448388269106781} a^{18} + \frac{19574621898982665778}{37069598407618264806701} a^{17} + \frac{6630}{14970967803637199} a^{16} - \frac{2177613265408711191}{1951031495137803410879} a^{15} - \frac{331500}{14970967803637199} a^{14} + \frac{2516726921352515212}{102685868165147547941} a^{13} - \frac{1199114849835843412}{102685868165147547941} a^{12} - \frac{32919552943047527}{5404519377113028839} a^{11} + \frac{102997961731509793}{5404519377113028839} a^{10} + \frac{5393745570648181}{284448388269106781} a^{9} - \frac{229772743951394689}{629974651319923607} a^{8} - \frac{3979651337974946}{14970967803637199} a^{7} + \frac{1034971159976113636}{5404519377113028839} a^{6} - \frac{3983689007158985}{14970967803637199} a^{5} - \frac{30520291157936402}{284448388269106781} a^{4} + \frac{8555921080556053}{284448388269106781} a^{3} + \frac{3281068081821192}{14970967803637199} a^{2} - \frac{7139850075311807}{14970967803637199} a + \frac{6252880273766472}{14970967803637199}$
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: | \( 20201504574439970000000000000 \) (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 | ||||||
| $19$ | 19.9.8.4 | $x^{9} + 311296$ | $9$ | $1$ | $8$ | $C_9$ | $[\ ]_{9}$ |
| 19.9.8.4 | $x^{9} + 311296$ | $9$ | $1$ | $8$ | $C_9$ | $[\ ]_{9}$ | |
| 19.9.8.4 | $x^{9} + 311296$ | $9$ | $1$ | $8$ | $C_9$ | $[\ ]_{9}$ | |