Normalized defining polynomial
\( x^{27} - 6 x^{26} - 119 x^{25} + 588 x^{24} + 6703 x^{23} - 23664 x^{22} - 229513 x^{21} + 474063 x^{20} + 5090845 x^{19} - 3919676 x^{18} - 73206350 x^{17} - 21588434 x^{16} + 656954365 x^{15} + 787530733 x^{14} - 3325690822 x^{13} - 7213234613 x^{12} + 6583428950 x^{11} + 29933674810 x^{10} + 11963110633 x^{9} - 47377292870 x^{8} - 61123319381 x^{7} - 6523878607 x^{6} + 31218812642 x^{5} + 18121240371 x^{4} - 246741993 x^{3} - 2622725154 x^{2} - 602443376 x - 28067027 \)
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: | \(82740369509121494587249825446059868376844092109571921823609=19^{24}\cdot 37^{18}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $152.10$ | 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: | $19, 37$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(703=19\cdot 37\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{703}(1,·)$, $\chi_{703}(195,·)$, $\chi_{703}(581,·)$, $\chi_{703}(454,·)$, $\chi_{703}(519,·)$, $\chi_{703}(137,·)$, $\chi_{703}(396,·)$, $\chi_{703}(334,·)$, $\chi_{703}(655,·)$, $\chi_{703}(593,·)$, $\chi_{703}(149,·)$, $\chi_{703}(343,·)$, $\chi_{703}(408,·)$, $\chi_{703}(100,·)$, $\chi_{703}(26,·)$, $\chi_{703}(158,·)$, $\chi_{703}(482,·)$, $\chi_{703}(676,·)$, $\chi_{703}(359,·)$, $\chi_{703}(232,·)$, $\chi_{703}(491,·)$, $\chi_{703}(556,·)$, $\chi_{703}(47,·)$, $\chi_{703}(112,·)$, $\chi_{703}(248,·)$, $\chi_{703}(121,·)$, $\chi_{703}(63,·)$$\rbrace$ | ||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $\frac{1}{191} a^{25} + \frac{72}{191} a^{24} - \frac{63}{191} a^{23} + \frac{83}{191} a^{22} - \frac{16}{191} a^{21} + \frac{85}{191} a^{20} - \frac{32}{191} a^{19} - \frac{78}{191} a^{18} + \frac{58}{191} a^{17} + \frac{76}{191} a^{16} + \frac{65}{191} a^{15} - \frac{50}{191} a^{14} + \frac{16}{191} a^{13} - \frac{79}{191} a^{12} + \frac{28}{191} a^{11} - \frac{24}{191} a^{10} - \frac{92}{191} a^{9} - \frac{78}{191} a^{8} + \frac{66}{191} a^{7} - \frac{46}{191} a^{6} - \frac{84}{191} a^{4} - \frac{87}{191} a^{3} - \frac{49}{191} a^{2} - \frac{65}{191} a - \frac{89}{191}$, $\frac{1}{1850970836601195828012047344435085839618546612394672758160875414450660827861621042140109313173609137774146452937831} a^{26} - \frac{3373750906768454240901935077594746292177699920808526758990825937498399065593367728702185227880302319915319391020}{1850970836601195828012047344435085839618546612394672758160875414450660827861621042140109313173609137774146452937831} a^{25} + \frac{448594004725605023607308570419577576893420466076555355798553639509270026061497786452330822796685722200982689708808}{1850970836601195828012047344435085839618546612394672758160875414450660827861621042140109313173609137774146452937831} a^{24} - \frac{759563557591690956690612699650060145953993249952027041898403959701699493112462180961528582330996821845528309111590}{1850970836601195828012047344435085839618546612394672758160875414450660827861621042140109313173609137774146452937831} a^{23} - \frac{850955941147913074342802530188017488123559419338636798602916952743110014831859773894428572928382350302436322594971}{1850970836601195828012047344435085839618546612394672758160875414450660827861621042140109313173609137774146452937831} a^{22} - \frac{452156306785894630081040457232283854865045710467550716631200949688940311863669795412508613336309092781053640498466}{1850970836601195828012047344435085839618546612394672758160875414450660827861621042140109313173609137774146452937831} a^{21} + \frac{307055396561062341588866376869905011347924640626726992712737883353991613181810443465742382631195258525484540296211}{1850970836601195828012047344435085839618546612394672758160875414450660827861621042140109313173609137774146452937831} a^{20} - \frac{710123501776445659352067264969904756518826984440887994612285067987723851616192460277881075468608281070233954251923}{1850970836601195828012047344435085839618546612394672758160875414450660827861621042140109313173609137774146452937831} a^{19} - \frac{630911654015114923846963904821642652220156359853555601023919605899548813144749504077414244716065206281908702749898}{1850970836601195828012047344435085839618546612394672758160875414450660827861621042140109313173609137774146452937831} a^{18} + \frac{15594701823471462502781864088996159821288863926945666777048914591567838453187538260298156576457657917219953681383}{1850970836601195828012047344435085839618546612394672758160875414450660827861621042140109313173609137774146452937831} a^{17} + \frac{626907728694877524563238677534802969253154505409758943470387328017029360632378360914001967397922741812317369279094}{1850970836601195828012047344435085839618546612394672758160875414450660827861621042140109313173609137774146452937831} a^{16} + \frac{160776251658152768766734234782081694949008438391987732087800513634388177560137354779799349376692116385281159089395}{1850970836601195828012047344435085839618546612394672758160875414450660827861621042140109313173609137774146452937831} a^{15} - \frac{236674381345268027256000014441814292676540349654690072123194689408821436622139859810850427810004286279846495480062}{1850970836601195828012047344435085839618546612394672758160875414450660827861621042140109313173609137774146452937831} a^{14} + \frac{682622526139572906394355315056034649845798899620111686912255342422972549492522874716370652920235396660948933920440}{1850970836601195828012047344435085839618546612394672758160875414450660827861621042140109313173609137774146452937831} a^{13} + \frac{482855762179249544371782670765790016894549500638500851444082104185048895577910201168509806017756328988895167936765}{1850970836601195828012047344435085839618546612394672758160875414450660827861621042140109313173609137774146452937831} a^{12} + \frac{433496568303621779291573678558970956672511735470777427601574895156723576800950576365517519771165473454520361627987}{1850970836601195828012047344435085839618546612394672758160875414450660827861621042140109313173609137774146452937831} a^{11} - \frac{357209502101491639817572859516957051647953787451982514092311773416657195417586130319070708902334789979418339919050}{1850970836601195828012047344435085839618546612394672758160875414450660827861621042140109313173609137774146452937831} a^{10} + \frac{220524792333951833026548220901145466792570163646130081607752384097824945002439425892631692586467746861143401628651}{1850970836601195828012047344435085839618546612394672758160875414450660827861621042140109313173609137774146452937831} a^{9} + \frac{117439942528846621508088790323000418845010300544548004385887598444217381129098837347873286647394156680974944015337}{1850970836601195828012047344435085839618546612394672758160875414450660827861621042140109313173609137774146452937831} a^{8} + \frac{710730889252711502423215814470960689147945549380676296451478784635716113291846261484152797494909923074554822576053}{1850970836601195828012047344435085839618546612394672758160875414450660827861621042140109313173609137774146452937831} a^{7} + \frac{83365801816800538521288394711894989049967951895035437413113170461069934915995751353385506396963032215324879417363}{1850970836601195828012047344435085839618546612394672758160875414450660827861621042140109313173609137774146452937831} a^{6} - \frac{544603795208597475816679373383664198844615188448820696870741783388516252179283365307432395981225524413524807968180}{1850970836601195828012047344435085839618546612394672758160875414450660827861621042140109313173609137774146452937831} a^{5} - \frac{661917913883382140596350441874714653043269253732372029534151621414058845126792334347540735937676602926419862460916}{1850970836601195828012047344435085839618546612394672758160875414450660827861621042140109313173609137774146452937831} a^{4} - \frac{105260395205875461527270564199717770025842301865873316224384884550091787402046596916594301775837073148850348156896}{1850970836601195828012047344435085839618546612394672758160875414450660827861621042140109313173609137774146452937831} a^{3} - \frac{426116294073709845695461391522359967371712674343395974948327492135584008582673017257258149715412805044033495477147}{1850970836601195828012047344435085839618546612394672758160875414450660827861621042140109313173609137774146452937831} a^{2} + \frac{642646452079570383802440900638983111787867943655236597293969653792319872867134774244082095246094227864001071552427}{1850970836601195828012047344435085839618546612394672758160875414450660827861621042140109313173609137774146452937831} a + \frac{199127527238226241301452478110147481370993785998606547933391123059809046971481399786175375452134599266473004538661}{1850970836601195828012047344435085839618546612394672758160875414450660827861621042140109313173609137774146452937831}$
Class group and class number
$C_{2}\times C_{2}$, which has order $4$ (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: | \( 149277476389860470000 \) (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
| 3.3.494209.2, 3.3.361.1, 3.3.1369.1, 3.3.494209.1, 9.9.120706859316371329.1, 9.9.43575176213210049769.2, \(\Q(\zeta_{19})^+\), 9.9.43575176213210049769.1 |
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 | ${\href{/LocalNumberField/2.9.0.1}{9} }^{3}$ | ${\href{/LocalNumberField/3.9.0.1}{9} }^{3}$ | ${\href{/LocalNumberField/5.9.0.1}{9} }^{3}$ | ${\href{/LocalNumberField/7.3.0.1}{3} }^{9}$ | ${\href{/LocalNumberField/11.3.0.1}{3} }^{9}$ | ${\href{/LocalNumberField/13.9.0.1}{9} }^{3}$ | ${\href{/LocalNumberField/17.9.0.1}{9} }^{3}$ | R | ${\href{/LocalNumberField/23.9.0.1}{9} }^{3}$ | ${\href{/LocalNumberField/29.9.0.1}{9} }^{3}$ | ${\href{/LocalNumberField/31.3.0.1}{3} }^{9}$ | R | ${\href{/LocalNumberField/41.9.0.1}{9} }^{3}$ | ${\href{/LocalNumberField/43.9.0.1}{9} }^{3}$ | ${\href{/LocalNumberField/47.9.0.1}{9} }^{3}$ | ${\href{/LocalNumberField/53.9.0.1}{9} }^{3}$ | ${\href{/LocalNumberField/59.9.0.1}{9} }^{3}$ |
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 |
|---|---|---|---|---|---|---|---|
| 19 | Data not computed | ||||||
| $37$ | 37.3.2.1 | $x^{3} - 37$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ |
| 37.3.2.1 | $x^{3} - 37$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
| 37.3.2.1 | $x^{3} - 37$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
| 37.3.2.1 | $x^{3} - 37$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
| 37.3.2.1 | $x^{3} - 37$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
| 37.3.2.1 | $x^{3} - 37$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
| 37.3.2.1 | $x^{3} - 37$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
| 37.3.2.1 | $x^{3} - 37$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
| 37.3.2.1 | $x^{3} - 37$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |