Normalized defining polynomial
\( x^{32} - 14 x^{31} + 90 x^{30} - 354 x^{29} + 1060 x^{28} - 3122 x^{27} + 9718 x^{26} - 27548 x^{25} + 68323 x^{24} - 162420 x^{23} + 388682 x^{22} - 885302 x^{21} + 1883356 x^{20} - 3859590 x^{19} + 7747823 x^{18} - 14894504 x^{17} + 27483236 x^{16} - 48703184 x^{15} + 83847191 x^{14} - 137630031 x^{13} + 219099037 x^{12} - 329935717 x^{11} + 485458653 x^{10} - 661629418 x^{9} + 894648443 x^{8} - 1078358227 x^{7} + 1337537651 x^{6} - 1342929253 x^{5} + 1540908056 x^{4} - 1158250822 x^{3} + 1251370063 x^{2} - 517957627 x + 555187123 \)
Invariants
| Degree: | $32$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[0, 16]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(11721315531921426838644724774450299033325694662265349722529=3^{16}\cdot 7^{16}\cdot 17^{30}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $65.26$ | 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: | $3, 7, 17$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(357=3\cdot 7\cdot 17\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{357}(1,·)$, $\chi_{357}(260,·)$, $\chi_{357}(13,·)$, $\chi_{357}(146,·)$, $\chi_{357}(20,·)$, $\chi_{357}(29,·)$, $\chi_{357}(167,·)$, $\chi_{357}(41,·)$, $\chi_{357}(43,·)$, $\chi_{357}(176,·)$, $\chi_{357}(307,·)$, $\chi_{357}(55,·)$, $\chi_{357}(62,·)$, $\chi_{357}(64,·)$, $\chi_{357}(197,·)$, $\chi_{357}(71,·)$, $\chi_{357}(202,·)$, $\chi_{357}(76,·)$, $\chi_{357}(335,·)$, $\chi_{357}(209,·)$, $\chi_{357}(218,·)$, $\chi_{357}(92,·)$, $\chi_{357}(349,·)$, $\chi_{357}(223,·)$, $\chi_{357}(106,·)$, $\chi_{357}(274,·)$, $\chi_{357}(125,·)$, $\chi_{357}(113,·)$, $\chi_{357}(118,·)$, $\chi_{357}(169,·)$, $\chi_{357}(253,·)$, $\chi_{357}(127,·)$$\rbrace$ | ||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $\frac{1}{957470290194758935608080898208915306548105739960506120121755369634977747289691520839904765159717506541554909643218103521959} a^{31} - \frac{13051584651746927644194251690537030361654471055134209529934911963003675556327672137056781988421084178167879512245611167216}{957470290194758935608080898208915306548105739960506120121755369634977747289691520839904765159717506541554909643218103521959} a^{30} + \frac{136782721226848880566249086605177087364081953055384465750012844410368133836413302724562276203792946768731314935151549955205}{957470290194758935608080898208915306548105739960506120121755369634977747289691520839904765159717506541554909643218103521959} a^{29} - \frac{36343252618320778597656177678149397917517706362002012241169466588876155771002269544075361178459128726828383768344827836605}{957470290194758935608080898208915306548105739960506120121755369634977747289691520839904765159717506541554909643218103521959} a^{28} + \frac{201488432505420264005590985268789111823519326660755728495404566769836460983908961433118258780089683874858185420125157363147}{957470290194758935608080898208915306548105739960506120121755369634977747289691520839904765159717506541554909643218103521959} a^{27} + \frac{248847700616002561557509086957136472384297888160680229857040643815210450658328962245225609800933830580727853854768626368142}{957470290194758935608080898208915306548105739960506120121755369634977747289691520839904765159717506541554909643218103521959} a^{26} + \frac{44687547202277011966421833128280967337837464230800538835796869681015217257955404234303508540073835980026550926480504756654}{957470290194758935608080898208915306548105739960506120121755369634977747289691520839904765159717506541554909643218103521959} a^{25} - \frac{179050030633740092890860573386739839992776971828095857972964999618867042053598853779131119831436648775438016389028425948851}{957470290194758935608080898208915306548105739960506120121755369634977747289691520839904765159717506541554909643218103521959} a^{24} - \frac{72125670528415820509655075026349597524069444195846921582737517613022437030321074736930001641691174743131889548338947703779}{957470290194758935608080898208915306548105739960506120121755369634977747289691520839904765159717506541554909643218103521959} a^{23} - \frac{78967686468495116145852211473217967485436273575432371866394348612337699707449606395924473566058521207046665908579206071429}{957470290194758935608080898208915306548105739960506120121755369634977747289691520839904765159717506541554909643218103521959} a^{22} - \frac{68189852673670976602007257632133647015909828798382287038069700915305114907615359694331790295556855088613187622745642556104}{957470290194758935608080898208915306548105739960506120121755369634977747289691520839904765159717506541554909643218103521959} a^{21} + \frac{115105370595581627875758851451609628324199791546061720048734073471347975702056199284169134322731941666428966873389562631380}{957470290194758935608080898208915306548105739960506120121755369634977747289691520839904765159717506541554909643218103521959} a^{20} + \frac{355143612288397221074703925527550622256696553727092368605643715984134418396716860809383729954435721142125662195578602268152}{957470290194758935608080898208915306548105739960506120121755369634977747289691520839904765159717506541554909643218103521959} a^{19} + \frac{60609628074245413699266932086932498687145600846649440540088690305874135487858944092374884488705520401298170890038982195361}{957470290194758935608080898208915306548105739960506120121755369634977747289691520839904765159717506541554909643218103521959} a^{18} - \frac{261361848310158231907567298956554850708888644263213926457666512086473286893640561365541808361670638301130823162373329264205}{957470290194758935608080898208915306548105739960506120121755369634977747289691520839904765159717506541554909643218103521959} a^{17} - \frac{191246934228883424261103644551646916534774445313696608282613513632433354299533808208441770792025749962071938792163659931198}{957470290194758935608080898208915306548105739960506120121755369634977747289691520839904765159717506541554909643218103521959} a^{16} + \frac{285833487865610492513850920435026040664946162779427822877912156469006816297973541633067372626228812215742525825530656940198}{957470290194758935608080898208915306548105739960506120121755369634977747289691520839904765159717506541554909643218103521959} a^{15} + \frac{441282508166861980498133546249576955733261881291329093440026846137463627662434011423696186397774738949070728759586356219565}{957470290194758935608080898208915306548105739960506120121755369634977747289691520839904765159717506541554909643218103521959} a^{14} + \frac{289511784117341567906392799788582259963710579884475926319940569206375617672209814855486724217356762654538220922258191236856}{957470290194758935608080898208915306548105739960506120121755369634977747289691520839904765159717506541554909643218103521959} a^{13} + \frac{450101043223055903500280351656718183009157966375345034363596332316638019972303981772835569943460363637570905769365519245283}{957470290194758935608080898208915306548105739960506120121755369634977747289691520839904765159717506541554909643218103521959} a^{12} + \frac{46925594582211642951283724617561188101786758942475251433786605587131872048238099914171347766943453648313437545907303451841}{957470290194758935608080898208915306548105739960506120121755369634977747289691520839904765159717506541554909643218103521959} a^{11} + \frac{391070901451509241710076711229544778506058758975729803945756190105063433625096620709576303796725847313972515155149839384126}{957470290194758935608080898208915306548105739960506120121755369634977747289691520839904765159717506541554909643218103521959} a^{10} - \frac{38281778097264262940079390191146026511528272455304152819135481743122956330593743468247174847664833911174623318666613310901}{957470290194758935608080898208915306548105739960506120121755369634977747289691520839904765159717506541554909643218103521959} a^{9} + \frac{455834920193777584759242865363566982148238031164452178191458406717657309755214963164428711640472305906461778310840298580501}{957470290194758935608080898208915306548105739960506120121755369634977747289691520839904765159717506541554909643218103521959} a^{8} + \frac{183749824077880317726244627227287145945979799708000213997225582235234136906928775883295236589585989066066545494366496512502}{957470290194758935608080898208915306548105739960506120121755369634977747289691520839904765159717506541554909643218103521959} a^{7} - \frac{391475364066960576746803808492461129558929122439288252278511768260823611822374413543612852208849078189870650737396833058519}{957470290194758935608080898208915306548105739960506120121755369634977747289691520839904765159717506541554909643218103521959} a^{6} + \frac{177305111310415642034728304152188937310618516819861055369207945091818705721709178602040856910994197550135312497049851288653}{957470290194758935608080898208915306548105739960506120121755369634977747289691520839904765159717506541554909643218103521959} a^{5} - \frac{312991947613111674027615088698938853092781628790579342580025896531247587652950137964652148363381740272455219357976759022986}{957470290194758935608080898208915306548105739960506120121755369634977747289691520839904765159717506541554909643218103521959} a^{4} - \frac{368355794059326291855661758852495160158080553587215207802689982773608894375801490654327291624886343211837330229439332739340}{957470290194758935608080898208915306548105739960506120121755369634977747289691520839904765159717506541554909643218103521959} a^{3} - \frac{385167718871683228132909599221693621405971361256386557647200393632921737396230781593518982566920829113182879336752505862140}{957470290194758935608080898208915306548105739960506120121755369634977747289691520839904765159717506541554909643218103521959} a^{2} + \frac{21442498795245008270482031121740146013659419991205150524505909796276746323162682517814782999171550780627267184621350642187}{957470290194758935608080898208915306548105739960506120121755369634977747289691520839904765159717506541554909643218103521959} a - \frac{43255730567731758703756888787230599813959023660579360373589581318492291181311978122340838675032725973644900232312616516717}{957470290194758935608080898208915306548105739960506120121755369634977747289691520839904765159717506541554909643218103521959}$
Class group and class number
Not computed
Unit group
| Rank: | $15$ | 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: | Not computed | magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
| |
| Regulator: | Not computed | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$C_2\times C_{16}$ (as 32T32):
| An abelian group of order 32 |
| The 32 conjugacy class representatives for $C_2\times C_{16}$ |
| Character table for $C_2\times C_{16}$ is not computed |
Intermediate fields
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 | ${\href{/LocalNumberField/2.8.0.1}{8} }^{4}$ | R | $16^{2}$ | R | $16^{2}$ | ${\href{/LocalNumberField/13.4.0.1}{4} }^{8}$ | R | ${\href{/LocalNumberField/19.8.0.1}{8} }^{4}$ | $16^{2}$ | $16^{2}$ | $16^{2}$ | $16^{2}$ | $16^{2}$ | ${\href{/LocalNumberField/43.8.0.1}{8} }^{4}$ | ${\href{/LocalNumberField/47.4.0.1}{4} }^{8}$ | ${\href{/LocalNumberField/53.8.0.1}{8} }^{4}$ | ${\href{/LocalNumberField/59.8.0.1}{8} }^{4}$ |
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 | Data not computed | ||||||
| 17 | Data not computed | ||||||