Normalized defining polynomial
\( x^{30} - x^{29} + 47 x^{28} - 48 x^{27} + 944 x^{26} - 993 x^{25} + 10687 x^{24} - 11729 x^{23} + 76154 x^{22} - 88925 x^{21} + 366388 x^{20} - 468468 x^{19} + 1294545 x^{18} - 1888754 x^{17} + 3897814 x^{16} - 6566292 x^{15} + 11930723 x^{14} - 20080376 x^{13} + 37185318 x^{12} - 47602046 x^{11} + 102379331 x^{10} - 79703953 x^{9} + 234755530 x^{8} - 89460368 x^{7} + 475554491 x^{6} - 73175697 x^{5} + 799313122 x^{4} - 123217303 x^{3} + 950484078 x^{2} - 224979702 x + 533593369 \)
Invariants
| Degree: | $30$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[0, 15]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(-252273899561549903518384292359448695620714577809320549079=-\,3^{15}\cdot 7^{25}\cdot 11^{27}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $75.87$ | 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: | $3, 7, 11$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(231=3\cdot 7\cdot 11\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{231}(64,·)$, $\chi_{231}(1,·)$, $\chi_{231}(130,·)$, $\chi_{231}(131,·)$, $\chi_{231}(4,·)$, $\chi_{231}(68,·)$, $\chi_{231}(194,·)$, $\chi_{231}(206,·)$, $\chi_{231}(16,·)$, $\chi_{231}(17,·)$, $\chi_{231}(67,·)$, $\chi_{231}(148,·)$, $\chi_{231}(227,·)$, $\chi_{231}(214,·)$, $\chi_{231}(215,·)$, $\chi_{231}(25,·)$, $\chi_{231}(101,·)$, $\chi_{231}(100,·)$, $\chi_{231}(163,·)$, $\chi_{231}(164,·)$, $\chi_{231}(37,·)$, $\chi_{231}(230,·)$, $\chi_{231}(167,·)$, $\chi_{231}(41,·)$, $\chi_{231}(173,·)$, $\chi_{231}(83,·)$, $\chi_{231}(190,·)$, $\chi_{231}(169,·)$, $\chi_{231}(58,·)$, $\chi_{231}(62,·)$$\rbrace$ | ||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $\frac{1}{43258643} a^{26} + \frac{402895}{43258643} a^{25} - \frac{17486816}{43258643} a^{24} - \frac{5386889}{43258643} a^{23} + \frac{13362742}{43258643} a^{22} - \frac{10277170}{43258643} a^{21} + \frac{2826428}{43258643} a^{20} - \frac{5267970}{43258643} a^{19} + \frac{19389099}{43258643} a^{18} - \frac{12751913}{43258643} a^{17} + \frac{8944771}{43258643} a^{16} + \frac{2040139}{43258643} a^{15} + \frac{8054836}{43258643} a^{14} + \frac{13225595}{43258643} a^{13} - \frac{9836040}{43258643} a^{12} + \frac{6625165}{43258643} a^{11} + \frac{136001}{43258643} a^{10} - \frac{13449709}{43258643} a^{9} - \frac{21066014}{43258643} a^{8} + \frac{505162}{43258643} a^{7} - \frac{11987940}{43258643} a^{6} + \frac{19433714}{43258643} a^{5} + \frac{15221665}{43258643} a^{4} - \frac{16705949}{43258643} a^{3} + \frac{17647747}{43258643} a^{2} - \frac{18084927}{43258643} a + \frac{16381897}{43258643}$, $\frac{1}{43258643} a^{27} + \frac{7819338}{43258643} a^{25} - \frac{16805407}{43258643} a^{24} - \frac{8630199}{43258643} a^{23} + \frac{5457948}{43258643} a^{22} - \frac{7557096}{43258643} a^{21} - \frac{18458698}{43258643} a^{20} + \frac{16102097}{43258643} a^{19} - \frac{8264649}{43258643} a^{18} - \frac{8320275}{43258643} a^{17} - \frac{10440862}{43258643} a^{16} + \frac{3728074}{43258643} a^{15} - \frac{19785408}{43258643} a^{14} + \frac{20452532}{43258643} a^{13} + \frac{16934378}{43258643} a^{12} - \frac{14409002}{43258643} a^{11} + \frac{1128077}{43258643} a^{10} + \frac{5526146}{43258643} a^{9} + \frac{3200449}{43258643} a^{8} - \frac{7316615}{43258643} a^{7} - \frac{13488222}{43258643} a^{6} - \frac{3114651}{43258643} a^{5} - \frac{14866657}{43258643} a^{4} + \frac{18929803}{43258643} a^{3} - \frac{255797}{43258643} a^{2} - \frac{13005429}{43258643} a + \frac{3063910}{43258643}$, $\frac{1}{412628557274744999} a^{28} - \frac{1175000021}{412628557274744999} a^{27} - \frac{2726930966}{412628557274744999} a^{26} - \frac{169104224082743680}{412628557274744999} a^{25} - \frac{199484867904625805}{412628557274744999} a^{24} + \frac{116116436099734233}{412628557274744999} a^{23} - \frac{25458845613662023}{412628557274744999} a^{22} + \frac{179384233187666941}{412628557274744999} a^{21} - \frac{173092200743578971}{412628557274744999} a^{20} + \frac{188623150278448814}{412628557274744999} a^{19} - \frac{171729404516971080}{412628557274744999} a^{18} + \frac{132924690319301574}{412628557274744999} a^{17} - \frac{40500614259964181}{412628557274744999} a^{16} + \frac{109341421857417661}{412628557274744999} a^{15} + \frac{65537447140471818}{412628557274744999} a^{14} + \frac{132563378433573336}{412628557274744999} a^{13} - \frac{12279091403854059}{412628557274744999} a^{12} - \frac{123964034258852032}{412628557274744999} a^{11} - \frac{2687997680814128}{412628557274744999} a^{10} - \frac{184351263554824672}{412628557274744999} a^{9} + \frac{1887090150086715}{412628557274744999} a^{8} + \frac{54475298656971384}{412628557274744999} a^{7} + \frac{163253658199995405}{412628557274744999} a^{6} + \frac{134950441639664197}{412628557274744999} a^{5} + \frac{33810191721096634}{412628557274744999} a^{4} - \frac{82161156964142840}{412628557274744999} a^{3} + \frac{116496203643371763}{412628557274744999} a^{2} + \frac{186298032228607969}{412628557274744999} a + \frac{162909984526291587}{412628557274744999}$, $\frac{1}{1245008304085496290008052634245615724210857798483514087550811534798731865142223438382517356418459} a^{29} + \frac{19344666632634339007603967466143204352402056071704319289915759463852955665327}{1245008304085496290008052634245615724210857798483514087550811534798731865142223438382517356418459} a^{28} - \frac{3669462866688677994295689928949301730347199814133563307146992706506030087041560969274423}{1245008304085496290008052634245615724210857798483514087550811534798731865142223438382517356418459} a^{27} - \frac{2559453580064789350136714973098872576135349376882971391338958958554673594133498742210583}{1245008304085496290008052634245615724210857798483514087550811534798731865142223438382517356418459} a^{26} - \frac{398406816588796548233801962847272278213661697966056454993628734828148312825882417133833757241009}{1245008304085496290008052634245615724210857798483514087550811534798731865142223438382517356418459} a^{25} - \frac{36519882661061001830248475598689486587662638093293516800296414442424220512365478035162798137821}{1245008304085496290008052634245615724210857798483514087550811534798731865142223438382517356418459} a^{24} + \frac{540916314751789892045365216295573467683930133344927412382677703934677305022054845098598877553988}{1245008304085496290008052634245615724210857798483514087550811534798731865142223438382517356418459} a^{23} - \frac{73251521388795688802510098762075032445253026888797115042770866834406457487396107572829728190250}{1245008304085496290008052634245615724210857798483514087550811534798731865142223438382517356418459} a^{22} - \frac{265043193577749240185570148822632032311056013426119976537973618835733751592877632165242294017528}{1245008304085496290008052634245615724210857798483514087550811534798731865142223438382517356418459} a^{21} - \frac{380994844236620526051716626988009648966927705891337977897720839734368136884151765163101101611012}{1245008304085496290008052634245615724210857798483514087550811534798731865142223438382517356418459} a^{20} - \frac{231955727086107549195498299046899348963259709578731891792745497629193555373807794856247084940459}{1245008304085496290008052634245615724210857798483514087550811534798731865142223438382517356418459} a^{19} + \frac{453816777595303442790536501748030321824848272906288798765515253565416127089301010033838106091105}{1245008304085496290008052634245615724210857798483514087550811534798731865142223438382517356418459} a^{18} - \frac{288733587025851802174014464609727154350276084909507132133870053551222659887781542186055761916513}{1245008304085496290008052634245615724210857798483514087550811534798731865142223438382517356418459} a^{17} - \frac{402638165747781159470312360277959699990283607490581344730280037774549287950546237722648045365197}{1245008304085496290008052634245615724210857798483514087550811534798731865142223438382517356418459} a^{16} - \frac{177836440076160704915407634504769626243414201310317798343171241147299146555684465731337522087211}{1245008304085496290008052634245615724210857798483514087550811534798731865142223438382517356418459} a^{15} - \frac{225082083230321902465089735202039582006733058783799527741015798739974546641726124772514501751401}{1245008304085496290008052634245615724210857798483514087550811534798731865142223438382517356418459} a^{14} + \frac{329682485192811346155126103243430834268202662660664881314936022977112611138363770516040493284934}{1245008304085496290008052634245615724210857798483514087550811534798731865142223438382517356418459} a^{13} - \frac{74859450380978510575555541610394879683762665847935233204953779204641063174218694315582701768727}{1245008304085496290008052634245615724210857798483514087550811534798731865142223438382517356418459} a^{12} - \frac{414978446790466992181179154283629176642195400922661579708185335100204684324075233082846528911627}{1245008304085496290008052634245615724210857798483514087550811534798731865142223438382517356418459} a^{11} - \frac{236465510913802270444798422803408816607249627401922046978914407573820956416765699124163080231}{2971380200681375393813968100824858530336176130032253192245373591405087983632991499719611829161} a^{10} + \frac{616468717979975653314554790640561549775428739158720007500520388004736105053632550950272721166821}{1245008304085496290008052634245615724210857798483514087550811534798731865142223438382517356418459} a^{9} - \frac{83670449541870658027951454691822998040521968690898338066312692863149796489857198226934388512419}{1245008304085496290008052634245615724210857798483514087550811534798731865142223438382517356418459} a^{8} - \frac{187348332248032024694301121667535323356648434148678927363379387594060327207087805941144599326148}{1245008304085496290008052634245615724210857798483514087550811534798731865142223438382517356418459} a^{7} - \frac{1550853606018604327495837443783962085500028910065124596887713537420538932795781849469875419465}{4055401641972300618918738222298422554432761558578221783553132035175022362026786444242727545337} a^{6} - \frac{187446259964799969190013405397114882492401432037422240351581820509368799988884107771755758324379}{1245008304085496290008052634245615724210857798483514087550811534798731865142223438382517356418459} a^{5} - \frac{159892352474162054691731237174150411916859490292766756533919227944581780260115078693116297581481}{1245008304085496290008052634245615724210857798483514087550811534798731865142223438382517356418459} a^{4} + \frac{391982282069742934389959960699786108445511465610642473447027612810182534662643732975767472723870}{1245008304085496290008052634245615724210857798483514087550811534798731865142223438382517356418459} a^{3} + \frac{405623069902795430891286019098042367956451861806662100512633682488139192256423165077795175610609}{1245008304085496290008052634245615724210857798483514087550811534798731865142223438382517356418459} a^{2} - \frac{619137244438158395405448499680487792456865684317508046534450820298275582151903974948243183889681}{1245008304085496290008052634245615724210857798483514087550811534798731865142223438382517356418459} a + \frac{187784318698826658646991939023748699854200788753290677457749990299604536650133141465277}{2333252953309276019147930292675012091859143396670497394338518353661797219864296024230611}$
Class group and class number
$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{29898}$, which has order $956736$ (assuming GRH)
Unit group
| Rank: | $14$ | 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: | \( 4697581952.048968 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A cyclic group of order 30 |
| The 30 conjugacy class representatives for $C_{30}$ |
| Character table for $C_{30}$ is not computed |
Intermediate fields
| \(\Q(\sqrt{-231}) \), \(\Q(\zeta_{7})^+\), \(\Q(\zeta_{11})^+\), 6.0.603993159.1, 10.0.9630096522760791.1, 15.15.886528337182930278529.1 |
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 | $15^{2}$ | R | $15^{2}$ | R | R | ${\href{/LocalNumberField/13.5.0.1}{5} }^{6}$ | $30$ | $15^{2}$ | ${\href{/LocalNumberField/23.6.0.1}{6} }^{5}$ | ${\href{/LocalNumberField/29.5.0.1}{5} }^{6}$ | $30$ | $15^{2}$ | ${\href{/LocalNumberField/41.10.0.1}{10} }^{3}$ | ${\href{/LocalNumberField/43.2.0.1}{2} }^{15}$ | $15^{2}$ | $30$ | $15^{2}$ |
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 | ||||||
| 11 | Data not computed | ||||||