Normalized defining polynomial
\( x^{24} - 5 x^{23} + 22 x^{22} - 51 x^{21} + 119 x^{20} - x^{19} - 821 x^{18} + 6142 x^{17} - 14334 x^{16} + 31930 x^{15} + 29369 x^{14} - 210769 x^{13} + 636038 x^{12} - 582811 x^{11} + 1635343 x^{10} - 2062773 x^{9} + 5475610 x^{8} - 1268874 x^{7} + 9361191 x^{6} + 7380279 x^{5} + 5795470 x^{4} + 15518693 x^{3} + 16763733 x^{2} - 12721577 x + 8245033 \)
Invariants
| Degree: | $24$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[0, 12]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(1220256264249613719359965066902998670022440897=3^{12}\cdot 7^{16}\cdot 17^{21}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $75.61$ | 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$$\chi_{357}(128,·)$, $\chi_{357}(1,·)$, $\chi_{357}(2,·)$, $\chi_{357}(67,·)$, $\chi_{357}(4,·)$, $\chi_{357}(134,·)$, $\chi_{357}(263,·)$, $\chi_{357}(8,·)$, $\chi_{357}(268,·)$, $\chi_{357}(205,·)$, $\chi_{357}(256,·)$, $\chi_{357}(16,·)$, $\chi_{357}(338,·)$, $\chi_{357}(212,·)$, $\chi_{357}(281,·)$, $\chi_{357}(155,·)$, $\chi_{357}(32,·)$, $\chi_{357}(64,·)$, $\chi_{357}(169,·)$, $\chi_{357}(106,·)$, $\chi_{357}(179,·)$, $\chi_{357}(53,·)$, $\chi_{357}(310,·)$, $\chi_{357}(319,·)$$\rbrace$ | ||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $\frac{1}{798953} a^{21} - \frac{195650}{798953} a^{20} - \frac{135856}{798953} a^{19} - \frac{379215}{798953} a^{18} - \frac{258006}{798953} a^{17} - \frac{345083}{798953} a^{16} + \frac{230389}{798953} a^{15} + \frac{90656}{798953} a^{14} - \frac{384702}{798953} a^{13} + \frac{85284}{798953} a^{12} - \frac{22730}{798953} a^{11} - \frac{323968}{798953} a^{10} + \frac{281147}{798953} a^{9} - \frac{303088}{798953} a^{8} + \frac{105509}{798953} a^{7} + \frac{142127}{798953} a^{6} - \frac{256904}{798953} a^{5} + \frac{360679}{798953} a^{4} - \frac{280111}{798953} a^{3} - \frac{23472}{798953} a^{2} - \frac{283905}{798953} a - \frac{218286}{798953}$, $\frac{1}{10386389} a^{22} - \frac{5}{10386389} a^{21} - \frac{4236641}{10386389} a^{20} - \frac{357931}{10386389} a^{19} + \frac{596805}{10386389} a^{18} - \frac{78413}{10386389} a^{17} - \frac{4401505}{10386389} a^{16} + \frac{124082}{798953} a^{15} + \frac{372653}{798953} a^{14} - \frac{369094}{10386389} a^{13} + \frac{3226810}{10386389} a^{12} - \frac{1960326}{10386389} a^{11} - \frac{3893582}{10386389} a^{10} - \frac{62728}{798953} a^{9} - \frac{53544}{10386389} a^{8} - \frac{2495088}{10386389} a^{7} - \frac{2178107}{10386389} a^{6} - \frac{329453}{798953} a^{5} + \frac{217830}{798953} a^{4} + \frac{3438874}{10386389} a^{3} + \frac{3114311}{10386389} a^{2} + \frac{797408}{10386389} a + \frac{2267098}{10386389}$, $\frac{1}{451997728695591943447411849425817581501042398811377809977902971696105780539660528391487} a^{23} + \frac{6553460558531966101698600850762558835243713641327452816607027525729895063461344}{451997728695591943447411849425817581501042398811377809977902971696105780539660528391487} a^{22} - \frac{27647800090705883298828093453349823903811158545691490697914161671301695019583829}{451997728695591943447411849425817581501042398811377809977902971696105780539660528391487} a^{21} + \frac{170909690637559797431901845969860358560975662226947513297051825136698924569677864350004}{451997728695591943447411849425817581501042398811377809977902971696105780539660528391487} a^{20} - \frac{112451133600628136912784392704471501210185986225255382855592294225675208580884540233773}{451997728695591943447411849425817581501042398811377809977902971696105780539660528391487} a^{19} + \frac{223012654296153408016000260914488606125668445809904800268585695299413714821462445393554}{451997728695591943447411849425817581501042398811377809977902971696105780539660528391487} a^{18} - \frac{135977005301853447996125958457093861038072886481475699057843712912022638819395962552448}{451997728695591943447411849425817581501042398811377809977902971696105780539660528391487} a^{17} + \frac{112323381354185296022311216480866028329541862372882482834356102255670546207986501859571}{451997728695591943447411849425817581501042398811377809977902971696105780539660528391487} a^{16} - \frac{10539149291202350845476868499875429501537880205827445890760094657250197361784884014559}{34769056053507072572877834571216737038541722985490600767530997822777367733820040645499} a^{15} - \frac{126488213374648011370644735632959947353771974479732340230046191661694233296124625519746}{451997728695591943447411849425817581501042398811377809977902971696105780539660528391487} a^{14} + \frac{100982699365766322007172789068726371377545915616655921366753478712258120922391946607234}{451997728695591943447411849425817581501042398811377809977902971696105780539660528391487} a^{13} - \frac{6554693327985708298543575067307105926630413669322208751219533205178419263400051770862}{34769056053507072572877834571216737038541722985490600767530997822777367733820040645499} a^{12} - \frac{176586361989118677143493073095322239399926806216607379363907746613742472903226627488425}{451997728695591943447411849425817581501042398811377809977902971696105780539660528391487} a^{11} - \frac{94384917125509114452636608599880505489166026849380304963977565695886939917952405210802}{451997728695591943447411849425817581501042398811377809977902971696105780539660528391487} a^{10} - \frac{5229884140324572203612502731335552891967499702047744639713801444295102984346510980587}{451997728695591943447411849425817581501042398811377809977902971696105780539660528391487} a^{9} - \frac{177526020562560968339631138566581774108076002669953751604808376385377263442948065436025}{451997728695591943447411849425817581501042398811377809977902971696105780539660528391487} a^{8} - \frac{96053171603746308467020131053210917137567697199862963431696497030254510411193097463451}{451997728695591943447411849425817581501042398811377809977902971696105780539660528391487} a^{7} - \frac{43393417677400589023969118561394300620422237400560683121992340515805245619389984450921}{451997728695591943447411849425817581501042398811377809977902971696105780539660528391487} a^{6} + \frac{12773476755091255770122956699155454254479886566153222665747087535492462471961490457261}{34769056053507072572877834571216737038541722985490600767530997822777367733820040645499} a^{5} + \frac{113939492761824665102583160822419871119180268005640778801010259746802504649141300681422}{451997728695591943447411849425817581501042398811377809977902971696105780539660528391487} a^{4} - \frac{220421977847683497296386598274759124846108302028603325347639960117499714716247488995978}{451997728695591943447411849425817581501042398811377809977902971696105780539660528391487} a^{3} + \frac{169935953866028367867120921130992876375608278106765480608597390907261907676477899104427}{451997728695591943447411849425817581501042398811377809977902971696105780539660528391487} a^{2} + \frac{12859781007374154195972135440524199607483435644770751542121118154339248235337654645791}{451997728695591943447411849425817581501042398811377809977902971696105780539660528391487} a + \frac{197208606574941514099983694870418109083950032609884819259776426234956929588831434850810}{451997728695591943447411849425817581501042398811377809977902971696105780539660528391487}$
Class group and class number
$C_{7}\times C_{1358}$, which has order $9506$ (assuming GRH)
Unit group
| Rank: | $11$ | 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: | \( 250243842.68845215 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A cyclic group of order 24 |
| The 24 conjugacy class representatives for $C_{24}$ |
| Character table for $C_{24}$ is not computed |
Intermediate fields
| \(\Q(\sqrt{17}) \), \(\Q(\zeta_{7})^+\), 4.4.4913.1, 6.6.11796113.1, 8.0.33237432513.1, 12.12.683635509017782097.1 |
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 | ${\href{/LocalNumberField/2.12.0.1}{12} }^{2}$ | R | $24$ | R | $24$ | ${\href{/LocalNumberField/13.2.0.1}{2} }^{12}$ | R | ${\href{/LocalNumberField/19.12.0.1}{12} }^{2}$ | $24$ | ${\href{/LocalNumberField/29.8.0.1}{8} }^{3}$ | $24$ | $24$ | ${\href{/LocalNumberField/41.8.0.1}{8} }^{3}$ | ${\href{/LocalNumberField/43.4.0.1}{4} }^{6}$ | ${\href{/LocalNumberField/47.3.0.1}{3} }^{8}$ | ${\href{/LocalNumberField/53.12.0.1}{12} }^{2}$ | ${\href{/LocalNumberField/59.12.0.1}{12} }^{2}$ |
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 | ||||||
| 17 | Data not computed | ||||||