Normalized defining polynomial
\( x^{24} - 8 x^{23} + 47 x^{22} - 214 x^{21} + 956 x^{20} - 3268 x^{19} + 8913 x^{18} - 24730 x^{17} + 64416 x^{16} - 93292 x^{15} + 242761 x^{14} - 883492 x^{13} + 426718 x^{12} + 715344 x^{11} + 7877438 x^{10} - 1257480 x^{9} - 24939133 x^{8} - 68610260 x^{7} - 28182313 x^{6} + 203608140 x^{5} + 626540010 x^{4} + 912724280 x^{3} + 976659364 x^{2} + 457755938 x + 161501293 \)
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: | \(771331169903802522095823068539961341836284345188352=2^{24}\cdot 13^{16}\cdot 17^{21}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $131.92$ | 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: | $2, 13, 17$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(884=2^{2}\cdot 13\cdot 17\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{884}(1,·)$, $\chi_{884}(451,·)$, $\chi_{884}(625,·)$, $\chi_{884}(263,·)$, $\chi_{884}(781,·)$, $\chi_{884}(399,·)$, $\chi_{884}(81,·)$, $\chi_{884}(835,·)$, $\chi_{884}(341,·)$, $\chi_{884}(87,·)$, $\chi_{884}(217,·)$, $\chi_{884}(859,·)$, $\chi_{884}(157,·)$, $\chi_{884}(287,·)$, $\chi_{884}(763,·)$, $\chi_{884}(807,·)$, $\chi_{884}(237,·)$, $\chi_{884}(477,·)$, $\chi_{884}(497,·)$, $\chi_{884}(627,·)$, $\chi_{884}(495,·)$, $\chi_{884}(373,·)$, $\chi_{884}(633,·)$, $\chi_{884}(315,·)$$\rbrace$ | ||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{52} a^{18} - \frac{3}{26} a^{17} - \frac{5}{52} a^{16} - \frac{7}{26} a^{15} - \frac{15}{52} a^{14} - \frac{7}{26} a^{13} + \frac{7}{52} a^{12} - \frac{1}{26} a^{11} - \frac{11}{26} a^{10} - \frac{5}{26} a^{9} + \frac{23}{52} a^{8} - \frac{11}{26} a^{7} + \frac{1}{52} a^{6} + \frac{1}{13} a^{5} + \frac{11}{52} a^{4} + \frac{6}{13} a^{3} + \frac{17}{52} a^{2} + \frac{11}{26} a + \frac{21}{52}$, $\frac{1}{52} a^{19} + \frac{11}{52} a^{17} + \frac{2}{13} a^{16} + \frac{5}{52} a^{15} - \frac{25}{52} a^{13} - \frac{3}{13} a^{12} + \frac{9}{26} a^{11} + \frac{7}{26} a^{10} + \frac{15}{52} a^{9} + \frac{3}{13} a^{8} + \frac{25}{52} a^{7} + \frac{5}{26} a^{6} - \frac{17}{52} a^{5} - \frac{7}{26} a^{4} + \frac{5}{52} a^{3} + \frac{5}{13} a^{2} - \frac{3}{52} a + \frac{11}{26}$, $\frac{1}{676} a^{20} + \frac{1}{338} a^{19} - \frac{1}{169} a^{18} - \frac{9}{169} a^{17} - \frac{41}{169} a^{16} - \frac{36}{169} a^{15} + \frac{11}{169} a^{14} + \frac{76}{169} a^{13} - \frac{163}{676} a^{12} - \frac{58}{169} a^{11} + \frac{61}{676} a^{10} - \frac{69}{169} a^{9} - \frac{35}{169} a^{8} + \frac{1}{26} a^{7} + \frac{23}{169} a^{6} + \frac{25}{169} a^{5} + \frac{18}{169} a^{4} + \frac{69}{338} a^{3} - \frac{161}{338} a^{2} + \frac{77}{338} a + \frac{93}{676}$, $\frac{1}{676} a^{21} + \frac{5}{676} a^{19} - \frac{1}{338} a^{18} - \frac{105}{676} a^{17} + \frac{79}{338} a^{16} + \frac{33}{676} a^{15} - \frac{87}{338} a^{14} - \frac{27}{169} a^{13} + \frac{30}{169} a^{12} + \frac{31}{676} a^{11} - \frac{28}{169} a^{10} - \frac{329}{676} a^{9} - \frac{73}{169} a^{8} - \frac{207}{676} a^{7} + \frac{18}{169} a^{6} - \frac{245}{676} a^{5} + \frac{49}{338} a^{4} + \frac{7}{52} a^{3} + \frac{37}{169} a^{2} + \frac{159}{338} a - \frac{15}{338}$, $\frac{1}{110452376319708052270006955879155206392948} a^{22} - \frac{10425722207550289322555048134537782883}{110452376319708052270006955879155206392948} a^{21} + \frac{27027060786010711917433202550376442219}{55226188159854026135003477939577603196474} a^{20} - \frac{651184959941726753724857343007599442763}{110452376319708052270006955879155206392948} a^{19} - \frac{629481831996550308457371818720932046377}{110452376319708052270006955879155206392948} a^{18} + \frac{44380566732233942592407869528838392411589}{110452376319708052270006955879155206392948} a^{17} + \frac{712877396922338329864604242463923271291}{2350050559993788346170360763386280987084} a^{16} - \frac{24009404122672920953804025925334927081581}{110452376319708052270006955879155206392948} a^{15} + \frac{13700115677142431017028893755393636914290}{27613094079927013067501738969788801598237} a^{14} + \frac{9449359592806246502135399649991485479905}{27613094079927013067501738969788801598237} a^{13} - \frac{13460693931931774259464653049730985419973}{55226188159854026135003477939577603196474} a^{12} + \frac{16196594522097490047968634699456769950111}{110452376319708052270006955879155206392948} a^{11} + \frac{590948221835064518019880546801061903753}{2124084159994385620577056843829907815249} a^{10} - \frac{12305892600927794410170603776628572861009}{110452376319708052270006955879155206392948} a^{9} - \frac{16898031689034606293120499412451611610489}{110452376319708052270006955879155206392948} a^{8} - \frac{41030194664506371109316790719593903720337}{110452376319708052270006955879155206392948} a^{7} + \frac{2259032888327947940762873282239327694273}{110452376319708052270006955879155206392948} a^{6} + \frac{16573201160493998428082142809209505782181}{110452376319708052270006955879155206392948} a^{5} - \frac{36156363103588579553085689834340637773017}{110452376319708052270006955879155206392948} a^{4} + \frac{22072656916139920472161653173726068712117}{110452376319708052270006955879155206392948} a^{3} - \frac{27147328809562101254474251382650369218063}{55226188159854026135003477939577603196474} a^{2} + \frac{25527023411879535261685832056445515076701}{55226188159854026135003477939577603196474} a + \frac{42359238121920513605168987261612604364745}{110452376319708052270006955879155206392948}$, $\frac{1}{185179765465295138645910495968447614148844610588012932164} a^{23} + \frac{365508705147125}{92589882732647569322955247984223807074422305294006466082} a^{22} + \frac{14631046605951844745693265209380996391753112184648033}{46294941366323784661477623992111903537211152647003233041} a^{21} + \frac{63912822138278287517792657196774846317760032067958217}{92589882732647569322955247984223807074422305294006466082} a^{20} + \frac{1171662779908709161017442141584041760402319947311105459}{185179765465295138645910495968447614148844610588012932164} a^{19} - \frac{297769194964647609159921085380712830401896990920420771}{92589882732647569322955247984223807074422305294006466082} a^{18} + \frac{12853151156646082003987155362510037408592408085513528397}{185179765465295138645910495968447614148844610588012932164} a^{17} - \frac{34197583048294226854870665830072272964475549911014468059}{92589882732647569322955247984223807074422305294006466082} a^{16} - \frac{15150640331877759840336094753287294341430708422844546721}{46294941366323784661477623992111903537211152647003233041} a^{15} + \frac{26110848269927011016013284587163203301501383417257626753}{92589882732647569322955247984223807074422305294006466082} a^{14} - \frac{26066690027416912815327392430625182222623110357918481467}{92589882732647569322955247984223807074422305294006466082} a^{13} - \frac{10092477118891371870793019373800525217435822435557486647}{46294941366323784661477623992111903537211152647003233041} a^{12} - \frac{17887547931933776302348745372936433764353677297287190205}{92589882732647569322955247984223807074422305294006466082} a^{11} - \frac{42127521642367295008916013488539265661821856387174221709}{92589882732647569322955247984223807074422305294006466082} a^{10} + \frac{67312493489103350616640334425647121795764838496706639149}{185179765465295138645910495968447614148844610588012932164} a^{9} - \frac{36375567324169682931809854444456181226958906364832511293}{92589882732647569322955247984223807074422305294006466082} a^{8} + \frac{52578177731419496528395791180715342121500295640133153431}{185179765465295138645910495968447614148844610588012932164} a^{7} + \frac{5261377249966059861380032067511528621762040681353727509}{92589882732647569322955247984223807074422305294006466082} a^{6} + \frac{75390894362215345364273020102394304974620096782271222743}{185179765465295138645910495968447614148844610588012932164} a^{5} - \frac{9555233329952933414198576372872777912729353783360768469}{92589882732647569322955247984223807074422305294006466082} a^{4} - \frac{12605472246158154321075107393326787616348985464922108567}{46294941366323784661477623992111903537211152647003233041} a^{3} - \frac{29655535917179970541733832537540415728501754029892126961}{92589882732647569322955247984223807074422305294006466082} a^{2} - \frac{4768751794024956677783259240838371561376386677849359565}{185179765465295138645910495968447614148844610588012932164} a + \frac{9521730699300457094691443552042222957127050854369695605}{92589882732647569322955247984223807074422305294006466082}$
Class group and class number
$C_{2}\times C_{530258}$, which has order $1060516$ (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: | \( 2336441224.9148784 \) (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
| \(\Q(\sqrt{17}) \), 3.3.169.1, 4.4.4913.1, 6.6.140320193.1, 8.0.105046700288.1, 12.12.96735773996756764337.1 |
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 | R | $24$ | ${\href{/LocalNumberField/5.8.0.1}{8} }^{3}$ | $24$ | $24$ | R | R | ${\href{/LocalNumberField/19.12.0.1}{12} }^{2}$ | $24$ | $24$ | ${\href{/LocalNumberField/31.8.0.1}{8} }^{3}$ | $24$ | $24$ | ${\href{/LocalNumberField/43.12.0.1}{12} }^{2}$ | ${\href{/LocalNumberField/47.1.0.1}{1} }^{24}$ | ${\href{/LocalNumberField/53.4.0.1}{4} }^{6}$ | ${\href{/LocalNumberField/59.12.0.1}{12} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
| $2$ | 2.12.12.25 | $x^{12} - 78 x^{10} - 1621 x^{8} + 460 x^{6} - 1977 x^{4} + 866 x^{2} + 749$ | $2$ | $6$ | $12$ | $C_{12}$ | $[2]^{6}$ |
| 2.12.12.25 | $x^{12} - 78 x^{10} - 1621 x^{8} + 460 x^{6} - 1977 x^{4} + 866 x^{2} + 749$ | $2$ | $6$ | $12$ | $C_{12}$ | $[2]^{6}$ | |
| $13$ | 13.6.4.3 | $x^{6} + 65 x^{3} + 1352$ | $3$ | $2$ | $4$ | $C_6$ | $[\ ]_{3}^{2}$ |
| 13.6.4.3 | $x^{6} + 65 x^{3} + 1352$ | $3$ | $2$ | $4$ | $C_6$ | $[\ ]_{3}^{2}$ | |
| 13.6.4.3 | $x^{6} + 65 x^{3} + 1352$ | $3$ | $2$ | $4$ | $C_6$ | $[\ ]_{3}^{2}$ | |
| 13.6.4.3 | $x^{6} + 65 x^{3} + 1352$ | $3$ | $2$ | $4$ | $C_6$ | $[\ ]_{3}^{2}$ | |
| 17 | Data not computed | ||||||