Normalized defining polynomial
\( x^{24} - 3 x^{23} - 12 x^{22} + 6 x^{21} + 282 x^{20} - 141 x^{19} - 370 x^{18} + 1773 x^{17} + 14088 x^{16} + 4028 x^{15} + 7050 x^{14} + 231120 x^{13} + 276543 x^{12} + 33774 x^{11} + 678828 x^{10} + 4932093 x^{9} + 2915523 x^{8} - 2760081 x^{7} + 10375899 x^{6} - 8875281 x^{5} + 23389131 x^{4} - 85081742 x^{3} - 113242662 x^{2} + 139342389 x + 160787897 \)
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: | \(13688036107155909050845006590269361308475240094281=3^{32}\cdot 41^{21}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $111.52$ | 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: | $3, 41$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(369=3^{2}\cdot 41\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{369}(1,·)$, $\chi_{369}(196,·)$, $\chi_{369}(325,·)$, $\chi_{369}(73,·)$, $\chi_{369}(202,·)$, $\chi_{369}(331,·)$, $\chi_{369}(109,·)$, $\chi_{369}(79,·)$, $\chi_{369}(208,·)$, $\chi_{369}(337,·)$, $\chi_{369}(355,·)$, $\chi_{369}(85,·)$, $\chi_{369}(214,·)$, $\chi_{369}(91,·)$, $\chi_{369}(286,·)$, $\chi_{369}(163,·)$, $\chi_{369}(40,·)$, $\chi_{369}(124,·)$, $\chi_{369}(301,·)$, $\chi_{369}(232,·)$, $\chi_{369}(178,·)$, $\chi_{369}(55,·)$, $\chi_{369}(247,·)$, $\chi_{369}(319,·)$$\rbrace$ | ||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{37} a^{17} + \frac{8}{37} a^{16} + \frac{10}{37} a^{15} - \frac{5}{37} a^{14} - \frac{1}{37} a^{13} - \frac{15}{37} a^{12} + \frac{1}{37} a^{10} - \frac{4}{37} a^{9} + \frac{1}{37} a^{8} + \frac{13}{37} a^{7} + \frac{2}{37} a^{6} + \frac{2}{37} a^{5} - \frac{4}{37} a^{4} + \frac{13}{37} a^{3} - \frac{10}{37} a^{2} + \frac{10}{37} a + \frac{15}{37}$, $\frac{1}{74} a^{18} + \frac{10}{37} a^{16} - \frac{11}{74} a^{15} + \frac{1}{37} a^{14} - \frac{7}{74} a^{13} + \frac{9}{74} a^{12} + \frac{1}{74} a^{11} - \frac{6}{37} a^{10} + \frac{33}{74} a^{9} + \frac{5}{74} a^{8} + \frac{9}{74} a^{7} - \frac{7}{37} a^{6} + \frac{17}{74} a^{5} - \frac{29}{74} a^{4} + \frac{17}{37} a^{3} - \frac{21}{74} a^{2} + \frac{9}{74} a - \frac{9}{74}$, $\frac{1}{74} a^{19} - \frac{23}{74} a^{16} + \frac{12}{37} a^{15} + \frac{19}{74} a^{14} + \frac{29}{74} a^{13} + \frac{5}{74} a^{12} - \frac{6}{37} a^{11} + \frac{13}{74} a^{10} + \frac{11}{74} a^{9} - \frac{11}{74} a^{8} + \frac{11}{37} a^{7} - \frac{23}{74} a^{6} + \frac{5}{74} a^{5} - \frac{17}{37} a^{4} + \frac{15}{74} a^{3} - \frac{13}{74} a^{2} + \frac{13}{74} a - \frac{2}{37}$, $\frac{1}{74} a^{20} - \frac{1}{74} a^{17} - \frac{11}{37} a^{16} + \frac{17}{74} a^{15} - \frac{7}{74} a^{14} - \frac{17}{74} a^{13} + \frac{14}{37} a^{12} + \frac{13}{74} a^{11} + \frac{33}{74} a^{10} - \frac{25}{74} a^{9} - \frac{15}{37} a^{8} - \frac{33}{74} a^{7} - \frac{25}{74} a^{6} + \frac{5}{37} a^{5} + \frac{1}{74} a^{4} - \frac{23}{74} a^{3} + \frac{15}{74} a^{2} - \frac{3}{37} a + \frac{17}{37}$, $\frac{1}{853738} a^{21} - \frac{391}{853738} a^{20} + \frac{2688}{426869} a^{19} + \frac{3151}{853738} a^{18} + \frac{4867}{853738} a^{17} - \frac{12675}{853738} a^{16} - \frac{19020}{426869} a^{15} - \frac{201341}{426869} a^{14} - \frac{160501}{853738} a^{13} - \frac{261215}{853738} a^{12} + \frac{150796}{426869} a^{11} - \frac{158938}{426869} a^{10} + \frac{401103}{853738} a^{9} - \frac{70249}{853738} a^{8} + \frac{109736}{426869} a^{7} + \frac{312913}{853738} a^{6} + \frac{360831}{853738} a^{5} + \frac{149960}{426869} a^{4} + \frac{140595}{426869} a^{3} - \frac{350767}{853738} a^{2} - \frac{127547}{426869} a + \frac{29458}{426869}$, $\frac{1}{853738} a^{22} + \frac{1238}{426869} a^{20} + \frac{5433}{853738} a^{19} + \frac{2449}{853738} a^{18} + \frac{9791}{853738} a^{17} - \frac{240721}{853738} a^{16} - \frac{127574}{426869} a^{15} - \frac{139547}{426869} a^{14} + \frac{10531}{426869} a^{13} - \frac{113557}{426869} a^{12} + \frac{44357}{426869} a^{11} + \frac{15252}{426869} a^{10} - \frac{186458}{426869} a^{9} - \frac{166033}{426869} a^{8} - \frac{331813}{853738} a^{7} + \frac{83041}{853738} a^{6} + \frac{184045}{426869} a^{5} - \frac{202189}{426869} a^{4} - \frac{32331}{426869} a^{3} + \frac{254493}{853738} a^{2} - \frac{60843}{853738} a + \frac{342877}{853738}$, $\frac{1}{9892681828201211617751481555208729497322891930753797955925077885467337879793641499246} a^{23} - \frac{86683869860717763720232314426778062502417613733620778052578109900962218454091}{9892681828201211617751481555208729497322891930753797955925077885467337879793641499246} a^{22} - \frac{772312873692921171984686444122510306223208046281171278584299870070532740189357}{4946340914100605808875740777604364748661445965376898977962538942733668939896820749623} a^{21} + \frac{1049999395457593419933512901237776492735031539989379721451756957394280922333637031}{9892681828201211617751481555208729497322891930753797955925077885467337879793641499246} a^{20} + \frac{11219347610196192837326433614705605275882595322475601081027424725463649754911380834}{4946340914100605808875740777604364748661445965376898977962538942733668939896820749623} a^{19} + \frac{711364817305406693634410998958337523160769206709356124423887334153463115613322411}{267369779140573286966256258248884581008726808939291836646623726634252375129557878358} a^{18} - \frac{571453620985475702366126404782651364444656475902637324668544322554698040713991783}{133684889570286643483128129124442290504363404469645918323311863317126187564778939179} a^{17} + \frac{893989218347979428486575740684582668864661577294334549915318205527848112280188957763}{9892681828201211617751481555208729497322891930753797955925077885467337879793641499246} a^{16} - \frac{4494116257282005249697868952564350910145090296923308242569792858118827634247465587097}{9892681828201211617751481555208729497322891930753797955925077885467337879793641499246} a^{15} + \frac{2189648853584777121274481936818017983534835489151314963849910316231277555784949337506}{4946340914100605808875740777604364748661445965376898977962538942733668939896820749623} a^{14} + \frac{1530198861655977136485960655497931421011041582098070262571550194927876032555886939913}{9892681828201211617751481555208729497322891930753797955925077885467337879793641499246} a^{13} + \frac{4228370060430046454758210795956586710395662997098817192522339237183735052302149588835}{9892681828201211617751481555208729497322891930753797955925077885467337879793641499246} a^{12} - \frac{2340885253721035840850336083068242646294953214471782913037892131386065284132603740151}{9892681828201211617751481555208729497322891930753797955925077885467337879793641499246} a^{11} - \frac{1483588796246554608886549013850896539879380965714546653567933323508446182439564181015}{4946340914100605808875740777604364748661445965376898977962538942733668939896820749623} a^{10} + \frac{3911039741815367427391002540816475836285559947741327059600830082808539296353437363351}{9892681828201211617751481555208729497322891930753797955925077885467337879793641499246} a^{9} + \frac{8192291144554282196171713382761311792438580490285033642619034925363751809637539147}{35585186432378459056660005594276005386053568096236683294694524767868121869761300357} a^{8} + \frac{838347659947379855318036731590339227933704805898241540909233353160217496758395314979}{9892681828201211617751481555208729497322891930753797955925077885467337879793641499246} a^{7} + \frac{1125685389857412734492076809799662542346827782807943166075183976711399693261254649327}{9892681828201211617751481555208729497322891930753797955925077885467337879793641499246} a^{6} + \frac{4725764542948199811068455565641487180350416405365043442373023048290543543795598691453}{9892681828201211617751481555208729497322891930753797955925077885467337879793641499246} a^{5} + \frac{125666828388955916120612400201258311595283067480625362553132438672955425877507079489}{9892681828201211617751481555208729497322891930753797955925077885467337879793641499246} a^{4} + \frac{350307491152668912171965979547917869528080869631482428617421086403879775192047192045}{9892681828201211617751481555208729497322891930753797955925077885467337879793641499246} a^{3} - \frac{2041722266695023620404027274185766646330126423851814048849259639157629991447089948859}{9892681828201211617751481555208729497322891930753797955925077885467337879793641499246} a^{2} + \frac{1052038309055763288964489612995996869376461671647342541728511997044154586688125194919}{9892681828201211617751481555208729497322891930753797955925077885467337879793641499246} a - \frac{785514049139614671513930392621998947087604102541111083665409182268341396467784505116}{4946340914100605808875740777604364748661445965376898977962538942733668939896820749623}$
Class group and class number
$C_{11}\times C_{11}\times C_{11}\times C_{11}$, which has order $14641$ (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: | \( 24556622934.637486 \) (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
| \(\Q(\sqrt{41}) \), \(\Q(\zeta_{9})^+\), 4.4.68921.1, 6.6.452190681.1, 8.0.194754273881.1, 12.12.14092718790297143251881.1 |
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 | ${\href{/LocalNumberField/5.12.0.1}{12} }^{2}$ | $24$ | $24$ | $24$ | ${\href{/LocalNumberField/17.8.0.1}{8} }^{3}$ | ${\href{/LocalNumberField/19.8.0.1}{8} }^{3}$ | ${\href{/LocalNumberField/23.6.0.1}{6} }^{4}$ | $24$ | ${\href{/LocalNumberField/31.6.0.1}{6} }^{4}$ | ${\href{/LocalNumberField/37.1.0.1}{1} }^{24}$ | R | ${\href{/LocalNumberField/43.12.0.1}{12} }^{2}$ | $24$ | ${\href{/LocalNumberField/53.8.0.1}{8} }^{3}$ | ${\href{/LocalNumberField/59.3.0.1}{3} }^{8}$ |
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 | ||||||
| 41 | Data not computed | ||||||