Normalized defining polynomial
\( x^{24} - 5 x^{23} - 61 x^{22} + 354 x^{21} + 1311 x^{20} - 9890 x^{19} - 9525 x^{18} + 138743 x^{17} - 50028 x^{16} - 1029530 x^{15} + 1192975 x^{14} + 3901011 x^{13} - 7038616 x^{12} - 6545957 x^{11} + 18290959 x^{10} + 2375261 x^{9} - 22525088 x^{8} + 4333001 x^{7} + 13458471 x^{6} - 4381142 x^{5} - 3710767 x^{4} + 1393042 x^{3} + 364409 x^{2} - 141483 x + 2243 \)
Invariants
| Degree: | $24$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[24, 0]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(19928975083988902198525863514102984930745573777=17^{21}\cdot 19^{16}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $84.95$ | 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: | $17, 19$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(323=17\cdot 19\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{323}(64,·)$, $\chi_{323}(1,·)$, $\chi_{323}(134,·)$, $\chi_{323}(140,·)$, $\chi_{323}(77,·)$, $\chi_{323}(144,·)$, $\chi_{323}(273,·)$, $\chi_{323}(83,·)$, $\chi_{323}(87,·)$, $\chi_{323}(26,·)$, $\chi_{323}(220,·)$, $\chi_{323}(30,·)$, $\chi_{323}(229,·)$, $\chi_{323}(49,·)$, $\chi_{323}(106,·)$, $\chi_{323}(172,·)$, $\chi_{323}(239,·)$, $\chi_{323}(305,·)$, $\chi_{323}(178,·)$, $\chi_{323}(115,·)$, $\chi_{323}(121,·)$, $\chi_{323}(315,·)$, $\chi_{323}(254,·)$, $\chi_{323}(191,·)$$\rbrace$ | ||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $\frac{1}{123594130698016265271219292510762198790596189145508119712395605519151} a^{23} - \frac{51994548174504053870693633798021197113354148136537173020780040510667}{123594130698016265271219292510762198790596189145508119712395605519151} a^{22} + \frac{8062355819304640776162346648543499718811722705532182853090896753296}{123594130698016265271219292510762198790596189145508119712395605519151} a^{21} + \frac{8304059954074466581995058791453092987521098043053335369102875368569}{123594130698016265271219292510762198790596189145508119712395605519151} a^{20} - \frac{736834869611026872976677557432810506200466068437949748903135217757}{123594130698016265271219292510762198790596189145508119712395605519151} a^{19} + \frac{28855079672138920296092515019619594750394687769750162037926265295516}{123594130698016265271219292510762198790596189145508119712395605519151} a^{18} + \frac{44918703320120467592325332951845690961526304838246246481684454737934}{123594130698016265271219292510762198790596189145508119712395605519151} a^{17} + \frac{10462692383764223950252466071482381350358559309328433425612394570297}{123594130698016265271219292510762198790596189145508119712395605519151} a^{16} + \frac{43630866580542104333181632335268429168434161597915438326550565712167}{123594130698016265271219292510762198790596189145508119712395605519151} a^{15} - \frac{45768653372328970909321119842488897378917328281542389884262102880842}{123594130698016265271219292510762198790596189145508119712395605519151} a^{14} - \frac{40522739428452493591567057075129566634562897407124579308435972218999}{123594130698016265271219292510762198790596189145508119712395605519151} a^{13} - \frac{14407725839382862278609657917990219022177480498733843801109673950108}{123594130698016265271219292510762198790596189145508119712395605519151} a^{12} - \frac{13512799361446751024126833953907713125381434771168841115217307641408}{123594130698016265271219292510762198790596189145508119712395605519151} a^{11} + \frac{59277660902723272795292725681448117566363946513046999809116295727891}{123594130698016265271219292510762198790596189145508119712395605519151} a^{10} + \frac{42986873068814330414793930220252078985862035195009106084490036836740}{123594130698016265271219292510762198790596189145508119712395605519151} a^{9} - \frac{584143841062943681238549894601577484823548269111691220450602380856}{123594130698016265271219292510762198790596189145508119712395605519151} a^{8} + \frac{27963834266445179742060583580267207631275116340701269579193953743317}{123594130698016265271219292510762198790596189145508119712395605519151} a^{7} + \frac{10074813782978644442961007135883489283895614440503185509476359251838}{123594130698016265271219292510762198790596189145508119712395605519151} a^{6} - \frac{15814164141531133929764229832328186503181590766888612075229076767693}{123594130698016265271219292510762198790596189145508119712395605519151} a^{5} - \frac{21458178353586486660414293866676252468590052422123028034850180760172}{123594130698016265271219292510762198790596189145508119712395605519151} a^{4} + \frac{6774611080002100633607700681609357448452725510565671115701946281686}{123594130698016265271219292510762198790596189145508119712395605519151} a^{3} - \frac{11017198045296846574856134070084280003680786318933906956476640325816}{123594130698016265271219292510762198790596189145508119712395605519151} a^{2} + \frac{33501414401410224024936252379867431479146342831700859348557195745742}{123594130698016265271219292510762198790596189145508119712395605519151} a + \frac{52040145054712868323196905063386707733316081189187175284933371900442}{123594130698016265271219292510762198790596189145508119712395605519151}$
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
| Rank: | $23$ | 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: | \( 2353165244433466.5 \) (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
| \(\Q(\sqrt{17}) \), 3.3.361.1, 4.4.4913.1, 6.6.640267073.1, \(\Q(\zeta_{17})^+\), 12.12.2014044676385121747377.1 |
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 | ${\href{/LocalNumberField/2.12.0.1}{12} }^{2}$ | $24$ | $24$ | ${\href{/LocalNumberField/7.8.0.1}{8} }^{3}$ | ${\href{/LocalNumberField/11.8.0.1}{8} }^{3}$ | ${\href{/LocalNumberField/13.6.0.1}{6} }^{4}$ | R | R | $24$ | $24$ | ${\href{/LocalNumberField/31.8.0.1}{8} }^{3}$ | ${\href{/LocalNumberField/37.8.0.1}{8} }^{3}$ | $24$ | ${\href{/LocalNumberField/43.12.0.1}{12} }^{2}$ | ${\href{/LocalNumberField/47.6.0.1}{6} }^{4}$ | ${\href{/LocalNumberField/53.12.0.1}{12} }^{2}$ | ${\href{/LocalNumberField/59.12.0.1}{12} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
| 17 | Data not computed | ||||||
| $19$ | 19.12.8.1 | $x^{12} - 114 x^{9} + 4332 x^{6} - 54872 x^{3} + 130321000$ | $3$ | $4$ | $8$ | $C_{12}$ | $[\ ]_{3}^{4}$ |
| 19.12.8.1 | $x^{12} - 114 x^{9} + 4332 x^{6} - 54872 x^{3} + 130321000$ | $3$ | $4$ | $8$ | $C_{12}$ | $[\ ]_{3}^{4}$ | |