Normalized defining polynomial
\( x^{24} - 8 x^{23} + 63 x^{22} - 310 x^{21} + 1456 x^{20} - 5348 x^{19} + 17865 x^{18} - 49630 x^{17} + 124148 x^{16} - 270256 x^{15} + 584017 x^{14} - 1171240 x^{13} + 2076162 x^{12} - 3016984 x^{11} + 4113398 x^{10} - 5646172 x^{9} + 7658667 x^{8} - 7777136 x^{7} + 1868535 x^{6} + 4655532 x^{5} + 13976710 x^{4} - 40286724 x^{3} + 84874236 x^{2} - 59364186 x + 23918117 \)
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: | \(38522626068874714759052304357183694398210179072=2^{24}\cdot 7^{16}\cdot 17^{21}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $87.31$ | 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: | $2, 7, 17$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(476=2^{2}\cdot 7\cdot 17\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{476}(1,·)$, $\chi_{476}(263,·)$, $\chi_{476}(137,·)$, $\chi_{476}(331,·)$, $\chi_{476}(205,·)$, $\chi_{476}(15,·)$, $\chi_{476}(81,·)$, $\chi_{476}(149,·)$, $\chi_{476}(151,·)$, $\chi_{476}(155,·)$, $\chi_{476}(225,·)$, $\chi_{476}(291,·)$, $\chi_{476}(421,·)$, $\chi_{476}(359,·)$, $\chi_{476}(169,·)$, $\chi_{476}(43,·)$, $\chi_{476}(429,·)$, $\chi_{476}(305,·)$, $\chi_{476}(179,·)$, $\chi_{476}(373,·)$, $\chi_{476}(219,·)$, $\chi_{476}(361,·)$, $\chi_{476}(127,·)$, $\chi_{476}(247,·)$$\rbrace$ | ||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{52} a^{18} - \frac{3}{26} a^{17} + \frac{7}{52} a^{16} - \frac{5}{26} a^{15} + \frac{1}{52} a^{14} - \frac{9}{26} a^{13} - \frac{9}{52} a^{12} - \frac{9}{26} a^{11} - \frac{3}{26} a^{10} - \frac{1}{26} a^{9} - \frac{5}{52} a^{8} - \frac{3}{26} a^{7} - \frac{15}{52} a^{6} - \frac{3}{13} a^{5} - \frac{25}{52} a^{4} - \frac{6}{13} a^{3} + \frac{1}{52} a^{2} + \frac{11}{26} a + \frac{17}{52}$, $\frac{1}{52} a^{19} + \frac{23}{52} a^{17} - \frac{5}{13} a^{16} - \frac{7}{52} a^{15} - \frac{3}{13} a^{14} - \frac{1}{4} a^{13} - \frac{5}{13} a^{12} - \frac{5}{26} a^{11} + \frac{7}{26} a^{10} - \frac{17}{52} a^{9} + \frac{4}{13} a^{8} + \frac{1}{52} a^{7} + \frac{1}{26} a^{6} + \frac{7}{52} a^{5} - \frac{9}{26} a^{4} + \frac{1}{4} a^{3} - \frac{6}{13} a^{2} - \frac{7}{52} a - \frac{1}{26}$, $\frac{1}{52} a^{20} + \frac{7}{26} a^{17} - \frac{3}{13} a^{16} + \frac{5}{26} a^{15} + \frac{4}{13} a^{14} - \frac{11}{26} a^{13} - \frac{11}{52} a^{12} + \frac{3}{13} a^{11} + \frac{17}{52} a^{10} + \frac{5}{26} a^{9} + \frac{3}{13} a^{8} - \frac{4}{13} a^{7} - \frac{3}{13} a^{6} - \frac{1}{26} a^{5} + \frac{4}{13} a^{4} + \frac{2}{13} a^{3} + \frac{11}{26} a^{2} + \frac{3}{13} a + \frac{25}{52}$, $\frac{1}{52} a^{21} + \frac{5}{13} a^{17} + \frac{4}{13} a^{16} + \frac{4}{13} a^{14} - \frac{19}{52} a^{13} - \frac{9}{26} a^{12} + \frac{9}{52} a^{11} - \frac{5}{26} a^{10} - \frac{3}{13} a^{9} + \frac{1}{26} a^{8} + \frac{5}{13} a^{7} - \frac{6}{13} a^{5} - \frac{3}{26} a^{4} - \frac{3}{26} a^{3} - \frac{1}{26} a^{2} - \frac{23}{52} a + \frac{11}{26}$, $\frac{1}{88164596448712662443112078934570979396} a^{22} - \frac{8612300778624798411782166962737821}{88164596448712662443112078934570979396} a^{21} - \frac{116682664323007183016383802588911047}{44082298224356331221556039467285489698} a^{20} + \frac{149718410608407699970508829666275685}{44082298224356331221556039467285489698} a^{19} - \frac{59195623357864541070782848145320857}{44082298224356331221556039467285489698} a^{18} + \frac{10546307702923446253408508309983389021}{44082298224356331221556039467285489698} a^{17} + \frac{9415889116232021594482641188153675447}{44082298224356331221556039467285489698} a^{16} - \frac{16419954400906894856922067535082718125}{44082298224356331221556039467285489698} a^{15} - \frac{19008853153405254966931666237327393689}{88164596448712662443112078934570979396} a^{14} - \frac{23287643208290890030244721860000873}{88164596448712662443112078934570979396} a^{13} - \frac{574353859860578606967055992968333485}{6781892034516358649470159918043921492} a^{12} - \frac{24859950818718270888170884990947282243}{88164596448712662443112078934570979396} a^{11} - \frac{10125988724970067489249202867590246715}{22041149112178165610778019733642744849} a^{10} - \frac{6421440042896415755373342258232019074}{22041149112178165610778019733642744849} a^{9} + \frac{8826366061262609364795537851896046082}{22041149112178165610778019733642744849} a^{8} + \frac{7842457267831021413155920917661829073}{44082298224356331221556039467285489698} a^{7} - \frac{17152505491682062274122944321622957405}{44082298224356331221556039467285489698} a^{6} - \frac{3886666933272278652684202010037983780}{22041149112178165610778019733642744849} a^{5} + \frac{10809090227025585477605679856806991157}{44082298224356331221556039467285489698} a^{4} + \frac{17360540097907711936357966994338877685}{44082298224356331221556039467285489698} a^{3} + \frac{3012837280207684589026942160224416381}{6781892034516358649470159918043921492} a^{2} + \frac{4507755281879417681151086263876164659}{88164596448712662443112078934570979396} a - \frac{384216543979417438594820106708278353}{44082298224356331221556039467285489698}$, $\frac{1}{10496005287942713630122407474718918517178819365492948} a^{23} - \frac{27092453521}{10280122711011472703352015156433808537883270681188} a^{22} - \frac{5830628467346524773459834787560796873415973555999}{10496005287942713630122407474718918517178819365492948} a^{21} - \frac{51053416909448088522034065190678525901476379475905}{10496005287942713630122407474718918517178819365492948} a^{20} - \frac{27451757686119987610701270655820620516303313367583}{10496005287942713630122407474718918517178819365492948} a^{19} - \frac{42560249069635504444088595101440216987392348139649}{10496005287942713630122407474718918517178819365492948} a^{18} - \frac{43610510424857070353466243876063015937419633135233}{10496005287942713630122407474718918517178819365492948} a^{17} - \frac{841369619551344420140748460247633747937538845082179}{10496005287942713630122407474718918517178819365492948} a^{16} - \frac{2413634624820185956154420218501452755549030433316545}{5248002643971356815061203737359459258589409682746474} a^{15} - \frac{281706508065580986336170802026326442569283146952272}{2624001321985678407530601868679729629294704841373237} a^{14} - \frac{4041583035627568205618789173252806123421100943252497}{10496005287942713630122407474718918517178819365492948} a^{13} + \frac{2224566372585839037364559107136590270162123331834179}{10496005287942713630122407474718918517178819365492948} a^{12} + \frac{2759409176708591086076380914055589334432516048064455}{10496005287942713630122407474718918517178819365492948} a^{11} - \frac{5026573005804151435360772302685927319884770118349753}{10496005287942713630122407474718918517178819365492948} a^{10} - \frac{1117366841743698411569271012396005130442820331151423}{10496005287942713630122407474718918517178819365492948} a^{9} - \frac{3478712606597178890089102417666366896564721703870245}{10496005287942713630122407474718918517178819365492948} a^{8} - \frac{1628050160708380073353841199359259112330715219991969}{10496005287942713630122407474718918517178819365492948} a^{7} - \frac{222439299643148616721016098240622410806414662818409}{10496005287942713630122407474718918517178819365492948} a^{6} + \frac{44734209917542732863542985342207047302846791423561}{10496005287942713630122407474718918517178819365492948} a^{5} - \frac{3546765600297735393235370871376267565020660122963083}{10496005287942713630122407474718918517178819365492948} a^{4} - \frac{1688670962121869071534810311941844768483697343932339}{5248002643971356815061203737359459258589409682746474} a^{3} + \frac{985797170325234795864697771716261421267548984352130}{2624001321985678407530601868679729629294704841373237} a^{2} + \frac{484506476590659402600440639348139816805775726396741}{5248002643971356815061203737359459258589409682746474} a - \frac{1466351559468202130142215592069333089139188645742703}{5248002643971356815061203737359459258589409682746474}$
Class group and class number
$C_{22}\times C_{2134}$, which has order $46948$ (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, 8.0.105046700288.1, 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 | R | $24$ | $24$ | 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 |
|---|---|---|---|---|---|---|---|
| $2$ | 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 | $x^{12} - 78 x^{10} - 1621 x^{8} + 460 x^{6} - 1977 x^{4} + 866 x^{2} + 749$ | $2$ | $6$ | $12$ | $C_{12}$ | $[2]^{6}$ | |
| 7 | Data not computed | ||||||
| 17 | Data not computed | ||||||