Normalized defining polynomial
\( x^{24} - 8 x^{23} + 20 x^{22} - 16 x^{21} + 146 x^{20} - 688 x^{19} + 432 x^{18} + 152 x^{17} + 8583 x^{16} - 7840 x^{15} - 26876 x^{14} - 71320 x^{13} + 104650 x^{12} + 445488 x^{11} + 538844 x^{10} - 1035552 x^{9} - 4034182 x^{8} - 5015240 x^{7} + 2705996 x^{6} + 21877704 x^{5} + 42388968 x^{4} + 47076680 x^{3} + 36958600 x^{2} + 14571392 x + 3601057 \)
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: | \(6589966906506961482901542164675154259132547072=2^{93}\cdot 13^{16}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $81.12$ | 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: | $2, 13$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(416=2^{5}\cdot 13\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{416}(1,·)$, $\chi_{416}(3,·)$, $\chi_{416}(211,·)$, $\chi_{416}(321,·)$, $\chi_{416}(393,·)$, $\chi_{416}(139,·)$, $\chi_{416}(209,·)$, $\chi_{416}(131,·)$, $\chi_{416}(313,·)$, $\chi_{416}(217,·)$, $\chi_{416}(27,·)$, $\chi_{416}(107,·)$, $\chi_{416}(289,·)$, $\chi_{416}(35,·)$, $\chi_{416}(243,·)$, $\chi_{416}(81,·)$, $\chi_{416}(105,·)$, $\chi_{416}(235,·)$, $\chi_{416}(113,·)$, $\chi_{416}(339,·)$, $\chi_{416}(9,·)$, $\chi_{416}(185,·)$, $\chi_{416}(347,·)$, $\chi_{416}(315,·)$$\rbrace$ | ||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $\frac{1}{47} a^{21} + \frac{11}{47} a^{20} + \frac{3}{47} a^{19} - \frac{4}{47} a^{18} + \frac{17}{47} a^{17} + \frac{13}{47} a^{16} - \frac{2}{47} a^{15} - \frac{8}{47} a^{14} + \frac{8}{47} a^{13} + \frac{1}{47} a^{12} - \frac{18}{47} a^{11} - \frac{2}{47} a^{10} + \frac{13}{47} a^{9} + \frac{23}{47} a^{8} - \frac{15}{47} a^{7} + \frac{23}{47} a^{6} + \frac{14}{47} a^{5} - \frac{10}{47} a^{4} - \frac{21}{47} a^{3} - \frac{11}{47} a^{2} + \frac{5}{47} a - \frac{12}{47}$, $\frac{1}{411206371929577896451293617943579329} a^{22} + \frac{1828461269617577328418751699970869}{411206371929577896451293617943579329} a^{21} + \frac{63736985822855809699700800715559748}{411206371929577896451293617943579329} a^{20} + \frac{1275549054308096888072795000316586}{411206371929577896451293617943579329} a^{19} + \frac{25716033608038004428742586884903317}{411206371929577896451293617943579329} a^{18} + \frac{36994092721845623110037551044381012}{411206371929577896451293617943579329} a^{17} - \frac{3026648889570355384634843180234796}{8749071743182508435133906764757007} a^{16} + \frac{169212796831555462718068439104000334}{411206371929577896451293617943579329} a^{15} + \frac{98975763205654352233912215505005983}{411206371929577896451293617943579329} a^{14} + \frac{121224212521335360682402995540813049}{411206371929577896451293617943579329} a^{13} - \frac{66420774588374206760122928465027555}{411206371929577896451293617943579329} a^{12} + \frac{39797690340327634686624495559829126}{411206371929577896451293617943579329} a^{11} - \frac{135928595399219828315610091360846162}{411206371929577896451293617943579329} a^{10} + \frac{23296113471428816494215995823657656}{411206371929577896451293617943579329} a^{9} - \frac{53894003677991473933467269424189574}{411206371929577896451293617943579329} a^{8} - \frac{15876478343102775418404935853510053}{411206371929577896451293617943579329} a^{7} - \frac{76908546142836649985574109772489399}{411206371929577896451293617943579329} a^{6} + \frac{33176827263103343232560077054646209}{411206371929577896451293617943579329} a^{5} + \frac{117018742419802332861479445626183550}{411206371929577896451293617943579329} a^{4} - \frac{163304756331898584910970309913239662}{411206371929577896451293617943579329} a^{3} - \frac{154873800201607064318242549789836159}{411206371929577896451293617943579329} a^{2} + \frac{129388780477621407290828382770009163}{411206371929577896451293617943579329} a - \frac{1004080841483452985195902403992227}{5205143948475669575332830606880751}$, $\frac{1}{358272739387083985529693628547415476706596403599} a^{23} - \frac{303240391385}{358272739387083985529693628547415476706596403599} a^{22} + \frac{876091670556568662013008778577153454358750816}{358272739387083985529693628547415476706596403599} a^{21} - \frac{75669948102705873603292126860471041076783885170}{358272739387083985529693628547415476706596403599} a^{20} + \frac{19983330629678934773559830549648250644727754415}{358272739387083985529693628547415476706596403599} a^{19} - \frac{28668616913207932495196447086922762786956119508}{358272739387083985529693628547415476706596403599} a^{18} + \frac{24253237681522185653857957009959180460166277560}{358272739387083985529693628547415476706596403599} a^{17} + \frac{126411265692400968483221037965866595128313315916}{358272739387083985529693628547415476706596403599} a^{16} - \frac{60257013853928893969463531516297848466342220}{358272739387083985529693628547415476706596403599} a^{15} + \frac{68556573464175215557126556355670892063366587953}{358272739387083985529693628547415476706596403599} a^{14} + \frac{155279460419644113535520161119614073446109826209}{358272739387083985529693628547415476706596403599} a^{13} - \frac{25500096005769884949853696162791122437010149496}{358272739387083985529693628547415476706596403599} a^{12} + \frac{157044184272686350263690491555181393940364860185}{358272739387083985529693628547415476706596403599} a^{11} + \frac{111843470874229987712497916194208520008516676971}{358272739387083985529693628547415476706596403599} a^{10} + \frac{97953159779914460661346810439493636865546875115}{358272739387083985529693628547415476706596403599} a^{9} - \frac{81604428252903018927841568553460064464233130398}{358272739387083985529693628547415476706596403599} a^{8} + \frac{73577877949234618959989227056662335172022157664}{358272739387083985529693628547415476706596403599} a^{7} - \frac{61677369687431209315832963259239133891589742799}{358272739387083985529693628547415476706596403599} a^{6} - \frac{125713756617397854618604638591531329049795320200}{358272739387083985529693628547415476706596403599} a^{5} - \frac{159298649472161080968317279998971814937569194349}{358272739387083985529693628547415476706596403599} a^{4} - \frac{46869792936225830957073573599717907767735129146}{358272739387083985529693628547415476706596403599} a^{3} + \frac{162945822145877988214017054240051281221266276497}{358272739387083985529693628547415476706596403599} a^{2} - \frac{2283048285681274541351043964504744085836600597}{7622824242278382670844545288242882483119072417} a + \frac{1029099292922582011096332148880467594544268187}{4535097966925113740882197829714119958311346881}$
Class group and class number
$C_{17401}$, which has order $17401$ (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: | \( 675765244.4059911 \) (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
| \(\Q(\sqrt{2}) \), 3.3.169.1, \(\Q(\zeta_{16})^+\), 6.6.14623232.1, 8.0.2147483648.1, 12.12.7007073538075000832.1 |
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 | $24$ | ${\href{/LocalNumberField/5.8.0.1}{8} }^{3}$ | ${\href{/LocalNumberField/7.12.0.1}{12} }^{2}$ | $24$ | R | ${\href{/LocalNumberField/17.6.0.1}{6} }^{4}$ | $24$ | ${\href{/LocalNumberField/23.12.0.1}{12} }^{2}$ | $24$ | ${\href{/LocalNumberField/31.2.0.1}{2} }^{12}$ | $24$ | ${\href{/LocalNumberField/41.12.0.1}{12} }^{2}$ | $24$ | ${\href{/LocalNumberField/47.1.0.1}{1} }^{24}$ | ${\href{/LocalNumberField/53.8.0.1}{8} }^{3}$ | $24$ |
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 | Data not computed | ||||||
| 13 | Data not computed | ||||||