Normalized defining polynomial
\( x^{24} - 5 x^{22} + 19 x^{20} - 66 x^{18} + 221 x^{16} - 358 x^{14} + 530 x^{12} - 723 x^{10} + 793 x^{8} - 157 x^{6} + 31 x^{4} - 6 x^{2} + 1 \)
Invariants
Degree: | $24$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[0, 12]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: |
\(2126907556454464000000000000000000\)
\(\medspace = 2^{24}\cdot 5^{18}\cdot 7^{16}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(24.47\) | sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
magma: Abs(Discriminant(OK))^(1/Degree(K));
oscar: (1.0 * dK)^(1/degree(K))
| |
Ramified primes: |
\(2\), \(5\), \(7\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q\) | ||
$\card{ \Gal(K/\Q) }$: | $24$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
This field is Galois and abelian over $\Q$. | |||
Conductor: | \(140=2^{2}\cdot 5\cdot 7\) | ||
Dirichlet character group: | $\lbrace$$\chi_{140}(1,·)$, $\chi_{140}(67,·)$, $\chi_{140}(71,·)$, $\chi_{140}(9,·)$, $\chi_{140}(11,·)$, $\chi_{140}(79,·)$, $\chi_{140}(81,·)$, $\chi_{140}(107,·)$, $\chi_{140}(121,·)$, $\chi_{140}(23,·)$, $\chi_{140}(29,·)$, $\chi_{140}(99,·)$, $\chi_{140}(37,·)$, $\chi_{140}(39,·)$, $\chi_{140}(43,·)$, $\chi_{140}(109,·)$, $\chi_{140}(93,·)$, $\chi_{140}(113,·)$, $\chi_{140}(51,·)$, $\chi_{140}(53,·)$, $\chi_{140}(137,·)$, $\chi_{140}(57,·)$, $\chi_{140}(123,·)$, $\chi_{140}(127,·)$$\rbrace$ | ||
This is a CM field. | |||
Reflex fields: | unavailable$^{2048}$ |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{164891}a^{18}+\frac{2495}{164891}a^{16}-\frac{40833}{164891}a^{14}+\frac{24303}{164891}a^{12}-\frac{43903}{164891}a^{10}+\frac{57450}{164891}a^{8}+\frac{47471}{164891}a^{6}+\frac{48407}{164891}a^{4}+\frac{75253}{164891}a^{2}-\frac{54614}{164891}$, $\frac{1}{164891}a^{19}+\frac{2495}{164891}a^{17}-\frac{40833}{164891}a^{15}+\frac{24303}{164891}a^{13}-\frac{43903}{164891}a^{11}+\frac{57450}{164891}a^{9}+\frac{47471}{164891}a^{7}+\frac{48407}{164891}a^{5}+\frac{75253}{164891}a^{3}-\frac{54614}{164891}a$, $\frac{1}{164891}a^{20}-\frac{57080}{164891}a^{10}+\frac{61964}{164891}$, $\frac{1}{164891}a^{21}-\frac{57080}{164891}a^{11}+\frac{61964}{164891}a$, $\frac{1}{164891}a^{22}-\frac{57080}{164891}a^{12}+\frac{61964}{164891}a^{2}$, $\frac{1}{164891}a^{23}-\frac{57080}{164891}a^{13}+\frac{61964}{164891}a^{3}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{3}$, which has order $3$ (assuming GRH)
Unit group
Rank: | $11$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
Torsion generator: |
\( -\frac{73393}{164891} a^{23} + \frac{366965}{164891} a^{21} - \frac{1394378}{164891} a^{19} + \frac{4843938}{164891} a^{17} - \frac{16219853}{164891} a^{15} + \frac{26274694}{164891} a^{13} - \frac{38898290}{164891} a^{11} + \frac{53094640}{164891} a^{9} - \frac{58200649}{164891} a^{7} + \frac{11522701}{164891} a^{5} - \frac{2275183}{164891} a^{3} + \frac{440358}{164891} a \)
(order $20$)
| sage: UK.torsion_generator()
gp: K.tu[2]
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
oscar: torsion_units_generator(OK)
| |
Fundamental units: |
$\frac{6079}{164891}a^{22}-\frac{23940}{164891}a^{20}+\frac{83160}{164891}a^{18}-\frac{278460}{164891}a^{16}+\frac{917429}{164891}a^{14}-\frac{749727}{164891}a^{12}+\frac{910980}{164891}a^{10}-\frac{999180}{164891}a^{8}+\frac{197820}{164891}a^{6}+\frac{4164412}{164891}a^{4}-\frac{824986}{164891}a^{2}+\frac{163631}{164891}$, $\frac{3315}{164891}a^{22}-\frac{12597}{164891}a^{20}+\frac{43758}{164891}a^{18}-\frac{146523}{164891}a^{16}+\frac{483135}{164891}a^{14}-\frac{351390}{164891}a^{12}+\frac{479349}{164891}a^{10}-\frac{525759}{164891}a^{8}+\frac{104091}{164891}a^{6}+\frac{2312194}{164891}a^{4}+\frac{3978}{164891}a^{2}-\frac{663}{164891}$, $\frac{597}{164891}a^{22}+\frac{220568}{164891}a^{12}+\frac{1870725}{164891}a^{2}$, $\frac{73393}{164891}a^{23}-\frac{366965}{164891}a^{21}+\frac{1394378}{164891}a^{19}-\frac{4843938}{164891}a^{17}+\frac{16219853}{164891}a^{15}-\frac{26274694}{164891}a^{13}+\frac{38898290}{164891}a^{11}-\frac{53094640}{164891}a^{9}+\frac{58200649}{164891}a^{7}-\frac{11522701}{164891}a^{5}+\frac{2275183}{164891}a^{3}-\frac{440358}{164891}a-1$, $\frac{73393}{164891}a^{23}-\frac{6079}{164891}a^{22}-\frac{366810}{164891}a^{21}+\frac{23940}{164891}a^{20}+\frac{1394378}{164891}a^{19}-\frac{83160}{164891}a^{18}-\frac{4843938}{164891}a^{17}+\frac{278460}{164891}a^{16}+\frac{16219853}{164891}a^{15}-\frac{917429}{164891}a^{14}-\frac{26274694}{164891}a^{13}+\frac{749727}{164891}a^{12}+\frac{38955004}{164891}a^{11}-\frac{910980}{164891}a^{10}-\frac{53094640}{164891}a^{9}+\frac{999180}{164891}a^{8}+\frac{58200649}{164891}a^{7}-\frac{197820}{164891}a^{6}-\frac{11522701}{164891}a^{5}-\frac{4164412}{164891}a^{4}+\frac{2275183}{164891}a^{3}+\frac{824986}{164891}a^{2}-\frac{69834}{164891}a+\frac{1260}{164891}$, $\frac{6521}{164891}a^{22}-\frac{23940}{164891}a^{20}+\frac{83160}{164891}a^{18}-\frac{278460}{164891}a^{16}+\frac{917429}{164891}a^{14}-\frac{585873}{164891}a^{12}+\frac{910980}{164891}a^{10}-\frac{999180}{164891}a^{8}+\frac{197820}{164891}a^{6}+\frac{4164412}{164891}a^{4}+\frac{675215}{164891}a^{2}+\frac{163631}{164891}$, $\frac{6145}{164891}a^{22}-\frac{23940}{164891}a^{20}+\frac{83160}{164891}a^{18}-\frac{278460}{164891}a^{16}+\frac{917429}{164891}a^{14}-\frac{724514}{164891}a^{12}+\frac{910980}{164891}a^{10}-\frac{999180}{164891}a^{8}+\frac{197820}{164891}a^{6}+\frac{4164412}{164891}a^{4}-\frac{362964}{164891}a^{2}+\frac{163631}{164891}$, $\frac{38730}{164891}a^{23}-\frac{71381}{164891}a^{22}-\frac{187195}{164891}a^{21}+\frac{350450}{164891}a^{20}+\frac{703595}{164891}a^{19}-\frac{1324030}{164891}a^{18}-\frac{2433426}{164891}a^{17}+\frac{4588392}{164891}a^{16}+\frac{8133300}{164891}a^{15}-\frac{15349171}{164891}a^{14}-\frac{12438785}{164891}a^{13}+\frac{133303}{911}a^{12}+\frac{18216010}{164891}a^{11}-\frac{35521040}{164891}a^{10}-\frac{24580640}{164891}a^{9}+\frac{48162100}{164891}a^{8}+\frac{26090063}{164891}a^{7}-\frac{51982306}{164891}a^{6}-\frac{961795}{164891}a^{5}+\frac{6088002}{164891}a^{4}+\frac{187195}{164891}a^{3}-\frac{1199376}{164891}a^{2}-\frac{32275}{164891}a+\frac{228181}{164891}$, $\frac{80}{911}a^{23}-\frac{58847}{164891}a^{22}-\frac{55179}{164891}a^{21}+\frac{294235}{164891}a^{20}+\frac{1056}{911}a^{19}-\frac{1118116}{164891}a^{18}-\frac{3536}{911}a^{17}+\frac{3883902}{164891}a^{16}+\frac{11647}{911}a^{15}-\frac{13005187}{164891}a^{14}-\frac{8480}{911}a^{13}+\frac{21067226}{164891}a^{12}+\frac{2037094}{164891}a^{11}-\frac{31188910}{164891}a^{10}-\frac{12688}{911}a^{9}+\frac{42540093}{164891}a^{8}+\frac{2512}{911}a^{7}-\frac{46665671}{164891}a^{6}+\frac{51687}{911}a^{5}+\frac{9238979}{164891}a^{4}+\frac{96}{911}a^{3}-\frac{1824257}{164891}a^{2}-\frac{373420}{164891}a+\frac{353082}{164891}$, $\frac{73703}{164891}a^{23}+\frac{287}{164891}a^{22}-\frac{389869}{164891}a^{21}+\frac{1503657}{164891}a^{19}-\frac{5252329}{164891}a^{17}+\frac{17629816}{164891}a^{15}-\frac{30870014}{164891}a^{13}+\frac{107140}{164891}a^{12}+\frac{45920783}{164891}a^{11}-\frac{63299266}{164891}a^{9}+\frac{71999151}{164891}a^{7}-\frac{26032727}{164891}a^{5}+\frac{2815084}{164891}a^{3}+\frac{1294568}{164891}a^{2}-\frac{545429}{164891}a$, $\frac{160669}{164891}a^{23}+\frac{40742}{164891}a^{22}-\frac{788799}{164891}a^{21}-\frac{203710}{164891}a^{20}+\frac{2979892}{164891}a^{19}+\frac{773943}{164891}a^{18}-\frac{10327892}{164891}a^{17}-\frac{2688972}{164891}a^{16}+\frac{34547813}{164891}a^{15}+\frac{9003982}{164891}a^{14}-\frac{54304836}{164891}a^{13}-\frac{14585636}{164891}a^{12}+\frac{79947102}{164891}a^{11}+\frac{21593260}{164891}a^{10}-\frac{108485808}{164891}a^{9}-\frac{29513180}{164891}a^{8}+\frac{116855970}{164891}a^{7}+\frac{32308406}{164891}a^{6}-\frac{13690055}{164891}a^{5}-\frac{6396494}{164891}a^{4}+\frac{2697017}{164891}a^{3}+\frac{1263002}{164891}a^{2}-\frac{513088}{164891}a-\frac{244452}{164891}$
| sage: UK.fundamental_units()
gp: K.fu
magma: [K|fUK(g): g in Generators(UK)];
oscar: [K(fUK(a)) for a in gens(UK)]
| |
Regulator: | \( 18292450.943147723 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{12}\cdot 18292450.943147723 \cdot 3}{20\cdot\sqrt{2126907556454464000000000000000000}}\cr\approx \mathstrut & 0.225240908955274 \end{aligned}\] (assuming GRH)
Galois group
$C_2\times C_{12}$ (as 24T2):
An abelian group of order 24 |
The 24 conjugacy class representatives for $C_2\times C_{12}$ |
Character table for $C_2\times C_{12}$ is not computed |
Intermediate fields
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 | ${\href{/padicField/3.12.0.1}{12} }^{2}$ | R | R | ${\href{/padicField/11.6.0.1}{6} }^{4}$ | ${\href{/padicField/13.4.0.1}{4} }^{6}$ | ${\href{/padicField/17.12.0.1}{12} }^{2}$ | ${\href{/padicField/19.6.0.1}{6} }^{4}$ | ${\href{/padicField/23.12.0.1}{12} }^{2}$ | ${\href{/padicField/29.2.0.1}{2} }^{12}$ | ${\href{/padicField/31.6.0.1}{6} }^{4}$ | ${\href{/padicField/37.12.0.1}{12} }^{2}$ | ${\href{/padicField/41.1.0.1}{1} }^{24}$ | ${\href{/padicField/43.4.0.1}{4} }^{6}$ | ${\href{/padicField/47.12.0.1}{12} }^{2}$ | ${\href{/padicField/53.12.0.1}{12} }^{2}$ | ${\href{/padicField/59.6.0.1}{6} }^{4}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\)
| Deg $24$ | $2$ | $12$ | $24$ | |||
\(5\)
| 5.12.9.2 | $x^{12} + 12 x^{10} + 12 x^{9} + 69 x^{8} + 108 x^{7} + 42 x^{6} - 396 x^{5} + 840 x^{4} + 252 x^{3} + 1476 x^{2} + 684 x + 1601$ | $4$ | $3$ | $9$ | $C_{12}$ | $[\ ]_{4}^{3}$ |
5.12.9.2 | $x^{12} + 12 x^{10} + 12 x^{9} + 69 x^{8} + 108 x^{7} + 42 x^{6} - 396 x^{5} + 840 x^{4} + 252 x^{3} + 1476 x^{2} + 684 x + 1601$ | $4$ | $3$ | $9$ | $C_{12}$ | $[\ ]_{4}^{3}$ | |
\(7\)
| 7.12.8.1 | $x^{12} + 15 x^{10} + 40 x^{9} + 84 x^{8} + 120 x^{7} + 53 x^{6} + 414 x^{5} - 1293 x^{4} - 1830 x^{3} + 10968 x^{2} - 13836 x + 12004$ | $3$ | $4$ | $8$ | $C_{12}$ | $[\ ]_{3}^{4}$ |
7.12.8.1 | $x^{12} + 15 x^{10} + 40 x^{9} + 84 x^{8} + 120 x^{7} + 53 x^{6} + 414 x^{5} - 1293 x^{4} - 1830 x^{3} + 10968 x^{2} - 13836 x + 12004$ | $3$ | $4$ | $8$ | $C_{12}$ | $[\ ]_{3}^{4}$ |