Normalized defining polynomial
\( x^{24} - 8 x^{23} - 44 x^{22} + 512 x^{21} + 226 x^{20} - 12464 x^{19} + 17024 x^{18} + 141816 x^{17} - 362121 x^{16} - 713488 x^{15} + 2995748 x^{14} + 677608 x^{13} - 11628982 x^{12} + 6093888 x^{11} + 21284236 x^{10} - 19658944 x^{9} - 18084118 x^{8} + 22241592 x^{7} + 7013084 x^{6} - 11297384 x^{5} - 996472 x^{4} + 2527320 x^{3} - 43816 x^{2} - 193168 x + 18817 \)
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: | \(2856585398747452538819699104507633428089224036352=2^{93}\cdot 19^{16}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $104.47$ | 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: | $2, 19$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(608=2^{5}\cdot 19\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{608}(1,·)$, $\chi_{608}(581,·)$, $\chi_{608}(577,·)$, $\chi_{608}(201,·)$, $\chi_{608}(77,·)$, $\chi_{608}(45,·)$, $\chi_{608}(273,·)$, $\chi_{608}(277,·)$, $\chi_{608}(505,·)$, $\chi_{608}(153,·)$, $\chi_{608}(349,·)$, $\chi_{608}(197,·)$, $\chi_{608}(353,·)$, $\chi_{608}(229,·)$, $\chi_{608}(305,·)$, $\chi_{608}(425,·)$, $\chi_{608}(429,·)$, $\chi_{608}(125,·)$, $\chi_{608}(49,·)$, $\chi_{608}(501,·)$, $\chi_{608}(457,·)$, $\chi_{608}(121,·)$, $\chi_{608}(381,·)$, $\chi_{608}(533,·)$$\rbrace$ | ||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $\frac{1}{31} a^{21} - \frac{11}{31} a^{20} - \frac{15}{31} a^{19} + \frac{4}{31} a^{18} - \frac{10}{31} a^{17} + \frac{8}{31} a^{16} - \frac{11}{31} a^{15} + \frac{10}{31} a^{14} + \frac{2}{31} a^{13} - \frac{12}{31} a^{12} + \frac{11}{31} a^{11} + \frac{9}{31} a^{10} + \frac{11}{31} a^{9} - \frac{1}{31} a^{8} - \frac{12}{31} a^{7} - \frac{1}{31} a^{6} + \frac{8}{31} a^{5} + \frac{5}{31} a^{4} + \frac{8}{31} a^{3} - \frac{7}{31} a^{2} + \frac{3}{31} a$, $\frac{1}{203970794458462585662903499361} a^{22} - \frac{913875642316726849290582616}{203970794458462585662903499361} a^{21} + \frac{3934531277319235599213762601}{203970794458462585662903499361} a^{20} + \frac{59135323387201609169690011312}{203970794458462585662903499361} a^{19} - \frac{40081777683207400815197660724}{203970794458462585662903499361} a^{18} - \frac{74273534591009474627992585493}{203970794458462585662903499361} a^{17} - \frac{89209288100216401716379853449}{203970794458462585662903499361} a^{16} - \frac{67083865351677241043298512437}{203970794458462585662903499361} a^{15} + \frac{24704595806215197683250788635}{203970794458462585662903499361} a^{14} - \frac{67554841523363732716478200437}{203970794458462585662903499361} a^{13} + \frac{25451668654449296699200535562}{203970794458462585662903499361} a^{12} - \frac{10904397546950377067886339156}{203970794458462585662903499361} a^{11} + \frac{72264652481713104736157516573}{203970794458462585662903499361} a^{10} - \frac{49835790739010536016141474903}{203970794458462585662903499361} a^{9} + \frac{42868917236690103140786985225}{203970794458462585662903499361} a^{8} - \frac{20452851854875989910464857223}{203970794458462585662903499361} a^{7} - \frac{29065950657558718154168636455}{203970794458462585662903499361} a^{6} + \frac{97505060914495255526765362974}{203970794458462585662903499361} a^{5} + \frac{78419920918221145336816430019}{203970794458462585662903499361} a^{4} + \frac{19162620353833487119238677828}{203970794458462585662903499361} a^{3} + \frac{70632823152406176705913572875}{203970794458462585662903499361} a^{2} + \frac{30519616595970126684832913669}{203970794458462585662903499361} a + \frac{1991383587318834476203567389}{6579703047047180182674306431}$, $\frac{1}{33874353211503395061163680103022877247} a^{23} + \frac{1507210}{33874353211503395061163680103022877247} a^{22} + \frac{292959990004649003327796520425393839}{33874353211503395061163680103022877247} a^{21} - \frac{11210001515634138258478400267768647072}{33874353211503395061163680103022877247} a^{20} + \frac{13716473541072850315045265016953791570}{33874353211503395061163680103022877247} a^{19} - \frac{8907999506409297145291390024683294457}{33874353211503395061163680103022877247} a^{18} + \frac{12406994946133585540283948526495356138}{33874353211503395061163680103022877247} a^{17} + \frac{162259895678177765216781047939422794}{1092721071338819195521409035581383137} a^{16} + \frac{7134862380279706983103783597972036694}{33874353211503395061163680103022877247} a^{15} - \frac{11827685841725494480113639975321779815}{33874353211503395061163680103022877247} a^{14} - \frac{1717709921492330093300431474381576514}{33874353211503395061163680103022877247} a^{13} - \frac{3098106466579762275977267192532012385}{33874353211503395061163680103022877247} a^{12} - \frac{4167953436190542679364018073771552973}{33874353211503395061163680103022877247} a^{11} + \frac{15617331251917199542118243467188998290}{33874353211503395061163680103022877247} a^{10} - \frac{159236620191848037430611574090535103}{1092721071338819195521409035581383137} a^{9} + \frac{10616810166578674738662326046773719200}{33874353211503395061163680103022877247} a^{8} + \frac{7385929262942640936056582331952377980}{33874353211503395061163680103022877247} a^{7} - \frac{1337196639737173246228733706697797865}{33874353211503395061163680103022877247} a^{6} - \frac{8609422172236387683771219293869277283}{33874353211503395061163680103022877247} a^{5} + \frac{5372287638598410339857278682988560416}{33874353211503395061163680103022877247} a^{4} - \frac{9418771548107413614918816232348224026}{33874353211503395061163680103022877247} a^{3} - \frac{5912887223192761295393321021439140218}{33874353211503395061163680103022877247} a^{2} + \frac{9535129595945976590197392251311746613}{33874353211503395061163680103022877247} a - \frac{294438989017323231717712039463478371}{1092721071338819195521409035581383137}$
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: | \( 59505566364712370 \) (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
| \(\Q(\sqrt{2}) \), 3.3.361.1, \(\Q(\zeta_{16})^+\), 6.6.66724352.1, \(\Q(\zeta_{32})^+\), 12.12.145887695661298614272.1 |
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 | $24$ | $24$ | ${\href{/LocalNumberField/7.4.0.1}{4} }^{6}$ | ${\href{/LocalNumberField/11.8.0.1}{8} }^{3}$ | $24$ | ${\href{/LocalNumberField/17.6.0.1}{6} }^{4}$ | R | ${\href{/LocalNumberField/23.12.0.1}{12} }^{2}$ | $24$ | ${\href{/LocalNumberField/31.1.0.1}{1} }^{24}$ | ${\href{/LocalNumberField/37.8.0.1}{8} }^{3}$ | ${\href{/LocalNumberField/41.12.0.1}{12} }^{2}$ | $24$ | ${\href{/LocalNumberField/47.6.0.1}{6} }^{4}$ | $24$ | $24$ |
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 | ||||||
| 19 | Data not computed | ||||||