Normalized defining polynomial
\( x^{24} + 186 x^{16} + 4725 x^{8} + 1 \)
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: | \(8750640431914714140510423543328014336=2^{72}\cdot 3^{32}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $34.61$ | 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: | $2, 3$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(144=2^{4}\cdot 3^{2}\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{144}(1,·)$, $\chi_{144}(67,·)$, $\chi_{144}(133,·)$, $\chi_{144}(7,·)$, $\chi_{144}(73,·)$, $\chi_{144}(139,·)$, $\chi_{144}(13,·)$, $\chi_{144}(79,·)$, $\chi_{144}(19,·)$, $\chi_{144}(85,·)$, $\chi_{144}(25,·)$, $\chi_{144}(91,·)$, $\chi_{144}(31,·)$, $\chi_{144}(97,·)$, $\chi_{144}(37,·)$, $\chi_{144}(103,·)$, $\chi_{144}(43,·)$, $\chi_{144}(109,·)$, $\chi_{144}(49,·)$, $\chi_{144}(115,·)$, $\chi_{144}(55,·)$, $\chi_{144}(121,·)$, $\chi_{144}(61,·)$, $\chi_{144}(127,·)$$\rbrace$ | ||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{65773} a^{16} - \frac{131}{3869} a^{8} - \frac{14910}{65773}$, $\frac{1}{65773} a^{17} - \frac{131}{3869} a^{9} - \frac{14910}{65773} a$, $\frac{1}{65773} a^{18} - \frac{131}{3869} a^{10} - \frac{14910}{65773} a^{2}$, $\frac{1}{65773} a^{19} - \frac{131}{3869} a^{11} - \frac{14910}{65773} a^{3}$, $\frac{1}{65773} a^{20} - \frac{131}{3869} a^{12} - \frac{14910}{65773} a^{4}$, $\frac{1}{65773} a^{21} - \frac{131}{3869} a^{13} - \frac{14910}{65773} a^{5}$, $\frac{1}{65773} a^{22} - \frac{131}{3869} a^{14} - \frac{14910}{65773} a^{6}$, $\frac{1}{65773} a^{23} - \frac{131}{3869} a^{15} - \frac{14910}{65773} a^{7}$
Class group and class number
$C_{39}$, which has order $39$ (assuming GRH)
Unit group
| Rank: | $11$ | magma: UnitRank(K);
sage: UK.rank()
gp: K.fu
| |
| Torsion generator: | \( -\frac{327}{65773} a^{19} - \frac{3591}{3869} a^{11} - \frac{1570184}{65773} a^{3} \) (order $16$) | 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: | \( 165705493.8155171 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
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 | R | ${\href{/LocalNumberField/5.12.0.1}{12} }^{2}$ | ${\href{/LocalNumberField/7.6.0.1}{6} }^{4}$ | ${\href{/LocalNumberField/11.12.0.1}{12} }^{2}$ | ${\href{/LocalNumberField/13.12.0.1}{12} }^{2}$ | ${\href{/LocalNumberField/17.1.0.1}{1} }^{24}$ | ${\href{/LocalNumberField/19.4.0.1}{4} }^{6}$ | ${\href{/LocalNumberField/23.6.0.1}{6} }^{4}$ | ${\href{/LocalNumberField/29.12.0.1}{12} }^{2}$ | ${\href{/LocalNumberField/31.6.0.1}{6} }^{4}$ | ${\href{/LocalNumberField/37.4.0.1}{4} }^{6}$ | ${\href{/LocalNumberField/41.6.0.1}{6} }^{4}$ | ${\href{/LocalNumberField/43.12.0.1}{12} }^{2}$ | ${\href{/LocalNumberField/47.6.0.1}{6} }^{4}$ | ${\href{/LocalNumberField/53.4.0.1}{4} }^{6}$ | ${\href{/LocalNumberField/59.12.0.1}{12} }^{2}$ |
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 | Data not computed | ||||||
| $3$ | 3.12.16.14 | $x^{12} + 72 x^{11} - 36 x^{10} + 108 x^{9} - 108 x^{8} + 54 x^{7} + 72 x^{6} - 81 x^{5} - 81 x^{4} - 81 x^{3} + 81 x^{2} - 81$ | $3$ | $4$ | $16$ | $C_{12}$ | $[2]^{4}$ |
| 3.12.16.14 | $x^{12} + 72 x^{11} - 36 x^{10} + 108 x^{9} - 108 x^{8} + 54 x^{7} + 72 x^{6} - 81 x^{5} - 81 x^{4} - 81 x^{3} + 81 x^{2} - 81$ | $3$ | $4$ | $16$ | $C_{12}$ | $[2]^{4}$ | |