Normalized defining polynomial
\( x^{24} + 33 x^{20} + 340 x^{16} + 1154 x^{12} + 1192 x^{8} + 136 x^{4} + 1 \)
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: | \(11935913115234869169771450911000887296=2^{48}\cdot 3^{12}\cdot 7^{20}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $35.06$ | 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: | $2, 3, 7$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(168=2^{3}\cdot 3\cdot 7\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{168}(1,·)$, $\chi_{168}(11,·)$, $\chi_{168}(79,·)$, $\chi_{168}(139,·)$, $\chi_{168}(13,·)$, $\chi_{168}(143,·)$, $\chi_{168}(17,·)$, $\chi_{168}(19,·)$, $\chi_{168}(149,·)$, $\chi_{168}(151,·)$, $\chi_{168}(25,·)$, $\chi_{168}(155,·)$, $\chi_{168}(157,·)$, $\chi_{168}(167,·)$, $\chi_{168}(41,·)$, $\chi_{168}(107,·)$, $\chi_{168}(29,·)$, $\chi_{168}(47,·)$, $\chi_{168}(115,·)$, $\chi_{168}(53,·)$, $\chi_{168}(89,·)$, $\chi_{168}(121,·)$, $\chi_{168}(61,·)$, $\chi_{168}(127,·)$$\rbrace$ | ||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $\frac{1}{17416349} a^{20} - \frac{3289265}{17416349} a^{16} + \frac{5876879}{17416349} a^{12} - \frac{8661}{17416349} a^{8} - \frac{4535794}{17416349} a^{4} + \frac{2092690}{17416349}$, $\frac{1}{17416349} a^{21} - \frac{3289265}{17416349} a^{17} + \frac{5876879}{17416349} a^{13} - \frac{8661}{17416349} a^{9} - \frac{4535794}{17416349} a^{5} + \frac{2092690}{17416349} a$, $\frac{1}{17416349} a^{22} - \frac{3289265}{17416349} a^{18} + \frac{5876879}{17416349} a^{14} - \frac{8661}{17416349} a^{10} - \frac{4535794}{17416349} a^{6} + \frac{2092690}{17416349} a^{2}$, $\frac{1}{17416349} a^{23} - \frac{3289265}{17416349} a^{19} + \frac{5876879}{17416349} a^{15} - \frac{8661}{17416349} a^{11} - \frac{4535794}{17416349} a^{7} + \frac{2092690}{17416349} a^{3}$
Class group and class number
$C_{24}$, which has order $24$ (assuming GRH)
Unit group
| Rank: | $11$ | magma: UnitRank(K);
sage: UK.rank()
gp: K.fu
| |
| Torsion generator: | \( \frac{1203986}{17416349} a^{22} + \frac{39756122}{17416349} a^{18} + \frac{410172887}{17416349} a^{14} + \frac{1397978225}{17416349} a^{10} + \frac{1466058174}{17416349} a^{6} + \frac{207497745}{17416349} a^{2} \) (order $4$) | 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: | \( 19752911.509995345 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$C_2^2\times C_6$ (as 24T3):
| An abelian group of order 24 |
| The 24 conjugacy class representatives for $C_2^2\times C_6$ |
| Character table for $C_2^2\times C_6$ 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 | ${\href{/LocalNumberField/5.3.0.1}{3} }^{8}$ | R | ${\href{/LocalNumberField/11.6.0.1}{6} }^{4}$ | ${\href{/LocalNumberField/13.2.0.1}{2} }^{12}$ | ${\href{/LocalNumberField/17.6.0.1}{6} }^{4}$ | ${\href{/LocalNumberField/19.6.0.1}{6} }^{4}$ | ${\href{/LocalNumberField/23.6.0.1}{6} }^{4}$ | ${\href{/LocalNumberField/29.2.0.1}{2} }^{12}$ | ${\href{/LocalNumberField/31.6.0.1}{6} }^{4}$ | ${\href{/LocalNumberField/37.6.0.1}{6} }^{4}$ | ${\href{/LocalNumberField/41.2.0.1}{2} }^{12}$ | ${\href{/LocalNumberField/43.2.0.1}{2} }^{12}$ | ${\href{/LocalNumberField/47.6.0.1}{6} }^{4}$ | ${\href{/LocalNumberField/53.6.0.1}{6} }^{4}$ | ${\href{/LocalNumberField/59.6.0.1}{6} }^{4}$ |
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 | Data not computed | ||||||
| $3$ | 3.12.6.2 | $x^{12} + 108 x^{6} - 243 x^{2} + 2916$ | $2$ | $6$ | $6$ | $C_6\times C_2$ | $[\ ]_{2}^{6}$ |
| 3.12.6.2 | $x^{12} + 108 x^{6} - 243 x^{2} + 2916$ | $2$ | $6$ | $6$ | $C_6\times C_2$ | $[\ ]_{2}^{6}$ | |
| $7$ | 7.12.10.1 | $x^{12} - 70 x^{6} + 35721$ | $6$ | $2$ | $10$ | $C_6\times C_2$ | $[\ ]_{6}^{2}$ |
| 7.12.10.1 | $x^{12} - 70 x^{6} + 35721$ | $6$ | $2$ | $10$ | $C_6\times C_2$ | $[\ ]_{6}^{2}$ | |