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);
| |
| Signature: | $[0, 12]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
| |
| Discriminant: | \(2126907556454464000000000000000000\)\(\medspace = 2^{24}\cdot 5^{18}\cdot 7^{16}\) | sage: K.disc()
gp: K.disc
magma: Discriminant(Integers(K));
| |
| Root discriminant: | $24.47$ | sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
| |
| Ramified primes: | $2, 5, 7$ | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(Integers(K)));
| |
| $|\Gal(K/\Q)|$: | $24$ | ||
| 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. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{164891} a^{18} + \frac{2495}{164891} a^{16} - \frac{40833}{164891} a^{14} + \frac{24303}{164891} a^{12} - \frac{43903}{164891} a^{10} + \frac{57450}{164891} a^{8} + \frac{47471}{164891} a^{6} + \frac{48407}{164891} a^{4} + \frac{75253}{164891} a^{2} - \frac{54614}{164891}$, $\frac{1}{164891} a^{19} + \frac{2495}{164891} a^{17} - \frac{40833}{164891} a^{15} + \frac{24303}{164891} a^{13} - \frac{43903}{164891} a^{11} + \frac{57450}{164891} a^{9} + \frac{47471}{164891} a^{7} + \frac{48407}{164891} a^{5} + \frac{75253}{164891} a^{3} - \frac{54614}{164891} a$, $\frac{1}{164891} a^{20} - \frac{57080}{164891} a^{10} + \frac{61964}{164891}$, $\frac{1}{164891} a^{21} - \frac{57080}{164891} a^{11} + \frac{61964}{164891} a$, $\frac{1}{164891} a^{22} - \frac{57080}{164891} a^{12} + \frac{61964}{164891} a^{2}$, $\frac{1}{164891} a^{23} - \frac{57080}{164891} a^{13} + \frac{61964}{164891} a^{3}$
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);
| |
| 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);
| |
| 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) | sage: UK.fundamental_units()
gp: K.fu
magma: [K!f(g): g in Generators(UK)];
| |
| Regulator: | \( 18292450.943147723 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
|
Class number formula
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 | ${\href{/LocalNumberField/3.12.0.1}{12} }^{2}$ | R | R | ${\href{/LocalNumberField/11.6.0.1}{6} }^{4}$ | ${\href{/LocalNumberField/13.4.0.1}{4} }^{6}$ | ${\href{/LocalNumberField/17.12.0.1}{12} }^{2}$ | ${\href{/LocalNumberField/19.6.0.1}{6} }^{4}$ | ${\href{/LocalNumberField/23.12.0.1}{12} }^{2}$ | ${\href{/LocalNumberField/29.2.0.1}{2} }^{12}$ | ${\href{/LocalNumberField/31.6.0.1}{6} }^{4}$ | ${\href{/LocalNumberField/37.12.0.1}{12} }^{2}$ | ${\href{/LocalNumberField/41.1.0.1}{1} }^{24}$ | ${\href{/LocalNumberField/43.4.0.1}{4} }^{6}$ | ${\href{/LocalNumberField/47.12.0.1}{12} }^{2}$ | ${\href{/LocalNumberField/53.12.0.1}{12} }^{2}$ | ${\href{/LocalNumberField/59.6.0.1}{6} }^{4}$ |
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 | ||||||
| $5$ | 5.12.9.2 | $x^{12} - 10 x^{8} + 25 x^{4} - 500$ | $4$ | $3$ | $9$ | $C_{12}$ | $[\ ]_{4}^{3}$ |
| 5.12.9.2 | $x^{12} - 10 x^{8} + 25 x^{4} - 500$ | $4$ | $3$ | $9$ | $C_{12}$ | $[\ ]_{4}^{3}$ | |
| $7$ | 7.12.8.1 | $x^{12} - 63 x^{9} + 637 x^{6} + 6174 x^{3} + 300125$ | $3$ | $4$ | $8$ | $C_{12}$ | $[\ ]_{3}^{4}$ |
| 7.12.8.1 | $x^{12} - 63 x^{9} + 637 x^{6} + 6174 x^{3} + 300125$ | $3$ | $4$ | $8$ | $C_{12}$ | $[\ ]_{3}^{4}$ | |