Normalized defining polynomial
\( x^{24} + 245 x^{16} + 7546 x^{8} + 2401 \)
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: | \(376808323956052112639025409344139165696=2^{72}\cdot 7^{20}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $40.49$ | 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: | $2, 7$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(112=2^{4}\cdot 7\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{112}(1,·)$, $\chi_{112}(3,·)$, $\chi_{112}(5,·)$, $\chi_{112}(65,·)$, $\chi_{112}(9,·)$, $\chi_{112}(79,·)$, $\chi_{112}(75,·)$, $\chi_{112}(13,·)$, $\chi_{112}(15,·)$, $\chi_{112}(81,·)$, $\chi_{112}(19,·)$, $\chi_{112}(23,·)$, $\chi_{112}(25,·)$, $\chi_{112}(27,·)$, $\chi_{112}(69,·)$, $\chi_{112}(101,·)$, $\chi_{112}(39,·)$, $\chi_{112}(95,·)$, $\chi_{112}(71,·)$, $\chi_{112}(45,·)$, $\chi_{112}(83,·)$, $\chi_{112}(57,·)$, $\chi_{112}(59,·)$, $\chi_{112}(61,·)$$\rbrace$ | ||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{7} a^{6}$, $\frac{1}{7} a^{7}$, $\frac{1}{7} a^{8}$, $\frac{1}{7} a^{9}$, $\frac{1}{7} a^{10}$, $\frac{1}{7} a^{11}$, $\frac{1}{49} a^{12}$, $\frac{1}{49} a^{13}$, $\frac{1}{49} a^{14}$, $\frac{1}{49} a^{15}$, $\frac{1}{26117} a^{16} + \frac{254}{3731} a^{8} - \frac{185}{533}$, $\frac{1}{26117} a^{17} + \frac{254}{3731} a^{9} - \frac{185}{533} a$, $\frac{1}{182819} a^{18} - \frac{116}{3731} a^{10} + \frac{202}{533} a^{2}$, $\frac{1}{182819} a^{19} - \frac{116}{3731} a^{11} + \frac{202}{533} a^{3}$, $\frac{1}{182819} a^{20} + \frac{254}{26117} a^{12} + \frac{202}{533} a^{4}$, $\frac{1}{182819} a^{21} + \frac{254}{26117} a^{13} + \frac{202}{533} a^{5}$, $\frac{1}{182819} a^{22} + \frac{254}{26117} a^{14} - \frac{185}{3731} a^{6}$, $\frac{1}{182819} a^{23} + \frac{254}{26117} a^{15} - \frac{185}{3731} a^{7}$
Class group and class number
$C_{2}\times C_{26}$, which has order $52$ (assuming GRH)
Unit group
| Rank: | $11$ | magma: UnitRank(K);
sage: UK.rank()
gp: K.fu
| |
| Torsion generator: | \( \frac{8}{26117} a^{22} + \frac{1965}{26117} a^{14} + \frac{8828}{3731} a^{6} \) (order $8$) | 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: | \( 215040822.68029645 \) (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 | ${\href{/LocalNumberField/3.12.0.1}{12} }^{2}$ | ${\href{/LocalNumberField/5.12.0.1}{12} }^{2}$ | R | ${\href{/LocalNumberField/11.12.0.1}{12} }^{2}$ | ${\href{/LocalNumberField/13.4.0.1}{4} }^{6}$ | ${\href{/LocalNumberField/17.6.0.1}{6} }^{4}$ | ${\href{/LocalNumberField/19.12.0.1}{12} }^{2}$ | ${\href{/LocalNumberField/23.6.0.1}{6} }^{4}$ | ${\href{/LocalNumberField/29.4.0.1}{4} }^{6}$ | ${\href{/LocalNumberField/31.6.0.1}{6} }^{4}$ | ${\href{/LocalNumberField/37.12.0.1}{12} }^{2}$ | ${\href{/LocalNumberField/41.1.0.1}{1} }^{24}$ | ${\href{/LocalNumberField/43.4.0.1}{4} }^{6}$ | ${\href{/LocalNumberField/47.6.0.1}{6} }^{4}$ | ${\href{/LocalNumberField/53.12.0.1}{12} }^{2}$ | ${\href{/LocalNumberField/59.12.0.1}{12} }^{2}$ |
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 | ||||||
| $7$ | 7.12.10.1 | $x^{12} - 70 x^{6} + 35721$ | $6$ | $2$ | $10$ | $C_6\times C_2$ | $[\ ]_{6}^{2}$ |
| 7.12.10.1 | $x^{12} - 70 x^{6} + 35721$ | $6$ | $2$ | $10$ | $C_6\times C_2$ | $[\ ]_{6}^{2}$ | |