Normalized defining polynomial
\( x^{16} - 2 x^{15} - 5 x^{14} + 28 x^{13} - 47 x^{12} - 2 x^{11} + 187 x^{10} - 464 x^{9} + 658 x^{8} - 606 x^{7} + 366 x^{6} - 126 x^{5} + 21 x^{4} - 18 x^{3} + 33 x^{2} - 18 x + 3 \)
Invariants
| Degree: | $16$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[0, 8]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(9440732714731831296=2^{24}\cdot 3^{14}\cdot 7^{6}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $15.34$ | 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 not Galois over $\Q$. | |||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{5} a^{11} - \frac{1}{5} a^{10} + \frac{1}{5} a^{7} + \frac{1}{5} a^{5} - \frac{1}{5} a^{2} + \frac{1}{5} a + \frac{1}{5}$, $\frac{1}{15} a^{12} - \frac{1}{15} a^{11} + \frac{1}{3} a^{9} + \frac{1}{15} a^{8} + \frac{2}{5} a^{6} - \frac{2}{5} a^{3} + \frac{2}{5} a^{2} + \frac{2}{5} a$, $\frac{1}{15} a^{13} - \frac{1}{15} a^{11} + \frac{1}{3} a^{10} + \frac{2}{5} a^{9} + \frac{1}{15} a^{8} + \frac{2}{5} a^{7} + \frac{2}{5} a^{6} - \frac{2}{5} a^{4} - \frac{1}{5} a^{2} + \frac{2}{5} a$, $\frac{1}{15} a^{14} + \frac{1}{15} a^{11} - \frac{2}{5} a^{10} + \frac{2}{5} a^{9} + \frac{7}{15} a^{8} + \frac{1}{5} a^{7} + \frac{2}{5} a^{6} + \frac{2}{5} a^{5} + \frac{2}{5} a^{3} + \frac{1}{5} a - \frac{1}{5}$, $\frac{1}{7357335} a^{15} - \frac{179122}{7357335} a^{14} - \frac{24322}{7357335} a^{13} + \frac{6674}{1471467} a^{12} + \frac{118332}{2452445} a^{11} - \frac{1851518}{7357335} a^{10} - \frac{3175937}{7357335} a^{9} - \frac{212114}{7357335} a^{8} + \frac{848118}{2452445} a^{7} - \frac{152793}{2452445} a^{6} + \frac{958038}{2452445} a^{5} - \frac{304084}{2452445} a^{4} + \frac{684593}{2452445} a^{3} + \frac{404279}{2452445} a^{2} + \frac{360513}{2452445} a + \frac{240729}{2452445}$
Class group and class number
Trivial group, which has order $1$
Unit group
| Rank: | $7$ | magma: UnitRank(K);
sage: UK.rank()
gp: K.fu
| |
| Torsion generator: | \( -\frac{1588}{571} a^{15} + \frac{38713}{8565} a^{14} + \frac{8890}{571} a^{13} - \frac{205656}{2855} a^{12} + \frac{890134}{8565} a^{11} + \frac{125748}{2855} a^{10} - \frac{1437384}{2855} a^{9} + \frac{9448078}{8565} a^{8} - \frac{4054146}{2855} a^{7} + \frac{660838}{571} a^{6} - \frac{1676668}{2855} a^{5} + \frac{75427}{571} a^{4} - \frac{32098}{2855} a^{3} + \frac{136538}{2855} a^{2} - \frac{214612}{2855} a + \frac{64463}{2855} \) (order $6$) | 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 | magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
| |
| Regulator: | \( 3737.55934116 \) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A solvable group of order 32 |
| The 14 conjugacy class representatives for $D_8:C_2$ |
| Character table for $D_8:C_2$ |
Intermediate fields
| \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-2}) \), \(\Q(\sqrt{6}) \), 4.0.189.1, 4.0.12096.1, \(\Q(\sqrt{-2}, \sqrt{-3})\), 8.0.146313216.6 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Galois closure: | data not computed |
| Degree 16 siblings: | data not computed |
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 | ${\href{/LocalNumberField/5.2.0.1}{2} }^{8}$ | R | ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/19.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ | ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$ |
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 | ||||||
| $3$ | 3.8.7.1 | $x^{8} + 3$ | $8$ | $1$ | $7$ | $QD_{16}$ | $[\ ]_{8}^{2}$ |
| 3.8.7.1 | $x^{8} + 3$ | $8$ | $1$ | $7$ | $QD_{16}$ | $[\ ]_{8}^{2}$ | |
| $7$ | 7.4.2.2 | $x^{4} - 7 x^{2} + 147$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ |
| 7.4.0.1 | $x^{4} + x^{2} - 3 x + 5$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 7.8.4.1 | $x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |