Normalized defining polynomial
\( x^{16} - 2 x^{14} + 3 x^{12} - 6 x^{10} + 9 x^{8} - 24 x^{6} + 48 x^{4} - 128 x^{2} + 256 \)
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: | \(7029206082976827375616=2^{16}\cdot 18097^{4}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $23.20$ | 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: | $2, 18097$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is not Galois over $\Q$. | |||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{4} a^{6} + \frac{1}{4} a^{4} - \frac{1}{2} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{7} - \frac{1}{4} a^{5} - \frac{1}{2} a^{4} - \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{8} - \frac{1}{2} a^{3} - \frac{1}{4} a^{2}$, $\frac{1}{4} a^{9} - \frac{1}{2} a^{4} - \frac{1}{4} a^{3}$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{8} a^{11} - \frac{1}{8} a^{7} - \frac{1}{4} a^{5} - \frac{1}{2} a^{4} + \frac{3}{8} a^{3}$, $\frac{1}{64} a^{12} - \frac{3}{32} a^{10} - \frac{5}{64} a^{8} - \frac{1}{32} a^{6} - \frac{1}{4} a^{5} + \frac{17}{64} a^{4} - \frac{1}{4} a^{3} - \frac{7}{16} a^{2} + \frac{1}{4} a + \frac{1}{4}$, $\frac{1}{128} a^{13} - \frac{3}{64} a^{11} - \frac{1}{8} a^{10} + \frac{11}{128} a^{9} + \frac{7}{64} a^{7} - \frac{1}{8} a^{6} + \frac{1}{128} a^{5} - \frac{1}{2} a^{4} - \frac{7}{32} a^{3} - \frac{1}{8} a^{2} - \frac{3}{8} a$, $\frac{1}{8448} a^{14} + \frac{3}{1408} a^{12} - \frac{1}{16} a^{11} + \frac{27}{256} a^{10} - \frac{1}{8} a^{9} - \frac{23}{1408} a^{8} + \frac{1}{16} a^{7} + \frac{315}{2816} a^{6} + \frac{1}{8} a^{5} - \frac{9}{64} a^{4} + \frac{7}{16} a^{3} + \frac{45}{176} a^{2} - \frac{1}{2} a + \frac{23}{66}$, $\frac{1}{16896} a^{15} + \frac{3}{2816} a^{13} + \frac{27}{512} a^{11} - \frac{1}{8} a^{10} + \frac{329}{2816} a^{9} - \frac{389}{5632} a^{7} - \frac{1}{8} a^{6} - \frac{25}{128} a^{5} - \frac{1}{2} a^{4} - \frac{131}{352} a^{3} + \frac{3}{8} a^{2} - \frac{43}{132} a - \frac{1}{2}$
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: | \( -\frac{5}{4224} a^{14} + \frac{7}{704} a^{12} + \frac{1}{128} a^{10} + \frac{5}{704} a^{8} + \frac{97}{1408} a^{6} - \frac{1}{16} a^{4} - \frac{2}{11} a^{2} + \frac{1}{66} \) (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 | magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
| |
| Regulator: | \( 41888.9764246 \) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$C_2\times C_2^3:S_4$ (as 16T747):
| A solvable group of order 384 |
| The 26 conjugacy class representatives for $C_2\times C_2^3:S_4$ |
| Character table for $C_2\times C_2^3:S_4$ is not computed |
Intermediate fields
| \(\Q(\sqrt{-1}) \), 4.4.289552.1, 8.0.83840360704.3, 8.8.83840360704.2, 8.0.83840360704.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Frobenius cycle types
| $p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 | 53 | 59 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/LocalNumberField/3.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/17.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{4}$ | ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/37.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{4}$ | ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/53.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{4}$ | ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
| $2$ | 2.8.8.1 | $x^{8} + 28 x^{4} + 144$ | $2$ | $4$ | $8$ | $C_4\times C_2$ | $[2]^{4}$ |
| 2.8.8.1 | $x^{8} + 28 x^{4} + 144$ | $2$ | $4$ | $8$ | $C_4\times C_2$ | $[2]^{4}$ | |
| 18097 | Data not computed | ||||||