Normalized defining polynomial
\( x^{16} - 6 x^{15} + 17 x^{14} - 28 x^{13} + 26 x^{12} - 10 x^{11} - 3 x^{10} + 2 x^{9} + x^{8} + 6 x^{7} - 10 x^{6} + 2 x^{5} + 6 x^{4} - 4 x^{3} - 2 x^{2} + 2 x + 1 \)
Invariants
Degree: | $16$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
| |
Signature: | $[0, 8]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
| |
Discriminant: | \(7771775074107392\)\(\medspace = 2^{16}\cdot 17^{9}\) | sage: K.disc()
gp: K.disc
magma: Discriminant(Integers(K));
| |
Root discriminant: | $9.84$ | 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: | $2, 17$ | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(Integers(K)));
| |
$|\Aut(K/\Q)|$: | $4$ | ||
This field is not Galois over $\Q$. | |||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{67} a^{15} - \frac{29}{67} a^{14} + \frac{14}{67} a^{13} - \frac{15}{67} a^{12} - \frac{31}{67} a^{11} + \frac{33}{67} a^{10} - \frac{25}{67} a^{9} - \frac{26}{67} a^{8} - \frac{4}{67} a^{7} + \frac{31}{67} a^{6} + \frac{14}{67} a^{5} + \frac{15}{67} a^{4} - \frac{4}{67} a^{3} + \frac{21}{67} a^{2} - \frac{16}{67} a - \frac{32}{67}$
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $7$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
| |
Torsion generator: | \( \frac{44}{67} a^{15} - \frac{137}{67} a^{14} + \frac{80}{67} a^{13} + \frac{546}{67} a^{12} - \frac{1699}{67} a^{11} + \frac{2390}{67} a^{10} - \frac{1971}{67} a^{9} + \frac{1067}{67} a^{8} - \frac{779}{67} a^{7} + \frac{694}{67} a^{6} - \frac{54}{67} a^{5} - \frac{546}{67} a^{4} + \frac{427}{67} a^{3} + \frac{53}{67} a^{2} - \frac{168}{67} a - \frac{68}{67} \) (order $4$) | 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 | sage: UK.fundamental_units()
gp: K.fu
magma: [K!f(g): g in Generators(UK)];
| |
Regulator: | \( 25.2889722607 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
|
Class number formula
Galois group
$C_4.D_4:C_4$ (as 16T260):
A solvable group of order 128 |
The 32 conjugacy class representatives for $C_4.D_4:C_4$ |
Character table for $C_4.D_4:C_4$ is not computed |
Intermediate fields
\(\Q(\sqrt{-1}) \), 4.0.272.1, 8.0.1257728.1 |
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 | $16$ | ${\href{/LocalNumberField/5.8.0.1}{8} }{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }^{2}$ | $16$ | $16$ | ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ | R | ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ | $16$ | ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ | $16$ | ${\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }^{2}$ | ${\href{/LocalNumberField/41.8.0.1}{8} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ | ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$ |
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 | ||||||
$17$ | 17.4.2.2 | $x^{4} - 17 x^{2} + 867$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ |
17.4.0.1 | $x^{4} - x + 11$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
17.8.7.4 | $x^{8} - 12393$ | $8$ | $1$ | $7$ | $C_8$ | $[\ ]_{8}$ |