Normalized defining polynomial
\( x^{16} - 3 x^{15} + 10 x^{14} - 13 x^{13} + 27 x^{12} - 24 x^{11} + 52 x^{10} - 30 x^{9} + 49 x^{8} - 32 x^{7} + 45 x^{6} - 22 x^{5} + 22 x^{4} - 18 x^{3} + 6 x^{2} - 6 x + 3 \)
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: | \(1779985907461265625=3^{12}\cdot 5^{6}\cdot 11^{8}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $13.82$ | 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: | $3, 5, 11$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{11} a^{12} + \frac{2}{11} a^{11} - \frac{4}{11} a^{10} - \frac{5}{11} a^{9} + \frac{4}{11} a^{8} + \frac{2}{11} a^{7} - \frac{2}{11} a^{6} + \frac{5}{11} a^{5} + \frac{5}{11} a^{4} - \frac{1}{11} a^{3} - \frac{1}{11} a - \frac{2}{11}$, $\frac{1}{11} a^{13} + \frac{3}{11} a^{11} + \frac{3}{11} a^{10} + \frac{3}{11} a^{9} + \frac{5}{11} a^{8} + \frac{5}{11} a^{7} - \frac{2}{11} a^{6} - \frac{5}{11} a^{5} + \frac{2}{11} a^{3} - \frac{1}{11} a^{2} + \frac{4}{11}$, $\frac{1}{55} a^{14} + \frac{2}{55} a^{13} + \frac{2}{55} a^{12} - \frac{4}{55} a^{11} - \frac{9}{55} a^{10} - \frac{17}{55} a^{9} - \frac{1}{5} a^{8} + \frac{6}{55} a^{7} - \frac{18}{55} a^{6} - \frac{3}{11} a^{5} + \frac{19}{55} a^{4} - \frac{7}{55} a^{3} + \frac{4}{11} a^{2} - \frac{17}{55} a + \frac{21}{55}$, $\frac{1}{307285} a^{15} + \frac{717}{307285} a^{14} - \frac{8928}{307285} a^{13} + \frac{8051}{307285} a^{12} - \frac{58429}{307285} a^{11} - \frac{77012}{307285} a^{10} - \frac{131701}{307285} a^{9} - \frac{80404}{307285} a^{8} - \frac{98903}{307285} a^{7} - \frac{27593}{61457} a^{6} + \frac{13279}{307285} a^{5} - \frac{76717}{307285} a^{4} - \frac{25210}{61457} a^{3} - \frac{6377}{307285} a^{2} - \frac{149769}{307285} a - \frac{14326}{61457}$
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: | \( -\frac{3}{11} a^{15} + \frac{37}{55} a^{14} - \frac{11}{5} a^{13} + \frac{94}{55} a^{12} - \frac{238}{55} a^{11} + \frac{37}{55} a^{10} - \frac{434}{55} a^{9} - \frac{152}{55} a^{8} - \frac{268}{55} a^{7} - \frac{166}{55} a^{6} - \frac{67}{11} a^{5} - \frac{167}{55} a^{4} - \frac{74}{55} a^{3} + \frac{1}{11} a^{2} + \frac{21}{55} a + \frac{47}{55} \) (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: | \( 1469.71837707 \) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$C_2\times D_8$ (as 16T29):
| A solvable group of order 32 |
| The 14 conjugacy class representatives for $C_2\times D_8$ |
| Character table for $C_2\times D_8$ |
Intermediate fields
| \(\Q(\sqrt{33}) \), \(\Q(\sqrt{-11}) \), \(\Q(\sqrt{-3}) \), 4.0.605.1, 4.0.5445.1, \(\Q(\sqrt{-3}, \sqrt{-11})\), 8.0.1334161125.2, 8.0.1334161125.1, 8.0.29648025.1 |
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 | ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ | R | R | ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ | R | ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/37.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{4}$ | ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
| $3$ | 3.4.3.1 | $x^{4} + 3$ | $4$ | $1$ | $3$ | $D_{4}$ | $[\ ]_{4}^{2}$ |
| 3.4.3.1 | $x^{4} + 3$ | $4$ | $1$ | $3$ | $D_{4}$ | $[\ ]_{4}^{2}$ | |
| 3.8.6.2 | $x^{8} + 4 x^{7} + 14 x^{6} + 28 x^{5} + 43 x^{4} + 44 x^{3} + 110 x^{2} + 92 x + 22$ | $4$ | $2$ | $6$ | $D_4$ | $[\ ]_{4}^{2}$ | |
| $5$ | 5.2.0.1 | $x^{2} - x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 5.2.0.1 | $x^{2} - x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 5.4.2.1 | $x^{4} + 15 x^{2} + 100$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 5.4.2.1 | $x^{4} + 15 x^{2} + 100$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 5.4.2.1 | $x^{4} + 15 x^{2} + 100$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| $11$ | 11.4.2.1 | $x^{4} + 143 x^{2} + 5929$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| 11.4.2.1 | $x^{4} + 143 x^{2} + 5929$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 11.4.2.1 | $x^{4} + 143 x^{2} + 5929$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 11.4.2.1 | $x^{4} + 143 x^{2} + 5929$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |