Normalized defining polynomial
\( x^{16} - x^{15} + x^{14} - 2 x^{13} + 3 x^{12} - 5 x^{11} + 4 x^{10} - 2 x^{9} + 2 x^{8} - x^{7} + 3 x^{6} + 5 x^{5} + 4 x^{4} + 2 x^{3} + 6 x^{2} - 2 x + 1 \)
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: | \(120573447359765625=3^{8}\cdot 5^{8}\cdot 19^{6}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $11.68$ | 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: | $3, 5, 19$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{17} a^{13} + \frac{4}{17} a^{12} + \frac{2}{17} a^{11} + \frac{4}{17} a^{10} + \frac{1}{17} a^{9} + \frac{1}{17} a^{7} + \frac{7}{17} a^{6} + \frac{1}{17} a^{5} - \frac{6}{17} a^{4} - \frac{1}{17} a^{3} - \frac{1}{17} a^{2} - \frac{6}{17} a + \frac{4}{17}$, $\frac{1}{85} a^{14} - \frac{2}{85} a^{13} + \frac{29}{85} a^{12} - \frac{8}{85} a^{11} - \frac{8}{17} a^{10} - \frac{23}{85} a^{9} - \frac{33}{85} a^{8} + \frac{18}{85} a^{7} - \frac{24}{85} a^{6} - \frac{29}{85} a^{5} + \frac{18}{85} a^{4} - \frac{12}{85} a^{3} + \frac{2}{5} a^{2} + \frac{6}{85} a - \frac{41}{85}$, $\frac{1}{5015} a^{15} - \frac{24}{5015} a^{14} - \frac{37}{5015} a^{13} - \frac{1511}{5015} a^{12} - \frac{1529}{5015} a^{11} - \frac{2303}{5015} a^{10} + \frac{2233}{5015} a^{9} - \frac{1211}{5015} a^{8} + \frac{438}{1003} a^{7} + \frac{664}{5015} a^{6} - \frac{814}{5015} a^{5} + \frac{2207}{5015} a^{4} - \frac{1}{295} a^{3} + \frac{983}{5015} a^{2} + \frac{997}{5015} a - \frac{218}{5015}$
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: | \( \frac{506}{1003} a^{15} - \frac{1602}{5015} a^{14} + \frac{849}{5015} a^{13} - \frac{3878}{5015} a^{12} + \frac{5806}{5015} a^{11} - \frac{1720}{1003} a^{10} + \frac{4016}{5015} a^{9} + \frac{276}{5015} a^{8} + \frac{2134}{5015} a^{7} - \frac{282}{5015} a^{6} + \frac{4518}{5015} a^{5} + \frac{15644}{5015} a^{4} + \frac{12804}{5015} a^{3} + \frac{6512}{5015} a^{2} + \frac{11193}{5015} a + \frac{1762}{5015} \) (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: | \( 179.27186754 \) | 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
| \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-15}) \), \(\Q(\sqrt{-3}) \), 4.2.475.1, 4.2.4275.1, \(\Q(\sqrt{-3}, \sqrt{5})\), 8.2.4286875.1, 8.2.347236875.1, 8.0.18275625.1 |
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 | ${\href{/LocalNumberField/7.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ | R | ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/31.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ | ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$ |
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 | Data not computed | ||||||
| $5$ | 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 | $x^{4} + 15 x^{2} + 100$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 5.4.2.1 | $x^{4} + 15 x^{2} + 100$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| $19$ | 19.2.0.1 | $x^{2} - x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 19.2.1.1 | $x^{2} - 19$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 19.2.0.1 | $x^{2} - x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 19.2.1.1 | $x^{2} - 19$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 19.4.2.1 | $x^{4} + 57 x^{2} + 1444$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 19.4.2.1 | $x^{4} + 57 x^{2} + 1444$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |