Normalized defining polynomial
\( x^{16} - 5 x^{15} + 8 x^{14} - x^{13} + 11 x^{11} - 38 x^{10} - 2 x^{9} + 58 x^{8} - 28 x^{7} - 59 x^{6} + 46 x^{5} + 66 x^{4} - 32 x^{3} - 55 x^{2} + 38 x + 1 \)
Invariants
| Degree: | $16$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[2, 7]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(-27187276612503486242087=-\,23^{7}\cdot 41^{8}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $25.24$ | 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: | $23, 41$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3} a^{8} - \frac{1}{3}$, $\frac{1}{3} a^{9} - \frac{1}{3} a$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{3}$, $\frac{1}{3} a^{12} - \frac{1}{3} a^{4}$, $\frac{1}{15} a^{13} - \frac{2}{15} a^{12} - \frac{1}{15} a^{11} + \frac{1}{15} a^{10} - \frac{2}{15} a^{9} - \frac{2}{15} a^{8} + \frac{2}{5} a^{7} - \frac{2}{5} a^{6} - \frac{1}{15} a^{5} - \frac{4}{15} a^{4} - \frac{2}{15} a^{3} - \frac{7}{15} a^{2} + \frac{1}{3} a - \frac{4}{15}$, $\frac{1}{135} a^{14} + \frac{2}{135} a^{13} + \frac{11}{135} a^{12} - \frac{2}{15} a^{11} - \frac{13}{135} a^{10} + \frac{2}{27} a^{9} + \frac{13}{135} a^{8} - \frac{4}{45} a^{7} + \frac{13}{27} a^{6} - \frac{8}{135} a^{5} + \frac{7}{135} a^{4} - \frac{1}{9} a^{3} - \frac{53}{135} a^{2} - \frac{49}{135} a - \frac{31}{135}$, $\frac{1}{715809015} a^{15} - \frac{174145}{143161803} a^{14} - \frac{48349}{47720601} a^{13} - \frac{52512146}{715809015} a^{12} - \frac{13012028}{143161803} a^{11} - \frac{55787491}{715809015} a^{10} - \frac{10982912}{79534335} a^{9} + \frac{95557436}{715809015} a^{8} + \frac{18730877}{715809015} a^{7} - \frac{102113836}{715809015} a^{6} - \frac{14706403}{47720601} a^{5} + \frac{230126654}{715809015} a^{4} + \frac{7169236}{715809015} a^{3} - \frac{319863014}{715809015} a^{2} - \frac{6969206}{238603005} a - \frac{52307014}{143161803}$
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
| Rank: | $8$ | magma: UnitRank(K);
sage: UK.rank()
gp: K.fu
| |
| Torsion generator: | \( -1 \) (order $2$) | 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 (assuming GRH) | magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
| |
| Regulator: | \( 151970.25308 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A solvable group of order 32 |
| The 11 conjugacy class representatives for $D_{16}$ |
| Character table for $D_{16}$ |
Intermediate fields
| \(\Q(\sqrt{41}) \), 4.2.38663.1, 8.2.34381034087.1 |
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 sibling: | 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 | ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/3.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/5.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{2}$ | $16$ | $16$ | ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ | $16$ | $16$ | R | ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/37.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ | R | ${\href{/LocalNumberField/43.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ | $16$ | ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
| $23$ | 23.2.0.1 | $x^{2} - x + 7$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 23.2.1.1 | $x^{2} - 23$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 23.4.2.1 | $x^{4} + 299 x^{2} + 25921$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 23.4.2.1 | $x^{4} + 299 x^{2} + 25921$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 23.4.2.1 | $x^{4} + 299 x^{2} + 25921$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| $41$ | 41.4.2.1 | $x^{4} + 943 x^{2} + 242064$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| 41.4.2.1 | $x^{4} + 943 x^{2} + 242064$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 41.4.2.1 | $x^{4} + 943 x^{2} + 242064$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 41.4.2.1 | $x^{4} + 943 x^{2} + 242064$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |