Normalized defining polynomial
\( x^{16} - 4 x^{15} + 10 x^{13} - 22 x^{12} + 42 x^{11} - 67 x^{10} + 82 x^{9} - 93 x^{8} + 82 x^{7} - 67 x^{6} + 42 x^{5} - 22 x^{4} + 10 x^{3} - 4 x + 1 \)
Invariants
| Degree: | $16$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[4, 6]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(9976868377454444544=2^{24}\cdot 3^{6}\cdot 13^{8}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $15.40$ | 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: | $2, 3, 13$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{9} - \frac{1}{3} a^{8} + \frac{1}{3} a^{6} + \frac{1}{3} a^{4} - \frac{1}{3} a^{2} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{11} + \frac{1}{3} a^{9} - \frac{1}{3} a^{8} + \frac{1}{3} a^{7} + \frac{1}{3} a^{6} + \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} + \frac{1}{3}$, $\frac{1}{21} a^{12} - \frac{1}{7} a^{11} - \frac{2}{21} a^{10} - \frac{1}{21} a^{9} - \frac{8}{21} a^{8} - \frac{8}{21} a^{7} - \frac{5}{21} a^{6} - \frac{8}{21} a^{5} - \frac{1}{21} a^{4} - \frac{8}{21} a^{3} - \frac{3}{7} a^{2} + \frac{4}{21} a - \frac{2}{7}$, $\frac{1}{21} a^{13} + \frac{1}{7} a^{11} - \frac{4}{21} a^{9} + \frac{10}{21} a^{8} + \frac{2}{7} a^{7} - \frac{2}{21} a^{6} + \frac{10}{21} a^{5} + \frac{10}{21} a^{4} - \frac{5}{21} a^{3} + \frac{5}{21} a^{2} - \frac{1}{21} a + \frac{1}{7}$, $\frac{1}{357} a^{14} + \frac{5}{357} a^{13} - \frac{1}{51} a^{12} + \frac{10}{357} a^{11} + \frac{58}{357} a^{10} + \frac{16}{51} a^{9} - \frac{6}{119} a^{8} - \frac{158}{357} a^{7} - \frac{23}{119} a^{6} - \frac{6}{17} a^{5} + \frac{41}{357} a^{4} + \frac{95}{357} a^{3} + \frac{163}{357} a^{2} - \frac{3}{17} a + \frac{103}{357}$, $\frac{1}{8211} a^{15} - \frac{5}{8211} a^{14} + \frac{4}{1173} a^{13} + \frac{97}{8211} a^{12} - \frac{671}{8211} a^{11} - \frac{9}{119} a^{10} - \frac{67}{8211} a^{9} + \frac{3116}{8211} a^{8} - \frac{3232}{8211} a^{7} + \frac{71}{8211} a^{6} + \frac{43}{119} a^{5} + \frac{397}{1173} a^{4} + \frac{1270}{8211} a^{3} - \frac{203}{1173} a^{2} + \frac{478}{8211} a - \frac{758}{8211}$
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
| Rank: | $9$ | 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: | \( 1479.07441648 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A solvable group of order 32 |
| The 14 conjugacy class representatives for $D_8:C_2$ |
| Character table for $D_8:C_2$ |
Intermediate fields
| \(\Q(\sqrt{26}) \), \(\Q(\sqrt{13}) \), \(\Q(\sqrt{2}) \), 4.2.507.1, 4.2.32448.2, \(\Q(\sqrt{2}, \sqrt{13})\), 8.4.1052872704.3 |
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 | R | R | ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/7.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ | R | ${\href{/LocalNumberField/17.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{4}$ | ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/23.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ | ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$ |
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 | ||||||
| $3$ | 3.4.2.2 | $x^{4} - 3 x^{2} + 18$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ |
| 3.4.0.1 | $x^{4} - x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 3.8.4.1 | $x^{8} + 36 x^{4} - 27 x^{2} + 324$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
| $13$ | 13.8.4.1 | $x^{8} + 26 x^{6} + 845 x^{4} + 6591 x^{2} + 114244$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |
| 13.8.4.1 | $x^{8} + 26 x^{6} + 845 x^{4} + 6591 x^{2} + 114244$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |