Normalized defining polynomial
\( x^{16} - 3 x^{15} + 3 x^{14} - 12 x^{13} + 33 x^{12} - 18 x^{11} + 45 x^{10} - 105 x^{9} + 15 x^{8} - 63 x^{7} + 144 x^{6} + 45 x^{5} + 81 x^{4} - 45 x^{3} - 9 x^{2} - 18 x + 9 \)
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: | \(10044752324575775625=3^{14}\cdot 5^{4}\cdot 7^{6}\cdot 13^{4}\) | 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: | $3, 5, 7, 13$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3} a^{8}$, $\frac{1}{3} a^{9}$, $\frac{1}{3} a^{10}$, $\frac{1}{3} a^{11}$, $\frac{1}{3} a^{12}$, $\frac{1}{39} a^{13} - \frac{1}{13} a^{12} + \frac{4}{39} a^{11} + \frac{5}{39} a^{10} + \frac{1}{13} a^{9} + \frac{2}{39} a^{8} + \frac{6}{13} a^{7} + \frac{3}{13} a^{6} - \frac{6}{13} a^{5} - \frac{2}{13} a^{4} - \frac{2}{13} a^{3} - \frac{3}{13} a^{2} - \frac{2}{13} a - \frac{6}{13}$, $\frac{1}{39} a^{14} - \frac{5}{39} a^{12} + \frac{4}{39} a^{11} + \frac{5}{39} a^{10} - \frac{2}{39} a^{9} - \frac{2}{39} a^{8} - \frac{5}{13} a^{7} + \frac{3}{13} a^{6} + \frac{6}{13} a^{5} + \frac{5}{13} a^{4} + \frac{4}{13} a^{3} + \frac{2}{13} a^{2} + \frac{1}{13} a - \frac{5}{13}$, $\frac{1}{701883} a^{15} + \frac{115}{233961} a^{14} - \frac{1972}{233961} a^{13} + \frac{21622}{233961} a^{12} - \frac{1426}{77987} a^{11} + \frac{1667}{77987} a^{10} + \frac{653}{233961} a^{9} + \frac{5748}{77987} a^{8} + \frac{13915}{233961} a^{7} - \frac{892}{5999} a^{6} - \frac{10063}{77987} a^{5} + \frac{31492}{77987} a^{4} - \frac{24944}{77987} a^{3} + \frac{30031}{77987} a^{2} - \frac{23467}{77987} a - \frac{31874}{77987}$
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: | \( \frac{7604}{33423} a^{15} - \frac{13609}{33423} a^{14} + \frac{698}{11141} a^{13} - \frac{88415}{33423} a^{12} + \frac{4490}{857} a^{11} + \frac{23939}{11141} a^{10} + \frac{146501}{11141} a^{9} - \frac{212731}{11141} a^{8} - \frac{158390}{11141} a^{7} - \frac{364558}{11141} a^{6} + \frac{313383}{11141} a^{5} + \frac{397476}{11141} a^{4} + \frac{608232}{11141} a^{3} + \frac{114207}{11141} a^{2} - \frac{726}{11141} a - \frac{98591}{11141} \) (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: | \( 1854.603895 \) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$C_2^2.C_2^5.C_2$ (as 16T511):
| A solvable group of order 256 |
| The 46 conjugacy class representatives for $C_2^2.C_2^5.C_2$ |
| Character table for $C_2^2.C_2^5.C_2$ is not computed |
Intermediate fields
| \(\Q(\sqrt{-3}) \), 4.0.189.1, 4.0.117.1, 4.0.2457.1, 8.0.18753525.1, 8.0.3169345725.2, 8.0.6036849.2 |
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 | ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ | R | R | R | ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ | R | ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$ |
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 | Data not computed | ||||||
| $5$ | 5.4.0.1 | $x^{4} + x^{2} - 2 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |
| 5.4.0.1 | $x^{4} + x^{2} - 2 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 5.8.4.1 | $x^{8} + 10 x^{6} + 125 x^{4} + 2500$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
| $7$ | 7.2.0.1 | $x^{2} - x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 7.2.0.1 | $x^{2} - x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 7.2.0.1 | $x^{2} - x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 7.2.0.1 | $x^{2} - x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 7.4.3.2 | $x^{4} - 7$ | $4$ | $1$ | $3$ | $D_{4}$ | $[\ ]_{4}^{2}$ | |
| 7.4.3.2 | $x^{4} - 7$ | $4$ | $1$ | $3$ | $D_{4}$ | $[\ ]_{4}^{2}$ | |
| $13$ | 13.2.0.1 | $x^{2} - x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 13.2.0.1 | $x^{2} - x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 13.2.0.1 | $x^{2} - x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 13.2.0.1 | $x^{2} - x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 13.4.2.1 | $x^{4} + 39 x^{2} + 676$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 13.4.2.1 | $x^{4} + 39 x^{2} + 676$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |