Normalized defining polynomial
\( x^{16} - 30 x^{14} - 6 x^{13} + 300 x^{12} + 24 x^{11} - 1353 x^{10} + 162 x^{9} + 2928 x^{8} - 792 x^{7} - 3074 x^{6} + 1194 x^{5} + 1426 x^{4} - 744 x^{3} - 191 x^{2} + 162 x - 23 \)
Invariants
| Degree: | $16$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[16, 0]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(714401597868117852160000=2^{24}\cdot 5^{4}\cdot 13^{8}\cdot 17^{4}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $30.97$ | 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: | $2, 5, 13, 17$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{92} a^{12} + \frac{3}{23} a^{11} - \frac{5}{23} a^{10} + \frac{17}{46} a^{9} - \frac{5}{92} a^{8} - \frac{1}{23} a^{7} - \frac{35}{92} a^{6} - \frac{9}{23} a^{5} - \frac{25}{92} a^{4} - \frac{3}{23} a^{3} - \frac{37}{92} a^{2} - \frac{15}{46} a - \frac{1}{4}$, $\frac{1}{92} a^{13} + \frac{5}{23} a^{11} - \frac{1}{46} a^{10} - \frac{45}{92} a^{9} - \frac{9}{23} a^{8} + \frac{13}{92} a^{7} + \frac{4}{23} a^{6} + \frac{39}{92} a^{5} + \frac{3}{23} a^{4} + \frac{15}{92} a^{3} - \frac{1}{2} a^{2} - \frac{31}{92} a$, $\frac{1}{184} a^{14} - \frac{1}{184} a^{12} + \frac{11}{92} a^{11} + \frac{7}{184} a^{10} + \frac{39}{92} a^{9} + \frac{13}{92} a^{8} - \frac{21}{46} a^{7} - \frac{27}{92} a^{6} + \frac{4}{23} a^{5} + \frac{10}{23} a^{4} + \frac{11}{92} a^{3} + \frac{5}{92} a^{2} - \frac{7}{92} a - \frac{3}{8}$, $\frac{1}{1146872} a^{15} - \frac{90}{143359} a^{14} + \frac{1031}{1146872} a^{13} + \frac{2817}{573436} a^{12} - \frac{197953}{1146872} a^{11} - \frac{86059}{573436} a^{10} + \frac{189657}{573436} a^{9} - \frac{34479}{286718} a^{8} - \frac{106815}{573436} a^{7} - \frac{48039}{286718} a^{6} + \frac{67202}{143359} a^{5} + \frac{68283}{573436} a^{4} + \frac{102585}{573436} a^{3} + \frac{55575}{573436} a^{2} - \frac{83733}{1146872} a - \frac{1410}{6233}$
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
| Rank: | $15$ | 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: | \( 5157242.87472 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$C_4^2:C_2^2$ (as 16T117):
| A solvable group of order 64 |
| The 28 conjugacy class representatives for $C_4^2:C_2^2$ |
| Character table for $C_4^2:C_2^2$ is not computed |
Intermediate fields
| \(\Q(\sqrt{13}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{26}) \), 4.4.183872.3, 4.4.183872.2, \(\Q(\sqrt{2}, \sqrt{13})\), 8.8.33808912384.1, 8.8.845222809600.1, 8.8.2924646400.1 |
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 | R | ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ | R | ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ | R | R | ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/23.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{8}$ | ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
| $2$ | 2.8.12.1 | $x^{8} + 6 x^{6} + 8 x^{5} + 16$ | $2$ | $4$ | $12$ | $C_4\times C_2$ | $[3]^{4}$ |
| 2.8.12.1 | $x^{8} + 6 x^{6} + 8 x^{5} + 16$ | $2$ | $4$ | $12$ | $C_4\times C_2$ | $[3]^{4}$ | |
| $5$ | 5.4.2.2 | $x^{4} - 5 x^{2} + 50$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ |
| 5.4.0.1 | $x^{4} + x^{2} - 2 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 5.4.0.1 | $x^{4} + x^{2} - 2 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 5.4.2.2 | $x^{4} - 5 x^{2} + 50$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ | |
| $13$ | 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 | $x^{8} + 26 x^{6} + 845 x^{4} + 6591 x^{2} + 114244$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
| $17$ | 17.2.0.1 | $x^{2} - x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 17.2.0.1 | $x^{2} - x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 17.2.0.1 | $x^{2} - x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 17.2.0.1 | $x^{2} - x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 17.4.2.1 | $x^{4} + 85 x^{2} + 2601$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 17.4.2.1 | $x^{4} + 85 x^{2} + 2601$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |