Normalized defining polynomial
\( x^{16} - 20 x^{14} - 4 x^{13} + 156 x^{12} + 8 x^{11} - 574 x^{10} + 260 x^{9} + 1005 x^{8} - 1288 x^{7} - 546 x^{6} + 1872 x^{5} - 1002 x^{4} - 104 x^{3} + 308 x^{2} - 72 x - 28 \)
Invariants
| Degree: | $16$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[8, 4]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(292946839147199189745664=2^{32}\cdot 17^{6}\cdot 41^{4}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $29.29$ | 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: | $2, 17, 41$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{6}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{7}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{5}$, $\frac{1}{14} a^{14} - \frac{1}{7} a^{13} - \frac{3}{14} a^{12} - \frac{3}{14} a^{11} - \frac{3}{14} a^{10} + \frac{3}{14} a^{9} - \frac{3}{14} a^{8} - \frac{3}{14} a^{7} + \frac{3}{7} a^{6} + \frac{5}{14} a^{5} - \frac{1}{7} a^{4} + \frac{1}{7} a^{3} + \frac{2}{7} a^{2} - \frac{1}{7} a$, $\frac{1}{134402670322231094} a^{15} - \frac{2006540587473958}{67201335161115547} a^{14} - \frac{15795870571234557}{67201335161115547} a^{13} + \frac{11699687815093289}{67201335161115547} a^{12} - \frac{1504193495815947}{134402670322231094} a^{11} - \frac{96632362313669}{1371455819614603} a^{10} + \frac{74615321475403}{19200381474604442} a^{9} - \frac{683526258422808}{67201335161115547} a^{8} + \frac{4627235168928293}{67201335161115547} a^{7} - \frac{19738809176775540}{67201335161115547} a^{6} - \frac{53780582999890611}{134402670322231094} a^{5} - \frac{200502745498790}{1371455819614603} a^{4} - \frac{21912571896225042}{67201335161115547} a^{3} - \frac{21024686910917087}{67201335161115547} a^{2} - \frac{13554091256044186}{67201335161115547} a - \frac{3508877222776487}{9600190737302221}$
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
| Rank: | $11$ | 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: | \( 1315836.80875 \) (assuming GRH) | 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
| \(\Q(\sqrt{2}) \), 4.4.178432.1, 4.4.4352.1, 4.4.2624.1, 8.4.541245636608.4, 8.4.321978368.1, 8.8.31837978624.2 |
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 | R | ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/7.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{4}$ | ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ | R | ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ | R | ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
| $2$ | 2.8.16.6 | $x^{8} + 4 x^{6} + 8 x^{2} + 4$ | $4$ | $2$ | $16$ | $C_2^3$ | $[2, 3]^{2}$ |
| 2.8.16.6 | $x^{8} + 4 x^{6} + 8 x^{2} + 4$ | $4$ | $2$ | $16$ | $C_2^3$ | $[2, 3]^{2}$ | |
| $17$ | 17.4.0.1 | $x^{4} - x + 11$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |
| 17.4.0.1 | $x^{4} - x + 11$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 17.8.6.2 | $x^{8} + 85 x^{4} + 2601$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ | |
| $41$ | 41.2.0.1 | $x^{2} - x + 12$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 41.2.0.1 | $x^{2} - x + 12$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 41.2.0.1 | $x^{2} - x + 12$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 41.2.0.1 | $x^{2} - x + 12$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 41.8.4.1 | $x^{8} + 57154 x^{4} - 68921 x^{2} + 816644929$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |