Normalized defining polynomial
\( x^{16} - 8 x^{15} + 12 x^{14} + 24 x^{13} - 64 x^{12} + 96 x^{11} + 296 x^{10} - 824 x^{9} - 1734 x^{8} + 1616 x^{7} + 4616 x^{6} - 240 x^{5} - 4516 x^{4} - 432 x^{3} + 2304 x^{2} + 176 x - 482 \)
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: | \(6471746811795667779846144=2^{56}\cdot 3^{12}\cdot 13^{2}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $35.54$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{65} a^{14} + \frac{28}{65} a^{13} + \frac{27}{65} a^{12} - \frac{28}{65} a^{11} + \frac{2}{65} a^{10} + \frac{22}{65} a^{9} + \frac{12}{65} a^{8} - \frac{8}{65} a^{7} - \frac{28}{65} a^{6} - \frac{28}{65} a^{5} + \frac{17}{65} a^{4} + \frac{31}{65} a^{3} - \frac{1}{65} a^{2} + \frac{14}{65} a + \frac{31}{65}$, $\frac{1}{92489729260027890989705} a^{15} - \frac{640781140455846725622}{92489729260027890989705} a^{14} + \frac{28548302521523407184427}{92489729260027890989705} a^{13} - \frac{1249135697641909498882}{3189301008966478999645} a^{12} + \frac{672969495774950674746}{1380443720298923746115} a^{11} + \frac{12594419219534954533397}{92489729260027890989705} a^{10} - \frac{15791802989073232298998}{92489729260027890989705} a^{9} - \frac{33559231659670845927778}{92489729260027890989705} a^{8} + \frac{17524262693882175211002}{92489729260027890989705} a^{7} + \frac{10642397332597102540212}{92489729260027890989705} a^{6} - \frac{1148390153074099780547}{3189301008966478999645} a^{5} + \frac{14382748291941611040621}{92489729260027890989705} a^{4} + \frac{431797677177163495821}{3189301008966478999645} a^{3} - \frac{42890843479862453143571}{92489729260027890989705} a^{2} + \frac{3944235612852371561766}{92489729260027890989705} a + \frac{6680987625586508863249}{18497945852005578197941}$
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: | \( 4338202.65384 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$C_2^3.C_2^4.C_2$ (as 16T608):
| A solvable group of order 256 |
| The 34 conjugacy class representatives for $C_2^3.C_2^4.C_2$ |
| Character table for $C_2^3.C_2^4.C_2$ is not computed |
Intermediate fields
| \(\Q(\sqrt{6}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{2}) \), 4.4.27648.1 x2, 4.4.13824.1 x2, \(\Q(\sqrt{2}, \sqrt{3})\), 8.8.3057647616.1 |
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 | R | ${\href{/LocalNumberField/5.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ | R | ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$ |
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 | Data not computed | ||||||
| 3 | Data not computed | ||||||
| $13$ | 13.2.0.1 | $x^{2} - x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 13.2.0.1 | $x^{2} - x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 13.4.0.1 | $x^{4} + x^{2} - x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 13.4.0.1 | $x^{4} + x^{2} - x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 13.4.2.1 | $x^{4} + 39 x^{2} + 676$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |