Normalized defining polynomial
\( x^{16} - x^{15} - 9 x^{14} - 18 x^{13} + 21 x^{12} + 112 x^{11} + 302 x^{10} + 239 x^{9} - 594 x^{8} - 1873 x^{7} - 2874 x^{6} - 1755 x^{5} + 164 x^{4} + 952 x^{3} + 616 x^{2} + 79 x - 1 \)
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: | \(394577369981946328515625=5^{8}\cdot 7^{4}\cdot 29^{10}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $29.84$ | 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: | $5, 7, 29$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{11} a^{14} - \frac{5}{11} a^{12} - \frac{1}{11} a^{11} - \frac{2}{11} a^{9} + \frac{3}{11} a^{8} + \frac{3}{11} a^{7} + \frac{4}{11} a^{6} + \frac{2}{11} a^{5} + \frac{4}{11} a^{4} - \frac{5}{11} a^{3} - \frac{1}{11} a^{2} - \frac{4}{11} a + \frac{3}{11}$, $\frac{1}{4644089126432325538661} a^{15} + \frac{184329749836194481795}{4644089126432325538661} a^{14} - \frac{1482315353896611623295}{4644089126432325538661} a^{13} + \frac{82066542562365527699}{4644089126432325538661} a^{12} + \frac{1430671349857641111252}{4644089126432325538661} a^{11} + \frac{1059652606290186063472}{4644089126432325538661} a^{10} + \frac{1638479738563137514683}{4644089126432325538661} a^{9} - \frac{974557021725225499643}{4644089126432325538661} a^{8} + \frac{921798720984433685752}{4644089126432325538661} a^{7} - \frac{1306602256223470397306}{4644089126432325538661} a^{6} - \frac{1943275048713174401472}{4644089126432325538661} a^{5} - \frac{2115708511692619292637}{4644089126432325538661} a^{4} + \frac{40532424215934428158}{663441303776046505523} a^{3} + \frac{11738447266602663534}{663441303776046505523} a^{2} + \frac{229323355881828282528}{663441303776046505523} a - \frac{1388796879508934848125}{4644089126432325538661}$
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: | \( 496811.690737 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$OD_{16}:C_2^2$ (as 16T106):
| A solvable group of order 64 |
| The 22 conjugacy class representatives for $OD_{16}:C_2^2$ |
| Character table for $OD_{16}:C_2^2$ is not computed |
Intermediate fields
| \(\Q(\sqrt{5}) \), \(\Q(\sqrt{29}) \), \(\Q(\sqrt{145}) \), 4.4.725.1 x2, 4.4.4205.1 x2, \(\Q(\sqrt{5}, \sqrt{29})\), 8.8.442050625.1 |
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 | ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ | R | R | ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ | R | ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
| $5$ | 5.8.4.1 | $x^{8} + 10 x^{6} + 125 x^{4} + 2500$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |
| 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 | $x^{2} - x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 7.4.2.1 | $x^{4} + 35 x^{2} + 441$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 7.4.2.1 | $x^{4} + 35 x^{2} + 441$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| $29$ | 29.8.4.1 | $x^{8} + 31958 x^{4} - 24389 x^{2} + 255328441$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |
| 29.8.6.2 | $x^{8} + 145 x^{4} + 7569$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ |