Normalized defining polynomial
\( x^{16} - x^{15} - 20 x^{14} + 8 x^{13} + 147 x^{12} + 4 x^{11} - 483 x^{10} - 127 x^{9} + 748 x^{8} + 266 x^{7} - 559 x^{6} - 205 x^{5} + 188 x^{4} + 60 x^{3} - 23 x^{2} - 4 x + 1 \)
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: | \(6401786224615359765625=5^{8}\cdot 29^{8}\cdot 181^{2}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $23.06$ | 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: | $5, 29, 181$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{52865167} a^{15} + \frac{11502639}{52865167} a^{14} - \frac{13039992}{52865167} a^{13} + \frac{4750228}{52865167} a^{12} - \frac{5245125}{52865167} a^{11} + \frac{10130090}{52865167} a^{10} + \frac{20594067}{52865167} a^{9} + \frac{21633270}{52865167} a^{8} - \frac{7606140}{52865167} a^{7} - \frac{1858008}{52865167} a^{6} + \frac{15382079}{52865167} a^{5} - \frac{15974279}{52865167} a^{4} - \frac{12070287}{52865167} a^{3} - \frac{7909351}{52865167} a^{2} - \frac{2307679}{52865167} a - \frac{7444359}{52865167}$
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: | \( 334223.730925 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$C_4^2:C_2^2.C_2$ (as 16T406):
| A solvable group of order 128 |
| The 26 conjugacy class representatives for $C_4^2:C_2^2.C_2$ |
| Character table for $C_4^2:C_2^2.C_2$ is not computed |
Intermediate fields
| \(\Q(\sqrt{29}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{145}) \), 4.4.4205.1 x2, 4.4.725.1 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 | ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ | R | ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
| $5$ | 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 | $x^{8} + 10 x^{6} + 125 x^{4} + 2500$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
| $29$ | 29.8.4.1 | $x^{8} + 31958 x^{4} - 24389 x^{2} + 255328441$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |
| 29.8.4.1 | $x^{8} + 31958 x^{4} - 24389 x^{2} + 255328441$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
| $181$ | 181.4.0.1 | $x^{4} - x + 54$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |
| 181.4.2.2 | $x^{4} - 181 x^{2} + 589698$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ | |
| 181.4.0.1 | $x^{4} - x + 54$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 181.4.0.1 | $x^{4} - x + 54$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |