Normalized defining polynomial
\( x^{16} - 6 x^{14} - 117 x^{12} - 160 x^{10} + 7741 x^{8} - 3630 x^{6} - 47493 x^{4} - 37752 x^{2} + 14641 \)
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: | \(2396072989114866073600000000=2^{32}\cdot 5^{8}\cdot 7^{4}\cdot 29^{6}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $51.43$ | 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: | $2, 5, 7, 29$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{5} a^{8} + \frac{2}{5} a^{6} + \frac{2}{5} a^{4} + \frac{1}{5} a^{2} - \frac{1}{5}$, $\frac{1}{5} a^{9} + \frac{2}{5} a^{7} + \frac{2}{5} a^{5} + \frac{1}{5} a^{3} - \frac{1}{5} a$, $\frac{1}{5} a^{10} - \frac{2}{5} a^{6} + \frac{2}{5} a^{4} + \frac{2}{5} a^{2} + \frac{2}{5}$, $\frac{1}{5} a^{11} - \frac{2}{5} a^{7} + \frac{2}{5} a^{5} + \frac{2}{5} a^{3} + \frac{2}{5} a$, $\frac{1}{65} a^{12} + \frac{3}{65} a^{10} - \frac{6}{65} a^{8} - \frac{27}{65} a^{6} + \frac{4}{65} a^{2} - \frac{4}{13}$, $\frac{1}{715} a^{13} + \frac{16}{715} a^{11} + \frac{59}{715} a^{9} - \frac{248}{715} a^{7} - \frac{18}{55} a^{5} - \frac{3}{13} a^{3} + \frac{71}{715} a$, $\frac{1}{2388384881583415} a^{14} - \frac{837485165652}{183721913967955} a^{12} + \frac{75011558577421}{2388384881583415} a^{10} + \frac{119104110851522}{2388384881583415} a^{8} + \frac{39921601755371}{477676976316683} a^{6} + \frac{1055547623027}{19738718029615} a^{4} - \frac{35487688882863}{77044673599465} a^{2} - \frac{1265350084152}{19738718029615}$, $\frac{1}{26272233697417565} a^{15} - \frac{837485165652}{2020941053647505} a^{13} + \frac{2463396440160836}{26272233697417565} a^{11} - \frac{358320758883042}{5254446739483513} a^{9} + \frac{3543346842993636}{26272233697417565} a^{7} - \frac{50265119253972}{217125898325765} a^{5} + \frac{57619988046978}{169498281918823} a^{3} - \frac{12885849555784}{43425179665153} a$
Class group and class number
$C_{6}$, which has order $6$ (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: | \( 9958003.99307 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$C_2\times D_4:D_4$ (as 16T265):
| A solvable group of order 128 |
| The 32 conjugacy class representatives for $C_2\times D_4:D_4$ |
| Character table for $C_2\times D_4:D_4$ is not computed |
Intermediate fields
| \(\Q(\sqrt{5}) \), \(\Q(\sqrt{10}) \), \(\Q(\sqrt{2}) \), 4.4.725.1, 4.4.46400.1, \(\Q(\sqrt{2}, \sqrt{5})\), 8.4.3059356160000.2, 8.4.191209760000.2, 8.8.2152960000.1 |
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 | ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ | R | R | ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ | R | ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{8}$ | ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/41.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ | ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$ |
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 | ||||||
| $5$ | 5.4.2.1 | $x^{4} + 15 x^{2} + 100$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| 5.4.2.1 | $x^{4} + 15 x^{2} + 100$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 5.4.2.1 | $x^{4} + 15 x^{2} + 100$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 5.4.2.1 | $x^{4} + 15 x^{2} + 100$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| $7$ | 7.4.0.1 | $x^{4} + x^{2} - 3 x + 5$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |
| 7.4.2.2 | $x^{4} - 7 x^{2} + 147$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ | |
| 7.4.2.2 | $x^{4} - 7 x^{2} + 147$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ | |
| 7.4.0.1 | $x^{4} + x^{2} - 3 x + 5$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| $29$ | 29.4.0.1 | $x^{4} - x + 19$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |
| 29.4.0.1 | $x^{4} - x + 19$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 29.8.6.2 | $x^{8} + 145 x^{4} + 7569$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ | |