Normalized defining polynomial
\( x^{16} - 32 x^{14} - 16 x^{13} + 359 x^{12} + 312 x^{11} - 1762 x^{10} - 2152 x^{9} + 3697 x^{8} + 6200 x^{7} - 2208 x^{6} - 7340 x^{5} - 1651 x^{4} + 2856 x^{3} + 1652 x^{2} + 164 x - 31 \)
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: | \(997947933825433600000000=2^{32}\cdot 5^{8}\cdot 29^{6}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $31.62$ | 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: | $2, 5, 29$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{7} a^{11} - \frac{2}{7} a^{9} + \frac{1}{7} a^{8} + \frac{3}{7} a^{7} - \frac{2}{7} a^{6} + \frac{1}{7} a^{5} - \frac{3}{7} a^{4} + \frac{3}{7} a^{3} + \frac{3}{7} a + \frac{2}{7}$, $\frac{1}{7} a^{12} - \frac{2}{7} a^{10} + \frac{1}{7} a^{9} + \frac{3}{7} a^{8} - \frac{2}{7} a^{7} + \frac{1}{7} a^{6} - \frac{3}{7} a^{5} + \frac{3}{7} a^{4} + \frac{3}{7} a^{2} + \frac{2}{7} a$, $\frac{1}{7} a^{13} + \frac{1}{7} a^{10} - \frac{1}{7} a^{9} - \frac{2}{7} a^{5} + \frac{1}{7} a^{4} + \frac{2}{7} a^{3} + \frac{2}{7} a^{2} - \frac{1}{7} a - \frac{3}{7}$, $\frac{1}{7} a^{14} - \frac{1}{7} a^{10} + \frac{2}{7} a^{9} - \frac{1}{7} a^{8} - \frac{3}{7} a^{7} - \frac{2}{7} a^{4} - \frac{1}{7} a^{3} - \frac{1}{7} a^{2} + \frac{1}{7} a - \frac{2}{7}$, $\frac{1}{631682880689} a^{15} - \frac{17191883687}{631682880689} a^{14} + \frac{34446543156}{631682880689} a^{13} - \frac{19643113992}{631682880689} a^{12} - \frac{35426041976}{631682880689} a^{11} - \frac{42336425634}{631682880689} a^{10} + \frac{292563849154}{631682880689} a^{9} + \frac{13634953502}{90240411527} a^{8} + \frac{275622385312}{631682880689} a^{7} + \frac{203181892850}{631682880689} a^{6} - \frac{115176946609}{631682880689} a^{5} - \frac{230188914159}{631682880689} a^{4} + \frac{38817726037}{90240411527} a^{3} + \frac{18421387755}{90240411527} a^{2} + \frac{23513821839}{90240411527} a + \frac{8311257298}{20376867119}$
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: | \( 5522915.69449 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A solvable group of order 32 |
| The 14 conjugacy class representatives for $D_8:C_2$ |
| Character table for $D_8:C_2$ |
Intermediate fields
| \(\Q(\sqrt{5}) \), \(\Q(\sqrt{10}) \), \(\Q(\sqrt{2}) \), 4.4.725.1, 4.4.46400.1, \(\Q(\sqrt{2}, \sqrt{5})\), 8.8.2152960000.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Galois closure: | data not computed |
| Degree 16 siblings: | data not computed |
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 | ${\href{/LocalNumberField/7.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ | R | ${\href{/LocalNumberField/31.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ | ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/41.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ | ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$ |
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 | ||||||
| $5$ | 5.8.4.1 | $x^{8} + 10 x^{6} + 125 x^{4} + 2500$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |
| 5.8.4.1 | $x^{8} + 10 x^{6} + 125 x^{4} + 2500$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
| $29$ | 29.4.2.2 | $x^{4} - 29 x^{2} + 2523$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ |
| 29.4.0.1 | $x^{4} - x + 19$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 29.8.4.1 | $x^{8} + 31958 x^{4} - 24389 x^{2} + 255328441$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |