Normalized defining polynomial
\( x^{16} - x^{15} + 3 x^{14} + 2 x^{13} - 6 x^{12} + 24 x^{11} - 32 x^{10} + 40 x^{9} - 55 x^{7} + 163 x^{6} - 233 x^{5} + 292 x^{4} - 262 x^{3} + 194 x^{2} - 100 x + 31 \)
Invariants
| Degree: | $16$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[0, 8]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(1524467112922265625=3^{8}\cdot 5^{8}\cdot 29^{6}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $13.69$ | 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: | $3, 5, 29$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{133} a^{14} - \frac{8}{133} a^{13} - \frac{2}{19} a^{12} + \frac{1}{7} a^{11} - \frac{48}{133} a^{10} + \frac{37}{133} a^{9} + \frac{3}{19} a^{8} - \frac{15}{133} a^{7} + \frac{5}{19} a^{6} - \frac{3}{133} a^{5} + \frac{23}{133} a^{4} - \frac{6}{19} a^{3} - \frac{29}{133} a^{2} - \frac{52}{133} a + \frac{15}{133}$, $\frac{1}{76668781} a^{15} + \frac{150301}{76668781} a^{14} + \frac{27437468}{76668781} a^{13} - \frac{222571}{4035199} a^{12} - \frac{12197516}{76668781} a^{11} + \frac{3671359}{10952683} a^{10} - \frac{32556878}{76668781} a^{9} + \frac{32599349}{76668781} a^{8} - \frac{13208347}{76668781} a^{7} - \frac{25672195}{76668781} a^{6} - \frac{16018554}{76668781} a^{5} - \frac{197110}{76668781} a^{4} - \frac{12277924}{76668781} a^{3} - \frac{31432227}{76668781} a^{2} - \frac{4906186}{10952683} a + \frac{1634956}{4035199}$
Class group and class number
Trivial group, which has order $1$
Unit group
| Rank: | $7$ | magma: UnitRank(K);
sage: UK.rank()
gp: K.fu
| |
| Torsion generator: | \( \frac{191889}{1564669} a^{15} - \frac{105684}{1564669} a^{14} + \frac{331734}{1564669} a^{13} + \frac{678039}{1564669} a^{12} - \frac{1131321}{1564669} a^{11} + \frac{3470598}{1564669} a^{10} - \frac{3106238}{1564669} a^{9} + \frac{2792540}{1564669} a^{8} + \frac{4852466}{1564669} a^{7} - \frac{10499636}{1564669} a^{6} + \frac{21035803}{1564669} a^{5} - \frac{22649472}{1564669} a^{4} + \frac{23940733}{1564669} a^{3} - \frac{15839643}{1564669} a^{2} + \frac{8651129}{1564669} a - \frac{936381}{1564669} \) (order $6$) | 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 | magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
| |
| Regulator: | \( 688.925584956 \) | 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
| \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-15}) \), \(\Q(\sqrt{-3}) \), 4.4.725.1, 4.0.6525.1, \(\Q(\sqrt{-3}, \sqrt{5})\), 8.0.42575625.1 |
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 | ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ | R | R | ${\href{/LocalNumberField/7.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/19.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ | ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ | R | ${\href{/LocalNumberField/31.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ | ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
| 3 | 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 | $x^{8} + 31958 x^{4} - 24389 x^{2} + 255328441$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |