Normalized defining polynomial
\( x^{16} - 2 x^{15} + 6 x^{14} - 49 x^{13} + 89 x^{12} - 183 x^{11} + 929 x^{10} - 1413 x^{9} + 685 x^{8} - 1722 x^{7} + 5533 x^{6} - 10619 x^{5} + 6070 x^{4} + 1512 x^{3} + 10887 x^{2} - 574 x + 39311 \)
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: | \(86380562306022715087890625=5^{12}\cdot 29^{12}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $41.79$ | 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: | $5, 29$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{19} a^{13} - \frac{4}{19} a^{12} + \frac{5}{19} a^{11} + \frac{9}{19} a^{10} - \frac{7}{19} a^{9} + \frac{5}{19} a^{8} - \frac{3}{19} a^{7} - \frac{4}{19} a^{6} + \frac{6}{19} a^{5} - \frac{8}{19} a^{4} + \frac{9}{19} a^{3} + \frac{1}{19} a^{2} - \frac{7}{19} a$, $\frac{1}{19} a^{14} + \frac{8}{19} a^{12} - \frac{9}{19} a^{11} - \frac{9}{19} a^{10} - \frac{4}{19} a^{9} - \frac{2}{19} a^{8} + \frac{3}{19} a^{7} + \frac{9}{19} a^{6} - \frac{3}{19} a^{5} - \frac{4}{19} a^{4} - \frac{1}{19} a^{3} - \frac{3}{19} a^{2} - \frac{9}{19} a$, $\frac{1}{3045855195882290996124606312278809} a^{15} - \frac{24687284789912198266439789095986}{3045855195882290996124606312278809} a^{14} + \frac{4185451903074050285268751448607}{160308168204331105059189805909411} a^{13} + \frac{366960342610470718193322118175411}{3045855195882290996124606312278809} a^{12} - \frac{428026281607767990574123970606482}{3045855195882290996124606312278809} a^{11} + \frac{832464006454636876379047445891421}{3045855195882290996124606312278809} a^{10} - \frac{196421624968835812331276060337953}{3045855195882290996124606312278809} a^{9} - \frac{157872724075252549771555710920501}{3045855195882290996124606312278809} a^{8} - \frac{908286376106040240610728244545992}{3045855195882290996124606312278809} a^{7} - \frac{565139410204062490614517901616085}{3045855195882290996124606312278809} a^{6} + \frac{233938341224562381609352687838421}{3045855195882290996124606312278809} a^{5} - \frac{444373343533003253890716299343331}{3045855195882290996124606312278809} a^{4} + \frac{1141551244732182147988386242981145}{3045855195882290996124606312278809} a^{3} + \frac{1163368319356805420176557417186208}{3045855195882290996124606312278809} a^{2} + \frac{788041305971513413777497468433776}{3045855195882290996124606312278809} a + \frac{60650879890806406823799004838536}{160308168204331105059189805909411}$
Class group and class number
$C_{2}\times C_{4}$, which has order $8$ (assuming GRH)
Unit group
| Rank: | $7$ | 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: | \( 142046.185687 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$C_4\wr C_2$ (as 16T28):
| A solvable group of order 32 |
| The 14 conjugacy class representatives for $C_4\wr C_2$ |
| Character table for $C_4\wr C_2$ |
Intermediate fields
| \(\Q(\sqrt{5}) \), 4.0.105125.2, 4.4.725.1, 4.0.3625.1, 8.0.11051265625.5 |
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 8 siblings: | data not computed |
| Degree 16 sibling: | 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 | ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ | R | ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/19.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ | ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ | R | ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
| 5 | Data not computed | ||||||
| $29$ | 29.4.3.4 | $x^{4} + 232$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ |
| 29.4.3.3 | $x^{4} + 58$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ | |
| 29.8.6.2 | $x^{8} + 145 x^{4} + 7569$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ | |