Normalized defining polynomial
\( x^{16} - 7 x^{15} + 14 x^{14} - 7 x^{13} - 32 x^{12} + 178 x^{11} - 259 x^{10} + 574 x^{9} - 1722 x^{8} - 462 x^{7} + 6204 x^{6} - 12056 x^{5} + 17028 x^{4} + 1936 x^{3} - 32549 x^{2} + 24079 x - 3509 \)
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: | \(146202125884644293212890625=5^{14}\cdot 11^{10}\cdot 31^{4}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $43.18$ | 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: | $5, 11, 31$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{11} a^{12} + \frac{4}{11} a^{11} + \frac{3}{11} a^{10} + \frac{4}{11} a^{9} + \frac{1}{11} a^{8} + \frac{2}{11} a^{7} + \frac{5}{11} a^{6} + \frac{2}{11} a^{5} + \frac{5}{11} a^{4}$, $\frac{1}{11} a^{13} - \frac{2}{11} a^{11} + \frac{3}{11} a^{10} - \frac{4}{11} a^{9} - \frac{2}{11} a^{8} - \frac{3}{11} a^{7} + \frac{4}{11} a^{6} - \frac{3}{11} a^{5} + \frac{2}{11} a^{4}$, $\frac{1}{209} a^{14} - \frac{6}{209} a^{13} - \frac{7}{209} a^{12} - \frac{2}{11} a^{11} + \frac{73}{209} a^{10} - \frac{53}{209} a^{9} + \frac{37}{209} a^{8} - \frac{4}{11} a^{7} + \frac{3}{209} a^{6} + \frac{54}{209} a^{5} + \frac{5}{11} a^{4} - \frac{2}{19} a^{3} + \frac{7}{19} a^{2} + \frac{4}{19} a - \frac{1}{19}$, $\frac{1}{1521938510954923867824145369} a^{15} + \frac{1314722846540405846788874}{1521938510954923867824145369} a^{14} + \frac{7523723909064354409005673}{1521938510954923867824145369} a^{13} - \frac{31283223268431109659308785}{1521938510954923867824145369} a^{12} - \frac{462453540797013920634083356}{1521938510954923867824145369} a^{11} - \frac{341932160573347448760713875}{1521938510954923867824145369} a^{10} + \frac{189081256450117232308782977}{1521938510954923867824145369} a^{9} - \frac{514007516665244502658554704}{1521938510954923867824145369} a^{8} - \frac{695112070828500641908134658}{1521938510954923867824145369} a^{7} - \frac{98139169862321498691613683}{1521938510954923867824145369} a^{6} + \frac{524886753968322959284991813}{1521938510954923867824145369} a^{5} - \frac{115502521784961085012203708}{1521938510954923867824145369} a^{4} + \frac{42862928064244362148261540}{138358046450447624347649579} a^{3} - \frac{40592800472313826433973877}{138358046450447624347649579} a^{2} - \frac{33633681145379735804582915}{138358046450447624347649579} a - \frac{11901840120784389172696871}{138358046450447624347649579}$
Class group and class number
$C_{2}$, which has order $2$ (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: | \( 9537582.38618 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A solvable group of order 1024 |
| The 43 conjugacy class representatives for t16n1161 |
| Character table for t16n1161 is not computed |
Intermediate fields
| \(\Q(\sqrt{5}) \), 4.4.15125.1, 8.4.78009078125.1 |
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 | ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ | R | ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/19.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{8}$ | ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ | R | ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$ |
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 | ||||||
| $11$ | 11.8.6.2 | $x^{8} - 781 x^{4} + 290521$ | $4$ | $2$ | $6$ | $D_4$ | $[\ ]_{4}^{2}$ |
| 11.8.4.1 | $x^{8} + 484 x^{4} - 1331 x^{2} + 58564$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
| 31 | Data not computed | ||||||