Normalized defining polynomial
\( x^{16} - 4 x^{14} - 36 x^{13} + 104 x^{12} + 456 x^{11} + 2346 x^{10} + 3756 x^{9} + 14201 x^{8} + 32856 x^{7} + 146358 x^{6} + 348468 x^{5} + 885224 x^{4} + 1326168 x^{3} + 2009246 x^{2} + 1544424 x + 1332841 \)
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: | \(8979181539709000089600000000=2^{32}\cdot 3^{8}\cdot 5^{8}\cdot 13^{8}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $55.86$ | 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: | $2, 3, 5, 13$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(1560=2^{3}\cdot 3\cdot 5\cdot 13\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{1560}(1,·)$, $\chi_{1560}(131,·)$, $\chi_{1560}(389,·)$, $\chi_{1560}(1351,·)$, $\chi_{1560}(1481,·)$, $\chi_{1560}(781,·)$, $\chi_{1560}(1039,·)$, $\chi_{1560}(1169,·)$, $\chi_{1560}(259,·)$, $\chi_{1560}(469,·)$, $\chi_{1560}(599,·)$, $\chi_{1560}(911,·)$, $\chi_{1560}(1249,·)$, $\chi_{1560}(1379,·)$, $\chi_{1560}(571,·)$, $\chi_{1560}(701,·)$$\rbrace$ | ||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{10} a^{10} + \frac{1}{5} a^{9} + \frac{1}{5} a^{8} - \frac{1}{5} a^{7} + \frac{1}{10} a^{6} + \frac{1}{5} a^{3} - \frac{1}{10} a^{2} + \frac{2}{5}$, $\frac{1}{10} a^{11} - \frac{1}{5} a^{9} - \frac{1}{10} a^{8} - \frac{1}{2} a^{7} - \frac{1}{5} a^{6} - \frac{3}{10} a^{4} - \frac{1}{2} a^{3} + \frac{1}{5} a^{2} + \frac{2}{5} a - \frac{3}{10}$, $\frac{1}{30} a^{12} + \frac{1}{30} a^{10} + \frac{1}{5} a^{8} + \frac{2}{5} a^{7} - \frac{7}{30} a^{6} + \frac{2}{5} a^{5} - \frac{2}{5} a^{3} + \frac{1}{30} a^{2} + \frac{2}{5} a + \frac{7}{30}$, $\frac{1}{30} a^{13} + \frac{1}{30} a^{11} + \frac{1}{5} a^{9} - \frac{1}{10} a^{8} - \frac{7}{30} a^{7} + \frac{2}{5} a^{6} + \frac{1}{10} a^{4} + \frac{1}{30} a^{3} + \frac{2}{5} a^{2} + \frac{7}{30} a - \frac{1}{2}$, $\frac{1}{5291370} a^{14} - \frac{9379}{755910} a^{13} - \frac{2897}{1058274} a^{12} - \frac{114104}{2645685} a^{11} + \frac{103483}{5291370} a^{10} + \frac{7421}{58793} a^{9} + \frac{580936}{2645685} a^{8} + \frac{194050}{529137} a^{7} - \frac{440761}{5291370} a^{6} + \frac{40492}{293965} a^{5} - \frac{1123204}{2645685} a^{4} + \frac{1271731}{2645685} a^{3} + \frac{427741}{2645685} a^{2} - \frac{339889}{755910} a - \frac{210541}{1058274}$, $\frac{1}{1546869277090136786860207890} a^{15} + \frac{7365813434392402492}{257811546181689464476701315} a^{14} - \frac{11296334575457913691672117}{773434638545068393430103945} a^{13} + \frac{6410574200212211688417824}{773434638545068393430103945} a^{12} - \frac{24551835928533017116375127}{515623092363378928953402630} a^{11} - \frac{55880467149854415591248627}{1546869277090136786860207890} a^{10} + \frac{260923421087943212883245777}{1546869277090136786860207890} a^{9} - \frac{12018764640120962176011583}{73660441766196989850486090} a^{8} + \frac{10507135887064614984137323}{34374872824225261930226842} a^{7} + \frac{304071200888277807910861253}{1546869277090136786860207890} a^{6} - \frac{89250457142099947302359801}{1546869277090136786860207890} a^{5} + \frac{1951171975872080955640439}{103124618472675785790680526} a^{4} - \frac{18602639380427185037207735}{154686927709013678686020789} a^{3} + \frac{1120712023453120118599253}{3589023844756697881346190} a^{2} - \frac{210530847856282790307247967}{515623092363378928953402630} a - \frac{131133880664052629670349723}{309373855418027357372041578}$
Class group and class number
$C_{2}\times C_{4}\times C_{8}\times C_{48}$, which has order $3072$ (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: | \( 15197.42445606848 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| An abelian group of order 16 |
| The 16 conjugacy class representatives for $C_2^4$ |
| Character table for $C_2^4$ |
Intermediate fields
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Frobenius cycle types
| $p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 | 53 | 59 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | R | R | ${\href{/LocalNumberField/7.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ | R | ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$ |
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 |
|---|---|---|---|---|---|---|---|
| $2$ | 2.8.16.6 | $x^{8} + 4 x^{6} + 8 x^{2} + 4$ | $4$ | $2$ | $16$ | $C_2^3$ | $[2, 3]^{2}$ |
| 2.8.16.6 | $x^{8} + 4 x^{6} + 8 x^{2} + 4$ | $4$ | $2$ | $16$ | $C_2^3$ | $[2, 3]^{2}$ | |
| $3$ | 3.4.2.1 | $x^{4} + 9 x^{2} + 36$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| 3.4.2.1 | $x^{4} + 9 x^{2} + 36$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 3.4.2.1 | $x^{4} + 9 x^{2} + 36$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 3.4.2.1 | $x^{4} + 9 x^{2} + 36$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| $5$ | 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 | $x^{4} + 15 x^{2} + 100$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| $13$ | 13.4.2.1 | $x^{4} + 39 x^{2} + 676$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| 13.4.2.1 | $x^{4} + 39 x^{2} + 676$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 13.4.2.1 | $x^{4} + 39 x^{2} + 676$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 13.4.2.1 | $x^{4} + 39 x^{2} + 676$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |