Normalized defining polynomial
\( x^{16} - 8 x^{15} + 44 x^{14} - 168 x^{13} + 684 x^{12} - 2284 x^{11} + 9374 x^{10} - 28984 x^{9} + 104261 x^{8} - 263116 x^{7} + 750810 x^{6} - 1433520 x^{5} + 3188158 x^{4} - 4245960 x^{3} + 7275852 x^{2} - 5355144 x + 6851556 \)
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: | \(186947103756597696921600000000=2^{32}\cdot 3^{8}\cdot 5^{8}\cdot 19^{8}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $67.53$ | 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: | $2, 3, 5, 19$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(2280=2^{3}\cdot 3\cdot 5\cdot 19\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{2280}(1,·)$, $\chi_{2280}(2051,·)$, $\chi_{2280}(1861,·)$, $\chi_{2280}(911,·)$, $\chi_{2280}(721,·)$, $\chi_{2280}(1559,·)$, $\chi_{2280}(1369,·)$, $\chi_{2280}(419,·)$, $\chi_{2280}(229,·)$, $\chi_{2280}(2279,·)$, $\chi_{2280}(2089,·)$, $\chi_{2280}(1139,·)$, $\chi_{2280}(1331,·)$, $\chi_{2280}(949,·)$, $\chi_{2280}(191,·)$, $\chi_{2280}(1141,·)$$\rbrace$ | ||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5}$, $\frac{1}{14} a^{10} + \frac{1}{7} a^{9} + \frac{1}{14} a^{8} - \frac{1}{7} a^{7} + \frac{1}{14} a^{6} + \frac{2}{7} a^{5} - \frac{1}{14} a^{4} - \frac{2}{7} a^{3} + \frac{2}{7} a^{2} - \frac{3}{7} a - \frac{2}{7}$, $\frac{1}{14} a^{11} - \frac{3}{14} a^{9} + \frac{3}{14} a^{8} + \frac{5}{14} a^{7} + \frac{1}{7} a^{6} + \frac{5}{14} a^{5} + \frac{5}{14} a^{4} - \frac{1}{7} a^{3} - \frac{3}{7} a - \frac{3}{7}$, $\frac{1}{42} a^{12} + \frac{1}{42} a^{10} - \frac{5}{21} a^{9} + \frac{3}{14} a^{8} - \frac{1}{7} a^{7} - \frac{19}{42} a^{6} - \frac{1}{7} a^{4} - \frac{8}{21} a^{3} + \frac{5}{21} a^{2} + \frac{2}{7} a + \frac{2}{7}$, $\frac{1}{42} a^{13} + \frac{1}{42} a^{11} - \frac{1}{42} a^{10} + \frac{1}{7} a^{9} + \frac{1}{14} a^{8} + \frac{5}{42} a^{7} + \frac{3}{14} a^{6} + \frac{3}{14} a^{5} + \frac{17}{42} a^{4} + \frac{8}{21} a^{3} + \frac{1}{7} a^{2} + \frac{1}{7}$, $\frac{1}{1816071408036942} a^{14} - \frac{1}{259438772576706} a^{13} - \frac{8108915964998}{908035704018471} a^{12} - \frac{16206197354143}{908035704018471} a^{11} - \frac{10186036756268}{908035704018471} a^{10} - \frac{200481114861521}{908035704018471} a^{9} + \frac{16545519668884}{908035704018471} a^{8} - \frac{79318650708664}{908035704018471} a^{7} - \frac{10232966439043}{259438772576706} a^{6} - \frac{51476566340803}{1816071408036942} a^{5} + \frac{201504609627634}{908035704018471} a^{4} - \frac{378641237051356}{908035704018471} a^{3} + \frac{51604584026129}{129719386288353} a^{2} + \frac{20697732071490}{43239795429451} a - \frac{92195732957669}{302678568006157}$, $\frac{1}{1102807546459024992558} a^{15} + \frac{303617}{1102807546459024992558} a^{14} + \frac{7310665714809513559}{1102807546459024992558} a^{13} - \frac{2176245742499447998}{183801257743170832093} a^{12} - \frac{15205358953567708654}{551403773229512496279} a^{11} + \frac{19176753277312536701}{1102807546459024992558} a^{10} - \frac{3724617064428906045}{183801257743170832093} a^{9} + \frac{12164998415672043709}{157543935208432141794} a^{8} - \frac{169104921097201055577}{367602515486341664186} a^{7} + \frac{476664592055719814}{10811838690774754829} a^{6} + \frac{64947289730906536097}{157543935208432141794} a^{5} - \frac{100568250794001690415}{1102807546459024992558} a^{4} - \frac{14973436129641139150}{78771967604216070897} a^{3} - \frac{51000562217952116003}{551403773229512496279} a^{2} - \frac{21833908319849374453}{183801257743170832093} a + \frac{664985794260799559}{183801257743170832093}$
Class group and class number
$C_{4}\times C_{8}\times C_{8}\times C_{48}$, which has order $12288$ (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 | ${\href{/LocalNumberField/7.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ | R | ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$ |
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 | $x^{4} + 15 x^{2} + 100$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| $19$ | 19.4.2.1 | $x^{4} + 57 x^{2} + 1444$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| 19.4.2.1 | $x^{4} + 57 x^{2} + 1444$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 19.4.2.1 | $x^{4} + 57 x^{2} + 1444$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 19.4.2.1 | $x^{4} + 57 x^{2} + 1444$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |