Normalized defining polynomial
\( x^{16} - 8 x^{15} + 38 x^{14} - 126 x^{13} + 392 x^{12} - 1078 x^{11} + 2868 x^{10} - 6508 x^{9} + 14535 x^{8} - 27518 x^{7} + 52626 x^{6} - 80626 x^{5} + 131167 x^{4} - 151166 x^{3} + 199323 x^{2} - 133920 x + 139681 \)
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: | \(22502537891856000000000000=2^{16}\cdot 3^{8}\cdot 5^{12}\cdot 11^{8}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $38.42$ | 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: | $2, 3, 5, 11$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(660=2^{2}\cdot 3\cdot 5\cdot 11\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{660}(1,·)$, $\chi_{660}(131,·)$, $\chi_{660}(197,·)$, $\chi_{660}(463,·)$, $\chi_{660}(593,·)$, $\chi_{660}(67,·)$, $\chi_{660}(529,·)$, $\chi_{660}(353,·)$, $\chi_{660}(419,·)$, $\chi_{660}(659,·)$, $\chi_{660}(551,·)$, $\chi_{660}(617,·)$, $\chi_{660}(43,·)$, $\chi_{660}(109,·)$, $\chi_{660}(241,·)$, $\chi_{660}(307,·)$$\rbrace$ | ||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{63396858157669} a^{14} - \frac{7}{63396858157669} a^{13} - \frac{14413803240103}{63396858157669} a^{12} + \frac{23085961283040}{63396858157669} a^{11} + \frac{5804567862393}{63396858157669} a^{10} + \frac{2377138531066}{63396858157669} a^{9} + \frac{9188758787082}{63396858157669} a^{8} + \frac{12737133305995}{63396858157669} a^{7} - \frac{1660448777708}{63396858157669} a^{6} - \frac{4614036627378}{63396858157669} a^{5} - \frac{27378723557140}{63396858157669} a^{4} + \frac{19901575564915}{63396858157669} a^{3} - \frac{3220263847495}{63396858157669} a^{2} - \frac{21807859284661}{63396858157669} a - \frac{12309323836118}{63396858157669}$, $\frac{1}{5382456654444255769} a^{15} + \frac{42443}{5382456654444255769} a^{14} + \frac{505131751996769339}{5382456654444255769} a^{13} + \frac{6494441537202320}{23504177530324261} a^{12} + \frac{1885628979815212058}{5382456654444255769} a^{11} + \frac{433300220745922128}{5382456654444255769} a^{10} - \frac{1893800025670358634}{5382456654444255769} a^{9} - \frac{1805547840929613582}{5382456654444255769} a^{8} + \frac{518564144893683561}{5382456654444255769} a^{7} + \frac{367201244070214798}{5382456654444255769} a^{6} - \frac{18071276310919208}{91228078888885691} a^{5} - \frac{1106544471962594703}{5382456654444255769} a^{4} - \frac{87051755592782376}{5382456654444255769} a^{3} - \frac{599075294729327428}{5382456654444255769} a^{2} + \frac{1781797082240032084}{5382456654444255769} a + \frac{1958378453440851877}{5382456654444255769}$
Class group and class number
$C_{2}\times C_{4}\times C_{4}\times C_{8}$, which has order $256$ (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: | \( 3121.7160225 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$C_2^2\times C_4$ (as 16T2):
| An abelian group of order 16 |
| The 16 conjugacy class representatives for $C_4\times C_2^2$ |
| Character table for $C_4\times C_2^2$ |
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 | ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ | R | ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/59.1.0.1}{1} }^{16}$ |
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 |
|---|---|---|---|---|---|---|---|
| $2$ | 2.8.8.1 | $x^{8} + 28 x^{4} + 144$ | $2$ | $4$ | $8$ | $C_4\times C_2$ | $[2]^{4}$ |
| 2.8.8.1 | $x^{8} + 28 x^{4} + 144$ | $2$ | $4$ | $8$ | $C_4\times C_2$ | $[2]^{4}$ | |
| $3$ | 3.8.4.1 | $x^{8} + 36 x^{4} - 27 x^{2} + 324$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |
| 3.8.4.1 | $x^{8} + 36 x^{4} - 27 x^{2} + 324$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
| $5$ | 5.8.6.1 | $x^{8} - 5 x^{4} + 400$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ |
| 5.8.6.1 | $x^{8} - 5 x^{4} + 400$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ | |
| $11$ | 11.4.2.1 | $x^{4} + 143 x^{2} + 5929$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| 11.4.2.1 | $x^{4} + 143 x^{2} + 5929$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 11.4.2.1 | $x^{4} + 143 x^{2} + 5929$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 11.4.2.1 | $x^{4} + 143 x^{2} + 5929$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |